calculus book - Mathematics and Computer Science

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$Math 241, Midterm Exam 3 Solutions Prof. Levandosky 1. Use the ...$

Contents Preface ix 1 The Language of Mathematics 1 1.1 The Nature of Mathematics . . . . . . . . . . . . . . . . 1 1.2 Sets and Operations . . . . . . . . . . . . . . . . . . . . 6 1.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Calculus and the “Real World” . . . . . . . . . . . . . . . 23 2 Numbers 33 2.1 Natural Numbers . . . . . . . . . . . . . . . . . . . . . . 34 2.2 Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3 Rational Numbers . . . . . . . . . . . . . . . . . . . . . . 53 2.4 Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . 67 2.5 Complex Numbers . . . . . . . . . . . . . . . . . . . . . 79 3 Functions 97 3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . 97 3.2 Basic Classes of Functions . . . . . . . . . . . . . . . . . 108 3.3 Composition, Iteration, and Inverses . . . . . . . . . . . 122 3.4 Linear Operators . . . . . . . . . . . . . . . . . . . . . . 131 4 Limits and Continuity 143 4.1 Order of Vanishing . . . . . . . . . . . . . . . . . . . . . 144 4.2 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.3 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.4 Sequences and Series . . . . . . . . . . . . . . . . . . . . 174 5 Continuity on Intervals 203 5.1 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . 204 5.2 Extrema of Continuous Functions . . . . . . . . . . . . . 209 i
Page 1: Calculus for Mathematicians, Comput
Page 5 and 6: CONTENTS iii 12.3 Properties of exp
Page 7 and 8: List of Figures 1.1 A Venn diagram
Page 9 and 10: LIST OF FIGURES vii 9.1 The mean va
Page 11 and 12: Preface Calculus is an important pa
Page 13 and 14: PREFACE xi To make one last analogy
Page 15 and 16: Chapter 1 The Language of Mathemati
Page 17 and 18: 1.1. THE NATURE OF MATHEMATICS 3 Ne
Page 19 and 20: 1.1. THE NATURE OF MATHEMATICS 5 Ma
Page 21 and 22: 1.2. SETS AND OPERATIONS 7 Sets can
Page 23 and 24: X Y X ∪ Y 1.3. LOGIC 9 X Y X ∩
Page 25 and 26: • If 1 = 0, then 1 2 = 0. (Invali
Page 27 and 28: 1.3. LOGIC 13 When you are asked to
Page 29 and 30: 1.3. LOGIC 15 over A. Wiles’ proo
Page 31 and 32: 1.3. LOGIC 17 Proof. Lemma 1.4 is e
Page 33 and 34: • m and n are even only if m + n
Page 35 and 36: 1.3. LOGIC 21 (m/n) 2 = 2 is imposs
Page 37 and 38: 1.4. CALCULUS AND THE “REAL WORLD
Page 47 and 48: Chapter 2 Numbers There is a pervas
Page 49 and 50: 2.1. NATURAL NUMBERS 35 () (()) •
Page 51 and 52: 2.1. NATURAL NUMBERS 37 empty. Othe
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2.1. NATURAL NUMBERS 39 m ∈ N. It
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2.1. NATURAL NUMBERS 41 On the basi
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2.1. NATURAL NUMBERS 43 n disk Towe
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2.1. NATURAL NUMBERS 45 The base ca
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n 6 5 4 3 2 1 • • • • •
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2.2. INTEGERS 49 natural numbers. T
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2.2. INTEGERS 51 Translating into t
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2.3 Rational Numbers 2.3. RATIONAL
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2.3. RATIONAL NUMBERS 55 “generat
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2.3. RATIONAL NUMBERS 57 difficult
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2.3. RATIONAL NUMBERS 59 Ordered Fi
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2.3. RATIONAL NUMBERS 61 (ii) If a
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2.3. RATIONAL NUMBERS 63 Proof. The
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2.3. RATIONAL NUMBERS 65 Consider t
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2.4. REAL NUMBERS 67 proving that y
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Infima 2.4. REAL NUMBERS 69 Everyth
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Completeness and Geometry 2.4. REAL
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2.4. REAL NUMBERS 73 A more natural
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2.4. REAL NUMBERS 75 Proof. Because
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2.4. REAL NUMBERS 77 3.14 + A(0.05)
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2.5. COMPLEX NUMBERS 79 founded on
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β i 0 1 α + β 2.5. COMPLEX NUMBE
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2.5. COMPLEX NUMBERS 83 Exercise 2.
