BowmansNotes.pdf - Math 117\/118 Honours Calculus John C Bowman University of Alberta Edmonton Canada c 2002\u201314 John C Bowman ALL RIGHTS RESERVED

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Unformatted text preview: Math 117/118: Honours Calculus John C. Bowman University of Alberta Edmonton, Canada May 19, 2017 c 2002–14 John C. Bowman ALL RIGHTS RESERVED Reproduction of these lecture notes in any form, in whole or in part, is permitted only for nonprofit, educational use. Contents 1 Real Numbers 1.A Elementary Concepts from Set Theory . . 1.B Hierarchy of Sets of Numbers . . . . . . . 1.C Algebraic Properties of the Real Numbers 1.D Absolute Value . . . . . . . . . . . . . . . 1.E Induction . . . . . . . . . . . . . . . . . . 1.F Binomial Theorem . . . . . . . . . . . . . 1.G Open and Closed Intervals . . . . . . . . . 1.H Lower and Upper Bounds . . . . . . . . . 1.I Supremum and Infimum . . . . . . . . . . 1.J Completeness Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 8 12 16 17 23 28 29 30 31 2 Sequences 2.A Limit of a Sequence . . . . . . 2.B Monotone Sequences . . . . . 2.C Subsequences . . . . . . . . . 2.D Bolzano–Weierstrass Theorem 2.E Cauchy Criterion . . . . . . . 3 Functions 3.A Examples of Functions . 3.B Trigonometric Functions 3.C Limit of a Function . . . 3.D Properties of Limits . . . 3.E Continuity . . . . . . . . 3.F One-Sided Limits . . . . 3.G Properties of Continuous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 32 40 43 48 49 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 54 56 64 68 71 74 75 . . . . 80 80 91 96 98 . . . . . . . . . . 4 Di↵erentiation 4.A The Derivative and Its Properties 4.B Maxima and Minima . . . . . . . 4.C Monotonic Functions . . . . . . . 4.D First Derivative Test . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 CONTENTS 4.E 4.F 4.G 4.H 4.I 4.J Second Derivative Test . . . . . . . . . . L’Hˆopital’s Rule . . . . . . . . . . . . . . Taylor’s Theorem . . . . . . . . . . . . . Convex and Concave Functions . . . . . Inverse Functions and Their Derivatives Implicit Di↵erentiation . . . . . . . . . . 5 Integration 5.A The Riemann Integral . . . . . . . 5.B Cauchy Criterion . . . . . . . . . . 5.C Riemann Sums . . . . . . . . . . . 5.D Properties of Integrals . . . . . . . 5.E Fundamental Theorem of Calculus 5.F Average Value of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 100 103 107 111 119 . . . . . . 121 121 127 130 132 135 141 6 Logarithmic and Exponential Functions 143 6.A Exponentials and Logarithms . . . . . . . . . . . . . . . . . . . . . . 143 6.B Logarithmic Di↵erentiation . . . . . . . . . . . . . . . . . . . . . . . . 152 6.C Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7 Techniques of Integration 7.A Change of Variables . . . . . . . . . . . . . . 7.B Integration by Parts . . . . . . . . . . . . . 7.C Integrals of Trigonometric Functions . . . . 7.D Partial Fraction Decomposition . . . . . . . 7.E Trigonometric & Hyperbolic Substitution . . 7.F Integration of Certain Irrational Expressions 7.G Strategy for Integration . . . . . . . . . . . . 7.H Numerical Approximation of Integrals . . . . 8 Applications of Integration 8.A Areas between Curves . . . . . . 8.B Arc Length . . . . . . . . . . . . 8.C Volumes by Cross Sections . . . . 8.D Volume by Shells . . . . . . . . . 8.E Work . . . . . . . . . . . . . . . . 8.F Hydrostatic Force . . . . . . . . . 8.G Surfaces of Revolution . . . . . . 8.H Centroids and Pappus’s Theorems 8.I Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 158 160 167 172 182 185 187 189 . . . . . . . . . 196 196 199 201 207 210 213 214 217 221 CONTENTS 9 Improper Integrals and Infinite 9.A Improper Integrals . . . . . . 9.B Infinite Series . . . . . . . . . 9.C Power Series . . . . . . . . . . 9.D Representation of Functions as 5 Series . . . . . . . . . . . . . . . . . . . . . . . . Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 224 232 246 250 A Complex Numbers 256 Bibliography 260 Index 261 Preface These notes were developed for a first-year honours-level mathematics course on di↵erential and integral calculus at the University of Alberta. The author would like to thank the many students who took Math 117/118 from September 2000–April 2003 and September 2005–April 2007 for their help in developing these notes. Particular thanks goes to Mande Leung for typesetting the original version of these notes, to Daniel Harrison for his careful proofreading, and to Andy Hammerlindl and Tom Prince for coauthoring the high-level graphics language Asymptote (freely available at ) that was used to draw the mathematical figures in this text. The code to lift TEX characters to three dimensions and embed them as surfaces in PDF files was developed in collaboration with Orest Shardt. 