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Algebraic geometry has a complicated, difficultlanguage. This bookcontains a definition, several references and the statements of the main theorems (without proofs) for every of the most common words in this subject. Some terms of relatedsubjects are included. It helps beginners that know some, but not all,basic facts of algebraic geometryto power series rings and rings of invariants of finite linear groups to the convolution algebra Weierstrass models and endomorphism algebras of abelian varieties to the generic Torelli... more...
In this book, Claire Voisin provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The volume is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the... more...
?Leading experts report on recent advances in higher-dimensional birational geometry, with special regard to arithmetic applications Highlights the tight connections between arithmetic and geometry Documents the central role of the theory of rational curves more...
This book is concerned with one of the most fundamental questions of mathematics: the relationship between algebraic formulas and geometric images.At one of the first international mathematical congresses (in Paris in 1900), Hilbert stated a special case of this question in the form of his 16th problem (from his list of 23 problems left over from the... more...
Algebraic geometry is a fascinating branch of mathematics that combines methods from both, algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. Moreover, Grothendieck's schemes invented in the late 1950s allowed the application of algebraic-geometric methods in fields that formerly... more...
Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry.The first part of the book studies classical problems and techniques that refer to polyhedral structures.... more...
| 677.169 | 1 |
Mathematics for Teachers: An Interactive Approach for Grades K-8 actively involves students in developing and explaining mathematical concepts and how the topics relate to NCTM Standards and Curriculum focal points. The text includes coverage of reasoning, sets, arithmetic, geometry, measurement, algebra, statistics, and probability. The carefully organized, interactive lesson format promotes student involvement and gradually leads the student to a deeper understanding of mathematical263.95
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More About
This Textbook
Overview
As in previous editions, the focus in INTERMEDIATE ALGEBRA remains on the Aufmann Interactive Method (AIM). Students are encouraged to be active participants in the classroom and in their own studies as they work through the How To examples and the paired Examples and You Try It problems. Student engagement is crucial to success. Presenting students with worked examples, and then providing them with the opportunity to immediately solve similar problems, helps them build their confidence and eventually master the concepts. Simplicity is key in the organization of this edition, as in all other editions. All lessons, exercise sets, tests, and supplements are organized around a carefully constructed hierarchy of objectives. Each exercise mirrors a preceding objective, which helps to reinforce key concepts and promote skill building. This clear, objective-based approach allows students to organize their thoughts around the content, and supports instructors as they work to design syllabi, lesson plans, and other administrative documents. New features like Focus on Success, Apply the Concept, and Concept Check add an increased emphasis on study skills and conceptual understanding to strengthen the foundation of student success. The Ninth Edition also features a new design, enhancing the Aufmann Interactive Method and making the pages easier for both students and instructors to follow. Available with InfoTrac Student Collections
Meet the Author
Richard Aufmann is the lead author of two bestselling developmental math series and a bestselling college algebra and trigonometry series, as well as several derivative math texts. He received a BA in mathematics from the University of California, Irvine, and an MA in mathematics from California State University, Long Beach. Mr. Aufmann taught math, computer science, and physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum, and the impact of technology on curriculum development.
Joanne Lockwood received a BA in English Literature from St. Lawrence University and both an MBA and a BA in mathematics from Plymouth State University. Ms. Lockwood taught at Plymouth State University and Nashua Community College in New Hampshire, and has over 20 years' experience teaching mathematics at the high school and college level. Ms. Lockwood has co-authored two bestselling developmental math series, as well as numerous derivative math texts and ancillaries. Ms. Lockwood's primary interest today is helping developmental math students overcome their challenges in learning
| 677.169 | 1 |
Mathematical Proofs: A Transition to Advanced Mathematics
K KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory. MARKET: For all readers interested in advanced mathematics and logic. ...more
Hardcover, Second Edition, 384 pages
Published
October 13th 2007
by Pearson
(first published May 28th 2002)
Community Reviews
example, by induction and the least element principle ), with many in-text exercises.
3- Equivalence Relations, functions and cardinalities of sets.
This part consists of three chapters and its the hardest, and important for any future investigation in pure mathematics.
Yay! Finally finished reading this book - and teaching it to my students. I really liked it actually, and yes, although I didn't teach sections 12.5 or 12.6 or chapter 13, I did actually read those as well. :-)
Clear, precise, and altogether excellent introduction of proofs and basic set theory. I'm glad historical context and facts about the development of logic were given (a move few maths textbooks have the balls to do). If you're looking to get into real maths, not the BS taught up to college, this is a great starting point.
Definitely one of the better Pearson text books ive read. Readin Pearson texts books is usually like standing in line at a government office. Mathematical Proofs really is a transition to advanced math, and I will definitely feel more complete studying advanced level calculus after reading this text. It offers a nice intro to set theory and logic that leads up to the basics of proving, and finishes off with the theoretically important proofs that found calculus, number theory and group theory.
I can't say enough good things about this textbooks -- it's definitely one of the best I have ever used. It's small and extremely concise and not burdened by tons of graphics and sidebars and sidenotes. Just exactly what you need to know, broken down into small pieces.
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More About
This Textbook
Overview
Winner of the 1983National Book Award!
"…a perfectly marvelous book about the Queen of Sciences, from which one will get a real feeling for what mathematicians do and who they are. The exposition is clear and full of wit and humor..." - The New Yorker (1983NationalEditorial Reviews
From the Publisher
From the reviews:
[The authors] have tried to provide a book usable in a course for liberal arts students and for future secondary teachers. They have done much more! This course should be required of every undergraduate major employing the mathematical sciences. It differs from the "mathematics appreciation" courses—courses that are merely a collection of amusing puzzles and toy problems giving an illusion of a mathematical encounter—presently found in many institutions. Students of this course are introduced to the context in which mathematics exists and the incredible magnitude of words devoted to communicating mathematics (hundreds of thousands of theorems each year). How much mathematics can there be? they are asked. Instructors in a "Mathematical Experience" course must be prepared to respond to questions from students concerning the fundamental nature of the whole mathematical enterprise. Stimulated by their reading of the text, students will ask about the underlying logical and philosophical issues, the role of mathematical methods and their origins, the substance of contemporary mathematical advances, the meaning of rigor and proof in mathematics, the role of computational mathematics, and issues of teaching and learning. How real is the conflict between "pure" mathematics, as represented by G.H. Hardy's statements, and "applied" mathematics? they may ask. Are there other kinds of mathematics, neither pure nor applied? This edition of the book provides a source of problems, collateral readings, references, essay and project assignments, and discussion guides for the course. I believe that it is likely that this course would be a challenge to many teachers and students alike, especially those teachers and students who are willing to follow their curiosity beyond the confines of this book and follow up on the many references that are provided.—Notices of the AMS (Kenneth C. Millett)
This beautifully written book can be recommended to any cultivated person with a certain sophistication of thought, and also to the practicing mathematician who will find here a vantage point from which to make a tour d'horizon of his science. —Publ. Math. Debrecen This is an unusual book, being more a book about mathematics than a mathematics book. It includes mathematical issues, but also questions from the philosophy of mathematics, the psychology of mathematical discovery, the history of mathematics, and biographies of mathematicians, in short, a book about the mathematical experience broadly considered… The book found its way into "Much for liberal arts students" courses and into education courses directed at future teachers. Term paper topics, essay assignments, problems, computer applications, and suggested readings are included. This new material should greatly enhance the usefulness of this very creative book. The range of topics covered is immense and the contents cannot easily be summarized. The book makes excellent casual reading, would make a good textbook, or could easily be used as a supplement to nearly any course concerned with mathematics. —Zentralblatt MATH
"This is a reprint of the 1995 edition … of a well-known and popular text. In a new edition, each of the authors added a brief essay in the end. … A warmly welcomed reprint of a very nice book that can be recommended for teaching, self-education, and simply as an entertaining reading." (Svitlana P. Rogovchenko, Zentralblatt MATH, Vol. 1230, 2012)
"The Mathematical Experience is a very interesting read – it provides a highly personal tour through aspects of mathematics, its history, its philosophy, and its relationship with the 'real' world. As such it provides a nice glimpse into how two mathematicians thought about their discipline as of some 30 years ago. … a worthy addition to the libraries of mathematicians interested in the scope and nature of our discipline. … I really enjoyed reading The Mathematical Experience and would recommend it for college and personal libraries." (Richard J. Wilders, The Mathematical Association of America, March, 2012)
New York Times Book Review
"A brilliant and engrossing view of the development of mathematics...wonderful at communicating its beauty and excitement to the general reader." --New York Times
Booknews
The addition of exercises and problems converts the 1981 edition into a textbook for math courses for liberal arts students and future secondary school math teachers, and courses in the appreciation of mathematics. Among the topics are the mathematical landscape, why math works and what it is good for, teaching and learning, certainty and fallibility, and mathematical reality. Includes a glossary without pronunciation
| 677.169 | 1 |
0883853Linear Algebra Problem Book (Dolciani Mathematical Expositions)
Can one learn linear algebra solely by solving problems? Paul Halmos thinks so, and you will too once you read this book. The Linear Algebra Problem Book is an ideal text for a course in linear algebra. It takes the student step by step from the basic axioms of a field through the notion of vector spaces, on to advanced concepts such as inner product spaces and normality. All of this occurs by way of a series of 164 problems, each with hints and, at the back of the book, full solutions. This book is a marvelous example of how to teach and learn mathematics by 'doing' mathematics. It will work well for classes taught in small groups and can also be used for self-study. After working their way through the book, students will understand not only the theorems of linear algebra, but also some of the questions which were asked which enabled the theorems to be discovered in the first place. They will gain confidence in their problem solving abilities and be better prepared to understand more advanced courses. As the author explains, 'I don't think I understand a subject until I know the questions ... I wrote this book to organize those questions, problems, in my own mind.' Try this book with your students and they too will be able to organize and understand the questions of linear algebra
| 677.169 | 1 |
Mathematics - Grade 9
MPM1D1 - Principles of Mathematics - This course enables students to develop understanding of mathematical concepts related to algebra, analytic geometry, and measurement and geometry through investigation, the effective use of technology, and abstract reasoning. Students will investigate relationships, which they will then generalize as equations of lines, and will determine the connections between different representations of a relationship. They will also explore relationships that emerge from the measurement of three-dimensional objects and two-dimensional shapes. Students will reason mathematically and communicate their thinking as they solve multi-step problems.
MFM1P1 - Foundations of Mathematics - This course enables students to develop understanding of mathematical concepts related to introductory algebra, proportional reasoning, and measurement and geometry through investigation, the effective use of technology, and hands-on activities. Students will investigate real-life examples to develop various representations of linear relationships, and will determine the connections between the representations. They will also explore certain relationships that emerge from the measurement of three-dimensional objects and two-dimensional shapes. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
MAT1L1 - Locally Developed - This course emphasizes further development of mathematical knowledge and skills to prepare students for success in their everyday lives, in the workplace, in the Grade 10 Locally Developed Compulsory Credit course, and in the Mathematics Grade 11 and Grade 12 Workplace Preparation courses. The course is organized by three strands related to money sense, measurement, and proportional reasoning. In all strands, the focus is on developing and consolidating key foundational mathematical concepts and skills by solving authentic, everyday problems. Students have opportunities to further develop their mathematical literacy and problem-solving skills and to continue developing their skills in reading, writing, and oral language through relevant and practical math activities.
| 677.169 | 1 |
These popular and proven workbooks help students build confidence before attempting end-of-chapter problems. They provide short problems and exercises that focus on developing a particular skill, often requiring students to draw or interpret sketches and graphs, or reason with math relationships. New to the Second Edition are exercises that provide guided practice for the textbook's Problem-Solving Strategies, focusing in particular on working symbolically
| 677.169 | 1 |
The heroic Dr. Ecco uncovers a fiendish plot in this collection of original puzzles inspired by research methods of computer science and mathematics. No sophisticated mathematical background necessary. Solutions....
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This fascinating and highly readable study by a noted historian uses maps, charts and diagrams to trace the development of the idea of a rational and interconnected material world across two and half millennia....
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Treasury of challenging brainteasers includes puzzles involving numbers, letters, probability, reasoning, more: The Enterprising Snail, The Fly and the Bicycles, The Lovesick Cockroaches, many others. No advanced...
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A coherent, well-organized look at the basis of quantum statistics' computational methods, the determination of the mean values of occupation numbers, the foundations of the statistics of photons and material...
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This comprehensive treatment features analytic formulas, enabling precise formulation of geometric facts, and it covers geometric manifolds and transformations, concluding with a systematic discussion of fundamentals....
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Readable, jargon-free book examines the earliest endeavors to count and record numbers, initial attempts to solve problems by using equations, and origins of infinite cardinal arithmetic. "Surprisingly exciting."...
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Only elementary math skills are needed to follow this manual, which covers many machines and their components, including hydrostatics and hydraulics, internal combustion engines, trains, and more. 204 black-and-white...
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Twenty-two essays examine the fourth dimension: how it may be studied, its relationship to non-Euclidean geometry, analogues to three-dimensional space, its absurdities and curiosities, and its simpler properties....
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Fundamentals Of Math Kit 2nd Edition Distributor Kit
Product description
Focuses on problem solving and real-life uses of math with special features in each chapter while reinforcing computational skills and building a solid math foundation. Dominion through Math problems regularly illustrate how mathematics can be used to manage God's creation to His glory.
Type: Boxed Set ()Category: > Home SchoolingISBN / UPC: 9780012517468/0012517461Publish Date: 6/1/2009Item No: 22448Vendor: Send The Light Distribution
| 677.169 | 1 |
9.
Preface
Basic Engineering Mathematics 5th Edition introduces and then consolidates basic mathematical principles and promotes awareness of mathematical concepts
for students needing a broad base for further vocational
studies.
In this fifth edition, new material has been added to many
of the chapters, particularly some of the earlier chapters, together with extra practical problems interspersed
throughout the text. The extent of this fifth edition
is such that four chapters from the previous edition
have been removed and placed on the easily available
website
The chapters removed to the website are 'Number
sequences', 'Binary, octal and hexadecimal', 'Inequalities' and 'Graphs with logarithmic scales'.
The text is relevant to:
• 'Mathematics for Engineering Technicians' for
BTEC First NQF Level 2 – Chapters 1–12, 16–18,
20, 21, 23 and 25–27 are needed for this module.
• The mandatory 'Mathematics for Technicians' for
BTEC National Certificate and National Diploma in
Engineering, NQF Level 3 – Chapters 7–10, 14–17,
19, 20–23, 25–27, 31, 32, 34 and 35 are needed
and, in addition, Chapters 1–6, 11 and 12 are helpful
revision for this module.
• Basic mathematics for a wide range of introductory/access/foundation mathematics courses.
• GCSE revision and for similar mathematics courses
in English-speaking countries worldwide.
Basic Engineering Mathematics 5th Edition provides a
lead into Engineering Mathematics 6th Edition.
Each topic considered in the text is presented in a
way that assumes in the reader little previous knowledge of that topic. Each chapter begins with a brief
outline of essential theory, definitions, formulae, laws
and procedures; however, these are kept to a minimum
as problem solving is extensively used to establish and
exemplify the theory. It is intended that readers will gain
real understanding through seeing problems solved and
then solving similar problems themselves.
This textbook contains some 750 worked problems,
followed by over 1550 further problems (all with
answers at the end of the book) contained within some
161 Practice Exercises; each Practice Exercise follows on directly from the relevant section of work. In
addition, 376 line diagrams enhance understanding of
the theory. Where at all possible, the problems mirror
potential practical situations found in engineering and
science.
Placed at regular intervals throughout the text are
14 Revision Tests (plus another for the website
chapters) to check understanding. For example, Revision Test 1 covers material contained in Chapters 1
and 2, Revision Test 2 covers the material contained
in Chapters 3–5, and so on. These Revision Tests do
not have answers given since it is envisaged that lecturers/instructors could set the tests for students to
attempt as part of their course structure. Lecturers/instructors may obtain a complimentary set of solutions of the Revision Tests in an Instructor's Manual,
available from the publishers via the internet – see
At the end of the book a list of relevant formulae contained within the text is included for convenience of
reference.
The principle of learning by example is at the heart of
Basic Engineering Mathematics 5th Edition.
JOHN BIRD
Royal Naval School of Marine Engineering
HMS Sultan, formerly University of Portsmouth
and Highbury College, Portsmouth
10.
Acknowledgements
The publisher wishes to thank CASIO Electronic Co.
Ltd, London for permission to reproduce the image of
the Casio fx-83ES calculator on page 23.
The publishers also wish to thank the Automobile Association for permission to reproduce a map of Portsmouth
on page 131.
11.
Instructor's Manual
Full worked solutions and mark scheme for all the
Assignments are contained in this Manual which is
available to lecturers only.
To download the Instructor's Manual visit http://
13.
Chapter 1
Basic arithmetic
1.1
Introduction
Whole numbers are called integers. +3, +5 and +72
are examples of positive integers; −13, −6 and −51
are examples of negative integers. Between positive
and negative integers is the number 0 which is neither
positive nor negative.
The four basic arithmetic operators are add (+), subtract
(−), multiply (×) and divide (÷).
It is assumed that adding, subtracting, multiplying and
dividing reasonably small numbers can be achieved
without a calculator. However, if revision of this area
is needed then some worked problems are included in
the following sections.
When unlike signs occur together in a calculation, the
overall sign is negative. For example,
3 + (−4) = 3 + −4 = 3 − 4 = −1
Problem 1. Determine 735 + 167
HTU
7 35
+1 67
9 02
1 1
(i) 5 + 7 = 12. Place 2 in units (U) column. Carry 1
in the tens (T) column.
(ii) 3 + 6 + 1 (carried) = 10. Place the 0 in the tens
column. Carry the 1 in the hundreds (H) column.
(iii) 7 + 1 + 1 (carried) = 9. Place the 9 in the hundreds column.
Hence, 735 + 167 = 902
and
(+5) × (−2) = −10
Like signs together give an overall positive sign. For
example,
3 − (−4) = 3 − −4 = 3 + 4 = 7
Problem 2. Determine 632 − 369
HTU
6 32
−3 69
2 63
and
(−6) × (−4) = +24
1.2 Revision of addition and
subtraction
You can probably already add two or more numbers
together and subtract one number from another. However, if you need a revision then the following worked
problems should be helpful.
DOI: 10.1016/B978-1-85617-697-2.00001-6
(i) 2 − 9 is not possible; therefore 'borrow' 1 from
the tens column (leaving 2 in the tens column). In
the units column, this gives us 12 − 9 = 3.
(ii) Place 3 in the units column.
(iii) 2 − 6 is not possible; therefore 'borrow' 1 from
the hundreds column (leaving 5 in the hundreds column). In the tens column, this gives us
12 − 6 = 6.
(iv) Place the 6 in the tens column.
16.
4 Basic Engineering Mathematics
(i) 8 × 4 = 32. Place the 2 in the units column and
carry 3 into the tens column.
(ii) 8 × 6 = 48; 48 + 3 (carried) = 51. Place the 1 in
the tens column and carry the 5 into the hundreds
column.
(iii) 8 × 7 = 56; 56 + 5 (carried) = 61. Place 1 in the
hundreds column and 6 in the thousands column.
(iv) Place 0 in the units column under the 2.
(v) 3 × 4 = 12. Place the 2 in the tens column and
carry 1 into the hundreds column.
(vi) 3 × 6 = 18; 18 + 1 (carried) = 19. Place the 9 in
the hundreds column and carry the 1 into the
thousands column.
(vii) 3 × 7 = 21; 21 + 1 (carried) = 22. Place 2 in the
thousands column and 2 in the ten thousands
column.
262
7 1834
(i) 7 into 18 goes 2, remainder 4. Place the 2 above
the 8 of 1834 and carry the 4 remainder to the
next digit on the right, making it 43.
(ii) 7 into 43 goes 6, remainder 1. Place the 6 above
the 3 of 1834 and carry the 1 remainder to the
next digit on the right, making it 14.
(iii) 7 into 14 goes 2, remainder 0. Place 2 above the
4 of 1834.
1834
Hence, 1834 ÷ 7 = 1834/7 =
= 262.
7
The method shown is called short division.
Problem 11.
Determine 5796 ÷ 12
(viii) 6112 + 22920 = 29032
Hence, 764 × 38 = 29032
Again, knowing multiplication tables is rather important
when multiplying such numbers.
It is appreciated, of course, that such a multiplication
can, and probably will, be performed using a calculator.
However, there are times when a calculator may not be
available and it is then useful to be able to calculate the
'long way'.
Problem 9.
Multiply 178 by −46
When the numbers have different signs, the result will
be negative. (With this in mind, the problem can now
be solved by multiplying 178 by 46). Following the
procedure of Problem 8 gives
×
178
46
1068
7120
8188
Thus, 178 × 46 = 8188 and 178 × (−46) = −8188
Problem 10.
Determine 1834 ÷ 7
483
12 5796
48
99
96
36
36
00
(i) 12 into 5 won't go. 12 into 57 goes 4; place 4
above the 7 of 5796.
(ii) 4 × 12 = 48; place the 48 below the 57 of 5796.
(iii) 57 − 48 = 9.
(iv) Bring down the 9 of 5796 to give 99.
(v) 12 into 99 goes 8; place 8 above the 9 of 5796.
(vi) 8 × 12 = 96; place 96 below the 99.
(vii) 99 − 96 = 3.
(viii) Bring down the 6 of 5796 to give 36.
(ix) 12 into 36 goes 3 exactly.
(x) Place the 3 above the final 6.
(xi) Place the 36 below the 36.
(xii) 36 − 36 = 0.
5796
Hence, 5796 ÷ 12 = 5796/12 =
= 483.
12
The method shown is called long division.
17.
Basic arithmetic
Now try the following Practice Exercise
Practice Exercise 2 Further problems on
multiplication and division (answers on
page 340)
Determine the values of the expressions given in
problems 1 to 9, without using a calculator.
1.
(a) 78 × 6
(b) 124 × 7
2.
(a) £261 × 7
(b) £462 × 9
3.
(a) 783 kg × 11
(b) 73 kg × 8
4.
(a) 27 mm × 13
(b) 77 mm × 12
5.
(a) 448 × 23
(b) 143 × (−31)
6.
(a) 288 m ÷ 6
(b) 979 m ÷ 11
7.
(a)
1813
7
(b)
8.
(a)
21424
13
(b) 15900 ÷ 15
9.
88737
(a)
11
(b) 46858 ÷ 14
896
16
is exactly divisible by each of two or more numbers is
called the lowest common multiple (LCM).
For example, the multiples of 12 are 12, 24, 36, 48,
60, 72,. . . and the multiples of 15 are 15, 30, 45,
60, 75,. . .
60 is a common multiple (i.e. a multiple of both 12 and
15) and there are no lower common multiples.
Hence, the LCM of 12 and 15 is 60 since 60 is the
lowest number that both 12 and 15 divide into.
Here are some further problems involving the determination of HCFs and LCMs.
Problem 12. Determine the HCF of the numbers
12, 30 and 42
Probably the simplest way of determining an HCF is to
express each number in terms of its lowest factors. This
is achieved by repeatedly dividing by the prime numbers
2, 3, 5, 7, 11, 13, … (where possible) in turn. Thus,
12 = 2 × 2 × 3
30 = 2
10. A screw has a mass of 15 grams. Calculate,
in kilograms, the mass of 1200 such screws
(1 kg = 1000 g).
1.4 Highest common factors and
lowest common multiples
When two or more numbers are multiplied together, the
individual numbers are called factors. Thus, a factor is a
number which divides into another number exactly. The
highest common factor (HCF) is the largest number
which divides into two or more numbers exactly.
For example, consider the numbers 12 and 15.
The factors of 12 are 1, 2, 3, 4, 6 and 12 (i.e. all the
numbers that divide into 12).
The factors of 15 are 1, 3, 5 and 15 (i.e. all the numbers
that divide into 15).
1 and 3 are the only common factors; i.e., numbers
which are factors of both 12 and 15.
Hence, the HCF of 12 and 15 is 3 since 3 is the highest
number which divides into both 12 and 15.
A multiple is a number which contains another number
an exact number of times. The smallest number which
×3×5
42 = 2
×3×7
The factors which are common to each of the numbers
are 2 in column 1 and 3 in column 3, shown by the
broken lines. Hence, the HCF is 2 × 3; i.e., 6. That is,
6 is the largest number which will divide into 12, 30
and 42.
Problem 13. Determine the HCF of the numbers
30, 105, 210 and 1155
Using the method shown in Problem 12:
30 = 2 × 3 × 5
105 =
3×5×7
210 = 2 × 3 × 5 × 7
1155 =
3 × 5 × 7 × 11
The factors which are common to each of the numbers
are 3 in column 2 and 5 in column 3. Hence, the HCF
is 3 × 5 = 15.
Problem 14. Determine the LCM of the numbers
12, 42 and 90
5
21.
Chapter 2
Fractions
2.1
Introduction
A mark of 9 out of 14 in an examination may be writ9
9
or 9/14.
is an example of a fraction. The
ten as
14
14
number above the line, i.e. 9, is called the numerator. The number below the line, i.e. 14, is called the
denominator.
When the value of the numerator is less than the
value of the denominator, the fraction is called a
9
proper fraction.
is an example of a proper
14
fraction.
When the value of the numerator is greater than the value
of the denominator, the fraction is called an improper
5
fraction. is an example of an improper fraction.
2
A mixed number is a combination of a whole number
1
and a fraction. 2 is an example of a mixed number. In
2
5
1
fact, = 2 .
2
2
There are a number of everyday examples in which
fractions are readily referred to. For example, three
people equally sharing a bar of chocolate would have
1
1
each. A supermarket advertises off a six-pack of
3
5
beer; if the beer normally costs £2 then it will now
3
of the employees of a company are
cost £1.60.
4
women; if the company has 48 employees, then 36 are
women.
Calculators are able to handle calculations with fractions. However, to understand a little more about fractions we will in this chapter show how to add, subtract,
multiply and divide with fractions without the use of a
calculator.
DOI: 10.1016/B978-1-85617-697-2.00002-8
Problem 1. Change the following improper
fractions into mixed numbers:
(a)
(a)
9
2
(b)
13
4
(c)
28
5
9
9
means 9 halves and = 9 ÷ 2, and 9 ÷ 2 = 4
2
2
and 1 half, i.e.
9
1
=4
2
2
(b)
13
13
means 13 quarters and
= 13 ÷ 4, and
4
4
13 ÷ 4 = 3 and 1 quarter, i.e.
13
1
=3
4
4
(c)
28
28
means 28 fifths and
= 28 ÷ 5, and 28 ÷ 5 =
5
5
5 and 3 fifths, i.e.
28
3
=5
5
5
Problem 2. Change the following mixed numbers
into improper fractions:
(a) 5
3
4
(b) 1
7
9
(c) 2
3
7
3
3
means 5 + . 5 contains 5 × 4 = 20 quarters.
4
4
3
Thus, 5 contains 20 + 3 = 23 quarters, i.e.
4
3 23
5 =
4
4
(a) 5
22.
10 Basic Engineering Mathematics
3
The quick way to change 5 into an improper
4
4 × 5 + 3 23
fraction is
= .
4
4
7 9 × 1 + 7 16
(b) 1 =
= .
9
9
9
3 7 × 2 + 3 17
(c) 2 =
= .
7
7
7
Problem 3. In a school there are 180 students of
which 72 are girls. Express this as a fraction in its
simplest form
72
The fraction of girls is
.
180
Dividing both the numerator and denominator by the
lowest prime number, i.e. 2, gives
36
72
=
180 90
Dividing both the numerator and denominator again by
2 gives
72
36 18
=
=
180 90 45
Addition and subtraction of fractions is demonstrated
in the following worked examples.
Problem 4.
1 1
+
3 2
(i) Make the denominators the same for each fraction. The lowest number that both denominators
divide into is called the lowest common multiple
or LCM (see Chapter 1, page 5). In this example,
the LCM of 3 and 2 is 6.
(ii) 3 divides into 6 twice. Multiplying both numera1
tor and denominator of by 2 gives
3
1 2
=
3 6
=
(iii) 2 divides into 6, 3 times. Multiplying both numer1
ator and denominator of by 3 gives
2
2 will not divide into both 18 and 45, so dividing both the
numerator and denominator by the next prime number,
i.e. 3, gives
72
36 18
6
=
=
=
180 90 45 15
Dividing both the numerator and denominator again by
3 gives
Simplify
1 3
=
2 6
=
1 1 2 3 5
+ = + =
3 2 6 6 6
+
(iv) Hence,
=
36 18
6
2
72
=
=
=
=
180 90 45 15 5
72
2
= in its simplest form.
180 5
2
Thus, of the students are girls.
5
So
2.2
Adding and subtracting fractions
When the denominators of two (or more) fractions to
be added are the same, the fractions can be added 'on
sight'.
2 5 7
3 1 4
+ = and + = .
9 9 9
8 8 8
In the latter example, dividing both the 4 and the 8 by
4 1
4 gives = , which is the simplified answer. This is
8 2
called cancelling.
Problem 5.
Simplify
7
3
−
4 16
(i) Make the denominators the same for each fraction. The lowest common multiple (LCM) of 4
and 16 is 16.
(ii) 4 divides into 16, 4 times. Multiplying both
3
numerator and denominator of by 4 gives
4
For example,
3 12
=
4 16
(iii)
=
7
already has a denominator of 16.
16
28.
Chapter 3
Decimals
3.1
Introduction
The decimal system of numbers is based on the digits
0 to 9.
There are a number of everyday occurrences in which we
use decimal numbers. For example, a radio is, say, tuned
to 107.5 MHz FM; 107.5 is an example of a decimal
number.
In a shop, a pair of trainers cost, say, £57.95; 57.95 is
another example of a decimal number. 57.95 is a decimal
fraction, where a decimal point separates the integer, i.e.
57, from the fractional part, i.e. 0.95
57.95 actually means (5 × 10) + (7 × 1)
1
1
+ 9×
+ 5×
10
100
3.2 Converting decimals to fractions
and vice-versa
Converting decimals to fractions and vice-versa is
demonstrated below with worked examples.
Problem 1. Convert 0.375 to a proper fraction in
its simplest form
0.375 × 1000
(i) 0.375 may be written as
i.e.
1000
375
0.375 =
1000
(ii) Dividing both numerator and denominator by 5
375
75
gives
=
1000 200
DOI: 10.1016/B978-1-85617-697-2.00003-X
(iii) Dividing both numerator and denominator by 5
15
75
=
again gives
200 40
(iv) Dividing both numerator and denominator by 5
15 3
again gives
=
40 8
Since both 3 and 8 are only divisible by 1, we cannot
3
'cancel' any further, so is the 'simplest form' of the
8
fraction.
3
Hence, the decimal fraction 0.375 = as a proper
8
fraction.
Problem 2.
Convert 3.4375 to a mixed number
0.4375 × 10000
(i) 0.4375 may be written as
i.e.
10000
4375
0.4375 =
10000
(ii) Dividing both numerator and denominator by 25
4375
175
gives
=
10000 400
(iii) Dividing both numerator and denominator by 5
175 35
gives
=
400 80
(iv) Dividing both numerator and denominator by 5
35
7
again gives
=
80 16
Since both 5 and 16 are only divisible by 1, we
7
cannot 'cancel' any further, so
is the 'lowest
16
form' of the fraction.
7
(v) Hence, 0.4375 =
16
7
as a mixed
Thus, the decimal fraction 3.4375 = 3
16
number.
29.
Decimals
7
Problem 3. Express as a decimal fraction
8
To convert a proper fraction to a decimal fraction, the
numerator is divided by the denominator.
0.8 7 5
8 7.0 0 0
(i) 8 into 7 will not go. Place the 0 above the 7.
(ii) Place the decimal point above the decimal point
of 7.000
(iii) 8 into 70 goes 8, remainder 6. Place the 8 above
the first zero after the decimal point and carry the
6 remainder to the next digit on the right, making
it 60.
(iv) 8 into 60 goes 7, remainder 4. Place the 7 above
the next zero and carry the 4 remainder to the next
digit on the right, making it 40.
(v) 8 into 40 goes 5, remainder 0. Place 5 above the
next zero.
7
Hence, the proper fraction = 0.875 as a decimal
8
fraction.
13
Problem 4. Express 5 as a decimal fraction
16
17
(vi) 16 into 80 goes 5, remainder 0. Place the 5 above
the next zero.
13
(vii) Hence,
= 0.8125
16
13
Thus, the mixed number 5 = 5.8125 as a decimal
16
fraction.
Now try the following Practice Exercise
Practice Exercise 8 Converting decimals to
fractions and vice-versa (answers on
page 341)
1.
Convert 0.65 to a proper fraction.
2.
Convert 0.036 to a proper fraction.
3.
Convert 0.175 to a proper fraction.
4.
Convert 0.048 to a proper fraction.
5.
Convert the following to proper fractions.
(a) 0.65
(b) 0.84
(c) 0.0125
(d) 0.282
(e) 0.024
6. Convert 4.525 to a mixed number.
7.
Convert 23.44 to a mixed number.
8.
Convert 10.015 to a mixed number.
9.
Convert 6.4375 to a mixed number.
10. Convert the following to mixed numbers.
For mixed numbers it is only necessary to convert the
proper fraction part of the mixed number to a decimal
fraction.
0.8 1 2 5
16 13.0 0 0 0
(a) 1.82
(d) 15.35
12.
(iii) 16 into 130 goes 8, remainder 2. Place the 8 above
the first zero after the decimal point and carry the
2 remainder to the next digit on the right, making
it 20.
(iv) 16 into 20 goes 1, remainder 4. Place the 1 above
the next zero and carry the 4 remainder to the next
digit on the right, making it 40.
(v) 16 into 40 goes 2, remainder 8. Place the 2 above
the next zero and carry the 8 remainder to the next
digit on the right, making it 80.
(c) 14.125
5
as a decimal fraction.
8
11
Express 6 as a decimal fraction.
16
7
as a decimal fraction.
Express
32
3
Express 11 as a decimal fraction.
16
9
Express as a decimal fraction.
32
11. Express
(i) 16 into 13 will not go. Place the 0 above the 3.
(ii) Place the decimal point above the decimal point
of 13.0000
(b) 4.275
(e) 16.2125
13.
14.
15.
3.3 Significant figures and decimal
places
A number which can be expressed exactly as a
decimal fraction is called a terminating decimal.
30.
18 Basic Engineering Mathematics
For example,
3
3
= 3.1825 is a terminating decimal
16
A number which cannot be expressed exactly as a decimal fraction is called a non-terminating decimal. For
example,
5
1 = 1.7142857 . . . is a non-terminating decimal
7
Note that the zeros to the right of the decimal point do
not count as significant figures.
Now try the following Practice Exercise
Practice Exercise 9 Significant figures and
decimal places (answers on page 341)
1. Express 14.1794 correct to 2 decimal places.
The answer to a non-terminating decimal may be
expressed in two ways, depending on the accuracy
required:
2. Express 2.7846 correct to 4 significant figures.
(a)
4. Express 43.2746 correct to 4 significant
figures.
correct to a number of significant figures, or
(b) correct to a number of decimal places i.e. the
number of figures after the decimal point.
The last digit in the answer is unaltered if the next digit
on the right is in the group of numbers 0, 1, 2, 3 or 4.
For example,
3. Express 65.3792 correct to 2 decimal places.
5. Express 1.2973 correct to 3 decimal places.
6. Express 0.0005279 correct to 3 significant
figures.
1.714285 . . . = 1.714 correct to 4 significant figures
= 1.714 correct to 3 decimal places
since the next digit on the right in this example is 2.
The last digit in the answer is increased by 1 if the next
digit on the right is in the group of numbers 5, 6, 7, 8 or
9. For example,
1.7142857 . . . = 1.7143 correct to 5 significant figures
= 1.7143 correct to 4 decimal places
since the next digit on the right in this example is 8.
Problem 5. Express 15.36815 correct to
(a) 2 decimal places, (b) 3 significant figures,
(c) 3 decimal places, (d) 6 significant figures
(a)
15.36815 = 15.37 correct to 2 decimal places.
(b) 15.36815 = 15.4 correct to 3 significant figures.
(c)
15.36815 = 15.368 correct to 3 decimal places.
(d) 15.36815 = 15.3682 correct to 6 significant
figures.
Problem 6. Express 0.004369 correct to
(a) 4 decimal places, (b) 3 significant figures
(a)
0.004369 = 0.0044 correct to 4 decimal places.
(b) 0.004369 = 0.00437 correct to 3 significant
figures.
3.4 Adding and subtracting decimal
numbers
When adding or subtracting decimal numbers, care
needs to be taken to ensure that the decimal points
are beneath each other. This is demonstrated in the
following worked examples.
Problem 7. Evaluate 46.8 + 3.06 + 2.4 + 0.09
and give the answer correct to 3 significant figures
The decimal points are placed under each other as
shown. Each column is added, starting from the
right.
46.8
3.06
2.4
+ 0.09
52.35
11 1
(i) 6 + 9 = 15. Place 5 in the hundredths column.
Carry 1 in the tenths column.
(ii) 8 + 0 + 4 + 0 + 1 (carried) = 13. Place the 3 in
the tenths column. Carry the 1 into the units
column.
(iii) 6 + 3 + 2 + 0 + 1 (carried) = 12. Place the 2 in
the units column. Carry the 1 into the tens column.
31.
Decimals
(iv) 4 + 1(carried) = 5. Place the 5 in the hundreds
column.
Hence,
46.8 + 3.06 + 2.4 + 0.09 = 52.35
= 52.4, correct to 3
significant figures
19
Now try the following Practice Exercise
Practice Exercise 10 Adding and
subtracting decimal numbers (answers on
page 341)
Determine the following without using a calculator.
1.
Evaluate 738.22 −349.38 −427.336 +56.779,
correct to 1 decimal place.
7.
(iii) 3 − 8 is not possible; therefore 'borrow' from the
hundreds column. This gives 13 − 8 = 5. Place
the 5 in the units column.
Evaluate 483.24 − 120.44 − 67.49, correct
to 4 significant figures.
6.
(ii) 3 − 7 is not possible; therefore 'borrow' 1 from
the units column. This gives 13 − 7 = 6. Place the
6 in the tenths column.
Evaluate 67.841 − 249.55 + 56.883, correct
to 2 decimal places.
5.
(i) 6 − 7 is not possible; therefore 'borrow' 1 from
the tenths column. This gives 16 − 7 = 9. Place
the 9 in the hundredths column.
Evaluate 68.92 + 34.84 − 31.223, correct to
4 significant figures.
4.
64.46
−28.77
35.69
Evaluate 378.1 − 48.85, correct to 1 decimal
place.
3.
As with addition, the decimal points are placed under
each other as shown.
Evaluate 37.69 + 42.6, correct to 3 significant figures.
2.
Problem 8. Evaluate 64.46 − 28.77 and give the
answer correct to 1 decimal place
Determine the dimension marked x in the
length of the shaft shown in Figure 3.1. The
dimensions are in millimetres.
82.92
(iv) 5 − 2 = 3. Place the 3 in the hundreds column.
27.41
8.32
x
34.67
Hence,
64.46 − 28.77 = 35.69
= 35.7 correct to 1 decimal place
Problem 9. Evaluate 312.64 − 59.826 − 79.66 +
38.5 and give the answer correct to 4 significant
figures
The sum of the positive decimal fractions
= 312.64 + 38.5 = 351.14.
The sum of the negative decimal fractions
= 59.826 + 79.66 = 139.486.
Taking the sum of the negative decimal fractions from
the sum of the positive decimal fractions gives
351.140
−139.486
211.654
Hence, 351.140 − 139.486 = 211.654 = 211.7, correct to 4 significant figures.
Figure 3.1
3.5 Multiplying and dividing decimal
numbers
When multiplying decimal fractions:
(a)
the numbers are multiplied as if they were integers,
and
(b) the position of the decimal point in the answer is
such that there are as many digits to the right of
it as the sum of the digits to the right of the decimal points of the two numbers being multiplied
together.
This is demonstrated in the following worked examples.
34.
Chapter 4
Using a calculator
4.1
Introduction
In engineering, calculations often need to be performed.
For simple numbers it is useful to be able to use mental arithmetic. However, when numbers are larger an
electronic calculator needs to be used.
There are several calculators on the market, many of
which will be satisfactory for our needs. It is essential
to have a scientific notation calculator which will have
all the necessary functions needed and more.
This chapter assumes you have a CASIO fx-83ES
calculator, or similar, as shown in Figure 4.1.
Besides straightforward addition, subtraction, multiplication and division, which you will already be able
to do, we will check that you can use squares, cubes,
powers, reciprocals, roots, fractions and trigonometric functions (the latter in preparation for Chapter 21).
There are several other functions on the calculator
which we do not need to concern ourselves with at this
level.
4.2 Adding, subtracting, multiplying
and dividing
Problem 1. Evaluate 364.7 ÷ 57.5 correct to 3
decimal places
(i) Type in 364.7
(ii) Press ÷.
(iii) Type in 57.5
3647
appears.
575
(v) Press the S ⇔ D function and the decimal answer
6.34260869 . . . appears.
(iv) Press = and the fraction
Alternatively, after step (iii) press Shift and = and the
decimal will appear.
Hence, 364.7 ÷ 57.5 = 6.343 correct to 3 decimal
places.
Problem 2.
Evaluate
significant figures
12.47 × 31.59
correct to 4
70.45 × 0.052
(i) Type in 12.47
(ii) Press ×.
(iii) Type in 31.59
Initially, after switching on, press Mode.
Of the three possibilities, use Comp, which is achieved
by pressing 1.
Next, press Shift followed by Setup and, of the eight
possibilities, use Mth IO, which is achieved by pressing 1.
By all means experiment with the other menu options –
refer to your 'User's guide'.
All calculators have +, −, × and ÷ functions and
these functions will, no doubt, already have been used
in calculations.
DOI: 10.1016/B978-1-85617-697-2.00004-1
(iv) Press ÷.
(v) The denominator must have brackets; i.e. press (.
(vi) Type in 70.45 × 0.052 and complete the bracket;
i.e. ).
(vii) Press = and the answer 107.530518. . . appears.
Hence,
figures.
12.47 × 31.59
= 107.5 correct to 4 significant
70.45 × 0.052
43.
Using a calculator
7.
8.
9.
Energy, E joules, is given by the formula
1
E = L I 2 . Evaluate the energy when
2
L = 5.5 and I = 1.2
The current I amperes in an a.c. circuit
V
. Evaluate the
is given by I =
2 + X 2)
(R
current when V = 250, R=11.0 and X =16.2
Distance s metres is given by the formula
1
s = ut + at 2. If u = 9.50, t = 4.60 and
2
a = −2.50, evaluate the distance.
12. Deduce the following information from the
train timetable shown in Table 4.1.
(a)
At what time should a man catch a train
at Fratton to enable him to be in London
Waterloo by 14.23 h?
(b) A girl leaves Cosham at 12.39 h and travels to Woking. How long does the journey take? And, if the distance between
Cosham and Woking is 55 miles, calculate the average speed of the train.
10. The area, A, of any triangle is given
√
by A = [s(s − a)(s − b)(s − c)] where
a +b+c
s=
. Evaluate the area, given
2
a = 3.60 cm, b = 4.00 cm and c = 5.20 cm.
(c)
A man living at Havant has a meeting
in London at 15.30 h. It takes around
25 minutes on the underground to reach
his destination from London Waterloo.
What train should he catch from Havant
to comfortably make the meeting?
11. Given that a = 0.290, b = 14.86, c = 0.042,
d = 31.8 and e = 0.650, evaluate v given that
ab d
−
v=
c
e
(d) Nine trains leave Portsmouth harbour
between 12.18 h and 13.15 h. Which
train should be taken for the shortest
journey time?
31
45.
