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Applications of Derivatives: Analyzing Functions and Optimization in Various Contexts

This chapter explores additional applications of derivatives, highlighting critical concepts in calculus through various figures and examples. It addresses military expenditure trends among former Soviet bloc countries, increasing and decreasing functions, and critical points on graphs. The chapter features graphs illustrating relative maxima and minima, concavity, absolute extrema, and the relationships between cost, revenue, and profit functions. It serves as a comprehensive guide for understanding the practical applications of derivatives in real-world scenarios including economics and engineering.

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Applications of Derivatives: Analyzing Functions and Optimization in Various Contexts

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  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

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