- 4 2 Exercises 1 The Graph Of A Function F Is Shown Verify That F Satisfies The Hypotheses Of Rolle S Theorem On The In 1 (133.64 KiB) Viewed 18 times
4.2 Exercises 1. The graph of a function f is shown. Verify that f satisfies the hypotheses of Rolle's Theorem on the interval [0, 8]. Then estimate the value(s) of c that satisfy the conclusion of Rolle's Theorem on that interval. 1 0 1 1 2. Draw the graph of a function defined on [0, 8] such that f(0) = f(8) = 3 and the function does not satisfy the con- clusion of Rolle's Theorem on [0, 8]. 3. The graph of a function g is shown. 0 1 y = f(x) y = g(x) x (a) Verify that g satisfies the hypotheses of the Mean Value Theorem on the interval [0, 8]. (b) Estimate the value(s) of c that satisfy the conclusion of the Mean Value Theorem on the interval [0, 8]. (c) Estimate the value(s) of that satisfy the conclusion of the Mean Value Theorem on the interval [2, 6]. 4. Draw the graph of a function that is continuous on [0, 8] where f(0) = 1 and f(8) = 4 and that does not satisfy the conclusion of the Mean Value Theorem on [0, 8]. 5-8 The graph of a function f is shown. Does f satisfy the hypotheses of the Mean Value Theorem on the interval [0, 5]? If so, find a value c that satisfies the conclusion of the Mean Value Theorem on that interval. 5. y 7. 1 0 ya 1 1 01 x 6. y 8. 1 0 ya 1 1 01 X 9-12 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. 9. f(x) = 2x² - 4x + 5, [-1,3] 10. f(x) = x³ 2x² - 4x + 2, [-2,2] 11. f(x) = sin(x/2), [T/2, 3π/2] 12. f(x) = x + 1/x, [1,2] X 13. Let f(x) = 1 - x2/3. Show that f(-1) = f(1) but there is no number c in (-1, 1) such that f'(c) = 0. Why does this not contradict Rolle's Theorem? 14. Let f(x) = tan x. Show that f(0) = f(7) but there is no num- ber c in (0,7) such that f'(c) = 0. Why does this not contra- dict Rolle's Theorem?
296 CHAPTER 4 Applications of Differentiation 15-18 Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. 15. f(x) = 2x²-3x+1, [0, 2] 16. f(x) = x³ 3x + 2, [-2,2] 17. f(x) = ln x, [1,4] 18. f(x)=1/x, [1,3]