Chapter 5
Percentages
5.1
Introduction
Percentages are used to give a common standard. The
use of percentages is very common in many aspects
of commercial life, as well as in engineering. Interest
rates, sale reductions, pay rises, exams and VAT are all
examples of situations in which percentages are used.
For this chapter you will need to know about decimals
and fractions and be able to use a calculator.
We are familiar with the symbol for percentage, i.e. %.
Here are some examples.
• Interest rates indicate the cost at which we can borrow money. If you borrow £8000 at a 6.5% interest
rate for a year, it will cost you 6.5% of the amount
borrowed to do so, which will need to be repaid along
with the original money you borrowed. If you repay
the loan in 1 year, how much interest will you have
paid?
• A pair of trainers in a shop cost £60. They are advertised in a sale as 20% off. How much will you
pay?
• If you earn £20 000 p.a. and you receive a 2.5%
pay rise, how much extra will you have to spend the
following year?
• A book costing £18 can be purchased on the internet
for 30% less. What will be its cost?
When we have completed his chapter on percentages
you will be able to understand how to perform the above
calculations.
Percentages are fractions having 100 as their denom40
is written as 40%
inator. For example, the fraction
100
and is read as 'forty per cent'.
The easiest way to understand percentages is to go
through some worked examples.
DOI: 10.1016/B978-1-85617-697-2.00005-3
5.2
Percentage calculations
5.2.1 To convert a decimal to a percentage
A decimal number is converted to a percentage by
multiplying by 100.
Problem 1. Express 0.015 as a percentage
To express a decimal number as a percentage, merely
multiply by 100, i.e.
0.015 = 0.015 × 100%
= 1.5%
Multiplying a decimal number by 100 means moving
the decimal point 2 places to the right.
Problem 2. Express 0.275 as a percentage
0.275 = 0.275 × 100%
= 27.5%
5.2.2 To convert a percentage to a decimal
A percentage is converted to a decimal number by
dividing by 100.
Problem 3. Express 6.5% as a decimal number
6.5% =
6.5
= 0.065
100
Dividing by 100 means moving the decimal point 2
places to the left.
47.
Percentages
10. Express as percentages, correct to 3 significant
figures,
19
11
7
(b)
(c) 1
(a)
33
24
16
11. Place the following in order of size, the smallest first, expressing each as a percentage
correct to 1 decimal place.
12
9
5
6
(a)
(b)
(c)
(d)
21
17
9
11
12. Express 65% as a fraction in its simplest form.
13. Express 31.25% as a fraction in its simplest
form.
14. Express 56.25% as a fraction in its simplest
form.
15. Evaluate A to J in the following table.
Decimal number Fraction Percentage
0.5
A
B
C
1
4
D
E
F
30
G
3
5
H
I
J
35
12.5
× 32
100
= 4 minutes
12.5% of 32 minutes =
Hence, new time taken = 32 − 4 = 28 minutes.
Alternatively, if the time is reduced by 12.5%, it now
takes 100% − 12.5% = 87.5% of the original time, i.e.
87.5
× 32
100
= 28 minutes
87.5% of 32 minutes =
Problem 12. A 160 GB iPod is advertised as
costing £190 excluding VAT. If VAT is added at
17.5%, what will be the total cost of the iPod?
17.5
× 190 = £33.25
100
Total cost of iPod = £190 + £33.25 = £223.25
VAT = 17.5% of £190 =
A quicker method to determine the total cost is: 1.175 ×
£190 = £223.25
85
5.3.2 Expressing one quantity as a
percentage of another quantity
To express one quantity as a percentage of another quantity, divide the first quantity by the second then multiply
by 100.
Problem 13. Express 23 cm as a percentage of
72 cm, correct to the nearest 1%
5.3
Further percentage calculations
5.3.1 Finding a percentage of a quantity
To find a percentage of a quantity, convert the percentage
to a fraction (by dividing by 100) and remember that 'of'
means multiply.
Problem 10. Find 27% of £65
27
× 65
27% of £65 =
100
= £17.55 by calculator
Problem 11. In a machine shop, it takes 32
minutes to machine a certain part. Using a new tool,
the time can be reduced by 12.5%. Calculate the
new time taken
23
× 100%
72
= 31.94444 . . .%
23 cm as a percentage of 72 cm =
= 32% correct to
the nearest 1%
Problem 14. Express 47 minutes as a percentage
of 2 hours, correct to 1 decimal place
Note that it is essential that the two quantities are in the
same units.
Working in minute units, 2 hours = 2 × 60
= 120 minutes
47 minutes as a percentage
47
× 100%
of 120 min =
120
= 39.2% correct to
1 decimal place
48.
36 Basic Engineering Mathematics
5.3.3 Percentage change
Percentage change is given by
new value − original value
× 100%.
original value
Problem 15. A box of resistors increases in price
from £45 to £52. Calculate the percentage change
in cost, correct to 3 significant figures
new value − original value
× 100%
original value
52 − 45
7
=
× 100% =
× 100
45
45
= 15.6% = percentage change in cost
% change =
Problem 16. A drilling speed should be set to
400 rev/min. The nearest speed available on the
machine is 412 rev/min. Calculate the percentage
overspeed
available speed − correct speed
×100%
% overspeed =
correct speed
12
412 − 400
×100% =
×100%
=
400
400
= 3%
Now try the following Practice Exercise
Practice Exercise 22 Further percentages
(answers on page 342)
1.
Calculate 43.6% of 50 kg.
2.
Determine 36% of 27 m.
3.
Calculate, correct to 4 significant figures,
(a) 18% of 2758 tonnes
6.
A block of Monel alloy consists of 70%
nickel and 30% copper. If it contains 88.2 g
of nickel, determine the mass of copper in the
block.
7.
An athlete runs 5000 m in 15 minutes 20
seconds. With intense training, he is able to
reduce this time by 2.5%. Calculate his new
time.
8.
A copper alloy comprises 89% copper, 1.5%
iron and the remainder aluminium. Find the
amount of aluminium, in grams, in a 0.8 kg
mass of the alloy.
9.
A computer is advertised on the internet at
£520, exclusive of VAT. If VAT is payable at
17.5%, what is the total cost of the computer?
10. Express 325 mm as a percentage of 867 mm,
correct to 2 decimal places.
11. A child sleeps on average 9 hours 25 minutes
per day. Express this as a percentage of the
whole day, correct to 1 decimal place.
12. Express 408 g as a percentage of 2.40 kg.
13. When signing a new contract, a Premiership
footballer's pay increases from £15 500 to
£21 500 per week. Calculate the percentage
pay increase, correct to 3 significant figures.
14. A metal rod 1.80 m long is heated and its
length expands by 48.6 mm. Calculate the
percentage increase in length.
15. 12.5% of a length of wood is 70 cm. What is
the full length?
16. A metal rod, 1.20 m long, is heated and
its length expands by 42 mm. Calculate the
percentage increase in length.
(b) 47% of 18.42 grams
(c)
147% of 14.1 seconds.
4.
When 1600 bolts are manufactured, 36 are
unsatisfactory. Determine the percentage that
is unsatisfactory.
5.
Express
(a) 140 kg as a percentage of 1 t.
5.4
More percentage calculations
5.4.1 Percentage error
Percentage error =
error
× 100%
correct value
(b) 47 s as a percentage of 5 min.
(c)
13.4 cm as a percentage of 2.5 m.
Problem 17. The length of a component is
measured incorrectly as 64.5 mm. The actual length
49.
Percentages
is 63 mm. What is the percentage error in the
measurement?
error
× 100%
correct value
64.5 − 63
× 100%
=
63
1.5
150
=
× 100% =
%
63
63
= 2.38%
37
Problem 20. A couple buys a flat and make an
18% profit by selling it 3 years later for £153 400.
Calculate the original cost of the house
% error =
In this case, it is an 18% increase in price, so we
new value
use
× 100, i.e. a plus sign in the
100 + % change
denominator.
new value
× 100
Original cost =
100 + % change
The percentage measurement error is 2.38% too high,
which is sometimes written as + 2.38% error.
=
=
Problem 18. The voltage across a component in
an electrical circuit is calculated as 50 V using
Ohm's law. When measured, the actual voltage is
50.4 V. Calculate, correct to 2 decimal places, the
percentage error in the calculation
error
% error =
× 100%
correct value
50.4 − 50
× 100%
=
50.4
0.4
40
=
× 100% =
%
50.4
50.4
= 0.79%
The percentage error in the calculation is 0.79% too
low, which is sometimes written as −0.79% error.
Problem 21. An electrical store makes 40% profit
on each widescreen television it sells. If the selling
price of a 32 inch HD television is £630, what was
the cost to the dealer?
In this case, it is a 40% mark-up in price, so we
new value
use
× 100, i.e. a plus sign in the
100 + % change
denominator.
new value
Dealer cost =
× 100
100 + % change
=
In this case, it is a 35% reduction in price, so we
new value
use
× 100, i.e. a minus sign in the
100 − % change
denominator.
new value
× 100
Original price =
100 − % change
149.5
× 100
=
100 − 35
149.5
14 950
=
× 100 =
65
65
= £230
630
× 100
100 + 40
=
new value
× 100%
100 ± % change
Problem 19. A man pays £149.50 in a sale for a
DVD player which is labelled '35% off'. What was
the original price of the DVD player?
1 5 340 000
153 400
× 100 =
118
118
= £130 000
5.4.2 Original value
Original value =
153 400
× 100
100 + 18
630
63 000
× 100 =
140
140
= £450
The dealer buys from the manufacturer for £450 and
sells to his customers for £630.
5.4.3 Percentage increase/decrease and
interest
New value =
100 + % increase
× original value
100
Problem 22. £3600 is placed in an ISA account
which pays 6.25% interest per annum. How much is
the investment worth after 1 year?
50.
38 Basic Engineering Mathematics
100 + 6.25
× £3600
100
106.25
=
× £3600
100
= 1.0625 × £3600
Value after 1 year =
annum. Calculate the value of the investment
after 2 years.
8.
An electrical contractor earning £36 000 per
annum receives a pay rise of 2.5%. He
pays 22% of his income as tax and 11%
on National Insurance contributions. Calculate the increase he will actually receive per
month.
9.
Five mates enjoy a meal out. With drinks, the
total bill comes to £176. They add a 12.5%
tip and divide the amount equally between
them. How much does each pay?
= £3825
Problem 23. The price of a fully installed
combination condensing boiler is increased by 6.5%.
It originally cost £2400. What is the new price?
100 + 6.5
× £2, 400
100
106.5
=
× £2, 400 = 1.065 × £2, 400
100
= £2,556
New price =
Now try the following Practice Exercise
Practice Exercise 23 Further percentages
(answers on page 342)
10. In December a shop raises the cost of a 40
inch LCD TV costing £920 by 5%. It does
not sell and in its January sale it reduces
the TV by 5%. What is the sale price of
the TV?
11. A man buys a business and makes a 20%
profit when he sells it three years later for
£222 000. What did he pay originally for the
business?
1.
A machine part has a length of 36 mm. The
length is incorrectly measured as 36.9 mm.
Determine the percentage error in the measurement.
12. A drilling machine should be set to
250 rev/min. The nearest speed available on
the machine is 268 rev/min. Calculate the
percentage overspeed.
2.
When a resistor is removed from an electrical circuit the current flowing increases from
450 μA to 531 μA. Determine the percentage
increase in the current.
13. Two kilograms of a compound contain 30%
of element A, 45% of element B and 25% of
element C. Determine the masses of the three
elements present.
3.
In a shoe shop sale, everything is advertised
as '40% off'. If a lady pays £186 for a pair of
Jimmy Choo shoes, what was their original
price?
4.
Over a four year period a family home
increases in value by 22.5% to £214 375.
What was the value of the house 4 years ago?
14. A concrete mixture contains seven parts by
volume of ballast, four parts by volume of
sand and two parts by volume of cement.
Determine the percentage of each of these
three constituents correct to the nearest 1%
and the mass of cement in a two tonne dry
mix, correct to 1 significant figure.
5.
An electrical retailer makes a 35% profit on
all its products. What price does the retailer
pay for a dishwasher which is sold for £351?
6.
The cost of a sports car is £23 500 inclusive
of VAT at 17.5%. What is the cost of the car
without the VAT added?
7.
£8000 is invested in bonds at a building society which is offering a rate of 6.75% per
15. In a sample of iron ore, 18% is iron. How
much ore is needed to produce 3600 kg of
iron?
16. A screw's dimension is 12.5 ± 8% mm. Calculate the maximum and minimum possible
length of the screw.
17. The output power of an engine is 450 kW. If
the efficiency of the engine is 75%, determine
the power input.
52.
Chapter 6
Ratio and proportion
6.1
Introduction
Ratio is a way of comparing amounts of something; it
shows how much bigger one thing is than the other.
Some practical examples include mixing paint, sand
and cement, or screen wash. Gears, map scales, food
recipes, scale drawings and metal alloy constituents all
use ratios.
Two quantities are in direct proportion when they
increase or decrease in the same ratio. There are several practical engineering laws which rely on direct
proportion. Also, calculating currency exchange rates
and converting imperial to metric units rely on direct
proportion.
Sometimes, as one quantity increases at a particular
rate, another quantity decreases at the same rate; this is
called inverse proportion. For example, the time taken
to do a job is inversely proportional to the number of
people in a team: double the people, half the time.
When we have completed this chapter on ratio and
proportion you will be able to understand, and confidently perform, calculations on the above topics.
For this chapter you will need to know about decimals
and fractions and to be able to use a calculator.
6.2
Ratios
Ratios are generally shown as numbers separated by a
colon (:) so the ratio of 2 and 7 is written as 2 :7 and we
read it as a ratio of 'two to seven.'
Some practical examples which are familiar include:
• Mixing 1 measure of screen wash to 6 measures of
water; i.e., the ratio of screen wash to water is 1 : 6
• Mixing 1 shovel of cement to 4 shovels of sand; i.e.,
the ratio of cement to sand is 1 : 4
• Mixing 3 parts of red paint to 1 part white, i.e., the
ratio of red to white paint is 3 :1
DOI: 10.1016/B978-1-85617-697-2.00006-5
Ratio is the number of parts to a mix. The paint mix is
4 parts total, with 3 parts red and 1 part white. 3 parts
red paint to 1 part white paint means there is
3
1
red paint to white paint
4
4
Here are some worked examples to help us understand
more about ratios.
Problem 1. In a class, the ratio of female to male
students is 6 :27. Reduce the ratio to its simplest
form
(i) Both 6 and 27 can be divided by 3.
(ii) Thus, 6 : 27 is the same as 2 : 9.
6 :27 and 2 :9 are called equivalent ratios.
It is normal to express ratios in their lowest, or simplest,
form. In this example, the simplest form is 2 : 9 which
means for every 2 females in the class there are 9 male
students.
Problem 2. A gear wheel having 128 teeth is in
mesh with a 48-tooth gear. What is the gear ratio?
Gear ratio = 128 :48
A ratio can be simplified by finding common factors.
(i) 128 and 48 can both be divided by 2, i.e. 128 :48
is the same as 64 :24
(ii) 64 and 24 can both be divided by 8, i.e. 64 :24 is
the same as 8 :3
(iii) There is no number that divides completely into
both 8 and 3 so 8 :3 is the simplest ratio, i.e. the
gear ratio is 8 : 3
53.
Ratio and proportion
Thus, 128 :48 is equivalent to 64 :24 which is equivalent
to 8 :3 and 8 : 3 is the simplest form.
Problem 3. A wooden pole is 2.08 m long. Divide
it in the ratio of 7 to 19
(i) Since the ratio is 7 :19, the total number of parts
is 7 + 19 = 26 parts.
(ii) 26 parts corresponds to 2.08 m = 208 cm, hence,
208
= 8.
1 part corresponds to
26
(iii) Thus, 7 parts corresponds to 7 × 8 = 56 cm and
19 parts corresponds to 19 × 8 = 152 cm.
Hence, 2.08 m divides in the ratio of 7 : 19 as 56 cm
to 152 cm.
(Check: 56 + 152 must add up to 208, otherwise an error
would have been made.)
Problem 4. In a competition, prize money of
£828 is to be shared among the first three in the
ratio 5 : 3 : 1
(i) Since the ratio is 5 : 3 :1 the total number of parts
is 5 + 3 + 1 = 9 parts.
(ii) 9 parts corresponds to £828.
828
= £92, 3 parts cor9
responds to 3 × £92 = £276 and 5 parts corresponds to 5 × £92 = £460.
(iii) 1 part corresponds to
Hence, £828 divides in the ratio of 5 : 3 : 1 as £460 to
£276 to £92. (Check: 460 + 276 + 92 must add up to
828, otherwise an error would have been made.)
Problem 5. A map scale is 1 : 30 000. On the map
the distance between two schools is 6 cm.
Determine the actual distance between the schools,
giving the answer in kilometres
Actual distance between schools
= 6 × 30 000 cm = 180 000 cm
=
180,000
m = 1800 m
100
1800
m = 1.80 km
1000
(1 mile ≈ 1.6 km, hence the schools are just over 1 mile
apart.)
=
41
Now try the following Practice Exercise
Practice Exercise 24
page 342)
Ratios (answers on
1. In a box of 333 paper clips, 9 are defective.
Express the number of non-defective paper
clips as a ratio of the number of defective paper
clips, in its simplest form.
2. A gear wheel having 84 teeth is in mesh with
a 24-tooth gear. Determine the gear ratio in its
simplest form.
3. In a box of 2000 nails, 120 are defective.
Express the number of non-defective nails as
a ratio of the number of defective ones, in its
simplest form.
4. A metal pipe 3.36 m long is to be cut into two
in the ratio 6 to 15. Calculate the length of each
piece.
5. The instructions for cooking a turkey say that
it needs to be cooked 45 minutes for every
kilogram. How long will it take to cook a 7 kg
turkey?
6. In a will, £6440 is to be divided among three
beneficiaries in the ratio 4 :2 : 1. Calculate the
amount each receives.
7. A local map has a scale of 1 :22 500. The distance between two motorways is 2.7 km. How
far are they apart on the map?
8. Prize money in a lottery totals £3801 and is
shared among three winners in the ratio 4 :2 : 1.
How much does the first prize winner receive?
Here are some further worked examples on ratios.
Problem 6. Express 45 p as a ratio of £7.65 in its
simplest form
(i) Changing both quantities to the same units, i.e. to
pence, gives a ratio of 45 :765
(ii) Dividing both quantities by 5 gives
45 : 765 ≡ 9 :153
(iii) Dividing both quantities by 3 gives
9 : 153 ≡ 3 : 51
(iv) Dividing both quantities by 3 again gives
3 : 51 ≡ 1 : 17
54.
42 Basic Engineering Mathematics
Thus, 45 p as a ratio of £7.65 is 1 : 17
45 : 765, 9 :153, 3 :51 and 1 : 17 are equivalent ratios
and 1 :17 is the simplest ratio.
2. In a laboratory, acid and water are mixed in the
ratio 2 :5. How much acid is needed to make
266 ml of the mixture?
Problem 7. A glass contains 30 ml of whisky
which is 40% alcohol. If 45 ml of water is added and
the mixture stirred, what is now the alcohol content?
3. A glass contains 30 ml of gin which is 40%
alcohol. If 18 ml of water is added and the
mixture stirred, determine the new percentage
alcoholic content.
(i) The 30 ml of whisky contains 40%
40
× 30 = 12 ml.
alcohol =
100
(ii) After 45 ml of water is added we have 30 + 45
= 75 ml of fluid, of which alcohol is 12 ml.
12
(iii) Fraction of alcohol present =
75
12
(iv) Percentage of alcohol present =
× 100%
75
4. A wooden beam 4 m long weighs 84 kg. Determine the mass of a similar beam that is 60 cm
long.
= 16%.
Problem 8. 20 tonnes of a mixture of sand and
gravel is 30% sand. How many tonnes of sand must
be added to produce a mixture which is 40% gravel?
(i) Amount of sand in 20 tonnes = 30% of 20 t
30
=
× 20 = 6 t.
100
(ii) If the mixture has 6 t of sand then amount of
gravel = 20 − 6 = 14 t.
(iii) We want this 14 t of gravel to be 40% of the
14
t and 100% of the
new mixture. 1% would be
40
14
mixture would be
× 100 t = 35 t.
40
(iv) If there is 14 t of gravel then amount of sand
= 35 − 14 = 21 t.
(v) We already have 6 t of sand, so amount of sand
to be added to produce a mixture with 40%
gravel = 21 − 6 = 15 t.
5. An alloy is made up of metals P and Q in the
ratio 3.25 :1 by mass. How much of P has to
be added to 4.4 kg of Q to make the alloy?
6. 15 000 kg of a mixture of sand and gravel is
20% sand. Determine the amount of sand that
must be added to produce a mixture with 30%
gravel.
6.3
Two quantities are in direct proportion when they
increase or decrease in the same ratio. For example,
if 12 cans of lager have a mass of 4 kg, then 24 cans of
lager will have a mass of 8 kg; i.e., if the quantity of cans
doubles then so does the mass. This is direct proportion.
In the previous section we had an example of mixing
1 shovel of cement to 4 shovels of sand; i.e., the ratio
of cement to sand was 1 : 4. So, if we have a mix of 10
shovels of cement and 40 shovels of sand and we wanted
to double the amount of the mix then we would need
to double both the cement and sand, i.e. 20 shovels of
cement and 80 shovels of sand. This is another example
of direct proportion.
Here are three laws in engineering which involve direct
proportion:
(a)
(Note 1 tonne = 1000 kg.)
Now try the following Practice Exercise
Practice Exercise 25 Further ratios
(answers on page 342)
1. Express 130 g as a ratio of 1.95 kg.
Direct proportion
Hooke's law states that, within the elastic limit of
a material, the strain ε produced is directly proportional to the stress σ producing it, i.e. ε ∝ σ (note
than '∝' means 'is proportional to').
(b) Charles's law states that, for a given mass of gas
at constant pressure, the volume V is directly proportional to its thermodynamic temperature T , i.e.
V ∝ T.
(c)
Ohm's law states that the current I flowing
through a fixed resistance is directly proportional
to the applied voltage V , i.e. I ∝ V .
55.
Ratio and proportion
Here are some worked examples to help us understand
more about direct proportion.
Problem 9. 3 energy saving light bulbs cost
£7.80. Determine the cost of 7 such light bulbs
(i) 3 light bulbs cost £7.80
7.80
= £2.60
3
Hence, 7 light bulbs cost 7 × £2.60 = £18.20
(ii) Therefore, 1 light bulb costs
Problem 10. If 56 litres of petrol costs £59.92,
calculate the cost of 32 litres
(i) 56 litres of petrol costs £59.92
59.92
= £1.07
56
Hence, 32 litres cost 32 × 1.07 = £34.24
(ii) Therefore, 1 litre of petrol costs
Problem 11. Hooke's law states that stress, σ , is
directly proportional to strain, ε, within the elastic
limit of a material. When, for mild steel, the stress
is 63 MPa, the strain is 0.0003. Determine (a) the
value of strain when the stress is 42 MPa, (b) the
value of stress when the strain is 0.00072
(a) Stress is directly proportional to strain.
(i) When the stress is 63 MPa, the strain is
0.0003
(ii) Hence, a stress of 1 MPa corresponds to a
0.0003
strain of
63
(iii) Thus, the value of strain when the stress is
0.0003
42 MPa =
× 42 = 0.0002
63
(b) Strain is proportional to stress.
(i) When the strain is 0.0003, the stress is
63 MPa.
(ii) Hence, a strain of 0.0001 corresponds to
63
MPa.
3
(iii) Thus, the value of stress when the strain is
63
0.00072 =
× 7.2 = 151.2 MPa.
3
Problem 12. Charles's law states that for a given
mass of gas at constant pressure, the volume is
directly proportional to its thermodynamic
temperature. A gas occupies a volume of 2.4 litres
43
at 600 K. Determine (a) the temperature when the
volume is 3.2 litres, (b) the volume at 540 K
(a)
Volume is directly proportional to temperature.
(i) When the volume is 2.4 litres, the temperature is 600 K.
(ii) Hence, a volume of 1 litre corresponds to a
600
temperature of
K.
2.4
(iii) Thus, the temperature when the volume is
600
× 3.2 = 800 K.
3.2 litres =
2.4
(b) Temperature is proportional to volume.
(i) When the temperature is 600 K, the volume
is 2.4 litres.
(ii) Hence, a temperature of 1 K corresponds to
2.4
a volume of
litres.
600
(iii) Thus, the volume at a temperature of
2.4
540 K =
× 540 = 2.16 litres.
600
Now try the following Practice Exercise
Practice Exercise 26 Direct proportion
(answers on page 342)
1. 3 engine parts cost £208.50. Calculate the cost
of 8 such parts.
2. If 9 litres of gloss white paint costs £24.75,
calculate the cost of 24 litres of the same paint.
3. The total mass of 120 household bricks is
57.6 kg. Determine the mass of 550 such
bricks.
4. A simple machine has an effort : load ratio of
3 :37. Determine the effort, in grams, to lift a
load of 5.55 kN.
5. If 16 cans of lager weighs 8.32 kg, what will
28 cans weigh?
6. Hooke's law states that stress is directly proportional to strain within the elastic limit of
a material. When, for copper, the stress is
60 MPa, the strain is 0.000625. Determine
(a) the strain when the stress is 24 MPa and
(b) the stress when the strain is 0.0005
56.
44 Basic Engineering Mathematics
7. Charles's law states that volume is directly
proportional to thermodynamic temperature
for a given mass of gas at constant pressure.
A gas occupies a volume of 4.8 litres at 330 K.
Determine (a) the temperature when the volume is 6.4 litres and (b) the volume when the
temperature is 396 K.
Here are some further worked examples on direct
proportion.
Problem 13. Some guttering on a house has to
decline by 3 mm for every 70 cm to allow rainwater
to drain. The gutter spans 8.4 m. How much lower
should the low end be?
(i) The guttering has to decline in the ratio 3 :700 or
3
700
(ii) If d is the vertical drop in 8.4 m or 8400 mm, then
d
the decline must be in the ratio d : 8400 or
8400
3
d
=
(iii) Now
8400 700
(ii) Hence, a current of 1 A corresponds to a
90
mV = 30 mV.
voltage of
3
(iii) Thus, when the current is 4.2 A, the
voltage = 30 × 4.2 = 126 mV.
Problem 15. Some approximate imperial to
metric conversions are shown in Table 6.1. Use the
table to determine
(a) the number of millimetres in 12.5 inches
(b) a speed of 50 miles per hour in kilometres
per hour
(c)
the number of miles in 300 km
(d) the number of kilograms in 20 pounds weight
(e)
the number of pounds and ounces in
56 kilograms (correct to the nearest ounce)
(f) the number of litres in 24 gallons
(g) the number of gallons in 60 litres
Table 6.1
length
1 mile = 1.6 km
weight
(iv) Cross-multiplying gives 700 ×d = 8400 ×3 from
8400 × 3
which,
d=
700
(a)
Current is directly proportional to the voltage.
(i) When voltage is 90 mV, the current is 3 A.
(ii) Hence, a voltage of 1 mV corresponds to a
3
current of
A.
90
(iii) Thus, when the voltage is 60 mV, the
3
current = 60 ×
= 2A.
90
(b) Voltage is directly proportional to the current.
(i) When current is 3 A, the voltage is 90 mV.
2.2 lb = 1 kg
(1 lb = 16 oz)
capacity
i.e. d = 36 mm, which is how much the lower end
should be to allow rainwater to drain.
Problem 14. Ohm's law state that the current
flowing in a fixed resistance is directly proportional
to the applied voltage. When 90 mV is applied
across a resistor the current flowing is 3 A.
Determine (a) the current when the voltage is
60 mV and (b) the voltage when the current is 4.2 A
1 inch = 2.54 cm
1.76 pints = 1 litre
(8 pints = 1 gallon)
(a)
12.5 inches = 12.5 × 2.54 cm = 31.75 cm
31.73 cm = 31.75 × 10 mm = 317.5 mm
(b) 50 m.p.h. = 50 × 1.6 km/h = 80 km/h
300
miles = 186.5 miles
1.6
20
kg = 9.09 kg
(d) 20 lb =
2.2
(e) 56 kg = 56 × 2.2 lb = 123.2 lb
(c)
300 km =
0.2 lb = 0.2 × 16 oz = 3.2 oz = 3 oz, correct to
the nearest ounce.
Thus, 56 kg = 123 lb 3 oz, correct to the nearest
ounce.
(f ) 24 gallons = 24 × 8 pints = 192 pints
192
litres = 109.1 litres
192 pints =
1.76
57.
Ratio and proportion
(g) 60 litres = 60 × 1.76 pints = 105.6 pints
105.6
105.6 pints =
gallons = 13.2 gallons
8
Problem 16. Currency exchange rates for five
countries are shown in Table 6.2. Calculate
(a) how many euros £55 will buy
60 mV is applied across a circuit a current of
24 μA flows. Determine (a) the current flowing
when the p.d. is 5 V and (b) the p.d. when the
current is 10 mA.
(c)
the number of pounds sterling which can be
exchanged for 6405 kronor
2.
The tourist rate for the Swiss franc is quoted in
a newspaper as £1 = 1.92 fr. How many francs
can be purchased for £326.40?
3.
If 1 inch = 2.54 cm, find the number of millimetres in 27 inches.
4.
If 2.2 lb = 1 kg and 1lb = 16 oz, determine the
number of pounds and ounces in 38 kg (correct
to the nearest ounce).
5.
If 1 litre = 1.76 pints and 8 pints = 1 gallon,
determine (a) the number of litres in 35 gallons
and (b) the number of gallons in 75 litres.
6.
Hooke's law states that stress is directly proportional to strain within the elastic limit of a
material. When for brass the stress is 21 MPa,
the strain is 0.00025. Determine the stress
when the strain is 0.00035.
7.
If 12 inches = 30.48 cm, find the number of
millimetres in 23 inches.
8.
(b) the number of Japanese yen which can be
bought for £23
The tourist rate for the Canadian dollar is
quoted in a newspaper as £1 = 1.84 fr. How
many Canadian dollars can be purchased for
£550?
(d) the number of American dollars which can be
purchased for £92.50
(e)
the number of pounds sterling which can be
exchanged for 2925 Swiss francs
45
Table 6.2
France
Japan
£1 = 185 yen
Norway
£1 = 10.50 kronor
Switzerland
£1 = 1.95 francs
USA
(a)
£1 = 1.25 euros
£1 = 1.80 dollars
£1 = 1.25 euros, hence £55 = 55 × 1.25 euros
= 68.75 euros.
(b) £1 = 185 yen, hence £23 = 23 × 185 yen
= 4255 yen.
(c)
6405
10.50
= £610.
£1 = 10.50 kronor, hence 6405 lira = £
(d) £1 = 1.80 dollars, hence
£92.50 = 92.50 × 1.80 dollars = $166.50
(e)
£1 = 1.95 Swiss francs, hence
2925
= £1500
2925 pesetas = £
1.95
Now try the following Practice Exercise
Practice Exercise 27 Further direct
proportion (answers on page 342)
1.
Ohm's law states that current is proportional
to p.d. in an electrical circuit. When a p.d. of
6.4
Inverse proportion
Two variables, x and y, are in inverse proportion to one
1
1
k
another if y is proportional to , i.e. y α or y = or
x
x
x
k = x y where k is a constant, called the coefficient of
proportionality.
Inverse proportion means that, as the value of one variable increases, the value of another decreases, and that
their product is always the same.
For example, the time for a journey is inversely proportional to the speed of travel. So, if at 30 m.p.h. a journey
is completed in 20 minutes, then at 60 m.p.h. the journey
would be completed in 10 minutes. Double the speed,
half the journey time. (Note that 30 × 20 = 60 × 10.)
In another example, the time needed to dig a hole is
inversely proportional to the number of people digging.
So, if 4 men take 3 hours to dig a hole, then 2 men
58.
46 Basic Engineering Mathematics
(working at the same rate) would take 6 hours. Half the
men, twice the time. (Note that 4 × 3 = 2 × 6.)
Here are some worked examples on inverse proportion.
Problem 17. It is estimated that a team of four
designers would take a year to develop an
engineering process. How long would three
designers take?
If 4 designers take 1 year, then 1 designer would take
4 years to develop the process. Hence, 3 designers would
4
take years, i.e. 1 year 4 months.
3
Problem 18. A team of five people can deliver
leaflets to every house in a particular area in four
hours. How long will it take a team of three people?
If 5 people take 4 hours to deliver the leaflets, then
1 person would take 5 × 4 = 20 hours. Hence, 3 peo2
20
hours, i.e. 6 hours, i.e. 6 hours
ple would take
3
3
40 minutes.
Problem 19. The electrical resistance R of a
piece of wire is inversely proportional to the
cross-sectional area A. When A = 5 mm2 ,
R = 7.02 ohms. Determine (a) the coefficient of
proportionality and (b) the cross-sectional area
when the resistance is 4 ohms
1.5 × 106 pascals, determine (a) the coefficient of
proportionality and (b) the volume if the pressure is
changed to 4 × 106 pascals
(a)
V ∝
1
k
i.e. V = or k = pV . Hence, the
p
p
coefficient of proportionality, k
= (1.5 × 106)(0.08) = 0.12 × 106
(b) Volume,V =
k
0.12 × 106
= 0.03 m3
=
p
4 × 106
Now try the following Practice Exercise
Practice Exercise 28 Further inverse
proportion (answers on page 342)
1. A 10 kg bag of potatoes lasts for a week with a
family of 7 people. Assuming all eat the same
amount, how long will the potatoes last if there
are only two in the family?
2. If 8 men take 5 days to build a wall, how long
would it take 2 men?
Rα
coefficient of proportionality, k = (7.2)(5) = 36
(a)
3. If y is inversely proportional to x and
y = 15.3 when x = 0.6, determine (a) the
coefficient of proportionality, (b) the value of
y when x is 1.5 and (c) the value of x when y
is 27.2
4. A car travelling at 50 km/h makes a journey in
70 minutes. How long will the journey take at
70 km/h?
1
k
, i.e. R =
or k = R A. Hence, when
A
A
R = 7.2 and A = 5, the
k
. Hence, when R = 4,
R
36
= 9 mm2
the cross sectional area, A =
4
(b) Since k = R A then A =
Problem 20. Boyle's law states that, at constant
temperature, the volume V of a fixed mass of gas is
inversely proportional to its absolute pressure p. If
a gas occupies a volume of 0.08 m3 at a pressure of
5. Boyle's law states that, for a gas at constant
temperature, the volume of a fixed mass of
gas is inversely proportional to its absolute
pressure. If a gas occupies a volume of 1.5 m3
at a pressure of 200 × 103 pascals, determine
(a) the constant of proportionality, (b) the
volume when the pressure is 800 × 103 pascals and (c) the pressure when the volume is
1.25 m3.
65.
Chapter 8
Units, prefixes and
engineering notation
8.1
Introduction
Table 8.1 Basic SI units
Quantity
Unit
Symbol
Length
metre
m (1 m = 100 cm
= 1000 mm)
Mass
kilogramkg (1 kg = 1000 g)
80 kV = 80 × 103 V, which means 80 000 volts
Time
second s
25 mA = 25 × 10−3 A,
which means 0.025 amperes
Electric current
ampere A
Thermodynamic
temperature
kelvin
Luminous
intensity
candela cd
Amount of
substance
mole
Of considerable importance in engineering is a knowledge of units of engineering quantities, the prefixes used
with units, and engineering notation.
We need to know, for example, that
and
and
−9
50 nF = 50 × 10 F,
which means 0.000000050 farads
This is explained in this chapter.
8.2
SI units
The system of units used in engineering and science
is the Système Internationale d'Unités (International
System of Units), usually abbreviated to SI units, and is
based on the metric system. This was introduced in 1960
and has now been adopted by the majority of countries
as the official system of measurement.
The basic seven units used in the SI system are listed in
Table 8.1 with their symbols.
There are, of course, many units other than these seven.
These other units are called derived units and are
defined in terms of the standard units listed in the table.
For example, speed is measured in metres per second,
therefore using two of the standard units, i.e. length and
time.
DOI: 10.1016/B978-1-85617-697-2.00008-9
K (K = ◦C + 273)
mol
Some derived units are given special names. For example, force = mass × acceleration has units of kilogram
metre per second squared, which uses three of the base
units, i.e. kilograms, metres and seconds. The unit of kg
m/s2 is given the special name of a Newton.
Table 8.2 contains a list of some quantities and their
units that are common in engineering.
8.3
Common prefixes
SI units may be made larger or smaller by using prefixes
which denote multiplication or division by a particular
amount.
66.
54 Basic Engineering Mathematics
Table 8.2 Some quantities and their units that are common in engineering
Quantity
Unit
Symbol
Length
metre
m
Area
square metre
m2
Volume
cubic metre
m3
Mass
kilogram
kg
Time
second
s
Electric current
ampere
A
Speed, velocity
metre per second
m/s
Acceleration
metre per second squared
m/s2
Density
kilogram per cubic metre
kg/m3
Temperature
kelvin or Celsius
K or ◦ C
Angle
radian or degree
rad or ◦
Angular velocity
radian per second
rad/s
Frequency
hertz
Hz
Force
newton
N
Pressure
pascal
Pa
Energy, work
joule
J
Power
watt
W
Charge, quantity of electricity
coulomb
C
Electric potential
volt
V
Capacitance
farad
F
Electrical resistance
ohm
Inductance
henry
H
Moment of force
newton metre
Nm
The most common multiples are listed in Table 8.3.
A knowledge of indices is needed since all of the prefixes
are powers of 10 with indices that are a multiple of 3.
Here are some examples of prefixes used with engineering units.
A frequency of 15 GHz means 15 × 109 Hz, which is
15 000 000 000 hertz,
i.e. 15 gigahertz is written as 15 GHz and is equal to 15
thousand million hertz.
(Instead of writing 15 000 000 000 hertz, it is much
neater, takes up less space and prevents errors caused
by having so many zeros, to write the frequency as
15 GHz.)
A voltage of 40 MV means 40 × 106 V, which is
40 000 000 volts,
i.e. 40 megavolts is written as 40 MV and is equal to 40
million volts.
An inductance of 12 mH means 12 × 10−3 H or
12
12
H, which is 0.012 H,
H or
103
1000
i.e. 12 millihenrys is written as 12 mH and is equal to
12 thousandths of a henry.
67.
55
Units, prefixes and engineering notation
Table 8.3 Common SI multiples
Prefix
Name
Meaning
G
giga
multiply by 109
i.e. × 1 000 000 000
M
mega
multiply by 106
i.e. × 1 000 000
103
i.e. × 1 000
k
kilo
multiply by
m
milli
multiply by 10−3
i.e. ×
1
1
=
= 0.001
103 1000
μ
micro
multiply by 10−6
i.e. ×
1
1
= 0.000001
=
106 1 000 000
n
nano
multiply by 10−9
i.e. ×
1
1
= 0.000 000 001
=
109 1 000 000 000
p
pico
multiply by 10−12
i.e. ×
1
1
=
=
12
10
1 000 000 000 000
0.000 000 000 001
150
A time of 150 ns means 150 × 10−9 s or 9 s, which
10
is 0.000 000 150 s,
i.e. 150 nanoseconds is written as 150 ns and is equal to
150 thousand millionths of a second.
A force of 20 kN means 20 × 103 N, which is 20 000
newtons,
i.e. 20 kilonewtons is written as 20 kN and is equal to
20 thousand newtons.
30
A charge of 30 μC means 30 × 10−6 C or 6 C, which
10
is 0.000 030 C,
i.e. 30 microcoulombs is written as 30 μC and is equal
to 30 millionths of a coulomb.
45
A capacitance of 45 pF means 45 × 10−12 F or 12 F,
10
which is 0.000 000 000 045 F,
i.e. 45 picofarads is written as 45 pF and is equal to 45
million millionths of a farad.
In engineering it is important to understand what such
quantities as 15 GHz, 40 MV, 12 mH, 150 ns, 20 kN,
30 μC and 45 pF mean.
3.
State the SI unit of area.
4.
State the SI unit of velocity.
5.
State the SI unit of density.
6.
State the SI unit of energy.
7.
State the SI unit of charge.
8.
State the SI unit of power.
9.
State the SI unit of angle.
10. State the SI unit of electric potential.
11. State which quantity has the unit kg.
12. State which quantity has the unit symbol
.
13. State which quantity has the unit Hz.
14. State which quantity has the unit m/s2 .
15. State which quantity has the unit symbol A.
16. State which quantity has the unit symbol H.
17. State which quantity has the unit symbol m.
Now try the following Practice Exercise
18. State which quantity has the unit symbol K.
19. State which quantity has the unit Pa.
Practice Exercise 32 SI units and common
prefixes (answers on page 343)
1.
State the SI unit of volume.
2.
State the SI unit of capacitance.
20. State which quantity has the unit rad/s.
21. What does the prefix G mean?
22. What is the symbol and meaning of the prefix
milli?
73.
Chapter 9
Basic algebra
9.1
Introduction
We are already familiar with evaluating formulae using
a calculator from Chapter 4.
For example, if the length of a football pitch is L and its
width is b, then the formula for the area A is given by
A= L ×b
This is an algebraic equation.
If L = 120 m and b = 60 m, then the area
A = 120 × 60 = 7200 m2.
The total resistance, RT , of resistors R1 , R2 and R3
connected in series is given by
RT = R1 + R2 + R3
This is an algebraic equation.
If R1 = 6.3 k , R2 = 2.4 k and R3 = 8.5 k , then
RT = 6.3 + 2.4 + 8.5 = 17.2 k
The temperature in Fahrenheit, F, is given by
9
F = C + 32
5
where C is the temperature in Celsius. This is an
algebraic equation.
9
If C = 100◦C, then F = × 100 + 32
5
= 180 + 32 = 212◦F.
If you can cope with evaluating formulae then you will
be able to cope with algebra.
9.2
Basic operations
Algebra merely uses letters to represent numbers.
If, say, a, b, c and d represent any four numbers then in
algebra:
DOI: 10.1016/B978-1-85617-697-2.00009-0
(a) a + a + a + a = 4a. For example, if a = 2, then
2 + 2 + 2 + 2 = 4 × 2 = 8.
(b) 5b means 5 × b. For example, if b = 4, then
5b = 5 × 4 = 20.
(c) 2a + 3b + a − 2b = 2a + a + 3b − 2b = 3a + b
Only similar terms can be combined in algebra.
The 2a and the +a can be combined to give 3a
and the 3b and −2b can be combined to give 1b,
which is written as b.
In addition, with terms separated by + and − signs,
the order in which they are written does not matter.
In this example, 2a + 3b + a − 2b is the same as
2a + a + 3b − 2b, which is the same as 3b + a +
2a − 2b, and so on. (Note that the first term, i.e.
2a, means +2a.)
(d) 4abcd = 4 × a × b × c × d
For example, if a = 3, b = −2, c = 1 and d = −5,
then 4abcd = 4 × 3 × −2 × 1 × −5 = 120. (Note
that − × − = +)
(e) (a)(c)(d) means a × c × d
Brackets are often used instead of multiplication
signs. For example, (2)(5)(3) means 2 × 5 × 3 =
30.
(f ) ab = ba
If a = 2 and b = 3 then 2 × 3 is exactly the same
as 3 × 2, i.e. 6.
(g) b2 = b × b. For
32 = 3 × 3 = 9.
example,
if
b = 3,
then
(h) a3 = a × a × a For example, if a = 2, then
23 = 2 × 2 × 2 = 8.
Here are some worked examples to help get a feel for
basic operations in this introduction to algebra.
85.
Chapter 11
Solving simple equations
11.1
Introduction
3x − 4 is an example of an algebraic expression.
3x − 4 = 2 is an example of an algebraic equation (i.e.
it contains an '=' sign).
An equation is simply a statement that two expressions
are equal.
