1. ESSENTIAL CALCULUS CH02 Derivatives

2. In this Chapter: Review 2.1 Derivatives and Rates of Change
2.2 The Derivative as a Function
2.3 Basic Differentiation Formulas
2.4 The Product and Quotient Rules
2.5 The Chain Rule
2.6 Implicit Differentiation
2.7 Related Rates
2.8 Linear Approximations and Differentials
Review

3. Chapter 2, 2.1, P73

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14. Provided that this limit exists.
1 DEFINITION The tangent line to the curve y=f(x) at the point P(a, f(a)) is the line through P with slope
m=line
Provided that this limit exists.
X→ a
Chapter 2, 2.1, P75

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17. 4 DEFINITION The derivative of a function f at a number a, denoted by f’(a), is
f’(a)=lim
if this limit exists.
h→ 0
Chapter 2, 2.1, P77

18. f’(a) =lim
x→ a
Chapter 2, 2.1, P78

19. The tangent line to y=f(X) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f’(a), the derivative of f at a.
Chapter 2, 2.1, P78

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23. 6. Instantaneous rate of change=lim
∆X→0
X2→x1
Chapter 2, 2.1, P79

24. The derivative f’(a) is the instantaneous rate of change of y=f(X) with respect to x when x=a.
Chapter 2, 2.1, P79

25. 9. The graph shows the position function of a car
9. The graph shows the position function of a car. Use the shape of the graph to explain your answers to the following questions
What was the initial velocity of the car?
Was the car going faster at B or at C?
Was the car slowing down or speeding up at A, B, and C?
What happened between D and E?
Chapter 2, 2.1, P81

26. 10. Shown are graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie.
(a) Describe and compare how the runners the race.
(b) At what time is the distance between the runners the greatest?
(c) At what time do they have the same velocity?
Chapter 2, 2.1, P81

27. 15. For the function g whose graph is given, arrange the following numbers in increasing order and explain your reasoning.
g’(-2) g’(0) g’(2) g’(4)
Chapter 2, 2.1, P81

28. the derivative of a function f at a fixed number a: f’(a)=lim
Chapter 2, 2.2, P83

29. f’(x)=lim
h→ 0
Chapter 2, 2.2, P83

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33. 3 DEFINITION A function f is differentiable a if f’(a) exists
3 DEFINITION A function f is differentiable a if f’(a) exists. It is differentiable on an open interval (a,b) [ or (a,∞) or (-∞ ,a) or (- ∞, ∞)] if it is differentiable at every number in the interval.
Chapter 2, 2.2, P87

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36. 4 THEOREM If f is differentiable at a, then f is continuous at a .
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41. (a) f’(-3) (b) f’(-2) (c) f’(-1) (d) f’(0) (e) f’(1) (f) f’(2)
(g) f’(3)
Chapter 2, 2.2, P91

42. 2. (a) f’(0) (b) f’(1) (c) f’’(2) (d) f’(3) (e) f’(4) (f) f’(5)
Chapter 2, 2.2, P91

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47. 33. The figure shows the graphs of f, f’, and f”
33. The figure shows the graphs of f, f’, and f”. Identify each curve, and explain your choices.
Chapter 2, 2.2, P93

48. 34. The figure shows graphs of f, f’, f”, and f”’
34. The figure shows graphs of f, f’, f”, and f”’. Identify each curve, and explain your choices.
Chapter 2, 2.2, P93

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51. 35. The figure shows the graphs of three functions
35. The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.
Chapter 2, 2.2, P94

52. The graph of f(X)=c is the line y=c, so f’(X)=0.
FIGURE 1
The graph of f(X)=c is the line y=c, so f’(X)=0.
Chapter 2, 2.3, P93

53. The graph of f(x)=x is the line y=x, so f’(X)=1.
FIGURE 2
The graph of f(x)=x is the line y=x, so f’(X)=1.
Chapter 2, 2.3, P95

54. DERIVATIVE OF A CONSTANT FUNCTION
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55. Chapter 2, 2.3, P95

56. THE POWER RULE If n is a positive integer, then
Chapter 2, 2.3, P95

57. THE POWER RULE (GENERAL VERSION) If n is any real number, then
Chapter 2, 2.3, P97

58. █GEOMETRIC INTERPRETATION OF THE CONSTANT MULTIPLE RULE
Multiplying by c=2 stretches the graph vertically by a factor of 2. All the rises have been doubled but the runs stay the same. So the slopes are doubled, too.
Chapter 2, 2.3, P97

