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Traditional Lecture
Traditional lecture is the mode of instruction with which most students are familiar.These courses work well for students with the following characteristics:
I learn best when a teacher demonstrates problems.
I can listen and take notes.
I like to work in groups.
I like a structured class.
Most lecture classes are 3 to 5 units and meet two or three times/week. A Math instructor teaches the class. Math 65 and Math 55 each take one semester to complete. These classes include one hour of lab per week in the Integrated Learning Center. Math 107 also takes 1 semester to complete and has several hours of lab work each week in the classroom.
Half-paced Lecture A&B
These courses are best for student wanting a traditional lecture for whom the following are true:
I need more time for Math concepts to sink in.
It's been forever since I studied Math, and I need a careful review of the basics.
These are 2.5 unit classes that meet two times/week. A Math instructor teaches the class. Math 65A&B and Math 55A&B each take 2 semesters to complete. Math 65A and Math 65B comprise Math 65; similarly Math 55A and 55B comprise Math 55. These classes include one hour of lab per week in the Integrated Learning Center.
Math X&Y
Math X & Math Y are designations used to describe courses which are offered in an independent study/self-paced mode. These courses work well for students with the following characteristics:
I am a self-motivated, dedicated, and organized student.
I like to work alone.
I need a quick brush up on my algebra skills.
Math 65X, 65Y, 55X, and 55Y are 2.5 units each; Math 107X and 107Y are 2 units each. Math 65X and Math 65Y comprise Math 65; similarly Math 55X and 55Y comprise Math 55 and M107X and Math 107Y comprise Math 107. Classes meet two times per week. A Math instructor is available to answer questions. There is no lecture. Assigned homework is completed by each student working independently.
Distance Education
Distance education refers to course in which all or part of the course is offered on-line. It works best for students to whom the following applies:
I am a self-motivated, dedicated, and organized student.
I like to work alone.
It's very hard for me to come to campus.
I'm computer savvy.
In a Math 65DE or Math 55DE course, all instruction is carried out on-line. Students come to to campus for orientation and to take exams. In a Math 65 Hybrid or Math 55 Hybrid course, part of the instruction is carried out in a face-to-face setting on campus and the rest of the instruction is on-line. Exams are given on campus at designated times. Like the traditional lecture offerings of 65 and 55, the distance education offerings are 5 unit courses and each take one semester to complete. The assigned instructor determines whether the TBA lab hour is to be carried out on-line or in the Integrated Learning Center.
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Discrete Mathematics With Application - 4th edition
Summary: Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, FOURTH EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such conce...show morepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses
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McDougal Littell Geometry TE
9780395977286
ISBN:
0395977282
Publisher: McDougal Littell
Summary: The theorems and principles of basic geometry are clearly presented in this workbook, along with examples and exercises for practice. All concepts are explained in an easy-to-understand fashion to help students grasp geometry and form a solid foundation for advanced learning in mathematics. Each page introduces a new concept, along with a puzzle or riddle which reveals a fun fact. Thought-provoking exercises encourag...e students to enjoy working the pages while gaining valuable practice in geometry.
Ray C. Jurgensen is the author of McDougal Littell Geometry TE, published under ISBN 9780395977286 and 0395977282. Two McDougal Littell Geometry TE textbooks are available for sale on ValoreBooks.com, and one used from the cheapest price of $165.60.[read more]
Ships From:Salem, ORShipping:Standard, ExpeditedComments:Has minor wear and/or markings. SKU:9780395977286-3-0-3 Orders ship the same or next business day... [more]
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Find an El Lago, TX GeometryThomas and received an A in the course. Linear Algebra is the study of matrices and their properties. The applications for linear algebra are far reaching whether you want to continue studying advanced algebra or computer science.
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The program is designed to support parallel computation, image processing, and rich data on astronomy, geography, life sciences, and the like. More than 4500 freely available examples of concepts in life sciences, physical sciences, engineering, creative arts, and many other areas have been published as part of the Wolfram Demonstrations Project. In addition, over 100,000 examples of Mathematica functions and
capabilities are provided in the Wolfram Mathematica Documentation Center.
"Mathematica is the perfect tool for a broader group -- from amateur scientists, to parents interested in introducing their children to concepts in math, science, finance, and other areas, to anyone who wants to make fully interactive models and simulations, analyze real-time data on stocks, weather, or stars,
transform and enhance images, or explore infinitely more activities," says Peter Overmann, Director of Software Technology at Wolfram Research.
Mathematica Home Edition is a 32-bit program available in the United States and Canada for Windows (2000/XP/Vista), Mac OS X (Intel), and Linux for a retail price of $295. Mathematica Home Edition is not licensed for commercial, nonprofit, academic, or government use. For more details, see
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Algebra and Trigonometry-Study Guide - 3rd edition
Summary: This carefully crafted learning resource helps students develop their problem-solving skills while reinforcing their understanding with detailed explanations, worked-out examples, and practice problems. Students will also find listings of key ideas to master. Each section of the main text has a corresponding section in the Study Guide.
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| 677.169 | 1 |
Undergraduate Algebraic Geometry
9780521356626
ISBN:
0521356628
Pub Date: 1988 Publisher: Cambridge University Press
Summary: Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics... and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.
Reid, Miles is the author of Undergraduate Algebraic Geometry, published 1988 under ISBN 9780521356626 and 0521356628. Five hundred six Undergraduate Algebraic Geometry textbooks are available for sale on ValoreBooks.com, one hundred fifty four used from the cheapest price of $17.60, or buy new starting at $38.03
| 677.169 | 1 |
Shipping prices may be approximate. Please verify cost before checkout.
About the book:
Ostebee and Zorn's approach applies reform principles to a rigorous calculus text. Conceptual understanding is the main goal of the text, and looking at mathematics from many representations (graphical, symbolic, numerical) is the main strategy for achieving this type of understanding. The key strengths of the text include combining symbolic manipulation with graphical and numerical representation, exercises of a varied nature and difficulty, and explanations written to be understandable to student readers.
A student-friendly and approachable tone, numerous examples, critical-thinking questions, and supportive details and commentary help students successfully read and use the text.
Representation of mathematical concepts through a variety of viewpoints supports different learning styles. Students see the math worked out through multiple representationsgraphically, numerically, and symbolicallyto enhance conceptual understanding.
Proofs presented at point of use contribute significantly to helping students understand rigorous calculus concepts and develop analytic skills.
Varied exercise sets offer instructors more options for creating homework assignments. Basic Exercises, which are straightforward and focus on a single idea, help students build basic skills.
Further Exercises are a little more ambitious and may require the synthesis of several ideas, a deeper or more sophisticated understanding of basic concepts, or the use of a computer algebra system such as Maple or Mathematica. These are available for professors to assign when they would like to challenge their students and incorporate technology into their course.
Answers to Select Exercises can be found in the back of the text, enabling students to get immediate feedback and assess their understanding of the material.
Interludes are brief project-oriented expositions, with related exercises, that extend the concepts presented in the chapter. Professors have the opportunity to include these topics found at the end of the chapter as independent work, group work, or as a classroom activity. The Interludes include theoretical problems and proofs intended to enhance student understanding of the key calculus concepts.
Softcover, ISBN 0618248579 Publisher: Houghton Mifflin (Academic), 2003 Usually dispatched within 1-2 business days, Dispatched from within the UK; please allow 9-13 working days for delivery. Prompt and Friendly customer service.
Softcover, ISBN 0618248579 Publisher: MCDOUGAL LITTEL, 200280618248575-5-0
Softcover, ISBN 0618248579 Publisher: MCDOUGAL LITTEL, 2002 Boulder, CO with delivery confirmation. Satisfaction guaranteed.
Softcover, ISBN 0618248579 Publisher: MCDOUGAL LITTEL, 200280618248575-4-0
Softcover, ISBN 0618248579 Publisher: Houghton Mifflin (Academic), 2003 Acceptable. US Edition. All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU:97806182485 0618248579 Publisher: Houghton Mifflin (Academic), 2003 Good. US Edition. May include moderately worn cover, writing, markings or slight discoloration. SKU:9780618248575-4-0-3 Orders ship the same or next business day. Expedited shipping within U.S. will arrive in 3-5 days. Hassle free 14 day return policy. Contact Customer Service for questions..
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Demos with Positive Impact is a collection of quick classroom demos that enhance the learning of mathematics content through...
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Demos with Positive Impact is a collection of quick classroom demos that enhance the learning of mathematics content through animations, experiments etc. Each demo comes with stated objective, prerequisites, instructor notes and platform info, plus the level of the demo and credits. This setup appears conducive to quick inclusion into a class. To view a video of the award winning author, go to target=״_blank״>Demos with Positive Impact - the Mathematics Award Winner 2008 videoThe author also participated in the MERLOT Classics Series on Elluminate: " target=״_blank״
GeoGebra is a free and multi-platform dynamic mathematics software for education in secondary schools that joins geometry,...
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GeoGebra is a free and multi-platform dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system--you can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards; on the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum. These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa.
This tutorial will help you determine how accurate a sample mean is likely to be, and how this accuracy is related to the...
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This tutorial will help you determine how accurate a sample mean is likely to be, and how this accuracy is related to the sample size. Brief reviews of the normal distribution and the Central Limit Theorem are included as supplemental materials.
The site includes guided interdisciplinary labs for first and second courses in statistics. As stated on the homepage for...
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The site includes guided interdisciplinary labs for first and second courses in statistics. As stated on the homepage for this site, "This site presents workbook-style, project-based material that emphasizes real world applications and conceptual understanding. This material is designed to give students a sense of the importance and allure of statistics early in their college career. By incorporating many of the successful reforms of the introductory statistics course into a wide range of more advanced topics we hope that students in any discipline can realize the intellectual content and broad applicability of statistics.״
A searchable database of more than 1000 test questions for introductory statistics concepts. The user is prompted to select...
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A searchable database of more than 1000 test questions for introductory statistics concepts. The user is prompted to select subject material and learning outcome expectations from a variety of question formats and then downloads the items and can edit the test with a word processor. IMPORTANT NOTE: When you initially click on the link to get to the ARTIST website, you may be directed to a page that indicates the need to accept a certificate in order to view the ARTIST site. This is due to some recent changes related to the server where ARTIST is currently housed. If you accept this certificate, you should then be able to get to the ARTIST site. Eventually, ARTIST will be housed on the CAUSEweb site.
Visual ANOVA is an interactive Flash program which demonstrates visually how variability between and within experimental...
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Visual ANOVA is an interactive Flash program which demonstrates visually how variability between and within experimental groups contributes to the F ratio in the Analysis of Variance. It is not a numerical calculator; rather it visually and holistically demonstrates the relations among important concepts. Visual ANOVA is supported by online instructions and by an extensive online lecture explaining the theory behind the Analysis of Variance. The online lecture is supported by two types of assignments: 1) Online computer-graded homework, and 2) A pdf file that gives students the opportunity to do handwritten homework problems with answer keys.
The Geometry Center contains a variety of appealing material, both textual and visual, including interactive exhibits,...
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The Geometry Center contains a variety of appealing material, both textual and visual, including interactive exhibits, simulations, graphics software and a library of reference materials pertaining to geometric tilings and polyhedra.
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More About
This Textbook
Overview
This tried-and-true text from Allyn Washington preserves the author's highly regarded approach to technical math, while enhancing the integration of technology. Appropriate for a two- to three optics, and the environment. Known for its exceptional problem sets and applied material, the book offers practice exercises, writing exercises, word problems, and practice tests. This edition features more technical applications, over 2300 new exercises, and additional graphing calculator screens.
Editorial Reviews
Booknews
A textbook intended primarily for students in technical and pre- engineering technology programs or other programs for which coverage of basic mathematics is required. There is an integrated treatment of mathematical topics, from algebra to calculus, with numerous applications from many fields of technology to indicate where and how mathematical techniques are used. For this edition (fifth was 1990), most sections have been rewritten to some degree to include additional or revised explanatory material, examples, and exercises
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Find a Lincolnwood CalculusYou should be familiar with the equations and graphs of basic functions (linear, quadratic, absolute value, logarithmic, exponential and trigonometric) and be able to write and graph their equations when they are subjected to transformations such as translation, scaling and reflection. The study...
| 677.169 | 1 |
You are here
Making Mathematics More Practical
Publisher:
World Scientific
Number of Pages:
148
Price:
38.00
ISBN:
9789814569071
This book is a summary of the "Mathematical Problem Solving for Everyone" (MProSE) program that was implemented in five secondary schools in Singapore. The purpose of the program is the universal one of improving the mathematical problem solving ability of students. The problem with all such initiatives is summarized nicely in the first paragraph of the preface. The designers of the program state that they knew the work would be "hard and unglamorous." That is the best description of curriculum modification that I have ever heard.
The program itself is logically executed. As described in Making Mathematics Practical, it is based on the standard four steps of problem solving as espoused by Pólya in his classic book on problem solving, published in 1945. In the five reports describing the implementation of the programs in five schools, there is nothing that veterans of the curriculum change process have not encountered. First and foremost, it is necessary to get the instructors to buy into the curriculum change and that includes the appropriate level of professional development. It also includes giving the educators the opportunity for constructive input, including criticism.
One significant problem was that the program was implemented as an extra-curricular activity and the students complained about it. No surprise here: if a program is designed to improve problem solving abilities, it should be part of the regular curriculum and not an extra that students can dismiss. Other difficulties, such as poor selection of problems (both too easy and too hard) are also described.
The goal of the educators in Singapore is of course an admirable one and something that is a regular feature of mathematics education. The problems of mathematics education are universal in both the geographic and temporal senses. The MProSE program does not solve them, but it is another step in the right direction.
Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.
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Elementary Number Theory and Its Applications is noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. Computational exercises and computer projects are also provided. In addition to years of use and professor feedback, the fifth edition of this text has been thoroughly checked to ensure the quality and accuracy of the mathematical content and the exercises.
The blending of classical theory with modern applications is a hallmark feature of the text. The Fifth Edition builds on this strength with new examples and exercises, additional applications and increased cryptology coverage. The author devotes a great deal of attention to making this new edition up-to-date, incorporating new results and discoveries in number theory made in the past few years.
Editorial Reviews
Booknews
New edition of a standard text. Integrates classical material with applications to cryptography and computer science. The author is with AT&T Bell Labs 31, 2005
there are better number theory books out there
there are better number theory books out there with better examples. my professors was rather dry .. with a better teacher this book may have seemed a little bit better to me. I ended up using the internet as a study source and a used copy i found online of Elementary Number Theory by David M. Burton
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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Discrete Mathematics
9780131593183
ISBN:
0131593188
Edition: 7 Pub Date: 2008 Publisher: Prentice Hall
Summary: This textbook provides an accessible introduction to discrete mathematics, using an algorithmic approach that focuses on problem-solving techniques. Each chapter has a special section dedicated to showing students how to attack and solve problems.
Johnsonbaugh, Richard is the author of Discrete Mathematics, published 2008 under ISBN 9780131593183 and 0131593188. Eight hundred twelve Discrete Mathematics text...books are available for sale on ValoreBooks.com, seventy nine used from the cheapest price of $83.95, or buy new starting at $149.84
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Using this virtual manipulative you may: graph a function; trace a point along the graph; dynamically vary function parameters; change the range of values displayed in the graph; graph multiple functi... More: lessons, discussions, ratings, reviews,...
Enter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each co... More: lessons, discussions, ratings, reviews,...
The NA_WorkSheet Demo (beta version) is a collective aggregation of algorithms coded in Java that implements various Numerical Analysis solutions/techniques in one easy to use open source tool. The to... More: lessons, discussions, ratings, reviews,...
On this online calculator calculate mathematical expressions and complex numbers. You can do matrix algebra and solve linear systems of equations and graph all 2D graph types. You can also calculate z... More: lessons, discussions, ratings, reviews,...
This is a Java graphing applet that can be used online or downloaded. The purpose it to construct dynamic graphs with parameters controlled by user defined sliders that can be saved as web pages or em... More: lessons, discussions, ratings, reviews,...
This tool lets you plot functions, polar plots, and 3D with just a suitable web browser (within the IE, FireFox, or Opera web browsers), and find the roots and intersections of graphs. In addition, yo... More: lessons, discussions, ratings, reviews,...
Commercial site with one free access per day. This graphs a log with bases from 1.1 to 15 and includes ln. Students can also type in the base, including base e, which will cause the function to change
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So, yesterday I finished off studying my course book differentiation, ( This means I have studied enough to get good marks in my annual examinations) although I am able to solve all the problems in the textbooks but I still don't even have the slightest idea of what differentiation is. I know that d/dx of xn is n.xn-1 , that d/dx of sin(x) is cos(x), that differentiation gives slope of function but honestly I don't know a bit about differentiation in practical sense. ( None of the teachers told us) All the books I referred to, start with formulaes and end with questions in which you have to apply those formulaes.
I find this way of learning horrible, whilst solving those questions I feel like I am dealing with some mechanical work, just putting things on their places. I would like to ask people from different countries about how they teach such topics there? Do you face/faced same problem ?
These problems are to be expected; I would hypothesize that most, if not all, first-time students of calculus experience this. The inherent issue is, ostensibly, that mathematics is 'to be used' but not necessarily understood; it is, in the hands of most, simply a tool. A carpenter who gazes at his hammer, his saw and his lathe and wonders how it is they came to be; what their underlying impetus is and how he might understand them more fully doesn't build many cabinets. However, this same carpenter, daydreaming about the foundational aspects of his tools might, instead, gain some insight into how to build better tools.
There is room for both, of course, but the high demand for cabinets in society ( due to a large population ) and the rapid dissemination of new techniques and tools makes the reflective carpenter a bit outmoded. Put more succinctly society needs one hundred or one thousand 'worker' carpenters for every 'reflective' one. This natural supply and demand problem is why, for example, engineers tend to find jobs more easily and be considered "more worthwhile" per-capita than, say, a pure mathematician, theoretical physicist, etc.
The upshot of this circuitousness response is simply "Yes, what you are experiencing is normal : the reason it is normal is due to the fact that introductory calculus courses are not designed for the reflective mind. They are, instead, designed to teach you to wield the information as a toolset in the same way you might use a hammer, a car or fly in an airplane while being ignorant of its more intricate machinations".
That being said : good on you for being curious. When I took my first calculus course at university I was actually convinced that everyone else in the course must have understood the material at a deeper level; that the professor must have made the connection as to "what" and "why" and I either missed it or was too daft to perceive it. It turns out, however, that I was wrong - little or none of this was actually taught in those courses.
The conceptual foundations of calculus are actually very simple, in hindsight, although understanding them and their respective interactions can be a meticulous, albeit rewarding, undertaking. To help you get there I would suggest this book; mainly for its ratio of cost to efficacy in relaying the historical and conceptual development of the calculus.
Now to answer the question you asked:
What is a derivative? Your answer is simply that it is a special kind of limit.
This begs the question : what is a limit? The simplistic answer here is that a limit is the embodiment of "what happens when we get 'closer and closer' ". For example, if you start out at one end of a room and walk half the distance to the wall .. if you continue this process ad-infinitum you could say that "the limit of my behavior of repeatedly halving the distance between myself and the wall is equal to.. the wall". We call the wall the limit because it is where you will "eventually" end up.
So now that we have the concepts of a limit down (and no doubt you remember how to calculate a limit of an expression, at least for a continuous function, from your maths course) lets talk about the concept of a derivative.
So lets use the time-worn example of distance and time. We might measure the displacement of an object in one direction, call it the 'x' direction, over a certain time interval. We would then say that the 'average speed' of the object was the distance the object traveled, delta(x), divided by the time it took to get there, delta(t).
So the derivative, in this case, is simply asking the question "What is the limit of this 'average speed' quantity we have defined (delta(x) / delta(t)) as we let the time span, delta(t), go to zero.
We simply observe the behavior of this defined quantity as what we control, delta(t), goes to zero. In the wall example you control yourself and the measuring quantity is your position in the room. So you say "What is the limit of my position in the room as I half the distance to the wall?" In the same way we ask "What is the average speed as delta(t) goes to zero?"
Then we DEFINE the derivative of a function f(x) to be simply this.. what is the behavior of the ratio how f(x) changes, i.e. f(x+delta(x))-f(x), and how my input (in this case, x) changes as the change in my input value approaches zero? . We are relating the change in f(x) to the change in its input, x.
So there you have it. What is a derivative? its just a special kind of limit. Its a limit where, specifically, we want to measure how an OUTPUT changes as the change in an INPUT goes to zero. Thats it :)
Now, as in all of mathematics, you can interpret a derivative differently depending on your point of view. From a geometric point of view you can relate it to the slope of a tangent line; from a differential geometry point of view you can relate it to an operator that acts on functions and gives you a vector field. There are a myriad of ways to interpret a single object : so don't get it stuck in your head that " a derivative is <this thing>". Rather, think of mathematical objects as abstract concepts; for example the abstract notion of "transportation" could mean a train, a bus, or your legs depending on the context.
Anyhow, these are exciting times for you; read as much as you can and don't discourage easily. Your perception that the conventional rote methodology of mathematical pedantry is lacking has lead you to a trail in the wilderness. It is a trail that many before you have traversed; along the way you will encounter steep cliffs and swirling, murky pools. However, your reward will be high vistas : beauty that few others will behold. Along the way there are guideposts, markers and journals to help you in your journey; these are the words of those who came before you and those you encounter on the trails with you. Happy traveling, adventurer.
As an engineer, I spent lots of time during my degree trying to understand the maths I was learning, instead of just using it. Unfortunately schools and universities don't always support this way of learning because there are, for better or worse, time constraints on doing work which require students to use mathematics before they have time to fully understand it. Personally I found this very frustrating, so it's nice to hear that I wasn't the only one with this experience.
I would offer that this is the result of the way it is taught, and not an intentional result as the previous poster suggests. Here is one small example of what researchers in this area have to say about this:
A central phenomenon of the twenty-first century will be change: economic,
social, and technological change. Indeed, engaging students in analysis of change
and variation is a central element of nearly every chapter in this book. Today,
however, the mathematics of change and variation (MCV), despite its importance
in understanding and controlling this ubiquitous phenomenon, is packed away in a
course, Calculus, that sits at the end of a long series of prerequisites that filter out
90% of the population. This is especially true for students from economically
poorer neighborhoods and families. And even the 10% who do have nominal
access to MCV in calculus courses develop mostly symbol manipulation skill but
little understanding (Tucker, 1990). The traditional curriculum thus excludes most
children from the concepts of rate of change, accumulation, approximation,
continuity, and limit (among others). These are the very concepts children most
need not only to participate in the physical, social, and life sciences of the
twenty-first century, but also to make informed decisions in their personal and
political lives. Even though MCV concepts were at the heart of mathematics and
science historically (Bochner, 1966), in education the opposite is more nearly true.
Conventional curricula neglect, delay, or deny students' access to MCV.
In high school we got to use a TI-92 graphing calculator with programming capabilities and since our maths teacher also was our IT teacher, we got to write some pretty nifty programs that did most, if not all of our problems by themselves. Solve for two variables? Press enter and get result and so on.
Now it became clear very quickly, that the standard way of doing maths i.e. get problem, learn how to solve it, rinse and repeat, wouldn't really fly with this setup, because the actual calculating part was now trivial. Instead he chose to teach us how shit actually worked and how those solutions were found. I remember this distinctly from differential equations for instance where he would describe the history of how people came up with near-solutions to those problems to the formulae and basic concepts behind them we use today.