Hence, A = πr 2 (where A is the area of a circle
of radius r)
9
F = C + 32 (which relates Fahrenheit and
5
Celsius temperatures)
and
y = 3x + 2 (which is the equation of a
straight line graph)
are all examples of equations.
11.2
Solving equations
To 'solve an equation' means 'to find the value of the
unknown'. For example, solving 3x − 4 = 2 means that
the value of x is required.
In this example, x = 2. How did we arrive at x = 2?
This is the purpose of this chapter – to show how to
solve such equations.
Many equations occur in engineering and it is essential
that we can solve them when needed.
Here are some examples to demonstrate how simple
equations are solved.
Problem 1. Solve the equation 4x = 20
Dividing each side of the equation by 4 gives
20
4x
=
4
4
i.e. x = 5 by cancelling, which is the solution to the
equation 4x = 20.
DOI: 10.1016/B978-1-85617-697-2.00011-9
The same operation must be applied to both sides of an
equation so that the equality is maintained.
We can do anything we like to an equation, as long
as we do the same to both sides. This is, in fact, the
only rule to remember when solving simple equations
(and also when transposing formulae, which we do in
Chapter 12).
Problem 2. Solve the equation
2x
=6
5
2x
= 5(6)
5
Cancelling and removing brackets gives 2x = 30
Multiplying both sides by 5 gives
5
Dividing both sides of the equation by 2 gives
2x
30
=
2
2
Cancelling gives
x = 15
2x
which is the solution of the equation
= 6.
5
Problem 3. Solve the equation a − 5 = 8
Adding 5 to both sides of the equation gives
a−5+5 = 8+5
i.e.
a = 8+5
i.e.
a = 13
which is the solution of the equation a − 5 = 8.
Note that adding 5 to both sides of the above equation
results in the −5 moving from the LHS to the RHS, but
the sign is changed to +.
Problem 4. Solve the equation x + 3 = 7
Subtracting 3 from both sides gives x + 3 − 3 = 7 − 3
i.e.
x = 7−3
i.e.
x =4
which is the solution of the equation x + 3 = 7.
86.
74 Basic Engineering Mathematics
Note that subtracting 3 from both sides of the above
equation results in the +3 moving from the LHS to the
RHS, but the sign is changed to –. So, we can move
straight from x + 3 = 7 to x = 7 − 3.
Thus, a term can be moved from one side of an equation to the other as long as a change in sign is made.
Problem 5.
Solve the equation 6x + 1 = 2x + 9
In such equations the terms containing x are grouped
on one side of the equation and the remaining terms
grouped on the other side of the equation. As in Problems 3 and 4, changing from one side of an equation to
the other must be accompanied by a change of sign.
6x + 1 = 2x + 9
Since
then
6x − 2x = 9 − 1
i.e.
By substituting p = 3 into the original equation, the
solution may be checked.
LHS = 4 − 3(3) = 4 − 9 = −5
RHS = 2(3) − 11 = 6 − 11 = −5
Since LHS = RHS, the solution p = 3 must be correct.
If, in this example, the unknown quantities had been
grouped initially on the LHS instead of the RHS, then
−3 p − 2 p = −11 − 4
Cancelling gives
8
4x
=
4
4
x=2
which is the solution of the equation 6x + 1 = 2x + 9.
In the above examples, the solutions can be checked.
Thus, in Problem 5, where 6x + 1 = 2x + 9, if x = 2,
then
LHS of equation = 6(2) + 1 = 13
RHS of equation = 2(2) + 9 = 13
Since the left hand side (LHS) equals the right hand
side (RHS) then x = 2 must be the correct solution of
the equation.
When solving simple equations, always check your
answers by substituting your solution back into the
original equation.
Problem 6.
Solve the equation 4 − 3 p = 2 p − 11
In order to keep the p term positive the terms in p are
moved to the RHS and the constant terms to the LHS.
Similar to Problem 5, if 4 − 3 p = 2 p − 11
then
4 + 11 = 2 p + 3 p
i.e.
Dividing both sides by 5 gives
Cancelling gives
15 = 5 p
15 5 p
=
5
5
3 = p or p = 3
which is the solution of the equation 4 − 3 p = 2 p − 11.
−5 p −15
=
−5
−5
p=3
from which,
and
as before.
It is often easier, however, to work with positive values
where possible.
4x = 8
Dividing both sides by 4 gives
−5 p = −15
i.e.
Problem 7.
Solve the equation 3(x − 2) = 9
Removing the bracket gives 3x − 6 = 9
Rearranging gives
3x = 9 + 6
i.e.
3x = 15
Dividing both sides by 3 gives
x=5
which is the solution of the equation 3(x − 2) = 9.
The equation may be checked by substituting x = 5 back
into the original equation.
Problem 8. Solve the equation
4(2r − 3) − 2(r − 4) = 3(r − 3) − 1
Removing brackets gives
8r − 12 − 2r + 8 = 3r − 9 − 1
Rearranging gives 8r − 2r − 3r = −9 − 1 + 12 − 8
3r = −6
−6
= −2
Dividing both sides by 3 gives r =
3
i.e.
which is the solution of the equation
4(2r − 3) − 2(r − 4) = 3(r − 3) − 1.
The solution may be checked by substituting r = −2
back into the original equation.
LHS = 4(−4 − 3) − 2(−2 − 4) = −28 + 12 = −16
RHS = 3(−2 − 3) − 1 = −15 − 1 = −16
Since LHS = RHS then r = −2 is the correct solution.
93.
Solving simple equations
7. An alloy contains 60% by weight of copper,
the remainder being zinc. How much copper
must be mixed with 50 kg of this alloy to give
an alloy containing 75% copper?
9. Applying the principle of moments to a beam
results in the following equation:
8. A rectangular laboratory has a length equal to
one and a half times its width and a perimeter
of 40 m. Find its length and width.
where F is the force in newtons. Determine
the value of F.
F × 3 = (5 − F) × 7
81
95.
Chapter 12
Transposing formulae
12.1
Introduction
V
In the formula I = , I is called the subject of the
R
formula.
Similarly, in the formula y = mx + c, y is the subject of
the formula.
When a symbol other than the subject is required to
be the subject, the formula needs to be rearranged to
make a new subject. This rearranging process is called
transposing the formula or transposition.
For example, in the above formulae,
if I =
y −c
m
How did we arrive at these transpositions? This is the
purpose of this chapter — to show how to transpose formulae. A great many equations occur in engineering and
it is essential that we can transpose them when needed.
12.2
Transposing formulae
There are no new rules for transposing formulae.
The same rules as were used for simple equations in
Chapter 11 are used; i.e., the balance of an equation
must be maintained: whatever is done to one side of
an equation must be done to the other.
It is best that you cover simple equations before trying
this chapter.
Here are some worked examples to help understanding
of transposing formulae.
Problem 1. Transpose p = q + r + s to make r
the subject
DOI: 10.1016/B978-1-85617-697-2.00012-0
q +r +s = p
(1)
From Chapter 11 on simple equations, a term can be
moved from one side of an equation to the other side as
long as the sign is changed.
Rearranging gives r = p − q − s.
Mathematically, we have subtracted q + s from both
sides of equation (1).
Problem 2. If a + b = w − x + y, express x as
the subject
V
then V = IR
R
and if y = mx + c then x =
The object is to obtain r on its own on the LHS of the
equation. Changing the equation around so that r is on
the LHS gives
As stated in Problem 1, a term can be moved from one
side of an equation to the other side but with a change
of sign.
Hence, rearranging gives x = w + y − a − b
Problem 3. Transpose v = f λ to make λ the
subject
v = f λ relates velocity v, frequency f and wavelength λ
Rearranging gives
Dividing both sides by f gives
Cancelling gives
fλ=v
v
fλ
=
f
f
v
λ=
f
Problem 4. When a body falls freely through a
height h, the velocity v is given by v 2 = 2gh.
Express this formula with h as the subject
102.
Chapter 13
Solving simultaneous
equations
13.1
Introduction
Only one equation is necessary when finding the value of
a single unknown quantity (as with simple equations
in Chapter 11). However, when an equation contains
two unknown quantities it has an infinite number of
solutions. When two equations are available connecting
the same two unknown values then a unique solution
is possible. Similarly, for three unknown quantities it is
necessary to have three equations in order to solve for a
particular value of each of the unknown quantities, and
so on.
Equations which have to be solved together to find
the unique values of the unknown quantities, which are
true for each of the equations, are called simultaneous
equations.
Two methods of solving simultaneous equations analytically are:
(a)
by substitution, and
(b) by elimination.
(A graphical solution of simultaneous equations is
shown in Chapter 19.)
Problem 1. Solve the following equations for x
and y, (a) by substitution and (b) by elimination
x + 2y = −1
4x − 3y = 18
(a)
(1)
(2)
By substitution
From equation (1): x = −1 − 2y
Substituting this expression for x into equation (2)
gives
4(−1 − 2y) − 3y = 18
This is now a simple equation in y.
Removing the bracket gives
−4 − 8y − 3y = 18
−11y = 18 + 4 = 22
y=
22
= −2
−11
Substituting y = −2 into equation (1) gives
x + 2(−2) = −1
x − 4 = −1
x = −1 + 4 = 3
13.2 Solving simultaneous equations
in two unknowns
The method of solving simultaneous equations is
demonstrated in the following worked problems.
DOI: 10.1016/B978-1-85617-697-2.00013-2
Thus, x = 3 and y = −2 is the solution to the
simultaneous equations.
Check: in equation (2), since x = 3 and y = −2,
LHS = 4(3) − 3(−2) = 12 + 6 = 18 = RHS
111.
Solving simultaneous equations
When c = 52,
T = 100, hence
52 = a + 100b
(1)
When c = 172, T = 400, hence
172 = a + 400b
(2)
Equation (2) – equation (1) gives
120 = 300b
from which,
b=
120
= 0.4
300
Substituting b = 0.4 in equation (1) gives
52 = a + 100(0.4)
a = 52 − 40 = 12
Hence, a = 12 and b = 0.4
Now try the following Practice Exercise
Practice Exercise 52 Practical problems
involving simultaneous equations (answers
on page 345)
1. In a system of pulleys, the effort P required to
raise a load W is given by P = aW + b, where
a and b are constants. If W = 40 when P = 12
and W = 90 when P = 22, find the values of
a and b.
2. Applying Kirchhoff's laws to an electrical
circuit produces the following equations:
5 = 0.2I1 + 2(I1 − I2 )
12 = 3I2 + 0.4I2 − 2(I1 − I2 )
Determine the values of currents I1 and I2
3. Velocity v is given by the formula v = u + at .
If v = 20 when t = 2 and v = 40 when t = 7,
find the values of u and a. Then, find the
velocity when t = 3.5
4. Three new cars and 4 new vans supplied to a
dealer together cost £97 700 and 5 new cars
and 2 new vans of the same models cost
£103 100. Find the respective costs of a car
and a van.
5.
y = mx + c is the equation of a straight line
of slope m and y-axis intercept c. If the line
passes through the point where x = 2 and
99
y = 2, and also through the point where x = 5
and y = 0.5, find the slope and y-axis intercept
of the straight line.
6. The resistance R ohms of copper wire at t ◦C
is given by R = R0 (1 + αt ), where R0 is the
resistance at 0◦ C and α is the temperature coefficient of resistance. If R = 25.44 at 30◦C
and R = 32.17 at 100◦ C, find α and R0
7. The molar heat capacity of a solid compound
is given by the equation c = a + bT . When
c = 60, T = 100 and when c = 210, T = 400.
Find the values of a and b.
8. In an engineering process, two variables p and
q are related by q = ap + b/ p, where a and b
are constants. Evaluate a and b if q = 13 when
p = 2 and q = 22 when p = 5.
9. In a system of forces, the relationship between
two forces F1 and F2 is given by
5F1 + 3F2 + 6 = 0
3F1 + 5F2 + 18 = 0
Solve for F1 and F2
13.6 Solving simultaneous equations
in three unknowns
Equations containing three unknowns may be solved
using exactly the same procedures as those used with
two equations and two unknowns, providing that there
are three equations to work with. The method is demonstrated in the following worked problem.
Problem 18. Solve the simultaneous equations.
x +y+z =4
(1)
2x − 3y + 4z = 33
(2)
3x − 2y − 2z = 2
(3)
There are a number of ways of solving these equations.
One method is shown below.
The initial object is to produce two equations with two
unknowns. For example, multiplying equation (1) by 4
and then subtracting this new equation from equation (2)
will produce an equation with only x and y involved.
114.
Chapter 14
Solving quadratic equations
14.1
Introduction
As stated in Chapter 11, an equation is a statement
that two quantities are equal and to 'solve an equation'
means 'to find the value of the unknown'. The value of
the unknown is called the root of the equation.
A quadratic equation is one in which the highest
power of the unknown quantity is 2. For example,
x 2 − 3x + 1 = 0 is a quadratic equation.
There are four methods of solving quadratic equations. These are:
(a)
by factorization (where possible),
(b) by 'completing the square',
(c)
by using the 'quadratic formula', or
(d) graphically (see Chapter 19).
14.2 Solution of quadratic equations
by factorization
Multiplying out (x + 1)(x − 3) gives x 2 − 3x + x − 3
i.e. x 2 − 2x − 3. The reverse process of moving from
x 2 − 2x − 3 to (x + 1)(x − 3) is called factorizing.
If the quadratic expression can be factorized this
provides the simplest method of solving a quadratic
equation.
For example, if x 2 − 2x − 3 = 0, then, by factorizing
(x + 1)(x − 3) = 0
Hence, either
or
(x + 1) = 0, i.e. x = −1
(x − 3) = 0, i.e. x = 3
Hence, x = −1 and x = 3 are the roots of the quadratic equation x 2 − 2x − 3 = 0.
The technique of factorizing is often one of trial and
error.
DOI: 10.1016/B978-1-85617-697-2.00014-4
Problem 1. Solve the equation x 2 + x − 6 = 0 by
factorization
The factors of x 2 are x and x. These are placed in
brackets:
(x
)(x
)
The factors of −6 are +6 and −1, or −6 and +1, or +3
and −2, or −3 and +2.
The only combination to give a middle term of +x is
+3 and −2,
i.e.
x 2 + x − 6 = (x + 3)(x − 2)
The quadratic equation x 2 + x − 6 = 0 thus becomes
(x + 3)(x − 2) = 0
Since the only way that this can be true is for either the
first or the second or both factors to be zero, then
either
(x + 3) = 0, i.e. x = −3
or
(x − 2) = 0, i.e. x = 2
Hence, the roots of x 2 + x − 6 = 0 are x = −3 and
x = 2.
Problem 2. Solve the equation x 2 + 2x − 8 = 0
by factorization
The factors of x 2 are x and x. These are placed in
brackets:
(x
)(x
)
The factors of −8 are +8 and −1, or −8 and +1, or +4
and −2, or −4 and +2.
The only combination to give a middle term of +2x is
+4 and −2,
i.e.
x 2 + 2x − 8 = (x + 4)(x − 2)
(Note that the product of the two inner terms (4x) added
to the product of the two outer terms (−2x) must equal
the middle term, +2x in this case.)
121.
Solving quadratic equations
Hence, the mass will reach a height of 16 m after
0.59 s on the ascent and after 5.53 s on the descent.
Problem 25. A shed is 4.0 m long and 2.0 m wide.
A concrete path of constant width is laid all the way
around the shed. If the area of the path is 9.50 m2 ,
calculate its width to the nearest centimetre
109
If A = 482.2 and l = 15.3, then
482.2 = πr(15.3) + πr 2
πr 2 + 15.3πr − 482.2 = 0
i.e.
r 2 + 15.3r −
or
482.2
=0
π
Using the quadratic formula,
Figure 14.1 shows a plan view of the shed with its
surrounding path of width t metres.
−15.3 ±
r=
(15.3)2 − 4
−482.2
π
2
√
−15.3 ± 848.0461 −15.3 ± 29.12123
=
=
2
2
t
t
2.0 m
4.0 m
(4.0 1 2t)
SHED
Figure 14.1
Area of path = 2(2.0 × t ) + 2t (4.0 + 2t )
Hence, radius r = 6.9106 cm (or −22.21 cm, which is
meaningless and is thus ignored).
Thus, the diameter of the base = 2r = 2(6.9106)
= 13.82 cm.
Now try the following Practice Exercise
Practice Exercise 57 Practical problems
involving quadratic equations (answers on
page 346)
1.
The angle a rotating shaft turns through in t
1
seconds is given by θ = ωt + αt 2. Deter2
mine the time taken to complete 4 radians if
ω is 3.0 rad/s and α is 0.60 rad/s2.
2.
The power P developed in an electrical circuit is given by P = 10I − 8I 2 , where I is
the current in amperes. Determine the current
necessary to produce a power of 2.5 watts in
the circuit.
3.
The area of a triangle is 47.6 cm2 and its
perpendicular height is 4.3 cm more than its
base length. Determine the length of the base
correct to 3 significant figures.
4.
The sag, l, in metres in a cable stretched
between two supports, distance x m apart, is
12
given by l =
+ x. Determine the distance
x
between the supports when the sag is 20 m.
5.
9.50 = 4.0t + 8.0t + 4t 2
i.e.
The acid dissociation constant K a of ethanoic
acid is 1.8 × 10−5 mol dm−3 for a particular solution. Using the Ostwald dilution law,
4t 2+ 12.0t − 9.50 = 0
or
Hence,
t=
−(12.0) ±
(12.0)2 − 4(4)(−9.50)
2(4)
√
−12.0 ± 296.0 −12.0 ± 17.20465
=
=
8
8
i.e.
t = 0.6506 m or − 3.65058 m.
Neglecting the negative result, which is meaningless,
the width of the path, t = 0.651 m or 65 cm correct to
the nearest centimetre.
Problem 26. If the total surface area of a solid
cone is 486.2 cm2 and its slant height is 15.3 cm,
determine its base diameter.
From Chapter 27, page 245, the total surface area A of
a solid cone is given by A = πrl + πr 2 , where l is the
slant height and r the base radius.
122.
110 Basic Engineering Mathematics
x2
Ka =
, determine x, the degree of
v(1 − x)
ionization, given that v = 10 dm3 .
6.
A rectangular building is 15 m long by 11 m
wide. A concrete path of constant width is
laid all the way around the building. If the
area of the path is 60.0 m2 , calculate its width
correct to the nearest millimetre.
7.
The total surface area of a closed cylindrical
container is 20.0 m3 . Calculate the radius of
the cylinder if its height is 2.80 m.
8.
The bending moment M at a point in a beam
3x(20 − x)
, where x metres
is given by M =
2
is the distance from the point of support.
Determine the value of x when the bending
moment is 50 Nm.
9.
A tennis court measures 24 m by 11 m. In the
layout of a number of courts an area of ground
must be allowed for at the ends and at the
sides of each court. If a border of constant
width is allowed around each court and the
total area of the court and its border is 950 m2,
find the width of the borders.
10. Two resistors, when connected in series, have
a total resistance of 40 ohms. When connected in parallel their total resistance is 8.4
ohms. If one of the resistors has a resistance
of Rx , ohms,
2
(a) show that Rx − 40Rx + 336 = 0 and
For a simultaneous solution the values of y must be
equal, hence the RHS of each equation is equated.
Thus,
5x − 4 − 2x 2 = 6x − 7
Rearranging gives
5x − 4 − 2x 2 − 6x + 7 = 0
−x + 3 − 2x 2 = 0
i.e.
2x 2 + x − 3 = 0
or
(2x + 3)(x − 1) = 0
3
x = − or x = 1
2
Factorizing gives
i.e.
In the equation y = 6x − 7,
3
− 7 = −16
2
3
when x = − ,
2
y=6 −
and when x = 1,
y = 6 − 7 = −1
(Checking the result in y = 5x − 4 − 2x 2 :
3
when x = − ,
2
y =5 −
=−
3
3
−4−2 −
2
2
2
9
15
− 4 − = −16, as above,
2
2
and when x = 1, y = 5 − 4 − 2 = −1, as above.)
Hence, the simultaneous solutions occur when
3
x = − , y = −16 and when x = 1, y = −1.
2
(b) calculate the resistance of each.
Now try the following Practice Exercise
14.6 Solution of linear and quadratic
equations simultaneously
Sometimes a linear equation and a quadratic equation
need to be solved simultaneously. An algebraic method
of solution is shown in Problem 27; a graphical solution
is shown in Chapter 19, page 160.
Problem 27. Determine the values of x and y
which simultaneously satisfy the equations
y = 5x − 4 − 2x 2 and y = 6x − 7
Practice Exercise 58 Solving linear and
quadratic equations simultaneously
(answers on page 346)
Determine the solutions of the following simultaneous equations.
1. y = x 2 + x + 1
2. y = 15x 2 + 21x − 11
y = 4−x
3.
2x 2 + y = 4 + 5x
x+y =4
y = 2x − 1
123.
Chapter 15
Logarithms
15.1
Introduction to logarithms
With the use of calculators firmly established, logarithmic tables are now rarely used for calculation. However,
the theory of logarithms is important, for there are several scientific and engineering laws that involve the rules
of logarithms.
From Chapter 7, we know that
16 = 24 .
The number 4 is called the power or the exponent or
the index. In the expression 24 , the number 2 is called
the base.
In another example, we know that
64 = 82.
In this example, 2 is the power, or exponent, or index.
The number 8 is the base.
of a', i.e.
if y = a x then x = loga y
In another example, if we write down that 64 = 82 then
the equivalent statement using logarithms is log8 64 = 2.
In another example, if we write down that log3 27 = 3
then the equivalent statement using powers is 33 = 27.
So the two sets of statements, one involving powers and
one involving logarithms, are equivalent.
15.1.2 Common logarithms
From the above, if we write down that 1000 = 103, then
3 = log10 1000. This may be checked using the 'log'
button on your calculator.
Logarithms having a base of 10 are called common
logarithms and log10 is usually abbreviated to lg. The
following values may be checked using a calculator.
lg 27.5 = 1.4393 . . .
15.1.1 What is a logarithm?
Consider the expression 16 = 24 .
An alternative, yet equivalent, way of writing this
expression is log2 16 = 4.
This is stated as 'log to the base 2 of 16 equals 4'.
We see that the logarithm is the same as the power
or index in the original expression. It is the base in
the original expression that becomes the base of the
logarithm.
The two statements 16 = 24 and
log2 16 = 4 are equivalent
If we write either of them, we are automatically implying the other.
In general, if a number y can be written in the form a x ,
then the index x is called the 'logarithm of y to the base
DOI: 10.1016/B978-1-85617-697-2.00015-6
lg 378.1 = 2.5776 . . .
lg 0.0204 = −1.6903 . . .
15.1.3 Napierian logarithms
Logarithms having a base of e (where e is a mathematical constant approximately equal to 2.7183) are called
hyperbolic, Napierian or natural logarithms, and
loge is usually abbreviated to ln. The following values
may be checked using a calculator.
ln 3.65 = 1.2947 . . .
ln 417.3 = 6.0338 . . .
ln 0.182 = −1.7037 . . .
Napierian logarithms are explained further in Chapter
16, following.
139.
Exponential functions
θ1 , correct to the nearest degree, when θ2 is
50◦C, t is 30 s and τ is 60 s and
(a)
(b) the time t , correct to 1 decimal place, for θ2 to
be half the value of θ1
and C is a constant. Find pressure p when
p0 = 1.012 × 105 Pa, height h = 1420 m and
C = 71500.
3.
(a) Transposing the formula to make θ1 the subject
gives
θ2
50
θ1 =
=
−t /τ
1 − e−30/60
1−e
=
i.e.
50
50
=
1 − e−0.5 0.393469 . . .
Rt
v = 200e− L , where R = 150
and
L = 12.5 × 10−3 H. Determine (a) the
voltage when t = 160 × 10−6 s and (b) the
time for the voltage to reach 85 V.
4.
The length l metres of a metal bar at temperature t ◦ C is given by l = l0 eαt , where l0
and α are constants. Determine (a) the value
of l when l0 = 1.894, α = 2.038 × 10−4 and
t = 250 ◦C and (b) the value of l0 when
l = 2.416, t = 310 ◦C and α = 1.682 × 10−4.
5.
The temperature θ2 ◦ C of an electrical conductor at time t seconds is given by
θ2 = θ1 (1 − e−t / T ), where θ1 is the initial temperature and T seconds is a constant. Determine (a) θ2 when θ1 = 159.9 ◦C,
t = 30 s and T = 80 s and (b) the time t for
θ2 to fall to half the value of θ1 if T remains
at 80 s.
6.
A belt is in contact with a pulley for a sector of
θ = 1.12 radians and the coefficient of friction between these two surfaces is μ = 0.26.
Determine the tension on the taut side of the
belt, T newtons, when tension on the slack
side is given by T0 = 22.7 newtons, given
that these quantities are related by the law
T = T0 eμθ .
7.
The instantaneous current i at time t is
given by i = 10e−t /CR when a capacitor
is being charged. The capacitance C is
7 ×10−6 farads and the resistance R is
0.3 × 106 ohms. Determine (a) the instantaneous current when t is 2.5 seconds and (b)
the time for the instantaneous current to fall to
5 amperes. Sketch a curve of current against
time from t = 0 to t = 6 seconds.
8.
The amount of product x (in mol/cm3 )
found in a chemical reaction starting
with 2.5 mol/cm3 of reactant is given by
x = 2.5(1 − e−4t ) where t is the time, in
minutes, to form product x. Plot a graph
at 30 second intervals up to 2.5 minutes and
determine x after 1 minute.
θ1 = 127 ◦C correct to the nearest degree.
(b) Transposing to make t the subject of the formula
gives
t
θ2
= 1 − e− τ
θ1
t
θ2
−
from which, e τ = 1 −
θ1
t
θ2
Hence,
− = ln 1 −
τ
θ1
t = −τ ln 1 −
i.e.
θ2
θ1
1
Since θ2 = θ1
2
t = −60 ln 1 −
1
2
= −60 ln 0.5
= 41.59 s
Hence, the time for the temperature θ2 to be one half
of the value of θ1 is 41.6 s, correct to 1 decimal place.
Now try the following Practice Exercise
Practice Exercise 66 Laws of growth and
decay (answers on page 347)
1.
2.
The voltage drop, v volts, across an inductor
L henrys at time t seconds is given by
The temperature, T ◦C, of a cooling object
varies with time, t minutes, according
to the equation T = 150 e−0.04 t. Determine the temperature when (a) t = 0,
(b) t = 10 minutes.
The pressure p pascals at height h metres
above ground level is given by p = p0 e−h/C ,
where p0 is the pressure at ground level
127
140.
128 Basic Engineering Mathematics
9.
The current i flowing in a capacitor at
time t is given by i = 12.5(1 − e−t /CR ),
where resistance R is 30 k
and the
capacitance C is 20 μF. Determine (a)
the current flowing after 0.5 seconds and
(b) the time for the current to reach
10 amperes.
10. The amount A after n years of a sum invested
P is given by the compound interest law
A = Pern/100 , when the per unit interest rate
r is added continuously. Determine, correct
to the nearest pound, the amount after 8 years
for a sum of £1500 invested if the interest
rate is 6% per annum.
142.
Chapter 17
Straight line graphs
17.1
Introduction to graphs
17.2
A graph is a visual representation of information,
showing how one quantity varies with another related
quantity.
We often see graphs in newspapers or in business
reports, in travel brochures and government publications. For example, a graph of the share price (in pence)
over a six month period for a drinks company, Fizzy
Pops, is shown in Figure 17.1.
Generally, we see that the share price increases to a high
of 400 p in June, but dips down to around 280 p in August
before recovering slightly in September.
A graph should convey information more quickly to the
reader than if the same information was explained in
words.
When this chapter is completed you should be able to
draw up a table of values, plot co-ordinates, determine
the gradient and state the equation of a straight line
graph. Some typical practical examples are included in
which straight lines are used.
Axes, scales and co-ordinates
We are probably all familiar with reading a map to locate
a town, or a local map to locate a particular street. For
example, a street map of central Portsmouth is shown in
Figure 17.2. Notice the squares drawn horizontally and
vertically on the map; this is called a grid and enables
us to locate a place of interest or a particular road. Most
maps contain such a grid.
We locate places of interest on the map by stating a letter
and a number – this is called the grid reference.
For example, on the map, the Portsmouth & Southsea
station is in square D2, King's Theatre is in square E5,
HMS Warrior is in square A2, Gunwharf Quays is in
square B3 and High Street is in square B4.
Portsmouth & Southsea station is located by moving
horizontally along the bottom of the map until the
square labelled D is reached and then moving vertically
upwards until square 2 is met.
The letter/number, D2, is referred to as co-ordinates;
i.e., co-ordinates are used to locate the position of
Fizzy Pops
400
350
300
250
Apr 07
Figure 17.1
DOI: 10.1016/B978-1-85617-697-2.00017-X
May 07
Jun 07
Jul 07
Aug 07
Sep 07
143.
Straight line graphs
131
1
2
3
4
5
A
B
C
D
E
F
Figure 17.2 Reprinted with permission from AA Media Ltd.
a point on a map. If you are familiar with using a
map in this way then you should have no difficulties
with graphs, because similar co-ordinates are used with
graphs.
As stated earlier, a graph is a visual representation
of information, showing how one quantity varies with
another related quantity. The most common method of
showing a relationship between two sets of data is to
use a pair of reference axes – these are two lines drawn
at right angles to each other (often called Cartesian or
rectangular axes), as shown in Figure 17.3.
The horizontal axis is labelled the x-axis and the vertical axis is labelled the y-axis. The point where x is 0
and y is 0 is called the origin.
x values have scales that are positive to the right of
the origin and negative to the left. y values have scales
that are positive up from the origin and negative down
from the origin.
Co-ordinates are written with brackets and a comma
in between two numbers. For example, point A is shown
with co-ordinates (3, 2) and is located by starting at the
y
4
B (24, 3)
3
A (3, 2)
2
Origin
1
24 23 22 21 0
21
C(23, 22)
1
2
3
4
x
22
23
24
Figure 17.3
origin and moving 3 units in the positive x direction
(i.e. to the right) and then 2 units in the positive y
direction (i.e. up).
When co-ordinates are stated the first number is
always the x value and the second number is always
144.
132 Basic Engineering Mathematics
the y value. In Figure 17.3, point B has co-ordinates
(−4, 3) and point C has co-ordinates (−3, −2).
17.3
Student task
The following table gives the force F newtons which,
when applied to a lifting machine, overcomes a
corresponding load of L newtons.
Straight line graphs
The distances travelled by a car in certain periods of time are shown in the following table of
values.
F (Newtons) 19
L (Newtons)
35
50
93 125 147
40 120 230 410 540 680
1. Plot L horizontally and F vertically.
Time (s)
10
20
30
40
50
60
Distance travelled (m) 50 100 150 200 250 300
We will plot time on the horizontal (or x) axis with a
scale of 1 cm = 10 s.
We will plot distance on the vertical (or y) axis with a
scale of 1 cm = 50 m.
(When choosing scales it is better to choose ones such as
1 cm = 1 unit, 1 cm = 2 units or 1 cm = 10 units because
doing so makes reading values between these values
easier.)
With the above data, the (x, y) co-ordinates become
(time, distance) co-ordinates; i.e., the co-ordinates are
(10, 50), (20, 100), (30, 150), and so on.
The co-ordinates are shown plotted in Figure 17.4
using crosses. (Alternatively, a dot or a dot and circle
may be used, as shown in Figure 17.3.)
A straight line is drawn through the plotted coordinates as shown in Figure 17.4.
300
Distance Travelled (m)
Horizontal axis (i.e. L): 1 cm = 50 N
Vertical axis (i.e. F): 1 cm = 10 N
3. Draw the axes and label them L (newtons) for
the horizontal axis and F (newtons) for the
vertical axis.
4. Label the origin as 0.
5. Write on the horizontal scaling at 100, 200, 300,
and so on, every 2 cm.
6. Write on the vertical scaling at 10, 20, 30, and
so on, every 1 cm.
7. Plot on the graph the co-ordinates (40, 19),
(120, 35), (230, 50), (410, 93), (540, 125) and
(680, 147), marking each with a cross or a dot.
8. Using a ruler, draw the best straight line through
the points. You will notice that not all of the
points lie exactly on a straight line. This is quite
normal with experimental values. In a practical situation it would be surprising if all of the
points lay exactly on a straight line.
Distance/time graph
250
9. Extend the straight line at each end.
200
10. From the graph, determine the force applied
when the load is 325 N. It should be close
to 75 N. This process of finding an equivalent
value within the given data is called interpolation. Similarly, determine the load that a force
of 45 N will overcome. It should be close to
170 N.
150
100
50
10
Figure 17.4
2. Scales are normally chosen such that the graph
occupies as much space as possible on the
graph paper. So in this case, the following
scales are chosen.
20
30
40
Time (s)
50
60
11. From the graph, determine the force needed to
overcome a 750 N load. It should be close to
161 N. This process of finding an equivalent
145.
Straight line graphs
133
y
value outside the given data is called extrapolation. To extrapolate we need to have extended
the straight line drawn. Similarly, determine the
force applied when the load is zero. It should be
close to 11 N. The point where the straight line
crosses the vertical axis is called the verticalaxis intercept. So, in this case, the vertical-axis
intercept = 11 N at co-ordinates (0, 11).
The graph you have drawn should look something
like Figure 17.5 shown below.
6
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
y ϭ 3x ϩ 2
4
2
Ϫ1
1
0
2
x
Figure 17.6 Graph of y/x
(c)
Graph of F against L
F (Newtons)
8
Choose scales so that interpolation is made as
easy as possible. Usually scales such as 1 cm =
1 unit, 1 cm = 2 units or 1 cm = 10 units are used.
Awkward scales such as 1 cm = 3 units or 1 cm = 7
units should not be used.
(d) The scales need not start at zero, particularly when
starting at zero produces an accumulation of points
within a small area of the graph paper.
(e)
The co-ordinates, or points, should be clearly
marked. This is achieved by a cross, or a dot and
circle, or just by a dot (see Figure 17.3).
(f ) A statement should be made next to each axis
explaining the numbers represented with their
appropriate units.
0
100
200
300 400 500
L (Newtons)
600
700
800
(g) Sufficient numbers should be written next to each
axis without cramping.
Figure 17.5
In another example, let the relationship between two
variables x and y be y = 3x + 2.
When x = 0, y = 0 + 2 = 2
When x = 1, y = 3 + 2 = 5
When x = 2, y = 6 + 2 = 8, and so on.
The co-ordinates (0, 2), (1, 5) and (2, 8) have been
produced and are plotted, with others, as shown in
Figure 17.6.
When the points are joined together a straight line
graph results, i.e. y = 3x + 2 is a straight line graph.
17.3.1 Summary of general rules to be
applied when drawing graphs
(a)
Give the graph a title clearly explaining what is
being illustrated.
(b) Choose scales such that the graph occupies as
much space as possible on the graph paper
being used.
Problem 1. Plot the graph y = 4x + 3 in the
range x = −3 to x = +4. From the graph, find
(a) the value of y when x = 2.2 and (b) the value
of x when y = −3
Whenever an equation is given and a graph is required,
a table giving corresponding values of the variable is
necessary. The table is achieved as follows:
When x = −3, y = 4x + 3 = 4(−3) + 3
= −12 + 3 = −9
When x = −2, y = 4(−2) + 3
= −8 + 3 = −5, and so on.
Such a table is shown below.
x
−3
−2 −1
0 1
y
−9
−5 −1
3 7
2
3
4
11 15 19
The co-ordinates (−3, −9), (−2, −5), (−1, −1), and
so on, are plotted and joined together to produce the
146.
134 Basic Engineering Mathematics
y
From the graph, find
20
(a)
15
11.8
10
(b) the value of y when x = −2.5
(c)
5
Ϫ1 0
Ϫ5
Ϫ3
1
3
2
4 x
2.2
10
3. Corresponding values obtained experimentally for two quantities are
x
−2.0 −0.5
0 1.0 2.5
3.0
5.0
y −13.0 −5.5 −3.0 2.0 9.5 12.0 22.0
Figure 17.7
straight line shown in Figure 17.7. (Note that the scales
used on the x and y axes do not have to be the same.)
From the graph:
(a)
the value of x when y = −6
(d) the value of x when y = 7
Ϫ1.5
Ϫ3 Ϫ2
the value of y when x = 1
when x = 2.2, y = 11.8, and
(b) when y = −3, x = −1.5
Now try the following Practice Exercise
Use a horizontal scale for x of 1 cm = 1 unit
2
and a vertical scale for y of 1 cm = 2 units and
draw a graph of x against y. Label the graph
and each of its axes. By interpolation, find
from the graph the value of y when x is 3.5.
4. Draw a graph of y − 3x + 5 = 0 over a range
of x = −3 to x = 4. Hence determine
(a) the value of y when x = 1.3
(b) the value of x when y = −9.2
Practice Exercise 67 Straight line graphs
(answers on page 347)
1. Assuming graph paper measuring 20 cm by
20 cm is available, suggest suitable scales for
the following ranges of values.
(a) Horizontal axis: 3 V to 55 V; vertical
axis: 10 to 180 .
(b) Horizontal axis: 7 m to 86 m; vertical
axis: 0.3 V to 1.69 V.
(c)
Horizontal axis: 5 N to 150 N; vertical
axis: 0.6 mm to 3.4 mm.
2. Corresponding values obtained experimentally for two quantities are
x
−5 −3 −1
5. The speed n rev/min of a motor changes when
the voltage V across the armature is varied.
The results are shown in the following table.
n (rev/min) 560 720 900 1010 1240 1410
V (volts)
80 100 120
140
160
180
It is suspected that one of the readings taken of
the speed is inaccurate. Plot a graph of speed
(horizontally) against voltage (vertically) and
find this value. Find also
(a) the speed at a voltage of 132 V.
(b) the voltage at a speed of 1300 rev/min.
0 2 4
y −13 −9 −5 −3 1 5
Plot a graph of y (vertically) against x (horizontally) to scales of 2 cm = 1 for the horizontal x-axis and 1 cm = 1 for the vertical y-axis.
(This graph will need the whole of the graph
paper with the origin somewhere in the centre
of the paper).
17.4 Gradients, intercepts and
equations of graphs
17.4.1 Gradients
The gradient or slope of a straight line is the ratio of
the change in the value of y to the change in the value of
x between any two points on the line. If, as x increases,
(→), y also increases, (↑), then the gradient is positive.
153.
Straight line graphs
Load, W (N)
5
15
20
25
60
Length, l (cm)
10
72
84
96
108
Plot a graph of load (horizontally) against
length (vertically) and determine
(a) the value of length when the load is 17 N.
(b) the value of load when the length is
74 cm.
(c) its gradient.
(d) the equation of the graph.
12. The following table gives the effort P to lift
a load W with a small lifting machine.
W (N) 10
P (N)
20
5.1
30
6.4
8.1
40
50
60
9.6 10.9 12.4
Plot W horizontally against P vertically and
show that the values lie approximately on a
straight line. Determine the probable relationship connecting P and W in the form
P = aW + b.
13. In an experiment the speeds N rpm of a flywheel slowly coming to rest were recorded
against the time t in minutes. Plot the results
and show that N and t are connected by
an equation of the form N = at + b. Find
probable values of a and b.
t (min)
2
4
6
8
10 12 14
values when plotted, this verifies that a law of the
form y = mx + c exists. From the graph, constants
m (i.e. gradient) and c (i.e. y-axis intercept) can be
determined.
Here are some worked problems in which practical
situations are featured.
Problem 12. The temperature in degrees Celsius
and the corresponding values in degrees Fahrenheit
are shown in the table below. Construct rectangular
axes, choose suitable scales and plot a graph of
degrees Celsius (on the horizontal axis) against
degrees Fahrenheit (on the vertical scale).
◦C
10
20
40
60
80
100
◦F
50
68
104
140
176
212
From the graph find (a) the temperature in degrees
Fahrenheit at 55◦C, (b) the temperature in degrees
Celsius at 167◦F, (c) the Fahrenheit temperature at
0◦ C and (d) the Celsius temperature at 230◦F
The co-ordinates (10, 50), (20, 68), (40, 104), and so
on are plotted as shown in Figure 17.17. When the
co-ordinates are joined, a straight line is produced.
Since a straight line results, there is a linear relationship
between degrees Celsius and degrees Fahrenheit.
y
240
230
Degrees Fahrenheit (8F)
11. A piece of elastic is tied to a support so that it
hangs vertically and a pan, on which weights
can be placed, is attached to the free end. The
length of the elastic is measured as various
weights are added to the pan and the results
obtained are as follows:
141
200
167
160
131
120
E
F
D
B
80
N (rev/min) 372 333 292 252 210 177 132
40
32
A
17.5 Practical problems involving
straight line graphs
0
20
G
40 55 60 75 80
100 110 120 x
Degrees Celsius (8C)
Figure 17.17
When a set of co-ordinate values are given or are
obtained experimentally and it is believed that they
follow a law of the form y = mx + c, if a straight line
can be drawn reasonably close to most of the co-ordinate
(a) To find the Fahrenheit temperature at 55◦C, a vertical line AB is constructed from the horizontal axis
to meet the straight line at B. The point where the
154.
142 Basic Engineering Mathematics
y
horizontal line BD meets the vertical axis indicates
the equivalent Fahrenheit temperature.
30
Hence, 55◦ C is equivalent to 131◦ F.
25
Volume (m3)
This process of finding an equivalent value in
between the given information in the above table
is called interpolation.
167◦F,
(b) To find the Celsius temperature at
a
horizontal line EF is constructed as shown in
Figure 17.17. The point where the vertical line FG
cuts the horizontal axis indicates the equivalent
Celsius temperature.
is seen to correspond to
70
75
80
85
Plot a graph of volume (vertical) against
temperature (horizontal) and from it find (a) the
temperature when the volume is 28.6 m3 and (b) the
volume when the temperature is 67◦C
If a graph is plotted with both the scales starting at zero
then the result is as shown in Figure 17.18. All of the
points lie in the top right-hand corner of the graph,
making interpolation difficult. A more accurate graph
is obtained if the temperature axis starts at 55◦C and
the volume axis starts at 24.5 m3 . The axes corresponding to these values are shown by the broken lines in
Figure 17.18 and are called false axes, since the origin
is not now at zero. A magnified version of this relevant
part of the graph is shown in Figure 17.19. From the
graph,
(a)
80
100 x
29
28.6
V m3 25.0 25.8 26.6 27.4 28.2 29.0
65
60
y
110◦ C.
When the volume is 28.6 m3 , the equivalent temperature is 82.5◦ C.
(b) When the temperature is 67◦C, the equivalent
volume is 26.1 m3 .
28
Volume (m3)
Problem 13. In an experiment on Charles's law,
the value of the volume of gas, V m3 , was measured
for various temperatures T ◦ C. The results are
shown below.
60
40
Figure 17.18
The process of finding equivalent values outside
of the given range is called extrapolation.
T ◦C
20
Temperature (8C)
From Figure 17.17, 0◦ C corresponds to 32◦ F.
(d)
10
0
If the graph is assumed to be linear even outside of
the given data, the graph may be extended at both
ends (shown by broken lines in Figure 17.17).
230◦F
15
5
Hence, 167◦ F is equivalent to 75◦C.
(c)
20
27
26.1
26
25
55
60
65 67 70
75
Temperature (ЊC)
80 82.5 85
x
Figure 17.19
Problem 14. In an experiment demonstrating
Hooke's law, the strain in an aluminium wire was
measured for various stresses. The results were:
Stress (N/mm2 ) 4.9
Strain
0.00007 0.00013
Stress (N/mm2 ) 18.4
Strain
8.7
0.00027
24.2
0.00034
15.0
0.00021
27.3
0.00039
155.