6 Chapter 1 Real Numbers 1.A Elementary Concepts from Set Theory Definition: A set is a collection of distinct objects. • Here are some examples of sets: {1, 2, 3}, {1, 2}, {1}, {book, pen}, N = {1, 2, 3, . . .}, the set of natural (counting) numbers, ; = {}, the empty set. Remark: Not all sets can be enumerated like this, as a (finite or infinite) list of elements. The set of real numbers is one such example.1 Remark: If we can write the elements of a set in a list, the order in which we list them is not important. Definition: We say that a set A is a subset of a set B if every element of A is also an element of B. We write A ⇢ B.2 Definition: We say that a set A contains a set B if every element of B is also an element of A. We write A B. Note that this definition implies that B ⇢ A. 1 See the excellent article on countability, “How do I love thee? Let me count the ways!” by L. Marcoux, , 2000. 2 Some authors write this as A ✓ B and reserve the notation A ⇢ B for the case where A is a subset of B but is not identical to B, that is, where A is a proper subset of B. In our notation, if we want to emphasize that A must be a proper subset of B, we explicitly write A ( B. 7 8 CHAPTER 1. REAL NUMBERS Definition: We say that two sets A and B are equal if A ⇢ B and B ⇢ A, that is, if every element in A is also in B and vice-versa, so that A and B contain exactly the same elements. We write A = B. • {1, 2} = {2, 1}. Definition: The set containing all elements of A and all elements of B (but no additional elements) is called the union of A and B and is denoted A [ B. Definition: The set containing exactly those elements common to both A and B is called the intersection of A and B and is denoted A \ B. These definitions are illustrated in Figure 1.1. • {1} [ {2} = {1, 2}. • {1, 2, 3} \ {1, 4} = {1}. • {1, 2} [ {2} = {1, 2}. A\B A B A[B Figure 1.1: Venn Diagram 1.B Hierarchy of Sets of Numbers We will find it useful to consider the following sets ( 2 means is an element of ): ; = {} the empty set, N = {1, 2, 3, . . .}, the set of natural (counting) numbers, Z = { n : n 2 N} [ {0} [ N, the set of integers, Q = { pq : p, q 2 Z, q 6= 0}, the set of rational numbers, R, the set of all real numbers. Notice that ; ⇢ N ⇢ Z ⇢ Q ⇢ R. 1.B. HIERARCHY OF SETS OF NUMBERS 9 Q. Why do we need the set R of real numbers to develop calculus? Why can’t we just use the set Q of rational numbers? One might try to argue, for example, that every number representable on a (finite-precision) digital computer is rational. If a subset of Q is good enough for computers, shouldn’t it be good enough for mathematicians, too? To answer this question, it will be helpful to recall Pythagoras’ Theorem, which states that the square of the length c of the hypotenuse of a right-angle triangle equals the sum of the squares of the lengths a and b of the other two sides. A simple geometric proof of this important result is illustrated in Figure 1.2. Four identical b c a a b Figure 1.2: Pythagoras’ Theorem copies of the triangle, each with area ab/2, are placed around a square of side c, so as to form a larger square with side a + b. The area c2 of the inner square is then just the area (a + b)2 = a2 + 2ab + b2 of the large square minus the total area 2ab of the four triangles. That is, c2 = a2 + b2 . Consider now the following problem. Suppose you draw a right-angle triangle having two sides of length one. 1 x 1 The Greek mathematicians of antiquity noticed that the length of the hypotenuse of such a triangle cannot possibly be a rational number; that is, it cannot be expressed as the ratio of two integers. Let us denote the length of the hypotenuse by x. From Pythagoras’ Theorem, we know that x2 = 12 + 12 = 2. Suppose that we could indeed write x = P/Q, where P and Q are integers (with Q 6= 0). By cancelling out any 10 CHAPTER 1. REAL NUMBERS common integer factors greater than one, it would then always be possible to find new integers p and q that are relatively prime (have no common factors) such that x = p/q. Then p2 2 = x2 = 2 ) p2 = 2q 2 ) p2 is even. q If p were an odd integer, say 2n + 1, then p2 = (2n + 1)2 = 4n2 + 4n + 1 could not be even. Thus, p must be even: that is, p = 2n for some integer n. Then (2n)2 = 2q 2 ) 4n2 = 2q 2 ) 2n2 = q 2 . This last result says that q 2 (and hence q) is also even, so now we know that both p and q are divisible by 2. But this contradicts