4. Applications of Derivatives

11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.

ƒ(x) = 1 - x²⸍³ ; [-1, 1]

11–18. Rolle’s Theorem Determine whether Rolle’s Theorem applies to the following functions on the given interval. If so, find the point(s) guaranteed to exist by Rolle’s Theorem.

g(x) = x³ - x² - 5x - 3; [-1, 3]

Let ƒ(x) = x²⸍³ . Show that there is no value of c in the interval (-1, 8) for which ƒ' (c) = (ƒ(8) - ƒ (-1)) / (8 - (-1)) and explain why this does not violate the Mean Value Theorem.

Running pace Explain why if a runner completes a 6.2-mi (10-km) race in 32 min, then he must have been running at exactly 11 mi/hr at least twice in the race. Assume the runner’s speed at the finish line is zero.

Mean Value Theorem for quadratic functions Consider the quadratic function f(x) = Ax² + Bx + C, where A, B, and C are real numbers with A ≠ 0. Show that when the Mean Value Theorem is applied to f on the interval [a,b], the number guaranteed by the theorem is the midpoint of the interval.

Means

b. Show that the point guaranteed to exist by the Mean Value Theorem for f(x) = 1/x on [a,b], where 0 < a < b, is the geometric mean of a and b; that is, c = √ab.

Equal derivatives Verify that the functions f(x) = tan² x and g(x) = sec² x have the same derivative. What can you say about the difference f - g? Explain.

100-m speed The Jamaican sprinter Usain Bolt set a world record of 9.58 s in the 100-meter dash in the summer of 2009. Did his speed ever exceed 30 km/hr during the race? Explain.

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = ln 2x; [1,e]

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.

a. What is the average rate of change in the population over the interval [0, 8]?

21–32. Mean Value Theorem Consider the following functions on the given interval [a, b].

a. Determine whether the Mean Value Theorem applies to the following functions on the given interval [a, b].

b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem.

ƒ(x) = x + 1/x; [1,3]

Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.

b. At what point of the interval [0, 8] is the instantaneous rate of change equal to the average rate of change?

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

a. A bamboo shoot was 500 cm tall at 10:00 A.M. and 515 cm tall at 3:00 P.M. Compute the average growth rate of the bamboo shoot in cm/hr over the period of time from 10:00 A.M. to 3:00 P.M.

Growth rate of bamboo Bamboo belongs to the grass family and is one of the fastest growing plants in the world.

b. Based on the Mean Value Theorem, what can you conclude about the instantaneous growth rate of bamboo measured in millimeters per second between 10:00 A.M. and 3:00 P.M.?

Suppose f'(x) < 2, for all x ≥ 2, and f(2) = 7. Show that f(4) < 11.