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## ESSENTIAL CALCULUS CH02 Derivatives

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**In this Chapter:**• 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**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**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**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. 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 h→ 0 Chapter 2, 2.2, P83**f’(x)=lim**h→ 0 Chapter 2, 2.2, P83**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**4 THEOREM If f is differentiable at a, then f is continuous**at a . Chapter 2, 2.2, P88**(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**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”’.**Identify each curve, and explain your choices. Chapter 2, 2.2, P93