Straight line graphs
Plot a graph of stress (vertically) against strain
(horizontally). Find (a) Young's modulus of
elasticity for aluminium, which is given by the
gradient of the graph, (b) the value of the strain at a
stress of 20 N/mm2 and (c) the value of the stress
when the strain is 0.00020
(c) The value of the stress when the strain is 0.00020
is 14 N/mm2.
Problem 15. The following values of resistance
R ohms and corresponding voltage V volts are
obtained from a test on a filament lamp.
R ohms
The co-ordinates (0.00007, 4.9), (0.00013, 8.7), and
so on, are plotted as shown in Figure 17.20. The
graph produced is the best straight line which can be
drawn corresponding to these points. (With experimental results it is unlikely that all the points will lie exactly
on a straight line.) The graph, and each of its axes, are
labelled. Since the straight line passes through the origin, stress is directly proportional to strain for the given
range of values.
y
28
Stress (N/mm2)
30
48.5
73
107
128
V volts
16
29
52
76
94
Choose suitable scales and plot a graph with R
representing the vertical axis and V the horizontal
axis. Determine (a) the gradient of the graph, (b) the
R axis intercept value, (c) the equation of the graph,
(d) the value of resistance when the voltage is 60 V
and (e) the value of the voltage when the resistance
is 40 ohms. (f) If the graph were to continue in the
same manner, what value of resistance would be
obtained at 110 V?
A
The co-ordinates (16, 30), (29, 48.5), and so on are
shown plotted in Figure 17.21, where the best straight
line is drawn through the points.
24
20
16
14
12
y
147
140
8
B
C
A
4
120
0.00005
0.00015
0.00025
0.00035
0.000285
x
Strain
Figure 17.20
(a) The gradient of the straight line AC is given by
28 − 7
21
AB
=
=
BC
0.00040 − 0.00010 0.00030
=
21
7
=
= 7 × 104
3 × 10−4 10−4
= 70 000 N/mm
Resistance R (ohms)
0
143
100
85
80
60
40
20
10
2
Thus, Young's modulus of elasticity for aluminium is 70 000 N/mm2. Since 1 m2 = 106 mm2 ,
70 000 N/mm2 is equivalent to 70 000 ×106 N/m2 ,
i.e. 70 × 109 N/m2 (or pascals).
From Figure 17.20,
(b) The value of the strain at a stress of 20 N/mm2 is
0.000285
0
B
C
20 24
40
60
80
Voltage V (volts)
100 110 120
x
Figure 17.21
(a) The slope or gradient of the straight line AC is
given by
AB
135 − 10 125
=
=
= 1.25
BC
100 − 0
100
156.
144 Basic Engineering Mathematics
y
(Note that the vertical line AB and the horizontal line BC may be constructed anywhere along
the length of the straight line. However, calculations are made easier if the horizontal line BC is
carefully chosen; in this case, 100.)
(c)
The equation of a straight line is y = mx + c, when
y is plotted on the vertical axis and x on the horizontal axis. m represents the gradient and c the
y-axis intercept. In this case, R corresponds to y,
V corresponds to x, m = 1.25 and c = 10. Hence,
the equation of the graph is R = (1.25 V + 10) .
8.50
8.36
6.76
When the resistance is 40 ohms, the voltage is 24V.
By extrapolation, when the voltage is 110 V, the
resistance is 147 .
B
6.50
0
(d) When the voltage is 60 V, the resistance is 85 .
(f )
7.50
7.00
From Figure 17.21,
(e)
A
8.00
Stress (N/cm2)
(b) The R-axis intercept is at R = 10 ohms (by extrapolation).
8.68
100
C
200 300 400 500
Temperature t (8C)
600
700 x
Figure 17.22
Rearranging σ = −0.0032t + 8.68 gives
Problem 16. Experimental tests to determine the
breaking stress σ of rolled copper at various
temperatures t gave the following results.
Stress σ (N/cm2 )
8.46
8.04
7.78
Temperature t (◦C)
70
200
7.37
7.08
6.63
Temperature t (◦C)
410
500
640
8.68 − σ
0.0032
Hence, when the stress, σ = 7.54 N/cm2 ,
280
Stress σ (N/cm2 )
0.0032t = 8.68 − σ, i.e. t =
Show that the values obey the law σ = at + b,
where a and b are constants, and determine
approximate values for a and b. Use the law to
determine the stress at 250◦C and the temperature
when the stress is 7.54 N/cm2 .
temperature, t =
Now try the following Practice Exercise
Practice Exercise 69 Practical problems
involving straight line graphs (answers on
page 347)
1.
The co-ordinates (70, 8.46), (200, 8.04), and so on, are
plotted as shown in Figure 17.22. Since the graph is a
straight line then the values obey the law σ = at + b,
and the gradient of the straight line is
a=
σ = −0.0032(250) + 8.68 = 7.88 N/cm2
The resistance R ohms of a copper winding
is measured at various temperatures t ◦ C and
the results are as follows:
R (ohms) 112 120 126 131 134
t ◦C
AB
8.36 − 6.76
1.60
=
=
= −0.0032
BC
100 − 600
−500
Vertical axis intercept, b = 8.68
Hence, the law of the graph is σ = 0.0032t + 8.68
When the temperature is 250◦C, stress σ is given by
8.68 − 7.54
= 356.3◦ C
0.0032
20
36
48
58
64
Plot a graph of R (vertically) against t (horizontally) and find from it (a) the temperature
when the resistance is 122 and (b) the
resistance when the temperature is 52◦ C.
2.
The speed of a motor varies with armature
voltage as shown by the following experimental results.
157.
Straight line graphs
n (rev/min)
285
V (volts)
517
60
95
750
917
1050
V (volts)
130
155
Temperature
110
n (rev/min)
175
Plot a graph of speed (horizontally) against
voltage (vertically) and draw the best straight
line through the points. Find from the graph
(a) the speed at a voltage of 145 V and (b) the
voltage at a speed of 400 rev/min.
3.
Stress σ (N/cm2 )
615
25
47
Load L (newtons)
5.
v (m/s)
5
430 550 700
15
18
23.2 26.0 28.1
6.
The mass m of a steel joist varies with length
L as follows:
mass, m (kg)
80
100 120 140 160
length, L (m) 3.00 3.74 4.48 5.23 5.97
(b) the F-axis intercept,
(c) the equation of the graph,
(d) the force applied when the load is
310 N, and
(e) the load that a force of 160 N will
overcome.
(f ) If the graph were to continue in the
same manner, what value of force will
be needed to overcome a 800 N load?
The following table gives the results of tests
carried out to determine the breaking stress
σ of rolled copper at various temperatures, t .
Plot a graph of mass (vertically) against
length (horizontally). Determine the equation of the graph.
7.
The crushing strength of mortar varies with
the percentage of water used in its preparation, as shown below.
Crushing strength,
8.51 8.07 7.80
75
220
310
1.67
1.40
1.13
6
9
12
0.86
0.59
0.32
15
18
21
F (tonnes)
% of water used, w%
Crushing strength,
Temperature t (◦C)
8
Plot v vertically and t horizontally and draw
a graph of velocity against time. Determine
from the graph (a) the velocity after 10 s,
(b) the time at 20 m/s and (c) the equation
of the graph.
(a) the gradient,
Stress σ (N/cm2 )
650
16.9 19.0 21.1
v (m/s)
Force F (newtons) 120 149 187
4.
2
t (seconds) 11
64
Choose suitable scales and plot a graph of
F (vertically) against L (horizontally). Draw
the best straight line through the points.
Determine from the graph
500
The velocity v of a body after varying time
intervals t was measured as follows:
t (seconds)
50 140 210
Load L (newtons)
420
Plot a graph of stress (vertically) against
temperature (horizontally). Draw the best
straight line through the plotted co-ordinates.
Determine the slope of the graph and the
vertical axis intercept.
The following table gives the force F
newtons which, when applied to a lifting
machine, overcomes a corresponding load of
L newtons.
Force F (newtons)
7.47 7.23 6.78
t (◦C)
F (tonnes)
% of water used, w%
145
158.
146 Basic Engineering Mathematics
Plot a graph of F (vertically) against w
(horizontally).
(a)
Interpolate and determine the crushing
strength when 10% water is used.
(b) Assuming the graph continues in the
same manner, extrapolate and determine the percentage of water used when
the crushing strength is 0.15 tonnes.
(c)
8.
9.
An experiment with a set of pulley blocks
gave the following results.
Effort, E (newtons)
Load, L (newtons)
9.0 11.0 13.6
15
25
38
Effort, E (newtons) 17.4 20.8 23.6
Load, L (newtons)
57
74
88
What is the equation of the graph?
In an experiment demonstrating Hooke's law,
the strain in a copper wire was measured for
various stresses. The results were
Plot a graph of effort (vertically) against load
(horizontally). Determine
(a)
the gradient,
(b) the vertical axis intercept,
Stress 10.6 × 106 18.2 × 106 24.0 × 106
(pascals)
Strain
Stress
(pascals)
Strain
0.00011
0.00019
0.00025
30.7 × 106
39.4 × 106
0.00032
0.00041
Plot a graph of stress (vertically) against
strain (horizontally). Determine
(a)
Young's modulus of elasticity for copper, which is given by the gradient of
the graph,
(c)
the law of the graph,
(d) the effort when the load is 30 N,
(e)
the load when the effort is 19 N.
10. The variation of pressure p in a vessel with
temperature T is believed to follow a law of
the form p = aT + b, where a and b are constants. Verify this law for the results given
below and determine the approximate values
of a and b. Hence, determine the pressures
at temperatures of 285 K and 310 K and the
temperature at a pressure of 250 kPa.
Pressure, p (kPa)
244 247 252
Temperature, T (K) 273 277 282
(b) the value of strain at a stress of
21 × 106 Pa,
(c)
the value of stress when the strain is
0.00030,
Pressure, p (kPa)
258 262 267
Temperature, T (K) 289 294 300
159.
Chapter 18
Graphs reducing non-linear
laws to linear form
18.1
Introduction
In Chapter 17 we discovered that the equation of a
straight line graph is of the form y = mx + c, where
m is the gradient and c is the y-axis intercept. This
chapter explains how the law of a graph can still be determined even when it is not of the linear form y = mx + c.
The method used is called determination of law and is
explained in the following sections.
18.2
Determination of law
Frequently, the relationship between two variables, say
x and y, is not a linear one; i.e., when x is plotted against
y a curve results. In such cases the non-linear equation
may be modified to the linear form, y = mx + c, so that
the constants, and thus the law relating the variables, can
be determined. This technique is called 'determination
of law'.
Some examples of the reduction of equations to linear
form include
(i)
(ii)
y = ax 2 + b compares with Y = m X + c, where
m = a, c = b and X = x 2 .
Hence, y is plotted vertically against x 2 horizontally to produce a straight line graph of gradient
a and y-axis intercept b.
a
1
+ b, i.e. y = a
+b
x
x
1
y is plotted vertically against horizontally to
x
produce a straight line graph of gradient a and
y-axis intercept b.
y=
DOI: 10.1016/B978-1-85617-697-2.00018-1
(iii)
y = ax 2 + bx
y
Dividing both sides by x gives = ax + b.
x
y
is
Comparing with Y = m X + c shows that
x
plotted vertically against x horizontally to proy
duce a straight line graph of gradient a and
x
axis intercept b.
Here are some worked problems to demonstrate determination of law.
Problem 1. Experimental values of x and y,
shown below, are believed to be related by the law
y = ax 2 + b. By plotting a suitable graph, verify
this law and determine approximate values of
a and b
x
1
y
9.8
2
3
4
5
15.2
24.2
36.5
53.0
If y is plotted against x a curve results and it is not
possible to determine the values of constants a and b
from the curve.
Comparing y = ax 2 + b with Y = m X + c shows that
y is to be plotted vertically against x 2 horizontally. A
table of values is drawn up as shown below.
x
1
2
3
4
5
x2
1
4
9
16
25
y
9.8
15.2
24.2
36.5
53.0
162.
150 Basic Engineering Mathematics
a and b. Determine the cross-sectional area
needed for a resistance reading of 0.50 ohms.
7. Corresponding experimental values of two
quantities x and y are given below.
x
1.5
3.0
4.5
6.0
7.5
9.0
y 11.5 25.0 47.5 79.0 119.5 169.0
By plotting a suitable graph, verify that y
and x are connected by a law of the form
y = kx 2 + c, where k and c are constants.
Determine the law of the graph and hence find
the value of x when y is 60.0
8. Experimental results of the safe load L kN,
applied to girders of varying spans, d m, are
shown below.
Span, d (m)
2.0
2.8
3.6
4.2
4.8
It is believed that the relationship between load
and span is L = c/d, where c is a constant.
Determine (a) the value of constant c and (b)
the safe load for a span of 3.0 m.
9. The following results give corresponding values of two quantities x and y which are
believed to be related by a law of the form
y = ax 2 + bx, where a and b are constants.
x 33.86 55.54 72.80 84.10 111.4 168.1
3.4
5.2
6.5
7.3
9.1
12.4
Verify the law and determine approximate values of a and b. Hence, determine (a) the value
of y when x is 8.0 and (b) the value of x when
y is 146.5
ln e = 1 and if, say, lg x = 1.5,
then x = 101.5 = 31.62
Further, if 3x = 7 then lg 3x = lg 7 and x lg 3 = lg 7,
lg 7
= 1.771
from which x =
lg 3
These laws and techniques are used whenever non-linear
laws of the form y = ax n , y = ab x and y = aebx are
reduced to linear form with the values of a and b needing
to be calculated. This is demonstrated in the following
section.
18.4 Determination of laws involving
logarithms
Examples of the reduction of equations to linear form
involving logarithms include
(a)
Load, L (kN) 475 339 264 226 198
y
Also,
y = ax n
Taking logarithms to a base of 10 of both sides
gives
lg y = lg(ax n )
= lg a + lg x n by law (1)
i.e.
lg y = n lg x + lg a by law (3)
which compares with Y = m X + c
and shows that lg y is plotted vertically against
lg x horizontally to produce a straight line graph
of gradient n and lg y-axis intercept lg a.
See worked Problems 4 and 5 to demonstrate how
this law is determined.
(b) y = abx
Taking logarithms to a base of 10 of both sides
gives
lg y = lg(ab x )
18.3
Revision of laws of logarithms
i.e.
lg y = lg a + lg b x by law (1)
lg y = lg a + x lg b by law (3)
The laws of logarithms were stated in Chapter 15 as
follows:
i.e.
log(A × B) = log A + log B
(1)
A
log
B
(2)
= log A − log B
logAn = n × logA
(3)
lg y = x lg b + lg a
or
lg y = (lg b)x + lg a
which compares with
Y = mX + c
163.
Graphs reducing non-linear laws to linear form
and shows that lg y is plotted vertically against
x horizontally to produce a straight line graph
of gradient lg b and lg y-axis intercept lg a.
151
A graph of lg P against lg I is shown in Figure 18.4
and, since a straight line results, the law P = RI n is
verified.
See worked Problem 6 to demonstrate how this law
is determined.
ln y = ln a + ln ebx
ln y = ln a + bx ln e
ln y = bx + ln a
i.e.
which compares with
i.e.
i.e.
by law (1)
by law (3)
since lne = 1
See worked Problem 7 to demonstrate how this law is
determined.
Problem 4. The current flowing in, and the
power dissipated by, a resistor are measured
experimentally for various values and the results are
as shown below.
Power, P (watts)
3.6
A
2.78
D
2.5
2.18
Y = mX + c
and shows that ln y is plotted vertically against
x horizontally to produce a straight line graph
of gradient b and ln y-axis intercept ln a.
Current, I (amperes) 2.2
3.0
2.98
lg P
(c) y = aebx
Taking logarithms to a base of e of both sides
gives
ln y = ln(aebx )
4.1
5.6
6.8
116 311 403 753 1110
Show that the law relating current and power is of
the form P = R I n , where R and n are constants,
and determine the law
2.0
0.30 0.40
0.50
0.60 0.70
lg L
0.80
0.90
Figure 18.4
AB
2.98 − 2.18
=
BC
0.8 − 0.4
0.80
=2
=
0.4
It is not possible to determine the vertical axis intercept
on sight since the horizontal axis scale does not start
at zero. Selecting any point from the graph, say point
D, where lg I = 0.70 and lg P = 2.78 and substituting
values into
lg P = n lg I + lg R
Gradient of straight line, n =
gives
2.78 = (2)(0.70) + lg R
from which,
lg R = 2.78 − 1.40 = 1.38
Hence,
Taking logarithms to a base of 10 of both sides of
P = R I n gives
lg P = lg(R I n ) = lg R + lg I n = lg R + n lg I
by the laws of logarithms
i.e.
lg P = n lg I + lg R
which is of the form Y = m X + c,
showing that lg P is to be plotted vertically against lg I
horizontally.
A table of values for lg I and lg P is drawn up as shown
below.
B
C
R = antilog 1.38 = 101.38 = 24.0
Hence, the law of the graph is P = 24.0I 2
Problem 5. The periodic time, T , of oscillation of
a pendulum is believed to be related to its length, L,
by a law of the form T = kL n , where k and n are
constants. Values of T were measured for various
lengths of the pendulum and the results are as
shown below.
Periodic time, T (s) 1.0 1.3 1.5 1.8 2.0 2.3
I
2.2
3.6
4.1
5.6
6.8
Length, L(m)
0.25 0.42 0.56 0.81 1.0 1.32
lg I
0.342
0.556
0.613
0.748
0.833
P
116
311
403
753
1110
lg P
2.064
2.493
2.605
2.877
3.045
Show that the law is true and determine the
approximate values of k and n. Hence find the
periodic time when the length of the pendulum is
0.75 m
166.
154 Basic Engineering Mathematics
Now try the following Practice Exercise
6. Experimental values of x and y are measured
as follows.
Practice Exercise 71 Determination of law
involving logarithms (answers on page 348)
In problems 1 to 3, x and y are two related variables and all other letters denote constants. For the
stated laws to be verified it is necessary to plot
graphs of the variables in a modified form. State
for each, (a) what should be plotted on the vertical
axis, (b) what should be plotted on the horizontal axis, (c) the gradient and (d) the vertical axis
intercept.
1.
y = ba x
2.
y = kx L
3.
y
= enx
m
4. The luminosity I of a lamp varies with
the applied voltage V and the relationship
between I and V is thought to be I = kV n .
Experimental results obtained are
I (candelas)
I (candelas)
4.32
9.72
40
V (volts)
1.92
60
90
23.52
30.72
115
V (volts)
15.87
140
160
Verify that the law is true and determine
the law of the graph. Also determine the
luminosity when 75 V is applied across the
lamp.
5. The head of pressure h and the flow velocity v are measured and are believed to be
connected by the law v = ah b , where a and
b are constants. The results are as shown
below.
x
0.4
y
8.35
0.9
10.6
13.4
17.2
24.6
29.3
v
9.77
11.0
12.44
14.88
16.24
Verify that the law is true and determine values
of a and b.
2.3
3.8
13.47 17.94 51.32 215.20
The law relating x and y is believed to be of
the form y = ab x , where a and b are constants.
Determine the approximate values of a and b.
Hence, find the value of y when x is 2.0 and
the value of x when y is 100.
7. The activity of a mixture of radioactive isotopes is believed to vary according to the law
R = R0 t −c , where R0 and c are constants.
Experimental results are shown below.
R 9.72 2.65 1.15 0.47 0.32 0.23
t
2
5
9
17
22
28
Verify that the law is true and determine
approximate values of R0 and c.
8. Determine the law of the form y = aekx which
relates the following values.
y 0.0306 0.285 0.841 5.21 173.2 1181
x
–4.0
5.3
9.8
17.4 32.0 40.0
9. The tension T in a belt passing round a pulley wheel and in contact with the pulley over
an angle of θ radians is given by T = T0 eμθ ,
where T0 and μ are constants. Experimental
results obtained are
T (newtons) 47.9 52.8 60.3 70.1 80.9
θ (radians)
h
1.2
1.12 1.48 1.97 2.53 3.06
Determine approximate values of T0 and μ.
Hence, find the tension when θ is 2.25 radians
and the value of θ when the tension is 50.0
newtons.
175.
Revision Test 7 : Graphs
This assignment covers the material contained in Chapters 17–19. The marks available are shown in brackets at the
end of each question.
1.
Determine the value of P in the following table of
values.
x
3.
1
4
y = 3x − 5 −5
2.
0
−2
P
6.
(2)
Assuming graph paper measuring 20 cm by 20 cm
is available, suggest suitable scales for the following ranges of values.
Horizontal axis: 5 N to 70 N; vertical axis: 20 mm
to 190 mm.
(2)
7.
Corresponding values obtained experimentally for
two quantities are:
x −5
−3
−1
0 2
4
y 33.9 55.5 72.8 84.1 111.4
y −17 −11 −5 −2 4 10
Plot a graph of y (vertically) against x (horizontally) to scales of 1 cm = 1 for the horizontal
x-axis and 1 cm = 2 for the vertical y-axis. From
the graph, find
(a) the value of y when x = 3,
(b) the value of y when x = −4,
(c) the value of x when y = 1,
(d) the value of x when y = −20.
(8)
4.
5.
If graphs of y against x were to be plotted for each
of the following, state (i) the gradient, and (ii) the
y-axis intercept.
(a) y = −5x + 3
(b) y = 7x
(c) 2y + 4 = 5x
(d) 5x + 2y = 6
y
7
(e) 2x − =
(10)
3 6
The resistance R ohms of a copper winding is measured at various temperatures t ◦C and the results
are as follows.
R ( ) 38 47 55 62
x and y are two related variables and all other
letters denote constants. For the stated laws to
be verified it is necessary to plot graphs of the
variables in a modified form. State for each
(a) what should be plotted on the vertical axis,
(b) what should be plotted on the horizontal axis,
(c) the gradient,
(d) the vertical axis intercept.
a
(i) y = p + rx 2
(4)
(ii) y = + bx
x
The following results give corresponding values
of two quantities x and y which are believed to be
related by a law of the form y = ax 2 + bx where
a and b are constants.
72
t (◦ C) 16 34 50 64 84
Plot a graph of R (vertically) against t (horizontally) and find from it
(a) the temperature when the resistance is 50 ,
(b) the resistance when the temperature is 72◦ C,
(c) the gradient,
(d) the equation of the graph.
(10)
x
3.4
5.2
6.5
7.3
9.1
168.1
12.4
Verify the law and determine approximate values
of a and b.
Hence determine (i) the value of y when x is 8.0
and (ii) the value of x when y is 146.5
(18)
8.
By taking logarithms of both sides of y = k x n ,
show that lg y needs to be plotted vertically and
lg x needs to be plotted horizontally to produce
a straight line graph. Also, state the gradient and
vertical-axis intercept.
(6)
9.
By taking logarithms of both sides of y = aek x
show that ln y needs to be plotted vertically and
x needs to be plotted horizontally to produce a
straight line graph. Also, state the gradient and
vertical-axis intercept.
(6)
10. Show from the following results of voltage V and
admittance Y of an electrical circuit that the law
connecting the quantities is of the form V = kY n
and determine the values of k and n.
Voltage
V (volts)
2.88 2.05 1.60 1.22 0.96
Admittance,
Y (siemens) 0.52 0.73 0.94 1.23 1.57
(12)
177.
Chapter 20
Angles and triangles
20.1
Introduction
• A straight line which crosses two parallel lines is
called a transversal (see MN in Figure 20.1).
Trigonometry is a subject that involves the measurement
of sides and angles of triangles and their relationship to
each other. This chapter involves the measurement of
angles and introduces types of triangle.
N
P
20.2
c
Angular measurement
h e
g f
R
An angle is the amount of rotation between two straight
lines. Angles may be measured either in degrees or in
radians.
If a circle is divided into 360 equal parts, then each part
is called 1 degree and is written as 1◦
i.e.
or
1 revolution = 360◦
1
1 degree is
th of a revolution
360
Some angles are given special names.
• Any angle between 0◦ and 90◦ is called an acute
angle.
• An angle equal to 90◦ is called a right angle.
• Any angle between 90◦ and 180◦ is called an obtuse
angle.
• Any angle greater than 180◦ and less than 360◦ is
called a reflex angle.
• An angle of 180◦ lies on a straight line.
• If two angles add up to 90◦ they are called complementary angles.
• If two angles add up to 180◦ they are called supplementary angles.
• Parallel lines are straight lines which are in the same
plane and never meet. Such lines are denoted by
arrows, as in Figure 20.1.
DOI: 10.1016/B978-1-85617-697-2.00020-X
d a
b
Q
S
M
Figure 20.1
With reference to Figure 20.1,
(a) a = c, b = d, e = g and f = h. Such pairs of
angles are called vertically opposite angles.
(b) a = e, b = f , c = g and d = h. Such pairs of
angles are called corresponding angles.
(c) c = e and b = h. Such pairs of angles are called
alternate angles.
(d) b + e = 180◦ and c + h = 180◦. Such pairs of
angles are called interior angles.
20.2.1 Minutes and seconds
One degree may be sub-divided into 60 parts, called
minutes.
i.e.
1 degree = 60 minutes
which is written as
1◦ = 60 .
183.
171
Angles and triangles
20.3
F
Triangles
308
A triangle is a figure enclosed by three straight lines.
The sum of the three angles of a triangle is equal
to 180◦ .
A
438
20.3.1 Types of triangle
An acute-angled triangle is one in which all the angles
are acute; i.e., all the angles are less than 90◦ . An
example is shown in triangle ABC in Figure 20.15(a).
A right-angled triangle is one which contains a right
angle; i.e., one in which one of the angles is 90◦. An
example is shown in triangle DEF in Figure 20.15(b).
828
758
758
E
558
B
G
C
(a)
A
D
(b)
Figure 20.17
548
508
A
c
678
598
B
C
E
F
(a)
(b)
Figure 20.15
B
C
a
An obtuse-angled triangle is one which contains an
obtuse angle; i.e., one angle which lies between 90◦
and 180◦. An example is shown in triangle PQR in
Figure 20.16(a).
An equilateral triangle is one in which all the sides and
all the angles are equal; i.e., each is 60◦. An example is
shown in triangle ABC in Figure 20.16(b).
A
P
22
b
408
608
Figure 20.18
With reference to Figure 20.18,
(a) Angles A, B and C are called interior angles of
the triangle.
(b) Angle θ is called an exterior angle of the triangle
and is equal to the sum of the two opposite interior
angles; i.e., θ = A + C.
(c) a + b + c is called the perimeter of the triangle.
8
A
1318
608
278
Q
(a)
R
B
608
(b)
C
Figure 20.16
An isosceles triangle is one in which two angles and
two sides are equal. An example is shown in triangle
EFG in Figure 20.17(a).
A scalene triangle is one with unequal angles and
therefore unequal sides. An example of an acute
angled scalene triangle is shown in triangle ABC in
Figure 20.17(b).
b
C
Figure 20.19
c
a
B
184.
172 Basic Engineering Mathematics
A right-angled triangle ABC is shown in Figure 20.19.
The point of intersection of two lines is called a vertex
(plural vertices); the three vertices of the triangle are
labelled as A, B and C, respectively. The right angle
is angle C. The side opposite the right angle is given
the special name of the hypotenuse. The hypotenuse,
length AB in Figure 20.19, is always the longest side of
a right-angled triangle. With reference to angle B, AC
is the opposite side and BC is called the adjacent side.
With reference to angle A, BC is the opposite side and
AC is the adjacent side.
Often sides of a triangle are labelled with lower case
letters, a being the side opposite angle A, b being the
side opposite angle B and c being the side opposite
angle C. So, in the triangle ABC, length AB = c, length
BC = a and length AC = b. Thus, c is the hypotenuse
in the triangle ABC.
∠ is the symbol used for 'angle'. For example, in the
triangle shown, ∠C = 90◦ . Another way of indicating
an angle is to use all three letters. For example, ∠ABC
actually means ∠B; i.e., we take the middle letter as the
angle. Similarly, ∠BAC means ∠A and ∠ACB means
∠C.
Here are some worked examples to help us understand
more about triangles.
Problem 18.
Figure 20.20
Name the types of triangle shown in
(d) Obtuse-angled scalene triangle (since one of the
angles lies between 90◦ and 180◦ ).
(e)
Isosceles triangle (since two sides are equal).
Problem 19. In the triangle ABC shown in
Figure 20.21, with reference to angle θ, which side
is the adjacent?
C
A
B
Figure 20.21
The triangle is right-angled; thus, side AC is the
hypotenuse. With reference to angle θ, the opposite side
is BC. The remaining side, AB, is the adjacent side.
Problem 20. In the triangle shown in
Figure 20.22, determine angle θ
2.6
2
2.1
2
398
2.8
2
(a)
(b)
568
518
(c)
2.5
2.1
1078
(d)
2.5
(e)
Figure 20.20
(a)
Equilateral triangle (since all three sides are
equal).
(b) Acute-angled scalene triangle (since all the
angles are less than 90◦).
(c)
Right-angled triangle (39◦ + 51◦ = 90◦ ; hence,
the third angle must be 90◦ , since there are 180◦
in a triangle).
Figure 20.22
The sum of the three angles of a triangle is equal to
180◦ .
The triangle is right-angled. Hence,
90◦ + 56◦ + ∠θ = 180◦
from which,
Problem 21.
Figure 20.23
∠θ = 180◦ − 90◦ − 56◦ = 34◦ .
Determine the value of θ and α in
187.
175
Angles and triangles
(a) the three sides of one are equal to the three sides
of the other (SSS),
688
c
d
568299
b
e
g
f
148419
1318
a
(a)
(b)
Figure 20.32
9.
Find the unknown angles a to k in
Figure 20.33.
b
228
a
d
c
g
j
(c) two angles of the one are equal to two angles of
the other and any side of the first is equal to the
corresponding side of the other (ASA), or
(d) their hypotenuses are equal and one other side of
one is equal to the corresponding side of the other
(RHS).
Problem 24. State which of the pairs of triangles
shown in Figure 20.35 are congruent and name their
sequence
f
1258 e
(b) two sides of one are equal to two sides of the other
and the angles included by these sides are equal
(SAS),
h
C
i
G
E
B
D
L
k 998
J
H
A
Figure 20.33
(a)
10. Triangle ABC has a right angle at B and
∠BAC is 34◦. BC is produced to D. If the
bisectors of ∠ABC and ∠ACD meet at E,
determine ∠BEC.
11. If in Figure 20.34 triangle BCD is equilateral,
find the interior angles of triangle ABE.
A
C
978
K
(b)
F
T
S
V
M
U
E
C
B
O
N
D
Q
P
(c)
R
W
X
(d)
A
(e)
Figure 20.35
B
E
I
F
D
(a) Congruent ABC, FDE (angle, side, angle; i.e.,
ASA).
(b) Congruent GIH, JLK (side, angle, side; i.e., SAS).
Figure 20.34
(c) Congruent MNO, RQP (right angle, hypotenuse,
side; i.e., RHS).
20.4
Congruent triangles
Two triangles are said to be congruent if they are equal
in all respects; i.e., three angles and three sides in one
triangle are equal to three angles and three sides in the
other triangle. Two triangles are congruent if
(d) Not necessarily congruent. It is not indicated that
any side coincides.
(e) Congruent ABC, FED (side, side, side; i.e.,
SSS).
190.
178 Basic Engineering Mathematics
Problem 29. A rectangular shed 2 m wide and
3 m high stands against a perpendicular building of
height 5.5 m. A ladder is used to gain access to the
roof of the building. Determine the minimum
distance between the bottom of the ladder and the
shed
14.58 mm
25.69 mm
111Њ
x
A side view is shown in Figure 20.43, where AF
is the minimum length of the ladder. Since BD
and CF are parallel, ∠ADB = ∠DFE (corresponding angles between parallel lines). Hence, triangles
BAD and EDF are similar since their angles are the
same.
4.74 mm
32Њ
y
37Њ
32Њ
7.36 mm
Figure 20.44
2.
PQR is an equilateral triangle of side 4 cm.
When PQ and PR are produced to S and T ,
respectively, ST is found to be parallel with
QR. If PS is 9 cm, find the length of ST. X
is a point on ST between S and T such that
the line PX is the bisector of ∠SPT. Find the
length of PX.
3.
In Figure 20.45, find
(a) the length of BC when AB = 6 cm,
DE = 8 cm and DC = 3 cm,
(b) the length of DE when EC = 2 cm,
AC = 5 cm and AB = 10 cm.
AB = AC − BC = AC − DE = 5.5 − 3 = 2.5 m
By proportion:
AB
BD
=
DE
EF
i.e.
2.5
2
=
3
EF
3
= 2.4 m = minimum distance
2.5
from bottom of ladder to the shed.
Hence, EF = 2
A
A
B
C
B
D
2m
3m
5.5 m
Shed
D
F
E
C
Figure 20.45
4.
Figure 20.43
E
In Figure 20.46, AF = 8 m, AB = 5 m and
BC = 3 m. Find the length of BD.
C
B
D
Now try the following Practice Exercise
Practice Exercise 80 Similar triangles
(answers on page 349)
1.
In Figure 20.44, find the lengths x and y.
A
Figure 20.46
E
F
191.
Angles and triangles
20.6
Construction of triangles
179
With reference to Figure 20.48:
To construct any triangle, the following drawing instruments are needed:
A
(a) ruler and/or straight edge
b 5 3 cm
(b) compass
608
(c) protractor
a 5 6 cm
B
(d) pencil.
C
Figure 20.48
Here are some worked problems to demonstrate triangle
construction.
(i) Draw a line BC, 6 cm long.
Problem 30. Construct a triangle whose sides are
6 cm, 5 cm and 3 cm
(ii) Using a protractor centred at C, make an angle of
60◦ to BC.
(iii) From C measure a length of 3 cm and label A.
(iv) Join B to A by a straight line.
D
G
C
F
Problem 32. Construct a triangle PQR given that
QR = 5 cm, ∠Q = 70◦ and ∠R = 44◦
E
Q9
A
6 cm
R9
B
P
Figure 20.47
With reference to Figure 20.47:
(i) Draw a straight line of any length and, with a pair
of compasses, mark out 6 cm length and label it
AB.
(ii) Set compass to 5 cm and with centre at A describe
arc DE.
(iii) Set compass to 3 cm and with centre at B describe
arc FG.
(iv) The intersection of the two curves at C is the vertex of the required triangle. Join AC and BC by
straight lines.
It may be proved by measurement that the ratio of the
angles of a triangle is not equal to the ratio of the sides
(i.e., in this problem, the angle opposite the 3 cm side is
not equal to half the angle opposite the 6 cm side).
Problem 31. Construct a triangle ABC such that
a = 6 cm, b = 3 cm and ∠C = 60◦
708
Q
448
5 cm
R
Figure 20.49
With reference to Figure 20.49:
(i) Draw a straight line 5 cm long and label it QR.
(ii) Use a protractor centred at Q and make an angle
of 70◦ . Draw QQ .
(iii) Use a protractor centred at R and make an angle
of 44◦ . Draw RR .
(iv) The intersection of QQ and RR forms the vertex
P of the triangle.
Problem 33. Construct a triangle XYZ given that
XY = 5 cm, the hypotenuse YZ = 6.5 cm and
∠X = 90◦
192.
180 Basic Engineering Mathematics
V
Z
P U
centred at Y and set to 6.5 cm, describe the
arc UV.
S
(iv) The intersection of the arc UV with XC produced,
forms the vertex Z of the required triangle. Join
YZ with a straight line.
C
R
B
A
Q
X
A9
Y
Figure 20.50
With reference to Figure 20.50:
(i) Draw a straight line 5 cm long and label it XY.
(ii) Produce XY any distance to B. With compass centred at X make an arc at A and A . (The length XA
and XA is arbitrary.) With compass centred at A
draw the arc PQ. With the same compass setting
and centred at A , draw the arc RS. Join the intersection of the arcs, C to X , and a right angle to
XY is produced at X . (Alternatively, a protractor
can be used to construct a 90◦ angle.)
(iii) The hypotenuse is always opposite the right angle.
Thus, YZ is opposite ∠X . Using a compass
Now try the following Practice Exercise
Practice Exercise 81 Construction of
triangles (answers on page 349)
In the following, construct the triangles ABC for
the given sides/angles.
1.
a = 8 cm, b = 6 cm and c = 5 cm.
2.
a = 40 mm, b = 60 mm and C = 60◦ .
3.
a = 6 cm, C = 45◦ and B = 75◦ .
4.
c = 4 cm, A = 130◦ and C = 15◦.
5.
a = 90 mm, B = 90◦ , hypotenuse = 105mm.
193.
Chapter 21
Introduction to trigonometry
21.1
If the lengths of any two sides of a right-angled triangle
are known, the length of the third side may be calculated
by Pythagoras' theorem.
Introduction
Trigonometry is a subject that involves the measurement
of sides and angles of triangles and their relationship to
each other.
The theorem of Pythagoras and trigonometric ratios
are used with right-angled triangles only. However,
there are many practical examples in engineering
where knowledge of right-angled triangles is very
important.
In this chapter, three trigonometric ratios – i.e. sine,
cosine and tangent – are defined and then evaluated
using a calculator. Finally, solving right-angled triangle problems using Pythagoras and trigonometric ratios
is demonstrated, together with some practical examples
involving angles of elevation and depression.
21.2
From equation (1):
Problem 1. In Figure 21.2, find the length of BC
C
b 5 4 cm
The theorem of Pythagoras states:
In any right-angled triangle, the square of the
hypotenuse is equal to the sum of the squares of the
other two sides.
In the right-angled triangle ABC shown in Figure 21.1,
this means
(1)
B
b
Figure 21.1
DOI: 10.1016/B978-1-85617-697-2.00021-1
a
A
a
c 5 3 cm
B
Figure 21.2
From Pythagoras, a 2 = b2 + c2
i.e.
a 2 = 42 + 32
= 16 + 9 = 25
√
a = 25 = 5 cm.
A
c
a2 + c2
Transposing equation (1) for a gives a 2 = b2 − c2 , from
√
which a = b2 − c 2
Transposing equation (1) for c gives c2 = b2 − a 2 , from
√
which c = b2 − a 2
Here are some worked problems to demonstrate the
theorem of Pythagoras.
The theorem of Pythagoras
b2 = a 2 + c 2
b=
C
Hence,
√
25 = ±5 but in a practical example like this an answer
of a = −5 cm has no meaning, so we take only the
positive answer.
Thus
a = BC = 5 cm.
195.
Introduction to trigonometry
nearest centimetre, runs from the top of the
tent to the peg?
6.
In a triangle ABC, ∠B is a right angle,
AB = 6.92 cm and BC = 8.78 cm. Find the
length of the hypotenuse.
183
14. Figure 21.8 shows a cross-section of a component that is to be made from a round bar.
If the diameter of the bar is 74 mm, calculate
the dimension x.
x
90◦ ,
7.
In a triangle CDE, D =
CD = 14.83 mm and CE = 28.31 mm.
Determine the length of DE.
8.
Show that if a triangle has sides of 8, 15 and
17 cm it is right-angled.
9.
Triangle PQR is isosceles, Q being a right
angle. If the hypotenuse is 38.46 cm find (a)
the lengths of sides PQ and QR and (b) the
value of ∠QPR.
10. A man cycles 24 km due south and then 20 km
due east. Another man, starting at the same
time as the first man, cycles 32 km due east
and then 7 km due south. Find the distance
between the two men.
11. A ladder 3.5 m long is placed against a perpendicular wall with its foot 1.0 m from the
wall. How far up the wall (to the nearest centimetre) does the ladder reach? If the foot of
the ladder is now moved 30 cm further away
from the wall, how far does the top of the
ladder fall?
r 516 mm
h
72 mm
Figure 21.8
21.3
Sines, cosines and tangents
With reference to angle θ in the right-angled triangle
ABC shown in Figure 21.9,
opposite side
hypotenuse
sine θ =
'Sine' is abbreviated to 'sin', thus sin θ =
BC
AC
C
12. Two ships leave a port at the same time. One
travels due west at 18.4 knots and the other
due south at 27.6 knots. If 1knot = 1 nautical
mile per hour, calculate how far apart the two
ships are after 4 hours.
13. Figure 21.7 shows a bolt rounded off at one
end. Determine the dimension h.
m
4m
7
se
nu
te
po
Hy
Opposite
A
Adjacent
B
Figure 21.9
Also, cosine θ =
R 5 45 mm
adjacent side
hypotenuse
'Cosine' is abbreviated to 'cos', thus cos θ =
Finally, tangent θ =
opposite side
adjacent side
AB
AC
BC
AB
These three trigonometric ratios only apply to rightangled triangles. Remembering these three equations
is very important and the mnemonic 'SOH CAH TOA'
is one way of remembering them.
'Tangent' is abbreviated to 'tan', thus tan θ =
Figure 21.7
203.
Introduction to trigonometry
3. A ladder rests against the top of the perpendicular wall of a building and makes an angle of
73◦ with the ground. If the foot of the ladder is
2 m from the wall, calculate the height of the
building.
pylon is 23◦. Calculate the height of the pylon to the
nearest metre
Figure 21.29 shows the pylon AB and the angle of
elevation of A from point C is 23◦.
A
4. Determine the length x in Figure 21.26.
C
10 mm
568
191
238
80 m
B
Figure 21.29
tan 23◦ =
x
AB
AB
=
BC
80
Hence, height of pylon AB = 80 tan 23◦
= 80(0.4245) = 33.96 m
= 34 m to the nearest metre.
Figure 21.26
21.6 Angles of elevation and
depression
If, in Figure 21.27, BC represents horizontal ground and
AB a vertical flagpole, the angle of elevation of the top
of the flagpole, A, from the point C is the angle that the
imaginary straight line AC must be raised (or elevated)
from the horizontal CB; i.e., angle θ.
Problem 24. A surveyor measures the angle of
elevation of the top of a perpendicular building as
19◦. He moves 120 m nearer to the building and
finds the angle of elevation is now 47◦. Determine
the height of the building
The building PQ and the angles of elevation are shown
in Figure 21.30.
P
A
h
478
C
B
Q
R
198
S
120
x
Figure 21.27
Figure 21.30
P
Hence,
Q
h
x + 120
h = tan 19◦(x + 120)
In triangle PQS, tan 19◦ =
R
i.e. h = 0.3443(x + 120)
Figure 21.28
If, in Figure 21.28, PQ represents a vertical cliff and
R a ship at sea, the angle of depression of the ship
from point P is the angle through which the imaginary
straight line PR must be lowered (or depressed) from
the horizontal to the ship; i.e., angle φ. (Note, ∠PRQ is
also φ − alternate angles between parallel lines.)
Problem 23. An electricity pylon stands on
horizontal ground. At a point 80 m from the base of
the pylon, the angle of elevation of the top of the
h
x
h = tan 47◦ (x) i.e. h = 1.0724x
(1)
In triangle PQR, tan 47◦ =
Hence,
Equating equations (1) and (2) gives
0.3443(x + 120) = 1.0724x
0.3443x + (0.3443)(120) = 1.0724x
(0.3443)(120) = (1.0724 − 0.3443)x
41.316 = 0.7281x
41.316
x=
= 56.74 m
0.7281
(2)
204.
192 Basic Engineering Mathematics
From equation (2), height of building,
h = 1.0724x = 1.0724(56.74) = 60.85 m.
Problem 25. The angle of depression of a ship
viewed at a particular instant from the top of a 75 m
vertical cliff is 30◦ . Find the distance of the ship
from the base of the cliff at this instant. The ship is
sailing away from the cliff at constant speed and 1
minute later its angle of depression from the top of
the cliff is 20◦. Determine the speed of the ship in
km/h
Figure 21.31 shows the cliff AB, the initial position of
the ship at C and the final position at D. Since the angle
of depression is initially 30◦, ∠ACB = 30◦ (alternate
angles between parallel lines).
