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# free221 - MATH 221 FIRST SEMESTER CALCULUS fall 2009...

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MATH 221 FIRST SEMESTER CALCULUS fall 2009 Typeset:June 8, 2010 1

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MATH 221 – 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2.0 (fall 2009) This is a self contained set of lecture notes for Math 221. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. The L A T E X and Python files which were used to produce these notes are available at the following web site http://www.math.wisc.edu/ ~ angenent/Free-Lecture-Notes They are meant to be freely available in the sense that “free software” is free. More precisely: Copyright (c) 2006 Sigurd B. Angenent. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ”GNU Free Documentation License”.
Contents Chapter 1. Numbers and Functions 5 1. What is a number? 5 2. Exercises 7 3. Functions 8 4. Inverse functions and Implicit functions 10 5. Exercises 13 Chapter 2. Derivatives (1) 15 1. The tangent to a curve 15 2. An example – tangent to a parabola 16 3. Instantaneous velocity 17 4. Rates of change 17 5. Examples of rates of change 18 6. Exercises 18 Chapter 3. Limits and Continuous Functions 21 1. Informal definition of limits 21 2. The formal, authoritative, definition of limit 22 3. Exercises 25 4. Variations on the limit theme 25 5. Properties of the Limit 27 6. Examples of limit computations 27 7. When limits fail to exist 29 8. What’s in a name? 32 9. Limits and Inequalities 33 10. Continuity 34 11. Substitution in Limits 35 12. Exercises 36 13. Two Limits in Trigonometry 36 14. Exercises 38 Chapter 4. Derivatives (2) 41 1. Derivatives Defined 41 2. Direct computation of derivatives 42 3. Differentiable implies Continuous 43 4. Some non-differentiable functions 43 5. Exercises 44 6. The Differentiation Rules 45 7. Differentiating powers of functions 48 8. Exercises 49 9. Higher Derivatives 50 10. Exercises 51 11. Differentiating Trigonometric functions 51 12. Exercises 52 13. The Chain Rule 52 14. Exercises 57 15. Implicit differentiation 58 16. Exercises 60 Chapter 5. Graph Sketching and Max-Min Problems 63 1. Tangent and Normal lines to a graph 63 2. The Intermediate Value Theorem 63 3. Exercises 64 4. Finding sign changes of a function 65 5. Increasing and decreasing functions 66 6. Examples 67 7. Maxima and Minima 69 8. Must there always be a maximum? 71 9. Examples – functions with and without maxima or minima 71 10. General method for sketching the graph of a function 72 11. Convexity, Concavity and the Second Derivative 74 12. Proofs of some of the theorems 75 13. Exercises 76 14. Optimization Problems 77 15. Exercises 78 Chapter 6. Exponentials and Logarithms (naturally) 81 1. Exponents 81 2. Logarithms 82 3. Properties of logarithms 83 4. Graphs of exponential functions and logarithms 83 5. The derivative of a x and the definition of e 84 6. Derivatives of Logarithms 85 7. Limits involving exponentials and logarithms 86 8.

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free221 - MATH 221 FIRST SEMESTER CALCULUS fall 2009...

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