Applied Calculus Math 215 - University of Hawaii

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c○ Copyright 1999 by the author. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author. Printed in the United States of America. This publication was typeset using AMS-TEX, the American Mathematical Society’s TEX macro system, and LATEX2ε. The graphics were produced with the help of Mathematica 1 . This book is dedicated to my wife Emily (Eun Hee) and my sons Christopher and Alexander. This is a draft which will undergo further changes. 1 Mathematica Version 2.2, Wolfram Research, Inc., Champaign, Illinois (1993).
Contents Preface iii 0 A Preview 1 1 Some Background Material 7 1.1 Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Parabolas and Higher Degree Polynomials . . . . . . . . . . . 13 1.3 The Exponential and Logarithm Functions . . . . . . . . . . . 20 1.4 Use of Graphing Utilities . . . . . . . . . . . . . . . . . . . . 31 2 The Derivative 35 2.1 Definition of the Derivative . . . . . . . . . . . . . . . . . . . 42 2.2 Differentiability as a Local Property . . . . . . . . . . . . . . 47 2.3 Derivatives of some Basic Functions . . . . . . . . . . . . . . 48 2.4 Slopes of Secant Lines and Rates of Change . . . . . . . . . . 57 2.5 Upper and Lower Parabolas . . . . . . . . . . . . . . . . . . . 61 2.6 Other Notations for the Derivative . . . . . . . . . . . . . . . 69 2.7 Exponential Growth and Decay . . . . . . . . . . . . . . . . . 70 2.8 More Exponential Growth and Decay . . . . . . . . . . . . . 76 2.9 Differentiability Implies Continuity . . . . . . . . . . . . . . . 81 2.10 Being Close Versus Looking Like a Line . . . . . . . . . . . . 83 2.11 Rules of Differentiation . . . . . . . . . . . . . . . . . . . . . 84 2.11.1 Linearity of the Derivative . . . . . . . . . . . . . . . . 85 2.11.2 Product and Quotient Rules . . . . . . . . . . . . . . . 87 2.11.3 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . 91 2.11.4 Derivatives of Inverse Functions . . . . . . . . . . . . . 101 2.12 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . 111 2.13 Related Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 2.14 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . 118 i
Page 1: Applied Calculus Math 215 Karl Hein
Page 5 and 6: Preface These notes are written for
Page 7 and 8: need the powerful machine developed
Page 9 and 10: Chapter 0 A Preview In this introdu
Page 11 and 12: scrutiny, you detect that the graph
Page 13 and 14: 0.8 0.6 0.4 0.2 -0.2 R(t) 1 2 3 4 t
Page 15 and 16: Chapter 1 Some Background Material
Page 17 and 18: 1.1. LINES 9 In application, we are
Page 19 and 20: 1.1. LINES 11 Exercise 3. Suppose a
Page 21 and 22: 1.2. PARABOLAS AND HIGHER DEGREE PO
Page 29 and 30: 1.3. THE EXPONENTIAL AND LOGARITHM
Page 39 and 40: 1.4. USE OF GRAPHING UTILITIES 31 E
Page 41 and 42: 1.4. USE OF GRAPHING UTILITIES 33 a
Page 43 and 44: Chapter 2 The Derivative The deriva
Page 45 and 46: straight line. There is still a dif
Page 47 and 48: 6 4 2 -2 -1 1 2 -2 -4 Figure 2.3: E
Page 49 and 50: Figure 2.4: The radial line is perp
Page 51 and 52: 2.1. DEFINITION OF THE DERIVATIVE 4
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2.1. DEFINITION OF THE DERIVATIVE 4
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2.2. DIFFERENTIABILITY AS A LOCAL P
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2.3. DERIVATIVES OF SOME BASIC FUNC
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2.4. SLOPES OF SECANT LINES AND RAT
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2.4. SLOPES OF SECANT LINES AND RAT
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2.5. UPPER AND LOWER PARABOLAS 61 o
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2.5. UPPER AND LOWER PARABOLAS 63 6
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2.5. UPPER AND LOWER PARABOLAS 65 2
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2.5. UPPER AND LOWER PARABOLAS 67 I
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2.6. OTHER NOTATIONS FOR THE DERIVA
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2.7. EXPONENTIAL GROWTH AND DECAY 7
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2.8. MORE EXPONENTIAL GROWTH AND DE
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2.8. MORE EXPONENTIAL GROWTH AND DE
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2.9. DIFFERENTIABILITY IMPLIES CONT
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2.10. BEING CLOSE VERSUS LOOKING LI
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2.11. RULES OF DIFFERENTIATION 85 t
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2.11. RULES OF DIFFERENTIATION 87 2
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2.11. RULES OF DIFFERENTIATION 89 E
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2.11. RULES OF DIFFERENTIATION 91 2
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2.11. RULES OF DIFFERENTIATION 93 f
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2.11. RULES OF DIFFERENTIATION 95 I
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2.11. RULES OF DIFFERENTIATION 97 T
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2.11. RULES OF DIFFERENTIATION 99 E
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2.11. RULES OF DIFFERENTIATION 101
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2.12. IMPLICIT DIFFERENTIATION 111
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2.12. IMPLICIT DIFFERENTIATION 113
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2.13. RELATED RATES 115 Exercise 80
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2.13. RELATED RATES 117 Exercise 86
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2.14. NUMERICAL METHODS 119 Your ca
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2.14. NUMERICAL METHODS 121 There i
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2.14. NUMERICAL METHODS 123 n xn f(
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2.14. NUMERICAL METHODS 125 Practic