A
208
75 m
208
308
C
B
x
D
Figure 21.31
75
AB
=
hence,
BC
BC
75
BC =
tan 30◦
= 129.9 m = initial position
of ship from base of cliff
In triangle ABD,
tan 20◦ =
Practice Exercise 86 Angles of elevation
and depression (answers on page 349)
1. A vertical tower stands on level ground. At
a point 105 m from the foot of the tower the
angle of elevation of the top is 19◦. Find the
height of the tower.
2. If the angle of elevation of the top of a vertical
30 m high aerial is 32◦ , how far is it to the
aerial?
3. From the top of a vertical cliff 90.0 m high
the angle of depression of a boat is 19◦ 50 .
Determine the distance of the boat from the
cliff.
4. From the top of a vertical cliff 80.0 m high the
angles of depression of two buoys lying due
west of the cliff are 23◦ and 15◦, respectively.
How far apart are the buoys?
308
tan 30◦ =
Now try the following Practice Exercise
AB
75
75
=
=
BD
BC + CD
129.9 + x
5. From a point on horizontal ground a surveyor
measures the angle of elevation of the top of
a flagpole as 18◦ 40 . He moves 50 m nearer
to the flagpole and measures the angle of elevation as 26◦22 . Determine the height of the
flagpole.
6. A flagpole stands on the edge of the top of a
building. At a point 200 m from the building
the angles of elevation of the top and bottom of the pole are 32◦ and 30◦ respectively.
Calculate the height of the flagpole.
7. From a ship at sea, the angles of elevation of
the top and bottom of a vertical lighthouse
standing on the edge of a vertical cliff are
31◦ and 26◦ , respectively. If the lighthouse is
25.0 m high, calculate the height of the cliff.
Thus, the ship sails 76.16 m in 1 minute; i.e., 60 s,
8. From a window 4.2 m above horizontal ground
the angle of depression of the foot of a building
across the road is 24◦ and the angle of elevation
of the top of the building is 34◦. Determine,
correct to the nearest centimetre, the width of
the road and the height of the building.
distance 76.16
=
m/s
Hence, speed of ship =
time
60
76.16 × 60 × 60
=
km/h = 4.57 km/h.
60 × 1000
9. The elevation of a tower from two points, one
due west of the tower and the other due east
of it are 20◦ and 24◦ , respectively, and the two
points of observation are 300 m apart. Find the
height of the tower to the nearest metre.
Hence,
from which
129.9 + x =
75
= 206.06 m
tan 20◦
x = 206.06 − 129.9 = 76.16 m
208.
196 Basic Engineering Mathematics
22.2
right-angled triangle 0AC. Then,
Angles of any magnitude
Figure 22.2 shows rectangular axes XX and YY
intersecting at origin 0. As with graphical work, measurements made to the right and above 0 are positive,
while those to the left and downwards are negative.
Let 0A be free to rotate about 0. By convention,
when 0A moves anticlockwise angular measurement is
considered positive, and vice versa.
+
=+
+
+
tan θ2 = = −
−
sin θ2 =
cos θ2 =
Let 0A be further rotated so that θ3 is any angle in the
third quadrant and let AD be constructed to form the
right-angled triangle 0AD. Then,
−
=−
+
−
tan θ3 = = +
−
sin θ3 =
908
Y
Quadrant 2
1808
1
Quadrant 1
1
0
A
2
X9
1
X
08
3608
2
2
Quadrant 3
Quadrant 4
Y9
2708
Figure 22.2
Let 0A be rotated anticlockwise so that θ1 is any
angle in the first quadrant and let perpendicular AB
be constructed to form the right-angled triangle 0AB
in Figure 22.3. Since all three sides of the triangle are positive, the trigonometric ratios sine, cosine
and tangent will all be positive in the first quadrant.
(Note: 0A is always positive since it is the radius of a
circle.)
Let 0A be further rotated so that θ2 is any angle in the
second quadrant and let AC be constructed to form the
−
=−
+
cos θ3 =
−
=−
+
Let 0A be further rotated so that θ4 is any angle in the
fourth quadrant and let AE be constructed to form the
right-angled triangle 0AE. Then,
−
=−
+
−
tan θ4 = = −
+
sin θ4 =
cos θ4 =
+
=+
+
The above results are summarized in Figure 22.4, in
which all three trigonometric ratios are positive in the
first quadrant, only sine is positive in the second quadrant, only tangent is positive in the third quadrant and
only cosine is positive in the fourth quadrant.
The underlined letters in Figure 22.4 spell the word
CAST when starting in the fourth quadrant and moving
in an anticlockwise direction.
908
Sine
All positive
08
3608
1808
908
Quadrant 2
A
D
C
2
1
2 2
3
1 1
4
E B
08
3608
2708
2
1
A
Quadrant 3
A
Quadrant 4
2708
Figure 22.3
1
1
0
Cosine
Tangent
A
1
1
1808
Quadrant 1
Figure 22.4
It is seen that, in the first quadrant of Figure 22.1,
all of the curves have positive values; in the second only
sine is positive; in the third only tangent is positive;
and in the fourth only cosine is positive – exactly as
summarized in Figure 22.4.
209.
Trigonometric waveforms
A knowledge of angles of any magnitude is needed
when finding, for example, all the angles between 0◦
and 360◦ whose sine is, say, 0.3261. If 0.3261 is entered
into a calculator and then the inverse sine key pressed (or
sin−1 key) the answer 19.03◦ appears. However, there
is a second angle between 0◦ and 360◦ which the calculator does not give. Sine is also positive in the second
quadrant (either from CAST or from Figure 22.1(a)).
The other angle is shown in Figure 22.5 as angle θ,
where θ = 180◦ − 19.03◦ = 160.97◦. Thus, 19.03◦
and 160.97◦ are the angles between 0◦ and 360◦ whose
sine is 0.3261 (check that sin 160.97◦ = 0.3261 on your
calculator).
908
whose sine is −0.4638 are 180◦ + 27.63◦ i.e. 207.63◦
and 360◦ – 27.63◦, i.e. 332.37◦ . (Note that if a calculator is used to determine sin−1 (−0.4638) it only gives
one answer: −27.632588◦.)
908
S
1808
A
T
08
3608
C
2708
Figure 22.7
S
Problem 2. Determine all of the angles between
0◦ and 360◦ whose tangent is 1.7629
A
1808
197
19.038
19.038
T
08
3608
C
A tangent is positive in the first and third quadrants –
see Figure 22.8.
y 5 tan x
y
2708
1.7629
Figure 22.5
Be careful! Your calculator only gives you one of
these answers. The second answer needs to be deduced
from a knowledge of angles of any magnitude, as shown
in the following worked problems.
Problem 1. Determine all of the angles between
0◦ and 360◦ whose sine is −0.4638
The angles whose sine is −0.4638 occur in the third
and fourth quadrants since sine is negative in these
quadrants – see Figure 22.6.
y
1.0
908
60.448
1808 2708
240.448
From Figure 22.9, θ = tan−1 1.7629 = 60.44◦ . Measured from 0◦ , the two angles between 0◦ and 360◦
whose tangent is 1.7629 are 60.44◦ and 180◦ + 60.44◦ ,
i.e. 240.44◦
908
A
S
908 1808
3608 x
Figure 22.8
y 5 sin x
207.638
0
20.4638
0
2708
332.378
1808
08
3608
3608 x
T
21.0
C
2708
Figure 22.9
Figure 22.6
From Figure 22.7, θ = sin−1 0.4638 = 27.63◦ . Measured from 0◦ , the two angles between 0◦ and 360◦
Problem 3. Solve the equation
cos−1 (−0.2348) = α for angles of α between 0◦
and 360◦
211.
Trigonometric waveforms
199
y
158 08 R
T
458
608
S
1.0
3308
3158
0.5
S9
y 5 cos x
2858
908
08
2558
O
O9
308 608
Angle x8
1208
1808
2408
3008
3608
20.5
1208
2258
1508
1808
2108
21.0
Figure 22.12
22.3.1 Sine waves
22.4.2 Amplitude
The vertical component TS may be projected across to
T S , which is the corresponding value of 30◦ on the
graph of y against angle x ◦ . If all such vertical components as TS are projected on to the graph, a sine wave
is produced as shown in Figure 22.11.
The amplitude is the maximum value reached in a half
cycle by a sine wave. Another name for amplitude is
peak value or maximum value.
A sine wave y = 5 sin x has an amplitude of 5, a
sine wave v = 200 sin 314t has an amplitude of 200 and
the sine wave y = sin x shown in Figure 22.11 has an
amplitude of 1.
22.3.2 Cosine waves
If all horizontal components such as OS are projected
on to a graph of y against angle x ◦ , a cosine wave is
produced. It is easier to visualize these projections by
redrawing the circle with the radius arm OR initially in
a vertical position as shown in Figure 22.12.
It is seen from Figures 22.11 and 22.12 that a cosine
curve is of the same form as the sine curve but is displaced by 90◦ (or π/2 radians). Both sine and cosine
waves repeat every 360◦ .
22.4 Terminology involved with sine
and cosine waves
Sine waves are extremely important in engineering, with
examples occurring with alternating currents and voltages – the mains supply is a sine wave – and with simple
harmonic motion.
22.4.1 Cycle
When a sine wave has passed through a complete
series of values, both positive and negative, it is said
to have completed one cycle. One cycle of a sine
wave is shown in Figure 22.1(a) on page 195 and in
Figure 22.11.
22.4.3 Period
The waveforms y = sin x and y = cos x repeat themselves every 360◦ . Thus, for each, the period is 360◦.
A waveform of y = tan x has a period of 180◦ (from
Figure 22.1(c)).
A graph of y = 3 sin 2A, as shown in Figure 22.13, has
an amplitude of 3 and period 180◦.
A graph of y = sin 3A, as shown in Figure 22.14, has an
amplitude of 1 and period of 120◦.
A graph of y = 4 cos 2x, as shown in Figure 22.15, has
an amplitude of 4 and a period of 180◦.
In general, if y = A sin px or y = A cos px,
360◦
amplitude = A and period =
p
y
y 5 3 sin 2A
3
0
23
Figure 22.13
908
1808
2708
3608
A8
212.
200 Basic Engineering Mathematics
v
y
10
y ϭ sin 3A
1.0
0
90Њ
180Њ
360Њ AЊ
270Њ
Ϫ1.0
10
20
t (ms)
210
Figure 22.17
Figure 22.14
y
4
0
22.4.5 Frequency
y ϭ 4 cos 2x
0
90Њ
The number of cycles completed in one second is called
the frequency f and is measured in hertz, Hz.
180Њ
270Њ
360Њ
xЊ
Figure 22.15
Sketch y = 2 sin
3
A over one cycle
5
360◦ × 5
360◦
= 600◦
Amplitude = 2; period = 3 =
3
5
3
A sketch of y = 2 sin A is shown in Figure 22.16.
5
y
2
0
1
1
or T =
T
f
Problem 5. Determine the frequency of the sine
wave shown in Figure 22.17
Ϫ4
Problem 4.
f=
y ϭ 2 sin
180Њ
360Њ
In the sine wave shown in Figure 22.17, T = 20 ms,
hence
frequency, f =
1
1
= 50 Hz
=
T
20 × 10−3
Problem 6. If a waveform has a frequency of
200 kHz, determine the periodic time
If a waveform has a frequency of 200 kHz, the periodic
time T is given by
1
1
periodic time, T = =
f
200 × 103
3
5A
540Њ
600Њ
AЊ
Ϫ2
Figure 22.16
22.4.4 Periodic time
In practice, the horizontal axis of a sine wave will be
time. The time taken for a sine wave to complete one
cycle is called the periodic time, T.
In the sine wave of voltage v (volts) against time t (milliseconds) shown in Figure 22.17, the amplitude is 10 V
and the periodic time is 20 ms; i.e., T = 20 ms.
= 5 × 10−6s = 5 μs
22.4.6 Lagging and leading angles
A sine or cosine curve may not always start at 0◦ .
To show this, a periodic function is represented by
y = A sin(x ± α) where α is a phase displacement
compared with y = A sin x. For example, y = sin A is
shown by the broken line in Figure 22.18 and, on
the same axes, y = sin(A − 60◦ ) is shown. The graph
y = sin(A − 60◦ ) is said to lag y = sinA by 60◦.
In another example, y = cos A is shown by the broken line in Figure 22.19 and, on the same axes, y =
cos(A + 45◦) is shown. The graph y = cos(A +45◦ ) is
said to lead y = cos A by 45◦ .
216.
204 Basic Engineering Mathematics
3. v = 300 sin(200πt − 0.412) V
4. A sinusoidal voltage has a maximum value of
120 V and a frequency of 50 Hz. At time t = 0,
the voltage is (a) zero and (b) 50 V. Express
the instantaneous voltage v in the form
v = A sin(ωt ± α).
7. The current in an a.c. circuit at any time t
seconds is given by
i = 5 sin(100πt − 0.432) amperes
Determine the
(a) amplitude, frequency, periodic time and
phase angle (in degrees),
5. An alternating current has a periodic time of
25 ms and a maximum value of 20 A. When
time = 0, current i = −10 amperes. Express
the current i in the form i = A sin(ωt ± α).
(b) value of current at t = 0,
6. An oscillating mechanism has a maximum displacement of 3.2 m and a frequency of 50 Hz.
At time t = 0 the displacement is 150 cm.
Express the displacement in the general form
A sin(ωt ± α)
(e)
(c)
value of current at t = 8 ms,
(d) time when the current is first a maximum,
time when the current first reaches 3A.
Also,
(f ) sketch one cycle of the waveform showing relevant points.
217.
Chapter 23
Non-right-angled triangles
and some practical
applications
23.1
The sine and cosine rules
To 'solve a triangle' means 'to find the values of
unknown sides and angles'. If a triangle is right-angled,
trigonometric ratios and the theorem of Pythagoras
may be used for its solution, as shown in Chapter 21.
However, for a non-right-angled triangle, trigonometric ratios and Pythagoras' theorem cannot be used.
Instead, two rules, called the sine rule and the cosine
rule, are used.
The rule may be used only when
(a)
1 side and any 2 angles are initially given, or
(b) 2 sides and an angle (not the included angle) are
initially given.
23.1.2 The cosine rule
With reference to triangle ABC of Figure 23.1, the
cosine rule states
a2 = b2 + c2 − 2bc cos A
or
or
23.1.1 The sine rule
With reference to triangle ABC of Figure 23.1, the sine
rule states
a
b
c
=
=
sin A sin B
sin C
b2 = a2 + c2 − 2ac cos B
c2 = a2 + b2 − 2ab cos C
The rule may be used only when
(a)
2 sides and the included angle are initially
given, or
(b) 3 sides are initially given.
A
c
B
Figure 23.1
DOI: 10.1016/B978-1-85617-697-2.00023-5
23.2
b
a
Area of any triangle
The area of any triangle such as ABC of Figure 23.1
is given by
C
(a)
1
× base × perpendicular height
2
223.
Non-right-angled triangles and some practical applications
211
D
28.5 m
728
34.6 m
758
52.4 m
488
C
Figure 23.14
448
5. Determine the length of members BF and EB
in the roof truss shown in Figure 23.15.
B
4m
F
2.5 m
508
A
5m
30.0 m
Figure 23.16
E
4m
A
B
D
2.5 m
508
5m C
For triangle ABC, using Pythagoras' theorem,
BC 2 = AB2 + AC 2
Figure 23.15
DC
tan 44◦
6. A laboratory 9.0 m wide has a span roof which
slopes at 36◦ on one side and 44◦ on the other.
Determine the lengths of the roof slopes.
7. PQ and QR are the phasors representing the
alternating currents in two branches of a circuit. Phasor PQ is 20.0 A and is horizontal.
Phasor QR (which is joined to the end of PQ
to form triangle PQR) is 14.0 A and is at an
angle of 35◦ to the horizontal. Determine the
resultant phasor PR and the angle it makes
with phasor PQ.
23.6 Further practical situations
involving trigonometry
Problem 11. A vertical aerial stands on
horizontal ground. A surveyor positioned due east
of the aerial measures the elevation of the top as
48◦. He moves due south 30.0 m and measures the
elevation as 44◦ . Determine the height of the aerial
DC 2
1
tan2 44◦
DC
tan 48◦
Similarly, from triangle BCD, BC =
DC
tan 44◦
1
tan2 48◦
= (30.0)2 +
DC
tan 48◦
= 30.02
DC 2 (1.072323 − 0.810727) = 30.02
30.02
= 3440.4
0.261596
√
Hence, height of aerial, DC = 3340.4 = 58.65 m.
DC 2 =
Problem 12. A crank mechanism of a petrol
engine is shown in Figure 23.17. Arm OA is
10.0 cm long and rotates clockwise about O. The
connecting rod AB is 30.0 cm long and end B is
constrained to move horizontally.
(a) For the position shown in Figure 23.17,
determine the angle between the connecting
rod AB and the horizontal, and the length
of OB.
(b) How far does B move when angle AOB
changes from 50◦ to 120◦?
In Figure 23.16, DC represents the aerial, A is the initial
position of the surveyor and B his final position.
DC
From triangle ACD, tan 48◦ =
from which
AC
AC =
−
2
m
A
30.0 c
10.0 cm
508
B
Figure 23.17
O
2
225.
Non-right-angled triangles and some practical applications
B
7. An idler gear, 30 mm in diameter, has to be
fitted between a 70 mm diameter driving gear
and a 90 mm diameter driven gear, as shown
in Figure 23.22. Determine the value of angle
θ between the centre lines.
408
A
C
Figure 23.20
90 mm
dia
4. From Figure 23.20, determine how far C
moves, correct to the nearest millimetre, when
angle CAB changes from 40◦ to 160◦, B
moving in an anticlockwise direction.
5. A surveyor standing W 25◦ S of a tower measures the angle of elevation of the top of
the tower as 46◦30 . From a position E 23◦S
from the tower the elevation of the top is
37◦15 . Determine the height of the tower if the
distance between the two observations is 75 m.
6. Calculate, correct to 3 significant figures, the
co-ordinates x and y to locate the hole centre
at P shown in Figure 23.21.
99.78 mm
30 mm
dia
70 mm
dia
Figure 23.22
8. 16 holes are equally spaced on a pitch circle
of 70 mm diameter. Determine the length of
the chord joining the centres of two adjacent
holes.
P
y
1408
1168
x
Figure 23.21
100 mm
213
226.
Chapter 24
Cartesian and polar
co-ordinates
24.1
Introduction
There are two ways in which the position of a point in
a plane can be represented. These are
(a)
y
x
y
r = x 2 + y 2 and θ = tan−1 are the two formulae
x
we need to change from Cartesian to polar co-ordinates.
The angle θ, which may be expressed in degrees or radians, must always be measured from the positive x-axis;
i.e., measured from the line OQ in Figure 24.1. It is
suggested that when changing from Cartesian to polar
co-ordinates a diagram should always be sketched.
from which θ = tan−1
Cartesian co-ordinates, i.e. (x, y).
(b) Polar co-ordinates, i.e. (r,
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have emphasized problem solving techniques rather than the systematic ... As Engineering students our courses on vectors and tensors were taught in the traditional ..... On the assumption that A, B ,and C are conformable for the indicated ...
CS1300 or equivalent with grade of C or better; PHYS2211 with grade of C or better. 3. Textbook: Engineering Computation with MATLAB Third Edition by David M. Smith, Pearson ... Create vectors, store, retrieve and manipulate data in vector form. • Apply conditionals, iteration and built in functions to solve problems.
reference is Engineering Problem Solving using Matlab, by D.M. Etter. (Prentice ...vectors and matrices, and familiarity with the standard vector and matrix operations is ..... from 0-100°C. Combine the three results into one matrix and display.
a · (b × c)=0 then the vectors are linearly dependent. a = λb + µc b a c n. • You can see this .... for civil engineers. Two long straight pipes are .... Use vectors and their algebra "constructively" to solve problems. (The elastic collision was a good
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More About
This Textbook
Overview
Module theory is an important tool for many different branches of mathematics, as well as being an interesting subject in its own right. Within module theory, the concept of injective modules is particularly important. Extending modules form a natural class of modules which is more general than the class of injective modules but retains many of its desirable properties. This book gathers together for the first time in one place recent work on extending modules. It is aimed at anyone with a basic knowledge of ring and module
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For the last few years Margaret has been working on using spreadsheets with her
classes, and she has three "units" that she thinks would make good additions to the Forum:
Linear transformations
This unit uses a spreadsheet to set up the multiplication of a transformation
matrix and column vectors to produce reflections, rotations, and dilatations
in a graph. It looks at the relation between the area of a transformed figure
and the determinant of the transformation matrix, the orientation of the figure
and the sign of the determinant, and the shape of the object and the dot product
of the basis vectors of the transformation. It is suitable for Canada's grade 13's
(last year of high school, first year college.) It was developed by a friend and
Margaret has used it successfully for four years (and made alterations
to it) so it's student approved. She uses ASEASYAS, but other spreadsheets
would also work.
Algebraic problem solving with grade 9's
This unit introduces algebraic problem solving using spreadsheet columns as the
variables. It emphasizes thinking about a problem and using numbers first to
"play around" with the ideas before writing formulas, but when the formulas
are written it doesn't limit students to one letter. They can write A+B+C = 32
and solve the problem by simple substitution.
An introduction to functions using EXCEL (grade 11 level)
This unit was sent to Margaret by a friend who knows she's interested in using
spreadsheets. She hasn't used the unit but says it looks fabulous and would
transfer to the online environment very well.
Margaret has also done some assignments using Sketchpad that she
thinks might give teachers some focus for a geometry unit. She writes,
Most Sketchpad books give very detailed info on how to do constructions and
they pose good questions, but I find that I don't have the time required
and the material doesn't always fit my curriculum. I want to telescope
many ideas into a few lab periods, get the most out of those sessions, then
head back to class and work on proof. In most cases I'm dealing with kids
who've never used a geometry program so an introduction needs to be built in.
For the last few years Margaret has taught an OAC Algebra and Geometry course
(similar to Advanced Placement in the U.S.). It includes
vector geometry, linear transformations, complex number theory
and three dimensional Cartesian graphing. For the independent study
associated with the course (worth about 10%), students usually do essays
or reports on topics. They use Spreadsheets,
Sketchpad, Maple, or a TI-83 to investigate a topic (the Koch snowflake,
Sierpinski triangle, Napoleon's theorem, Polar graphing, the Glad proof, etc.).
They must produce a 2-minute display of their work for the class and hand in
a short outline of the mathematics behind the display. Margaret writes that
last year the "show and tell" [display] session even brought non-math people
in from the corridor. The students learned quite a bit about using their
chosen program (which was the whole point!) and the displays were great.
This idea for an independent study could be put on the Forum so teachers
could use it in their classes, but I think it could also be run as an
online "show and tell." Since attachments are easier to send these days,
maybe we could have a contest where the best demonstrations are put online,
with their accompanying math explanations.
Margaret says many parents ask what can be done when their kids are falling
behind in math - especially in grade 11. She thinks there could be online
practice of some sort to fill in the gaps where students are missing chunks
of knowledge, and would be interested in exploring such a possibility.
This past year, as Margaret has continued her doctoral work in
Computer Applications and Mathematics, she has been taking some courses
online in Computer Mediated Communication and online course design. She
hopes to use the knowledge she has gained to get involved in developing
Web resources.
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The purpose of A Guide to Real Variables is to provide an aid and conceptual support for the student studying for the qualifying exam in real variables. Beginning with the foundations of the subject, the text moves rapidly but thoroughly through basic topics like completeness, convergence, sequences, series, compactness, topology and the like. All the basic examples like the Cantor set, the Weierstrass nowhere differentiable function, the Weierstrass approximation theory, the Baire category theorem, and the Ascoli-Arzela theorem are treated.
The book contains over 100 examples, and most of the basic proofs. It illustrates both the theory and the practice of this sophisticated subject. Graduate students studying for the qualifying exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.
Steven Krantz is well-known for his skill in expository writing and this volume confirms it. He is the author of more than 50 books, and more than 150 scholarly papers. The MAA has awarded him both the Beckenbach Book Prize and the Chauvenet Prize.
Table of Contents
Preface
1. Basics
2. Sequences
3. Series
4. The Topology of the Real Line
5. Limits and the Continuity of Functions
6. The Derivative
7. The Integral
8. Sequences and Series of Functions
9. Advanced Topics
Glossary of Terms from Real Variable Theory
Bibliography
Index
About the Author
About the Author
Steven G. Krantz was born in San Francisco, California and grew up in Redwood City, California. He received his undergraduate degree from the University of California at Santa Cruz and the PhD from Princeton University. Krantz has held faculty positions at UCLA, Princeton University, Penn State University, and Washington University in St. Louis. He is currently Deputy Director of the American Institute of Mathematics.
Krantz has written 160 scholarly papers and over 50 books. At least five of the latter are about aspects of complex analysis. Krantz is the holder of the Chauvenet Prize and the Beckenbach Book Award, both awarded by the Mathematical Association of America. He won the UCLA Alumni Association Distinguished Teaching Award. He is the author of How to Teach Mathematics. He has directed 16 PhD students. Krantz serves on the editorial boards of six journals and is Editor-in-Chief of two.
MAA Review
This is the third book in the imaginatively devised series 'MAA Guides'. These are not textbooks in the usual sense, and neither are they 'handbooks' consisting of lists of formulae, tables and definitions. They are intended for mathematics students in general and graduates and faculty in particular. Each book in the series provides an overview of a particular mathematical topic.
For example, what Steven Krantz has compiled here is a remarkably lucid and concise résumé of the main ideas and methods of real analysis. It begins with a review of set theory, functions, real numbers, and functions and it concludes with an outline of the main concepts and methods of metric space topology. Intermediate chapters cover a range of standard topics, such as sequences, series, topology of the real line, limits and continuity, the derivative and integral. Continued...
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Main menu
Math Problem Solver
Many high school and college students experience difficulties with solving math problems, because they are not very good at math. Unfortunately, some students hate math, but need to study it in order to complete high school or their degree requirements and thus suffer the consequences. Every year, thousands of students across the world face difficulties with mathematics, calculus, statistics, algebra and geometry, and because of this Math Problem Online Solver decided to help.
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151 – Intermediate Algebra
This course prepares students for courses that require algebraic skills beyond those taught in Elementary Algebra. Topics include equations, inequalities, linear systems in two and three variables, complex numbers and applications of functions: linear, exponential, logarithmic, quadratic, polynomial, rational and radical. In addition, the course provides a basic introduction to right triangle trigonometry, vectors, and the Laws of Sines and Cosines. Problems are approached from a variety of perspectives, including graphical, numerical, verbal and algebraic. A graphing calculator is required – the specific model is determined by the department.
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0792339401
9780792339403
This text covers the basic elements of difference equations and the tools of difference and sum calculus necessary for studying and solving, primarily, ordinary linear difference equations. It is lucidly written and carefully motivated with examples from various fields of applications.
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MATH 10A - STRUCTURE AND CONCEPTS IN MATHEMATICS I 3 units, 3 lecture hours PREREQUISITE: Math 103 or equivalent. ADVISORY: Eligibility for English 125 and 126 or English 153 or ESL 67 and 68 recommended.
Development of computational and problem solving skills, number sequences, logic, set theory, functions, ancient numeration systems, real world applications, and technology. Students should plan time outside of class to take basic skills course component. Designed for elementary credential candidates. (CAN MATH 4) (A, CSU-GE, UC, I)
Methodology of solving math-based real-world application problems from pre-calculus reinforced through a small group collaborative learning environment, emphasizing the algebraic, graphic, numeric, and symbolic methodologies. Targets at-risk and underrepresented students. Not open to students who have satisfactorily completed Mathematics 4B. (A, CSU)
Methodology of solving math-based real-world application problems from first semester calculus reinforced through a small group collaborative learning environment, emphasizing the algebraic, graphic, numeric, and symbolic methodologies. Targets at-risk and underrepresented students. Not open to students who have satisfactorily completed Mathematics 5A. (A, CSU)
Methodology for solving math-based, real-world application problems reinforced through a small group collaborative learning environment, emphasizing the algebraic, graphic, numeric, and symbolic methodologies. Targets at-risk and underrepresented students. not open to students who have satisfactorily completed Math 5B. (A, CSU)
Arithmetic operations of whole numbers, fractions, and decimals; ratio and proportions; percents; order of operations; the metric system; word problems and applications of arithmetic. Designed as a quick refresher of college arithmetic.
MATH 255 - PRE-ALGEBRA 5 units, 5 lecture hours, (Formerly Math 55) PREREQUISITE: Math 260D or a test score to place out of or above Math 260D. ADVISORY: Eligibility for English 125 and 126 or English 153 or ESL 67 and 68 recommended.
Designed to increase the student success in Elementary Algebra; ideal for students who have math anxiety or who do not feel ready for the fast pace of Mathematics 101. And introduction to selected topics that are often difficult for Mathematics 101 students (e.g., signed numbers, simplification, equations, word problems, factoring, and graphing).
Review of the mechanics of arithmetic involving computing with whole numbers, fractions, decimals, and percents. Includes word problems and applications of arithmetic. Designed for students with severe disabilities and/or learning disabilities.
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Synopses & Reviews
Publisher Comments:
CliffsStudySolver Basic Math and Pre-Algebra is for students who want to reinforce their knowledge with a learn-by-doing approach. Inside, you'll get the practice you need to tackle numbers with problem-solving tools such as
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Find a Scarsdale PrecalThis course is designed to build on algebraic and geometric concepts. It develops advanced algebra skills such as systems of equations, advanced polynomials, imaginary and complex numbers, quadratics, and concepts and includes the study of trigonometric functions. Algebra 2 is vital for students' success on the ACT, SAT 2 Math, and college mathematics entrance exams
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The purpose of the theory of each concept is to organize the precise definition of the concept, examples of it, its substructures, the ways to relate different examples of the concept algebraically (these are called morphisms in some cases), and the concept's applications, both inside its own theory and outside in other areas of mathematics.
During history, different fields of mathematics have used algebras. Algebras are about finding or specifying rules on how to calculate with certain mathematical formulas and expressions. Another algebra (which is not abstract) is elementary algebra, for example.
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Linear Algebra (CA Textbook)
Resource Type(s): Free Textbooks, Reference / Free Tool
This textbook is an introduction to Linear Algebra for advanced high school students and college undergraduates. It covers the topics of a standard US sophomore college course: Gaussian reduction, vector spaces, matrices and maps between spaces, determinants, and eigenvalues. The exercises have been developed to help students internalize the concepts and master the computations. This digital textbook was reviewed for its alignment with the content standards only; California's Social Content Review criteria were not applied. Districts, schools, and individuals planning to take advantage of this free textbook are reminded to conduct their own review to determine whether this resource meets their instructional needs. Publisher: Jim Hefferon
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A Problem-Solving Model of Quadratic Min Values Using Computer
Choi-Koh, Sang Sook, International Journal of Instructional Media
INTRODUCTION
There are two facets in the direction of reforming mathematics education. First, it is a trend that many educators in mathematics education have paid attention to problem solving since 1980's. The National Council of Teachers of Mathematics' (1980) recommendations to make problem solving the focus of school mathematics posed fundamental questions about the nature of school. The aspects of mathematics education proposed by the NCTM (1989) emphasized fostering students' problem-solving ability as one of faculties students should develop. After students graduate from schools and enter job markets, they face highly developed scientific industry in the era of information and should be able to apply what they learned through various activities for problem solving in schools. Obviously, it is necessary that mathematics instruction should be designed so that the students can have a sufficient experience of problem solving in school mathematics. So, mathematics educators are interested in operation and creative activities related to problem solving such as exploration, investigation and observation.
Second, there has been a classroom renovation with advanced multi-media facility including a computer in Korea. Also, the development of industry related to the computer changes contents and methods in teaching and learning mathematics in classrooms. Through this diffusion of the advanced computer in every area of modern society, broad and profound mathematical knowledge is required. In past, mathematical knowledge was only for a few elite of society and scholars, but nowadays, administrators in a company need manpower well prepared with more mathematical knowledge than ever before. Since computer software provides a convenient tool in teaching mathematics, various teaching methods need to be developed. Mathematical properties and principles, which used to be hardly explained and observed, can be visualized at ease, as well as complicated calculations time-consuming by paper-and-pencil can be done at once on the monitor of computer. This changing circumstance calls for reforming school mathematics.
There has been some research (cf. Roman, 1975; Roman & Laudato, 1974) using computer software for drill and practice in computer assisted instruction (CAI), however, few research (cf. Bobango, 1988; Schwartz & Yerushalmy, 1987) focused students problem-solving performance using dynamic software. As mentioned earlier, mathematics classrooms in Korea have been equipped with a pentium computer, a 43-inch TV projector, and other high-tech educational materials. However, existing software such as Geometer's Sketchpad and Mathematica written in English have difficulties being used in Korean secondary-school classrooms due to a large number of students. In addition, using computer software as a tool, teachers lack the knowledge on how to use computers and how to integrate it into regular math programs. Thus, a new computer program needs to put the computer between teachers and students for problem solving as an instructional goal and helps to lessen these difficulties. The purpose of the study is to introduce a problem-solving model using computer as a tool and discuss how effectively it integrates mathematics instruction of problem solving into a regular math program of quadratic minimum value for fostering students' problem solving ability.
BACKGROUND
Visualization
Zimmermann and Cunningham (1991) stated that visualization is the process of producing or using geometrical or graphical representations of mathematical concepts, principles or problems, whether hand drawn or computer generated (p. 1). Reinforcing the visual and intuitive side of mathematics opens a new possibility for mathematical work, especially now that visualizing figures related to a problem has enough power and resolution to support it with accurate representations of problems and their solutions. The benefits of visualization include the ability to focus on specific components and details of very complex problems, to show the dynamics of systems and processes, and to increase intuition and understanding of mathematical problems and processes (Cunningham, 1991
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Undergraduate Project Documentation
This page describes what is required for an undergraduate Mathematics project, at BSc and MMath levels.
Undergraduate Project Document (2014/2015) - updated 30/10/2014
Aims and objectives
The aim of this option is to give third and fourth year students an opportunity to research a chosen mathematical topic in some depth and to improve their communication skills through producing a written account and giving a short verbal presentation on the topic. Every MMath student must take this option in the 4th year of study. All students may take this option in the 3rd year of study. It provides opportunities to develop transferable communication and time-and task-management skills, through researching the topic and organising and producing a written account and a short presentation.
Supervision
The role of the supervisor is to give guidance, initially and as the project develops, to make you aware of the standard and quantity of work desired, to comment on the general shape of your report and to give a certain amount of detailed feedback, for instance on a sample or draft chapter. For 2 semester projects Each project is different, and the frequency of meetings should be determined between you and your supervisor as the project progresses. It is usual to meet with your supervisor every two weeks initially to discuss progress, ideas and methods. However, you are encouraged to work independently and show initiative and creativity and the main responsibility for progress lies with you. If you are stuck or unclear about where you should be heading then you should contact your supervisor: do not postpone this because the deadline seems far away.
General guidelines
Different types of project:
Broadly, projects can be divided into several types:
Reading several sources and presenting some mathematical ideas. The mathematics is expected to be correct and substantial, and the presentation coherent.
Investigating a mathematical or statistical model using numerical/statistical methods.
Developing some new mathematical ideas or details, where statements and/or proofs are not in the literature. Compared to the previous types, the amount of mathematics can be less, but should be no less accurate.
Essay style projects - for example a historical project. The amount of actual mathematics would be lower, but there should be a correspondingly greater amount of analysis and criticism.
Other types of project are also possible, and many projects will be a combination of more than one of these aspects. A mark of 100% would be obtainable for a perfectly written project which a student has done mostly independently and is sufficiently novel that the content could be published in a respectable journal (probably after being suitably rewritten).
What is expected
This varies according to the type of project.
Length: There is no set length, and it depends on the 'density of the content'. As a rough guide, projects tend to be about 25 pages per 10 credits. If there are many diagrams, or much computer code then this should be increased by a corresponding amount, and essay style projects should also be a little longer.
Correct English: Grammar, punctuation and spelling are important. Notice that in all books and research papers you read, the mathematics is punctuated properly, and displayed equations end in a full stop where appropriate. The book by Higham [1] (listed in the bibliography below) is an excellent manual for writing mathematics.
Typesetting: All projects must be typeset on the computer although diagrams may be added by hand. You will also be expected to submit a pdf version, which may exclude the diagrams if they are hand-drawn. See below for more details.
Plagiarism
Plagiarism is simply passing someone else's work off as your own, and is considered a serious offence. In mathematics, copying a definition or the statement of a theorem is not considered plagiarism. But for a long proof, it is much better to read it, absorb it and then write it in your own words, perhaps adding extra details. If there is something you want to copy more or less verbatim (perhaps a proof), then make sure you quote the source so you are not passing it off as your own. The electronic online submission will be used to check for plagiarism. See University guidelines on plagiarism.
Originality
Writing a project is like telling a story, and there are various ways you can put a bit of originality into a mathematics project.
There might be details in a proof you don't understand at first reading, so when you write it add some details which would have helped you. Also where the original author has written "clearly, ..." you could add a justification of this point - why is it clear?
A simple way to add something of your own is to add examples illustrating a definition or a theorem, or showing that a particular hypothesis is needed.
If you are reading from a text book that has exercises, then solve some of these exercises at the appropriate place in your project.
In numerical work, you can investigate a system or aspects of a system that have not been studied before.
Like telling a story, it's how the facts fit together in a narrative that makes it your own.
Finally there is the indisputable originality of proving a new result that cannot be found in the literature, or giving a significantly different proof of an existing result. Such a possibility is most likely to arise from suggestions by the supervisor.
Word Processing
There are two types of software suitable for writing the project. Firstly the wysiwyg type such as MS Word or OpenOffice.org, both of which have equation editors though both have their limitations. The other type is LaTeX, which is the ideal for writing a large amount of mathematics - it has a steeper learning curve than the wysiwyg variety, but is usually worth the effort. If you are thinking of working in a research environment, then it is even more worthwhile investing the time to learn LaTeX. The School also has a limited number of licences for Scientific Word, a package providing a user-friendly front end to LaTeX. Whatever software is used to write the project it must be capable of producing machine readable pdf, needed for the online submission.
Structure
The project must begin with a title page showing the title, author (you!), your student ID number, your supervisor's name, and the course code.
A table of contents while not essential is very helpful for the reader.
An introduction, giving an overview of the project and its context, and perhaps mentioning prerequisites (such as saying, "the reader should be familiar with a first course in linear Algebra"). Often an introduction will contain a paragraph or so describing briefly what is done in each chapter.
The main body should be divided into sections or chapters, rather than being a continuous stream of ideas.
Possibly a conclusion, summing up the most important aspects. This is often a good place to show an overall understanding.
'Appendices'' if relevant, giving for example computer code.
Bibliography. This should include in all the texts you have made use of during your project, including websites. Reference to websites should include the date of access, just as reference to a book should include the edition number if there's more than one. It is a good idea to collect this information as you progress, rather than trying to remember at the end which sources you used (usually an impossible task). There are several different acceptable styles for bibliographies, and looking in books or research papers will help.
Oral presentation and examination
There will be an oral examination for every project and this will take place after the submission deadline and will be scheduled by your supervisor. This exam should begin by the student giving a short (10-15 minute) presentation on the project, which is followed by questions from the examiners. The main purpose is to test understanding. The presentation can be delivered with chalk and blackboard, with overhead transparencies or with a computer presentation. The latter two need a bit more organization, so please ensure you give the supervisor adequate notice of which method of delivery you prefer. In such a short presentation, you will not be able to cover all the details of the project, so do not try. It is better to give a short overview describing what you find are the most interesting points, and perhaps selected details.
Students who fail to attend the oral examination without good reason, will see a reduction of marks for Understanding.
Awards of marks
Marks for all projects are awarded under 5 principal categories (but not every criterion here is relevant to every project):
Structure (10%) Well written introduction and possibly conclusion; bibliography; overall organization of material
Presentation (25%) Precise and effective communication; Clarity of writing and exposition; explanation and coherent use of notation; clearly written equations
Accuracy (20%) Precise mathematical arguments; consideration of accuracy in use of numerical methods;
Understanding (25%) Appreciation of the meaning, context and significance of the work.
While 'quantity' is not explicit in this list, lack of content would be reflected in low marks across all 5 categories.
Regulations
Who can take a project:
Level 3: Students may either do a one semester (10-credit) or a two semester (20-credit) project. Normally, a student would not undertake a one semester project in each semester, however permission to do so can be granted by the Senior Tutor. A first semester project may be converted to a two-semester project up to the ninth week of the first semester, and only with the agreement of the supervisor. A joint honours student will not be permitted to do a project if, by doing so, the mathematics content of the third year would amount to less than 40 credits. On some joint degree programmes this might mean that students cannot do a project in both subjects.
Level 4: All MMath students must take 30-credits worth of projects, which can be a two semester 30-credit project or a one semester project (15-credit) in each semester. Students who wish to take the latter option are required to arrange supervisors for both projects at the start of the academic year.
MPhys: Note that MPhys students on a Mathematics and Physics joint degree are governed by regulations in the School of Physics. It is usual to do two projects, one in mathematics (of either 15 or 20 credits) and one in physics (of 20 credits), with one in each semester. There is a possibility of one combined project in an appropriate topic in mathematical physics. The course codes for the 20-credit Mathematics projects are MATH40031 in the first semester and MATH40032 in the second (these are only available on the Mathematics and Physics joint degree).
Overlap
A student may do a project and also take a course that covers related material but the overlap must not be large since, in terms of content, the project will be judged on the non-overlapping material. Check with the supervisor and plan ahead to avoid any such conflicts.
Registration
You must register by filling in the registration form (available here) to confirm you have the agreement of the supervisor. The completed form should be taken to the Teaching and Learning Office, where a member of staff will register you on the system. If, having agreed to do a project, you later decide not to proceed, please inform the supervisor (since the supervisor might then be able to take on another student) and also the Teaching and Learning Office.
Submission
For Full Year Projects
Two paper copies (with the exception of the Full Year Interim Report, which requires one paper copy) must be submitted to the Teaching and Learning Office by the following deadlines:
Semester 1 Projects and Full Year Interim Report – 12th January 2015
Full Year Projects – 1st May 2015
Semester 2 Projects - 14th May 2015
An electronic version in machine readable pdf format should be submitted online (via Blackboard) by the same deadline. Late submissions will be subject to a penalty of a 10% reduction of the final mark for each weekday late.
The front page of your project should contain at least the following information: your name and student number; the name of your project supervisor; the relevant course code and the title of the project.
A Project Supervision Questionnaire will be sent to you after the oral examination has taken place. You should also keep a copy of the project for your own use at the oral examination.
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Mathematics in Computing: An Accessible Guide to Historical, Foundational and Application Contexts This clearly written and enlightening textbook provides a concise, introductory guide to the keymathematical concepts and techniques used by computer scientists. Topics and features: ideal for self-study, offering many pedagogical features such as chapter-opening key topics, chapter introductions and summaries, review questions, and a glossary; places our current state of knowledge within the context of the contributions made by early civilizations, such as the ancient Babylonians, Egyptians and Greeks; examines the building blocks of mathematics, including sets, relations and functions; presents an introduction to logic, formal methods and software engineering; explains the fundamentals of number theory, and its application in cryptography; describes the basics of coding theory, language theory, andgraph theory; discusses the concept of computability and decideability; includes concise coverage ofcalculus, probability and statistics, matrices, complex numbers and quaternions.
Part I Energy Efficiency Part II Networking Part III Cloud Computing Part IV Hardware Part V Modeling and Simulation Part VI Security Part VII Data Services Part VIII Monitoring Part IX Resource ManagementPart I: Getting Started with QBO and QBOA Chapter 1: Introducing QBO and QBOA Chapter 2: Embracing the QBO/QBOA Format
Part II: Managing the Books for the End User Chapter 3: Creating a Client Company in QBO Chapter 4: Managing List Information Chapter 5: Dealing with the Outflow of Money Chapter 6: Managing the Inflow of Money Chapter 7: Working in Registers Chapter 8: Handling Bank and Credit Card Transactions Chapter 9: Paying Employees Chapter 10: How's the Business Doing?