the fact that p and q are relatively prime! Hence our original assumption that x = P/Q must be false; that is, x cannot be represented as a rational number. Remark: This style of mathematical proof is known as a proof by contradiction. By assuming that there are integers p and q such that (p/q)2 = 2 we have produced two contradictory statements: p and q are relatively prime and p and q are both even. Remark: If A and B are two statements, the notation A)B says that if A holds, then B must also hold; that is, “A only if B.” The notation A(B says that if B holds, then A must also hold; that is, “A if B.” If A and B are equivalent to each other, we write A () B, which means “A if and only if B.” Thus, the length of the hypotenuse of a right-angle triangle with unit sides cannot be expressed as a rational number. Mathematicians have invented a new number system, the real numbers, precisely to circumvent this kind of deficiency with the rational numbers Q. The real numbers, denoted bypR, include all rational numbers plus the curious “missing” irrational numbers (like 2). In particular, the length of any line segment is contained in the set of real numbers. This means that there are no “holes” in the real line. Mathematicians express this fact by saying that the real numbers are complete. Another important property of real numbers is that they can be written in a prescribed order on a horizontal number line, in such a way that every nonzero number is either to the right of the position occupied by the real number 0 (so that its negative is to the left of 0), or to the left of 0 (so that its negative is to the right of 0), and such that the sum and product of two numbers to the right of zero will also appear to the right of zero. Mathematicians express this particular property of the set of real numbers by saying that it can be ordered . 1.B. HIERARCHY OF SETS OF NUMBERS 11 Remark: It is easy to see that the decimal expansion of a rational number must end in a repeating pattern (which could be all zeros, in which case the rational number can be represented exactly as a decimal number with a finite number of digits). When we divide the integer p by the natural number q, the remainder can only take on one of q di↵erent values, namely 0, 1, . . . (q 1). If the number can be represented exactly with finitely many digits, then the decimal expansion will end with the repeating pattern 000 . . . (which we represent using the notation 0). Otherwise, we can never obtain the remainder 0, and only q 1 values of the remainder are possible. Upon doing q steps of long division, we will therefore encounter a repeated remainder, by the Pigeon-Hole Principle.3 At the second occurrence of the repeated remainder, the pattern of digits in the quotient will then begin to repeat itself. For q > 1, there will never be more than q 1 digits in this pattern. For example, when computing 1/7 by long division, the pattern of quotient digits will start repeating at the second occurrence of the remainder 1. In this example, the maximum possible number of digits in the pattern, q 1 = 6, is actually achieved. Problem 1.1: Show that the converse of the above remark holds; that is, if the decimal expansion of a number eventually ends in a repeating pattern of digits, the number must be rational. Problem 1.2: Show that every real number may be approximated by a rational number as accurately as desired. This shows that the rationals densely cover the real line. We say that the rationals are dense in R. Problem 1.3: Prove that p Problem 1.4: Prove that p 3 3 is an irrational number. 2 is an irrational number. Suppose that there existed integers p and q such that p3 = 2q 3 . Without loss of generality we may assume that p and q are not both even (otherwise we could cancel out the common factor of 2). We note that p3 is even. Express p = 2n + r where r = 0 or 1. Then p3 = 8n3 + 12n2 r + 6nr2 + r3 . This is even only if r = 0, that is, if p is even. (Alternatively, consider the prime factorization of p. Since 2 is prime, the only way it can be a factor of p2 is if it is also a factor of p.) Hence 8n3 = 2q 3 , or 4n3 = q 3 , so that q 3 is even. Replacing p by q in the above argument, we see that q is also even. This contradicts the fact that p and q are not both even. 3 The Pigeon-Hole Principle [Fomin et al. 1996, pp. 31–37] (also known as Dirichlet’s Box Principle) states that if you try to stu↵ more than n pigeons into n holes, at least one hole must contain two (or more) pigeons! 