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Math 222 Second Semester Calculus

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Math 222 Second Semester Calculus MATH 222 SECOND SEMESTER CALCULUS spring 2007 1 2 Math 222 – 2nd Semester Calculus Lecture notes version 1.2 (spring 2007) 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. Version 1.2 is a revision of Version 1.1. It retains the revisions in Version 1.11 and adds some material at the end. The LATEXfiles, aswell asthe XFIG and OCTAVE files which were used to produce these notes are available at the following web sites: www.math.wisc.edu/˜angenent/Free-Lecture-Notes (Version 1.1) www.math.wisc.edu/˜robbin/222dir/222.html (Version 1.11) They are meant to be freely available for non-commercial use, in the sense that “free soft- ware” is free. More precisely: This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA. 3 CONTENTS I. Methods of Integration 5 1. The indefinite integral 5 2. Standard Integrals 6 3. Method of substitution 7 4. The double angle trick 9 5. Integration by Parts 9 6. Reduction Formulas 11 7. Partial Fraction Expansion 13 8. Problems 17 II. Taylor’s Formula and Infinite Series 27 9. Taylor Polynomials 27 10. Some special Taylor polynomials 30 11. The Remainder Term 31 12. Lagrange’sFormulafortheRemainderTerm 33 13. Thelimitas x 0, keeping n fixed 35 14. Little-oh → 35 15. ComputationswithTaylorpolynomials 37 16. Differentiating Taylor polynomials 41 17. The limit n , keeping x fixed 42 18. Leibniz’ formulas→ ∞ for ln 2 and π/4 46 19. Someproofs 47 20. Problems 49 III. Complex Numbers 55 21. Complex numbers 55 22. Argument and Absolute Value 56 23. Geometry of Arithmetic 57 24. Applications in Trigonometry 59 25. Calculus of complex valued functions 60 26. The Complex Exponential Function 61 27. Otherhandythingsyoucandowithcomplexnumbers 62 28. Problems 65 IV. Differential Equations 69 29. What is aDiffEq? 69 30. First Order Separable Equations 69 31. First Order Linear Equations 71 32. Dynamical Systems and Determinism 72 33. Higher order equations 74 34. ConstantCoefficientLinearHomogeneousEquations 76 35. Inhomogeneous Linear Equations 79 36. Variation of Constants 79 37. ApplicationsofSecondOrderLinearEquations 83 38. Problems 86 4 V. Vectors 93 39. Introduction to vectors 93 40. Parametricequationsforlinesandplanes 97 41. Vector Bases 100 42. DotProduct 101 43. Cross Product 108 44. Afewapplicationsofthecrossproduct 111 45. Notation 114 46. Problems 114 VI. Vector Functions and Parametrized Curves 121 47. Parametric Curves 121 48. Examples of parametrized curves 122 49. The derivative of a vector function 124 50. Higher derivatives and product rules 125 51. Interpretation of ~x 0(t) asthevelocityvector 127 52. Acceleration and Force 128 53. Tangents and the unit tangent vector 130 54. Sketching a parametric curve 133 55. Lengthofa curve 135 56. Curvatureandtheunitnormalvector 138 57. GraphsinCartesianandinPolarCoordinates 141 58. Problems 143 VII. If There’s Time 149 59. Conic Sections 149 60. The Kepler Problem 150 5 I. Methods of Integration 1. The indefinite integral We recall some facts about integration from first semester calculus. Definition 1.1. A function y = F (x) is called an antiderivative of another function y = f(x) if F 0(x)= f(x) for all x. 2 J 1.2 Example. F1(x)= x is an antiderivative of f(x)=2x. 2 F2(x)= x + 2007 is also an antiderivative of f(x)=2x. 1 I 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 antiderivatives: (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. b By definitionR f(x)dx is any function a f(x)dRx was defined in terms of Rie- of x whose derivative is f(x). mann sums and can be interpreted as R “areaR under the graph of y = f(x)”, at least when f(x) > 0. x is not a dummy variable, for example, x is a dummy variable, for example, 2xdx = x2 + C and 2tdt = t2 + C 1 1 0 2xdx = 1, and 0 2tdt = 1, so are functions of diffferent variables, so 1 2xdx = 1 2tdt. theyR are not equal. R R0 0 R R R 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 6 F 0(x) turns out to be equal to f(x), then your F (x) is indeed an antiderivative and your computation isn’t important anymore. J 1.3 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. I − R 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. Theorem 1.4. 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)= F (x)+ C. ≤ ≤ 1 2 Proof. Consider the difference G(x)= F (x) F (x). Then G0(x)= F 0(x) F 0(x)= 1 − 2 1 − 2 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 F (x) can both equal f(x) dx without equaling each other. When this happens, they 2 R (F1 and F2) differ by a constant. This can sometimes lead to confusing situations, e.g. you can check that R 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 antiderivative. 2. Standard Integrals Here is a list of the standardderivativesand hencethe standard integrals everyone should know. 1He integrated by parts. 7 xn+1 xn dx = + C (n = 1) n +1 6 − Z 1 dx = ln x + C x | | Z sin x dx = cos x + C − Z cos x dx = sin x + C Z tan x dx = lncos x + C − Z 1 dx = arctan x + C 1+ x2 Z 1 π dx = arcsin x + C = arccos x + C √1 x2 2 − Z dx− 1 1 + sin x π π = ln + C for <x< . cos x 2 1 sin x − 2 2 Z − 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). − 3. Method of substitution The chain rule says that dF (G(x)) = F 0(G(x)) G0(x), dx · so that F 0(G(x)) G0(x) dx = F (G(x)) + C. · Z J 3.1 Example. Consider the function f(x)=2x sin(x2 + 3). It does not appear in the list 2 d 2 of standard integrals we know by heart. But we do notice that 2x = dx (x + 3). So let’s call G(x)= x2 +3, and F (u)= cos u, then − F (G(x)) = cos(x2 + 3) − and dF (G(x)) = sin(x2 + 3) 2x = f(x), dx · 0 F 0(G(x)) G (x) so that | {z } |{z} 2x sin(x2 +3)dx = cos(x2 +3)+ C. − Z I 2 You will start noticing things like this after doing several examples.
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