That was exactly my experience throughout high school. I had an amazing Math lecturer in first year university for a calculus course, and it changed my perception of mathematics for good. What was once a trial now comes much more easily.
I took a calculus-based physics class and it really helped me see the applications of calc.
You can derive physics formulas very easily with calc, if you start with the definition that acceleration is the first derivative of velocity with respect to time you have:
a=dv/dt
<=> dv=adt
<=> INT(v0,vf)dv = INT(t0,tf)adt
<=>vf-v0 = a(tf-t0)
<=>vf = v0 + a(tf-t0)
So we've shown that velocity is equal to initial speed (v0) plus acceleration multiplied by change in time. We can keep going with this by realizing that velocity is just the derivative of distance with respect to time:
v=dx/dt
<=> dx = vdt
<=> dx=(v0 + at)dt
<=> INT(x0,xf)dx = INT(t0,tf)(v0 + at)dt
<=> xf-x0 = v0(tf-t0) + 1/2a(tf-t0)2
<=> xf = x0 + v0(tf-t0) + 1/2a(tf-t0)2
So there ya go, fuck formula sheets when you have calculus you can derive all the things.
(sorry for the horrible formatting, using INT to represent an integral and INT(0,1) would be an integral from 0 to 1)
Am I the only one who thought this didn't clear anything up? All I clearly read from this is that a derivative is a way to see how a function will give different outputs from different inputs. I tried following the example as if it were a graph (as I know that's how a lot of it's used), but I'm tripping over the variables he uses. Is X=y and T=x on a conventional graph? Which axis is approaching zero? In what way is it approaching zero? Perhaps the people who are saying this cleared things up are understanding something else about this than what I have questions about.
Neither axis is going to zero. The function can range all over the place as long as it's continuous.
A derivative considers what happens when you change the input (the x in f(x)) a tiny tiny bit: what does the output do? If we consider smaller and smaller input changes we get closer and closer to the true slope (rate of change) at that point. The change is what goes to zero.
You can simulate it for yourself with a curvy function like sin(x). Consider some point like x=0. We will consider smaller displacements from this point. Calculate sin(0.1), sin(0.01), sin(0.001) and so on...it will quickly converge to whatever you put in...which acts like plain ol' y=x with a slope of 1. Do that for all the points and it traces out cos(x).
Ok, that's definitely making progress for me. Could you explain what sort of result we're actually looking for in the situation of the third sentence?
Or maybe I just had an epiphany moment (a small one). You're saying the change in results/outputs approaches zero, right? Not the change in input? Is (what the difference between outputs becomes) what is considered the limit?
Also, thank you for spending any time to read and answer what I said now and before.
Both actually. In fact you want to see how the output changes for a given very small change of input. Make that small change arbitrarily small (very close to zero, call it epsilon because why not) and see how much the output changes.
The output might change not at all (slope 0, flat). It might change exactly in line with the input ( slope 1) or some multiple of the input (slope >1). Or maybe it goes down (slope < 0).
The derivative at a point is the RATIO of the change in output to the change in input. BOTH approach zero, but (usually) the RATIO does not.
For example, for y = f(x) = x2 , the derivative f'(x) is 2x. So if you are interested in, say, x = 3, f'(x) = 6. As you make a very small increase in x value, the change in y value will be 6 times the change in x value.
I'm glad you ask : your question allows a more direct broach of the importance of historical context in understanding mathematics more fully.
The oft purported meaning of an integral is, simply put, the "inverse" operation of a derivative; this is true in some cases, most notably those which arise in the context of an introductory calculus course. However, one must delve a bit deeper in order to grasp the quintessential essence of an integral; thereby allowing these notions to be connected in a meaningful way to the theory of derivatives. Before I commence with the following allow me, for those who are more well versed in mathematical history, to apologize for any inaccuracies; I eschew the claim of complete historical precision though my intentions are to retain the vestiges of such to the degree of totality that my memory permits.
The tale begins, as far as I am aware, circa 250 B.C. in ancient greece; Archimedes of Syracuse, no doubt hunched over a stack of documents lit dimly by the wafting yellow-gold of candlelight, explores a method for approximation of the value we now call 'Pi'. He employs a limiting technique whereby he evaluates the circumscribed area of a circle via successive approximation by other n-gons of which the area was known. On some accounts he proceeds by approximation outside the circular form and, by others, he proceeds by approximation from its interior. In this way, by form but not by name, he introduces the concept of a limit.
Archimedes' gifts to the engineering and mathematical world did not, however, attenuate at this accomplishment. He purportedly devised another method of successive approximation of areas via their subdivision into elementary elements of known area; one manifestation of this technique would be, again, the calculation of the area of a circle. To accomplish this a circle was approximated via an inscribed regular n-gon; this n-gon can be viewed as being decomposed into triangular elements of known area. Thus, adding the areas of the individual triangular elements we arrive at an approximation for the area of a circle; this process was repeated with a more accurate n-gon inscription. Repeating this process a number of times one could then define the area of the circle as the act of passing to the limit (to employ a modern term) of the process of adding the individual areas of the triangles composing the series of inscribed n-gons. From this precipice one might conjecture that the modern notion of an integral, utilized in coursework today, was conspicuous or even imminent. As most tales of the evolution of scientific knowledge, however, human progress proves a capricious beast; alas, this connection is not to be made for centuries still. This is not, of course, to suggest that these notions were not immeasurably useful, however; the applications of the intellectual developments of Archimedes rang heavily throughout the Greek and, in turn, Roman world.
In the time of Isaac Newton, when the notion of derivatives was put forth jointly by himself and Gottfried Leibniz, the modern concept of the integral utilized in today's calculus textbooks was still not in place. Isaac Newton noticed the connection between the process of the operation of "successive approximation of area beneath a curve" (what we call integration) and that of differentiation. More precisely : Isaac newton noticed, basically, that "if one considers the 'derivative' of the process of successive approximation one recovers the underlying function". Isaac Newton offered not a proof of this observation in rigorous form due to two facts : the notion of 'proof' at the time (what was considered an acceptable form of argument for establishing scientific fact) was in flux (one must forgive this transgression and keep in mind that Sir Newton was in the midst of the enlightenment) and the vernacular in which to frame the ideas rigorously and succinctly did not yet exist. The original 'proof' by Newton of this concept was but a picture; he was, however, correct; at least within the context of the assumptions implicit in his medium and one must admire is perspicacity nonetheless. ( a light-hearted read on this can be found here)
As one moves forward in time we encounter some of the titans of more modern mathematics : names such as Baron Augustin-Louis Cauchy, Karl Weierstrass and Bernhard Riemann pepper the literature. At this time, the evolutionary goal of fixing the calculus upon a rigorous stage was in full swing. Most notably, for your inquiry, it was Bernhard Riemann who put forth the notion of the integral utilized in your beginner calculus texts. Conspicuous in the foundations of Riemann's integral theory are the roots of Archimedes' momentous contributions; one need not look further than "upper and lower rectangular sums" to bear witness to ancestry.
More specifically, Riemann suggested we proceed by, essentially, considering rectangular sections above and below the graph of a function : estimation of the area beneath the graph proceeds by considering the 'upper estimate', of sums of areas of the rectangles above the graph, and the 'lower estimate' of sums of areas of rectangles below. We then consider another approximation consisting of upper rectangles and lower rectangles with 'smaller bases' than before and, hence, refine our respective estimates. We call a function f(x) Riemann integrable if the limit of the upper sums and the limit of the lower sums coincide in value.
It is instructive, mind you, to note that not all functions are riemann integrable; that is to say, there exists functions g(x) whose upper sum limit and lower sum limit do not agree at all.
So there you have it in bare detail : the Riemann integral is simply the intellectual child of Archimedes'. This is, of course, if one takes the glib perspective that the efforts put forth to allow for the rigorous and definitive realization of the notion of 'integral' are vestigial to its quintessential notion (this vantage point will not gain you any camaraderie in a mathematics department).
We make the final press now to alleviate the notion that "the integral is the inverse of differentiation and vice versa". For his previously noted observation Isaac Newton is typically accredited with what is called the Fundamental Theorem of Calculus; this accreditation is as warranted as crediting Archimedes with the integral. It is precisely due to the existence of the "Fundamental Theorem of Calculus" that high school teachers and early-calculus professors muck about the classroom spreading the mantra that "integration and differentiation are 'opposite' or 'inverse' operations". In general this is simply not true. One should, instead, tell their students that "Integration is an operation and so is differentiation and sometimes the two are related in a convenient inverse-like way; in particular in this course (assuming a basic calculus course) they will always be related in this way".
So what does the Fundamental theorem of calculus say? In an unburdened dialect it says, essentially:
If f(x) is a continuous function then it has an antiderivative, F(x), defined by F(x) = integral(from a, to x) f(t) dt. Thus defined F(x) satisfies F'(x) = f(x)
If F(x) is any antiderivative of f(x) then the value of integral(from a, to b) f(x) dx coincides with F(b) - F(a).
Notice that the fundamental theorem of calculus does not say that an antiderivative, F(x), of a continuous function f(x) is unique : only that one exists and if you happen to have an explicit formula for any antiderivative of f(x) in hand then you can evaluate a definite integral of f(x) in a simple manner; namely F(b) - F(a). The existence of the first part of the fundamental theorem of calculus, establishing F'(x)=f(x), is the culprit for the aforementioned mantra of derivatives and integrals being "inversely related"; one can see directly, from the statement, that f(x) being a continuous function is an elemental assumption.
I hope this rather extensive tirade of detail proved helpful to you; thank you for the opportunity to expound upon the subject. The history and development of the calculus is a remarkable tale of human achievement; in it we see the the embodiment of the inherent altruistic struggle for truth. Embroiled in the development of the calculus we also observe the ebb and flow of ideas; the interplay between competing theories, the rise and collapse of individual egos and the ultimate victory which stems from perseverance. As it goes with mankind : if one desires to understand the present we must look to the past; it is an important and constant reminder of the humanity and capability of us all.
Is there an intuitive way of understanding the antiderivative or integral in the same way that the derivative is understood as the ratio of the change in output to the change in input at a given point? It's one thing to see all the rectangles, but it's much more difficult to understand why they work the way they do. Could it maybe be something like, for a given set of inputs, the aggregate sum of the differences between outputs and a constant?
The first encountered analogy for the integral is the famous 'area under the curve'; this is where the 'rectangles' relation arises. You can think of an integral more abstractly, however, as the "sum of the output of a function * the 'weight' of the input".
Allow me to elaborate with a more abstract example as impetus : your very life is an integral. You can think of an individual's life as the sum of the actions performed each day. If every action that happened each second, or millisecond, was to be given equal significance then you could represent the weight of that action by the number of seconds (or milliseconds) it took to complete. E.g. you could write a day as the sum of events * length of the event; summing over all events that occurred on every day would give you the 'integral' of your life.
This is precisely how the 'rectangles' arise : we are approximating the 'output' of f(x) over a small interval, delta(x), by a single value (the height of the rectangle), and multiplying by the length of the base of the rectangle (which is delta(x)). Adding all such rectangles gives you the area under the curve. I hope this assists you in your understanding.
Naturally, one might think "the integral is simple : its just the area under a curve" much in the same way it is often said "the derivative is simple : it is the slope of a line". The author's of these statements betray their lack of depth of education on the topic; granted, these are the concepts that arise in elementary explanations but these notions are more subtle in general.
Your question is actually a bit deep : though the depth of your question is masked by your choice of phrasing. To answer your question correctly it would be necessary to discuss the lebesgue integral which is an extension on the concepts of riemann (traditional) integration. Don't stop asking questions such as this; dive, swim below the surface and you uncover much that is hidden from the light of day.
Thanks! I do enjoy math, but never really had a head for arithmetic or showing work. Thus, math education was always a struggle.
Riemann integrals had always bothered me when I took introductory calculus back in high school, given their lack of precision. Same with the "area under a curve" conception - useful, but it tells me nothing about how I can be certain that the result of the integration describes what it is supposed to.
When you say "weight" of the input, that refers to the influence that a certain input has over the entire set of inputs, if I'm not mistaken? Would that be akin to the effect of the magnitude of a particular vector in a field upon a line integral curve? (I'm way out of my league here, so I could be completely wrong).
While I lack the eloquence of tbthomps, I would like to help in some way.
An integral is sometimes called an antiderivative, because they are 'opposite' operations in a sense. The easiest way to understand an integral is in the geometric sense - it is the area under a curve, as shown in the graphic. The curve (or function) is represented by the black line.
Intuitively, how would you find the approximate area between the curve and the x-axis? You could line up a bunch of rectangles of various sizes next to each other, with their top right corners touching the curve and their bottoms touching the x-axis (the top left picture). Then, you multiply the length of each rectangle by its width and you get each rectangle's area. Sum them all up, and that's an approximation of the area under the curve. You could do the same thing with the middle of each rectangle touching the curve, or the left corner of each rectangle touching the curve, etc. or even with other shapes like trapezoids or triangles. Rectangles are just the most convenient since the formula for the area of a rectangle is simple.
In the wiki graphic, if you go from left to right across the top, and then from left to right across the bottom row, you'll see the rectangles getting thinner and thinner and more numerous. If you found the total area of the rectangles in the bottom right graph, would it be closer to the true area under the curve than the first one we looked at (top left)?
Yes, because the thinner rectangles are 'hugging' the curve much more closely, so there is less error. How far could we take this? At what point would the rectangles be thin enough that the area of them all would be exactly equal to the true area under the curve?
Now you need to understand what a limit is, because if we let the thickness of every rectangle shrink to 0, the sum of all the areas becomes the integral of the function over the range we specified. Of course, it's ridiculous to think of a rectangle having zero width, because that's just a line segment. You can think of the rectangles as being infinitesimally thin and infinitely numerous to fill in all the space under the curve exactly. The thickness of each rectangle approaches 0 but does not quite reach 0. Thus, the limit of the total sum of the rectangle areas, as the rectangle thickness goes to 0, is the integral of the function (over whatever range we picked).
That's one way of looking at it, and probably the simplest: an integral is just a sum of an infinite number of infinitesimally small segments which gives you the exact area under some function (curve). It is the limit of starting with a finite number of segments of finite size and going to the extreme, since the notion of an infinite number of things of infinitely small size can only exist in the abstract sense.
We call the wall the limit because it is where you will "eventually" end up.
Or to put it another way, if you took infinite steps, you'd eventually end up at the wall. If you only took finite steps, though, you'd get closer and closer to the wall till the difference was barely noticeable.
However, this same carpenter, daydreaming about the foundational aspects of his tools might, instead, gain some insight into how to build better tools.
Yes. To complete the analogy for this specific case, Issac Newton was the carpenter, who was trying to use Maths to describe acceleration, speed, and position. The tools that relate these observations of nature didn't exist, so he created them. You can use the tools that he created to predict fairly accurately where something will be in space if you have some pieces of the puzzle, like initial position, initial velocity, etc.
You will get a MUCH better understanding of derivatives when you take a physics class on classical mechanics. Newton created both together, and separating them doesn't make much sense - the math becomes abstract, confusing, and meaningless.
The why is directly related to Newton's laws of motion - study that, and you will see why differentiation exists and what it means in the physical sense that necessitated its creation.
I wish they would have been honest about the purpose of school math. They wanted you to become masters of following rules, remember definitions, and avoiding your analytical exploration, and questioning of things.
My problem was that I loved strategy and logic and felt the rules and definitions were "boring", and I pretty much became disenfranchised with math.
If someone told me that school wasn't a place to learn what math is, but only a place to learn rules and regulations, I would of dropped out of high school immediately and do what I'm doing now, studying analysis :D
They wanted you to become masters of following rules, remembering definitions, avoiding your analytical exploration, and [avoiding] questioning of things.
That is a compelling indictment of whatever school you went to. I don't know if this in the US or elsewhere, but let me assure you, it absolutely did NOT use to be that way, at least in most highs schools in the US. I don't want to condemn based on one person's experiences, but I have heard essentially the same story many times before.
Let us be very clear: any school like the one you describe is a travesty. Any high school teacher following such a policy in mathematics is incompetent and a disgrace.
Very nice. Judging from the replies, it seems that this foundational stuff isn't being taught sufficiently at the high school or college level. I remember struggling at the beginning of calculus with the epsilon-delta definition of a limit and my math teacher saying something like "Yeah this is hard, but this is important and many classes won't teach this. You'll appreciate it later in college."
A carpenter who gazes at his hammer, his saw and his lathe and wonders how it is they came to be; what their underlying impetus is and how he might understand them more fully doesn't build many cabinets.
Uhhh, I'm pretty sure that it's not too hard to figure out everything there is to know about a hammer.
My point was it was simple enough to work without being confusing, especially given that the OP isn't a native English speaker. Also, i hope you're not actually trying to say that your example and the hammer metaphor make the same level of "sense"; most craftsmen don't take the time to look at their tools, but those that do often can find ways to improve them. A skill saw or router might have been a better example, but you're suggesting that the metaphor doesn't work at all. By the way, would you like to explain to me why the claw of the hammer is curved just that way, or why the striking end is tapered, or why the handle is wood and not a polymer? Just because it's simple doesn't mean it's not worthwhile to think about, and if that's your attitude, at least keep it to yourself so those of us who care about learning (and not nit-picking others' work) can help those who want to learn.
Ok, now I'm genuinely baffled. What the hell in that post would lead you to that conclusion? I never mentioned physics, and just because I disagree with you doesn't mean I don't think prose and rhetoric are valuable. I also very sincerely hope you're not trying to suggest that you're better at either - if that's the case, perhaps you need to peruse your old posts a bit.
Physics explains why, for instance, you use a wood handle for a hammer (to damp vibrations, for one thing).
If someone had given a long explanation of, say, an astronomical phenomenon, but had screwed up the math in the beginning, and I'd pointed out these issues, the general reaction would have been positive. When I point out a rhetorical error, the response is negative, the difference being the ways in which the audience values finer points in mathematics and composition.
I also very sincerely hope you're not trying to suggest that you're better at either - if that's the case, perhaps you need to peruse your old posts a bit.
Superiority is no prerequisite for criticism. (If it were, top athletes wouldn't have coaches!) Consequently, an ad hominem argument is properly regarded as a fallacy.
A metaphor is a figure of speech that equates two generally unrelated objects by identifying a property they share. For instance, "My love is as a fever, longing still for that which longer nurses the disease."
The OP's metaphor equates engineers and carpenters by insinuating that neither understands the tools of their trade beyond a superficial level. The metaphor is only as successful as its central conceit. If it evokes a picture in your head of a carpenter thinking, "Fucking hammers, how do they work?" then it's done its job. But I'm not feeling it -- only an idiot would fail to understand the fundamental principles underlying hammering.
Is it though? You might say "a hammer works because it is hard and has a flat surface and a handle, etc..." But why is the hammer hard? What material is it? What makes that material hard? Is it the elements that make it up? Why are those elements, arranged in that particular way, what we call "hard"? You may know less about a hammer than you think.
I know the answers to all these questions, but the people who invented the hammer didn't. Importantly, they're really at the wrong level of abstraction to understand hammers. Given that hardness exists and there is a wide spectrum of it, a better understanding of hardness is not going to help you build a better hammer, except insofar as it lets you build novel materials of different hardness. "Black boxes" aren't evil in and of themselves; they're only problematic when they go beyond simplifying assumptions and into the realm of rote calculation without understanding.
In principle, you could understand economics via statistical mechanics, since all the people participating in an economy are just aggregations of particles. In reality, though, this isn't the right way to build a mathematical model of the situation. The character of macroscale behavior is often so different from the character of microscale behavior that understanding the one gives you no insight whatsoever into the other.
That may well be true, I don't actually know much about economics.
I just wanted to make the point that statistical mechanics can be applied far more generally. However, I do find it very disputable that economies tend toward some equilibrium (from my limited understanding of the matter).
Well, it's not statistical mechanics that's being applied, since the underlying assumptions aren't mechanical in nature; it's just statistics. Though, outside of research in mathematical statistics, most people just mean vanilla hypothesis testing when they talk about statistics.
There are plenty of example of "econophysics"-type research essentially reverse-engineering the interpolation method that a firm is using and claiming it represents some basic economic law, though, which is why they're not really regarded as equal players.
The reason you have no idea what differentiation is is primarily the education system. This is not just india, but around the world the way math is taught in high school is depressing. Many end up either hating the subject, or just not understanding it enough to appreciate what it actually is.
In general, the masses learn the motivation behind high school calculus and calculus theorems (intermediate value theorem, mean value theorem etc) in university in classes called analysis. As you may or may not imagine, analysis is the proper way to teach single variable calculus. You need to understand that knowing what differentiation is is not sufficient. You need to know why such a vital concept was even developed, the reason you need to know why it was developed is because you wouldnt know why the aforementioned theorems exist. Analysis is a logical process, and starting from scratch with a particular motivation (primarily functional analysis) is what drives understanding.
You must be asking, why do i have to wait until this accused high school class is over so i can learn some real mathematics? The answer is, of course, the education system. It is not our fault if education systems fail extremely at teaching the most important subject in school, other than language (maybe). If youre really interested, pick up Calculus by Michael Spivak, and read on from chapter 0. Good luck.
I don't think analysis can be taught in schools. The reason is taht most children would find it too difficult and wouldn't appreciate its importance. Although I'm not sure if you're saying it should. Many high school students find it hard enough using differentiation just as a tool to solve problems. They might have a vague idea of the slope of a tangent to the curve or the rate of change of a function.
I guess you studied maths at university like a lot of people in this subreddit. The thing is most students studying calculus at school aren't going to study mathematics full time at university. They might study a science, which involves a bit of maths though. I think most of them don't need to have studied a rigorous course in analysis
It's unclear why anyone would need to use derivatives to "solve a problem" (read: plug things into formulas) without understanding the meaning of a derivative. Humans are totally useless for routine calculation anyway now that we have computers.
By solve a problem I mean find a solution to a differential equation. If they study a physics or biology course or something like that they may be faced with differential equations. But they won't need to know things like the intermediate value theorem. Obviously, in a job they'd need a computer to find the solution.
I'm not talking about knowing the specific details of convergence, I'm just talking about intuitively understanding the fact that derivatives are instantaneous rates of change, that the instantaneous rate of change is the limit of the average rates of change, etc. These are the things people aren't learning in calculus classes, and without these a differential equation is just a block of symbols.
While there probably isn't a big need for an analysis class in high schools, it certainly can still be taught. There are a good many high school students who take Calculus as a sophomore or junior (this varies wildly depending on location) and many of these students end up taking analysis classes at a community college or local university since they have exhausted their own school's math curriculum.
Yes. The foundation needed for solving theorems in FoA is a few definitions from predicate logic and he assumes you knows these, i.e. existence and uniqueness, the transformation rules, etc..
None of these principles are difficult or hard to learn! If students don't get these intuitively they just need to practice them like they would practice getting better at a game. Students in America already waste thousands of hours practicing route drills because American schools are mainly about training you to operate in a drone like job.
If all those thousands of hours were spent thinking about logic and math from from an analytical point of view, many students would be extremely successful in understanding math on a more generalized and rigorous level.
I have to disagree to this. Actually understanding the principles behind a lot of the mathematics we learn at school can go a long way.
For most of junior/senior school I hated mathematics. Then in year 9 our family moved and I ended up in a school which didn't have such a difficult syllabus. I started having plenty of time to think about the stuff that I learned. That's whan I started loving mathematics. In less than two years i was converted into an unbearable math nut.
