MATH 222 SECOND SEMESTER CALCULUS Spring 2011 1 2 Math 222 – 2nd Semester Calculus Lecture notes version 1.7(Spring 2011) This is a self contained set of lecture notes for Math 222. The notes were written by Sigurd Angenent, starting from an extensive collection of notes and problems compiled by Joel Robbin. Some problems were contributed by A.Miller. The LATEX files, as well as the Xfig and Octave files which were used to produce these notes are available at the following web site www.math.wisc.edu/~angenent/Free-Lecture-Notes They are meant to be freely available for non-commercial use, 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: Methods of Integration 3 1. The indefinite integral 3 2. You can always check the answer 4 3. About “+C” 4 4. Standard Integrals 5 5. Method of substitution 5 6. The double angle trick 7 7. Integration by Parts 7 8. Reduction Formulas 9 9. Partial Fraction Expansion 12 10. PROBLEMS 16 Chapter 2: Taylor’s Formulaand Infinite Series 27 11. Taylor Polynomials 27 12. Examples 28 13. Some special Taylor polynomials 32 14. The Remainder Term 32 15. Lagrange’s Formula for the Remainder Term 34 16. The limit as x 0, keeping n fixed 36 → 17. The limit n , keeping x fixed 43 →∞ 18. Convergence of Taylor Series 46 19. Leibniz’ formulas for ln 2 and π/4 48 20. Proof of Lagrange’s formula 49 21. Proof of Theorem 16.8 50 22. PROBLEMS 51 Chapter 3: Complex Numbers and the Complex Exponential 56 23. Complex numbers 56 24. Argument and Absolute Value 57 25. Geometry of Arithmetic 58 26. Applications in Trigonometry 60 27. Calculus of complex valued functions 61 3 28. The Complex Exponential Function 61 29. Complex solutions of polynomial equations 63 30. Other handy things you can do with complex numbers 65 31. PROBLEMS 67 Chapter 4: Differential Equations 72 32. What is a DiffEq? 72 33. First Order Separable Equations 72 34. First Order Linear Equations 73 35. Dynamical Systems and Determinism 75 36. Higher order equations 77 37. Constant Coefficient Linear Homogeneous Equations 78 38. Inhomogeneous Linear Equations 81 39. Variation of Constants 82 40. Applications of Second Order Linear Equations 85 41. PROBLEMS 89 Chapter 5: Vectors 97 42. Introduction to vectors 97 43. Parametric equations for lines and planes 102 44. Vector Bases 104 45. Dot Product 105 46. Cross Product 112 47. A few applications of the cross product 115 48. Notation 118 49. PROBLEMS 118 Chapter 6: Vector Functions and Parametrized Curves 124 50. Parametric Curves 124 51. Examples of parametrized curves 125 52. The derivative of a vector function 127 53. Higher derivatives and product rules 128 54. Interpretation of x′(t) as the velocity vector 129 55. Acceleration and Force 131 56. Tangents and the unit tangent vector 133 57. Sketching a parametric curve 135 58. Length of a curve 137 59. The arclength function 139 60. Graphs in Cartesian and in Polar Coordinates 140 61. PROBLEMS 141 GNU Free Documentation License 148 1. APPLICABILITY AND DEFINITIONS 148 2. VERBATIM COPYING 149 3. COPYING IN QUANTITY 149 4. MODIFICATIONS 149 5. COMBINING DOCUMENTS 150 6. COLLECTIONS OF DOCUMENTS 150 7. AGGREGATION WITH INDEPENDENT WORKS 150 8. TRANSLATION 150 9. TERMINATION 150 10. FUTURE REVISIONS OF THIS LICENSE 151 4 11. RELICENSING 151 5 Chapter 1: Methods of Integration 1. The indefinite integral We recall some facts about integration from first semester calculus. 