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2.14. NUMERICAL METHODS 127 Solutio
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2.14. NUMERICAL METHODS 129 t y(t)
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2.14. NUMERICAL METHODS 131 stabili
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2.14. NUMERICAL METHODS 133 peratur
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2.15. SUMMARY 135 • Quotient rule
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Chapter 3 Applications of the Deriv
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3.2. CAUCHY’S MEAN VALUE THEOREM
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3.3. THE FIRST DERIVATIVE AND MONOT
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3.4. THE SECOND AND HIGHER DERIVATI
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3.5. THE SECOND DERIVATIVE AND CONC
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Page 173 and 174:
3.6. LOCAL EXTREMA AND INFLECTION P
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3.7. THE FIRST DERIVATIVE TEST 167
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3.7. THE FIRST DERIVATIVE TEST 169
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3.8. THE SECOND DERIVATIVE TEST 171
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3.9. EXTREMA OF FUNCTIONS 173 and a
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3.9. EXTREMA OF FUNCTIONS 175 200 1
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3.9. EXTREMA OF FUNCTIONS 177 P (25
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3.9. EXTREMA OF FUNCTIONS 179 We le
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3.9. EXTREMA OF FUNCTIONS 181 We de
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3.10. DETECTION OF INFLECTION POINT
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3.10. DETECTION OF INFLECTION POINT
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3.11. OPTIMIZATION PROBLEMS 187 3 2
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3.11. OPTIMIZATION PROBLEMS 189 sha
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3.11. OPTIMIZATION PROBLEMS 191 is
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3.11. OPTIMIZATION PROBLEMS 193 (
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3.11. OPTIMIZATION PROBLEMS 195 or
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3.11. OPTIMIZATION PROBLEMS 197 Exe
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3.12. SKETCHING GRAPHS 199 is conca
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3.12. SKETCHING GRAPHS 201 The valu
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Chapter 4 Integration We will intro
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0.6 0.5 0.4 0.3 0.2 0.1 1 2 3 4 5 6
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4.1. UPPER AND LOWER SUMS 207 So, i
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4.1. UPPER AND LOWER SUMS 209 1.25
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4.1. UPPER AND LOWER SUMS 211 4 3 2
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4.2. INTEGRABILITY AND AREAS 213 4.
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4.2. INTEGRABILITY AND AREAS 215 Pr
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4.2. INTEGRABILITY AND AREAS 217 th
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4.3. SOME ELEMENTARY OBSERVATIONS 2
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4.3. SOME ELEMENTARY OBSERVATIONS 2
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4.4. INTEGRABLE FUNCTIONS 223 4.4 I
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4.5. ANTI-DERIVATIVES 225 In this e
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4.6. THE FUNDAMENTAL THEOREM OF CAL
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4.7. SUBSTITUTION 235 We explain th
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4.7. SUBSTITUTION 237 Solution: We
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4.7. SUBSTITUTION 239 The first ide
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4.7. SUBSTITUTION 241 u = π/2. (Fo
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4.8. AREAS BETWEEN GRAPHS 243 x-coo
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4.9. NUMERICAL INTEGRATION 245 In a
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4.9. NUMERICAL INTEGRATION 247 In t
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4.9. NUMERICAL INTEGRATION 249 an a
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4.10. APPLICATIONS OF THE INTEGRAL
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4.10. APPLICATIONS OF THE INTEGRAL
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4.11. THE EXPONENTIAL AND LOGARITHM
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Page 271 and 272:
Chapter 5 Prerequisites from Precal
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5.1. THE REAL NUMBERS 265 A typical
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5.2. INEQUALITIES AND ABSOLUTE VALU
Page 277 and 278:
5.3. FUNCTIONS, DEFINITION AND NOTA
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5.4. GRAPHING EQUATIONS 275 Example
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5.5. TRIGONOMETRIC FUNCTIONS 277 1
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5.5. TRIGONOMETRIC FUNCTIONS 279 40
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5.5. TRIGONOMETRIC FUNCTIONS 281 op
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5.5. TRIGONOMETRIC FUNCTIONS 283 an
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5.5. TRIGONOMETRIC FUNCTIONS 285 A
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5.6. INVERSE FUNCTIONS 287 Observe
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5.6. INVERSE FUNCTIONS 289 5 4 3 2
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5.6. INVERSE FUNCTIONS 291 Exercise
Page 301 and 302:
5.7. NEW FUNCTIONS FROM OLD ONES 29
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Applied Calculus Math 215 - University of Hawaii

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