12 1.C CHAPTER 1. REAL NUMBERS Algebraic Properties of the Real Numbers [Spivak 1994, pp. 3–10] We now list the algebraic properties of the real numbers that we will use in our development of calculus. (P1) If a, b, and c are any real numbers, then a + (b + c) = (a + b) + c. (associative) (P2) There is a real number 0 (the additive identity) such that for any real number a, a + 0 = 0 + a = a. (identity) Problem 1.5: Show that the additive identity is unique, that is, if a + ✓ = ✓ + a = a for all a, then ✓ = 0. Hint: set a = ✓ in one pair of equalities, set a = 0 in the other. (P3) Every real number a has an additive inverse a such that a + ( a) = ( a) + a = 0. (inverse) Problem 1.6: Show that postulates (P1–P3) imply that every number has a unique additive inverse. That is, if a + b = 0, show that b = a. . . Definition: We define a b = a + ( b). (We use the symbol = to emphasize a definition, although the notation := is more common.) Problem 1.7: If a b = 0, show that a = b. (P4) If a and b are real numbers, then a + b = b + a. (commutative) Remark: Not all operations have this property. Can you give an example of an noncommutative operation? 1.C. ALGEBRAIC PROPERTIES OF THE REAL NUMBERS 13 (P5) If a, b, and c are any real numbers, then a · (b · c) = (a · b) · c. (associative) (P6) There is a real number 1 6= 0 (the multiplicative identity) such that if a is any real number, a · 1 = 1 · a = a. (identity) (P7) If a, b, and c are any real numbers, then a · (b + c) = a · b + a · c. (distributive) Remark: a · 0 = 0 for all real a. Proof: a+a·0=a·1+a·0 = a · (1 + 0) =a·1 = a. ) a · 0 = 0. Note: the symbol ) means therefore. (P8) For any real number a 6= 0, there is a real number a a·a 1 =a 1 · a = 1. 1 such that (inverse) Q. Why do we restrict a 6= 0 here? Problem 1.8: Show that both the multiplicative identity 1 and the multiplicative inverse a 1 of any real number a is unique. (P9) If a and b are real numbers, then a · b = b · a. Definition: If a b > 0, we write a > b. Similarly, if a (commutative) b < 0, we write a < b. 14 CHAPTER 1. REAL NUMBERS (P10) Given two real numbers a and b, exactly one of the following relationships holds: a < b, a = b, a > b. (Trichotomy Law) (P11) a > 0 and b > 0 ) a + b > 0. (closure under +) a > 0 and b > 0 ) a · b > 0. (closure under ·) (P12) Definition: If a < b or a = b we write a b. If a > b or a = b we write a b. Q. Is it correct to write 1 2? Why or why not? Q. Let x = 1, y = 2. Is it correct to write x y? Remark: All the elementary rules of algebra and inequalities follow from these twelve properties. • To see that ab = ( a)b, we use the distributive property: ( a)b + ab = ( a + a) · b = 0 · b = 0. • Likewise, we see that ( a)( b) = ab ( a)( b) ab = ( a) · ( b + b) = ( a) · 0 = 0, so ( a)( b) = ab. • If a < 0 and b < 0, then a > 0 and b > 0 ) ab = ( a)( b) > 0. Remark: By setting a = b in the above example and in (P12), we see that the square of any nonzero number is positive. • If a > b and b > c, then a b > 0 and b c>0 ) a c>0 ) a > c. by (P11) (transitive) 1.C. ALGEBRAIC PROPERTIES OF THE REAL NUMBERS 15 • If a > b and c > 0, then a b > 0 and c > 0 ) ac i.e. ac > bc. bc > 0 by (P12) and (P7) • If a > b and c < 0, then a b > 0 and i.e. ac < bc. c>0 ) ac + bc > 0 by (P12) • If a > b, c 2 R, then (a + c) (b + c) = a b > 0 ) a + c > b + c. ( a > 0 and b > 0, • ab > 0 ) or a < 0 and b < 0. Proof: a = 0 or b = 0 ) ab = 0 contradicts ab > 0. Also a > 0, b < 0 ) a > 0, b > 0 ) ab > 0 contradicts ab > 0. Likewise, a < 0, b > 0 contradicts ab > 0. Problem 1.9: If a < b and c < d, show that a + c < b + d. Problem 1.10: If 0 < a < b and 0 < c < d, show that ac < bd. Definition: If a < x and x < b, we write a < x < b and say x is between a and b. Lemma 1.1 (Midpoint Lemma): a<b ) a< a+b < b. 2 Proof: a<b ) a+a<a+b<b+b ) a= a+a a+b b+b < < = b. 2 2 2 Remark: This lemma (small theorem) establishes that there is no least positive number. Moreover, between any two distinct numbers there exists another one. Q. What about 1 0.9? 16 CHAPTER 1. REAL NUMBERS 1.D Absolute Value [Muldowney 1990, pp. 11–13] The fact that for any nonzero real number either x > 0 or x > 0 makes it convenient to define an absolute value function: ⇢ x if x 0, |x| = x if x < 0. Properties: Let x and y be any real numbers. (A1) |x| 0. (A2) |x| = 0 () x = 0. (A3) | x| = |x|. (A4) |xy| = |x| |y|. (A5) If c 0, then |x| c () c x c. Proof: |x| c () 0 x c or 0 < () c x c. (A6) xc |x| x |x|. Proof: Apply (A5) with c = |x|. (A7) |x| |y| |x ± y| |x| + |y| . (Triangle Inequality) Proof: RHS: (A6) ) ⇢ ) |x| x |x| |y| y |y| (|x| + |y|) x + y |x| + |y| = c (A5...
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