1.1. Definition. A function y = F (x) is called an antiderivative of another function y = f(x) if F ′(x)= f(x) for all x. 2 1.2. Example. F1(x)= x is an antiderivative of f(x)=2x. 2 F2(x)= x + 2004 is also an antiderivative of f(x)=2x. 1 G(t)= 2 sin(2t + 1) is an antiderivative of g(t) = cos(2t + 1). The Fundamental Theorem of Calculus states that if a function y = f(x) is continuous on an interval a x b, then there always exists an antiderivative F (x) of f, and one has ≤ ≤ b (1) f(x) dx = F (b) F (a). Za − The best way of computing an integral is often to find an antiderivative F of the given function f, and then to use the Fundamental Theorem (1). How you go about finding an antiderivative F for some given function f is the subject of this chapter. The following notation is commonly used for antiderivates: (2) F (x)= f(x)dx. Z The integral which appears here does not have the integration bounds a and b. It is called an indefinite integral, as opposed to the integral in (1) which is called a definite integral. It’s important to distinguish between the two kinds of integrals. Here is a list of differences: Indefinite integral Definite integral b f(x)dx is a function of x. a f(x)dx is a number. R b R By definition f(x)dx is any func- a f(x)dx was defined in terms of tion of x whoseR derivative is f(x). RRiemann sums and can be inter- preted as “area under the graph of y = f(x)”, at least when f(x) > 0. x is not a dummy variable, for exam- x is a dummy variable, for example, ple, 2xdx = x2 + C and 2tdt = 1 1 0 2xdx = 1, and 0 2tdt = 1, so t2 +CRare functions of diffferentR vari- R 1 1 R 0 2xdx = 0 2tdt. ables, so they are not equal. R R 6 2. You can always check the answer Suppose you want to find an antiderivative of a given function f(x) and after a long and messy computation which you don’t really trust you get an “answer”, F (x). You can then throw away the dubious computation and differentiate the F (x) you had found. If F ′(x) turns out to be equal to f(x), then your F (x) is indeed an antiderivative and your computation isn’t important anymore. 2.1. Example. Suppose we want to find ln x dx. My cousin Bruce says it might be F (x)= x ln x x. Let’s see if he’s right:R − d 1 (x ln x x)= x +1 ln x 1 = ln x. dx − · x · − Who knows how Bruce thought of this1, but he’s right! We now know that ln xdx = x ln x x + C. − R 3. About “+C” Let f(x) be a function defined on some interval a x b. If F (x) is an ≤ ≤ antiderivative of f(x) on this interval, then for any constant C the function F˜(x)= F (x)+ C will also be an antiderivative of f(x). So one given function f(x) has many different antiderivatives, obtained by adding different constants to one given antiderivative. 3.1. Theorem. If F1(x) and F2(x) are antiderivatives of the same function f(x) on some interval a x b, then there is a constant C such that F (x) = ≤ ≤ 1 F2(x)+ C. Proof. ′ ′ Consider the difference G(x)= F1(x) F2(x). Then G (x)= F1(x) ′ − − F2(x)= f(x) f(x) = 0, so that G(x) must be constant. Hence F1(x) F2(x)= C for some constant.− − It follows that there is some ambiguity in the notation f(x) dx. Two functions F1(x) and F2(x) can both equal f(x) dx without equalingR each other. When this happens, they (F1 and F2) differR by a constant. This can sometimes lead to confusing situations, e.g. you can check that 2 sin x cos x dx = sin2 x Z 2 sin x cos x dx = cos2 x Z − are both correct. (Just differentiate the two functions sin2 x and cos2 x!) These − two answers look different until you realize that because of the trig identity sin2 x+ cos2 x = 1 they really only differ by a constant: sin2 x = cos2 x + 1. − To avoid this kind of confusion we will from now on never forget to include the “arbitrary constant +C” in our answer when we compute an antideriv- ative. 1He integrated by parts. 7 4. Standard Integrals Here is a list of the standard derivatives and hence the standard integrals everyone should know. f(x) dx = F (x)+ C Z xn+1 xn dx = + C for all n = 1 Z n +1 − 1 dx = ln x + C Z x | | sin x dx = cos x + C Z − cos x dx = sin x + C Z tan x dx = ln cos x + C Z − 1 dx = arctan x + C Z 1+ x2 1 π dx = arcsin x + C (= arccos x + C) Z √1 x2 2 − − dx 1 1 + sin x π π = ln + C for <x< . Z cos x 2 1 sin x − 2 2 − All of these integrals are familiar from first semester calculus (like Math 221), except a for the last one. You can check the last one by differentiation (using ln b = ln a ln b simplifies things a bit). − 5. Method of substitution The chain rule says that dF (G(x)) ′ ′ = F (G(x)) G (x), dx · so that ′ ′ F (G(x)) G (x) dx = F (G(x)) + C.