It's been a whilesince high school (New York public school), but I remember basically what you're describing, with one difference; we were given instruction on limits and had to solve
lim(x2->x1) (y(x2) - y(x1))/(x2-x1)
which gives dy/dx. Honestly, differentiation really is just the slope of a function. The only intuition I've ever needed was that, if the object is the coordinate of a moving object verses time, then your slope is the velocity at any given moment.
From this point on, you'll probably run into subjects that you don't understand up front. I know people who, for example, didn't understand matrices and linear algebra until they had to use them for physics, and I didn't understand graphs until I started using them in AI.
This very well explains the motivation of the definition. I feel like the other possible gap is that in high school you generally learn the definition, and then start memorizing rules about derivatives like d/dx xn = nxn-1 and the derivative of sin is cos. What is important to remember is that all of those rules come from some calculation from that limit, for example:
So for for any monomial xn the slope at a given point is nxn-1. Every other derivative rule is derived in a similar though sometimes more complicated way, e.g. chain rule. number420pencil's post gives you why the definition exists, and derivations like this give how it results in the calculations you do. The only other thing I can think of that may help you better grasp differentiation is seeing some of the closely related math such as integration and differential equations.
I just remember getting basic derivatives proved to us through an infinitely small right angled triangle. We were taught this at the beginning of calculus, and it stuck. I believe we also proved the sin and cos ones too (it was a while ago)...
y = x2
y + dy = (x + dx)2
x2 + dy = x2 + 2xdx + dx2
dy = 2xdx + dx2
dy/dx = 2x + dx
As dx becomes infinitely small, we end up with
dy/dx = 2x
I used to love seeing simple proofs like that at school - Euclid's Theorem in particular was very satisfying :D
Did I personally understand differentiation in 12th grade? Yes, I understood it long before that, but that's not the point.
One great motivation is the intuitive idea of speed. Everybody (well, almost) has traveled in cars and sees the speedometer. As you accelerate (you can feel acceleration) the needle moves up. It shows the velocity (speed) you are traveling at any instant. So, at any time t there is a function v(t) that gives the velocity (or speed) at that time.
How to compute this? Cars also have an odometer, the distance you have traveled at any time. That's a function of t also, let's call it s(t). Suppose the speedometer is broken but you have a watch and the odometer. How do you find speed?
To make a long story short, you could note a time t and the odometer reading s(t). Then let a small amount of time pass, say 1 minute. Note that odometer reading. Speed is just distance divided by time, so the speed over that interval of time is the difference in the odometer readings divided by the difference in times.
Instead of 1 minute, let's call the interval h. So we have
v(t) = [ s(t+h) - s(t) ]/ h
I called it v(t), but actually it's the average velocity over the period from t to t+h.
Now the coolest part: as h -> 0, we get the exactly correct velocity at time t.
v(t) = lim (h -> 0) [ s(t+h) - s(t) ]/ h
That's a mind-expanding idea. But there you go, that's the idea of derivative. velocity is the derivative of position.
Don't ever rely on teachers. If you want to learn a lot, you have to get good books and read them. And don't rely too heavily on online forums.
When I was in grad school, a fellow PhD student would continually whine about how he's not learning much because the professors were poor lecturers. Well, yes - they were poor lecturers. But why isn't he reading the books? He'll get his PhD, but has no future in research with that attitude.
I remember when I was an undergrad, one professor pointed out that the lecture he gives should be the least of one's source of learning. First would be reading the book(s). Next would be stuff like discussing with your peers and talking to the professor during office hours. The lecture was merely supposed to be a review after one has read.
So, long story short: Expect education systems to suck - even going to a top university will not prevent you from having poor textbooks and poor teachers. Never use them as an excuse for not learning well enough. Go read material on your own, and find like minded souls so that you can bump ideas off of them.
I was taught it as instantaneous slope, which was sufficient intuition to understand application to word problems. I still don't know it too much better than that, but I hate calculus.
This is a legitimate issue in many areas though, there's more concern for shuffling students through with ease than emphasizing understanding. I'd say it's more excusable at a high school level where students are picking their courses from an extremely limited number of electives, but it still should be considered a problem. Determinants are what did it for me, everywhere I saw them in undergrad was basically a magic trick of taking numbers in a box and spitting out a new one, didn't learn about multilinear forms or exterior algebra until about when I graduated, and I only saw the derivation for how they relate to volume after I had passed the point where they were used a lot. When I found Axler's "Down with Determinants" paper while searching for an explanation I felt so vindicated.
Your plight is a good example of the difference between educational styles across the world. I studied maths and calculus at the high school level in both the U.S. and the U.K. in the 1980s. In the U.S. ("A.P. calculus AB"), we worked from fundamentals - introducing the idea of a continuous function, and how to take a formal limit, and the limit as h->0 of the slope of a line intersecting the subject curve at two points x and x+h. Then we learned the polynomial rules, the product and quotient rules, and the chain rule. Finally we moved on to Taylor expansion and used that to derive the rote rules for differentiation for a zillion different elementary functions.
In the second year ("A.P. calculus BC"), we did almost the same thing with integration: building up Riemann integration from limits, then moving on to things like how to have hunches for integral substitution, how to integrate by parts, and such.
By contrast, in the U.K. (A-level track at Winchester College, a feeder for one of the Oxford schools) nearly every 16 year old could differentiate and integrate like a whiz, but most of my cohort didn't know what a limit was or how to take one. The curriculum there went more like this: define briefly what a derivative is (the slope of a curve, calculated at a single point); introduce the polynomial rules and linearity; introduce the product and quotient rules; introduce rules for myriad elementary functions like sin and cos; drill on differentiating compound functions; then do the same sort of thing for integration. In other words, the focus was on learning how to take derivatives and integrals, rather than on what they were or how they came to be.
I think that many places, especially those influenced by the U.K. (like India) teach integration and differentiation that way - more or less how we all learn our multiplication tables, by memorization of a gazillion products so that multiplication would be easy.
It's really a question of focus. If you want to use differentiation but don't particularly care about how it came to be, the way you are being taught is the most expedient. If you want to understand how differentiation came to be, you want a slightly different approach that is less efficient at educating most people (since most people don't care how it came to be).
Of course, there's some danger in teaching that way, since a derivative is many things to many people. After taking linear algebra in university you will learn a completely different approach! (differentiation is a linear operator on vectors in Hilbert space...). If you take a course in signal processing you'll learn another approach for it. Likewise, simple Riemann integration (what most people call "integration") is just the beginning. There are several other interesting ways to take integrals, including Lebesgue integration, some of which work on functions that you can't integrate the Riemann way. All this is a way of saying there's a lot of really cool stuff coming your way if you choose to start looking in depth.
Oddly (or perhaps not-so-oddly) Wikipedia) has a very nice treatment of the roots of what a derivative "is", from the standpoint you currently care about. I'm sure others can point you to even better online references.
Edit: I also like tbthomps' reply -- we seem to have been typing at the same time.
It's really a question of focus. If you want to use differentiation but don't particularly care about how it came to be, the way you are being taught is the most expedient.
People say this, but how does someone who doesn't intuitively understand what a derivative is make any use of the formal rules for computing derivatives? Unless you spoon-feed someone a problem and ask them to take the derivative, they won't know when to take a derivative in the first place. In real-world situations where you have a set of instructions ("differentiate, set equal to zero, solve."), these are going to be performed by a computer, not by a human.
Sure, except that the second includes the word "limit", which bright 16-year-old U.K. students were not taught much about.
Further, those two statements ("a derivative is the slope of a curve" and "the gangent line is the limit of the secan lines" are the pretty much the same from the standpoint of mathematical definitions but very different from the standpoint of pedagogy.
"Slope of a curve at a point" doesn't really make sense, though -- sure, maybe as an intuitive idea, but there's no way to convince yourself that such a thing really exists or is well defined. Limits, on the other hand, can easily be half-assed in such a way that it's clear there's no issue with formalizing them. Actually, they're a required part of the curriculum in the US, though they're generally given such short shrift that nobody ends up understanding them.
I think you're missing the point of my original post, which echoes that of the top ranked article in this discussion: you can teach how to use calculus or how to derive calculus and those are different skill sets. I happen to have experienced two educational systems with very different approaches at the secondary level. Apparently I did not succeed in communicating those differences.
I learned how to derive them in a U.S. school, but by the time I got to the U.K. my cohort had memorized them blindly. There was some handwaving to show that the functional forms were right, but they (at 15 or 16) were expected to just know that d(sin(x))/dx is cos(x), and such.
The year they were 17 (the second year of 6th form for them - the year they took A levels and applied to university) we covered Taylor expansion and they learned retroactively how to derive the formulae for sin and cos.
Download geogebra and illustrate all of the important concepts for yourself. It's very easy to get a hang of.
Also, Calculus Made Easy by Silvanus P. Thompson is worth reading. Download the PDF here (from Project Gutenberg). This is a funny and extremely well written book focusing on the concepts behind the computation.
Chapter one is titled: "To Deliver You From The Preliminary Terrors." Here is the prologue:
Prologue
Considering how many fools can calculate, it is surprising that it
should be thought either a difficult or a tedious task for any other fool
to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult.
The fools who write the textbooks of advanced mathematics—and they
are mostly clever fools—seldom take the trouble to show you how easy
the easy calculations are. On the contrary, they seem to desire to
impress you with their tremendous cleverness by going about it in the
most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach
myself the difficulties, and now beg to present to my fellow fools the
parts that are not hard. Master these thoroughly, and the rest will
follow. What one fool can do, another can.
This reminds me of an anecdote from Surely You're Joking, Mr. Feynman. (Article)
tl;dr: Feynman goes to Brazil to teach some students. He soon finds that the students have memorized everything, but lack an intuitive understanding of what the information actually means. This meant that, though the students could quote the definitions of properties and perform all the relevant calculations flawlessly, they couldn't apply this knowledge to real world problems because the problems weren't stated the way the students were used to. In essence, they couldn't form a connection between theory and practice.
Calc I introduces differentiation as a measure of changing rates. For example, say you have a tub filled with water. A hose is connected to one side of the tub near the bottom, so the tub drains by the force of gravity pulling down on the water (Like the spigot on the side of an ice chest). How long will it take for the tub to empty?
Let's think about this. First of all, the flow rate is based on the pressure of the water in the hose, so if that pressure changes, the flow rate will change. So where does that pressure come from? It comes from the weight of the water in the tub. So the flow rate is dependent on the amount of water left in the tub (Mostly. We'll assume the other factors are constant). Since the whole point is that the tub is being drained, we know that the weight of the water, and thus the pressure and flow rate, will change. As more water leaves the tub, the flow rate will become less and less. Obviously, if we base our estimation on the initial flow rate, it will be way off. So how do we come up with a more accurate estimate?
Differentiation. If we take the derivative of the flow rate over some measure of time, we are essentially finding out how the flow rate changes over time. With this information, we can create a model that predicts when the flow rate will become zero, or at least very close to zero. At that point, the tub will be as empty as it's going to get.
I went to a public high school in the US, I had the same problem. Though, to be fair, every thing I learned in math was purely mechanical. It wasn't until I studied math at university that I started to understand
If you take a function f(x) and there exists a function g(x) such that the integral from 0 to x g(x) = f(x) then the derivative of f(x) at x is g(x).
That might not have helped.
A derivative is a way of expressing how fast a function changes at a point. On the function f(x)=x2 , the derivative at f(0) is 0. At f(4) it is 8. At f(x) it is 2x.
Differentiation is just the method of finding the derivative at every point of a function.
To try an analogy, imagine I was trying to explain a certain function to you.
I could say, "Well, when x=0, f(x)=1, and when x=1, f(x)=2, and when x=2...." and so forth. It's much quicker and provides more information to just say f(x)=x+1.
Similarly, you might ask me what the instantaneous slope is when x=3, or when x=4, or when x=5. When you differentiate, it allows you to find the instantaneous slope at each point.
You need to understand that the rules for differentiation come from the limit rules that were probably briefly encountered near the beginning of the course. They only hold because they follow from the limits.
If for any continuous function, you were to go from negative infinity to infinity, and you took the instantaneous slope, and put the value of the slope on each respective point (e.g. if on x = a, the instantaneous slope is b, you would make another graph so that when x = a, y = b). If you do this you'll see that the derivative is merely a function demonstrating how the slope changes.
I live in the US, I learned derivatives in 10th grade. It is finding the slope of a line at any point. So, with a linear equation (f(x)=mx+b) the slope will never change, and the derivative=m. As you get to higher degrees, the slope will change more and more drastically, which is accounted for by only losing one power with a derivative. Hope this helps!
If you want to learn more deeply about the math behind the "math" you're being taught, look at some of the older textbooks. The 2 volumes by Tom Apostol are pretty good this way.
My experience with high school calculus was like yours. Lots of formulas and basic concepts, hardly any real understanding. I got come of the deeper math in college, but by then, I was more focused on learning about programming that I didn't have much patience with all the theoretical math we were being taught.
One thing you want to realize with math is that there is generally never one way of viewing or thinking about something, and doing so will just further constrain you. So try to come up with YOUR OWN EXPLANATION of what the derivative means to you, and see if your explanation is universally true in various situations you can think up.
A slow read through a good textbook should make it clear what differentiation is. It is the measure of "instantaneous" change, as in. Here I am measuring different points on a curve and averaging them together. What happens when I bring the points closer to a single point and I take fewer samples. What value do they approach? The value they approach is called the derivative at that point.
Another way to look at it, is the slope of the line tangent to a point.
Yet another way (my favorite way) is to define it as the limit of the slope of the secant line as the points used to define the secant are brought infinitely close.
Just in general, I find that some of these concepts take a bit of time to really "soak in." Not saying don't try, but sometimes after summer break, you come back, and have that "aha moment." Try not to stress about it!
A lot of mathematics is taught as procedure rather than being taught for understanding. This does a disservice to people, because some of these concepts can be understood without necessarily knowing how to execute the procedures.
Some people survive the traditional mathematics curriculum and also get to a point of understanding, despite the traditional methods of instruction.
In short: yes. The way of learning mathematics as procedures is, as you put it, "horrible" for people who want understanding. And understanding the mathematics of change and variation is something useful to anyone who wants to understand a dynamic world.
I was in the same boat as you until I started taking university math during my senior year of high school (my high school didn't offer much math). That's when I found out there was a reason for d/dx of xn = nxn-1 . It was the same way with physics, I had to wait for college to learn what the reasoning was for all the stuff I'd been taught in high school. I imagine it's the same story for a lot of people.
By the way, to understand what a derivative actually is, think about the formula for slope: (change in y) / (change in x). This formula can give you the slope between any two points on a curve. If these changes in x and y are infinitesimal, then the two points virtually coincide into one point, and suddenly you're looking at the slope of a tangent line.
Very simply put, differentiation is a tool that allows us to examine how a function changes.
For example, the velocity of something tells us about how it's position changes over time ( v = dx/dt ). The acceleration of something tells us about how it's velocity changes over time ( a = dv/dt ). We can differentiate with respect to something other than time (for instance, the rate of change of the area of a circle with respect to it's circumference, or the area of a polygon to it's perimeter). We can also differentiate with respect to multiple variables. But, underneath it all is the idea that we can derive useful information about a function by examining how it changes, and differentiation is the tool we use to do that.
Differentiation is an operation on a function that returns another function, which describes the slope of the original function at any point. Consider the linear function f(x) = x. The slope of the line is everywhere 1. Therefore, the derivative f'(x)=1. The product f'(x) and an infintesimal change in the variable x gives the infinitesimal change in the function value.
Another way of thinking about it is through the concept of a limit. The derivative of any function is given by
f'(x) = lim_{h->0} (f(x+h)-f(x))/h for all x where that the limit exists.
Hiya there, I was wondering which board does your school follow. I did CBSE and things like limits and differentiation and integration from first principles were taught to me. I do agree though that they didn't focus on much visualization of what it all meant. It also didn't help that a good number of students did their math quite mechanically. I got a better idea though, once I started looking at the calculus we did in 12th grade physics. Maybe that might help you. (Of course, I presume that if you have math at 12th grade, you took the Science stream)
Yes, calculus was very natural for me, relatively speaking. Had much more trouble with algebra because my first algebra teacher in middle school didn't do shit but sit at her desk and ignore us. Totally screwed me over when I took real algebra my first year of high school.
Luckily I had a good high school teacher who taught differentiation conceptually, without rote memorization. It clicked fairly well without too much effort. Just think of differentiation as an operator that acts on a function, and spits out a new function that shows how the old function changes with respect to some variable. If it changes a lot as it moves along the x axis for example, the new function will have a high y value because it changed a lot. If it doesnt change at all it will have a zero y value. If it changes in a negative way, then y is negative.
tl;dr: Differentiating a function produces a new function that represents how the old function changed along some axis.
I first understood it by thinking about it like this: Imagine a point moving through 2 dimensional space, that gets from point a to point b with a tiny jet engine. If the point is traveling in along a curve, the jet points tangent to that curve. It is the jet that determines how the point moves, so in a sense it "derives" the motion. Therefore, a description of how the jet decides which way to point is called the "derivative" of the function describing the point's motion.
Another way to think of it is as a rate of change. Again, picture a point moving in 2D space, but for this point we also consider time. We can describe this motion in terms of the point's position on the x and y axes at a given time. Therefore it is a function of three variables: f(t,x,y).
dx/dt describes how the x position changes with time. It tells you the rate of change in x as t changes. Similarly, dy/dx tells you the rate of change in y as x changes, and so on. Derivatives describe how one variable depends on another.
One way of putting it is that its finding the slope of a curve. For a line, of course, the slope is just a constant, and its true for all points on the line. But when you have a curve you need a formula to find the slope for any point, and thats the derivative, which is a function of the points x-value. In fact, the derivative works for finding the slope of lines at any point too: just take the derivative of f(x) = mx + b and you will see that it equals m. So it works for lines too.
There are other ways of seeing it but this is one correct way and it works for me: derivatives are a generalization of slope.
I took AP Calc and AP physics in 12th grade and learned more calculus in Physics in the first month of school than I did in actual Calculus. It made a lot of sense when we went over differentiation of position functions. I feel like applying calculus as I learned it helped me understand it better.
I didn't quite understand what it meant until introduction to quantum physics in my physics degree. I do recall have to integrate and differentiate for solutions until then but it was only when using it in a "practical" thought process scenario that helped it make sense.
You are not alone. I saw the end result of what you are talking about in grad school. I worked with some brilliant foreign exchange students from India while I was there and what I consistently observed from the Indian students was exactly what you describe - they were amazing at solving calculus problems, but they had trouble connecting what they were doing with what it meant in the real world. Understanding the mechanics of calculus is very important in science/engineering, but understand 'what' the calculus means is much more useful in the end. Keep heading down the road you are starting down. If you can master both sides, the mechanics and the meaning, you will be a force to reckon with if you go into science or engineering - especially if you decide to study in the US.
okay its totally normal to not REALY understand math in school. But to give u 1 example. We can approx linear functions by looking at one a and then say f(x)=f(a)+d/dx(f(a))(x-a). Now the question is if that is also possible for lets say f(x)=x2. u clearly see that f(x)~=~0+0(x-a) is wrong. because the derivative of x2 is not a constant but a linear function, now since we can approx linear functions as shown before, we could sayf(x)~=f(0)+2(x-0)~=~f(0)+x. This ofcourse is not extremely close to x2, but if we look at it in the intervall [0,1] it is decently good. since its the same at both intervall ends and all of x2 are also expressed my x in that intervall. And if we look at a very small intervall lets say I=[0,e] with e >0 the difference between x2 and x is maximal e-e2=e*(1-e) < e so in that small thing x2 is approxed by x. What that means is that u can approx highly complex functions by easier ones , since x is definatly easier to look at than x2. SO as a mathematician u basically need derivatives to approximate complex functions to spare time. There are for sure many more uses like f.e. if u want to find the limes of the quotient of 2 functions that both go to 0 u cant look at the derivatives so u find a valuable limes and not "0/0", but i think the approximation of functions is the most important that i know for now.
I'll try to answer this from a physics perspective, which might help to understand the significance of the derivative. A lot of things we observe around us vary. Some observations vary with time (the speed of a car when it accelerates), others with temperature, pressure etc.
But we need absolute values for calculations. If I want to know how much energy is imparted by a car crash, I need to know the velocity of the car precisely at the moment of impact. And therein lies our problem. We know that the faster a car travels, the more distance it can cover in the same time. Velocity is change of distance with time. That means the distance the car traveled in a given time interval. But how fast was it going at a precise instant in time?
Obviously, since there is no change in time at that instant, there must be no change in distance. So, velocity is 0? The problem is that we're trying to fit in 0 into our equations. (by trying to use zero time interval). Now, zero is a strange number and doesn't play well when it is in the denominator. So we can't use it. How then do we define a moment in time (i.e. where time changes by 0)?
This was the problem that Isaac Newton faced when he was originally working on the laws of motion. He needed a way to define instantaneous velocity using equations, and derivatives are what he came up with.
If you can't measure the distance over zero time interval, what you do is measure it over an extremely small interval. You get a value. Then you measure it over an even smaller interval. You get another value. And you repeat it forever. What you'll notice is that each successive value you observe, converges ever closer towards a precise value. You can basically extrapolate your data and determine the precise value over zero change. Which is just what differentiation is.
I took it in college. Differentials themselves I think are no big deal, but the underlying Algebra to really master Calculus is a bear to really get right.
Conceptually you do get the idea that if one has an equation that yields your speed at a given time, then the derivative is an equation that tells you the change in speed over a given time, yes? So if you are travelling at 85mph (constant), then the derivative is zero (not changing). Obviously more complex conditions are more challenging to understand, but that's the gist.
tbthomps's long explanation is good. for the lazy, though, think about derivatives simply as a way of describing rates of change.
you know speed right? it's a way of talking about how distance changes over time. well, your speed can be different, and this eventually affects how the distance changes. Differentiation is just a mathematical way of modeling the pattern of how a value like distance changes over time.
Remember that the result of differentiation gives you an equation. Technically, it doesn't give you a single value as the slope of a function. With an equation, you can plug in values for the variables and calculate the numeric value of the equation. If you start with a function f(x) and you find the derivative f '(x), you now have a new function, an equation, that will tell you the numeric value of the slope of the line tangent to the curve of f(x) at any point on the curve of f(x). Notice, not just for a single point, but for any point on the curve.
Imagine that you can shrink yourself down and walk along a curve of a function. Now, think of having the derivative in your pocket as sort of like having a magic handheld device; it's sort of like having a GPS built into your cell phone. As you walk along the curve, you can look at your "GPS" and it will give you not your latitude and longitude, but the value of the slope of the line tangent to the curve at that point. That can sometimes come in handy to help you get your "bearings," when you need that kind of information.
As I understand it, it's more or less a way to divide a function's y axis by it's x axis (or vice-versa, depending on what you differentiate with respect to).
When you do this, the end result is another function, where for each point on the graph the y-coordinate is equal to the slope of the original function at f(x) where x is the x-coordinate.
So, lets say you've got a function in the variable x (which is in seconds), that outputs variable y (which is in meters). You can differentiate with respect to x and essentially divide your meters by seconds, which giving you a function where the y coordinate is meters per seconds.
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Factoring Polynomials Study Guide
Student Contributed
Factoring Polynomials Study Guide
This study guide reviews methods to factor polynomials, including identifying special products, factoring monomials, and tips for factoring polynomials completely. It also looks at how factoring can be used to solve polynomial equations.
