Yet Another Calculus Text, 2007a

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ii This work is licensed under a Creative Commons License. c○Dan Sloughter 2007
Preface I intend this book to be, firstly, a introduction to calculus based on the hyperreal number system. In other words, I will use infinitesimal and infinite numbers freely. Just as most beginning calculus books provide no logical justification for the real number system, I will provide none for the hyperreals. The reader interested in questions of foundations should consult books such as Abraham Robinson’s Non-standard Analysis or Robert Goldblatt’s Lectures on the Hyperreals. Secondly, I have aimed the text primarily at readers who already have some familiarity with calculus. Although the book does not explicitly assume any prerequisites beyond basic algebra and trigonometry, in practice the pace is too fast for most of those without some acquaintance with the basic notions of calculus. iii
Page 1: Yet Another Calculus Text A Short I
Page 5 and 6: Contents Preface Contents iii v 1 D
Page 7 and 8: Chapter 1 Derivatives 1.1 The arrow
Page 9 and 10: 1.2. RATES OF CHANGE 3 determinate
Page 11 and 12: 1.2. RATES OF CHANGE 5 Exercise 1.2
Page 13 and 14: 1.3. THE HYPERREALS 7 To find the r
Page 15 and 16: 1.4. CONTINUOUS FUNCTIONS 9 the dis
Page 17 and 18: 1.4. CONTINUOUS FUNCTIONS 11 Simila
Page 19 and 20: 1.5. PROPERTIES OF CONTINUOUS FUNCT
Page 29 and 30: 1.6. THE DERIVATIVE 23 y 10 8 6 4 y
Page 31 and 32: 1.6. THE DERIVATIVE 25 y 1 y = x 2
Page 33 and 34: 1.7. PROPERTIES OF DERIVATIVES 27 1
Page 35 and 36: 1.7. PROPERTIES OF DERIVATIVES 29 a
Page 37 and 38: 1.7. PROPERTIES OF DERIVATIVES 31 T
Page 39 and 40: 1.7. PROPERTIES OF DERIVATIVES 33 1
Page 41 and 42: 1.7. PROPERTIES OF DERIVATIVES 35 o
Page 43 and 44: 1.7. PROPERTIES OF DERIVATIVES 37 E
Page 45 and 46: 1.8. A GEOMETRIC INTERPRETATION OF
Page 47 and 48: 1.9. INCREASING, DECREASING, AND LO
Page 49 and 50: 1.9. INCREASING, DECREASING, AND LO
Page 51 and 52: 1.10. OPTIMIZATION 45 y 10 y = x +2
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1.10. OPTIMIZATION 47 y 6 5 4 3 2 y
Page 55 and 56:
1.10. OPTIMIZATION 49 y 5 4 3 2 1 4
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1.10. OPTIMIZATION 51 y 4 3 y = x 2
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1.11. IMPLICIT DIFFERENTIATION AND
Page 61 and 62:
1.11. IMPLICIT DIFFERENTIATION AND
Page 63 and 64:
1.12. HIGHER-ORDER DERIVATIVES 57 D
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1.12. HIGHER-ORDER DERIVATIVES 59 1
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1.12. HIGHER-ORDER DERIVATIVES 61 y
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Chapter 2 Integrals 2.1 Integrals W
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2.1. INTEGRALS 65 From our rules fo
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2.1. INTEGRALS 67 Exercise 2.1.2. F
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2.2. DEFINITE INTEGRALS 69 2.2 Defi
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2.2. DEFINITE INTEGRALS 71 integral
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2.3. PROPERTIES OF DEFINITE INTEGRA
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2.4. THE FUNDAMENTAL THEOREM OF INT
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2.4. THE FUNDAMENTAL THEOREM OF INT
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2.5. APPLICATIONS OF DEFINITE INTEG
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2.6. SOME TECHNIQUES FOR EVALUATING
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2.7. THE EXPONENTIAL AND LOGARITHM
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Answers to Exercises 1.2.1. (a) 32
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1.7.11. dy dx∣ = 216 x=1 131 1.7.
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1.11.2. dy dx∣ = − 7 (x,y)=(2,
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135 2.5.9. 8π 3 2.5.10. 2π 3 2.5.
Page 143 and 144:
137 2.6.22. ∫ π 2 0 sin(x)sin(2x
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139 2.7.20. ∫ 1 0 x +14 x 2 dx =5
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INDEX 141 partial fractions, 123, 1
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Yet Another Calculus Text, 2007a

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