MATH 221 FIRST SEMESTER CALCULUS fall 2009 Typeset:June 8, 2010 1 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 LATEX 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 3. Exercises 64 4. Finding sign changes of a function 65 5. Increasing and decreasing functions 66 Chapter 1. Numbers and Functions5 6. Examples 67 1. What is a number?5 7. Maxima and Minima 69 2. Exercises7 8. Must there always be a maximum? 71 3. Functions8 9. Examples { functions with and without maxima or 4. Inverse functions and Implicit functions 10 minima 71 5. Exercises 13 10. General method for sketching the graph of a function 72 Chapter 2. Derivatives (1) 15 11. Convexity, Concavity and the Second Derivative 74 1. The tangent to a curve 15 12. Proofs of some of the theorems 75 2. An example { tangent to a parabola 16 13. Exercises 76 3. Instantaneous velocity 17 14. Optimization Problems 77 4. Rates of change 17 15. Exercises 78 5. Examples of rates of change 18 Chapter 6. Exponentials and Logarithms (naturally) 81 6. Exercises 18 1. Exponents 81 2. Logarithms 82 Chapter 3. Limits and Continuous Functions 21 3. Properties of logarithms 83 1. Informal definition of limits 21 4. Graphs of exponential functions and logarithms 83 2. The formal, authoritative, definition of limit 22 5. The derivative of ax and the definition of e 84 3. Exercises 25 6. Derivatives of Logarithms 85 4. Variations on the limit theme 25 7. Limits involving exponentials and logarithms 86 5. Properties of the Limit 27 8. Exponential growth and decay 86 6. Examples of limit computations 27 9. Exercises 87 7. When limits fail to exist 29 8. What's in a name? 32 Chapter 7. The Integral 91 9. Limits and Inequalities 33 1. Area under a Graph 91 10. Continuity 34 2. When f changes its sign 92 11. Substitution in Limits 35 3. The Fundamental Theorem of Calculus 93 12. Exercises 36 4. Exercises 94 13. Two Limits in Trigonometry 36 5. The indefinite integral 95 14. Exercises 38 6. Properties of the Integral 97 7. The definite integral as a function of its integration Chapter 4. Derivatives (2) 41 bounds 98 1. Derivatives Defined 41 8. Method of substitution 99 2. Direct computation of derivatives 42 9. Exercises 100 3. Differentiable implies Continuous 43 4. Some non-differentiable functions 43 Chapter 8. Applications of the integral 105 5. Exercises 44 1. Areas between graphs 105 6. The Differentiation Rules 45 2. Exercises 106 7. Differentiating powers of functions 48 3. Cavalieri's principle and volumes of solids 106 8. Exercises 49 4. Examples of volumes of solids of revolution 109 9. Higher Derivatives 50 5. Volumes by cylindrical shells 111 10. Exercises 51 6. Exercises 113 11. Differentiating Trigonometric functions 51 7. Distance from velocity, velocity from acceleration 113 12. Exercises 52 8. The length of a curve 116 13. The Chain Rule 52 9. Examples of length computations 117 14. Exercises 57 10. Exercises 118 15. Implicit differentiation 58 11. Work done by a force 118 16. Exercises 60 12. Work done by an electric current 119 Chapter 5. Graph Sketching and Max-Min Problems 63 Chapter 9. Answers and Hints 121 1. Tangent and Normal lines to a graph 63 2. The Intermediate Value Theorem 63 GNU Free Documentation License 125 3 1. APPLICABILITY AND DEFINITIONS 125 2. VERBATIM COPYING 125 3. COPYING IN QUANTITY 125 4. MODIFICATIONS 125 5. COMBINING DOCUMENTS 126 6. COLLECTIONS OF DOCUMENTS 126 7. AGGREGATION WITH INDEPENDENT WORKS 126 8. TRANSLATION 126 9. TERMINATION 126 10. FUTURE REVISIONS OF THIS LICENSE 126 11. RELICENSING 126 4 CHAPTER 1 Numbers and Functions The subject of this course is \functions of one real variable" so we begin by wondering what a real number \really" is, and then, in the next section, what a function is. 1. What is a number? 