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Events, Sample Spaces and Probability
Finding the sample space is the key for student success in computing probabilities. Frequently problems arise when possible outcomes are missed, or double counted, or otherwise. That is to say, there are lots of less than intuitive possibilities so tools to support students finding correct answers are helpful. One of those tools is listed in the text, where a table is made with the different outcomes. Lists are also good, especially for sample spaces that only involve one item, like a single die. There is also no shame in drawing pictures or working with manipulative. I frequently visualize a deck of cards in my mind when working with those probabilities, but as a bridge player I'm comfortable with cards, while a student might not be. Don't hesitate to give students cards to work with.
I encourage my students to never work in percentages. I realize that it's kind of picky, but I find it useful to drive home that probabilities must be between zero and one inclusive. The strong restriction is very useful as it frequently provides feedback for mistakes. If you are computing any probability and it ends up outside of that interval, then something is wrong. It's also a great tool to use to eliminate answers on a multiple choice test. I like to have a mantra to really drive it home. Between and and all possibilities add to .
Compound Events
Elementary set operations are a fundamental part of mathematics, but one that is taught in inconsistent places. I've always marveled at the fact that nearly all of my college textbooks, including my graduate level texts, start with a preliminary chapter on set theory. This has always led me to believe that set theory is either not taught, or is given an incomplete treatment in a lot of classes. There really isn't any reason to give a deep treatment to it here. Sometimes sticking too much to the strict notation is going to cause more problems than it's worth, since the ideas here are intuitive and students have worked with them in venn diagrams and other problems in the past.
An important thing to remember is that set operations are binary operators. That is, even if there are more symbols, only two sets can be operated on at a time . Due to the fact that the combination of unions and intersections is not associative (that is to say: I always include parenthesis if there is more than one operator, even if they are all the same operation, just to avoid confusion. This is, and matrices, are some of the first structures students encounter that do not follow all the classic rules of real numbers that students are used to. It can be an opportunity to push a talented class, but it is really an extra topic that has limited utility to the ultimate goal for a stats class.
The Complement of an Event
The classic example of value here is the birthday problem: In a given room, what is the probability that at least a pair of people have the same birthday? This is a great problem, as it shows how working smarter using some principles of probability makes a seemingly tough problem easy. It also has a result that is fairly counter intuitive; the probability of at least a match is much higher than one would presume. Both are key ideas to drive across to students studying probability for the first time. Asking for "at least one pair" means that if you were to directly calculate the probability it would take a very, very long time. There are just too many ways to get a match once the number of people in the room exceeds 4 or 5. However, asking the question "what is the probability that no two people share the same birthday" is logically equivalent to the first question and much easier to compute. Subtracting this quantity from 1 (finding the complement) then yields the answer.
The key here might be in re-writing the question in terms of the complement. Students probably don't see what the big deal is in calculating complements, and that is really simple. However, making the complement work for you requires seeing where to apply it, and then what exactly you are looking for. The key hints are when a question is asking for a probability where multiple situations are possible. In the birthday problem, asking for exactly a single pair should be calculated directly, but at least a pair dictates that the complement is easier. Students should practice identifying and re-writing questions to make it so that time isn't wasted attempting monumental calculations.
Conditional Probability
Students are going to get tripped up with the order of conditional probabilities. The more intuitive way of thinking of conditional situations is "if then" as opposed to "given". While I am nearly always in favor of understanding the concept and avoiding formulae, in this case the formula is great. Because the order is a little strange, and frequently is mixed up, this is one of the few formulae that I put on a poster and ask students to commit to memory. Application is easy once the spaces are put in their proper place.
Additive and Multiplicative Rules
This is probably student's first exposure to the dreaded problem of double counting. In this case, in contrast to the previous section, I don't emphasize the formula here. Finding values or probabilities that are double counted is a huge part of statistics, and one that most stats students have memories of sitting in a group, having problems with getting the correct answer, pulling hairs out, only to then have someone say "double counting!" For this reason, I really try hard to get my students to understand where double counting occurs and try to train them to always be aware of where the error is likely to occur.
Basic Counting Rules
Combinatorics are the foundation of lots of basic probability questions. Cards are a great way to do look at problems, especially with the recent attention paid to poker. The television broadcasts will show probabilities for winning in "real time". This can be used to practice and find those same percentages, which are relatively simple to compute. The foundation of each probability is finding the different cards that will allow for a win, against the total number of possible cards left. This is a nice extension of finding general probabilities for each type of poker hand.
Whenever combinitorics and probability comes up gambling is not far behind. This can cause problems in some cases considering the ethics of teaching typical gambling games in a classroom to students who are not legally able to wager bets. A couple of thoughts on the subject, and a general defense. First, it may be useful to make the distinction that while games are being taught and talked about, at no point is gambling going to occur. This is similar to a health class where effects of drugs are being discussed, but clearly no endorsement, or use, of drugs is happening. There is also historical context, as the earliest theories and work on probability was, in fact, motivated by gambling. The legacy remains, even in situations where gambling is no longer associated. For example, a hand with no high card points in bridge is called a Yarborough, named in honor of a lord who would offer his opponents a payout if no points were dealt (odds of getting a yarborough are , so the Earl made quite a profit on this). Second, I would promote the idea that gambling institutions profit from a lack of knowledge of probability. Like the Earl of Yarborough, the idea of casinos is to present a situation that looks favorable to the gambler where in actuality the advantage is firmly in the direction of the house. Knowing exactly how much of an advantage is disheartening to a gambler, and in many cases will cause a loss of interest in gaming.
While it is useful to compute a couple of combinations and permutations by hand to get a sense for how they work, I quickly pull out the calculators. In fact permutations and combinations might be the functions I use most often on the calculator right behind the trig functions. There is no benefit to spending any extra time with the tedium of not using the calculator functions.
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Trigonometric functions, such as sines and cosines
Graphs and trigonometric identities
Vectors, polar coordinates, and complex numbers
Inverse functions and equations
You can use CliffsQuickReview Trigonometry in any way that fits your personal style for study and review — you decide what works best with your needs. You can read the book from cover to cover or just look for the information you want and put it back on the shelf for later. Here are just a few ways you can search for topics:
With titles available for all the most popular high school and college courses, CliffsQuickReview guides are a comprehensive resource that can help you get the best possible grades.
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At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented writers who know how to cut to the chase—and zero in on the essential information you need to succeed.
Master the basics—fast
Complete coverage of core concepts
Accessible, topic-by-topic organization
Free pocket guide for easy reference
About the Author
DAVE KAY is a writer, engineer, and aspiring naturalist and artist. He has written or cowritten more than a dozen computer books.
Most Helpful Customer Reviews
This book is good for one thing, a Quick Review. I suppose I can't blame CliffsNotes, since that is given in the title of the book. I bought this book when I was beginning to learn Trigonometry, along with another textbook. I found that a lot of the things the textbook went over was not even mentioned in this Quick Review book. If you have taken Trigonometry in the past and want to refresh some of the basic trigonometry concepts, then this book is for you. However, if you know nothing about Trigonometry and want to learn about it, this book will do nothing but CONFUSE you.
Most people that will look into these books are people who want to learn the subject for their first time, so take my advice: instead of buying this book, save that money towards a better, complete basic trigonometry textbook.
I'm a (returning :P) university Freshman preparing for the College Board CLEP tests. I was already familiar with the material covered in this book, but needed to refresh my memory. This review turned out to be *exactly* what I needed. The author's ability to explain the material to the student are just shy of enlightening. The discussions & theorem proofs are written in a very concise, clear style. I'm a big advocate of the Cliff's QuickReview series. Intended as a course supplement, these books are also *GREAT* for students wanting to refine their skills. Most of them are also very accessible to students with less familiarity on the subject; trying to learn it for the first time. After reading this, I bought the Calculus & Differential Equations QuickReviews & I'm looking forward to reading them!
After several years in a corporate engineering job, I started moonlighting as a math tutor. The Cliff's Quick Review Guides are wonderful to have in my "back pocket" when I need to quickly look something up that is covered in dust in the "archives of my brain."
Trig is something you always have to practice if you want to remain competent. When practicing basic trig problems (identities, equations, vectors, graphs, angles, cmplx.#'s....) this book gives me just enough explanation with the info I need. But like others said, you should already have some trig under your belt before purchasing it.
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Algebra Puzzles
Games
Price:$13.99 Available Qty: 42
Qty:
Build Pre-Algebra and Algebra Skills through Puzzles and Problems. Not your typical algebra workbook, Algebra Puzzles uses games, puzzles, and other problem-solving activities to give students fresh, new ways of exploring learned concepts. While reviewing essential concepts and vocabulary for pre-algebra and algebra, the book helps students visualize and think more deeply about these abstract ideas. The perfect antidote to algebra anxiety.
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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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Elementary Number Theory and Its Applications
9780321237071
ISBN:
0321237072
Edition: 5 Pub Date: 2004 Publisher: Addison-Wesley
Summary: Elementary Number Theory and Its Applicationsis noted for its outstanding exercise sets, including basic exercises, exercises designed to help students explore key concepts, and challenging exercises. Computational exercises and computer projects are also provided. In addition to years of use and professor feedback, the fifth edition of this text has been thoroughly checked to ensure the quality and accuracy of the m...athematical content and the exercises. The blending of classical theory with modern applications is a hallmark feature of the text. The Fifth Edition builds on this strength with new examples and exercises, additional applications and increased cryptology coverage. The author devotes a great deal of attention to making this new edition up-to-date, incorporating new results and discoveries in number theory made in the past few years.
Rosen, Kenneth H. is the author of Elementary Number Theory and Its Applications, published 2004 under ISBN 9780321237071 and 0321237072. Twenty Elementary Number Theory and Its Applications textbooks are available for sale on ValoreBooks.com, fifteen used from the cheapest price of $34.29, or buy new starting at $187.24.[read more
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The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom
| 677.169 | 1 |
Classical Geometry – Lecture Notes.pdf
Classical Geometry, is a lecture notes in pdf file that has been prepared by Danny Calegari. Taken from Appendix – What Is Geometry?: Geometry is a beast that can be approached from many angles. Four of the most important concepts that arise from our different primitive intuitions of geometry are symmetry, measurement, analysis, and continuity. We briefly discuss these four faces of geometry, and mention some fundamental concepts in each. Don't worry if these concepts seem very technical or abstract-think of this section as an abstraction of the concrete notions found in the main body of the text.
Contents:
A CRASH COURSE IN GROUP THEORY A group is an algebraic object which formalizes the mathematical notion which expresses the intuitive idea of symmetry
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Elementary and Intermediate Algebra: A Combined Approach, 5th Edition
Master algebraic fundamentals with Kaufmann/Schwitters ELEMENTARY AND INTERMEDIATE ALGEBRA 5e. Learn from clear and concise explanations, multiple examples and numerous problem sets in an easy-to-read format. The text's 'learn, use & apply' formula helps you learn a skill, use the skill to solve equations, then apply it to solve application problems. With this simple, straightforward approach, you will grasp and apply key problem-solving skills necessary for success in future mathematics303.95
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Synopses & Reviews
Publisher Comments:
Many Algebra I students have trouble relating to the material and concepts. Teachers tell us, 'they just don't think it's relevant.
That's why we created 100 Lesson Starters for Algebra I. It's a new program, in a new format, perfect for starting off a class or in-between time.
The program comes packaged on a CD-ROM with 100 slides in a PowerPoint presentation. Each slide is set in a student-oriented context, such as a local mall, a live concert, or a school classroom. Against this background, a problem is presented - a problem that requires Algebra I content to be solved. 100 Lesson Starters for Algebra I is rigorous enough to require true problem-solving and accessible enough to allow all students to progress towards a solution.
The teacher notes on each slide contain:References to Algebra I curriculum outcomes and the Glencoe Algebra I text
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As more and
more states raise their graduation standards, there are an increasing
number of students who find themselves faced with the task of completing
algebra 2. By developing CORD Algebra
2: Learning in Context, CORD has expanded its
ground-breaking series of contextual-based math textbooks. More importantly,
teachers are provided a tool to help the student who would traditionally
struggle with math succeed and reach their full potential. By combining
real-world applications and lab activities with the mathematics being
taught, students receive a more meaningful education and are better
prepared for state testing…and life.
NEW!
Common Core Standard Supplements
-Makes All CORD textbooks Common Core compliant
- Available on-line and new book reprints!
Takes math from
abstract concepts to concrete applications to which students can better
relate.
Workplace Applications,
real-world examples, labs and activities fit perfectly with the Common
Core Standards mission statement of: " The standards are designed
to be robust and relevant to the real world, reflecting the knowledge
and skills that our young people need for success in college and careers."
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More About
This Textbook
Overview
There is a gap between the mathematics taught to engineering students and the mathematics that engineers use in practice. The mathematics presented to undergraduates is often already in modelled form, there is always enough data, the solution is known, there is a preferred technique, and the students are expected to work by themselves. In practice the engineer will work on an ill-formulated problem and be expected to produce the model, will never have enough data, will never be perfectly sure if the solution is correct, and will be part of a team. Written by internationally renowned engineering scientists and researchers, this volume in the NATO Special Programme on Advanced Educational Technology addresses this gap by discussing the design of modelling courses. Fluid flow and heat transfer serve as
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Roads to Geometry
9780130413963
0130413968
Summary: Clarifying, extending and unifying concepts discussed in basic high school geometry courses, this text gives readers a comprehensive introduction to plane geometry.
Wallace, Edward is the author of Roads to Geometry, published 2003 under ISBN 9780130413963 and 0130413968. Four hundred sixty five Roads to Geometry textbooks are available for sale on ValoreBooks.com, one hundred eighteen used from the cheapest... price of $28.88, or buy new starting at $78
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eBooks
George Thomas' clear, precise calculus text with superior applications defined the modern-day, three-semester or four-quarter calculus course. The ninth edition of this proven text has been carefully revised to give students the solid base of material they will need to succeed in math, science, and engineering programs. This edition includes recent innovations in teaching and learning that involve technology, projects, and group work.
Practice makes perfect—and helps deepen your understanding of
calculus
1001 Calculus Practice Problems For Dummies takes you
beyond the instruction and guidance offered in Calculus For
Dummies, giving you 1001 opportunities to practice solving
problems from the major topics in your calculus course. Plus, an
online component provides you with a collection of calculus
problems presented in multiple-choice format to further help you
test... more...
Essential Calculus: Early Transcendental Functions responds to the growing demand for a more streamlined and faster paced text at a lower price for students. This text continues the Larson tradition by offering instructors proven pedagogical techniques and accessible content and innovative learning resources for student success.
Calculus and Its Applications, Tenth Edition, remains a best-selling text because of its accessible presentation that anticipates student needs. The writing style is ideal for today's readers, providing intuitive explanations that work with the carefully crafted artwork to help them visualize new calculus concepts. Additionally, the text's numerous and up-to-date applications from business, economics, life sciences, and social sciences help... more...
The... more...
Calculus Made Easy has long been the most popular calculus primer, and this major revision of the classic math text makes the subject at hand still more comprehensible to readers of all levels. With a new introduction, three new chapters, modernized language and methods throughout, and an appendix of challenging and enjoyable practice problems, Calculus Made Easy has been thoroughly updated for the modern reader.
Slay the calculus monster with this user-friendly guide
Calculus For Dummies, 2nd Edition makes
calculus manageable—even if you're one of the many students who
sweat at the thought of it. By breaking down differentiation and
integration into digestible concepts, this guide helps you build
a stronger foundation with a solid understanding of the big ideas
at work. This user-friendly math book leads you step-by-step
through each concept,... more...
The New Fifth Edition Of Complex Analysis For Mathematics And
Engineering Presents A Comprehensive, Student-Friendly Introduction
To Complex Analysis Concepts. Its Clear, Concise Writing Style And
Numerous Applications Make The Foundations Of The Subject Matter
Easily Accessible To Students. Believing That Mathematicians,
Engineers, And Scientists Should Be Exposed To A Careful
Presentation Of Mathematics, The Authors Devote Attention To
Important... more...
This book contains a detailed account of the bigeometric calculus,
a non-Newtonian calculus in which the power functions play the role
that the linear functions play in the classical calculus of Newton
and Leibniz. This nonlinear system provides mathematical tools for
use in science, engineering, and mathematics. It appears to have
considerable potential for use as an alternative to the classical
calculus. It may well be that the bigeometric calculus
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Mathematica Cookbook helps you master the application's core principles by walking you through real-world problems. Ideal for browsing, this book includes recipes for working with numerics, data structures, algebraic equations, calculus, and statistics. You'll also venture into exotic territory with recipes for data visualization using 2D and 3D graphic tools, image processing, and music.
Although Mathematica 7 is a highly advanced computational platform, the recipes in this book make it accessible to everyone -- whether you're working on high school algebra, simple graphs, PhD-level computation, financial analysis, or advanced engineering models.
Learn how to use Mathematica at a higher level with functional programming and pattern matching
Delve into the rich library of functions for string and structured text manipulation
Learn how to apply the tools to physics and engineering problems
Draw on Mathematica's access to physics, chemistry, and biology data
Get techniques for solving equations in computational finance
Learn how to use Mathematica for sophisticated image processing
Process music and audio as musical notes, analog waveforms, or digital sound samples
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--Mike Riley - Dr Dobbs Code Talk
The Mathematica Cookbook does a good job of showing the wide range of capabilities of the Mathematica program... --Jerry Pournelle, Chaos Manor, The User's Column, August 2010, Column 360
[Mathematica CookBook] supplies a number of very nice examples with which to extend user expertise. --John A. Wass, Ph.D., Scientific Computing
About the Author
Sal Mangano has been developing software since the days Borland Turbo C and has worked with an eclectic mix of programming languages and technologies. Sal worked on many mission-critical applications, especially in the area of financial-trading applications. In his day job, he works mostly with mainstream languages like C++ and Java so he chooses to play with more interesting technology whenever he gets a chance.
Sal's two books (XSLT Cookbook and Math Mathematica Cookbook) may seem to be an odd pair of technologies for a single author but there is a common theme that reflects his view at what makes a language powerful. Both Mathematica and XSLT rest on the idea of pattern matching and transformation. They may use these patterns in different ways and transformations to achieve different ends but they are both good at what they do and interesting to program in for a common reason. Sal's passion for these languages and ideas comes through in both these cookbooks. He also likes to push technologies as far as they can go and into every nook and cranny of application. This is reflected in the wide mix of recipes he assembled for these books.
Sal has a Master's degree in Computer Science from Polytechnic University.
More About the Author
I fell in love with science at a very young age but got hooked on computers and mathematics only much later. I have had most of my professional experience programming complex trading systems in C++ but have more of a personal passion for AI, Genetic Algorithms, pure Computer Science and advanced software development paradigms and certain areas of theoretical math (although my ability on the mathematical side is not quite on par with my passion).
My two books XSLT Cookbook and Mathematica Cookbook are about very different technologies but there is a common theme that runs through both XSLT and Mathematica - pattern matching and transformation. This is one of the most powerful paradigms in computer science.
I like the cookbook format because word for word, cookbooks are the most useful of all technical books. Cookbooks teach by example and that is how people learn. Cookbooks are about getting things done.
Both XSLT and Mathematica are sort off the beaten path type languages and that tells you a bit about me.
XSLT is a very particle skill to have if you find yourself needing the deal with XML a lot. If you manipulate XML using straight DOM programming you are really doing way to much work in many cases. Give XSLT a try.
Mathematica is probably the single most useful system there is for experimental use of a computer. If you work in the IT industry chance are slim you will ever need Mathematica skills. BUT, if you like to tinker with ideas and data, if you like to explore mathematical and scientific concepts, if you want to know how it feels to discover beauty in a few lines of code THEN you really ought to give Mathematica a try. This used to be an expensive proposition but Wolfram has a fully functional HOME Edition of Mathematica. At about $300 it is probably the single best software investment you will ever make. If you are a student, it is even less.
Most Helpful Customer Reviews
1. If you are new to Mathematica, this book is not for you. I have used Mathematica off and on for 10 years, so I know some of the basics that are omitted in this text.
2. This is for version 7. The current version is number 9. There are DEFINATELY compatibility issues in the examples. These can be overcome, but it will be frustrating if you don't have other books to refer to. If wolfram is going to change the syntax every 2-3 years, and the publishers keep selling old versions, people are going to get frustrated. Even microsoft-platform authors clearly put the version number on the cover. How much would you pay for a "new copy" of a software manual for Windows 3.1? Clearly, I should have looked more carefully when I bought it, but the publishers must know this is an issue.
3. The text says the code examples can be downloaded for free from the website. The website says you can download them for free from O'reilly. O'reilly wants $4.99. It shouldn't be this hard to get the support materials on a $50 book.
That stated, there are many nice examples in this text. It certainly beats the Rose and Smith Book (Version 4 from 2002).
This book was very, very helpful in a place where mathematica can be lacking. Example code. I was writing database code for mining data. This was going extremely poorly. This book got me going with it. I have referred back to it several times. It has been a real gem. I had the PDF for free and still bought the hard cover. Seriously, the PDF was so helpful that I bought the book. There are two other must haves for use of Mathematica:
I would read An Introduction to Programming with Mathematica first. Excellent book. If you understand 50% of the book, you will fly with Mathematica. The material is very accessible. Then read/skim the Cookbook. Then skim the Navigator so you know what they have. Seriously, good books.
I am a brand new baby in using Mathematica, and am using Mathematica 8. This book starts off throwing out code, consisting of, relative to the absolute new guy, advanced work.
This book persistently uses the "cookbook" theme, so I'll give my review with this metaphor. I feel like the kitchen that the author expects is much more than is in the kitchen of a new programmer. I fortunately have taken a course in C, otherwise, I would have been utterly lost.
The other reviews of this book encourage that novice programmers and Mathematica users use this book as an advanced supplement with another more basic book on Mathematica. I fully agree.
However, the sections on plotting, customizing plots, and mathematical applications are very approachable and accessible. This book is not intended to teach one how to use Mathematica, but to add fine-tuned finishing touches.
An extremely attractive title to even one who is an absolute dud in cooking, like me. But I was glad that it did not let me down at all. Though there are a lot of books on Mathematica arranged in several topics, this one suits an engineer like me who uses Mathematica as a programming language to solve problems in a specific domain.
If one is adequately motivated, even without much experience in Mathematica, this book can be a wonderful starting point. I would say I benefited much from the Functional Programming chapter. For instance, specifying default values for arguments to a function was not something I had known before. It is a pretty thick book so it would take a while to use up all the recipes in the book. For future editions, I would suggest to Sal Mangano to add material relating to Wolfram Workbench.
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Word Problems - 1 edition
Summary: Research by cognitive psychologists and mathematics educators has often been compartmentalized by departmental boundaries. Word Problems integrates this research to show its relevance to the debate on the reform of mathematics education. Beginning with the different knowledge structures that represent rule learning and conceptual learning, the discussion proceeds to the application of these ideas to solving word problems. This is followed by chapters on elementary, multistep, and algebra proble...show morems, which examine similarities and differences in the cognitive skills required by students as the problems become more complex. The next section, on abstracting, adapting, and representing solutions, illustrates different ways in which solutions can be transferred to related problems. The last section focuses on topics emphasized in the NCTM Standards and concludes with a chapter that evaluates some of the programs on curriculum reform. ...show less
New Book. Shipped from UK within 4 to 14 business days. Established seller since 2000.
$67.09
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This course allows you to truly explore your understanding of functions. Whether they are linear, quadratic, rational, polynomial or exponential. Depending on the course, many teachers also include trigonometry.
