C H A P T E R 3

1. C H A P T E R 3 Additional Applicationsof the Derivative

2. Figure 3.1 Military expenditure of former Soviet bloc countries as a percentage of GDP. 3-1-65

3. Figure 3.2 Increasing and decreasing functions. 3-1-66

4. Figure 3.3 The graph of f(x) = 2x3 + 3x2 – 12x - 7. 3-1-67

5. Figure 3.4 The graph of 3-1-68

6. Figure 3.5 A graph with various kindsof “peaks” and “valleys.” 3-1-69

7. Figure 3.6 Three critical points where f’(x) = 0: (a) relative maximum, (b) relative minimum, and (c) not a relative extremum. 3-1-70

8. Figure 3.7 Three critical points where f’(x) is undefined: (a) relative maximum, (b) relative minimum, and (c) not a relative extremum. 3-1-71

9. Figure 3.8 The graph of f(x) = x4 + 8x3 + 18x2 – 8. 3-1-72

10. Figure 3.9 The graph of g(t) = 3-1-73

11. Figure 3.10 The graph of R(x) = for 0  x 63. 3-1-74

12. Figure 3.11 The output of a factory worker. 3-2-75

13. Figure 3.12 Concavity and the slopeof the tangent. 3-2-76

14. Figure 3.13 Possible combinations of increase, decrease, and concavity. 3-2-77

15. Figure 3.14 The graph of f(x) = x4. 3-2-78

16. Figure 3.15 The graph off(x) = 3x4– 2x3– 12x2 + 18x + 15. 3-2-79

17. Figure 3.16 A possible graph of f(x). 3-2-80

18. Figure 3.17 The second derivative test. 3-2-81

19. Figure 3.18 The graph of f(x) = 2x3 – 3x2 – 12x – 7. 3-2-82

20. Figure 3.19 Three functions whose first and second derivatives are zero at x = 0. 3-2-83

21. Figure 3.20 The production of an average worker. 3-2-84

22. Figure 3.21 A graphical illustrationof limits involving infinity. 3-3-85

23. Figure 3.22 The graph of f(x) = 3-3-86

24. Figure 3.23 The graph of f(x) = 3-3-87

25. Figure 3.24 The graph of f(x) = 3-3-88

26. Figure 3.25 The graph of f(x) = 3-3-89

27. Figure 3.26 The graph of f(x) = 3-3-90

28. Figure 3.27 The average cost 3-3-91

29. Figure 3.28 Absolute extrema. 3-4-92

30. Figure 3.29 Absolute extrema of a continuous function on a closed interval: (a) the absolute maximum coincides with a relative maximum, (b) the absolute maximum occurs at an endpoint,(c) the absolute minimum coincides with a relative minimum, and (d) the absolute minimum occurs at an endpoint. 3-4-93

31. Figure 3.30 The absolute extrema ofy = 2x3 + 3x2 – 12x – 7 on –3 x  0. 3-4-94

32. Figure 3.31 Traffic speedS(t) = t3 – 10.5t2 + 30t + 20. 3-4-95

33. Figure 3.32 The speed of air during a coughS(r) = ar2(r0 – r). 3-4-96

34. Figure 3.33 Extrema of functions on unbounded intervals: (a) no absolute maximum for x > 0 and (b) no absolute minimum for x  0. 3-4-97

35. Figure 3.34 The function f(x) = x2 + on the interval x > 0. 3-4-98

36. Figure 3.35 The relative minimum is notthe absolute minimum because of theeffect of another critical point. 3-4-99

37. Figure 3.36 Graphs of profit, average cost, and marginal cost for Example 4.5. 3-4-100

38. Figure 3.37 Rectangular picnic area. 3-5-101

39. Figure 3.38 The graph of F(x) = x + For x > 0. 3-5-102

40. Figure 3.39 A cylinder of radius r and height hhas lateral (curved) area A = 2rhand volume V = r2h. 3-5-103

41. Figure 3.40 The cost function for r > 0. 3-5-104

42. Figure 3.41 The profit functionP(x) = 400(15 – x)(x – 2). 3-5-105

43. Figure 3.42 Relative positions of factory, river, and power plant. 3-5-106

44. Figure 3.43 Two choices for the variable x. 3-5-107

45. Figure 3.44 The revenue functionR(x) = (35 + x)(60 – x). 3-5-108

46. Figure 3.45 Inventory graphs: (a) actual inventory graph and (b) constant inventory of tires. 3-5-109

47. Figure 3.46 Total cost C(x) = 0.48x + 3-5-110

48. Figure 3.47 Elasticity in relationto a revenue curve. 3-5-111