Part III: Managing the Books for the Accountant Chapter 11: Setting Up Shop in QBOA Chapter 12: Adding Companies to the QBOA Client List Chapter 13: Exploring a Client's Company from QBOA Chapter 14: Working in a Client's Company Chapter 15: Using Accountant Tools
Part IV: The Part of Tens Chapter 16: Almost Ten Things about the Chrome Browser Interface Chapter 17: Ten Ways to Use Chrome Effectively
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CY900 Foundations in Scientific Computing
Current Lecturer:
Teaching Assistants:
Current Course Homepage:
Aims:
This module aims to ensure that all students embarking on this MSc degree in Scientific Computing will have a common set of basic computational and mathematic skills that will enable full participation in the taught modules that follow this Foundations module. This module will be assessed as P/F. Students must pass this Foundations module to proceed with the rest of the course.
Learning Outcomes:
By the end of the module the student should be able to
competently manipulate files, directories and utilities under the linux operating system
demonstrate a familiarity with the emacs editor
write simple code in a scientific programming language
demonstrate a familiarity with compiling code
create and manipulate simple makefiles
demonstrate a familiarity with basic linear algebra
demonstrate a familiarity with the solving of basic differential equations
use MATLAB software for simply symbolic algebra applications
demonstrate a familiarity with basic statistics Use R (or S) for standard applications in statistics
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You are here
Elements of Abstract Algebra
Publisher:
Dover
Number of Pages:
205
Price:
12.95
ISBN:
978-0486647258
This is a concrete introduction to abstract algebra, focusing on number theory and solvability by radicals. It is not a survey course, but it covers selected important portions of the standard topics of set theory, group theory, ring theory, and field theory. It emphasizes normal subgroups, permutation groups, and composition series in group theory, polynomial rings and ideals in ring theory, and algebraic numbers and Galois theory of field theory. Its concision (only 200 pages) comes from its selectivity and from having only a few worked examples.
This is a 1984 corrected reprint of a 1971 text. The book is modular: it is organized as a series of short (1–3 page) articles, with each article explaining some idea and giving a handful of exercises on that idea. The exercises are quite difficult; a few deal with concrete or numerical examples, but most ask for proofs and they often introduce additional concepts that are not used in the main narrative.
The applications treated include solvability of polynomial equations by radicals (in particular there is an example of a quintic that is not solvable), impossibility of the classical Greek construction problems (trisecting an angle, doubling the cube, and squaring the circle), and constructibility of regular polygons (including an explicit construction of a regular 5-gon and the numerical work to construct a regular 17-gon). Algebraic numbers are used to solve some classical number theory problems, including representation of integers as a sum of two squares and Fermat's Last Theorem for exponent 3. The ideal theory section is the only one that doesn't seem closely tied to particular problems; it is motivated by Fermat's Last Theorem and the failure of unique factorization in number fields, but having re-established unique factorization through ideals, we don't do anything with it.
Although it is oriented towards several classical problems, the motivation often feels weak because the exposition is organized by subject area and not by problem. For example, composition series, the Jordan–Hölder Theorem, and solvable groups are defined and their properties derived on pp. 53–57, but we don't get around to using them until the Fundamental Theorem of Galois Theory on p. 112.
This book is not very similar to any other on the market. Another bargain-priced Dover book that is more accessible (due to having more and easier exercises), but without much broader coverage, is Pinter's A Book of Abstract Algebra. The classical survey work here is Birkhoff & Mac Lane's A Survey of Modern Algebra, but that is a genuine survey book and much more comprehensive than the present book. The present book has enough coverage for an undergraduate course, but will probably be too difficult for most undergraduate students because of the lack of examples and of easy exercises. It might be useful for an honors course, or for a course that emphasizes inquiry learning. I've found it very useful as a reference, because it is easy to find things in it and it is clearly written and well cross-referenced.
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Math 251
Ordinary and Partial Differential Equations
Course Description
In calculus courses like MATH 140 and 141, students learn to calculate the derivative of a function and to use derivatives in simple applications. In MATH 251, students will learn how derivatives commonly appear in equations used to describe the world. Equations involving derivatives are called differential equations. Differential equations play an important role in modeling the real world. Newton's laws, Maxwell's laws of electromagnetism, Einstein's equations of general relativity, Euler and Bernoulli's beam equation, the Black-Scholes equation from finance, Perelson's viral-dynamics equations in biology, and the million-dollar Navier-Stokes equations are all differential equations used daily in their respective disciplines. Today, differential equations are one of the fundamental mathematical tools for the study of systems that change over time, and are used in most areas of science, engineering, and mathematics.
MATH 251 is an introductory course on ordinary differential equations and partial differential equations. It is a 1-credit extension of MATH 250; this extra credit is used to allow the coverage of partial differential equations (which are not covered in MATH 250 because of time limitations). Partial differential equations are differential equations that involve derivatives with respect to more than one independent variable. Such equations are needed to understand phenomena like the vibration of a guitar string, the failure of an I-beam, and the diffusion of particles in fluid. The goal of this course is to teach the students some elementary techniques of ordinary and partial differential equations. Some of the topics covered in this course include first order ordinary differential equations, second order ordinary differential equations with constant coefficients, 2 × 2 linear systems with constant coefficients, stability of equilibrium solutions, Laplace transforms, Fourier series, partial differential equations which include the heat equation, wave equation and the Laplace equation.
MATH 251 fulfills all the differential equations training requirements fulfilled by MATH 250 and some majors require 251 in place of 250. On completing MATH 251, students may enroll in MATH 405, 411, 412, 417, and 419.
Class size, frequency of offering, and evaluation methods will vary by location and instructor. For these details check the specific course syllabus.
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Atherton AlgebraAfter learning the basic skills, application becomes very important. But the depth of understanding in the course by a student leads to a better prepared thinker on a higher level. You learn to think for yourself, evaluate and not simply memorize
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think, making math accessible to them. KEY TOPICS: MARKET: For all readers interested in algebra.
think, making math accessible to them. KEY TOPICS: MARKET:
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Mathematics
The mission of the Enterprise State Community College Mathematics Department is to provide an educational experience in mathematics that helps students prepare for successful roles in an ever changing society. This will be accomplished through our commitment to excellent teaching, a well-designed curriculum, and a supportive environment for all our students.
To support this mission we will be committed to professional development, updating the curriculum, making real world connections, and incorporating technology. We will employ an assortment of assessment techniques, provide a variety of teaching styles, and maintain an intervention plan for students that might be having difficulty.
Students will be challenged to develop skills in analysis, reasoning, creativity, collaborative learning, and self-expression as they gain knowledge of mathematics. We will maintain high academic and behavioral expectations and try to make every classroom minute count. Efforts will be made to direct students to realize their full potential.
Division Chair
This email address is being protected from spambots. You need JavaScript enabled to view it.
(334) 347-2623 ext. 2245
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3 Identify, interpret, and evaluate pertinent data and previous experience to reach conclusions.
8. Mathematical Reasoning
Definition:
Understanding and applying concepts of mathematics and logical reasoning in a variety of contexts, both academic and non-academic.
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Eola, IL Algebra 2Furthermore, many of my college and grad school classes used discrete math as the foundation of the topic (computer operations, quantum mechanics, etc). I have taken differential equations courses as an undergraduate, and diff-eq has been a part of the physics and engineering I've done as a gradu...
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CORE VETERINARY SKILLS: MATHEMATICAL CALCULATIONS
It's about 11 am and it has been a busy morning. A case is admitted that needs some special attention and the doctor's orders indicate the patient is to be given an oral medication available at 125mg/mL. The dosage is 50mg/kg per day split into 2 doses. Can you calculate the correct dosage? Can all the medical members of your team?
Administration of an overdose of a drug may cause side effects, or in the worst-case scenario, cost an animal its life. Conversely, under dosage could also cause great problems, for example, where an antibiotic under-dose has consequently created a bacterial mutation which is now resistant to that particular antibiotic. If the members of the veterinary medical team are skilled at calculating dosages for their patients, miscalculations are more likely to be noticed and rectified long before a drug is administered to a patient incorrectly.
Veterinary technicians, veterinary assistants and students alike will find this 6-week course an invaluable reference source, whether performing relevant veterinary calculations or studying for the VTNE. Participants are not required to work in practice to complete this course.
PLEASE NOTE: This course contains flash based Articulate presentations. iPhones and iPads are not natively flash compatible but if a flash compatible browser has been installed on the iPad the presentations will play correctly. This link Flash on iPads provides information on flash compatible browsers. The presentations may also be viewed on a Mac.
This course was previously titled Basic Principles of Veterinary Calculations.
Continuing Education Credits
Course meets the requirements for 15 RACE hours of continuing education credit for veterinary technicians in jurisdictions which recognize AAVSB's RACE approval. However, participants should be aware that some boards have limitations on the number of hours accepted in certain categories and/or restrictions on certain methods of delivery.
This course is an interactive online course that meets RACE requirements; program number 57-10522.
Course Content
Core Veterinary Skills: Mathematical Calculations is a 6-week course designed to provide an overview of basic concepts related to veterinary calculations. Numerous worked examples are included to develop the student's confidence in carrying out the procedures involved.
Each type of calculation has its own separate section and is created to present the concepts in the clearest method possible. The course is structured in such a way that the student can progress from a simple explanation of the arithmetic principles involved to the application of these principles essential to veterinary calculations.
Course Focus and Learning Objectives
This course is appropriate for all team members desiring to increase their skill in veterinary medical math.
After enrolling, please allow up to 48 hours for course activation. Each participant will have a personal start and end date that begins upon activation.
Participant Feedback
...The whole course was very beneficial for me. It's been a while since I have done dosage and it was an excellent review. I liked that the Radiography and anesthetic gases and flow rates were in the course as well.
...Review of fluid needs and drip rates was clear, concise, and understandable.
Review of dilutions/concentrations was very handy.
The quizzes and final exam were comprehensive and of adequate challenge level.
...The step by step methods the course was laid out in was very helpful. I used the course to improve my confidence in math used in every day clinic use. I wanted to also increase my knowledge in areas we don't use math as frequently, due to charts pre done in our clinic. I achieved this goals in the majority of the areas I was seeking to do so.
...I liked the interactive presentations of the material, it required me to be engaged during each module which ensured that I learned the material. I also liked the variation in assignments it kept the work interesting especially since the module's each had multiple assignments.
...The course was great! The teacher took the time to help me understand what I was doing wrong.
...I liked how each Module had example problems to work through during the lesson so you had an idea of what to do when it came time to do the assignments and exams.
Course Completion Requirements
Completed students are awarded a certificate of completion. Completion requirements include:
Examinations: All examinations must be submitted with a score of 80% or better
Course survey
Course Instructor
Pat Telschow, BS, LVT
Pat started her career in veterinary medicine as a receptionist for a small animal practice. Through the encouragement of a veterinarian, she pursued her associate's degree in veterinary science technology at Suffolk County Community College and graduated with high honors. She also has a B.S. in Education from Adelphi University.
Pat has worked for many years at the Miller Place Animal Hospital on Long Island, NY as a licensed veterinary technician and truly enjoys the "team" environment. Pat has been a VetMedTeamer since its inception in 1999. She has enjoyed working in various positions with the administrative team and the members of VetMedTeam. She started with welcoming new members in 2000 and moved up to CE Director in 2003. Pat has decided to slow down her work schedule, but will still be part of the team assisting in course development. She is also the Program Facilitator for the VetMedTeam Veterinary Assistant Program.
Pat has been married to her husband, Glenn, for more than 40 years. They have 2 adult children, Kristian and Meredith. Kristian and his wife have blessed them with 2 perfect grandsons. Meredith shares her mother's passion for veterinary medicine and is also an LVT. Pat and Glenn's "empty nest" is filled with 3 kitties, a red slider turtle, a beautiful English Pointer, Willow, and the little clown (Pekingese/Shih Tzu/Chihuahua mix), Luci.
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TI-89 Graphing Calculator for Dummies
by C. C. Edwards Publisher Comments
You'll be inspired by this low-stress guide to a high-tech math learning tool! Your TI-Nspire is unlike any mathematical tool you've ever seen, so you'll really appreciate this plain-English guide to what it can do and how to do it. From loading the... (read more)
Ti-nspire for Dummies (2ND 11 Edition)
by Jeff Mccalla Publisher Comments
Your TI-Nspire has had a makeover! Learn to use all its amazing, updated features Your TI-Nspire is simply one of the most fantastic math tools in existence. Now with color, touchpad control, and other upgrades, it has even more to offer! Whether you're... (read more)
Using the Ti-83 Plus/Ti-84 Plus
by Christopher R. Mitchell Publisher Comments
Summary Using the TI-83 Plus/TI-84 Plus is a hands-on guide to these powerful graphing calculators. This easy-to-follow book includes terrific tutorials and plenty of exercises and examples that let you learn by doing. It starts by giving you a hands-on... (read more)
Slide Rules: Their History, Models & Makers
by Peter M Hopp Book News Annotation
In the hopes of "preserving these delightful devices for future
generations," this collector of slide rules covers everything one
could possibly want to know about this crude form of analog computer:
from its invention in the 17th century to... (read more)
Slide Rules: A Journey Through Three Centuries
by Dieter von Jezierski Description
A work for the serious collector, historian of technology, or widget
fanatic, this is a newly revised and translated edition of a labor of
love first published in German in 1977. It traces the history of the
slide rule from its beginnings in the 17th
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Applied Mathematics - 4th editionur
burb ...show less
1118475801
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Math Links
Ever want to know the definition of a
mathematics term? See the math
dictionary. Click on any of the textbooks shown on the main page
and then click "glossary" in the left margin.
Do you have questions
about a particular "math" problem or a mathematics procedure or topic?
Ask Dr. Math. or check out
Big Chalk Do you need to review the
"basics"" Check out aaa math.
How about algebra? Log on to
purple math.
"M10: A Student's First Course in
Mathematica," a video-based training course that allows
students to access more in-depth Mathematica training at
their own time and pace. Students can work alongside the
videos with the included exercises to practice the skills
they've learned. For more information and to order this
course, visit:
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Basic Linear Equation Practice in Elementary Algebra, Grades 4-5
Grade 4 Math · Grade 5 Math
Ned Tarrington
This book is available for download with iBooks on your Mac or iOS device, and with iTunes on your computer. Books can be read with iBooks on your Mac or iOS device.
Description
Practice solving linear equations with these fifty basic problems in elementary algebra. The student selects a single variable linear equation, solves for the variable, and checks the answer by viewing the step-by-step solution. Links allow navigation from the problem list to the answer, and back to the list.
Instead of merely viewing an answer, seeing the steps made to solve a problem can often enhance the learning process. Problems start with low difficulty and gradually increase to challenging. Most appropriate for 4th and 5th grade students.
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MA212 Calculus and Analytic Geom III
for U1AA 2005
Mission Statement: The mission of Park University, an entrepreneurial institution of learning, is to
provide access to academic excellence, which will prepare learners to think critically, communicate effectively
and engage in lifelong learning while serving a global community.
Vision Statement: Park University will be a renowned international leader in providing innovative
educational opportunities for learners within the global society.
Additional Resources:
Various Shaum's Outline Series books are great for review or supplemental resources.
Course Description:
The algebra and calculus of vectors and vector functions, constant termed sequences and series, power series and
convergence criteria. Pre-requisite: MA211. 3:0:3
Educational Philosophy:
Success in mathematics requires a lot of practice, and students are expected to work sufficient numbers of
problems outside of class to attain an adequate level of proficiency. Math courses are not exercises in
"memorization", rather, upon completion of the course the student should have a thorough understanding of the
fundamentals and be able to effectively apply the learned concepts to new problems and accurately interpret
the results. Students must take responsibility for their own learning.
Learning Outcomes:
Upon completion of the course, the student should understand and be able to solve problems involving the following
mathematical concepts:
- vectors and vector valued functions
- scalar (dot) and vector (cross) products
- vector and non-vector based mathematical descriptions of lines, planes, and surfaces
- limits, derivatives, and integrals of vector valued functions
- curvalinear motion and the concept of curvature
- convergence/divergence of sequences and convergent sequence limits
- sequences of partial sums and their use in describing infinite series
- power series representation of functions
Course Assessment:
- Daily mini-quizzes/participation:
Most class periods will include a mini-quiz. These 10 minute quizzes cover fundamental concepts and are intended
to provide both the student and the instructor feedback on the students' progress. The mini-quizzes also
encourage the students to stay caught up...falling behind in math classes can be disasterous! Students fully
participating in the class and working the recommended homework problems will easily excel on the mini-quizzes.
On class days without a quiz, students will receive a participation grade, equal in weight to a mini-quiz. The
participation grade will be adjusted according to the student's participation (e.g., absence=0, leaving midway
through class=50%, etc.). Missing class detracts from the learning experience (whether the absence is legitimate
or not), so there will be NO make-up's for mini-quizzes/participation grades under any circumstances.
Final Grades will be assigned based on:
90 to 100 A
80 to < 90 B
70 to < 80 C
60 to < 70 D
< 60 F
Late Submission of Course Materials:
Assignments should be turned in on the specified due date. A penalty of 10% per class period beyond the due date
will be assessed on late work, at the discretion of the instructor.
Classroom Rules of Conduct:
Students will be courteous and respectful to each other and the instructor at all times. Disruptions of the
learning environment will not be tolerated. Students will not eat or drink (except water) in the classrooms at
any time.
Academic Honesty:
Academic integrity is the foundation of the academic community. Because each student has the primary
responsibility for being academically honest, students are advised to read and understand all sections of this
policy relating to standards of conduct and academic life. Park University
2004-2005 Undergraduate Catalog Page 101
Plagiarism:
Plagiarism involves the use of quotations without quotation marks, the use of quotations without indication of the
source, the use of another's idea without acknowledging the source, the submission of a paper, laboratory report,
project, or class assignment (any portion of such) prepared by another person, or incorrect paraphrasing. Park University 2004-2005 Undergraduate Catalog Page 101
The instructor may excuse absences for valid reasons, but missed work must be made up within
the semester/term of enrollment.
Work missed through unexcused absences must also be made up within the
semester/term of enrollment, but unexcused absences may carry further penalties.
In the event of two
consecutive weeks of unexcused absences in a semester/term of enrollment, the student will be administratively
withdrawn, resulting in a grade of "WH". A "Contract for Incomplete" will not be issued to a student who has
unexcused or excessive absences recorded for a course. Students receiving Military Tuition Assistance or Veterans
Administration educational benefits must not exceed three unexcused absences in the semester/term of enrollment.
Excessive absences will be reported to the appropriate agency and may result in a monetary penalty to the student.
Report of a "F" grade (attendance or academic) resulting from excessive absence for those students who are
receiving financial assistance from agencies not mentioned in item 5 above will be reported to the appropriate
agency.
Disability Guidelines:
Park University is committed to meeting the needs of all students that meet the criteria for special assistance.
These guidelines are designed to supply directions to students concerning the information necessary to accomplish
this goal. It is Park University's policy to comply fully with federal and state law, including Section 504 of the
Rehabilitation Act of 1973 and the Americans with Disabilities Act of 1990, regarding students with disabilities.
In the case of any inconsistency between these guidelines and federal and/or state law, the provisions of the law
will apply. Park University is committed to meeting the needs of all learners that meet the criteria for special
assistance. These guidelines are designed to supply directions to learners concerning the information necessary
to accomplish this goal. It is Park University's policy to comply fully with federal and state law, including
Section 504 of the Rehabilitation Act of 1973 and the American with Disabilities Act of 1990, regarding learners
with disabilities and, to the extent of any inconsistency between these guidelines and federal and/or state law,
the provisions of the law will apply. Additional information concerning Park University's policies and procedures
related to disability can be found on the Park University web page: .
Copyright:
This material is copyright and can not be reused without author permission.
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The best way to master math is to practice, practice, practice?and 1,001 Math Problems offers ?mathophobes? and others who just need a little math tutoring the practice they need to succeed. Whether students need help calculating a tip or facing a standardized math test that could determine their future, the 1,001 math questions in this useful... more...
Presents mathematical results and open problems in a simple and self-contained manner Contains new results in rapidly progressing areas of research in mathematics, engineering, cryptography, mathematical physics and economics Provides a historical overview of analytic number theory, approximation theory and special functions more...
The book provides a systematic application of Lie groups to difference equations, difference meshes, and difference functionals. Besides the well-explained theoretical background and motivations, there is also a large number of concrete examples discussed in reasonable details. Due to the fairly broad introductory part, the book is indeed self-contained....Mark Zegarelli is a math and test prep tutor and instructor in SanFrancisco and New Jersey. He is the author of Basic Math & Pre-Algebra For Dummies, SAT Math For Dummies, ACT Math For Dummies, Logic For Dummies, and Calculus II For Dummies . more...
Boundary Value Problems is a translation from the Russian of lectures given at Kazan and Rostov Universities, dealing with the theory of boundary value problems for analytic functions. The emphasis of the book is on the solution of singular integral equations with Cauchy and Hilbert kernels. Although the book treats the theory of boundary value problems,... more...
The CliffsTestPrep series offers full-length practice exams that simulate the real tests; proven test-taking strategies to increase your chances at doing well; and thorough review exercises to help fill in any knowledge gaps. CliffsTestPrep California High School Exit Exam: Mathematics can help you pass this critical competency exam necessary for... more...
This book consists of lecture notes for a semester-long introductory graduate course on dynamical systems and chaos taught by the authors at Texas A&M University and Zhongshan University, China. There are ten chapters in the main body of the book, covering an elementary theory of chaotic maps in finite-dimensional spaces. The topics include one-dimensional... more...
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1. Produce writing which reflects increasing proficiency through
planning, writing, revising, and editing and which is specific to audience and purpose.
2. Communicate ideas for a variety of school and other life
situations through listening, speaking, and reading aloud.
3. Read, evaluate, and use print, non-print, and technological
sources to research issues and problems, to present information, and to complete projects.
4. Work individually and as a member of a team to analyze and
interpret information, to make decisions, to solve problems, and to reflect, using
increasingly complex and abstract thinking.
5. Complete oral and written presentations which exhibit
interaction and consensus within a group.
6. Explore cultural contributions to the history of the English
language and its literature.
7. Discover the power and effect of language by reading and
listening to selections from various literary genres.
8. Read, discuss, analyze, and evaluate literature from various
genres and other written material.
9. Sustain progress toward fluent control of grammar,
mechanics, and usage of standard English in the context of writing and speaking.
10. Use language and critical thinking strategies to serve as
tools for learning.
ALGEBRA I
1. Describe the real number system using a diagram to show the
relationships of component sets of numbers that make up the set of real numbers.
2. Model the properties and equivalence relationships of real
numbers using manipulative materials and demonstrate these properties using algebraic
expressions.
3. Convert repeating decimals to rational numbers and explain the
process used; estimate decimal representations of irrational numbers and verify these
representations; apply these concepts in real life situations; use calculators/computers
where appropriate.
4. Analyze relationships using real life examples, identifying
domain and range; explain how a change in one quantity may result in a change in another;
determine whether or not each relationship is a function.
5. Identify the algebraic representation for a rule that defines
a given sequence and find the sequence represented by a given algebraic representation of
a rule; apply such patterns to graphs, spreadsheets, etc.
6. Solve systems of equations in two variables by various
methods, including the use of matrices; apply this concept to real life problem
situations; use calculators/computers where appropriate.
7. Use the language of algebra to communicate, in writing or by
example, the following concepts: variable, term, constant, coefficient, exponent, base,
factor, monomial, polynomial, linear expression/ equation/inequality, quadratic
expression/equation/inequality.
17. Determine the distance between two points in a Cartesian
coordinate plane and find the midpoint of the segment, with and without the use of
calculators/computers; apply distance and midpoint concepts to real life problems.
18. Perform operations on numbers written in scientific notation;
apply to real life problems; explain processes used and solutions obtained; use
calculators/computers as appropriate.
19. Investigate measurement as a tool for solving problems in the
scientific realm as well as in other real world applications.
20. Use area models and other techniques to show factoring
olynomials of the following types: greatest common factor, the difference of two squares,
and special quadratic-type trinomials; justify the non-factorability of non-factorable
polynomials.
21. Interpret the slope of a line, using algebraic and geometric
interpretations; apply to real life situations.
22. Graph linear equations and inequalities on the Cartesian
coordinate plane using both calculator/computer and paper-and-pencil methods; describe the
graphical interpretations reached and investigate effects of changes in coefficients and
constants; apply these concepts to real life situations.
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1934968390","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":7.95,"ASIN":"1933241586","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":7.95,"ASIN":"1933241578","moqNum":1,"isPreorder":0}],"shippingId":"1934968390::WOPsmK5Jb8Ls7ombCIpHKvGdDj2sTeWHToKFz1c0cMTsnI3GT43vP6A792DxHonvtstc5ZU8JGjPUSenM5ufLmFwQzuF%2BxKfZbPC3%2BH7PtA%3D,1933241586::zBkYN1AX9ruQUgT9q1j4BonvH%2FmXuGagMtD9FnEMegCA7Bu9CtnX1VzRtwnt1zVQOg9y2zQlYtSdMMjnUkk3S0%2FuC%2FqnJHSU3ueeqsANzHw%3D,1933241578::srgb2c8q8G6zsF4FDxBsIAm%2BIohbtViL%2FR5aYEeuIJdka29jOhNwUOOEZS5%2FrwiZ1cBaOhi8R%2B%2BTOsws6bvFphUBUAfX91x9ZykB6ysnNS word problems workbook, I would have expected problems that require more concept-level thinking (what operation is needed here?). The problems provide plenty of repeat practice for basic math functions, but there's not much mix of types of problems in each lesson, so the student isn't really required to do much critical thinking in figuring out what approach to use in solving the problem. Once the student solves the first problem or two, the rest of the problems are pretty predictable. I would prefer to have quicker math functions involved if it meant my daughter had to spend more time critically evaluating the path to the solution.
Will write a second review later; but, just wanted to let parents/readers know that I browsed through the book and it looks great. My daughter did 2 pages of it and the questions in books are already picking her brain!!! It is less boring (better than plain old additions/substractions/multiplication/division exercise rut)! She loves to check on the net to see how many meters are in a kilometer to solve the problems in the book. So far so good; keeping my fingers crossed.
I got this to help my daughter keep up on her math during the summer. I have used the Kumon workbooks since my daughter was in Kindergarten. She is a fifth grader now and I think these books are a great helper in addition to her school studies.
i bought this series of books last year for my fourth grader.Though the way of explaining was good. The problems were too easy for fourth grader and they don't have to use any critical thinking to solve the problems. So my main mission was to teach her some tough problems or develop her brain in competitive way but she was like, its too easy and repetitive.
I love the kumon books great work and I recommend it to kids who need practice my son loves the little pictures but its mostly little examples and worksheets I bought almost all they offer in math and love all of them
If your kid has been to Kumon, you know what is involved - yes, the daily grind for the kid and the parents. If you like that grind (each of the Kumon franchise will tell you how essential it is) then of course you do not have to buy this book.
However, for rest of us, who questions that grind, this is an easier way to get the juice of the course but without the endless repetitions - at a fraction of the cost. The problems are not same as what a Kumon franchise offers. But that really did not matter. All problems are very nicely created. The difficulty level progresses through the course. And you can always go back to a previous level if you need to. I suggest you do not let your kid to mark the book to make this last part easier.
The price of the book is right. However, I wish the book should have been thicker with more problems - of course a 80 page book cannot be sufficient for the entire year even if the price goes up a bit.
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Introductory Linear Algebra An Applied First Course
9780131437401
ISBN:
0131437402
Edition: 8 Pub Date: 2004 Publisher: Prentice Hall
Summary: This book presents an introduction to linear algebra and to some of its significant applications. It covers the essentials of linear algebra (including Eigenvalues and Eigenvectors) and shows how the computer is used for applications.Emphasizing the computational and geometrical aspects of the subject, this popular book covers the following topics comprehensively but not exhaustively: linear equations and matrices an...d their applications; determinants; vectors and linear transformations; real vector spaces; eigenvalues, eigenvectors, and diagonalization; linear programming; and MATLAB for linear algebra.Its useful and comprehensive appendices make this an excellent desk reference for anyone involved in mathematics and computer applications.
Kolman, Bernard is the author of Introductory Linear Algebra An Applied First Course, published 2004 under ISBN 9780131437401 and 0131437402. One hundred ninety four Introductory Linear Algebra An Applied First Course textbooks are available for sale on ValoreBooks.com, four used from the cheapest price of $24.17, or buy new starting at $77
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Math Resources
For help in your math classes at MxCC, the math department has created a video channel on YouTube with step-by-step instructions on how to solved different types of equations. Check it out here.
Taking a standardized placement test for college-level mathematics courses can be very helpful in selecting the appropriate classes to get you started and the best course sequence for you. Visit the links below for some help in the process of completing the Placement Test.
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2014-15 Course Catalog
Mathematics Classes
MA101 Algebra for College Students I. 3 hours. A university common core Math course in the study of operations on real
numbers and algebraic expressions, polynomials, factoring, radicals and rational exponents
and graphs functions and models from a problem-solving perspective. All students having
an ACT Math subscore 19 or below are required to take this course. This course does
not count toward the analytical skills general education requirement. Successfully
completing this course with a grade of C or better and then passing MA102 Algebra
for College Students II satisfies the Math literacy common core competency requirements
and is equivalent to passing MA103 College Algebra.
MA102 Algebra for College Students II. 3 hours. A university common core Math course which is a continuation of MA101 Algebra
for College Students I with emphasis on graphs and functions, matrices, and analytical
geometry from a problem-solving perspective. Successfully completing this course with
a grade of C or better satisfies the Math literacy common core competency requirements
and is equivalent to passing MA103 College Algebra. Prerequisite: A grade of C or
better in MA101.
MA103 College Algebra. 3 hours. A study of equations and inequalities, functions and graphs, and systems
of equations and inequalities. Prerequisite: ACT Math subscore ≥ 20.
MA104 Analytic Geometry and Trigonometry. 3 hours. Theory and application of the trigonometric functions. Prerequisite: MA103
(or MA101/102) or equivalent. Primarily for students preparing for calculus or physics.
Spring.
MA112 Selected Topics in Calculus. 3 hours. An introduction to the basic concepts of calculus with business and social
science applications. Prerequisite: One and one-half years of high school algebra
or MA103 (or MA101/102). Not open to students having credit in MA118 or its equivalent.
Even-numbered Springs.
MA118 Calculus and Analytic Geometry I. 5 hours. The differentiation and integration of algebraic functions and transcendental
functions of a single variable, and an introduction to analytic geometry. Prerequisites:
ACT math subscore ≥ 20 or having passed MA103 (or MA101/102). Students having MA112
credit receive 3 hours for this course. Fall.
MA224 Mathematics for Elementary and Middle Grade Teachers. 4 hours. Mathematics central to a comprehensive elementary and middle school mathematics
curriculum in a problem solving context. Includes the development of the real numbers
as a mathematical system and an informal introduction to geometric concepts. Only
Early Childhood majors may count this course for the General Education Common Core.
Fall and Spring.
MA303 History of Mathematics. 3 hours. A study of the history of mathematics. Prerequisite: MA112 or MA118 or instructor's
permission. Even-numbered Falls.
MA305 Statistics II. 3 hours. This is a continuation of the study of statistics that began in MA105. Topics
include but are not limited to experimental design, non-parametric techniques, regression
analysis, and ANOVA. Prerequisite: A grade of "C" or better in MA105. Spring.
MA319 College Geometry. 4 hours. The rigorous development of geometry from foundational axioms, with consideration
of absolute, Euclidean, and some non-Euclidean geometry. Prerequisite: MA118 or MA112
or permission of instructor and division chair. Spring.
MA320 Writing Mathematical Documents. 2 hours. This course concerns creating mathematical documents. Students will learn
to read and write in the language of mathematics, including all the symbols and notations
commonly found in the field of mathematics, by creating original documents and interpreting
replicating existing documents. Prerequisites: MA118 and MA209 or equivalent, or by
instructor's permission.
MA480 Senior Projects (Capstone). 3 hours. A course tailored to the individual student's needs. Special projects will
be designed to extend each student's area of interest. Prerequisite: Math major with
Junior or Senior standing or with permission of the instructor.
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Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more
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Linear Algebra and Its Applications, 4th Edition
9780321385178
ISBN:
0321385179
Edition: 4th Pub Date: 2011 Publisher: Pearson
Summary: Linear assimi...late. Since they are fundamental to the study of linear algebra, students'understandingDavid C. Lay is the author of Linear Algebra and Its Applications, 4th Edition, published 2011 under ISBN 9780321385178 and 0321385179. Three hundred twenty four Linear Algebra and Its Applications, 4th Edition textbooks are available for sale on ValoreBooks.com, nine used from the cheapest price of $64.97, or buy new starting at $89The Class was Introduction to Linear Algebra. It was about learning the basics of linear algebra.
The book was very helpful for studying for the tests and trying to get a better understanding of what was being taught in class. The step by step processes outlined in the book definitely helped to make the material easier
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Has the content in this course allowed you to think of math as a useful tool? If so, how? What concepts investigated in this course can apply to your personal and professional life? In what ways did you use MyMathLab® for extra support?
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Browse related Subjects ...
Read More and features exceptional exercises based on real-world applications; the authors provide alternative avenues through which students can understand the material; each topic is presented four ways - geometrically, numerically, analytically, and verbally; students are encouraged to interpret answers and explain their reasoning throughout the book, which the author considers a unique concept compared to other books; and many of the real-world problems are open-ended, meaning that there may be more than one approach and more than one solution, depending on the student's analysis. Solving a problem often relies on the use of common sense and critical thinking skills. Students are encouraged to develop estimating and approximating skills. The book presents the main ideas of calculus in a clear, simple manner to improve students' understanding and encourage them to read the examples. Technology is used as a tool to help students visualize the concepts and learn to think mathematically. Graphics calculators, graphing software, or computer algebra systems perfectly complement this book but the emphasis is on the calculus concepts rather than the technology. (Textbook ISBN: 0471207926) Student Solutions Manual: Provides complete solutions to every odd exercise in the text. These solutions will help you develop the strong foundation you need to succeed in your Calculus class and allow you to finish the course with the foundation that you need to apply the calculus you learned to subsequent courses. (Solutions Manual ISBN: 0471213624
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Prentice Hall Math Course 1 Student Workbook 2007c
Browse related Subjects ...
Read More understanding of rational number operations preparing them to apply these skills to algebraic equations. Activity Labs throughout the text provide hands-on, minds-on experiences reaching all types of learners
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Mathematical Skills with Geometry, 8/e by Baratto/Bergman is part of the latest offerings in the successful Hutchison Series in Mathematics. The eigth edition continues the hallmark approach of encouraging the learning of mathematics by focusing its coverage on mastering math through practice. This worktext seeks to provide carefully detailed explanations and accessible pedagogy to introduce basic mathematical skills and put the content in context. The authors use a three-pronged approach (I. Communication, II. Pattern Recognition, and III. Problem Solving) to present the material and stimulate critical thinking skills. Items such as Math Anxiety boxes, Check Yourself exercises, and Activities represent this approach and the underlying philosophy of mastering math through practice. The exercise sets have been expanded, organized, and clearly labeled. Vocational and professional-technical exercises have been added throughout. Repeated exposure to this consistent structure should help advance the student's skills in relating to mathematics. The book is designed for a one-semester basic math course and is appropriate for lecture, learning center, laboratory, or self-paced courses. It is accompanied by numerous useful supplements, including McGraw-Hill's online homework management system, MathZone.
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":166.94,"ASIN":"0077354745","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":13.75,"ASIN":"1591941946","moqNum":1,"isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":30.2,"ASIN":"1591942004","moqNum":1,"isPreorder":0}],"shippingId":"0077354745::OUdrycJymv%2FSZAVkIRBV3%2FTU3x5TrAjTp5dz1zX0IqwICrC9NC02wqTMqF%2FHlzCqJYUx%2BG5mJGFrYB%2BQpmZ5%2FPa2iJM7BIO2L0POu0hCLeKGC3yg68CkyQ%3D%3D,1591941946::pLrZbQ7JtS1RfiJ%2FSAfRFaf5Mj4Hkgnmnclpx57TPDXsu8toPwqOZmjp6l%2BkcsBLM3J9tgT2D0pPVtUP1Jx1FcW7mngsFMOUZJlEt1BeiPPc6MISRNz3Ww%3D%3D,1591942004::XttyNFIesxxSt7fJSQc1F%2B88rlwVDqMmPj4KUkmtV%2B0NCPr9%2BJkEynejxMl8v8yxQ69jPPayR%2BqNg7q71%2FZYsQNmMY4O2fjbQjq%2BDulqVifzgfs7%2Bv%2BPyStefan began teaching math and science in New York City middle schools. He also taught math at the University of Oregon, Southeast Missouri State University, and York County Technical College. Currently, Stefan is a member of the mathematics faculty at Clackamas Community College where he has found a niche, delighting in the CCC faculty, staff, and students. Stefan's own education includes the University of Michigan (BGS, 1988), Brooklyn College (CUNY), and the University of Oregon (MS, 1996).
Stefan is currently serving on the AMATYC Executive Board as the organization's Northwest Vice President. He has also been involved with ORMATYC, NEMATYC, NCTM, and the State of Oregon Math Chairs group, as well as other local organizations. He has applied his knowledge of math to various fi elds, using statistics, technology, and web design. More personally, Stefan and his wife, Peggy, try to spend time enjoying the wonders of Oregon and the Pacifi c Northwest. Their activities include scuba diving, self-defense training, and hiking.
Barry has enjoyed teaching mathematics to a wide variety of students over the years. He began in the fi eld of adult basic education and moved into the teaching of high school mathematics in 1977. He taught high school math for 11 years, at which point he served as a K-12 mathematics specialist for his county. This work allowed him the opportunity to help promote the emerging NCTM standards in his region.
In 1990, Barry began the next portion of his career, having been hired to teach at Clackamas Community College. He maintains a strong interest in the appropriate use of technology and visual models in the learning of mathematics.
Throughout the past 32 years, Barry has played an active role in professional organizations. As a member of OCTM, he contributed several articles and activities to the group's journal. He has presented at AMATYC, OCTM, NCTM, ORMATYC, and ICTCM conferences. Barry also served 4 years as an offi cer of ORMATYC and participated on an AMATYC committee to provide feedback to revisions of NCTM's standards.
Don began teaching in a preschool while he was an undergraduate. He subsequently taught children with disabilities, adults with disabilities, high school mathematics, and college mathematics. Although each position offered different challenges, it was always breaking a challenging lesson into teachable components that he most enjoyed.
It was at Clackamas Community College that he found his professional niche. The community college allowed him to focus on teaching within a department that constantly challenged faculty and students to expect more. Under the guidance of Jim Streeter, Don learned to present his approach to teaching in the form of a textbook. Don has also been an active member of many professional organizations. He has been president of ORMATYC, AMATYC committee chair, and ACM curriculum committee member. He has presented at AMATYC, ORMATYC, AACC, MAA, ICTCM, and a variety of other conferences.
Above all, he encourages you to be involved, whether as a teacher or as a learner. Whether discussing curricula at a professional meeting or homework in a cafeteria, it is the process of communicating an idea that helps one to clarify it.
Most Helpful Customer Reviews
i've taught from this text for 2 semesters. the students seem to like it and performed well in the class. to be honest, i don't have much to compare it to since i haven't taught developmental math classes like this. it's very easy to read and includes jillions of sample problems and homework problems, summary exams for each chapter and for the entire part of the book as you progress. i'm not in any hurry to go find a better text since i like saving my students money by making used texts available.
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Gifts by Price
Algebra 1 Teachers Edition 2nd Edition
Product description
Present algebra topics in a logical order. The text develops an understanding of algebra by justifying methods and by explaining how to do the problems. It introduces graphing, solving systems of equations, operations with polynomials and radicals, factoring polynomials, solving rational equations, and graphing quadratic functions. Biographical sketches of mathematicians are included, as well as features on probability and statistics, Algebra Around Us, and Algebra and Scripture. The Teacher's Edition includes student pages with notes and answers (including solutions) and sections with common student errors and one-on-one activities.
Type: Spiral Bound (Teacher's Guide)Category: > Home SchoolingISBN / UPC: 9781579243265/1579243266Publish Date: 1/1/2000Item No: 114383Vendor: Bob Jones University Press
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GOAL AREA 4: MATHEMATICS
Requirements
One course
Three credits
Goals
To increase students' knowledge about mathematical and logical modes of thinking.
Knowledge of mathematics will enable students to appreciate the breadth of applications
of mathematics, evaluate arguments, and detect fallacious reasoning. Students
will learn to apply mathematics, logic, and/or statistics to help them make
decisions in their lives and careers. Minnesota's public higher education systems
have agreed that developmental mathematics includes the first three years of
a high school mathematics sequence through intermediate algebra.
Critical Thinking
The practice of critical thinking skills necessary for mathematical and logical
reasoning, including analysis, identification of appropriate problem-solving
techniques, search for valid proofs.
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Discrete MathematicsDiscrete Mathematics, by Washburn, Marlowe, and Ryan, is now available for your students. This new textbook excels at integrating the topics that make up a discrete mathematics course, creating a cohesive presentation for your students.Discrete Mathematicscombines classic, historical material and cutting-edge computer science applications in a clear, high-quality format. The exercise sets, including basic exercises, advanced exercises, and computer exercises, are designed to allow your students to master what they have learned before moving on to more difficult material. With its highly flexible organization, and unique grade of difficulty,Discrete Mathematicssuccessfully fits either the freshman-sophomore course or a more advanced junior-senior course, and is accessible to both computer scientists and mathematicians.
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Beginning and Intermediate Algebra (5th Edition
Is there anything more beautiful than an "A" in Algebra? Not to the Lial team! Marge Lial, John Hornsby, and Terry McGinnis write their textbooks and accompanying resources with one goal in mind: giving students all the tools they need to achieve success.
With this revision, the Lial team has further refined the presentation and exercises throughout the text. They offer several exciting new resources for students that will provide extra help when needed, regardless of the learning environment (classroom, lab, hybrid, online, etc)-new study skills activities in the text, an expanded video program available in MyMathLab and on the Video Resources on DVD, and more!
This site does not store any files on its server. We only index and link to content provided by other sites.
If you have any doubts about legality of content or you have another suspicions,
feel free to Contact Us.
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We will explore the mathematical strategies behind popular games, toys, and puzzles. Topics covered will combine basic...
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We will explore the mathematical strategies behind popular games, toys, and puzzles. Topics covered will combine basic fundamentals of game theory, probability, group theory, and elementary programming concepts. Each week will consist of a lecture and discussion followed by game play to implement the concepts learned268 The Mathematics in Toys and Games (MIT) to your Bookmark Collection or Course ePortfolio
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"Getting an education at MIT is like trying to drink from a firehose." — folk saying The Torch or The Firehose: A Guide to...