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GED Math Problem Solver - 2nd edition
Summary: The GED Math Problem Solverintegrates problem-solving and reasoning strategies with mathematical skills using problems encountered in everyday life. This text builds understanding of mathematical relationships by focusing on problem-solving skills, developing estimation and mental math strategies, and integrating algebra, geometry, and data analysis with arithmetic.FEATURES 25 lessons combining instruction, practice, and review Complete answer key, including solutions Cumulative...show more review and GED practice at the end of each lesson Test-taking lessons and practice Exercises using data and graphs collected in the appendix Calulator exploration using the Casiofx-260 Full-length GED Mathematics practice test0072527552 Item in good condition and ready to ship!
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Bay City Books Benicia, CA
some pages bent. front cover bent at the edge ACCEPTABLE with noted wear to cover and pages. Binding intact. May contain highlighting, inscriptions or notations. We offer a no-hassle guarantee on all ...show moreour items. Orders generally ship by the next business day. Default Text ...show less
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Helping Hands Global Kansas city, MO
2002 Paperback This book is in good condition with minimum amount of wear. Same day shipping. Thank you.
$12.44 +$3.99 s/h
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Mary Phipps FL Rockford, IL
2002-03-01 Paperback New NEW-IT IS BRAND NEW-clean text, tight binding, It is free from any foreign markings.
$1225.58
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MATH 314 - Mathematical Circles (SL) (4)
An introduction to the Eastern European Mathematical Circles culture. Students will learn mathematical folklore and problem-solving methods drawn from geometry and discrete mathematics, and will both observe and teach students in several mathematical circles in the Bay Area. In addition to the mathematics and pedagogy, students will explore issues of equity in educational opportunity. This is a service earning course designed for math, physics, or computer science majors who are interested in teaching. Prerequisite: MATH 110 or permission of instructor.
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MyWorkBook for Algebra for College Students
MyWorkBook provides extra practice exercises for every chapter of the text. MyWorkbook can be packaged with the textbook or with the MyMathLab ...Show synopsisMyWorkBook provides extra practice exercises for every chapter of the text. MyWorkbook can be packaged with the textbook or with the MyMathLab access kit and includes the following resources for every section: Key vocabulary terms, and vocabulary practice problems Guided Examples with stepped-out solutions and similar Practice Exercises, keyed to the text by Learning Objective References to textbook Examples and Section Lecture Videos for additional help Additional Exercises with ample space for students to show their work, keyed to the text by Learning ObjectiveFair. Hardcover. Workbook. All text is legible, may contain...Fair. Hardcover. Workbook. All text is legible, may contain markings, cover wear, loose/torn pages or staining and much writing. SKU: 9780321715524.
Description:Very Good. 0321715527 Great used condition. We are a tested and...Very Good. 0321715527 MyWorkBook for Algebra for College Students. This book is...Good. MyWorkBook for Algebra for College Students
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This package consists of the textbook plus an access kit for MyMathLab/MyStatLab.
Michael Sullivan's Fundamentals of Statistics, Third Edition, was written to address the everyday challenges Mike faces teaching statistics. Almost every aspect of the book was tested in his classroom to ensure that it truly helps students learn better. Mike skillfully connects statistical concepts to readers' lives, helping them to think critically, become informed consumers, and make better decisions.
If you are looking for a streamlined textbook, which will help you think statistically and become a more informed consumer through analyzing data, then Sullivan's Fundamentals of Statistics, Third Edition, is the book for you.
MyMathLab provides a wide range of homework, tutorial, and assessment tools that make it easy to manage your course online
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CAT in Mathematics
The CUNY Assessment Test in Mathematics (also known as the CAT in Mathematics, or the COMPASS Math test) is an untimed, multiple-choice, computer-based test designed to measure students' knowledge of a number of topics in mathematics. The test draws questions from four sections: numerical skills / pre-algebra,
algebra, college algebra, and trigonometry. Numerical skills / pre-algebra questions range from basic math concepts and skills (integers, fractions, and decimals) to the knowledge and skills that are required in an entry-level algebra course (absolute values, percentages, and exponents). The algebra items are questions from elementary and intermediate algebra (equations, polynomials, formula manipulations, and algebraic expressions). The college algebra section includes questions that measure skills required to perform operations with functions, exponents, matrices, and factorials. The trigonometry section addresses topics such as trigonometric functions and identities, right-triangle trigonometry, and graphs of trigonometric functions. No two tests are the same; questions are assigned randomly from the four sections, adapting to your test-taking experience.
Placement into CUNY's required basic math courses is based on results of the numerical skills/pre-algebra and algebra sections. The test covers progressively advanced topics with placement into more advanced mathematics or mathematics-related courses based on results of the last three sections of the test.
Students are permitted to use only the Microsoft Windows calculator while taking the test. CAT in Mathematics Practice Materials
Below are some sample tests and websites containing more samples and information about the CAT in Mathematics and related materials. Special software may be needed to view some of these files; check under our Software section to get them.
Calculus Boot Camp, an initiative of Interim Chancellor William Kelly, has launched at six CUNY campuses. In July and August 2014, hundreds of students across Baruch College, Brooklyn College, City College, LaGuardia Community College, Lehman College and NYC College of … Continue reading →
City Tech professor Janet Liou-Mark was presented with the Distinguished Teaching Award by the Mathematical Association of America's New York Section at their annual meeting on May 3rd. As her colleague, I can confirm that her creativity, positivity, enduring belief … Continue reading →
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Items tagged with MATHThis course is structured around the creation and completion of a real-life data analysis project that allows participants to apply knowledge and skills from other mathematical strands. Key concepts such as data collection, graphical representations of data, and measures of center are highlighted.
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Textbook
Overview
Now in its 8th edition, MATHEMATICS FOR PLUMBERS AND PIPEFITTERS delivers the essential math skills necessary in the plumbing and pipefitting professions. Starting with a thorough math review to ensure a solid foundation, the book progresses into specific on-the-job applications, such as pipe length calculations, sheet metal work, and the builder's level.Broad-based subjects like physics, volume, pressures, and capacities round out your knowledge, while a new chapter on the business of plumbing invites you to consider an exciting entrepreneurial venture. Written by a Master Plumber and experienced vocational educator, MATHEMATICS FOR PLUMBERS AND PIPEFITTERS, 8th Edition includes a multitude of real-world examples, reference tables, and formulas to help you build a rewarding career in the plumbing and pipefitting trade.
Related Subjects
Meet the Author
Lee Smith is a Registered Master Plumber and certified vocational math instructor. His many years of practical and teaching experience in the pipe trades includes domestic and industrial work in the areas of refrigeration and air conditioning, deep sea diving vehicles, and nuclear piping. In addition to several certificates in the trades, Mr. Smith has held Master Plumber credentials for forty years. He earned his A.A.in applied science and technology from Thomas A. Edison State College, and his B.A. and M.A. degrees in education from
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FKP bmfp
24 Pages
School: UTEM Chile
Section 1.4
Complex Numbers
The Imaginary Unit i
The Imaginary Unit i
The imaginary unit i is defined as
i = -1, where i 2 = 1.
Complex Numbers and Imaginary Numbers
The set of all numbers in the form
a+bi
with real numbers a and b, and i, the imaginary u
35 Pages
School: UTEM Chile
Section 1.5
Quadratic Equations
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written
in the general form
ax 2 + bx + x = 0,
where a, b, and c are real numbers, with a 0. A quadratic
equation in x is also called a
29 Pages
School: UTEM Chile
Section 1.6
Other Types of Equations
Polynomial Equations
A polynomial equation is the result of setting two
polynomials equal to each other. The equation is in
general form if one side is 0 and the polynomial on
the other side is in descending powers of
Section 6.4
Product-to-Sum and
Sum-to-Product Formulas
The Product-to-Sum Formulas
Example
Express each of the following products as a sum or difference:
a. sin 2x cos 3x
b. sin 3x sin 2x
Example
Express each of the following as a sum or difference:
a. 2
MOST POPULAR MATERIALS FROM
FKP
4 Pages
School: UTEM Chile
Section 8.3
Geometric Sequences and Series
Geometric Sequences
Example
In a geometric sequence the first term is 5 and the
second term is 10, what is the common ratio? To find
a2
the common ratio r = .
a1
Example
If in a geometric sequence the first term
Tutorial 3: Single Side Band Discuss Title:
Group 3 4 5 6 1 2
Title Single Side Band System Comparison of SSB to conventional AM SSB Generation SSB Transmitter SSB Receiver FDM
Note: 1. every group present their discussion result 2. two students of each g
15 Pages
School: UTEM Chile
Section 5.8
Applications of Trigonometric
Functions
Solving Right Triangles
Example
Find the height of the tower given that it is 150 feet
from the line of sight of the individual who is looking up
at the top of the tower at a 60 degree angle.
600
150 fee
30 Pages
School: UTEM Chile
Section 6.4
Multiplicative Inverses of Matices
and Matrix Equations
The Multiplicative Identity
Matrix
1 0
The Multiplicative Identity matrix is I=
for 2 2 matrices.
0 1
That means that AI=A and IA=A
The Multiplicative Inverse
of a Matrix
If a square ma
31 Pages
School: UTEM Chile
Section 7.3
The Parabola
Definition of a Parabola
Example
Find the vertex and the axis of symmetry of the parabola given
by y = 2( x 3) 2 4. Does it open up or down?
Standard Form of the Equation
of a Parabola
Example
Find the focus and the directrix of t
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CAT: Quantitative Aptitude Common Admission Test For Admission Into IIMs is a comprehensive resource manual for students preparing for the mathematics section of the CAT.
Summary Of The Book
Sarvesh K. Verma presents an all-inclusive book for students who aspire to get a seat in leading Indian institutes for higher studies in subjects like management. Quantum CAT: Quantitative Aptitude Common Admission Test For Admission Into IIMs includes the topics pertaining to mathematics that need to be covered for the Common Admission Test (CAT). CAT is a computer-based examination conducted by the Indian Institutes of Management (IIM) to select students for their management programs. It tests the Verbal Ability, Quantitative Ability, Logical Reasoning, and Data Interpretation skills of a candidate.
The book provides exhaustive coverage of all the aspects of the Quantitative Ability section for the examination. Verma wraps up several of the core concepts of the subject such as Probability, Theory of Equations, Alligations, Permutation and Combination, Logarithm, Time and Work, Functions and Graph, Percentages, Averages Co-ordinate Geometry, Algebra, Series and Sequence, Profit, Loss and Discount, Set Theory, and Trigonometry.
Besides providing in-depth explanations for each topic being discussed, the book also consists of numerous solved examples to ensure that the students grasp the fundamentals clearly. Each chapter comprises questions from last years' papers to help students check their knowledge. One of the salient features of this book is that it offers solved papers for several other management-related exams besides CAT. So, students preparing for IRMA, FMS, MAT, SNAP, NMAT, TISS, and JMET can benefit from this reference book.
To provide a complete understanding of all the topics of the quantitative section, the book includes over 300 core concepts, 800 plus illustrated examples, and more than 4000 solved practice problems. It also discusses innovative methods to solve numerous problems. Additionally, the author incorporates a variety of questions to check ingenuity, pace of solving problems, and comprehension of the students.
Sarvesh has taken care of the needs of students with no mathematical background and for that he has provided exclusive tips and solutions for such readers. To help one's mathematical thought process, the book provides logical and diagrammatic solutions. Also those who are repeating their examination can learn more about next generation approaches along with exclusive score boosting techniques.
Quantum CAT: Quantitative Aptitude Common Admission Test For Admission Into IIMs also contains the solved CAT papers from 2003 to 2008.
About Sarvesh K. Verma
Sarvesh Kumar Verma is a bestselling writer famously known for his reference books written for the CAT.
The other book written by him is Latest, CATest & Quickest A Practice Book for Prometric CAT 2012.
Verma's style of using analogies and illustrations to simplify complex concepts and make it easy for the readers has made his books popular among students.
Verma is a management graduate in Accounts and Finance. He has helped numerous students prepare themselves for various entrance examinations. His areas of expertise are quality innovations and re-engineering in business. He is also a consultant for startups and entrepreneurs.
The 'certified buyer' badge indicates that this user has purchased this product on flipkart.com.
A lengthy saga(Expand)ence type questions that appear on the CAT. This is particularly worrying because, there were a lot of instances where I wasn't able to understand what the question was or what the author intended to convey.
Now, let me explain the structure of the book. It starts with a detailed - yet grammatically incorrect - analysis of the CAT, a few strategies for preparation, and the criteria which had been specified by the IIM's for CAT'12. Solved papers of XAT'13, SNAP'12, IIFT'12 are also included. The book contains 21 chapters, each of which is very important for the CAT and the chapter 'Number Systems' has been included under the name 'Fundamentals'. Each chapter has a few introductory exercises, level-1 problems, level-2 problems and a 'final round', each of which are of increasing levels of difficulty. Extensive coverage has been given to all kinds of problems and detailed solutions have been provided to almost every problem at the end of the corresponding chapter.
Additionally, in the final pages of the book, previous year papers of the CAT examination (from 2003-2008) have been included. It would be wise to assume that a person who has solved the problems in this book atleast twice may score somewhere around 90%-95% in the original exam. However, this is a real challenge. Given the number of problems and solved examples, it becomes extremely difficult to solve the entire book in a short span of time. The book is too lengthy and it is tough to solve it twice or thrice within the limited time.
In conclusion, I am of the opinion that despite its shortcomings, this is a good book for beginners; repeaters might feel frustrated while solving the basics but revising them will only strengthen their fundamentals. Also, the authors should edit the book wisely and crop out all the grammatical errors and should try to make the book 'error-free'. I am giving 3 stars out of 5 for this gargantuan compilation of problems
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Great Book, you need not look any further e... edition, so you will not get precise dates for the 2013 papers.
2. Rather lengthy, thought CAT is all about practice so its not necessarily a con.
3. As the book is over 800 pages, to keep it as thin as possible, the paper quality is very thin.
I hope my review has helps people. This book is a must buy and hopefully will lead to success for many students like me who do NOT want to join coaching classes.
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Thumbs down(Expand) time and work, speed-time, pnc, probability etc are too messed up. It doesn't live up to the Arihant Publications standards.
Recent Top Reviews
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Loved it!
I had purchased this book for competitive exam study and I have to say it made many concepts very clear. The book is detailed with lots of questions that help in making the concepts firmly instilled in your brain so that you never forget them again. If you are able to get through this book then a... (View complete review)
★★★★★
★★★★★
aditi sharma
15 Aug 2014
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GOOD book for competitive exam except CAT
it helps you build concepts,provide sufficient exercise questions, illustrated examples, shortcuts are also explained in lucid manner. chapters and shortcuts on percentage, Number system are very good. Geometry is also covered in lucid manner.
Answers are correct with explaination unlike others.... (View complete review)
★★★★★
★★★★★
Subhanjan Dutta
28 Jul 2014
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Surely Excellent!
Books with very well explained topics and lots of examples and exercises which will make you ready for CAT and by completing this book from top to bottom, one has a 100% chance of getting good percentile in CAT provided they have completed other sections too. Moreover the exercises are also solve... (View complete review)
★★★★★
★★★★★
Sandesh Kumar
26 Jul 2014
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Great Book
Bought it 8 months ago and still helping me...great explanations with illustrations making it easy for everyone. (View complete review)
★★★★★
★★★★★
B
03 Jul 2014
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Good
Good Book for practice. Well suited foe scenarios when you already know the topic and need one for practice. More emphasis on tricks than methods which I think is a demerit. For practice, excellent. (View complete review)
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Summary: - By Judith A. Penna - Contains keystroke level instruction for the Texas Instruments TI-83 Plus, TI-84 Plus, and TI-89 - Teaches students how to use a graphing calculator using actual examples and exercises from the main text - Mirrors the topic order to the main text to provide a just-in-time mode of instruction - Automatically ships with each new copy of the text
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Abstract: To test the vision of Standards–based mathematics education, we conducted a comparative study of the effects of the Core-Plus Mathematics Project (CPMP) curriculum and more conventional curricula on growth of student understanding, skill, and problem-solving ability in algebra. Results indicate that the CPMP curriculum is more effective than conventional curricula in developing student ability to solve algebraic problems when those problems are presented in realistic contexts and when students are allowed to use graphing calculators. Conventional curricula are more effective than the CPMP curriculum in developing student skills in manipulation of symbolic expressions in algebra when those expressions are presented free of application context and when students are not allowed to use graphing calculators.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Academic Support Center: offers support for students to improve individual writing, reading, statistics and freshman math skills.
Center for Mathematics and Statistics SupportThe Center for Math & Statistics, C4MS2,
is a walk-in tutoring service available to all UHD students. The center is staffed with mathematics and statistics faculty and student peer tutors. Tutoring is available for mathematics courses (numbered through 1XXX - this includes Developmental Courses and classes such as MATH 1301 College Algebra and Math 1306 Applied Calculus) and for statistics courses.
Final Exam Review Information:
The UHD Algebra Student webbsite: provides additional resources and links to help you succeed in Math 0300, 1300, 1301, and other courses too.
Math Bypass Exam Review: The Department of Computer and Mathematical Sciences allows students to bypass the following courses, 0300, 1300, 1302, and 1404, provided they pass the corresponding bypass exam. The exams are given several times a year.
MS Student Organization
The Mathematical Association of America is the largest professional society that focuses on mathematics accessible at the undergraduate level. Our members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; read more
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It was written by high school and middle school teachers in the Arlington Central School District and is being used by students in our district. We had good success with it last year, as measured by the amazingly generous June Algebra Regents exam.
Still, there are about 120 lessons and homeworks available on topics ranging from classic algebra (equations, systems, quadratics, rationals) to right triangle trig, to measurement and error. Feel free to download and use these anyway you like, as review or even as a primary text as we do.
I know this has been posted before, but I thought I would give an update in case anyone had missed it before now.
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Tips For Success
Attend all classes.
There is an extremely large correlation between attendance and passing this course. Each class utilizes tools and concepts learned from previous classes. In the rare event that you must be absent from class, be sure to complete the material covered during your absence so that you will be able to accomplish the tasks asked of you when you return to class:
Obtain the notes from a peer;
Read and complete the activity from your text that was covered in class;
Outline the important information discovered while doing the activity;
Compare your outline with your peers class notes;
Complete the associated homework;
After you have reviewed the notes and worked on homework, go to the tutor lab or make an appointment with your instructor to discuss any questions you have about the material.
Actively participate during classes. Do not wait for your peers to hand you the solutions.
After completing an in-class task ask your self what the goal of the task was.
Why did the instructor ask you to do this specific task?
What did you learn from the task?
How did this build off of or extend a previously known skill or concept?
Constantly ask yourself and others "why".
Why did you think to start the problem in this way? What was your intuition?
Why is this step valid (mathematically legal)?
Why is this middle step helpful? In other words, how will this middle step help us finish the problem?
Fill out a 3x5 card with any questions about the material covered that day or a mathematical revelation on your part for the day. Your instructor will have 3x5 cards available every day and these can be filled out anonymously if you prefer.
Take notes during class. Be sure to take notes on what is said as well as what is written on the board. Keep in mind that it is not humanly possible to capture everything and pay attention, so it is extremely important to fine-tune your notes after class. If you need help developing your note taking skills, visit the Academic Skills Center. They will be more than happy to help and have a College Success Seminar devoted entirely to note taking. All College Success Seminars are free to WMU students.
Fine-tune your class notes as soon after class as possible. Do not wait more than a day. The longer you wait the less you will recall of important classroom discussions. These rewritten notes will be more organized than in class notes and will allow you to study for exams more efficiently.
Summarize the activities and discussions from class.
Indicate and explain important ideas/concepts.
Provide connections between this class and previous classes.
Phrase questions on topics unclear to you. These questions might be answered while you outline your text. If not, be sure to go to the tutor lab, your instructor, or the SI leader for assistance.
Read and outline your textbook.
Be sure to rewrite definitions in your own words and create/present examples to clarify the definitions.
Reread examples to determine what point the authors are trying to make with the examples given.
The Academic Skills Center provides a College Success Seminar devoted to reading a text, so utilize this free service for additional help.
Talk to peers to fill in each other's gaps.
Do all assigned homework.
If you struggle with any of the assigned problems, find a similar problem in the text or online homework (MathXL or WileyPlus) for extra practice.
Be sure to show all work, justify each step, and write down any formula you use.
If you use a graphing calculator write down the equation(s) you typed into the calculator.
Check your solutions. Many assigned problems will not be collected. You are responsible for determining whether your solutions are correct. This may mean discussing your homework with your peers.
After we complete a chapter, look over your notes (rewritten class notes, text notes, and comments from peers). Summarize the content of these and look for connections within the chapter and connections between the new material with previous knowledge.
Your teaching team (your instructor, SI leader and tutors in the tutor lab) is here for you. Utilize us as need be, but realize that you will need to be able to fly solo during exam time and knowledge of these skills and concepts will be expected in your next class.
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The Biological World
The Biological World provides opportunities to apply mathematics to human physiology and health, botany, zoology, and ecology.
Solving problems from human physiology and health often requires collecting and analyzing data, using statistical concepts, making measurements, and applying ratio and proportion and percents in the analysis of data.
What are the nutritional values of school lunches?
What kinds of growth in heights do students undergo during a school year?
Exploring questions from botany can lead to applying mathematical topics from geometry, probability, measurement, and statistics.
What are characteristics of leaves that grow on the outer parts of an oak tree versus those that grow on the inner parts of the tree?
What can genetics reveal about the colors of the kernels of the offspring of Indian corn?
Zoology presents opportunities to make measurements; to use proportional reasoning; to collect, organize, and interpret data; and to develop hypotheses or theories.
What can be learned about animals from the imprints their feet make on a soft surface?
What is the nutritional value of a locust?
What are the differences between Africanized or "killer" bees and European honeybees?
Ecology has become a very important science in environmental studies, and mathematical topics are essential in drawing conclusions from those studies.
How are matrices and matrix operations used to explore the effects on the animal food chain of an ecosystem in which a wildfire has destroyed the grasses?
What can mathematical models predict about the reproduction rates of animals in an ecosystem?
How can mathematical chaos help to explain the difficulties of reintroducing an endangered species into the wild?
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Mathematical Induction
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Document Description
Mathematical induction is a method of mathematical proof typically used to
establish that a given statement is true of all natural numbers (non-negative integers). It is done by proving that the rst statement in the innite sequence of statements is true, and then proving that if any one statement in
the infinite sequence of statements is true, then so is the next one.
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Mathematical induction is a method of mathematical proof typically used to
establish that a given statement is true for all natural numbers (positive integers).
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In mathematics, the concept linear algebra can be considered as a part of algebra.
Linear function is a mathematical concept which deals with two different concepts
that is first degree of polynomial ...
Slope Of The Tangent Line
In today session we discuss the tangent line approximation. Before going on the main topic
we discuss about slope of line or tangent slope. The trigonometrically tangent of ...
Slope Of The Tangent Line
In today session we discuss the tangent line approximation. Before going on the main topic
we discuss about slope of line or tangent slope.
The trigonometrically tangent of ...
A function is a relation between a set of inputs and a set of permissible outputs
with the property that each input is related to exactly one output. An example is
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ntroduction to English for 8th Grade In this segment of English lesson for 8th grade Math students you will
learn about interrogative sentences and its usage Lesson for 8th Grade - Interrogative .. ...
