calculus book - Mathematics and Computer Science

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

ii CONTENTS 5.3 Continuous Functions and Intermediate Values . . . . . . 210 5.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 214 6 What is Calculus? 223 6.1 Rates of Change . . . . . . . . . . . . . . . . . . . . . . . 223 6.2 Total Change . . . . . . . . . . . . . . . . . . . . . . . . 226 6.3 Notation and Infinitesimals . . . . . . . . . . . . . . . . . 226 7 Integration 229 7.1 Partitions and Sums . . . . . . . . . . . . . . . . . . . . 231 7.2 Basic Examples . . . . . . . . . . . . . . . . . . . . . . . 235 7.3 Abstract Properties of the Integral . . . . . . . . . . . . 240 7.4 Integration and Continuity . . . . . . . . . . . . . . . . . 244 7.5 Improper Integrals . . . . . . . . . . . . . . . . . . . . . 250 8 Differentiation 261 8.1 The Derivative . . . . . . . . . . . . . . . . . . . . . . . 262 8.2 Derivatives and Local Behavior . . . . . . . . . . . . . . 273 8.3 Continuity of the Derivative . . . . . . . . . . . . . . . . 277 8.4 Higher Derivatives . . . . . . . . . . . . . . . . . . . . . 279 9 The Mean Value Theorem 285 9.1 The Mean Value Theorem . . . . . . . . . . . . . . . . . 285 9.2 The Identity Theorem . . . . . . . . . . . . . . . . . . . 287 9.3 Differentiability of Inverse Functions . . . . . . . . . . . 293 9.4 The Second Derivative and Convexity . . . . . . . . . . . 295 9.5 Indeterminate Limits . . . . . . . . . . . . . . . . . . . . 302 10 The Fundamental Theorems 311 10.1 Integration and Differentiation . . . . . . . . . . . . . . . 311 10.2 Antidifferentiation . . . . . . . . . . . . . . . . . . . . . 315 11 Sequences of Functions 325 11.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . 326 11.2 Series of Functions . . . . . . . . . . . . . . . . . . . . . 334 11.3 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . 339 11.4 Approximating Sequences . . . . . . . . . . . . . . . . . 348 12 Log and Exp 359 12.1 The Natural Logarithm . . . . . . . . . . . . . . . . . . . 360 12.2 The Natural Exponential . . . . . . . . . . . . . . . . . . 361
CONTENTS iii 12.3 Properties of exp and log . . . . . . . . . . . . . . . . . . 362 13 The Trigonometric Functions 373 13.1 Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . 373 13.2 Auxiliary Trig Functions . . . . . . . . . . . . . . . . . . 382 13.3 Inverse Trig Functions . . . . . . . . . . . . . . . . . . . 386 13.4 Geometric Definitions . . . . . . . . . . . . . . . . . . . . 390 14 Taylor Approximation 397 14.1 Numerical Approximation . . . . . . . . . . . . . . . . . 397 14.2 Function Approximation . . . . . . . . . . . . . . . . . . 400 15 Elementary Functions 419 15.1 A Short Course in Complex Analysis . . . . . . . . . . . 420 15.2 Elementary Antidifferentiation . . . . . . . . . . . . . . . 431 Postscript 457 Bibliography 463 Index 465
Page 1: Calculus for Mathematicians, Comput
Page 6 and 7: iv CONTENTS
Page 8 and 9: vi LIST OF FIGURES 3.11 A piecewise
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Page 12 and 13: x PREFACE trustworthy tells you wha
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40 CHAPTER 2. NUMBERS Assume A is n
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46 CHAPTER 2. NUMBERS Note that the
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50 CHAPTER 2. NUMBERS It remains to
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52 CHAPTER 2. NUMBERS A.4 (Commutat
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60 CHAPTER 2. NUMBERS Proof. Suppos
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82 CHAPTER 2. NUMBERS Exercise 2.2
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84 CHAPTER 2. NUMBERS (b) x2 − 1
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86 CHAPTER 2. NUMBERS Then prove yo
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88 CHAPTER 2. NUMBERS (d) Consider
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90 CHAPTER 2. NUMBERS (b) Show that
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92 CHAPTER 2. NUMBERS non-terminati
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94 CHAPTER 2. NUMBERS with ak integ
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96 CHAPTER 2. NUMBERS
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98 CHAPTER 3. FUNCTIONS “potentia
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100 CHAPTER 3. FUNCTIONS f(t0) f(t1
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102 CHAPTER 3. FUNCTIONS The Vertic
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104 CHAPTER 3. FUNCTIONS 3.3 y = x
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106 CHAPTER 3. FUNCTIONS Y X × Y N
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108 CHAPTER 3. FUNCTIONS |x| = √
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110 CHAPTER 3. FUNCTIONS If we pick