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"Getting an education at MIT is like trying to drink from a firehose." — folk saying The Torch or The Firehose: A Guide to Section Teaching, by MIT Mathematics Professor Arthur Mattuck, is a guide to recitation teaching at MIT. During a typical recitation section, a teaching assistant (TA) meets with a small group of students to review the most recent lecture, expand on the concepts, work through practice problems, and conduct a discussion with the students. With good humor and sound advice, Professor Mattuck offers both novice and seasoned recitation instructors guidelines on how sections can best serve as a complement to lectures, how to help students become better learners, and how to enjoy their experience as recitation teachers. Lecturers claim they have learned something from it, too. This content was first published as a printed booklet in 1981. This is the second edition. It has had a wide distribution, both at MIT and other universities, since it first appeared. It is finally available in digital format to allow broader distribution and use of this valuable material. If any significant changes are required to adapt it to the needs of another institution, please clearly notify readers that the work is modified from the orginal version and provide a link to this web site. For archival purposes, translators should notify MIT OpenCourseWare of their version Torch or The Firehose: A Guide to Section Teaching to your Bookmark Collection or Course ePortfolio
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Intermediate Algebra, the Language and Symbolism of Mathematics with MathZone 1st edition
0073229687
9780073229683
Details about Intermediate Algebra, the Language and Symbolism of Mathematics with MathZone:
Intended for schools that want a single text covering the standard topics from Intermediate Algebra. The topics in this book are organized not following the historical pattern, but by using as the guiding principles, the AMATYC standards as outlined in Crossroads in Mathematics. Use of a graphing calculator is assumed.
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First-year math class teaches more than formulas
By Sally Pobojewski News and Information Services
A funny thing happened to 1,700 first-year students who took Math 115 Calculus this term. Along with derivatives, integrals and functions, they learned something even more valuable. They learned how to think.
The learning process wasn't always easy, according to Pat Shure, lecturer in mathematics. Shure is part of a team of U-M educators who have spent the past three years redesigning the Math 115 program.
With its emphasis on open-ended problems, written and oral communication, cooperative learning and group homework, Math 115 forces first-year students to learn calculus in a radically different way than the "plug-numbers-into-formulas" approach taught in most high schools.
Shure says she has come to expect a litany of student complaints at the beginning of each term, including:
-- "This takes too much time."
-- "If there's no right answer, how are we supposed to know if we solved the problem?"
-- "Why do we have to write complete sentences? This is a math class, not English!"
-- "I know what calculus is and this isn't calculus."
"The first two months are difficult for everyone," agrees David Burkam, who teaches Math 115 in the Residential College. "It's especially tough for students who realize that their high school study patterns aren't working anymore."
Burkam and Shure don't enjoy making life tougher for first-year students. They, along with others at the U-M and many U.S. colleges and universities today, just believe the old ways of teaching calculus don't work.
"The traditional approach to calculus instruction was based on lecture and formula memorization," says Beverly Black, from the Center for Research on Learning and Teaching, who is the program's instructional consultant.
"Students worked in isolation plugging in numbers that produced the 'right' answer, but with no understanding of why it was correct," Black says. "Now we're focusing on learning how to think through and solve problems that connect calculus to everyday life."
During the first two years of reform, Math 115 was taught by a small group of instructors who volunteered for the job. This year, however, Math 115 is being taught by a wide cross-section of faculty and teaching assistants.
Since teaching Math 115 is very different than teaching a traditional calculus class, Black worked with Shure to improve the Department of Mathematics' one-week instructional development program. All those teaching Math 115 are required to attend. Instructors also meet in small groups throughout the term to discuss teaching issues and midterm feedback from students.
"The adjustment is often more difficult for the instructor than the student," Shure says. "It's hard to give up the control of traditional teaching and make the transition to cooperative learning, where the instructor's role is more like a coach."
Ever since Math 115 began changing in the fall of 1992, Burkam has been analyzing student performance and compiling data for a formal evaluation of the new program's effectiveness. His report for LS&A and the National Science Foundation (NSF), the primary funding source for the program, is due this summer. Morton Brown, professor of mathematics, is NSF principal investigator for the overall project.
Burkam says the biggest change in students who take Math 115 is in their attitude toward mathematics. "Students have a deeper understanding of the diversity of mathematical solutions and see math as more relevant to other
classes and to 'real world' problems," Burkam observes. "They express a greater interest in mathematics and have more confidence in their own mathematical abilities."
Burkam also points out the nonquantifiable benefits students mention in essays they complete at the end of the term, such as:
-- "Group homework forced me to open myself to others' methods and develop friendships with people I would have never met outside of class."
-- "The most important thing I learned in calculus was how to work as a member of a team."
NSF is watching the U-M calculus program closely, according to Shure, to see whether the cooperative learning approach can work with large groups of students and instructors of diverse backgrounds and educational levels.
"Everyone's looking at Michigan now," Burkam says. "I don't believe there's any other institution this size attempting to implement such an innovative program that reaches all first-year students. We're setting precedents for future reform—not just in calculus, but in larger questions of emphasis on undergraduate education and new roles for faculty at major research institutions."
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JavaView Cycloid Curves to your Bookmark Collection or Course ePortfolio
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This is a simple illustration of the principle of least squares. The learner has to fit a line to three data points by...
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This is a simple illustration of the principle of least squares. The learner has to fit a line to three data points by adjusting the slope and intercept of the line.The main point of the illustration is that we visualise the SQUARE of the residuals, and the total sum of squares is made available to the learner to guide the fit.A simple tickbox allows the least squares calculated line of best fit to be superimposedast Squares as a square (geogebra illustration) to your Bookmark Collection or Course ePortfolio
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Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Logger
Select this link to open drop down to add material Logger 95 Coin Problem to your Bookmark Collection or Course ePortfolio
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By means of three spin buttons, the user specifies the parameters of a parabola: the quadratic- and linear-term coefficients...
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By means of three spin buttons, the user specifies the parameters of a parabola: the quadratic- and linear-term coefficients and the vertical intercept. An accompanying chart updates automatically. Seven additional spin buttons give the user control of the chart and three buttons reset the values, place the vertex in the chart and return a point on the parabola to the vertex :: conic sections :: parabola to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Math :: conic sections :: parabola
Select this link to open drop down to add material Math :: conic sections :: parab and Music to your Bookmark Collection or Course ePortfolio
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By means of four spin buttons, the user specifies the directrix, the focus and the eccentricity for a conic section; an...
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By means of four spin buttons, the user specifies the directrix, the focus and the eccentricity for a conic section; an accompanying chart updates automatically. Six additional spin buttons give the user control of the chart and three command buttons respectively initialize an ellipse, hyperbola and parabola: conic sections; eccentricity to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Math: conic sections; eccentricity
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Gives a clear view of how the change in bias voltages causes the MOSFET to operate in different regions of operation. Along...
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Gives a clear view of how the change in bias voltages causes the MOSFET to operate in different regions of operation. Along with the variation along the IV characteristic curve the applet also shows how the channel length vary with the change in voltages Metal Oxide Semiconductor FET Operation to your Bookmark Collection or Course ePortfolio
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Möbius Transformations Revealed is a short video by Douglas Arnold and Jonathan Rogness which depicts the beauty of Möbius...
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Möbius Transformations Revealed is a short video by Douglas Arnold and Jonathan Rogness which depicts the beauty of Möbius transformations and shows how moving to a higher dimension reveals their essential unity. It was one of the winners in the 2007 Science and Visualization Challenge and was featured along with the other winning entries in the September 28, 2007 issue of journal Science. The video, which was first released on YouTube in June 2007, has been watched there by more than a million viewers and classified as a "Top Favorite of All Time" first in the Film & Animation category and later in the Education category. It has been selected for inclusion in MathFilm Festival 2008ebius Transformations Revealed to your Bookmark Collection or Course ePortfolio
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Whole numbers are no better than any others! Practice plotting values on the number line as a passionate activist rises up...
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Whole numbers are no better than any others! Practice plotting values on the number line as a passionate activist rises up and demands equity for all numbers, including fractions and decimals.Number Rights addresses number and operations standards as well as the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It supports students in comparing and ordering fractions, decimals, and percents efficiently and finding their approximate locations on a number line Number Rights to your Bookmark Collection or Course ePortfolio
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Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x , and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential... more...
This classic text by a distinguished mathematician and former Professor of Mathematics at Harvard University, leads students familiar with elementary calculus into confronting and solving more theoretical problems of advanced calculus. In his preface to the first edition, Professor Widder also recommends various ways the book may be used as a text... more...
Modern conceptual treatment of multivariable calculus, emphasizing the interplay of geometry and analysis via linear algebra and the approximation of nonlinear mappings by linear ones. At the same time, ample attention is paid to the classical applications and computational methods. Hundreds of examples, problems and figures. 1973 edition. more...
"This book is an excellent classroom text, since it is clearly written, contains numerous problems and exercises, and at the end of each chapter has a summary of the significant results of the chapter." ? Quarterly of Applied Mathematics . Fundamental introduction for beginning student of absolute differential calculus and for those interested... more...
This book by Robert Weinstock was written to fill the need for a basic introduction to the calculus of variations. Simply and easily written, with an emphasis on the applications of this calculus, it has long been a standard reference of physicists, engineers, and applied mathematicians. The author begins slowly, introducing the reader to the calculus... more...
Comprehensive but concise, this introduction to differential and integral calculus covers all the topics usually included in a first course. The straightforward development places less emphasis on mathematical rigor, and the informal manner of presentation sets students at ease. Many carefully worked-out examples illuminate the text, in addition to... more...
Students progressing to advanced calculus are frequently confounded by the dramatic shift from mechanical to theoretical and from concrete to abstract. This text bridges the gap, offering a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces. The first five chapters consist... more...
This concise introduction to Lebesgue integration is geared toward advanced undergraduate math majors and may be read by any student possessing some familiarity with real variable theory and elementary calculus. The self-contained treatment features exercises at the end of each chapter that range from simple to difficult. The approach begins with... more...
This text for advanced undergraduate and graduate students presents a rigorous approach that also emphasizes applications. Encompassing more than the usual amount of material on the problems of computation with series, the treatment offers many applications, including those related to the theory of special functions. Numerous problems appear throughout... more...
Written by a great English mathematician, this classic text begins with the differences of elementary functions and explores interpolation, mechanical quadrature, finite integration, and the summation of series. Several useful tests for the convergence and divergence of series are developed, as is a method for finding the limits of error in series... more...
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Details about Laboratories in Mathematical Experimentation:
The text is composed of a set of sixteen laboratory investigations which allow the student to explore rich and diverse ideas and concepts in mathematics. The approach is hands-on, experimental, an approach that is very much in the spirit of modern pedagogy. The course is typically offered in one semester, at the sophomore (second year) level of college. It requires completion of one year of calculus. The course provides a transition to the study of higher, abstract mathematics. The text is written independent of any software. Supplements will be available on the projects' web site.
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This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
Cool-er Reader
Nook
Kobo Reader
iRiver Story
Precise numerical analysis may be defined as the study of computer methods for solving mathematical problems either exactly or to prescribed accuracy. This book explains how precise numerical analysis is constructed. The book also provides exercises which illustrate points from the text and references for the methods presented.
All disc-based content for this title is now available on the Web.
· Clearer, simpler descriptions and explanations of the various numerical methods · Two new types of numerical problems; accurately solving partial differential equations with the included software and computing line integrals in the complex plane.
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This site is devoted to learning mathematics through practice. Many dozens of practice problems are provided in Precalculus,...
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This site is devoted to learning mathematics through practice. Many dozens of practice problems are provided in Precalculus, Calculus I - III, Linear Algebra, Number Theory, and Abstract Algebra. The last two subject areas -- referred to as "books" on the site -- are under construction. To each topic within a book (for example, Epsilon and Delta within Calculus I) there is a "module" of approximately 20 to 30 problems. Each module also includes a help page of background material. The modules are interactive to some extent and often provide suggestions when wrong answers are enteredOW -- Calculus on the Web to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material COW -- Calculus on the Web
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This is a collection of activities for use in precalculus and single variable calculus. It is prefaced by a brief summary of...
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This is a collection of activities for use in precalculus and single variable calculus. It is prefaced by a brief summary of what I know about group learning and how I use the activities. Many activities are quick combinations of discovery and practice. The statistics gets a bit lengthy, but I thought I'd include it anyway. As far as I recall, my text is only mentioned once and this posting should not be considered a commercial. Use the activities any way you want Activities for Calculus to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Classroom Activities for Calculus
Select this link to open drop down to add material Classroom Activities for Calculus Graphic derivative to your Bookmark Collection or Course ePortfolio
Select this link to close drop down of your Bookmark Collection or Course ePortfolio for material Graphic derivative
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This is an activity that provides the student with the graph of the derivative of a function and asks the student to use the...
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This is an activity that provides the student with the graph of the derivative of a function and asks the student to use the mouse to sketch the graph of the original function that passes through the origin an Anti-Derivative of a Function to your Bookmark Collection or Course ePortfolio
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Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra,...
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Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra, Trigonometry, PreCalculus, Calculus or Statistics and enter a problem. The computer solves the problem and shows the steps for the solution. It also has a worksheet generatorway to your Bookmark Collection or Course ePortfolio
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Select this link to open drop down to add material Graphing Calculator HDral of Absolute Value to your Bookmark Collection or Course ePortfolio
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| 677.169 | 1 |
Overview
Editorial Reviews
Booknews
Bittinger (mathematics, Indiana U. and Purdue U.) uses a five step problem solving approach with real data applications to make algebra both straightforward and connected to everyday life. Detailed graphs and color drawings and photographs also help students to visualize mathematical concepts. The book is designed to assist in every step of curriculum, from review exercises with answers, to pre and post-tests. There are also a number of supplemental materials available for the instructor to use in conjunction with this text. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Meet the Author
Marvin Bittinger has been teaching math at the university level for more than thirty-eight years. Since 1968, he has been employed at Indiana University Purdue University Indianapolis, and is now professor emeritus of mathematics education. Professor Bittinger has authored over 190 publications on topics ranging from basic mathematics to algebra and trigonometry to applied calculus. He received his BA in mathematics from Manchester College and his PhD in mathematics education from Purdue University. Special honors include Distinguished Visiting Professor at the United States Air Force Academy and his election to the Manchester College Board of Trustees from 1992 to 1999. Professor Bittinger has also had the privilege of speaking at many mathematics conventions, most recently giving a lecture entitled "Baseball and Mathematics." His hobbies include hiking in Utah, baseball, golf, and bowling. In addition, he also has an interest in philosophy and theology, in particular, apologetics. Professor Bittinger currently lives in Carmel, Indiana with his wife Elaine. He has two grown and married sons, Lowell and Chris, and four granddaughters.
Systems of Equations in Two Variables. Solving by Substitution or Elimination. Solving by Applications: Systems of Two Equations. Systems of Equations inThree Variables. Solving Applications: Systems of Three Equations. Systems of Linear Inequalities in Two Variables. Business and Economic Applications
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Algebra for College Students 6th edition
0534384862
9780534384869
Details about Algebra for College Students:
Aims to prepare college students for their next mathematics course, presenting all the topics associated with a first course in algebra. The structure of each section, the structure of the book and the various pedagogical features all help students to master the text.
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Rent Algebra for College Students 6th edition today, or search our site for R. David textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Brooks/Cole.
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Learn more about Beginning Algebra:
From the Publisher: Contains fully worked-out solutions to all of the odd-numbered exercises in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer
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Would it be possible to know to what extent are computers, graphing
calculators and other packages used in the teaching of mathematics nowadays
in schools (especially in high schools)? Is the use compulsory or optional?
Lastly, if these menas are not available, how desirable would it be to
introduce computer packages in a mathematics class?
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Key To Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language and examples are easy to follow. Word problems relate algebra to familiar situations, helping students understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system. This kit contains only Books 1-10. Answers Notes for Books 1-4 Books 5-7 and Books 8-10 are available separately, as well as the Key to Algebra Reproducible Tests.
Critical Thinking Co.'s Understanding Algebra I---(part of the Mathematical Reasoning series)---is a one-year Algebra I course for upper middle school and high school students that teaches basic algebraic concepts and skills. Presenting algebra as generalized arithmetic, this course was designed to help students see the connection between the math they already know and algebra. Over 100 engaging, concept-based activity sheets cover sets and set notation, evaluating expressions and solving inequalities, solving algebraic word problems, polynomials, factoring, linear functions, and more. 362 reproducible pages, softcover. Detailed answer key included in the back of the book. Grades 7-9.
Life of Fred Pre-Algebra O with Physics was formerly-titled Elementary Physics. The content has remained the same and it is still the first book in the Life of Fred Pre-Algebra Getting Ready for High School Math Series which is designed for students who have completed the elementary, intermediate, and arithmetic series books. Most students who use this series are in middle school. While most schools teach physics in twelfth grade, this age-appropriate book provides a template to learning about physics before algebra is learned; it should be used before the (sold- separately) texts Pre-Algebra 1 with Biology and Pre-Algebra 2 with Economics. Forty chapters are included, each of which ends with a Your Turn to Play segment with a small number of thought- provoking questions. Answers are...
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Kobo eBooksCurrent Algebras and Their Applications: International Series of Monographs in N... eBook
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Buy Current Algebras and Their Applications: International Series of Monographs in Natural Philosophy by Renner, B. and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Algebra 1/2 Home Study Kit includes the hardcover student text, softcover answer key and softcover test booklet. Containing 123 lessons, this text is the culmination of prealgebra mathematics, a full pre-algebra course and an introduction to geometry and discrete mathematics. Some topics covered include Prime and Composite numbers; fractions & decimals; order of operations, coordinates, exponents, square roots, ratios, algebraic phrases, probability, the Pythagorean Theorem and more. Utilizing an incremental approach to math, your students will learn in small doses at their own pace, increasing retention of knowledge and satisfaction!
An understanding of the principle elements of algebra is essential to upper-level math and good standardized test scores. Introduce your junior high students to advanced math with this kit's 160 colorful lessons. The colorful student workbook reviews basic math skills before introducing algebra, geometry, and trigonometry concepts like absolute value, transformations and nets, compound interest, permutations, combinations, two variable equations, volume and surface area of solids, four operations with monomials and polynomials, representations of data, trigonometric ratios and more. Grade 7. The student workbook includes a set of college test prep questions that follows each block of ten lessons; a new collection of math-minute interviews help students understand how ordinary people use...
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Advanced Algebra: Expanded Edition is the second book in the Life of Fred High School Mathematics Series, and is designed for students in 10th grade who have already finished the preceding Beginning Algebra, Expanded Edition. This new edition of Algebra replaces the both the earlier Life of Fred Advanced Algebra and Fred's Home Companion Advanced Algebra books; it also contains all problems completely worked out. The sold-separately Zillions of Practice Problems for Advanced Algebra book is an optional resource for students who want more practice opportunities. Ten chapters with multiple sub-lessons (105 total) are included. Each lesson ends with a Your Turn to Play segment with a small number of thought- provoking questions. Answers are provided on the next page for students to go over...
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Buy Basic Modern Algebra with Applications by Avishek Adhikari, Mahima Ranjan Adhikari and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones! The Essentials For Dummies Series Dummies is proud to present our new series, The Essentials For Dummies. Now students...
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Buy Relation Algebras by Games by I. Hodkinson, R. Hirsch and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Pre-Algebra 2 with Economics is the third book in the Life of Fred Pre-Algebra Series (the fifth and last by highlighting the importance of math in understanding economics (and the importance of understanding economics!) Thirty-four chapters are included, each of which ends with a Your Turn to Play segment with three or four questions. Answers are provided on the next page for students to go over themselves after...
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Buy Operator Algebras and Applications by Evans, David E. and Read this Book on Kobo's Free Apps. Discover Kobo's Vast Collection of Ebooks Today - Over 3 Million Titles, Including 2 Million Free Ones!
Beginning Algebra: Expanded Edition is the first book in the Life of Fred High School Mathematics Series, and is designed for students in 9th grade. This expanded edition of Algebra replaces the both the earlier Life of Fred Beginning Algebra and Fred's Home Companion Algebra books; it also contains all problems completely worked out. The sold-separately Zillions of Practice Problems for Beginning Algebra book is an optional add-on for students who want more practice opportunities. Twelve chapters with multiple sub-lessons are included. Each lesson ends with a Your Turn to Play segment with a small number of thought- provoking questions. Answers are provided on the next page for students to go over themselves after attempting to solve the problems. Chapters conclude with three problem...
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Pre-Algebra 1 with Biology is the second book in the Life of Fred Pre-Algebra Series (the fourth-the interplay between science and math. Age-appropriate, this book does not discuss reproduction, the digestive system, or evolution. Forty-six chapters are included, each of which ends with a Your Turn to Play segment with a small number of thought- provoking questions. Answers are provided on the next page for students to...
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This Saxon Algebra 2 Home Study Kit includes the Student Textbook, Testing Book and Answer Key. Traditional second-year algebra topics, as well as a full semester of informal geometry, are included with both real-world, abstract and interdisciplinary applications. Topics include geometric functions like angles, perimeters, and proportional segments; negative exponents; quadratic equations; metric conversions; logarithms; and advanced factoring. Student Text is 558 pages, short answers for problem/practice sets, an index and glossary are included; hardcover. The Test book contains both student tests and solutions with work shown along with the final answer. 32 tests are included. The Answer key shows only the final solution for the practice and problem sets found in the student text. 44
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The key to mastering Algebra 2 is becoming familiar with the shapes of graphs, and knowing how the graphs evolve as we tinker with the different aspects of each function. Calculus is a challenging subject, even for experts. The key to mastering Calculus is to properly "clean up" problems with the appropriate substitutions and cancellations.
...I can program in about 6 different computer languages, and did computer programming for Ticket Master and Card Service International, a credit card processing company which was bought out by First Data Corp. That said, I am also a trained educator. So I won't jump right into using words that yo...
...For example, one of my tips is: if there is any way an answer can be wrong, it is wrong. This sounds obvious and nearly tautological, but once students find themselves agonizing over a decision between two answers for the first time, they will understand exactly what I mean by this tip. Each qu...
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Fundamentals of Complex Analysis with Applications to Engineering and Science, 3e
This text provides a comprehensive introduction to complex variable theory and its application to engineering problems. Modeled after standard calculus books, both in layout and level of exposition, this text incorporates physical applications throughout.
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Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting... more...
Another Calculus book? As long as students find calculus scary, the failure rate in mathematics is higher than in all other subjects, and as long as most people mistakenly believe that only geniuses can learn and understand mathematics, there will always be room for a new book of Calculus. We call it Calculus Light. This book is designed for a one... more...
This new work by Wilfred Kaplan, the distinguished author of influential mathematics and engineering texts, is destined to become a classic. Timely, concise, and content-driven, it provides an intermediate-level treatment of maxima, minima, and optimization. Assuming only a background in calculus and some linear algebra, Professor Kaplan presents topics... more...
Nonconvex Optimization is a multi-disciplinary research field that deals with the characterization and computation of local/global minima/maxima of nonlinear, nonconvex, nonsmooth, discrete and continuous functions. Nonconvex optimization problems are frequently encountered in modeling real world systems for a very broad range of applications including... more...
Advances on Fractional Inequalities use primarily the Caputo fractional derivative, as the most important in applications, and presents the first fractional differentiation inequalities of Opial type which involves the balanced fractional derivatives. The book continues with right and mixed fractional differentiation Ostrowski inequalities in the univariate... more...
What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area? By combining the mathematical history of extrema with contemporary... more...
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9780471089759
ISBN:
0471089753
Publisher: Wiley & Sons, Incorporated, John
Summary: Combining standard Volumes I & II into one soft cover edition, this helpful book explains how to solve mathematical problems in a clear, step-by-step progression. It shows how to think about a problem, how to look at special cases, & how to devise an effective strategy to attack & solve the problem. Covers arithmetic, algebra, geometry, & some elementary combinatorics. Includes an updated bibliography & newly expande...d index.
Pólya, George is the author of Mathematical Discovery On Understanding, Learning, and Teaching Problem Solving, published under ISBN 9780471089759 and 0471089753. One hundred eighty four Mathematical Discovery On Understanding, Learning, and Teaching Problem Solving textbooks are available for sale on ValoreBooks.com, fifty eight used from the cheapest price of $0.01, or buy new starting at $52
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Course Description
Prerequisites
Students must have earned at least a "C" in a course
equivalent to MAT 271-Foundations of Higher Mathematics before
enrolling in this course. A semester of linear algebra (MAT
331) is recommended but not required.
understand and use modular arithmetic and the Chinese
remainder theorem
state major theorems (e.g., the division algorithm, the
unique factorization theorem, the remainder theorem, the
factor theorem, the first, second, and third isomorphism
theorems, classification of cyclic groups, Cayley's theorem)
and be able to identify the structures to which each theorem
applies (e.g. the integers, integral domains, polynomial
rings k[x] where k is a field, groups, etc.)
find examples of objects that satisfy given algebraic
properties (a noncommutative ring, a commutative ring but not
an integral domain, etc)
determine whether a given conjecture is true or false,
then prove or disprove it, constructing examples where
appropriate
prove that two rings or groups are, or are not,
isomorphic
prove that a given set is, or is not, a subring
(subgroup, subfield, ...) of a given ring (group,
field,...)
prove that congruence classes (or cosets) in a set S do,
or do not, inherit given properties from S
write proofs of other simple propositions using basic
definitions and theorems
use the techniques of abstract algebra to solve applied
problems, as appropriate.
Expected outcomes
Students should be able to demonstrate through written
assignments, tests, and/or oral presentations, that they
have achieved the objectives of MAT 333.
Method of Evaluating Outcomes
Evaluations are based on homework, class participation,
short tests and scheduled examinations covering students'
understanding of topics that are covered in MAT 333.
Irreducibility in Polynomial Rings over the
Rationals, Reals, and Complex Fields
Congruence in F[x]
Congruence-Class Arithmetic in F[x]
The Structure of F[x]/(p(x)) when p(x) is
Irreducible
Ideals and Quotient Rings
Ideals and Congruence
Quotient Rings and Homomorphisms
The Structure of R/I when I is Prime or Maximal
Groups
Definitions, Examples, and Properties of Groups
Subgroups
Isomorphisms and Homomorphisms
Additional Topics as time permits
Grading Policy
Students' grades are based on homework, class participation,
short tests, and scheduled examinations covering students'
understanding of the topics covered in MAT 333. The instructor
determines the relative weights of these factors.
Attendance Requirements
Attendance policy is set by the instructor.
Policy on Due Dates and Make-Up Work
Due dates and policy regarding make-up work are set by
the instructor.
Schedule of Examinations
The instructor sets all test dates except the date of the
final exam. The final exam is given at the date and time
announced in the Schedule of Classes.
Academic Integrity
The mathematics department does not tolerate cheating.
Students who have questions or concerns about academic
integrity should ask their professors or the counselors in the
Student Development Office, or refer to the University Catalog
for more information. (Look in the index under "academic
integrity".)
Accomodations for Students with Disabilities
Cal State Dominguez Hills adheres to all applicable federal, state, and local laws, regulations, and guidelines with respect to providing reasonable accommodations for students with temporary and permanent disabilities. If you have a disability that may adversely affect your work in this class, I encourage you to register with Disabled Student Services (DSS) and to talk with me about how I can best help you. All disclosures of disabilities will be kept strictly confidential. Please note: no accommodation may be made until you register with the DSS in WH B250. For information call (310) 243-3660 or to use telecommunications Device for the Deaf, call (310) 243-2028.
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97807645868CliffsAP Calculus AB and BC is for students who are enrolled in AP Calculus AB and/or BC or who are preparing for the Advanced Placement Examination in these areas. The Calculus BC exam includes all of the material in the Calculus AB exam plus additional selected topics, notably on sequences and series. Inside, you ll find test-taking strategies, a clear explanation of the exam format, a look at how exams are graded, and more:
A topic-by-topic look at what s on the exam
Tips for test preparation
Suggested approaches to free-response and multiple-choice questions
Two full-length practice tests
Answers to frequently asked questions about the exam
Sample questions (and answers!) and practice tests reinforce what you ve learned in areas such as limits and continuity, antiderivatives and definite integrals, and polynomial approximations. CliffsAP Calculus AB and BC also includes information on the following:
Trigonometric functions
Algebraic techniques for finding limits
Derivatives of exponential functions
Differential equations and slope fields
Radius and interval of convergence of power series
Numerical solutions to differential equations: Euler's Method
This comprehensive guide offers a thorough review of key concepts and detailed answer explanations. It s all you need to do your best and get the college credits you deserve.
*Advanced Placement Program and AP are registered trademarks of the College Board, which was not involved in the production of, and does not endorse this product
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Grade 7
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives — operating computer equipment, planning timelines and schedules, reading and interpreting data, comparing prices, managing personal finances, and completing other problem-solving tasks. What they learn in mathematics and how they learn it will provide an excellent preparation for a challenging and everchanging future. The state of Indiana has established the following mathematics standards to make clear to teachers, students, and parents what knowledge, understanding, and skills students should acquire in Grade 7:
Standard 1 — Number Sense
Understanding the number system is the basis of mathematics. Students extend this understanding to include irrational numbers, such as π and the square root of 2. They compare and order rational and irrational numbers and convert terminating decimals into fractions. They also use exponents to write whole numbers in scientific notation and to write the prime factorizations of numbers.
Standard 2 — Computation
Fluency in computation is essential. Students add, subtract, multiply, and divide integers, fractions, and decimals. They solve problems using percentages, including calculating discounts, markups, and commissions. They use mental arithmetic to compute with simple fractions, decimals, and powers.
Standard 3 — Algebra and Functions
Algebra is a language of patterns, rules, and symbols. Students at this level use variables and other symbols to translate verbal descriptions into equations and formulas. They write and solve linear equations and inequalities, and write and use formulas to solve problems. They also use properties of the rational numbers to evaluate and simplify algebraic expressions, and they further extend their understanding of graphs by investigating rates of change for linear and nonlinear functions and by developing and using the concept of the slope of a straight line.
Standard 4 — Geometry
Students learn about geometric shapes and develop a sense of space. They link geometry to coordinate graphs, using them to plot shapes, calculate lengths and areas, and find images under transformations. They understand the Pythagorean Theorem and use it to find lengths in right triangles. They also construct nets (two-dimensional patterns) for three-dimensional objects, such as prisms, pyramids, cylinders, and cones.
Standard 5 — Measurement
The study of measurement is essential because of its uses in many aspects of everyday life. Students measure in order to compare lengths, areas, volumes, weights, times, temperatures, etc. They develop the concept of similarity and use it to make scale drawings and scale models and to solve problems relating to these drawings and models. They find areas and perimeters of two-dimensional shapes and volumes and surface areas of three-dimensional shapes, including irregular shapes made up of more basic shapes.
Standard 6 — Data Analysis and Probability
Data are all around us — in newspapers and magazines, in television news and commercials, in quality control for manufacturing — and students need to learn how to understand data. At this level, they learn how to display data in bar, line, and circle graphs and in stem-and-leaf plots. They analyze data displays to find whether they are misleading and analyze the wording of survey questions to tell whether these could influence the results. They find the probability of disjoint events. They also find the number of arrangements of objects using a tree diagram.
Standard 7 — Problem Solving
In a general sense, mathematics is problem solving. In all mathematics, students use problem-solving skills: they choose how to approach a problem, they explain their reasoning, and they check their results. As they develop their skills with irrational numbers, analyzing graphs, or finding surface areas, for example, students move from simple ideas to more complex ones by taking logical steps that build a better understanding of mathematics. As part of their instruction and assessment, students should also develop the following learning skills by Grade 12 that are woven throughout the mathematics standards:
Communication
The ability to read, write, listen, ask questions, think, and communicate about math will develop and deepen students' understanding of mathematical concepts. Students should read text, data, tables, and graphs with comprehension and understanding. Their writing should be detailed and coherent, and they should use correct mathematical vocabulary. Students should write to explain answers, justify mathematical reasoning, and describe problem-solving strategies.
Reasoning and Proof
Mathematics is developed by using known ideas and concepts to develop others. Repeated addition becomes multiplication. Multiplication of numbers less than ten can be extended to numbers less than one hundred and then to the entire number system. Knowing how to find the area of a right triangle extends to all right triangles. Extending patterns, finding even numbers, developing formulas, and proving the Pythagorean Theorem are all examples of mathematical reasoning. Students should learn to observe, generalize, make assumptions from known information, and test their assumptions.
Representation
The language of mathematics is expressed in words, symbols, formulas, equations, graphs, and data displays. The concept of one-fourth may be described as a quarter, 1 4 , one divided by four, 0.25, 1 + 1 , 25 percent, or an appropriately shaded portion of a pie graph. Higher-level mathematics 8 8 involves the use of more powerful representations: exponents, logarithms, π, unknowns, statistical representation, algebraic and geometric expressions. Mathematical operations are expressed as representations: +, =, divide, square. Representations are dynamic tools for solving problems and communicating and expressing mathematical ideas and concepts.
Connections
Connecting mathematical concepts includes linking new ideas to related ideas learned previously, helping students to see mathematics as a unified body of knowledge whose concepts build upon each other. Major emphasis should be given to ideas and concepts across mathematical content areas that help students see that mathematics is a web of closely connected ideas (algebra, geometry, the entire number system). Mathematics is also the common language of many other disciplines (science, technology, finance, social science, geography) and students should learn mathematical concepts used in those
disciplines. Finally, students should connect their mathematical learning to appropriate real-world contexts.
Standard 1 Number Sense
Students understand and use scientific notation* and square roots. They convert between fractions and decimals. 7.1.1 7.1.2 Read, write, compare, and solve problems using whole numbers in scientific notation. Example: Write 300,000 in scientific notation. Compare and order rational* and common irrational* numbers and place them on a number line. Example: Place in order: -2, 5 8 , -2.45, 0.9, π, -1 3 4 . Identify rational and common irrational numbers from a list. Example: Name all the irrational numbers in the list: -2, 5 8 , -2.45, 0.9, π, -1 3 4 . Understand and compute whole number powers of whole numbers. Example: 35 = 3 3 3 3 3 = ? Find the prime factorization* of whole numbers and write the results using exponents. Example: 24 = 2 2 2 3 = 23 3. Understand and apply the concept of square root. Example: Explain how you can find the length of the hypotenuse of a right triangle with legs that measure 5 cm and 12 cm. Convert terminating decimals* into reduced fractions. Example: Write 0.95 as a fraction.
* * * * * * scientific notation: a shorthand way of writing numbers using powers of ten (e.g., 300,000 = 3 105) irrational number: a real number that cannot be written as a ratio of two integers (e.g., π, 3 , 7π) prime factors: e.g., prime factors of 12 are 2 and 3, the two prime numbers that divide 12 terminating decimals: decimals that do not continue indefinitely (e.g., 0.362, 34.1857)
7.1.3 7.1.4 7.1.5 7.1.6
7.1.7
Standard 2 Computation
Students solve problems involving integers*, fractions, decimals, ratios, and percentages. 7.2.1 Solve addition, subtraction, multiplication, and division problems that use integers, fractions, decimals, and combinations of the four operations. Example: The temperature one day is 5º. It then falls by 3º each day for 4 days and, after that, rises by 2º each day for 3 days. What is the temperature on the last day? Explain your method.
7.2.2
Calculate the percentage increase and decrease of a quantity. Example: The population of a country was 36 million in 1990 and it rose to 41.4 million during the 1990s. What was the percentage increase in the population? Solve problems that involve discounts, markups, and commissions. Example: A merchant buys CDs for $11 wholesale and marks up the price by 35%. What is the retail price? Use estimation to decide whether answers are reasonable in problems involving fractions and decimals. Example: Your friend says that 3 3 8 2 2 9 = 10. Without solving, explain why you think the answer is wrong. Use mental arithmetic to compute with simple fractions, decimals, and powers. Example: Find 34 without using pencil and paper.
* integers: …, -3, -2, -1, 0, 1, 2, 3, …
7.2.3
7.2.4
7.2.5
Standard 3 Algebra and Functions
Students express quantitative relationships using algebraic terminology, expressions, equations, inequalities, and graphs. 7.3.1 Use variables and appropriate operations to write an expression, a formula, an equation, or an inequality that represents a verbal description. Example: Write in symbols the inequality: 5 less than twice the number is greater than 42. Write and solve two-step linear equations and inequalities in one variable and check the answers. Example: Solve the equation 4x – 7 = 12 and check your answer in the original equation. Use correct algebraic terminology, such as variable, equation, term, coefficient*, inequality, expression, and constant. Example: Name the variable, terms, and coefficient in this equation: 7x + 4 = 67. Evaluate numerical expressions and simplify algebraic expressions by applying the correct order of operations and the properties of rational numbers* (e.g., identity, inverse, commutative*, associative*, distributive properties*). Justify each step in the process. Example: Simplify 3(4x + 5x – 1) + 2(x + 3) by removing the parentheses and rearranging. Explain each step you take. Solve an equation or formula with two variables for a particular variable. Example: Solve the formula C = 2πr for r. Define slope as vertical change per unit of horizontal change and recognize that a straight line has constant slope or rate of change. Example: Examine a table of values and make a conjecture about whether the table represents a linear function. Find the slope of a line from its graph. Example: Draw the graph of y = 2x – 1. Choose two points on the graph and divide the
7.3.2
7.3.3
7.3.4
7.3.5 7.3.6
7.3.7
change in y-value by the change in x-value. Repeat this for other pairs of points on the graph. What do you notice? 7.3.8 Draw the graph of a line given the slope and one point on the line, or two points on the line. Example: Draw the graph of the equation with slope of 3 and passing through the point with coordinates (0, -2). Identify functions as linear or nonlinear and examine their characteristics in tables, graphs, and equations. Example: A plant is growing taller according to the formula H = 2d + 3, where H is the height after d days. Draw the graph of this function and explain what the point where it meets the vertical axis represents. Is this graph linear or nonlinear? Identify and describe situations with constant or varying rates of change and know that a constant rate of change describes a linear function. Example: In the last example, how will the graph be different if the plant's speed of growth changes?
* * * * * coefficient: e.g., 7 is the coefficient in 7x commutative property: the order when adding or multiplying numbers makes no difference (e.g., 5 + 3 = 3 + 5), but note that this is not true for subtraction or division associative property: the grouping when adding or multiplying numbers makes no difference (e.g., in 5 + 3 + 2, adding 5 and 3 and then adding 2 is the same as 5 added to 3 + 2), but note that this is not true for subtraction or division distributive property: e.g., 3(5 + 2) = (3 5) + (3 2)
7.3.9
7.3.10
*
Standard 4 Geometry
Students deepen their understanding of plane and solid geometric shapes by constructing shapes that meet given conditions and by identifying attributes of shapes. 7.4.1 Understand coordinate graphs and use them to plot simple shapes, find lengths and areas related to the shapes, and find images under translations (slides), rotations (turns), and reflections (flips). Example: Draw the triangle with vertices (0, 0), (3, 0), and (0, 4). Find the lengths of the sides and the area of the triangle. Translate (slide) the triangle 2 units to the right. What are the coordinates of the triangle? new Understand that transformations such as slides, turns, and flips preserve the length of segments, and that figures resulting from slides, turns, and flips are congruent* to the original figures. Example: In the last example, find the lengths of the sides and the area of the new triangle. Discuss your results. Know and understand the Pythagorean Theorem and use it to find the length of the missing side of a right triangle and the lengths of other line segments. Use direct measurement to test conjectures about triangles.
7.4.2
7.4.3
Example: Use the length and width of your classroom to calculate the distance across the room diagonally. Check by measuring. 7.4.4 Construct two-dimensional patterns (nets) for three-dimensional objects, such as right prisms*, pyramids, cylinders, and cones. Example: Draw a rectangle and two circles that will fit together to make a cylinder.
* * congruent: the term to describe two figures that are the same shape and size right prism: a three-dimensional shape with two congruent ends that are polygons and all other faces are rectangles
Standard 5 Measurement
Students compare units of measure and use similarity* to solve problems. They compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less regular objects. 7.5.1 Compare lengths, areas, volumes, weights, capacities, times, and temperatures within measurement systems. Example: The area of the school field is 3 acres. How many square yards is that? Explain your method. Use experimentation and modeling to visualize similarity problems. Solve problems using similarity. Example: At a certain time, the shadow of your school building is 36 feet long. At the same time, the shadow of a yardstick held vertically is 4 feet long. How high is the school building? Read and create drawings made to scale, construct scale models, and solve problems related to scale. Example: On a plan of your school, your classroom is 5 cm long and 3 cm wide. The actual classroom is 10 m long. How wide is it? Explain your answer. Use formulas for finding the perimeter and area of basic two-dimensional shapes and the surface area and volume of basic three-dimensional shapes, including rectangles, parallelograms*, trapezoids*, triangles, circles, right prisms*, and cylinders. Example: Find the surface area of a cylindrical can 15 cm high and with a diameter of 8 cm. Estimate and compute the area of more complex or irregular two-dimensional shapes by dividing them into more basic shapes. Example: A room to be carpeted is a rectangle 5 m 4 m. A semicircular fireplace of diameter 1.5 m takes up some of the floor space. Find the area to be carpeted. Use objects and geometry modeling tools to compute the surface area of the faces and the volume of a three-dimensional object built from rectangular solids. Example: Build a model of an apartment building with blocks. Find its volume and total surface area.
* similar: the term to describe figures that have the same shape but may not have the same size
7.5.2
7.5.3
7.5.4
7.5.5
7.5.6
* * *
parallelogram: a four-sided figure with both pairs of opposite sides parallel trapezoid: a four-sided figure with one pair of opposite sides parallel right prism: a three-dimensional shape with two congruent ends that are polygons and all other faces are rectangles
Standard 6 Data Analysis and Probability
Students collect, organize, and represent data sets and identify relationships among variables within a data set. They determine probabilities and use them to make predictions about events. 7.6.1 Analyze, interpret, and display data in appropriate bar, line, and circle graphs and stem-andleaf plots* and justify the choice of display. Example: You survey the students in your school to find which of three designs for a magazine cover they prefer. To display the results, which would be more appropriate: a bar chart or a circle graph? Explain your answer. Make predictions from statistical data. Example: Record the temperature and weather conditions (sunny, cloudy, or rainy) at 1 p.m. each day for two weeks. In the third week, use your results to predict the temperature from the weather conditions. Describe how additional data, particularly outliers, added to a data set may affect the mean*, median*, and mode*. Example: You measure the heights of the students in your grade on a day when the basketball team is playing an away game. Later you measure the players on the team and include them in your data. What kind of effect will including the team have on the mean, median, and mode? Explain your answer. Analyze data displays, including ways that they can be misleading. Analyze ways in which the wording of questions can influence survey results. Example: On a bar graph of a company's sales, it appears that sales have more than doubled since last year. Then you notice that the vertical axis starts at $5 million and can see that sales have in fact increased from $5.5 million to $6.2 million. Know that if P is the probability of an event occurring, then 1 – P is the probability of that event not occurring. Example: The weather forecast says that the probability of rain today is 0.3. What is the probability that it won't rain? Understand that the probability of either one or the other of two disjoint events* occurring is the sum of the two individual probabilities. Example: Find the probability of rolling 9 with two number cubes. Also find the probability of rolling 10. What is the probability of rolling 9 or 10? Find the number of possible arrangements of several objects using a tree diagram. Example: A state's license plates contain 6 digits and one letter. How many different license
7.6.2
7.6.3
7.6.4
7.6.5
7.6.6
7.6.7
plates can be made if the letter must always be in the third position and the first digit cannot be a zero?
* stem-and-leaf plot: e.g., this one shows 62, 63, 67, 71, 75, 75, 76, etc. Stem Leaf 6 2 3 7 7 1 5 5 6 8 9 8 0 1 1 2 3 5 5 7 8 8 9 1 2 2 3 3 4 mean: the average obtained by adding the values and dividing by the number of values median: the value that divides a set of data, written in order of size, into two equal parts mode: the most common value in a given data set disjoint events: events that cannot happen at the same time
* * * *
Standard 7 Problem Solving
Students make decisions about how to approach problems and communicate their ideas. 7.7.1 Analyze problems by identifying relationships, telling relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Example: Solve the problem: ―The first three triangular numbers are shown in the diagram below. Find an expression to calculate the nth triangular number.‖
7.7.2
1 3 6 Decide to look for patterns. Make and justify mathematical conjectures based on a general description of a mathematical question or problem. Example: In the first example, notice that three dots make an equilateral triangle for the number 3 and six dots make the next equilateral triangle. Decide when and how to divide a problem into simpler parts. Example: In the first example, decide to make a diagram for the fourth and fifth triangular numbers.
