ESSENTIAL CALCULUS CH02 Derivatives

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

Chapter 2, 2.1, P73

Chapter 2, 2.1, P73

Chapter 2, 2.1, P73

Chapter 2, 2.1, P74

Chapter 2, 2.1, P74

Chapter 2, 2.1, P74

Chapter 2, 2.1, P74

Chapter 2, 2.1, P74

Chapter 2, 2.1, P74

Chapter 2, 2.1, P75

Chapter 2, 2.1, P75

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

Chapter 2, 2.1, P76

Chapter 2, 2.1, P76

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

f’(a) =lim x→ a Chapter 2, 2.1, P78

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

Chapter 2, 2.1, P78

Chapter 2, 2.1, P79

Chapter 2, 2.1, P79

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

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

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

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

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

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

f’(x)=lim h→ 0 Chapter 2, 2.2, P83

Chapter 2, 2.2, P84

Chapter 2, 2.2, P84

Chapter 2, 2.2, P84

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

Chapter 2, 2.2, P88

Chapter 2, 2.2, P88

4 THEOREM If f is differentiable at a, then f is continuous at a . Chapter 2, 2.2, P88

Chapter 2, 2.2, P89

Chapter 2, 2.2, P89

Chapter 2, 2.2, P89

Chapter 2, 2.2, P89

(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

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

Chapter 2, 2.2, P92

Chapter 2, 2.2, P92

Chapter 2, 2.2, P93

Chapter 2, 2.2, P93

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

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

Chapter 2, 2.2, P93

Chapter 2, 2.2, P93

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

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

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

DERIVATIVE OF A CONSTANT FUNCTION Chapter 2, 2.3, P95

Chapter 2, 2.3, P95

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

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

█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

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

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

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

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

Chapter 2, 2.3, P100

Chapter 2, 2.3, P100

Chapter 2, 2.3, P101

THE PRODUCT RULE If f and g are both differentiable, then Chapter 2, 2.4, P106

THE QUOTIENT RULE If f and g are differentiable, then Chapter 2, 2.4, P109

Chapter 2, 2.4, P110

DERIVATIVE OF TRIGONOMETRIC FUNCTIONS Chapter 2, 2.4, P111

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) Chapter 2, 2.4, P112

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

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

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 Chapter 2, 2.5, P115

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

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

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

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

█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

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

Chapter 2, 2.8, P133

f(x) ~ f(a)+f”(a)(x-a) ~ Is called the linear approximation or tangent line approximation of f at a. Chapter 2, 2.8, P133

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

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

relative error Chapter 2, 2.8, P136

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

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

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

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