1. Copyright © 2000 by the McGraw-Hill Companies, Inc. C H A P T E R 3 Additional Applications of the Derivative

2. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.1 Military expenditure of former Soviet bloc countries as a percentage of GDP. 3-1-65

3. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.2 Increasing and decreasing functions. 3-1-66

4. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.3 The graph of f(x) = 2x 3 + 3x 2 – 12x - 7. 3-1-67

5. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.4 The graph of 3-1-68

6. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.5 A graph with various kinds of “peaks” and “valleys.” 3-1-69

7. Copyright © 2000 by the McGraw-Hill Companies, Inc. 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. Copyright © 2000 by the McGraw-Hill Companies, Inc. 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. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.8 The graph of f(x) = x 4 + 8x 3 + 18x 2 – 8. 3-1-72

10. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.9 The graph of g(t) = 3-1-73

11. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.10 The graph of R(x) = for 0  x  63. 3-1-74

12. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.11 The output of a factory worker. 3-2-75

13. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.12 Concavity and the slope of the tangent. 3-2-76

14. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.13 Possible combinations of increase, decrease, and concavity. 3-2-77

15. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.14 The graph of f(x) = x 4. 3-2-78

16. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.15 The graph of f(x) = 3x 4 – 2x 3 – 12x 2 + 18x + 15. 3-2-79

17. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.16 A possible graph of f(x). 3-2-80

18. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.17 The second derivative test. 3-2-81

19. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.18 The graph of f(x) = 2x 3 – 3x 2 – 12x – 7. 3-2-82

20. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.19 Three functions whose first and second derivatives are zero at x = 0. 3-2-83

21. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.20 The production of an average worker. 3-2-84

22. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.21 A graphical illustration of limits involving infinity. 3-3-85

23. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.22 The graph of f(x) = 3-3-86

24. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.23 The graph of f(x) = 3-3-87

25. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.24 The graph of f(x) = 3-3-88

26. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.25 The graph of f(x) = 3-3-89

27. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.26 The graph of f(x) = 3-3-90

28. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.27 The average cost 3-3-91

29. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.28 Absolute extrema. 3-4-92

30. Copyright © 2000 by the McGraw-Hill Companies, Inc. 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. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.30 The absolute extrema of y = 2x 3 + 3x 2 – 12x – 7 on –3  x  0. 3-4-94

32. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.31 Traffic speed S(t) = t 3 – 10.5t 2 + 30t + 20. 3-4-95

33. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.32 The speed of air during a cough S(r) = ar 2 (r 0 – r). 3-4-96

34. Copyright © 2000 by the McGraw-Hill Companies, Inc. 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. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.34 The function f(x) = x 2 + on the interval x > 0. 3-4-98

36. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.35 The relative minimum is not the absolute minimum because of the effect of another critical point. 3-4-99

37. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.36 Graphs of profit, average cost, and marginal cost for Example 4.5. 3-4-100

38. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.37 Rectangular picnic area. 3-5-101

39. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.38 The graph of F(x) = x + For x > 0. 3-5-102

40. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.39 A cylinder of radius r and height h has lateral (curved) area A = 2  rh and volume V =  r 2 h. 3-5-103

41. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.40 The cost function for r > 0. 3-5-104

42. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.41 The profit function P(x) = 400(15 – x)(x – 2). 3-5-105

43. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.42 Relative positions of factory, river, and power plant. 3-5-106

44. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.43 Two choices for the variable x. 3-5-107

45. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.44 The revenue function R(x) = (35 + x)(60 – x). 3-5-108

46. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.45 Inventory graphs: (a) actual inventory graph and (b) constant inventory of tires. 3-5-109

47. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.46 Total cost C(x) = 0.48x + 3-5-110

48. Copyright © 2000 by the McGraw-Hill Companies, Inc. Figure 3.47 Elasticity in relation to a revenue curve. 3-5-111