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Plane Geometry
A Beka Book's Plane Geometry, a one-year course, uses traditional methods, including formal proofs. The emphasis is on logical, systematic thinking skills. The course is primarily theoretical rather than practical. (An example of a more practical course would be Discovering Geometry.) Order the student text and the solution key, which demonstrates solutions to problems. A Student Test Booklet and Student Quiz Booklet are also available. This text is equivalent to the typical high school geometry course.
A Beka Book
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Glencoe Algebra 2 is a key program in our vertically aligned high school mathematics series developed to help all students achieve a better understanding of mathematics and improve their mathematics scores on today's high-stakes assessments
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Course Syllabus
Overview
By the end of this course, participants will:
represent data with plots on the real number line.
use statistics appropriate to the shape of the data distribution to compare center.
interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of extreme data points.
investigate the learning progression for statistics.
view and create histograms.
calculate standard deviation.
examine the normal distribution and its properties.
explore limits of sequences by using graphs of discrete functions.
summarize categorical data for two categories in two-way frequency tables.
interpret relative frequencies in the content of the data (including joint, marginal, and conditional relative frequencies).
recognize possible associations and trends in the data.
represent data on two quantitative variables on a scatter plot and describe how the variables are related.
fit a linear function to given data.
use functions fitted to data to solve problems in the context of the data.
interpret the slope (rate of change) and y-intercept (constant term) of a linear model in the context of the data.
compute (using technology) and interpret the correlation coefficient of a linear fit.
represent data on two quantitative variables on a scatter plot and describe how the variables are related.
fit an exponential function to the data.
use exponential functions fitted to data to solve problems in the context of the data.
showcase and discuss student work from selected course activities.
create a student assessment for Math I Statistics.
Course Organization
This course includes several different activity components, all of which are described below. During each session, you will participate in a unique collection of these activity components, depending on the particular focus of that session.
Read
When you see this icon you will be reading relevant articles, resources, and instructional materials that will help inform your online course development process.
Activities
When you see this icon you will be completing activity-based curriculum and inputting various components of your course content into course project.
Discuss
When you see this icon you will be using the online discussion board to share ideas, resources, and thoughtful conversation with your fellow course participants and facilitator.
Prerequisites
This course is designed for High School Math 1 teachers.
High School Mathematics I Course Description (graphic)
The fundamental purpose of Mathematics I is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, organized into units, deepen and extend understanding of linear relationships, in part by contrasting them with exponential phenomena, and in part by applying linear models to data that exhibit a linear trend. Mathematics 1 uses properties and theorems involving congruent figures to deepen and extend understanding of geometric knowledge from prior grades. The final unit in the course ties together the algebraic and geometric ideas studied. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.
Although not a requirement, high speed Internet access definitely enhances the online experience. Participants should be proficient with using email, browsing the Internet and navigating through computer files. Access to Microsoft Office is recommended. Participants who do not have Microsoft Office should click here for some available options.
Prior to Session One course participants at to secure:
1. a WVDE WebTop account.
2. a SAS Curriculum Pathways teacher account.
3. the SAS Curriculum Pathways school account for students.
4. graphing calculators, TI-84 or TI-Inspire, for student and teacher use. Online graphing calculator information is provided within the Session One readings/resource section.
5. a scanner for scanning course assignments to be uploaded to the course drop box.
Format and Requirements
This workshop is divided into six one-week sessions which each include readings, activities, and an online discussion among workshop participants. The time necessary to complete each session is estimated to be seven to eight hours.
The outline for the workshop is as follows:
Session One
Measures of Central Tendency and Graphic Representations
Session Two
Histograms, Standard Deviation and the Normal Distribution
Session Three
Categorical Data
Session Four
Fitting Data with Linear Functions
Session Five
Fitting Data with Exponential Functions
Session Six
Connect, Reflect and Assess
Session One: Measures of Central Tendency and Graphic Representations
Students will have studied measures of center and graphic representations in middle school. This session reviews the content that is included in the middle school, as well as extends the material by using these measures and graphic representations to further describe and understand the data in real-life applications. Course participants will investigate the statistics standards and explore their importance.
Session Two: Histograms, Standard Deviation and the Normal Distribution
Participants will use and create histograms based on real world data. Participants will apply sequences and series in an activity with standard deviation. Participants will also examine normally distributed data, calculate standard deviation, and answer questions about the data based on their calculations.
Session Three: Categorical Data
Participants will summarize, represent, and interpret data for two categorical variables. They will read about quantitative versus numerical data and two-way tables before completing activities organizing and analyzing categorical data. Participants will use the proper vocabulary related to two-way tables.
Session Four: Fitting Data with Linear Functions
High school Math I students should be very comfortable with linear functions and will extend this knowledge to use of scatterplots to find relationships between quantities. Participants will use linear regression to fit a function to given data, and interpret attributes of a graph based on the context of the situation. They will become familiar with the usefulness of the correlation coefficient and residuals to analyze the goodness of fit for a regression equation.
Session Five: Fitting Data with Exponential Functions
Students in Math I will focus on functions that are both linear and exponential. Participants will examine multiple representations of exponential functions and fit exponential functions to real world data. The readings and activities highlight various applications that can be modeled with exponential functions.
Session Six: Connect, Reflect and Assess
Participants will make connections among the different Math I credential courses and sessions within those courses with the development of a Math 1 portfolio and comprehensive student assessment/solution key with work.
Assessment
Workshop participants will complete an orientation and then six weekly workshop sessions that include readings, activities, and online discussions. There will be a pre-workshop survey and a post-workshop survey. In addition, participants will incrementally develop a Math I portfolio components throughout the duration of the course. The time for completing each session is estimated to be six to seven hours weekly.
Math 1 Portfolio:
Course participants will choose three activities to use in their classroom from those suggested throughout this Math I Functions course. The activities must be from three different sessions throughout Math I Functions. Portfolio items are to be selected from the following course activities:
Complete the following for each activity to be submitted to your portfolio:
Submit two samples of your students' work that show different levels of understanding.
Explain why you selected each sample of student work and describe the level of understanding demonstrated by each student.
Discuss how this particular lesson or activity has helped develop your students' understanding of the mathematics involved.
A total of six student samples are to be included your portfolio for Math I Functions.
Culminating Assessment:
Course participants will create a Math 1 Functions culminating assessment for students along with a solution key, complete with all work. This assessment may be used with Math I students to evaluate their conceptual understanding of the Next Generation CSO's that were targeted in this course. Categories and indicators from the Student Assessment Rubric are to be used to guide the development of the Math 1 Functions culminating assessment for students.
Discussion Participation
Participants will be evaluated on the frequency and quality of their participation in the discussion forum. Participants are required to post a minimum of one substantial original posting each session reflecting on the question for that session. They are to read all original postings made by the other workshop participants and reply thoughtfully to at least two of them per session. Participants' original postings will be evaluated on their relevance, demonstrated understanding of workshop concepts, examples cited, and overall quality. Postings that respond to other participants will be evaluated on relevance, degree to which they extend the discussion, and tone.
Certificate of Completion
Upon successful completion of this course, Math 1 Functions, and the successful panel portfolio review participants will receive a Certificate of Completion documenting successful completion of the course requirements.
Non-Degree Graduate Credit Information
Participants in this course are eligible to receive non-degree graduate credits from either West Virginia University, Marshall University or Concord University. Credits will be awarded at the end of the semester in which the course occurs. Additional information is available on the course News/Welcome Page.
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Basic Math and Pre-Algebra - 2nd edition
Delivers the appropriate amount of expert coverage on basic math topics for consumers who need a supplement to a beginning basic math text or class; who need to prepare for exams; or who need to brush up on the fundamentals of basic math
Chapter Check-In gives readers an overview of what they'll learn in the chapter
Chapter Check-Out reviews the chapter to enforce the items learned and help with comprehensio...show moren
Review section is a summary test on all chapter topics in the book -- great tool for teachers and students
Resource Center directs reader to additional information available for the subject such as books and websites
500 practice questions available online at CliffsNotes.com and directly ties to each chapter in the bookVitalbooks PA Southampton, PA
PAPERBACK Very Good 0470880406
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Intelligent Tutor Math Educational Software, Inc.
The programs in the Intelligent Tutor software series contain tutorials designed to help students learn and master Grades 7-12 math subjects. The programs are designed with school versions for use in schools, and home versions for individuals in self-study
...more>>
IXL Learning
IXL provides free online practice, organized by grade levels K-8 and aligned to state standards. Choose one math skill from among the hundreds offered, and submit your answer. Respond incorrectly, and click on the "explanation" button to learn more; get
...more>>
James Franklin
Research activities, books, articles, opinions, and more from the author of What Science Knows and The Science of Conjecture: Evidence and Probability Before Pascal. Franklin researches and teaches about the philosophy of mathematics in the School of
...more>>
James Propp
James Propp studies tilings, games, and other aspects of combinatorics, probability, and dynamical systems. Many of his articles are available for download in PostScript and gzipped PostScript formats. Code for C programs related to tilings and cellular
...more>>
Java at Xanadu - Jim Carlson, University of Utah
Applets for calculus and probability/statistics. There are calculus applets for experimenting with graphs of cubic polynomials, the arc length of graph of a cubic polynomial, and the area under the graph of a cubic polynomial. Probability and statistics
...more>>
Jim Loy's Mathematics Page - Jim Loy
This is an informal collection of articles on many different mathematics problems. Topics include Algebra, Geometry, Calendars, Number Theory, and reviews of mathematics-related books. See also Jim Loy's Puzzle Page and Jim Loy's Logic Page.
...more>>
Jim Loy's Puzzle Page - Jim Loy
This is a group of number, geometry, and logic puzzles with solutions, including "The Missing Dollar," "The Monty Hall Trap," and comments on the Tower of Hanoi.
...more>>
John Allen Paulos
John Allen Paulos is a mathematician and author of the books Mathematics and Humor, I Think, Therefore I Laugh, Innumeracy - Mathematical Illiteracy and Its
Consequences, Beyond Numeracy - Ruminations of a Numbers Man, A Mathematician Reads the Newspaper,and
...more>>
Johnnie's Math Page - Johnnie Wilson
A directory of interactive math activities and tools for elementary and middle school children. The site is organized and presented so that children can easily find engaging and worthwhile math activities. The directory is organized by content: number,
...more>>
Joseph Mazur
Books by the professor emeritus of mathematics at Marlboro College: Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Powers; Euclid in the Rainforest: Discovering Universal Truth in Mathematics; The Motion Paradox: The 2,500-Year
...more>>
Journal Homepages - Kluwer Academic Publishers
A search page for finding journals published by Kluwer Academic Publishers. Each includes past, present, and future article listings, aims and scope, and submission and subscription information. Search the complete catalogue by keyword or view journals
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Knotted Walks - Ivars Peterson (MathTrek)
Given that it normally takes some effort to create a knot, the spontaneous formation of knots in ropes and strings can appear rather puzzling. Having no obvious explanation of the effect, frustrated users can't help but acknowledge this knotting phenomenon
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Lavarand - LavaLite
Lavarand is a system by which a pseudo-random number generator is seeded by the digital output of a photograph of six Lava Lite® lamps.
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Learning Math - Annenberg Media
The Annenberg Foundation and the Corporation for Public Broadcasting (CPB) use media and telecommunications to advance excellent teaching in American schools. Video programs with coordinated Web and print materials are offered for the professional development
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Let 'Em Roll™ Simulation - Matthew Carpenter
The goal of this activity is to demonstrate a real world situation where the probabilities of mutually exclusive and independent events occur. Each student is asked to calculate the experimental and theoretical probabilities of these events. Students
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Let's Do Math! - Cynthia Lanius
Maria and Cassie love doing math. The first problem they solve together is counting all the squares on a checkerboard - not just the individual squares, but all the different-sized squares that could be counted. Other Do Math problems include probability
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Let's Play Math! - Denise Gaskins
"As a homeschool mom who loves math, I want to help other homeschoolers see the variety and richness of the subject.... I hope this blog will be a place where we can play around with ideas about learning, teaching, and understanding math. (For me, it
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Library Video Company
Library Video Company offers a searchable, browseable catalog of video and CD-ROM products for mathematics instruction on the K-12 level, from addition to variables. This resource for primary and high school math educators looking for curriculum enhancers
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LiveMath - Theorist Interactive, LLC
Web-Shareable computer algebra and graphing software (formerly MathView/Expressionist / Theorist / MathPlus). LiveMath notebooks may be shared via the Web. They are similar to spreadsheets in that a change in one value will ripple throughout the calculations.
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Losing to Win - Ivars Peterson (MathTrek)
Researchers have demonstrated that two games of chance, each guaranteed to give a player a predominance of losses in the long term, can add up to a winning outcome if the player alternates randomly between the two games. This striking new result in game
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Lottery Mania - Keith Devlin (Devlin's Angle)
The math behind the odds of winning the Big Game, a lottery open to 7 states (Massachusetts, Maryland, Georgia, Illinois, Michigan, New Jersey, Virginia,), and how to understand it ("the psychology of lotteries has a logic all of its
own").
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MAA Bookstore - Mathematical Association of America
A searchable list of books, with descriptions, in the following categories: Algebra; Analysis; Applied Mathematics; Calculus; Career Information; Computing and Computers; Elementary Models; Games, Puzzles, and Popular Exposition; Geometry and Topology;
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More About
This Textbook
Overview
Occupations like those of machinists, tool and die makers, pattern makers, drafters, and designers require a fundamental knowledge of general math as well as of more advanced topics like oblique trigonometry, compound angles, and numerical control. This updated edition of Mathematics for Machine Technology promotes an understanding of all the mathematical concepts necessary for success in the machine trades and manufacturing fields. The author effectively combines math concepts with the relevant machine applications so that readers fully understand the value of what they are learning. Industry-specific examples, realistic illustrations, and actual applications further enhance the theory-to-application connection
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MATLAB is an interactive system for numerical computation that is widely used for teaching and research in industry and academia. It provides a modern programming language and problem solving environment, with powerful data structures, customizable graphics, and easy-to-use editing and debugging tools. This second edition of MATLAB Guide completely revises and updates the best-selling first edition and is more than 30% longer. The book remains a lively, concise introduction to the most popular and important features of MATLAB 7 and the Symbolic Math Toolbox.
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Offering
8 subjects
including calculus
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Sharing great ideas and resources with maths teachers around the world
Main menu
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GCSE exam questions by topic- bland.in
A colleague of mine recently stumbled across the website of Peter Bland, a maths tutor. It contains some excellent GCSE revision resources in the form on booklets of exam questions on particular topic. They are all available form his website here, but for your convenience I have also linked to his hosted resources by topic below:
Please don't bother buying the answers. It costs £25 and I found 6 errors in the first 4 topics I tried. Actually knocked my daughters confidence as she couldn't work out why she was going wrong. Have emailed pointing out errors and requesting refund but he hasn't replied.
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These authors understand what it takes to be successful in mathematics, the skills that students bring to this course, and the way that technology can be used to enhance learning without sacrificing math skills. As a result, they have created a textbook with an overall learning system involving preparation, practice, and review to help students get the most out of the time they put into studying. In sum, Sullivan and Sullivan's College Algebra: Enhanced with Graphing Utilities gives students a model for success in mathematicsWhiteboard Compatible (WBC)
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Foundations of Mathematics
Publisher:
John Wiley
Number of Pages:
392
Price:
160.95
ISBN:
9780470085011
Between graduate school, full-time teaching positions at three colleges, and a brief temporary gig as an adjunct, I have taught 86 different courses, with #87 and 88 on the fall schedule. I have never taught anything along the lines of "Transition to Advanced Mathematics," the kind of course that is supposed to bridge the apparent gap between computational and theoretical mathematics classes. I'm not inherently opposed to the idea, I just have never been at a college where such a course is offered — my two tenure-track appointments have been at colleges where discrete mathematics fills that role.
Nonetheless, I am certainly aware of such courses, for which this book would be a fine choice as a textbook. My primary criterion concerns the far side of the transition: how well does a "transitions math" book give a good sense of where a math major taking the class is headed, in the form of a meaningful introduction to at least some of the ideas of analysis, algebra, and other advanced topics? Most of the competing books take on these important topics, but it's easy to get caught up in the standard material on sets, logic, functions, and relations to an extent that the student doesn't get a meaningful glimpse of what lies ahead.
On this score, Sibley has succeeded. The chapters on abstract algebra and real analysis provide a good overview of these important subjects and connect nicely to the preliminary material. A final chapter on the philosophy of mathematics ties the rest of the book together and provides some interesting questions to challenge the prospective mathematics major. Sibley's book succeeds on two fronts: as a capstone to lower-division mathematics and as an introduction to the upper-division.
Mark Bollman (mbollman@albion.edu) is associate professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.
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Contents
The design and development of space systems is a part of engineering in general, which in turn relies on the knowledge base of mathematics and the sciences. Therefore we will start the first part of this book with a review of the fundamentals of these fields. We will pay the most attention to the parts that apply to space systems, but that is by no means the whole of those fields of knowledge. A non-technical reader can get a general idea of the concepts and projects presented in this book, but it is mainly aimed at people who want a deeper understanding of, or to actually work with future space projects. To do that, a proper foundation in mathematics and the sciences is needed at a secondary education (high school) graduate/first year university science or engineering level. If you do not have such a background, there are open source textbooks available online, such as those from the CK-12 Foundation, as well as video lectures from the Khan Academy, and traditional books and classes.
Our discussion in this book is thus at an introductory engineering level. It is not a complete survey on any topic. In many cases there is simply too much detail to fit it all. In others the technical level is too advanced, and, in the case of some future methods, the ideas have not been fully developed yet. Other books, articles, and materials are linked throughout the book, especially in the References section, and also in our online Library. Readers are encouraged to delve deeper into any topics that interest them. The next few sections will give a more detailed summary of the background of mathematics and science, and how it relates to engineering and key design principles for space systems.
The importance of mathematics to science and engineering can be summarized in one sentence:
Our Universe appears to follow mathematical laws.
By that we mean mathematical formulas and calculations produce results which match what we see when we look at the real world. This is a very powerful circumstance, because we can do the calculations before we look, even before something exists, and thus predict the future. Why mathematics works so well in describing reality is a philosophical question to which we don't have a good answer. This was pointed out by Eugene Wigner in 1960 in an article entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences (which is also discussed in a Wikipedia article). Regardless of why, it does work in practice. That allows us, among other things, to design systems that will work as intended.
The correspondence of mathematical predictions to the real world is not just a general one. In many cases it can be astoundingly exact. One of the earliest examples of prediction is the motion of the Sun, Moon, and planets in the sky. Even in ancient times people were able to predict where they would be in the future. Those predictions allowed knowing useful things, like when to plant crops, because of the linkage of the Earth's motion around the Sun to the seasons. Nowadays we can predict the motion of objects in the Solar System to fractional parts per million accuracy. An example of this accuracy was the 2012 landing of the Curiosity rover on Mars within 2 km of the intended location, after a trip of 566 million km. This could not have been done without predicting both the spacecraft trajectory and the future location of the landing point on a moving and rotating planet to 4 parts per billion. Further examples of using mathematics in design are all around you. Every tall building and bridge relies on the simple mathematical relationship that the strength be greater than the sum of all the loads. When you design such structures you calculate the strength, and calculate the loads, and then make sure the first is greater than the second. Proof that this method works is that tall buildings and bridges rarely fall down.
Like other engineering fields, space systems engineering relies on using such formulas and calculations. They are derived either from the sciences or practical experience and measurements within engineering. We present many of these formulas and calculations in this book. Therefore as a minimum you should understand the following mathematics topics:
Algebra - How to manipulate algebraic formulas and how to obtain a numeric answer given input values, the relationship of formulas and functions to graphs, and exponents and polynomials.
Geometry - The types of geometric shapes and angles, and how the dimensions of two and three dimensional shapes (perimeter, area, and volume) are calculated.
More advanced topics, such as Mathematical Analysis, Calculus and beyond are helpful in understanding how the formulas are derived, or in solving the more complex problems in engineering. They are mostly not needed for an introductory level book such as this one.
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. The predictions are often embodied in the results of mathematical formulas and calculations which relate to the real world. It is pursued partly on the basis that knowledge about the Universe satisfies human interest and curiosity. It also often turns out that the knowledge is useful in some practical way. We do not know in advance what knowledge will turn out to be useful, so scientists as a group study everything. Knowledge is a seamless whole, but from its history, and for the purposes of teaching and study, it is conventionally divided into branches according to the object of study. The ones most relevant to space systems include:
Physics - This is the study of the forces and interactions of matter and energy, the results of those interactions, and the fundamental laws and components of which things are made.
Astronomy - This is the study of objects and phenomena outside the Earth's atmosphere. This is the same location as all space systems operate in, so is highly relevant. Planetary Science in particular studies condensed objects orbiting stars.
Chemistry - This is the study of matter at the atomic, molecular, and larger scales as far as how they react and their physical properties.
Other fields besides these three will also prove useful depending on the type of project. At least a basic understanding of these areas of science is needed to work with space systems, since engineering of those systems is derived from that knowledge. Beyond the branches of science, you should have an understanding of the Scientific Method, by which ideas are generated, experiments and observations are made to test those ideas, and thus they are validated or rejected. Peer review, statistics, and repeatability are among the methods used to ensure observations and conclusions are reliable. Absolute truth is never reached in science, merely increasing confidence in a given explanation, which is known as a Theory. Sufficiently well tested ideas join the body of knowledge considered settled, but they are always subject to revision and new ideas are constantly proposed.
Mathematics and science are developed for their own sake and for their ability to predict the future. Engineering then applies accumulated knowledge from the sciences and from experience towards useful ends by designing, building, and operating systems to perform intended functions. When the systems are complex, a method called Systems Engineering is used across an entire project to organize and optimize the resulting design. This method can coordinate the work of thousands of people. Systems Engineering is described in more detail later in Part 1.
The total of accumulated engineering knowledge is too vast for any single person to know more than a small part of it. Therefore engineering in general is divided into major fields of specialization, each of which has it's own training path. It starts with a common basis in science and mathematics, then concentrates on particular areas of application, such as Mining, Chemical, Mechanical, and Electrical Engineering. Working engineers often further specialize their study and experience. They, or the organizations they work for, are called on as needed for each project. This is more efficient than keeping full time staff for every possible subject area. The specialists who are called on also have more experience in their area from having worked on many similar projects. Since the teams working on a project are not permanent, how you manage their interaction then becomes important. Project organization is also covered later in this part of the book.
Aerospace Engineering is the specialty field within which space systems fall. Space systems are projects which happen to operate in the space environment in the same way that ships and airplanes happen to operate in the marine or atmospheric environment. Although the particular environment imposes some differences in how things are designed, they all rely on the same base of knowledge in subjects like mechanics, materials science, and thermodynamics. Therefore a complex project will use engineers from many of the specialty areas such as Mechanical, Chemical, and Electrical engineering, as well as Aerospace Engineers specialized in the methods and environments that apply to space. We will identify the other specialties later in Part 1 of this book, but will concentrate on the methods that apply to space. There are many existing books and articles about the other specialties for those who are interested.
Through training and experience, engineers develop a sense of what will work or not, and how to optimize a design. Partly this is through broad principles that apply in their specialty. We note a few of the more important ones that apply to space systems here. These and others will appear throughout the book and we will try to highlight them:
Earth vs Space - On Earth, transport involves friction of various kinds, and most things are moving slowly in relation to each other. Therefore energy and cost are proportional to distance, but not time. Space is a nearly frictionless medium, and things are moving at relatively high velocity with respect to each other. So difficulty and cost are more related to kinetic and potential energy, which governs the paths you follow. It also depends more on the time you start, since your destination does not stay in the same relative location, than to absolute distance.