7.7.3
Students use strategies, skills, and concepts in finding and communicating solutions to problems. 7.7.4 7.7.5 Apply strategies and results from simpler problems to solve more complex problems. Example: In the first example, list the differences between any two triangular numbers. Make and test conjectures by using inductive reasoning. Example: In the first example, predict the difference between the fifth and sixth numbers and use this to predict the sixth triangular number. Make a diagram to test your conjecture.
7.7.6
Express solutions clearly and logically by using the appropriate mathematical terms and notation. Support solutions with evidence in both verbal and symbolic work. Example: In the first example, use words, numbers, and tables to summarize your work with triangular numbers. Recognize the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy. Example: Calculate the amount of aluminum needed to make a can with diameter 10 cm that is 15 cm high and 1 mm thick. Take π as 3.14 and give your answer to appropriate accuracy. Select and apply appropriate methods for estimating results of rational-number computations. Example: Measure the dimensions of a swimming pool to find its volume. Estimate an answer by working with an average depth. Use graphing to estimate solutions and check the estimates with analytic approaches. Example: Use a graphing calculator to find the crossing point of the straight lines y = 2x + 3 and x + y = 10. Confirm your answer by checking it in the equations. Make precise calculations and check the validity of the results in the context of the problem. Example: In the first example, check that your later results fit with your earlier ones. If they do not, repeat the calculations to make sure.
7.7.7
7.7.8
7.7.9
7.7.10
Students determine when a solution is complete and reasonable and move beyond a particular problem by generalizing to other situations. 7.7.11 Decide whether a solution is reasonable in the context of the original situation. Example: In the first example, calculate the 10th triangular number and draw the triangle of dots that goes with it. Note the method of finding the solution and show a conceptual understanding of the method by solving similar problems. Example: Use your method from the first example to investigate pentagonal numbers.
7.7.12
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Course Description:(Developing Thinking in Algebra)
The course is designed to help mathematics teachers in
their teaching of secondary algebra. Emphasis will also
be placed on learning theories associated with algebra
instruction, aligning the algebra curriculum with
national, state and local standards, and assessment.
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Orange, CAFrom fractions and ratios, to integers, decimals, factorization, negative numbers, natural numbers. All that good stuff that makes mathematics work! This subject helps build a foundation in algebra and trigonometry to prepare you for calculus, which uses these concepts and expects a certain level of competence in them.
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More About
This Textbook
Overview
COLLEGE ALGEBRA AND CALCULUS: AN APPLIED APPROACH, Second Edition provides a comprehensive resource for college algebra and applied calculus courses. The mathematical concepts and applications are consistently presented in the same tone and pedagogy to promote confidence and a smooth transition from one course to the next. The consolidation of content for two courses in a single text saves instructors time in their course--and saves students the cost of an extra textbook.Anne Hodgkins received her Ed.D. in higher education from Texas A&M University at Commerce in 1990. She has taught at the secondary, community college and university levels. She currently teaches at Phoenix College in Phoenix, Arizona. She especially enjoys teaching intermediate and college algebra as well as calculus. Her interests include math education and research in mathematics teaching strategies
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Mathematica in K–12 and Community College Education: Part 2
Cliff Hastings
This is the second video in a series showing examples of Mathematica features that are especially useful for K–12 and community college educators. In this video, you'll discover how easy it is to create interactive Demonstrations, lessons, quizzes, and instructional handouts with Mathematica.
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I've been programming C++ now for exactly 4 months, so it's safe to say that I'm a beginner. I'm picking up the language quite well (so far) and for the most part feel rather confident, but I do have a serious weakness. My math skills are lacking
Could anyone recommend any good books to bone up on my math? Preferably one oriented towards programming.
04-13-2008
Dino
What flavor math?
04-13-2008
NeonBlack
What you can learn really depends on what you already know.
So, what do you know? What is your mathematical background?
04-13-2008
grok
Calculus would probably be the best place to start, then move from there. I've had math up through calculus, but I was pretty indifferent to it and it was a few years ago.
-edit- reading up on some floating point math and matrices probably wouldn't hurt either.
04-13-2008
Yarin
>> Need to bone up on my math
Make a 3D space shooter using math.h as your only math helping library :D
I am in no way even close to a math geek. I'm terrible, but I've noticed that programming doesn't challenge how much math you know as much as it challenges how you know how to use what math you do know.
04-13-2008
grok
Quote:
Originally Posted by Yarin
>>I've noticed that programming doesn't challenge how much math you know as much as it challenges how you know how to use what math you do know.
I think this is exactly my problem. I figure that reading up on math I haven't done in years will help some, but i guess just doing it is the best way to figure things out.
04-13-2008
m37h0d
what sort of math do you really need?
i cannot honestly imagine a single scenario under which you'd need calculus to do any programming. i do process control engineering for a living, so i personally use it all the time, but never has my need for it arisen from anything to do with programming. linear algebra? possibly, but probably not. logic and arithmetic, yes; absolutely. cannot survive without them. some number theory might also be helpful.
seriously, if you don't need it, don't waste your time relearning calculus. integrals, derivatives, and differential equations have little to nothing to do with programming. even if in some bizarre application they did somehow, everything in programming is discrete, so it all ends up boiling down to algebraic expressions anyway!
04-13-2008
Raigne
Algebra, Geometry, and Trigonometry seem to be the most important in programming to me. Although I may be mistaken.
04-13-2008
grok
I have no problems with algebra or geometry, perhaps some trig, and as I said earlier, matrices I seriously need to look into. I mean, we didn't even get into those when I went through college in the 90s.
04-14-2008
master5001
The matrix didn't come out until the late 90's so you may have finished school too early to learn about matrices. Ok, in all seriousness (what? I thought it was funny...) I think the only time I've had to use some hardcore calculus in a program is when writing software for analyzing algorithms. All things equal, read up a bit on linear algebra and you should be ok.
Math is not (only) the study of rules, procedures, functions, and formulas, but of how to think. And as much as finding applications for the rules, procedures, functions, and formulas an excellent study of mathematics includes finding out how those rules, procedures, functions, and formulas were developed so that you can reproduce the thought processes yourself.
04-14-2008
dudeomanodude
You might not do a lot of actual boolean algebra as a programmer, but you will encounter a lot of boolean logic (any if statement or other conditional is an example of this). You might think thats trivial but it will help you, trust me.
Geometry and Trig are musts. Usually, most people can intuitively infer geometry things, and go look up trig things on the web and apply them when needed.
Calculus? Probably best to understand some of the basics (taking the derivative of a polynomial and very basic integration) just in case you encounter something where it's needed.
Discrete math. That usually includes the boolean logic stuff, but also includes algorithm analysis (like Big-O) plus mathematical induction, matrices (adjacency and connectivity matrices), maps and many other odds and ends things like the pigeon-hole principle, etc.
And if you're brave, regular expressions are in and of themselves a mathematical discipline.
04-14-2008
kcpilot
If you plan on doing any kind of serious or semi-serious graphics programming, then trigonometry is an absolute must. And if you're doing graphics in 3-space, you might even need a little bit of analytic geometry. But that is probably down the road for your programming needs for now.
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Math 110 Techniques of Calculus I
Calendar - Summer 2012
Updated 05/05/2012
Date
Day
Reading Assignments,
Homework Problems, and Exam
Dates
WEEK#01 05/14/12
Monday
General information
about the course: Required texts for the course, the course syllabus will be
distributed, and how your course grade will be determined. An overview
of what you will learn in this course will be discussed. Problem
solving techniques will be the focus of this course along with applications
related to the real world. We start by reviewing the Real Number System.
Section 1-1, Real Numbers, Inequalities, and Lines (pp. 4-15). Section
1-2, Exponents (pp. 21-30).
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Buy PDF
For the first time in science education, the subject of multiple solution methods is explored in book form. While a multiple method teaching approach is utilized extensively in math education, there are very few journal articles and no texts written on this topic in science. Teaching multiple methods to science students in order to solve quantitative word problems is important for two reasons. First it challenges the practice by teachers that one specific method should be used when solving problems. Secondly, it calls into question the belief that multiple methods would confuse students and retard their learning. Using a case study approach and informed by research conducted by the author, this book claims that providing students with a choice of methods as well as requiring additional methods as a way to validate results can be beneficial to student learning. A close reading of the literature reveals that time spent on elucidating concepts rather than on algorithmic methodologies is a critical issue when trying to have students solve problems with understanding. It is argued that conceptual understanding can be enhanced through the use of multiple methods in an environment where students can compare, evaluate, and verbally discuss competing methodologies through the facilitation of the instructor. This book focuses on two very useful methods: proportional reasoning (PR) and dimensional analysis (DA). These two methods are important because they can be used to solve a large number of problems in all of the four academic sciences (biology, chemistry, physics, and earth science). This book concludes with a plan to integrate DA and PR into the academic science curriculum starting in late elementary school through to the introductory college level. A challenge is presented to teachers as well as to textbook writers who rely on the single-method paradigm to consider an alternative way to teach scientific problem solving.
Multiple Solution Methods for Teaching Science in the Classroom: Improving Quantitative Problem Solving Using Dimensional Analysis and Proportional Re PDF (Adobe DRM) can be read on any device that can open PDF (Adobe DRM) files.
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Main Content
Course Catalog: Mathematics
A weeklong course intended for students who placed into MT*0103. The Pre-Algebra Bridge is designed to assist in improving placement scores by focusing on test taking skills and targeted remediation in mathematics. Topics include whole numbers, integers, fractions, decimals, percents, ratios, and proportions. MT*0030 utilizes a departmental workbook and a computer learning management system. At the end of weeklong bridge, students will retake the mathematics portion of the placement exam.
A weeklong course intended for students who placed into MT*0204. The Introductory Algebra Bridge is designed to assist in improving placement scores by focusing on test taking skills and targeted remediation in mathematics. Topics include solving linear and absolute value equations and inequalities, graphing on the Cartesian Coordinate system, slope and equations of lines, graphing linear inequalities, solving systems of equations with graphing, rules of exponents, and multiplication/division of polynomials. MT*0040 utilizes a departmental woorkbook and a computer learning management system. At the end of weeklong bridge, students will retake the mathematics portion of the placement exam.
Pre-Algebra is an entry level course for students in preparation for the Introductory/Intermediate algebra sequence. Students will gain a background in arithmetic and algebraic topics by means of various presentation styles and group work. Topics to be covered include: arithmetic operations on the set of whole numbers, integers, and rational numbers, including decimals, exponents and percents, solving linear equations, various applications in problem solving, the coordinate system, and basic graphing. Placement in Pre-Algebra is determined by a student's score on the math placement test and/or his/her mathematical background. When successfully completed (grade C or better), the course satisfies the college's prerequisite for Introductory Algebra. DEV
Prerequisite: None
Introductory Algebra is the first course in a two course algebra sequence designed to prepare students for coursework in college level mathematics. Students will be introduced to basic algebra topics and the application of technology to those topics. The course will begin with a review of pertinent pre-algebra topics including fractions, decimals, and signed numbers. Other topics to be covered at the introductory algebra level include: arithmetic skills, solving equations and inequalities, exponents, linear equations and expressions, and the coordinate plane. Many of these topics will be developed more in subsequent courses. Successful completion with a grade of "C" or higher will satisfy the prerequisite for Intermediate Algebra.
Prerequisite: Minimum grade of C in MT 0103 or appropriate math placement score.
A developmental mathematics course designed to prepare students for coursework in college level mathematics alternate to algebra based courses. Students will gain a background in geometry, analysis, and reasoning. Topics to be covered include introductions to statistics, geometry, functions, unit analysis, consumer mathematics, and reasoning mathematically. Application based problems and learning will be infused throughout the course. Successful completion with a grade of "C" or higher will satisfy the prerequisite for Math for Art and Design or any other non-algebraic intensive college level mathematics course at a similar level. Not intended for transfer. DEV
Prerequisite: MT 0103 or appropriate math placement; minimum grade C
A course in business and financial applications of mathematics such as discounts, markups, interest, installment buying and credit cards, payroll, depreciation, taxes, etc. Intended for students in several AAS degree programs.
Prerequisite: Minimum grade of "C" in MT 0103 or appropriate math placement score to enter MT 0204.
Applied algebra and trigonometry is a study of applied topics in algebra such as equations: linear and quadratic, graphs and equation solving combined with a study of topics in geometry and trigonometry such as angles, triangles, and vectors.
Prerequisite: Minimum grade or "C" in MT 0203 of MT 0204 or appropriate math placement score to enter MT 1303.
Intermediate Algebra is the second course in a two-course sequence designed to prepare students for additional coursework in mathematics. Topics covered include: polynomial operations, factoring (including sum and difference of cubes and solving quadratics using factoring), rational expressions and equations, and radical expressions. When successfully completed (grade C or better), the course satisfies the college's prerequisite for College Mathematics and/or College Algebra or any other college level mathematics course at a similar level.
Prerequisite: Minimum grade of "C" in MT 0203 or MT 0204 or appropriate math placement score to enter MT 1303.
This course is primarily designed to provide the student enrolled in an Allied Health program with a review of basic mathematics as well as methods of medication dosage calculations using metric, apothecary, and household measures for adult and pediatric medication administration. Intravenous (IV) calculations for simple and complex administration to adult and pediatric patients is introduced. Not intended for transfer. Prequisite: Minimum grade of "C" in MT 1303 or higher.
Real Number System is intended for elementary education majors and is designed to familiarize potential elementary school teachers with the various mathematical topics taught in an elementary school environment. Topics covered will include: sets, logic, number theory, the development of the set of real numbers and real number operations, number bases and various algorithms. HOT, MTH
Prerequisite: Minimum grade of "C" in MT 1303 or appropriate math placement score.
Metric and Non-Metric Geometry is designed to familiarize potential elementary school teachers with the various mathematical topics taught in an elementary school environment. Topics covered will include measurements, plane and solid geometry, statistics and probability. HOT, MTH
Prerequisite: Minimum grade of "C" in MT 1303 or appropriate math placement score.
This course is designed for Art and Graphic Design majors The goal of the course is to study connections between mathematics and art and design. Students will see how mathematics is not just about formulas and logic, but about patterns, symmetry, structure, shape and beauty. Students will study topics like tilings, polyhedra and perspective. HOT, MAI, MTH
Prerequisite: MT 1303; minimum grade C or successful placement scores into MT 1343
College Algebra is designed to meet the needs of the student wishing to satisfy the general education math requirement or planning to enroll in additional mathematics courses. Topics covered in MT 1403 include: functions, domain, range, complex numbers, logs and exponents, polynomials, rational expressions, radicals, solving equations and inequalities, graphing equations and inequalities. The use of the graphing calculator and its application to the topics of College Algebra will be illustrated. HOT, MAI, MTH
Prerequisite: MT 1303; minimum grade C or successful placement scores into MT 1403. Graphing calculator required.
Pre-calculus Mathematics is designed to meet the needs of the student planning to enroll in mathematics courses numbered 1600 or above. MT 1505 is a unified study of College Algebra and Trigonometry, with particular emphasis given to the preparation of the student for the study of the Calculus. Topics covered include: sets, complex numbers, logs and exponents, polynomials, rational expressions, radicals, solving equations and inequalities, graphing equations and inequalities, and the study of the trigonometric functions. The use of a graphing calculator and its application to the topics of PreCalculus will be discussed. HOT, MAI, MTH
Prerequisite: Minimum grade of "B" in MT 1303 or minimum grade of "C" in MT 1403 or appropriate math placement score.
A first course in a sequence of courses including analytic geometry, differential calculus, and integral calculus. This series is recommended for majors in engineering, the physical sciences, and mathematics. Topics include: properties of real numbers, introduction to analytic geometry, functions, limits, continuity, the derivative, differentiation of functions, applications of the derivative, antiderivatives, and the definite integral. HOT, MAI, MTH
Prerequisite: Minimum grade of "C" in MT 1505 or appropriate math placement score.
Survey Calculus is an introductory study of the techniques of differential and integral calculus. The focus of the course will be on functions and the applications of the calculus to the life, social, and managerial sciences, with particular emphasis on business applications. Trigonometry is not used. HOT, MTH, MAI
Prerequisite: MT 1403 or MT 1505; minimum grade C or appropriate math placement score.
A continuation of MT 2105 including the following topics: vectors and surfaces in three-dimensional space, solid analytic geometry, differential calculus of functions of several variables, and multiple integration.
Prerequisite: Minimum grade of 'C' in MT 2105.
A specialized program of study directly related to the department's area of expertise. The course is arranged between a faculty member and student and takes into consideration the needs, interests and background of the student.
Prerequisite: Consent of instructor
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Shipping prices may be approximate. Please verify cost before checkout.
About the book:
Based linear algebra as a prerequisite, but takes the reader quickly from the basics to topics of recent research, including a number of unanswered questions. The lectures - introduce the basic facts about polytopes, with an emphasis on the methods that yield the results (Fourier-Motzkin elimination Schlegel diagrams, shellability, Gale transforms, and oriented matroids) - discuss important examples and elegant constructions (cyclic and neighbourly polytopes, zonotopes, Minkowski sums, permutahedra and associhedra, fiber polytopes, and the Lawrence construction) - show the excitement of current work in the field (Kalai's new diameter bounds, construction of non-rational polytopes, the Bohne-Dress tiling theorem, the upper-bound theorem). They should provide interesting and enjoyable reading for researchers as well as students.
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Math Tutorial Lab
The Math Lab is a drop-in center for students studying mathematics at GRCC. The lab provides student-oriented services designed to assist during the learning experience. This is a free service. No appointment is necessary.
Resources
The Math Lab provides a variety of resources including practice tests, formula handouts, videos, and other useful information. For the quickest access to these resources, visit the Mathematics Tutorial Lab Blackboard site.
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Oftentimes, when it comes to math, some people struggle with the concepts, finding it too difficult. But follow these few steps and Math will be easy in every class.
Ad
Steps
1
Master Your Basics: The number one reason that people struggle in math class is because their basics and their fundamentals are not fully developed. Algebra and Geometry are the building blocks for the more advanced math later on (Calculus, Differential Equations, etc...).
Ad
2
Get Ahead: Most schools give you a textbook for math and it's a pretty big book. What you can do is, study ahead. Whenever you have time, you can look a section ahead, and be prepared for tomorrow's material.
3
Self-Study: This is the most efficient way of studying math. I would recommend you to buy math textbooks from a local bookstore. You can also search on the internet for great math books.Don't get a book that is very short (100 pages) for a topic like Geometry. Get a textbook or a few workbooks on the topic. It's good to buy more than one book, since some books leave out certain things.
4
Studying: When you self-study, it's good to have the book and a notebook with you, college ruled preferably. Write down all the vocabulary and terms and the example problems. You don't have to do each and every single practice problem if you find it repetitive, just have an intuitive answer. (As long as you know the process of solving it) It's also good to get into a habit of working on more Word Problems, which can help you apply the concept into real-life situations.
5
Competition: If you do enough self-studying, and you look through your studying notes when you have free time, you should already have a very good basis in math. If you're a fast learner, then it would be even better since you can learn the higher level concepts quicker. If your school has a math club or team that you can join, go for it! Chances are, you'll meet individuals who are very talented in math and can help you expand your knowledge by attending competitions.
6
Loving Math: Once you do this part, math would be no challenge whatsoever. Once you get good in math, help others, it's okay to show off you knowledge, in a good way. Once you start to take interest in math and start studying it and attend math competitions and expand your knowledge on math, you will love it. Once you have a passion for math, you will want to learn more, achieve more, and become the mathematician you've never imagined
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History of Mathematics : An Introduction - 2nd edition
add addAncient Mathematics The Beginnings of Mathematics in Greece Archimedes and Apollonius Mathematical Methods in Hellenistic Times The Final Chapters of Greek Mathematics
PART II: MEDIEVAL MATHEMATICS: 500-1400
Medieval China and India The Mathematics of Islam Mathematics in Medieval Europe Mathematics Around the World
PART III: EARLY MODERN MATHEMATICS: 1400-1700
Algebra in the Renaissance Mathematical Methods in the Renaissance Geometry, Algebra, and Probability in the Seventeenth Century The Beginnings of Calculus
PART IV: MODERN MATHEMATICS: 1700-2000
Analysis in the Eighteenth Century Probability, Algebra, and Geometry in the Eighteenth Century Algebra in the Nineteenth Century Analysis in the Nineteenth Century Geometry in the Nineteenth Century Aspects of the Twentieth Century
Answers to Selected Problems General References in the History of Mathematics Index and Pronunciation Guide95 +$3.99 s/h
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SellBackYourBook Aurora, IL
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Used, Acceptable Condition, may show signs of wear and previous use. Please allow 4-14 business days for delivery. 100% Money Back Guarantee, Over 1,000,000 customers served.
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HPB-Wisconsin Wauwatosa, WI
1998 Hardcover Good Hardcover Item may show signs of shelf wear. Pages may include limited notes and highlighting. Includes supplemental or companion materials if applicable. Access codes may or m...show moreay not work. Connecting readers since 1972. Customer service is our top priority. ...show less
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Kings Ridge Media Appleton, WI
1998-03
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Undergraduate Algebra is a text for the standard undergraduate algebra course. It concentrates on the basic structures and results of algebra, discussing groups, rings, modules, fields, polynomials, finite fields, Galois Theory, and other topics. The author has also included a chapter on groups of matrices which is unique in a book at this level. Throughout... more...
The Second Waterloo Workshop on Computer Algebra was dedicated to the 70th birthday of combinatorics pioneer Georgy Egorychev. This book of formally-refereed papers submitted after that workshop covers topics closely related to Egorychev s influential works. more...
Mary Jane Sterling has been teaching algebra, business calculus, geometry, and finite mathematics at Bradley University in Peoria, Illinois, for more than 35 years. She is the author of many For Dummies math titles including Algebra I For Dummies and Algebra Workbook For Dummies . more...
This volume is a compilation of lectures on algebras and combinatorics presented at the Second International Congress in Algebra and Combinatorics. It reports on not only new results, but also on open problems in the field. The proceedings volume is useful for graduate students and researchers in algebras and combinatorics. Contributors include eminent... more...
The book is an introduction to the foundations of Mathematics. The use of the constructive method in Arithmetic and the axiomatic method in Geometry gives a unitary understanding of the backgrounds of geometry, of its development and of its organic link with the study of real numbers and algebraic structures. more...
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The Britannica Guide to Geometry by P-EncyclopaediaBrit Britannica Guide to Geometry
Math Explained
Author: Britannica Educational Publishing
Editor: William L. Hosch
Edition: 1st Edition
Age Group: Ages 14 and Up
Description
The universal language of numbers has allowed individuals to transcend cultural differences and make
collaborative efforts to comprehend the world mathematically. Though many of these mathematicians
may never have met the predecessors who made their own work possible, their collective works form the
foundations of mathematics as it is known today. The books in this series introduce students not only to
the theories and formulas that form the basis of each field of mathematics, but to the individuals who
dedicated their lives to pushing numerical boundaries. Detailed diagrams provide visual summaries of
complex concepts and make these books an asset to both lovers of math and those who may find math
challenging.
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More About
This Textbook
Overview
Topics in Commutative Ring Theory is a textbook for advanced undergraduate students as well as graduate students and mathematicians seeking an accessible introduction to this fascinating area of abstract algebra.
Commutative ring theory arose more than a century ago to address questions in geometry and number theory. A commutative ring is a set-such as the integers, complex numbers, or polynomials with real coefficients—with two operations, addition and multiplication. Starting from this simple definition, John Watkins guides readers from basic concepts to Noetherian rings-one of the most important classes of commutative rings—and beyond to the frontiers of current research in the field. Each chapter includes problems that encourage active reading—routine exercises as well as problems that build technical skills and reinforce new concepts. The final chapter is devoted to new computational techniques now available through computers. Careful to avoid intimidating theorems and proofs whenever possible, Watkins emphasizes the historical roots of the subject, like the role of commutative rings in Fermat's last theorem. He leads readers into unexpected territory with discussions on rings of continuous functions and the set-theoretic foundations of mathematics.
Written by an award-winning teacher, this is the first introductory textbook to require no prior knowledge of ring theory to get started. Refreshingly informal without ever sacrificing mathematical rigor, Topics in Commutative Ring Theory is an ideal resource for anyone seeking entry into this stimulating field of study
What People Are Saying
Karen Smith
A very elementary introduction to commutative ring theory, suitable for undergraduates with little background. It is written with great care, in a conversational and engaging style that I think will appeal to students. Essentially every detail is made explicit, and readers are admonished to beware typical pitfalls. The book is also peppered with very nice detours into the history of mathematics.
— Karen Smith, University of Michigan
Editorial Reviews
Choice
- D.V. Feldman
As an honest, focused treatment of an important subject packaged into an attractive, slender volume that average undergraduates may reasonably hope to master in one semester, this book will find its niche and its admirers.
Mathematical Reviews
- Marco Fontana
A highlight is the attention given here to the ring of continuous functions, an important non-standard example of a commutative ring that shows the profound relationship between algebra and topology.
From the Publisher
"As an honest, focused treatment of an important subject packaged into an attractive, slender volume that average undergraduates may reasonably hope to master in one semester, this book will find its niche and its admirers."—D.V. Feldman, Choice
"A highlight is the attention given here to the ring of continuous functions, an important non-standard example of a commutative ring that shows the profound relationship between algebra and topology."—Marco Fontana, Mathematical Reviews
Choice
As an honest, focused treatment of an important subject packaged into an attractive, slender volume that average undergraduates may reasonably hope to master in one semester, this book will find its niche and its admirers.
— D.V. Feldman
Mathematical Reviews
A highlight is the attention given here to the ring of continuous functions, an important non-standard example of a commutative ring that shows the profound relationship between algebra and topology.
— Marco Fontana
Related Subjects
Meet the Author
John J. Watkins is professor of mathematics at Colorado College. He is the author of "Across the Board: The Mathematics of Chessboard Problems" (Princeton) and the coauthor of "Graphs: An Introductory Approach
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Web Site Webmath.com This is a dynamic math website where students enter problems and where the site's math engine solves the problem. Students in most cases are given a step-by-... Curriculum: Mathematics Grades: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
8.
Web Site Prentice Hall Math Textbook Resources This site has middle school and high school lesson quizzes, vocabulary, chapter tests and projects for most chapters in each textbook. In some sections, ther... Curriculum: Mathematics Grades: 6, 7, 8, 9, 10, 11, 12
9.
Web Site Dave's Short Trig Course Check out the short trigonometry course and learn the new way of learning trig. This short course breaks into sections and allows user to learn at his/her o... Curriculum: Mathematics Grades: 9, 10, 11, 12
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Cartoon Guide to Calculus - 12 edition teach
es all of the essentials, with numerous examples and problem sets. For the curious and confused alike, The Cartoon Guide to Calculus is the perfect combination of entertainment and education---a valuable supplement for any student, teacher, parent, or professional. teaches all of the essentials, with numerous examples and problem sets. For the curious and confused alike, The Cartoon Guide to Calculus is the perfect combination of entertainment and education---a valuable supplement for any student, teacher, parent, or professional. ...show less
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Contribution Description
This activity is designed to help students gain a qualitative intuitive understanding of Fourier Analysis. It was developed to be used in a sophomore level modern physics course, after lecture instruction on atomic models. a sophomore level modern physics course. For more details about the course, please see:
It is not as inquiry based as the guidelines recommend because the course requires computer grading, so the questions are very directed.
Level
Undergraduate - Advanced, Undergraduate - Intro
Type
Homework
Subject
Mathematics, Physics
Answers Included
No
Language
English
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Use Wolfram|Alpha to Solve Calculus Problems and…...
Use Wolfram|Alpha to Solve Calculus Problems and… Everything Else.
Wolfram|Alpha is like Google on crack. However, it is not technically a search engine; it is a "computational knowledge" engine. They use a huge collection of trustworthy, built-in data to get the user the information or knowledge they are looking for. When you search for an item, Wolfram|Alpha gives you all of the relevant knowledge they have on that specific search query. For example, here is the results for the search "when did the Beatles break up?" Not only do you get the date the Beatles broke up, you also get how long away that date is from today and other noteworthy events that occurred on the same day. Here is another example, for the search "carbon footprint driving 536 miles at 32mpg" that tells you the amount of fuel consumed and the amount of c02 and carbon emitted.
Because Wolfram|Alpha is just retrieving answers from its huge database of information and formulas, you have to be specific and ask non-opinionated questions. For example, the website does not know which Lil Wayne song is the best. However, it does know things that are not opinions, like the nutritional facts of 10,000 big macs and how many planes are currently flying directly over you.
I find Wolfram|Alpha to be better than Google when I am quickly looking for specific answers. I just typed in "Countries that border France" on both Wolfram|Alpha and Google. Wolfram|Alpha quickly showed me a list of the 8 countries and a map with of France with its bordering countries highlighted. Google on the other hand sent me over to Yahoo Answers…
Other than a fun search engine, Wolfram|Alpha can also be used as a highly effective tool for college. Like the title mentions, the knowledge engine can in fact solve any calculus problem. It can easily solve any math problem thrown its way, from a basic algebra problem to whatever this is.
Wolfram|Alpha can also be used for many other college courses such as biology, astronomy, history, etc.
As Wolfram|Alpha can be kind of confusing and hard to get the hang of at first, I suggest going through this short tour and looking at some examples to help give you a better sense of how to use it. Even if you find it a little bit confusing at first, keep trying because Wolfram|Alpha really is a great way to "hack college."
ABOUT THE AUTHOR
Logan James Ivey is a sophomore at the two year school of Sierra College located in the beautiful foothills of California. His plans are to transfer to the University of California, Davis as an international relations major and perhaps double major or minor in economics. In High School he enjoyed making yearbooks and short films in his Multi Media class. Logan's other interests include rivers, the Internet, becoming strong and listening to hip hop music. You can learn more about this amazing boy by following him on Twitter or Google +
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24, based on Pólya's method of problem solving, aids students in their transition to higher-level mathematics. It begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics and ends by providing projects for independent study. Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them.
Editorial Reviews
Review
From the reviews:
U. Daepp and P. Gorkin
Reading, Writing, and Proving
A Closer Look at Mathematics
"Aids students in their transition from calculus (or precalculus) to higher-level mathematics . . . The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations."
—L'ENSEIGNEMENT MATHEMATIQUE
"Daepp and Gorkin (both, Bucknell Univ.) offer another in the growing genre of books designed to teach mathematics students the rigor required to write valid proofs … . The book is well written and should be easy for a first- or second- year college mathematics student to read. There are many 'tips' offered throughout, along with many examples and exercises … . A book worthy of serious consideration for courses whose goal is to prepare students for upper-division mathematics courses. Summing Up: Highly recommended." (J.R. Burke, CHOICE, 2003)
"The book Reading, Writing, and Proving … provides a fresh, interesting, and readable approach to the often-dreaded 'Introduction to Proof' class. … RWP contains more than enough material for a one-semester course … . I was charmed by this book and found it quite enticing. … My students found the overall style, the abundance of solved exercises, and the wealth of additional historical information and advice in the book exceptionally useful. … well-conceived, solidly executed, and very useful textbook." (Maria G. Fung, MAA online, December, 2004)
"The book is intended for undergraduate students beginning their mathematical career or attending their first course in calculus. … Throughout the book … students are encouraged to 1) learn to understand the problem, 2) devise a plan to solve the problem, 3) carry out that plan, and 4) look back and check what the results told them. This concept is very valuable. … The book is written in an informal way, which will please the beginner and not offend the more experienced reader." (EMS Newsletter, December, 2005)
From the Back Cover
This book, which is based on Pólya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study.
Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.
Most Helpful Customer Reviews
I have used the Daepp, Gorkin text twice, for an introduction to proofs type of course. This course is usually taken by Math and Computer Science majors after Calculus and either with or after a course in Linear Algebra. This type of course was not in existence when I was a student, in the 70's. In those days, there was some proofs in Calculus (certainly some delta-epsilon type arguments were given) and Linear Algebra was much more proof oriented. Hence, most math majors picked up the ability to read and learn abstract mathematics during the first two years. These days, Linear Algebra has become a course in row-reducing matrices and very little abstraction takes place. Hence there is a real need for a course (and texts) to pave the way for courses in Analysis and Abstract Algebra. The Daepp, Gorkin text compares favorably to all similar texts I have looked at and it is priced reasonably. I passed on another text that I liked because it was $125, which is ridiculous for studenst who are not wealthy. On the plus side, this text covers all the material you would need in such a course and, in fact, there are several avenues open to the instructor of a one semester course. Besides the usual material on sets and mappings, there are chapters on cardinality issues, intoductory analysis ideas and slightly more advanced topics in number theory. The chapters are short and "digestable." There are some possible independent research topics at the end of the text. On the negative side, the examples given in the text are mostly all drawn from the standard number systems. This makes it harder to motivate basic concepts of sets and mappings. Why not give some examples from sets of mappings (e.g.Read more ›
First of all I am a student of Computer Science at a US university. This book was used for an intermediate level mathematics course intended to provide a bridge between Calculus and higher level courses. For that purpose this book was well suited.
Like many (probably most) students my high school and early college mathematics courses taught me a great deal about using a graphing calculator to guess-and-check but much less about the fundamentals of mathematics. The writing is generally clear and concise and avoids leaving "obvious" (read: very difficult) theorems as exercises to the reader. Like the reviewer Jerry D. Rosen I think some of the exercises are "odd", for lack of a better term. I think the authors tried to avoid the "question 1, parts a-f [trivial exercise]" format of many mathematics textbooks but the mix of questions did not always come out well. Also note that there are no answers provided so this book is not well suited to self-study.
The greatest virtue of this book is that that one gets the sense of *liking* mathematics, and that is contagious. I found this book well suited to the course I took it for and it has proven even more valuable for getting through Linear Algebra and Discrete math courses with truly horrible texts.
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Journey into Mathematics : An Introduction to Proofs - 06 edition
This 3-part treatment begins with the mechanics of writing proofs, proceeds to considerations of the area and circumference of circles, and concludes with examinations of complex numbers and their application, via De Moivre's theorem, to real numbers.
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This eBook introduces the subject of handling data, considers both information and data, probability and chance as well as the various styles of tables, pictograms, graphs, charts along with the concept of average.
This concise article takes you on a short tour on how to plot graphs using MATLAB. The presentation covers both two-dimensional and three-dimensional graphs. This article is taken form the bestselling book "MATLAB for Beginners: A Gentle Approach."
This eBook introduces co-ordinate geometry and graphs, ranging from finding the equations of the straight-line joining two points for which the co-ordinates are known, to calculating both the mid-point and length of a line between two known co-ordinates to plotting equations of the form y = kx^n where n is even or odd for various values of k, as well as y = k√(x) where x is positive.
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About the book:
Provides a world view of mathematics, balancing ancient, early modern and modern history. Problems are taken from their original sources, enabling students to understand how mathematicians in various times and places solved mathematical problems. In this new edition a more global perspective is taken, integrating more non-Western coverage including contributions from Chinese/Indian, and Islamic mathematics and mathematicians. An additional chapter covers mathematical techniques from other cultures. *Up to date, uses the results of very recent scholarship in the history of mathematics. *Provides summaries of the arguments of all important ideas in the field.
Softcover, ISBN 0321016181 Publisher: Addison Wesley21016181 Publisher: Addison-Wesley Pub (Sd21016181 Publisher: Addison-Wesley Pub (Sd), 1998 Good. hardcover Item may show signs of shelf wear. Pages may include limited notes and highlighting. Includes supplemental or companion materials if applicable. Access codes may or may not work. Connecting readers since 1972. Customer service is our top priority. . hardcover Item may show signs of shelf wear. Pages may include limited notes and ...
Hardcover, ISBN 0321016181 Publisher: Addison Wesley longman, 1998 Used - Good. hardcover016181 Publisher: Pearson, 1998 Used - Acceptable. This is an ex-library book and may have the usual library/used-book markings inside.This book has hardback covers. In fair condition, suitable as a study copy. No dust jacket. (MATHEMATICS)
Softcover, ISBN 0321016181 Publisher: Addison-Wesley Pub (Sd), 0321016181 Publisher: Addison-Wesley Pub (Sd), 1998 This book is in good condition. We are longtime sellers of books online and guarantee prompt shipment and top service. CDs and other peripherals such as access codes may not be included as is typically the case with used books and textbooks.
Softcover, ISBN 0321016181 Publisher: Addison-Wesley Pub (Sd), 1998
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Description
A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight. In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out—from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft—indeed, brilliant—instructions on stripping away irrelevancies and going straight to the heart of the problem.
About the author
George Polya (1887–1985) was one of the most influential mathematicians of the twentieth century. His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was a teacher par excellence who maintained a strong interest in pedagogical matters throughout his long career. Even after his retirement from Stanford University in 1953, he continued to lead an active mathematical life. He taught his final course, on combinatorics, at the age of ninety. John H. Conway is professor emeritus of mathematics at Princeton University. He was awarded the London Mathematical Society's Polya Prize in 1987. Like Polya, he is interested in many branches of mathematics, and in particular, has invented a successor to Polya's notation for crystallographic groups How to Solve It: A New Aspect of Mathematical Method Finished this book in couple hours. This book is more suitable for teacher who want to improve their teaching skills
Ashish Sharma, possibly, one of the best books on logic I have ever read. I wish I had read this in college. Hell, I wish I had read this in high school. It has given me a new perspective and understanding problems out there. The author has Hailed as the classic guide to problem solving, this book did quite a good job at categorizing the ways of looking at a problem, and some general methods of solving and treating them. However, I think ...
Review: How to Solve It: A New Aspect of Mathematical Method The information in this book was extremely useful and extremely difficult to approach. I read the Kindle edition which may be part of the problem, but the majority of the book is structured as aSimilar construction of geometric proofs and presents criteria useful for determining whether a proof is logically correct and whether it actually constitutes proof. It features sample invalid proofs, in which the errors are explained and corrected. Mistakes in Geometric Proofs, the second book in this compilation, consists chiefly of examples of faulty proofs. Some illustrate mistakes in reasoning students might be likely to make, and others are classic sophisms. Chapters 1 and 3 present the faulty proofs, and chapters 2 and 4 offer comprehensive analyses of the errors.
This easy-to-read 2010 book demonstrates how a simple geometric idea reveals fascinating connections and results in number theory, the mathematics of polyhedra, combinatorial geometry, and group theory. Using a systematic paper-folding procedure it is possible to construct a regular polygon with any number of sides. This remarkable algorithm has led to interesting proofs of certain results in number theory, has been used to answer combinatorial questions involving partitions of space, and has enabled the authors to obtain the formula for the volume of a regular tetrahedron in around three steps, using nothing more complicated than basic arithmetic and the most elementary plane geometry. All of these ideas, and more, reveal the beauty of mathematics and the interconnectedness of its various branches. Detailed instructions, including clear illustrations, enable the reader to gain hands-on experience constructing these models and to discover for themselves the patterns and relationships they unearth.
This single-volume compilation of three books centers on Hyperbolic Functions, an introduction to the relationship between the hyperbolic sine, cosine, and tangent, and the geometric properties of the hyperbola. Configuration Theorems concerns theories on collinear points and concurrent lines, and Equivalent and Equidecomposable Figures examines the dissection and reassembly of polyhedrons. 1963 edition.
A noted educator's account of some of the more stimulating and surprising branches of mathematics, this volume profiles the mathematical mind and the aims of mathematics. Five introductory chapters offer conceptual groundwork, and subsequent chapters present lucid, accessible explorations of non-Euclidean geometry, matrices, determinants, group theory, and related topics. 1955 edition.
This introductory text is designed to help undergraduate students develop a solid foundation in geometry. Early chapters progress slowly, cultivating the necessary understanding and self-confidence for the more rapid development that follows. The extensive treatment can be easily adapted to accommodate shorter courses. Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter includes a problem set arranged in order of increasing difficulty as well as review exercises and annotated references suggesting sources for further study. In addition to three helpful Appendixes, the book concludes with answers and hints for selected problems.
The new edition of this classic book describes and provides a myriad of examples of the relationships between problem posing and problem solving, and explores the educational potential of integrating these two activities in classrooms at all levels. The Art of Problem Posing, Third Edition encourages readers to shift their thinking about problem posing (such as where problems come from, what to do with them, and the like) from the "other" to themselves and offers a broader conception of what can be done with problems. Special features include: an exploration of the logical relationship between problem posing and problem solving; sketches, drawings, and diagrams that illustrate the schemes proposed; and a special section on writing in mathematics.
In the updated third edition, the authors specifically: *address the role of problem posing in the NCTM Standards; *elaborate on the concept of student as author and critic; *include discussion of computer applications to illustrate the potential of technology to enhance problem posing in the classroom; *expand the section on diversity/multiculturalism; and *broaden discussion of writing as a classroom enterprise.
This book offers present and future teachers at the middle school, secondary school, and higher education levels ideas to enrich their teaching and suggestions for how to incorporate problem posing into a standard mathematics curriculum.
The Second International Workshop on Automated Deduction in Geometry (ADG '98) was held in Beijing, China, August 1–3, 1998. An increase of interest in ADG '98 over the previous workshop ADG '96 is represented by the notable number of more than 40 participants from ten countries and the strong tech- cal program of 25 presentations, of which two one-hour invited talks were given by Professors Wen-tsun ̈ Wu and Jing-Zhong Zhang. The workshop provided the participants with a well-focused forum for e?ective exchange of new ideas and timely report of research progress. Insight surveys, algorithmic developments, and applications in CAGD/CAD and computer vision presented by active - searchers, together with geometry software demos, shed light on the features of this second workshop. ADG '98 was hosted by the Mathematics Mechanization Research Center (MMRC) with ?nancial support from the Chinese Academy of Sciences and the French National Center for Scienti?c Research (CNRS), and was organized by the three co-editors of this proceedings volume. The papers contained in the volume were selected, under a strict refereeing procedure, from those presented at ADG '98 and submitted afterwards. Most of the 14 accepted papers were carefully revised and some of the revised versions were checked again by external reviewers. We hope that these papers cover some of the most recent and signi?cant research results and developments and re?ect the current state-of-the-art of ADG.
With a standard program committee and a pre-review process, the Third - ternational Workshop on Automated Deduction in Geometry (ADG 2000) held in Zurich, Switzerland, September 25–27, 2000 was made more formal than the previous ADG '96 (Toulouse, September 1996) and ADG '98 (Beijing, August 1998). The workshop program featured two invited talks given by Christoph M. Ho?mann and Jurgen ¨ Bokowski, one open session talk by Wen-tsun ¨ Wu, 18 regular presentations, and 7 short communications, together with software demonstrations (see Some of the most recent and signi?cant research developments on geometric deduction were - ported and reviewed, and the workshop was well focused at a high scienti?c level. Fifteen contributions (out of the 18 regular presentations selected by the program committee from 31 submissions) and 2 invited papers were chosen for publication in these proceedings. These papers were all formally refereed and most of them underwent a double review-revision process. We hope that this volume meets the usual standard of international conference proceedings, rep- sentsthecurrentstateoftheartofADG,andwillbecomeavaluablereferencefor researchers, practitioners, software engineers, educators, and students in many ADG-related areas from mathematics to CAGD and geometric modeling. ADG2000washostedbytheDepartmentofComputerScience,ETHZurich.
This book constitutes the thoroughly refereed post-proceedings of the 4th International Workshop on Automated Deduction in Geometry, ADG 2002, held at Hagenberg Castle, Austria in September 2002.
The 13 revised full papers presented were carefully selected during two rounds of reviewing and improvement. Among the issues addressed are theoretical and methodological topics, such as the resolution of singularities, algebraic geometry and computer algebra; various geometric theorem proving systems are explored; and applications of automated deduction in geometry are demonstrated in fields like computer-aided design and robotics
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