Yet Another Calculus Text, 2007a

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38 CHAPTER 1. DERIVATIVES Hence 1 − cos(dx) dx is an infinitesimal. Moreover, from (1.5.36), we know that (1.7.57) Hence sin(dx) dx ≃ 1. (1.7.58) dy ≃ cos(x)(1) − sin(x)(0) = cos(x) (1.7.59) dx and dw ≃−sin(x)(1) + cos(x)(0) = − sin(x) (1.7.60) dx Thatis,wehaveshownthefollowing. Theorem 1.7.11. For all real values x, d sin(x) =cos(x) (1.7.61) dx and d cos(x) =− sin(x). (1.7.62) dx Example 1.7.22. Using the chain rule, d dx cos(4x) =− sin(4x) d (4x) =−4sin(4x). dt Example 1.7.23. If f(t) =sin 2 (t), then, again using the chain rule, f ′ (t) =2sin(t) d sin(t) =2sin(t) cos(t). dt Example 1.7.24. If g(x) =cos(x 2 ), then g ′ (x) =− sin(x 2 )(2x) =−2x cos(x 2 ). Example 1.7.25. If f(x) =sin 3 (4x), then, using the chain rule twice, f ′ (x) =3sin 2 (4x) d dx sin(4x) = 12 sin2 (4x)cos(4x). Exercise 1.7.19. Find the derivatives of y =cos(3t +6)andw =sin 2 (t)cos 2 (4t).
1.8. A GEOMETRIC INTERPRETATION OF THE DERIVATIVE 39 Exercise 1.7.20. Verify the following: (a) d dt tan(t) =sec2 (t) (c) d sec(t) = sec(t) tan(t) dt (d) (b) d dt cot(t) =− csc2 (t) d csc(t) =− csc(t) cot(t) dt Exercise 1.7.21. Exercise 1.7.22. Find the derivative of y = sec 2 (3t). Find the derivative of f(t) =tan 2 (3t). 1.8 A geometric interpretation of the derivative Recall that if y = f(x), then, for any real number Δx, Δy f(x +Δx) − f(x) = (1.8.1) Δx Δx is the average rate of change of y with respect to x over the interval [x, x +Δx] (see (1.2.7)). Now if the graph of y is a straight line, that is, if f(x) =mx + b for some real numbers m and b, then(1.8.1) ism, the slope of the line. In fact, astraightlineischaracterizedbythefactthat(1.8.1) is the same for any values of x and Δx. Moreover, (1.8.1) remains the same when Δx is infinitesimal; that is, the derivative of y with respect to x is the slope of the line. For other differentiable functions f, thevalueof(1.8.1) depends upon both x and Δx. However, for infinitesimal values of Δx, the shadow of (1.8.1), that is, the derivative dy dy dx ,dependsonx alone. Hence it is reasonable to think of dx as the slope of the curve y = f(x) atapointx. Whereas the slope of a straight line is constant from point to point, for other differentiable functions the value of the slope of the curve will vary from point to point. If f is differentiable at a point a, we call the line with slope f ′ (a) passing through (a, f(a)) the tangent line to the graph of f at (a, f(a)). That is, the tangent line to the graph of y = f(x) atx = a is the line with equation y = f ′ (a)(x − a)+f(a). (1.8.2) Hence a tangent line to the graph of a function f is a line through a point on the graph of f whose slope is equal to the slope of the graph at that point. Example 1.8.1. If f(x) =x 5 − 6x 2 +5,then f ′ (x) =5x 4 − 12x. In particular, f ( ′ − 2) 1 = 101 16 , and so the equation of the line tangent to the graph of f at x = − 1 2 is y = 101 ( x + 1 ) + 111 6 2 32 . See Figure 1.8.1
Page 1 and 2: Yet Another Calculus Text A Short I
Page 3 and 4: Preface I intend this book to be, f
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: 1.7. PROPERTIES OF DERIVATIVES 37 E
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
Page 53 and 54: 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
Page 57 and 58: 1.10. OPTIMIZATION 51 y 4 3 y = x 2
Page 59 and 60: 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
Page 65 and 66: 1.12. HIGHER-ORDER DERIVATIVES 59 1
Page 67 and 68: 1.12. HIGHER-ORDER DERIVATIVES 61 y
Page 69 and 70: Chapter 2 Integrals 2.1 Integrals W
Page 71 and 72: 2.1. INTEGRALS 65 From our rules fo
Page 73 and 74: 2.1. INTEGRALS 67 Exercise 2.1.2. F
Page 75 and 76: 2.2. DEFINITE INTEGRALS 69 2.2 Defi
Page 77 and 78: 2.2. DEFINITE INTEGRALS 71 integral
Page 79 and 80: 2.3. PROPERTIES OF DEFINITE INTEGRA
Page 81 and 82: 2.4. THE FUNDAMENTAL THEOREM OF INT
Page 83 and 84: 2.4. THE FUNDAMENTAL THEOREM OF INT
Page 85 and 86: 2.5. APPLICATIONS OF DEFINITE INTEG
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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.
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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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