Non-Linearity - Many of the formulas and variables related to space systems have values raised to a power or an exponential. So the difficulty of a task does not have a one-to-one relation to the magnitude of the desired goal. This is called a non-linear system. Understanding the direction and amount of the non-linearity is important, as this can greatly help or hinder a given task. One of many examples is atmospheric pressure, which decreases exponentially with altitude, thus decreasing aerodynamic drag proportionally.
Uncertainty and Margins - Although some values, like the orbit of a planet, are known quite accurately, no physical parameter is known with absolute accuracy. Anything built by humans will deviate by some amount from the ideal item embodied in the design drawings. The natural environment can fluctuate over time, and be uneven from measured averages. So all engineering designs need to account for the uncertainties in the physical data they are based on and production variations. One method to do this is to introduce Design Margin above the expected conditions that is larger than the uncertainties. How much margin to use is based on cost, experience, and the use to which the design is put. For example, a passenger airplane would generally have higher margins than a drone with no crew, even though both are aircraft.
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Fourier Analysis: An Introduction
Publisher:
Princeton University Press
Number of Pages:
320
Price:
49.95
ISBN:
0-691-11384-X
Every once in a while, I am struck by how often mathematics textbooks sound just like each other. Glance at the table of contents of your typical analysis or algebra textbook, and you can be 90% sure of seing exactly the same sequence of topics each time. There are honorable exceptions of course, and I'm always glad to see one, because they indicate that (at least some) mathematicians are still actively thinking about what should be taught, in what order, and how.
So here is the first volume in the Princeton Lectures on Analysis, entitled Fourier Analysis: an Introduction and written by Elias M. Stein and Rami Shakarchi. The series wants to serve as an integrated introduction to "the core areas in analysis." The following volumes will treat complex analysis (volume 2), measure theory, integration, and hilbert spaces (volume 3), and selected other topics (volume 4). The basic pre-requisite for the series seems to be a standard undergraduate introduction to analysis covering the basic theory of convergence, derivatives, and the Riemann integral. Some basic familiarity with the complex numbers and elementary functions (e.g., complex exponentials) is assumed. So the book is aimed at graduate students and maybe advanced undergraduates.
The new series begins with Fourier analysis because the authors feel that this subject plays a central role in modern analysis and because it has played an important historical role. It is also much more concrete than abstract measure theory or functional analysis. Finally, the authors plan to use results from volume one in the following volumes, emphasizing that analysis is a coherent whole rather than a collection of disjointed topics.
The first book covers the basic theory of Fourier series, Fourier transforms in one and more dimensions, and finite Fourier analysis. The last topic allows the authors to present, as an application, the proof of Dirichlet's theorem on primes in arithmetic progressions. The result would make a great book for independent study courses with advanced undergraduates, and, I think, would also be useful for graduate courses. It's definitely worth a look.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College, where he occasionally gets to teach some analysis.
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Geometry's hard? Geometry's boring? No way, Math Fans! Geometry doesn't have to be hard or boring. Watch as Zero attacks each Geometry subject and problem with passion and humor to keep students interest level high. Designed to make learning Geometry fun, students will enjoy the skits, fun humor along with the solid teaching style of veteran High School math teacher – Lowell Irving. The Zero the Math Hero series is an energetic and fun series that teaches students how to learn geometry concepts, recognize and apply myriad geometry terms, functions and procedures. It incorporates two important and highly effective teaching methods - problem practice and repetition both during the lesson and with the accompanying problem practice worksheets. Each problem is presented using step-by-step instruction and should be repeated frequently to help student's retention and problem solving skills. Online Teachers Guide & Resource Guide available for download directly on our website. Each guide includes definitions, subject matter revision, practice problems and student quizzes. This program teaches three important geometric concepts and an explanation of the different angle types based on their measures, using units called degrees. Students will Learn: Segments: identify segments, name segments, use correct symbol for segments. Rays: identify rays, name rays, use correct symbol for rays. Angles: identify angles, name angles, use correct symbol for angles.
This product is manufactured on demand using DVD-R recordable media. Amazon.com's standard return policy will apply.
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Hmm, well, I wouldn't bother with the Intro to Programming I or II books (ITI 1120/1). Those classes the textbook is more a companion to the class and not actually used.
Calculus (MAT 1320) you need the textbook for sure, but you can probably torrent it if you're comfortable with that. Calculus II uses the same textbook and you'll need it.
For discrete math (MAT 1348) you don't need the textbook, but I did find a torrent for it. My professor had all his notes online and the assignments didn't come from the textbook either.
Digital Systems (ITI 1100) you'll probably need the textbook as they assign homework out of it that you'll do as assignments. I found a torrent for that one as well.
Generally, I would say hold off on buying the textbook until you go to the class. The professor always outlines what's happening in the course the first day and will most likely tell you if you'll need the textbook or not. And when you do go to buy the textbooks, if you can't get them used from an older student and need to buy them new, if you're gonna go to the campus bookstore, get there early, like when it opens. It gets packed quick and the line stretches throughout the store all day. For the first couple weeks of every semester
Hmm, I really can't think for most of my courses. For first semester, there's Calculus (MAT 1320), which the entire class came out of the textbook. There's Intro to Programming I (ITI 1120), which is basic java syntax, if you look online and find somewhere to learn Java, you'll do fine. If you know programming already and just not Java, even better because then all you're learning is syntax. There's intro to linear algebra (MAT 1341), which is another math class. And then second semester there's intro to programming II (ITI 1121), which I actually have my teacher's website for:
On that page, there's all his lecture notes, sorted by week. But I'd at least download and save them all soon, because he may take them down to refresh his site for next year or whatever. That's why I can't give you anything else really, a lot of teachers have taken their notes off their sites now. There's discrete mathematics (MAT 1348) and digital systems (ITI 1100), those notes were taken down. There's calculus II (MAT 1322), another class right out of the textbook. And I took a second year course, probability and statistics for engineers (MAT 2377), another course where the notes were taken down.
Sorry I really can't help that much! I mean, I can answer questions about the classes if you'd like? I just don't have any specific notes or anything for them.
Computer Science here, I'm only going into second year though. First year all we used was Java, in Intro to Computing I & II. And you start bare basics Java syntax first semester, no prior knowledge required. Second semester you get into more general programming concepts, still using Java, but I've heard it's a bit of a jump for people who aren't ready for it.
Haha, I'm living in Toronto for the summer, but I'm visiting Ottawa at the end of June/beginning of July. The show is July 2nd in Toronto, while I'm in Ottawa, then it's in Ottawa on July 3rd, when I get back to Toronto o.O Just bad luck, I suppose! :P
I play League of Legends every day, so I when I changed all my passwords for Heartbleed, I made a bunch of different ones for all my accounts with important personal information. Then, I would change my League of Legends password to those, so every day when I log on, I have to use it, I'll probably have to check it for the first while. But after a couple weeks of using it every day, I've got it memorized and I change my League password to the next one I need to memorize.
Edit: But I do have them written down, just to make sure I can remember them, then once I've got it memorized, I scrap that.
I think I would use the opportunity to motivate myself to do something new. If you don't read much, start reading, maybe you can start working out. Get out of your routine. I find that by doing something new, it doesn't really give my mind a chance to linger on the break up. If I'm doing something I always use to do when we were dating, it's easier for my mind to wander back to them.
Personally, I use breakups as motivation to better myself. In the past that's meant working out, or just going for runs. Or perhaps investing my time in learning something new that's going to help me in the future. For me, I'm going into Computer Science and programming, so I start looking into a new programming language or library.
Through high school I was on the cross country and nordic ski teams, but I never did really well beyond "zone" rounds. In the 9th grade I was the fastest in both and by 12th grade, as my closest friends and most competitive teammates kept training in the off-season, I was the slowest. Now I've just finished my first year of university and though I'm a little late, I want to get back into it and accomplish something they never did. Put myself back on top ;) So I'm training for a half-marathon in September and a marathon next May.
Can't say much for MAT 1339, since I had taken it in high school, but I can say so long as you're keeping up, you'll do fine. Do the homework, study for tests, like any other class. And as for french, I don't speak any at all either and it's no big deal. Anybody who knows french knows english as well, and if you ever come up to people speaking in french, just ask and they'll almost always be fine to switch to english for you. Everything on campus is in both english and french, no worries there.
I just finished my first year living in Stanton. I applied for a double room because I knew I was the type to not really be able to make friends that well, so I figured being forced to live with someone would give me a better shot at being their friend. It did work out, him and I are going to be living together probably for the rest of our time at uOttawa.
Like /u/IAmKhaleesi said, you risk having a roommate that you don't like very much, but I don't think that happens too much. On our floor, there was one girl who didn't like her roommate (or anyone else on the floor, for that matter), so she ended up getting a room change with one of the other girls on the floor. The girl who didn't like her roommate moved into other girl's single room and then the other girl moved into her double. These things are usually pretty easy to work out if you don't like your roommate and can find somebody willing to switch with you.
The first day I showed up to my room, my roommate had already moved in and had done what /u/IAmKhaleesi said and moved the wardrobe you get into the middle of the room, so that it was split down the middle and we had quite a bit of privacy compared to most other people. It wasn't because we already didn't like each other or anything, it was just for the privacy and it worked out really well. If we had wanted to, we could have ignored each other all year. If we wanted to talk, we just went over to the other's side. So it is pretty effective if you want to get away from a roommate as well.
As for a single room, if you're a social person, you'll be fine. Frosh week (or 101 week) is a great opportunity to just go up to people and start talking, especially on the first night. Chances are a lot of the friends you'll make live somewhere in one of the residences on campus. You can make friends with the people on your floor by just spending time in the common room. If you're in there working, people will come and go and you can talk to them. I never really made any good friends on my floor, I'm too shy a person to really start a conversation, but I could make small talk with any of them, and everybody was pretty kind. If I had tried and made friends, I definitely could have just by spending time in the common room early on in the year.
So yeah, it's really up to you, think about if you're a social enough person to make friends on your own, if not, you can be friends with a roommate and team up to go try and talk to people. Having a roommate is pretty great, as long as you can communicate any issues, like bringing girls/guys over and such.
Yeah, if you go with this schedule again, you'll be fine because the walk from one end of campus to the middle is manageable, I did it quite often between classes. If you were happy with this schedule before, then distance shouldn't change that, since a walk halfway across campus is kind of unavoidable.
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on teaching and learning proof and proving has expanded in recent decades. This reflects the growth of mathematics education research in general, but also an increased emphasis on proof in mathematics education. This development is a welcome one for those interested in the topic, but also poses a challenge, especially to teachers and new scholars. It has become more and more difficult to get an overview of the field and to identify the key concepts used in research on proof and proving. This book is intended to help teachers, researchers and graduate students to overcome the difficulty of getting an overview of research on proof and proving. It reviews the key findings and concepts in research on proof and proving, and embeds them in a contextual frame that allows the reader to make sense of the sometimes contradictory statements found in the literature. It also provides examples from current research that explore how larger patterns in reasoning and argumentation provide insight into teaching and learning.
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Preview The MathResource, a comprehensive interactive mathematical learning software based on The Collins Dictionary of Mathematics and powered by Maple, but requiring no programming by the user. Also preview Let's Do Math: Tools and Things; Let's Do Math: Graphing and Calculating; and The MRI Graphing Calculator Software, which provides graphic and calculating functionality for the Casio Cassiopeia A-21S (Student) and A-22T (Teacher) versions of the Computer Extender plus the E-100 Palm-Size PC.
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Elementary Algebra for College Students (8th Edition)
9780321620934
ISBN:
0321620933
Edition: 8 Pub Date: 2010 Publisher: Prentice Hall
Summary: Angel, Allen R. is the author of Elementary Algebra for College Students (8th Edition), published 2010 under ISBN 9780321620934 and 0321620933. Four hundred eighty six Elementary Algebra for College Students (8th Edition) textbooks are available for sale on ValoreBooks.com, two hundred eighty used from the cheapest price of $7.61, or buy new starting at $67Woodinville, WAShipping:Standard, ExpeditedComments:Brand new book. STUDENT US EDITION. Never used. Only Two Very Minor Marks On Bottom Side of the B... [more]Brand new book. STUDENT US EDITION. Never used. Only Two Very Minor Marks On Bottom Side of the Book. Nice gift. Best buy. Shipped promptly and packaged carefullythe class that required me to use this book was math 101 at Rockland community college. the class was very effective especially with the professor who taught us each topic. it was a very cooperative class.
there is nothing i would change about this book. it offered problems to do and even showed exactly how to do them with examples provided.
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More About
This Textbook
Overview
Computer algebra systems have revolutionized the use of computers within mathematics research, and are currently extending that revolution to the undergraduate mathematics curriculum. But the power of such systems goes beyond simple algebraic or numerical manipulation. In this practical resource Roman Maeder shows how computer-aided mathematics has reached a level where it can support effectively many of the computations in science and engineering. Besides treating traditional computer science topics, he demonstrates how scientists and engineers can use these computer-based tools to do scientific computations. A valuable text for computer science courses for scientists and engineers, this book will also prove useful to Mathematica users at all levels. Covering the latest release of Mathematica, the book includes useful tips and techniques to help even seasoned users
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The graphs of sin, cos and tan are generated by the user by means of a scroll bar which controls the motion of a point on the...
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The graphs of sin, cos and tan are generated by the user by means of a scroll bar which controls the motion of a point on the unit circle. Emphasis is laid on the relation between the geometrically intuitive form of the graphs and the corresponding numerical values
This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. ...
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This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. There are currently 15 modules in Mathematics and 6 modules in Science; also, there are approximately two dozen additional modules that have been created by instructors and/or Education students.The learning modules here are web-based, technology intensive lessons focusing on mathematics and science in an applied context. They have been developed for teachers, by teachers, aligned with the Illinois State Learning Standards and the National Council for Teachers of Mathematics (NCTM) Standards. Some of the lessons are designed to last over several days, some only for a class period.
The MSTE lessons site contains a collection of excellent high school/lower college division math lessons (and a limited...
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The MSTE lessons site contains a collection of excellent high school/lower college division math lessons (and a limited number of science lessons) that use the Internet in significant ways. The lessons have been developed with descriptions of the problem, connections to standards, examples of use, references, and more. Java source codes are often available. The Office for Mathematics, Science, and Technology Education (MSTE) is a division of the College of Education at the University of Illinois at Urbana-Champaign.
Quadratic Functions contains two applets that allow the user to change the coefficients of a quadratic equation and observe...
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Quadratic Functions contains two applets that allow the user to change the coefficients of a quadratic equation and observe the change in the corresponding graph. One applet uses the standard form of a quadratic equation to investigate the role of ?a?, ?b?, and ?c?? and the second uses the standard form for a parabola to study the role of ?a?, ?h?, and ?k??.
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simplifying expressions, evaluating and solving equations and inequalities, and graphing linear and quadratic functions and relations. Real world applications are presented within the course content and a function's approach is emphasized. This course builds on algebraic and geometric conceptsI can help you untangle the features of the language and develop your software program whether it be a toy hobbyist application or a custom business application tied into Microsoft Office. I graduated from Northwestern University in 1990 with a doctorate in Theoretical and Applied Mechanics. Applied mechanics bridges the gap between physical theory and its application to technology.
...I held supplemental instruction sessions 3 times per week with anywhere from 10-60 students. This experience allowed me to see material and be able to present it in many different ways. Variety of the presentation of material may be all a student needs to succeed.
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More About
This Textbook
Overview
Developed for the "transition" course for mathematics majors moving beyond the primarily procedural methods of their calculus courses toward a more abstract and conceptual environment found in more advanced courses, A Transition to Mathematics with Proofs emphasizes mathematical rigor and helps students learn how to develop and write mathematical proofs. The author takes great care to develop a text that is accessible and readable for students at all levels. It addresses standard topics such as set theory, number system, logic, relations, functions, and induction in at a pace appropriate for a wide range of readers. Throughout early chapters students gradually become aware of the need for rigor, proof, and precision, and mathematical ideas are motivated through
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Beginning Algebra - With CD - 5th edition
Summary: KEY MESSAGE:Elayn Martin-Gay'sdevelopmental math textbooks and video resources are motivated by her firm belief that every student can succeed. Martin-Gay's focus on the student shapes her clear, accessible writing, inspires her constant pedagogical innovations, and contributes greatly to the popularity and effectiveness of her video resources. This revision of Martin-Gay's algebra series continues this focus on students and what they need to be successful. Martin-Gay also strives t...show moreo ...show less
Former Library book. With CD! Great
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Mathematical Ideas (12th Edition)
9780321693815
ISBN:
0321693817
Edition: 12 Pub Date: 2011 Publisher: Addison Wesley
Summary: Mathematical Ideas offers students a comprehensive understanding of how they can relate math to everyday situations and even more unique situations such as those from film and television. It uses an innovative approach to guide students through the complex mathematical concepts through relatively easy to understand approaches that are easy to apply. These methods form part of a very readable and accessible textbook. ...It also offers excellent study tools to aid subject comprehension. We offer many mathematics textbooks of this calibre to buy brand new or to rent in good condition. We also offer a buyback service for those with used textbooks to sell.
Miller, Charles David is the author of Mathematical Ideas (12th Edition), published 2011 under ISBN 9780321693815 and 0321693817. One thousand one hundred twenty Mathematical Ideas (12th Edition) textbooks are available for sale on ValoreBooks.com, four hundred eighty three used from the cheapest price of $61.59, or buy new starting at $135.85 This is a loose leaf edition textbook (same content, just cheaper). May contain highlighting/writing. May not contain supplementary materials such as access codes or [more]
ALTERNATE EDITION: This is a loose leaf edition textbook (same content, just cheaper). May contain highlighting/writing. May not contain supplementary materials such as access codes or CDs. Second day shipping available, ships same or next business day. Get Bombed!!This is the U.S. student edition as pictured. [less]
The book clearly showed the breakdown for all of the different types of problems that were introduced and had plenty of practice problems to work on, which was very helpful for studying before tests especially with a teacher that did not always explain the problems very well.
It was one of the basic concepts of math class that students had to chose from.
I used this book for a mathematical course for liberal arts degree. It was the second part and the course number was 152. We studied line graphs, geometry, statistics, and probability. I felt that the material was well described both in the book and in class, the online tutorials were overkill.
I felt that this book only partially prepared me for the GRE exam, I did study for the test using this particular text, and I did complete the practice quizzes, but I did not do quite as well as I had hoped. Overall I thought that it prepared pretty well, but could use some improvement in the English and Math sections.
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Culinary Calculations - 2nd edition
Summary: The math skills needed for a successful foodservice career-now in a new edition Culinary Calculations, Second Edition provides the mathematical knowledge and skills that are essential for a successful career in today's competitive foodservice industry. This user-friendly guide starts with basic principles before introducing more specialized topics like recipe conversion and costing, AP/EP, menu pricing, and inventory costs. Written in a nontechnical, easy-to-understand style, the ...show morebook features a running case study that applies math concepts to a real-world example: opening a restaurant. This revised and updated Second Edition of Culinary Calculations covers relevant math skills for four key areas: Basic math for the culinary arts and foodservice industry Math for the professional kitchen Math for the business side of the foodservice industry Computer applications for the foodservice industry Each chapter is rich with resources, including learning objectives, helpful callout boxes for particular concepts, example menus and price lists, and information tables. Review questions, homework problems, and the case study end each chapter. Also included is an answer key for the even-numbered problems throughout the book. Culinary Calculations, Second Edition provides readers with a better understanding of the culinary math skills needed to expand their foodservice knowledge and sharpen their business savvy as they strive for success in their careers in the foodservice industry. ...show less
May have minimal notes/highlighting, minimal wear/tear. Please contact us if you have any Questions.
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HPB-Ohio Columbus, OH047174816143 +$3.99 s/h
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indoo Avenel, NJ
BRAND NEW
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MATH TREK Algebra 1
04/01/04
For curriculum-based algebra instruction, teachers and students can use MATH TREK Algebra 1. The multimedia program includes tutorials, assessments and student tracking. Students can use the program's scientific calculator, glossary and journal to help them complete the various exercises and activities. The assessment and student-tracking features provide immediate feedback to students so that they can stay on top of their progress. This engaging program, complete with sound, animation and graphics, can be used on stand-alone computers or a network. NECTAR Foundation, (613) 224-3031
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Data Analysis and Probability (focus 11/08 MT)
List Price
$12.95
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$10.36
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Stock #
13382
ISBN #
Published
Pages
Grades
Grades 9-12,
See What's Inside
Product Description
Containing many lessons, activities, teaching strategies and in-depth articles, the Data Analysis and Probability focus issue of MT is a great resource for high school teachers, preservice teachers and teacher educators. Featured articles include:
The development of statistical reasoning must be a high priority for school mathematics. This book offers a
blueprint for emphasizing statistical reasoning and sense making in the high
school curriculum.
This little book presents eleven short discussions of some of the most frequently asked questions about statistics, questions that are consistently raised by statistics students and by classroom teachers alike.
This book focuses on algebra as a language of process, expands the notion
of variable, develops ideas about the representation of functions, and extends
students' understanding of algebraic equivalence and change.
Cartoon Corner provides cartoons collected and adapted from the popular "Cartoon Corner" in Mathematics Teaching in the Middle School, adding notes from teachers who field-tested the questions and solutions with their students.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Elementary Statistics - With CD - 6th edition
Summary: Elementary Statistics is appropriate for a one-semester introductory statistics course, with an algebra prerequisite. ES has a reputation for being thorough and precise, and for using real data extensively. Students find the book readable and clear, and the math level is right for the diverse population that takes the introductory statistics course. The text thoroughly explains and illustrates concepts through an abundance of worked out examples.
Book has a small amount of wear visible on the binding, cover, pages00 +$3.99 s/h
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Wonder Book Frederick, MD
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$4.79
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What Is Discrete Mathematics?
Discrete Mathematics is a rapidly growing and increasingly used area of mathematics, with many practical and relevant applications.
Because it is grounded in real-world problems, discrete mathematics lends itself easily to implementing the recommendations fo the National Council of Teachers of Mathematics (NCTM) standards. (The recently published Standards and Principles for School Mathematics notes that "As an active branch of contemporary mathematics that is widely used in business and industry, discrete mathematics should be an integral part of the school mathematics curriculum.")
Because many discrete math problems are simply stated and have few mathematical prerequisites, they can be introduced at all grade levels, even with children who are not yet fluent readers.
Discrete mathematics will make math concepts come alive for your students. It's an excellent tool for improving reasoning and problem-solving skills, and is appropriate for students at all levels and of all abilities. Teachers have found that discrete mathematics offers a way of motivating unmotivated students while challenging talented students at the same time.
Because many discrete math problems are
simply stated and have few mathematical
prerequisites, they can be easily be
introduced at the middle school grade level.
EXAMPLE: Linear Programming
Minimize C = 3x + 2y on the given feasible
set.
Students spent a lot of time graphing lines
without seeing how it can be useful.
Linear programming is a powerful tool for
finding the optimal value of a linear function
on some feasible set. The feasible set is
created by solving a system of linear
inequalities. Solutions can be found
graphically so even students who have not
studied systems of equations can solve
these problems.
EXAMPLE: Systematic Listing & Counting
There are 45 creatures here. How many of them
are fish?
