Applicatinns of Differntiation Chapter 178 CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises Mean V 3.2 Exercises a) Find (1 29. Vertical Motion The height of a ball t seconds after it is thrown upward from a height of 6 feet and with an initial (b) Use inter CONCEPT CHECK velocity of 48 feet per second is the In your own words, describe 1. Rolle's Theorem f(t) 16/2 48t + 6 Rolle's Theorem. (c) Finc In your own words, describe (a) Verify that f(1) = f(2). (b) According to Rolle's Theorem, what must the velocity be at some time in the interval (1, 2)? Find that time 2. Mean Value Theorem (d) Use tang the Mean Value Theorem. 30. Reorder Costs The ordering and transportation costC for components used in a manufacturing process is Writing In Exercises 3-6, explain why Rolle's Theorem does not apply to the function even though there exist a and b such that f(a) f(b). approximated by ( [п, Зл] 2' 4. f(x)= cot 3. f(x) -1, 1] 1 C(x) = 10 + x 6. f(x) (2-x/) 5. f(x)= 1- x - 1, where C is measured in thousands of dollars and x is the ordia + [-1, 1] [0, 2] -4 size in hundreds. Using Rolle's Theorem In Exercises 7-10, find the two x-intercepts of the function f and show that f'(x) = 0 at some point between the two x-intercepts. (a) Verify that C(3) = C(6). Fig (b) According to Rolle's Theorem, the rate of change of the cost must be 0 for some order size in the interval (3.6 38. Mean f(x) = Find that order size. 8. f(x) = x2 +6x 10. f(x) = -3xVx+ 1 7. f(x) = x2 - x - 2 (а) Fi (- Mean Value Theorem In Exercises 31 and 32, copy the graph and sketch the secant line to the graph through the points (a,f (a)) and (b,f(b). Then sketch any tangent lines to the graph for each value of c guaranteed by the Mean Value Theorem. To print an enlarged copy of the graph, go to MathGraphs.com. 9. f(x) = xx + 4 (b) U in Using Rolle's Theorem In Exercises 11-24, determine whether Rolle's Theorem can be applied to f on the closed interval [a, b]. If Rolle 's Theorem can be applied, find all values of c in the open interval (a, b) such that f'(c) = 0. If Rolle's Theorem cannot be applied, explain why not. th (c) F AP (d) U 31. y ta y 32. 11. f(x) x2 +3x, [0, 3] 12. f(x) x2-8x + 5, [2,6] 13. f(x) (x 1)x 2)(x - 3), [1, 3] 14. f(x) (x - 4) (x + 2)2, [-2,4] 15. f(x) x2/3- 1, [-8, 8] 16. f(x) 3- x -3, [0, 6] a b b a x2 2x- 3 x2-4 Writing In Exercises 33-36, explain why the Mean Value Theorem does not apply to the function f on the interval (0,6 17. f(x) = 18. f(x) 11 x + 2 x 1 [-2, 2] [-1, 3] 33. у 19. f(x) = sin x, [0, 2T] 34. 20. f(x) = cos x, [T, 371] 6- 6 21. f(x) = cos Tx, [0, 2] 39. f(x T 7TT 2' 6 22. f(x) = sin 3x, 5 5 41. f(x 4 23. f(x) = tan x, [0, T] 4. f 24. f(x) sec x, [T, 2] 3 - 42. f(x AS Using Rolle's Theorem In Exercises 25-28, use a graphing utility to graph the function on the closed interval [a, b]. Determine whether Rolle's Theorem can be applied to f on the interval and, if so, find all values of c in the open interval (a, b) such that f(c) =0. 43. f(x -+ x 1 2 3 4 5 6 45. f(x I 2 3 45 6 1 35. f(x) = 46. f 3 X 25, f(a)x - 1, 1-1,1] 47. fo 26. f)x- /, [0, 1] 36. fx) x - 3 48. f( 27 fl si1,0] 28. /) - tan m, [- LX 3 2 1