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(c) ∞ (0, 1/n] = ∅. n=1 2.5. CO
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2.5. COMPLEX NUMBERS 87 Use the def
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2.5. COMPLEX NUMBERS 89 Intuitively
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2.5. COMPLEX NUMBERS 91 (d) Decimal
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2.5. COMPLEX NUMBERS 93 (a) Set r0
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2.5. COMPLEX NUMBERS 95 (a) Let x =
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Chapter 3 Functions The concept of
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3.1. BASIC DEFINITIONS 99 The set
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3.1. BASIC DEFINITIONS 101 A functi
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3.1. BASIC DEFINITIONS 103 • X =
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Y B 3.1. BASIC DEFINITIONS 105 X ×
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3.1. BASIC DEFINITIONS 107 conventi
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3.2. BASIC CLASSES OF FUNCTIONS 109
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3.3. COMPOSITION, ITERATION, AND IN
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3.4 Linear Operators 3.4. LINEAR OP
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3.4. LINEAR OPERATORS 133 Example 3
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3.4. LINEAR OPERATORS 135 even func
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3.4. LINEAR OPERATORS 137 Similarly
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x 2 + y 2 = 1 3.4. LINEAR OPERATORS
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3.4. LINEAR OPERATORS 141 −1 0 1
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Chapter 4 Limits and Continuity The
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4.1. ORDER OF VANISHING 145 There a
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2 a 4.1. ORDER OF VANISHING 147 2 3
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4.1. ORDER OF VANISHING 149 This is
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4.1. ORDER OF VANISHING 151 The rul
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4.2. LIMITS 153 vacuous assertions
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4.2. LIMITS 155 The real number ℓ
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4.2. LIMITS 157 In other words 1/g
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4.2. LIMITS 159 If the hypothesis i
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4.2. LIMITS 161 0 1 Proof. Recall t
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4.2. LIMITS 163 lim(f, a + ) a lim(
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4.2. LIMITS 165 with y < x. (We don
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4.2. LIMITS 167 the first equality
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4.2. LIMITS 169 If f can be made ar
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4.3. CONTINUITY 171 is indeterminat
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thus g ◦ f is continuous at a. 4.
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4.4. SEQUENCES AND SERIES 175 inter
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4.4. SEQUENCES AND SERIES 177 inequ
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4.4. SEQUENCES AND SERIES 179 by wh
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4.4. SEQUENCES AND SERIES 181 Algeb
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4.4. SEQUENCES AND SERIES 183 terms
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4.4. SEQUENCES AND SERIES 185 Theor
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4.4. SEQUENCES AND SERIES 187 howev
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4.4. SEQUENCES AND SERIES 189 Final
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4.4. SEQUENCES AND SERIES 191 Examp
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4.4. SEQUENCES AND SERIES 193 Proof
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4.4. SEQUENCES AND SERIES 195 If A
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4.4. SEQUENCES AND SERIES 197 Exerc
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4.4. SEQUENCES AND SERIES 199 (b) P
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4.4. SEQUENCES AND SERIES 201 (a) F
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Chapter 5 Continuity on Intervals T
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5.1. UNIFORM CONTINUITY 205 If f is
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f(I) 5.1. UNIFORM CONTINUITY 207 I
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5.2. EXTREMA OF CONTINUOUS FUNCTION
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5.3. CONTINUOUS FUNCTIONS AND INTER
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5.3. CONTINUOUS FUNCTIONS AND INTER
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5.4. APPLICATIONS 215 existence of
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5.4. APPLICATIONS 217 as an provide
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5.4. APPLICATIONS 219 Exercise 5.6
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5.4. APPLICATIONS 221 (b) Prove tha
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Chapter 6 What is Calculus? Calculu
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6.1. RATES OF CHANGE 225 it at two
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6.3. NOTATION AND INFINITESIMALS 22
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Chapter 7 Integration We begin the
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7.1. PARTITIONS AND SUMS 231 These
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7.1. PARTITIONS AND SUMS 233 The up
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7.2. BASIC EXAMPLES 235 integrals d
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7.2. BASIC EXAMPLES 237 are ti−1
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so the geometric sum formula U(f, P
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7.3. ABSTRACT PROPERTIES OF THE INT
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7.3. ABSTRACT PROPERTIES OF THE INT
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7.4. INTEGRATION AND CONTINUITY 245
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7.5. IMPROPER INTEGRALS 251 Again b
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7.5. IMPROPER INTEGRALS 253 Make a
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7.5. IMPROPER INTEGRALS 255 (b)For
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7.5. IMPROPER INTEGRALS 257 (f) In
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(a) If lim δ→0 1 δ 7.5. IMPROP
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Chapter 8 Differentiation Integrati
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8.1. THE DERIVATIVE 263 How should
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Derivatives and o Notation 8.1. THE
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8.1. THE DERIVATIVE 267 Since 0 ≤