1.1. Different kinds of numbers. The simplest numbers are the positive integers 1; 2; 3; 4; ··· the number zero 0; and the negative integers ··· ; −4; −3; −2; −1: Together these form the integers or \whole numbers." Next, there are the numbers you get by dividing one whole number by another (nonzero) whole number. These are the so called fractions or rational numbers such as 1 1 2 1 2 3 4 ; ; ; ; ; ; ; ··· 2 3 3 4 4 4 3 or 1 1 2 1 2 3 4 − ; − ; − ; − ; − ; − ; − ; ··· 2 3 3 4 4 4 3 By definition, any whole number is a rational number (in particular zero is a rational number.) You can add, subtract, multiply and divide any pair of rational numbers and the result will again be a rational number (provided you don't try to divide by zero). One day in middle school you were told that there are other numbers besides the rational numbers, and the first example of such a number is the square root of two. It has been known ever since the time of the m greeks that no rational number exists whose square is exactly 2, i.e. you can't find a fraction n such that m 2 = 2; i.e. m2 = 2n2: n Nevertheless, if you compute x2 for some values of x between 1 and 2, and check if you x x2 get more or less than 2, then it looks like there should be some number x between 1:4 and 1:2 1:44 1:5 whose square is exactly 2. So,p we assume that there is such a number, and we call it 1:3 1:69 the square root of 2, written as 2. This raises several questions. How do we know there 1:4 1:96 < 2 really is a number between 1:4 and 1:5 for which x2 = 2? How many other such numbers 1:5 2:25 > 2 are we going to assume into existence? Do these new numbers obey the same algebra rules 1:6 2:56 (likep a + b = b + a) as the rational numbers? If we knew precisely what these numbers (like 2) were then we could perhaps answer such questions. It turns out to be rather difficult to give a precise description of what a number is, and in this course we won't try to get anywhere near the bottom of this issue. Instead we will think of numbers as “infinite decimal expansions" as follows. One can represent certain fractions as decimal fractions, e.g. 279 1116 = = 11:16: 25 100 5 1 Not all fractions can be represented as decimal fractions. For instance, expanding 3 into a decimal fraction leads to an unending decimal fraction 1 = 0:333 333 333 333 333 ··· 3 1 It is impossible to write the complete decimal expansion of 3 because it contains infinitely many digits. But we can describe the expansion: each digit is a three. An electronic calculator, which always represents 1 numbers as finite decimal numbers, can never hold the number 3 exactly. Every fraction can be written as a decimal fraction which may or may not be finite. If the decimal expansion doesn't end, then it must repeat. For instance, 1 = 0:142857 142857 142857 142857 ::: 7 Conversely, any infinite repeating decimal expansion represents a rational number. A real number is specified by a possibly unending decimal expansion. For instance, p 2 = 1:414 213 562 373 095 048 801 688 724 209 698 078 569 671 875 376 9 ::: Of course you can never write all the digits in the decimal expansion, so you only write thep first few digits and hide the others behind dots. To give a precise description of a real number (such as 2) you have to explain how you could in principle compute as many digits in the expansion as you would like. During the next three semesters of calculus we will not go into the details of how this should be done. p 1.2. A reason to believe in 2. The Pythagorean theorem says that thep hy- potenuse of a right triangle with sides 1 and 1 must be a line segment of length 2. In middle or high school you learned somethingp similar to the following geometric construction of a line segment whose length is 2. Take a square with side of length 1, and construct a new square one of whose sides is the diagonal of the first square. The figure you get consists of 5 triangles of equal area and by counting triangles you see that the larger square has exactly twice the areap of the smaller square. Therefore the diagonal of the smaller square, being the side of the larger square, is 2 as long as the side of the smaller square.