59. █ Using prime notation, we can write the Sum Rule as (f+g)’=f’+g’
Chapter 2, 2.3, P97

60. THE CONSTANT MULTIPLE RULE If c is a constant and f is a differentiable function, then
Chapter 2, 2.3, P97

61. THE SUM RULE If f and g are both differentiable, then
Chapter 2, 2.3, P97

62. THE DIFFERENCE RULE If f and g are both differentiable, then
Chapter 2, 2.3, P98

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66. THE PRODUCT RULE If f and g are both differentiable, then
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67. THE QUOTIENT RULE If f and g are differentiable, then
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69. DERIVATIVE OF TRIGONOMETRIC FUNCTIONS
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70. 43. If f and g are the functions whose graphs are shown, left u(x)=f(x)g(X) and v(x)=f(X)/g(x)
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71. 44. Let P(x)=F(x)G(x)and Q(x)=F(x)/G(X), where F and G and the functions whose graphs are shown.
Chapter 2, 2.4, P112

72. F’(x)=f’(g(x))‧g’(x)
THE CHAIN RULE If f and g are both differentiable and F =f。g is the composite function defined by F(x)=f(g(x)), then F is differentiable and F’ is given by the product
F’(x)=f’(g(x))‧g’(x)
In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then
Chapter 2, 2.5, P114

73. F (g(x) = f’ (g(x)) ‧ g’(x)
outer evaluated derivative evaluated derivative
function at inner of outer at inner 　 of inner
function function function function
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74. 4. THE POWER RULE COMBINED WITH CHAIN RULE If n is any real number and u=g(x) is differentiable, then
Alternatively,
Chapter 2, 2.5, P116

75. 49. A table of values for f, g, f’’, and g’ is given
If h(x)=f(g(x)), find h’(1)
If H(x)=g(f(x)), find H’(1).
Chapter 2, 2.5, P120

76. 51. IF f and g are the functions whose graphs are shown, let u(x)=f(g(x)), v(x)=g(f(X)), and w(x)=g(g(x)). Find each derivative, if it exists. If it dose not exist, explain why.
u’(1) (b) v’(1) (c)w’(1)
Chapter 2, 2.5, P120

77. 52. If f is the function whose graphs is shown, let h(x)=f(f(x)) and g(x)=f(x2).Use the graph of f to estimate the value of each derivative.
(a) h’(2) (b)g’(2)
Chapter 2, 2.5, P120

78. █WARNING A common error is to substitute the given numerical information (for quantities that vary with time) too early. This should be done only after the differentiation.
Chapter 2, 2.7, P129

79. Steps in solving related rates problems:
Read the problem carefully.
Draw a diagram if possible.
Introduce notation. Assign symbols to all quantities that are functions of time.
Express the given information and the required rate in terms of derivatives.
Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3).
Use the Chain Rule to differentiate both sides of the equation with respect to t.
Substitute the given information into the resulting equation and solve for the unknown rate.
Chapter 2, 2.7, P129

80. Chapter 2, 2.8, P133

81. f(x) ~ f(a)+f”(a)(x-a) ~
Is called the linear approximation or tangent line approximation of f at a.
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82. The linear function whose graph is this tangent line, that is ,
is called the linearization of f at a.
L(x)=f(a)+f’(a)(x-a)
Chapter 2, 2.8, P133

83. The differential dy is then defined in terms of dx by the equation.
So dy is a dependent variable; it depends on the values of x and dx. If dx is given a specific value and x is taken to be some specific number in the domain of f, then the numerical value of dy is determined.
dy=f’(x)dx
Chapter 2, 2.8, P135

84. relative error
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85. 1. For the function f whose graph is shown, arrange the following numbers in increasing order:
Chapter 2, Review, P139

86. 7. The figure shows the graphs of f, f’, and f”
7. The figure shows the graphs of f, f’, and f”. Identify each curve, and explain your choices.
Chapter 2, Review, P139

87. 50. If f and g are the functions whose graphs are shown, let P(x)=f(x)g(x), Q(x)=f(x)/g(x), and C(x)=f(g(x)). Find (a) P’(2), (b) Q’(2), and (c)C’(2).
Chapter 2, Review, P140

88. 61. The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.
Chapter 2, Review, P141