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ESSENTIAL CALCULUS CH02 Derivatives

ESSENTIAL CALCULUS CH02 Derivatives

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ESSENTIAL CALCULUS CH02 Derivatives

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  1. ESSENTIAL CALCULUSCH02 Derivatives

  2. 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

  3. Chapter 2, 2.1, P73

  4. Chapter 2, 2.1, P73

  5. Chapter 2, 2.1, P73

  6. Chapter 2, 2.1, P74

  7. Chapter 2, 2.1, P74

  8. Chapter 2, 2.1, P74

  9. Chapter 2, 2.1, P74

  10. Chapter 2, 2.1, P74

  11. Chapter 2, 2.1, P74

  12. Chapter 2, 2.1, P75

  13. Chapter 2, 2.1, P75

  14. 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

  15. Chapter 2, 2.1, P76

  16. Chapter 2, 2.1, P76

  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

  20. Chapter 2, 2.1, P78

  21. Chapter 2, 2.1, P79

  22. Chapter 2, 2.1, P79

  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. 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. 0 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 h→ 0 Chapter 2, 2.2, P83

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

  30. Chapter 2, 2.2, P84

  31. Chapter 2, 2.2, P84

  32. Chapter 2, 2.2, P84

  33. 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

  34. Chapter 2, 2.2, P88

  35. Chapter 2, 2.2, P88

  36. 4 THEOREM If f is differentiable at a, then f is continuous at a . Chapter 2, 2.2, P88

  37. Chapter 2, 2.2, P89

  38. Chapter 2, 2.2, P89

  39. Chapter 2, 2.2, P89

  40. Chapter 2, 2.2, P89

  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

  43. Chapter 2, 2.2, P92

  44. Chapter 2, 2.2, P92

  45. Chapter 2, 2.2, P93

  46. Chapter 2, 2.2, P93

  47. 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”’. Identify each curve, and explain your choices. Chapter 2, 2.2, P93

  49. Chapter 2, 2.2, P93

  50. Chapter 2, 2.2, P93