Systematic listing and counting are crucial
analytical skills which play a fundamantal
role in many areas of mathematics, in
particular probability. There are many
nuances of counting which are often missed
in elementary courses. One of our goals is
to shed light on this topic by exploring many
examples and employing a variety of
learning styles.
EXAMPLE: How Many Possibilities?
Combinations and permutations can range from simple to highly complex
problems, and the concepts used are relevant to everyday life.
Problems and solution methods can range so much that these
mathematical ideas can be used with students from elementary school to
high school.
Even young students with limited reading skills can solve problems
with combinations of small numbers of items. For example, given that
a classmate has two shirts and three pairs of pants, students can
determine that there are six possible outfits. They can reason about
this problem and even draw out the different options.
For older students, more advanced solution strategies can allow them
to handle more complex problems, such as the following:
I have a 6-CD player in my car and I own 100 CD's.
How many different ways can I load 6 CD's into my player?
1st Slot
2nd Slot
3rd Slot
4th Slot
5th Slot
6th Slot
EXAMPLE: Which pizza place is closest?
Voronoi diagrams allow students and teachers to explore a technique
that is used in a variety of applications, while at the same time
employing critical thinking skills and geometric concepts.
These types of diagrams allow you to map out the areas in a given
space that are closest to one specified point or another. For
example, if there are 17 ice cream shops of equal quality in your
town, a Voronoi diagram can show you which one is the closest for each
region of town. This example is shown in the picture below.
This technique is used in biology, chemistry, geology, forestry, and
more, as well as in resource planning and placement. This last topic
is easily familiar to students as it can include determining placement
for a new cell phone tower. or a new pizza place!
These diagrams are constructed using perpendicular bisectors, but
students can approach these either strictly through geometric
constructions, or through a more algebraic approach for more advanced
students.
You are lucky enough to live in a town with 17 ice cream shops, each
as good as the next. On the town map below, the ice cream shops are
marked with letters. For each numbered point, which ice cream shop or
shops should you frequent?
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In Algebra students will learn to solve equations (1st and 2nd degree) by connecting and disconnecting numbers and letters that represent numbers. I will ask them to "al-jabr".
It was once known as the Cossic Art. "Coss" is Latin for "the thing". In modern Algebra...
read more
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Worksheets for Classroom or Lab Practice for Elementary Algebra
9780321523105
ISBN:
0321523105
Edition: 1 Pub Date: 2008 Publisher: Pearson
Summary: Pearson Education is the author of Worksheets for Classroom or Lab Practice for Elementary Algebra, published 2008 under ISBN 9780321523105 and 0321523105. Two hundred seven Worksheets for Classroom or Lab Practice for Elementary Algebra textbooks are available for sale on ValoreBooks.com, fifty five used from the cheapest price of $0.44, or buy new starting at $25
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ALEX Lesson Plans
Title: Determining Percent of Seed Germination-Enhancing mathematics in the career/technical classroom and providing relevance in the mathematics classroom
Description:
InStandard(s): [AFN] J01 (9-12) 10: Determine characteristics and functions of plants. [AFN] J01 (9-12) 10: Determine characteristics and functions of plants. [AFN] J01 (9-12) 10: Determine characteristics and functions of plants. Agriculture, Food, and Natural Resources (9 - 12), or Mathematics (9 - 12) Title: Determining Percent of Seed Germination-Enhancing mathematics in the career/technical classroom and providing relevance in the mathematics classroom Description: In
Title: Math is Functional
Description:
ThisStandard(s): ALC (9-12) 12: Create a model of a set of data by estimating the equation of a curve of best fit from tables of values or scatter plots. (Alabama) AL1 (9-12) 37: Distinguish between situations that can be modeled with linear functions and with exponential functions. [F-LE1TC2] CA2 (9-12) 5: Utilize advanced features of spreadsheet software, including creating charts and graphs,
sorting and filtering data, creating formulas, and applying functions. [TC2] CA2 (9-12) 14: Use digital tools to defend solutions to authentic problems. [MA2013] AL1 (9-12) 13: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]
Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: Math is Functional Description: This twoTitle: Swimming Pool Math
Description:
Students will use a swimming pool example to practice finding perimeter and area of different rectangles.
Standard(s): ALC (9-12) 12: Create a model of a set of data by estimating the equation of a curve of best fit from tables of values or scatter plots. (Alabama) [MA2013] ALC (9-12) 1: Create algebraic models for application-based problems by developing and solving equations and inequalities, including those involving direct, inverse, and joint variation. (Alabama) [MA2013] AL1 (9-12) 34: Write a function that describes a relationship between two quantities.* [F-BF1 GEO (9-12) 12: Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. [G-CO12] [MA2013] ALC (9-12) 11: Use ratios of perimeters, areas, and volumes of similar figures to solve applied problems 5: Define appropriate quantities for the purpose of descriptive modeling. [N-Q2]8) 24: Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems. [8-G9]
Thinkfinity Lesson PlansUsing this reproducible chart, from an Illuminations lesson, students record the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. This resource can also be used as an overhead transparency so students can share their results with their classmates
Subject: Mathematics Title: Graph Chart Description: Using this reproducible chart, from an Illuminations lesson, students record the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. This resource can also be used as an overhead transparency so students can share their results with their classmates. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Title: Automobile Mileage: Years Since 1990 vs. Mileage
Description:
This
Standard(s): [MA2013] AL1 (9-12) 13: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2]
Subject: Mathematics Title: Automobile Mileage: Years Since 1990 vs. Mileage Description: This Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 exponential function Finding Our Top Speed
Description:
This Illuminations lesson sets the stage for a discussion of travel in the solar system. By situations Age vs. Mileage
Description:
In Title: Automobile Mileage: Age vs. Mileage Description: In Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 a features symbolically own features ALTTitle: Investigating Pick's Theorem
Description:
In 19: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. [A-REI5] [MA2013] AL1 (9-12) 24: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. [A-REI12 Investigating Pick's Theorem Description: In Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Title: Escape from the Tomb Activity
Description:
This ALC (9-12) 2: Solve application-based problems by developing and solving systems of linear equations and inequalities. (Alabama)
Subject: Mathematics Title: Escape from the Tomb Activity Description: This Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Web Resources
Interactives/Games
Title: Online Algebra Problem Solver
Description:
Online Algebra Solver allows studentsLearning Activities
Title: Online Algebra Problem Solver
Description:
Online Algebra Solver allows studentsThinkfinity Learning Activities
Title: Flowing Through Mathematics
Description:
This student interactive, from Illuminations, simulates water flowing from a tube through a hole in the bottom. The diameter of the hole can be adjusted and data can be gathered for the height or volume of water in the tube at any time GEO (9-12) 36: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [G-GMD3 [MA2013] ALC (9-12) 3: Use formulas or equations of functions to calculate outcomes of exponential growth or decay Flowing Through Mathematics Description: This student interactive, from Illuminations, simulates water flowing from a tube through a hole in the bottom. The diameter of the hole can be adjusted and data can be gathered for the height or volume of water in the tube at any time. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
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More About
This Textbook
Overview
This is the fourth edition of Serge Lang's Complex Analysis. The first part of the book covers the basic material of complex analysis, and the second covers many special topics, such as the Riemann Mapping Theorem, the gamma function, and analytic continuation. Power series methods are used more systematically than in other texts, and the proofs using these methods often shed more light on the results than the standard proofs do. The first part of Complex Analysis is suitable for an introductory course on the undergraduate level, and the additional topics covered in the second part give the instructor of a graduate course a great deal of flexibility in structuring a more advanced course.
Editorial Reviews
Booknews
In this new edition of a standard textbook, new exercises have been added, in addition to more material on the Borel theorem, Picard's theorem, and J.D. Newman's proof of the prime number theorem. The treatment of gamma and zeta functions has been expanded and an appendix has been added which includes material not usually included in standard texts. The first part of the book is an introduction to complex analysis, while the second covers many special topics which may be used in an advanced course. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
"The very understandable style of explanation, which is typical for this author, makes the book valuable for both students and teachers."
EMS Newsletter, Vol. 37, Sept. 2000
Fourth Edition
S. Lang
Complex Analysis
"A highly recommendable book for a two semester course on complex analysis
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Seat Pleasant, MD TrigonometryLinear algebra applies math to such matrices as any Excel spreadsheet, and using the power of matrices, you can calculate millions of values or solve billions of equations simultaneously. I have used linear algebra in such advanced math classes as Multivariate Calculus 3, Differential Equations,...
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A+ National Pre-apprenticeship Maths and Literacy for Hospitality by Andrew Spencer
Book Description
Pre-apprenticeship Maths and Literacy for Hospitality is a write-in workbook that helps to prepare students seeking to gain a Hospitality Apprenticeship. It combines practical, real-world scenarios and terminology specifically relevant to the Hospitality industry, and provides students with the mathematical skills they need to confidently pursue a career in the Hospitality trade. Mirroring the format of current apprenticeship entry assessments, Pre-apprenticeship Maths and Literacy for Hospitality includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Hospitality Apprenticeship. Pre-apprenticeship Maths and Literacy for Hospitality also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-apprenticeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level.
Buy A+ National Pre-apprenticeship Maths and Literacy for Hospitality book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books.
You might also like...
Your guide to becoming an effective hospitality manager The hospitality industry is a "people" business. Whether dealing with guests or customers, managers or coworkers, those who work in this industry interact with other people perhaps more than in any other.
Written for the professional bartender, this book contains nearly 1,500 different cocktails and shooters. It provides tips and tricks, bar terminology, measurements, how to set up a bar, glassware, responsible serving issues, garnishes, bar games and tricks, famous toasts, and much more. It includes a section on non-alcoholic drinks.
Drawing on a large number of interviews with renowned chefs, diners, and Michelin inspectors, this book provides an unprecedented insight into Michelin-starred restaurants in Britain and Germany. Restaurants are viewed not simply as businesses but as cultural enterprises that shape our taste in food, ambience, and sociality.
Books By Author Andrew Spencer
Current approaches to morphology, Andrew Spencer argues, are flawed. He uses intermediate types of lexical relatedness in different languages to develop a morphologically-informed model of the lexical entry. He uses this to build a model of lexical relatedness consistent with paradigm-based models. A book for all morphologists and lexicographers.
Helps learners' improve their Maths and English skills and help prepare for Level 1 and Level 2 Functional Skills exams. This title enables learners to improve their maths and English skills and real-life questions and scenarios are written with an automotive context to help learners find essential Maths and English theory understandable Hairdressing context beauty therapy context
| 677.169 | 1 |
A+ National Pre-apprenticeship Maths and Literacy for Hospitality by Andrew Spencer
Book Description
Pre-apprenticeship Maths and Literacy for Hospitality is a write-in workbook that helps to prepare students seeking to gain a Hospitality Apprenticeship. It combines practical, real-world scenarios and terminology specifically relevant to the Hospitality industry, and provides students with the mathematical skills they need to confidently pursue a career in the Hospitality trade. Mirroring the format of current apprenticeship entry assessments, Pre-apprenticeship Maths and Literacy for Hospitality includes hundreds of questions to improve students' potential of gaining a successful assessment outcome of 75-80% and above. This workbook will therefore help to increase students' eligibility to obtain a Hospitality Apprenticeship. Pre-apprenticeship Maths and Literacy for Hospitality also supports and consolidates concepts that students studying VET (Vocational Educational Training) may use, as a number of VCE VET programs are also approved pre-apprenticeships. This workbook is also a valuable resource for older students aiming to revisit basic literacy and maths in their preparation to re-enter the workforce at the apprenticeship level.
Buy A+ National Pre-apprenticeship Maths and Literacy for Hospitality book by Andrew Spencer from Australia's Online Bookstore, Boomerang Books.
You might also like...
With articles and photographs from the Sheffield Star and Sheffield Telegraph, Peter Tuffrey has recorded subjects and incidents ranging from pub closures to murders, from retirements to renovations and from pub bombings to pub ghosts. Many traditional pubs are pictured and documented in decline or just before demolition.
This title contains activities throughout each chapter that ask students to develop and extend their understanding of the content and concepts. Special focus interpretation studies are included at the end of each chapter, providing real information on guiding for a range of Australian situations.
Drawing on a large number of interviews with renowned chefs, diners, and Michelin inspectors, this book provides an unprecedented insight into Michelin-starred restaurants in Britain and Germany. Restaurants are viewed not simply as businesses but as cultural enterprises that shape our taste in food, ambience, and sociality.
Books By Author Andrew Spencer
Current approaches to morphology, Andrew Spencer argues, are flawed. He uses intermediate types of lexical relatedness in different languages to develop a morphologically-informed model of the lexical entry. He uses this to build a model of lexical relatedness consistent with paradigm-based models. A book for all morphologists and lexicographers.
Helps learners' improve their Maths and English skills and help prepare for Level 1 and Level 2 Functional Skills exams. This title enables learners to improve their maths and English skills and real-life questions and scenarios are written with an automotive context to help learners find essential Maths and English theory understandable Hairdressing context beauty therapy context
| 677.169 | 1 |
Elementary Algebra for College Students (8th Edition)
9780321620934
ISBN:
0321620933
Edition: 8 Pub Date: 2010 Publisher: Prentice Hall
Summary: Angel, Allen R. is the author of Elementary Algebra for College Students (8th Edition), published 2010 under ISBN 9780321620934 and 0321620933. Three hundred forty seven Elementary Algebra for College Students (8th Edition) textbooks are available for sale on ValoreBooks.com, one hundred forty used from the cheapest price of $17.82, or buy new starting at $67Woodinville, WAShipping:Standard, ExpeditedComments:Brand new book. STUDENT US EDITION. Never used. Only Two Very Minor Marks On Bottom Side of the B... [more]Brand new book. STUDENT US EDITION. Never used. Only Two Very Minor Marks On Bottom Side of the Book. Nice gift. Best buy. Shipped promptly and packaged carefullythe class that required me to use this book was math 101 at Rockland community college. the class was very effective especially with the professor who taught us each topic. it was a very cooperative class.
there is nothing i would change about this book. it offered problems to do and even showed exactly how to do them with examples provided.
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Metadata
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.
In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information.
The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines.
Objectives of this module: be able to relate solutions to a linear equation to lines, know the general form of a linear equation, be able to construct the graph of a line using the intercept method, be able to distinguish, by their equations, slanted, horizontal, and vertical lines.
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Merchandising Mathematics for Retailing Good(1 Copy):
Good 0132724162179.30
FREE
About the Book
Written by experienced retailers, "MECHANDISING MATH FOR RETAILING, 5/e" introduces students to the essential principles and techniques of merchandising mathematics, and explains how to apply them in solving everyday retail merchandising problems. Instructor- and student-friendly, it features clear and concise explanations of key concepts, followed by problems, case studies, spreadsheets, and summary problems using realistic industry figures. Most chapters lend themselves to spreadsheet use, and skeletal spreadsheets are provided to instructors. This edition is extensively updated to reflect current trends, and to discuss careers from the viewpoint of working professionals. It adds 20+ new case studies that encourage students to use analytic skills, and link content to realistic retail challenges. This edition also contains a focused discussion of profitability measures, and an extended discussion of assortment planning.
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ocuses on word problems to demonstrate practical applicability of algebra skills to real-world problems and standardized test problems. Each chapter covers one important concept, with the emphasison hands-on learning for problem-solving and mastering algebra skills.
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295 insight into mathematical ideas. The authors' attention to detail and clarity, as in James Stewart's market-leading Calculus text, is what makes this text the market leader.
Table of Contents
Fundamentals
Overview
Real Numbers
Exponents and Radicals
Algebraic Expressions
Discovery Project: Visualizing a Formula
Fractional Expressions
Equations
Modeling with Equations
Discovery Project: Equations through the Ages
Inequalities
Coordinate Geometry
Graphing Calculators: Solving Equations and Inequalities Graphically
Lines
Modeling Variation
Review
Test
Focus on Problem Solving: General Principles
Functions
Overview
What is a Function? Graphs of Functions
Discovery Project: Relations and Functions
Increasing and Decreasing Functions: Average Rate of Change
Transformations of Functions
Quadratic Functions: Maxima and Minima
Modeling with Functions
Combining Functions
Discovery
Project: Iteration and Chaos
One-to-One Functions and Their Inverses Review
Test
Focus on Modeling: Fitting Lines to Data
Polynomial and Rational Functions
Overview
Polynomial Functions and Their Graphs
Dividing Polynomials
Real Zeros of Polynomials
Discovery Project: Zeroing in on a Zero
Complex Numbers
Complex Zeros and the Fundamental Theorem of Algebra
Rational Functions
Review
Test
Focus on Modeling: Fitting Polynomials to Data
Exponential and Logarithmic Functions
Overview
Exponential Functions
Discovery Project: Exponential Explosion
Logarithmic Functions
Laws of Logarithms
Exponential and Logarithmic Equations
Modeling with Exponential and Logarithmic Functions
Review
Test
Focus on Modeling: Fitting Exponential and Power Curves to Data
Trigonometric Functions of Real Numbers
Overview
The Unit Circle
Trigonometric Functions of Real Numbers
Trigonometric Graphs
Discovery Project: Predator-Prey Models
More Trigonometric Graphs
Modeling Harmonic Motion
Review
Test
Focus on Modeling: Fitting Sinusoidal Curves to Data
Trigonometric Functions of Angles
Overview
Angle Measure
Trigonometry of Right Triangles
Discovery Project: Similarity
Trigonometric Functions of Angles
The Law of Sines
The Law of Cosines
Review
Test
Focus on Modeling:Surveying
Analytic Trigonometric
Overview
Trigonometric Identities
Addition and Subtraction Formulas
Double-Angle, Half-Angle, and Sum-Product Identities
Inverse Trigonometric Functions
Discovery Project: Where to Sit at the Movies
Trigonometric Equations
Review
Test
Focus on Modeling: Traveling and Standing Waves
Polar Coordinates and Vectors
Overview
Polar Coordinates
Graphs of Polar Equations
Polar Form of Complex Numbers
DeMoivre's Theorem
Discovery Project: Fractals
Vectors
The Dot Product
Discovery Project: Sailing Against the Wind
Review
Test
Focus on Modeling: Mapping the World
Systems of Equations and Inequalities
Overview
Systems of Equations
Systems of Linear Equations in Two Variables
Systems of Linear Equations in Several Variables
Discovery Project: Best Fit versus Exact Fit
Systems of Linear Equations: Matrices
The Algebra of Matrices
Discovery Project: Will the Species Survive? Inverses of Matrices and Matrix Equations
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Thomas' Calculus - 11th edition
Summary: The new edition of Thomas is a return to what Thomas has always been: the book with the best exercises. For the 11th edition, the authors have added exercises cut in the 10th edition, as well as, going back to the classic 5th and 6th editions for additional exercises and examples.
The book's theme is that Calculus is about thinking; one cannot memorize it all. The exercises develop this theme as a pivot point between the lecture in class, and the understand...show moreing that comes with applying the ideas of Calculus.
In addition, the table of contents has been refined to match the standard syllabus. Many of the examples have been trimmed of distractions and rewritten with a clear focus on the main ideas. The authors have also excised extraneous information in general and have made the technology much more transparent.
The ambition of Thomas 11e is to teach the ideas of Calculus so that students will be able to apply them in new and novel ways, first in the exercises but ultimately in their careers. Every effort has been made to insure that all content in the new edition reinforces thinking and encourages deep understanding of the
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9780534495015
ISBN:
053449501X
Pub Date: 2005 Publisher: Brooks/Cole
Summary: An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems. Based on their teaching experiences, the authors offer an accessible text that emphasizes the fundamentals of discrete mathematics and its advanced topics. This text shows how to express precise ideas in clear mathematical language. Students discover the importance of discrete ma...thematics in describing computer science structures and problem solving. They also learn how mastering discrete mathematics will help them develop important reasoning skills that will continue to be useful throughout their careers.
Schlipf, John is the author of Discrete Mathematics For Computer Science With Student Solutions Manual on CDROM, published 2005 under ISBN 9780534495015 and 053449501X. Four hundred forty Discrete Mathematics For Computer Science With Student Solutions Manual on CDROM textbooks are available for sale on ValoreBooks.com, fifty three used from the cheapest price of $76.85, or buy new starting at $53.49
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Roads to Geometry
9780130413963
0130413968
Summary: Clarifying, extending and unifying concepts discussed in basic high school geometry courses, this text gives readers a comprehensive introduction to plane geometry.
Wallace, Edward is the author of Roads to Geometry, published 2003 under ISBN 9780130413963 and 0130413968. Five hundred sixty Roads to Geometry textbooks are available for sale on ValoreBooks.com, one hundred twelve used from the cheapest price ...of $36.44, or buy new starting at $78.24
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Algebra Camp Gr. 6-8, 9-12
Develop an understanding of some basic algebra concepts as you build models and use exercises to practice your algebraic-thinking skills. No previous knowledge of algebra is necessary for this fun and informal course
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Find a South OrangeThe idea of a function and its inverse is introduced. Extensive use is made of exponential and logarithmic functions, including graphing and solving equations. Applications include compound interest problems and radioactive decay
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College Algebra - 6th edition
Summary: Accessible to students and flexible for instructors, College Algebra, Sixth Edition graphing calculators. Additional...show more program components that support student success include Eduspace tutorial practice, online homework, SMARTHINKING Live Online Tutoring, and Instructional DVDs.
The
Updated! End-of-chapter exercises--Assessing Concepts--have been revised to include more question types including fill-in-the-blank, multiple choice, and matching.
Revised! Prepare for This Section exercises, formerly Prepare for the Next Section, have been moved from the end of each chapter to the beginning of each chapter and afford students the opportunity to test their understanding of prerequisite skills about to be covered.
New! Calculus Connection icons have been added to indicate topics that will be revisited in subsequent courses, laying the groundwork for further study.
Applications require students to use problem-solving strategies and new skills to solve practical problems. Covering topics from many disciplines, including agriculture, business, chemistry, education, and sociology, these problems demonstrate to students the practicality and value of algebra.
Noted by a pie chart icon, Real Data examples and exercises require students to analyze and construct mathematical models from actual situations.Appearing throughout the text, Integrating Technology notes offer relevant information about using graphing calculators as an alternative way to solve a problem. Step-by-step instructions allow students to use technology with confidence.
Exploring Concepts with Technology, an optional end-of-chapter feature, uses technology (graphing calculators, CAS, etc.) to explore ideas covered in the chapter. These investigations can be used in a variety of ways, such as group projects or extra-credit assignments. Together with Integrating Technology tips, this feature makes the text appropriate for courses that allow the use of graphing calculators99 +$3.99 s/h
Good
HPB-Beavercreek Beavercreek, OH
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How Well Aligned Are Textbooks to the Common Core Standards in Mathematics?
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Onward Courses, Labs and Degree Credit - Mathematics
Overview
Onward Mathematics courses are designed to give you the basic skills needed for success in future math courses at UMaine and for many personal and professional situations as well. Credit for these courses will not count towards your bachelor's degree (except for MAT 111), but they do count for financial aid, and all UMaine add/drop and withdrawal policies apply.
This course covers the basic topics in algebra needed to enter a mathematics course at the precalculus level. The covered topics include a brief review of the real number system (including absolute value, exponents, roots, and radicals), linear equations and inequalities, quadratic equations, graphs, functions (primarily linear and other polynomial), factoring, rational and radical expressions. Note: This course counts for 3 degree credits, but it does not satisfy UM's General Education Mathematics Requirement. Prerequisite: ONM 12 or permission. 3 degree
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