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112 CHAPTER 3. FUNCTIONS see Figure
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114 CHAPTER 3. FUNCTIONS is the lar
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116 CHAPTER 3. FUNCTIONS This is th
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118 CHAPTER 3. FUNCTIONS Every rati
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120 CHAPTER 3. FUNCTIONS infinite o
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122 CHAPTER 3. FUNCTIONS To convey
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124 CHAPTER 3. FUNCTIONS you run er
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126 CHAPTER 3. FUNCTIONS solve for
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128 CHAPTER 3. FUNCTIONS to the set
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130 CHAPTER 3. FUNCTIONS The situat
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132 CHAPTER 3. FUNCTIONS customary
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134 CHAPTER 3. FUNCTIONS There is a
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136 CHAPTER 3. FUNCTIONS By inducti
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138 CHAPTER 3. FUNCTIONS (a) For k
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140 CHAPTER 3. FUNCTIONS Exercise 3
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142 CHAPTER 3. FUNCTIONS Exercise 3
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144 CHAPTER 4. LIMITS AND CONTINUIT
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204 CHAPTER 5. CONTINUITY ON INTERV
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224 CHAPTER 6. WHAT IS CALCULUS? Su
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226 CHAPTER 6. WHAT IS CALCULUS? 6.
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228 CHAPTER 6. WHAT IS CALCULUS? wh
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230 CHAPTER 7. INTEGRATION y = f(x)
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232 CHAPTER 7. INTEGRATION each sub
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252 CHAPTER 7. INTEGRATION 0 k k +
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254 CHAPTER 7. INTEGRATION Exercise
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256 CHAPTER 7. INTEGRATION Hint: Co
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258 CHAPTER 7. INTEGRATION Exercise
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260 CHAPTER 7. INTEGRATION
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262 CHAPTER 8. DIFFERENTIATION The
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264 CHAPTER 8. DIFFERENTIATION zero
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276 CHAPTER 8. DIFFERENTIATION Opti
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280 CHAPTER 8. DIFFERENTIATION Poly
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282 CHAPTER 8. DIFFERENTIATION (c)
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284 CHAPTER 8. DIFFERENTIATION
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286 CHAPTER 9. THE MEAN VALUE THEOR
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312 CHAPTER 10. THE FUNDAMENTAL THE
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326 CHAPTER 11. SEQUENCES OF FUNCTI
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360 CHAPTER 12. LOG AND EXP 12.1 Th
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362 CHAPTER 12. LOG AND EXP log x 1
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364 CHAPTER 12. LOG AND EXP differe
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366 CHAPTER 12. LOG AND EXP Two Rep
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368 CHAPTER 12. LOG AND EXP 1 2 3 4
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370 CHAPTER 12. LOG AND EXP Exercis
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372 CHAPTER 12. LOG AND EXP (d) How
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374 CHAPTER 13. THE TRIGONOMETRIC F
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398 CHAPTER 14. TAYLOR APPROXIMATIO
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420 CHAPTER 15. ELEMENTARY FUNCTION
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458 APPENDIX progress on a question
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460 APPENDIX ingredient labelling o
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462 APPENDIX . POSTSCRIPT
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464 BIBLIOGRAPHY
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466 INDEX of singleton, 253 Charlie
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468 INDEX and integral of power fun
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470 INDEX Natural logarithm functio
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472 INDEX no limit at zero, 159 Sim
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