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8.1. THE DERIVATIVE 269 Theorem 8.7
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8.1. THE DERIVATIVE 271 Proof. The
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8.2. DERIVATIVES AND LOCAL BEHAVIOR
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8.2. DERIVATIVES AND LOCAL BEHAVIOR
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8.3. CONTINUITY OF THE DERIVATIVE 2
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8.4. HIGHER DERIVATIVES 279 While f
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8.4. HIGHER DERIVATIVES 281 determi
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8.4. HIGHER DERIVATIVES 283 Exercis
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Chapter 9 The Mean Value Theorem Th
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9.2. THE IDENTITY THEOREM 287 Next
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9.2. THE IDENTITY THEOREM 289 The e
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9.2. THE IDENTITY THEOREM 291 Proof
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9.3. DIFFERENTIABILITY OF INVERSE F
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9.4. THE SECOND DERIVATIVE AND CONV
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9.5. INDETERMINATE LIMITS 303 The C
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9.5. INDETERMINATE LIMITS 305 Theor
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9.5. INDETERMINATE LIMITS 307 Exerc
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9.5. INDETERMINATE LIMITS 309 Exerc
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Chapter 10 The Fundamental Theorems
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10.1. INTEGRATION AND DIFFERENTIATI
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10.2. ANTIDIFFERENTIATION 315 An in
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10.2. ANTIDIFFERENTIATION 317 tives
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10.2. ANTIDIFFERENTIATION 319 of th
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10.2. ANTIDIFFERENTIATION 321 Exerc
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10.2. ANTIDIFFERENTIATION 323 Exerc
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Chapter 11 Sequences of Functions A
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Pointwise Convergence 11.1. CONVERG
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11.1. CONVERGENCE 329 (see Exercise
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11.1. CONVERGENCE 331 f+ f f− Fig
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11.1. CONVERGENCE 333 Proof. The st
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of partial sums by (11.7) sn(x) = 1
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11.2. SERIES OF FUNCTIONS 337 −2
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11.3. POWER SERIES 339 is an intege
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11.3. POWER SERIES 341 Example 11.1
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11.3. POWER SERIES 343 and the sequ
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11.3. POWER SERIES 345 Proof. Real
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Corollary 11.21. If ∞ 11.3. POWER
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11.4. APPROXIMATING SEQUENCES 349 o
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11.4. APPROXIMATING SEQUENCES 351 M
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11.4. APPROXIMATING SEQUENCES 353 F
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11.4. APPROXIMATING SEQUENCES 355 s
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11.4. APPROXIMATING SEQUENCES 357 S
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Chapter 12 Log and Exp Aside from
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12.2. THE NATURAL EXPONENTIAL 361 y
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12.3. PROPERTIES OF EXP AND LOG 363
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Chapter 13 The Trigonometric Functi
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13.1. SINE AND COSINE 375 Proof. Se
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13.1. SINE AND COSINE 377 Further,
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13.1. SINE AND COSINE 379 A physici
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13.1. SINE AND COSINE 381 and since
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− 3π 2 13.2. AUXILIARY TRIG FUNC
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e.g. 13.2. AUXILIARY TRIG FUNCTIONS
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13.3. INVERSE TRIG FUNCTIONS 387 Th
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13.3. INVERSE TRIG FUNCTIONS 389 Th
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13.4. GEOMETRIC DEFINITIONS 391 to
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13.4. GEOMETRIC DEFINITIONS 393 θ
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(a) tan(x + y) = 13.4. GEOMETRIC DE
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Chapter 14 Taylor Approximation Arm
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14.1. NUMERICAL APPROXIMATION 399 p
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14.2. FUNCTION APPROXIMATION 401
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14.2. FUNCTION APPROXIMATION 403 a
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14.2. FUNCTION APPROXIMATION 405
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14.2. FUNCTION APPROXIMATION 407 Co
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14.2. FUNCTION APPROXIMATION 409 un
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14.2. FUNCTION APPROXIMATION 411 Co
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14.2. FUNCTION APPROXIMATION 413 De
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14.2. FUNCTION APPROXIMATION 415 Th
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14.2. FUNCTION APPROXIMATION 417 It
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Chapter 15 Elementary Functions Rec
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15.1. A SHORT COURSE IN COMPLEX ANA
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15.2. ELEMENTARY ANTIDIFFERENTIATIO
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Postscript In the landmark essay Th
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POSTSCRIPT 459 functions, was very
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POSTSCRIPT 461 future teachers, gro
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Bibliography [1] Lars V. Ahlfors, C
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Index := (is defined to be), 7 A no
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patching, 292-293 of polynomial, 26
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15 game, 82 of ordered fields, 130
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Rational numbers, 53-56 countabilit
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of convolution with δ-function, 35
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