Question 1

Which of the following is a necessary condition for Rolle's Theorem to apply to a function $$f on$$ $$[a, b]$$?

Accepted Answer

The function $$f$$ must satisfy $$f(a) = f(b).$$

Question 2

Given $$f(x) = x(x-1)^2$ on $$[0,1]$$, which value of c$ in (0$$,1)$$ is guaranteed by Rolle's Theorem such that f$$'(c) = 0$$?

Accepted Answer

$$c = \frac{1}{3}$$

Question 3

For the function $$f(x) = 3x + x$$^{5/3}$$ on $$[-1, 1]$$, why does Rolle's Theorem not apply?

Accepted Answer

Because $$f is $$not differentiable at $$x = 0.$$

Question 4

Which equation represents the conclusion of the Mean Value Theorem for a function $$f$$ continuous on $$[a, b]$$ and differentiable on $$(a, b)$$?

Accepted Answer

$$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Question 5

Suppose a car travels from point A to point B, 120 miles apart, in 2 hours. Which theorem guarantees that at some instant, the car's instantaneous speed was exactly 60 mph?

Accepted Answer

Mean Value Theorem

Question 6

Let $$f(x) = 7 - x^2$ on $$[-1, 2]$$. Find the value of c$ guaranteed by the Mean Value Theorem.

Accepted Answer

$$c = \frac{1}{2}$$

Question 7

For $$f(x) = x + \frac{1}{x} on$$ $$[1, 3]$$, does the Mean Value Theorem apply? Why?

Accepted Answer

Yes, because $$f is $$continuous and differentiable on $$[1, 3]$$ and $$0 is $$not in the interval.

Question 8

A chemical reaction's temperature $$T(t) is $$measured continuously over time $$t$$ from $$t = 0 to$$ $$t = 10$$ minutes. If $$T(0) = 20^	ext{o}C $$and $$T(10) = 50^	ext{o}C$$, what does the Mean Value Theorem guarantee?

Accepted Answer

There is a time $$c$$ where $$T'(c) = 3$$ $$^	ext{o}C/$$min.

Question 9

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, and $$f(a) = f(b)$$, what does Rolle's Theorem guarantee?

Accepted Answer

There is a point $$c in$$ $$(a, b)$$ where $$f'(c) = 0.$$

Question 10

A runner completes a 5 km race in 25 minutes. Which statement is true according to the Mean Value Theorem?

Accepted Answer

At some instant, the runner's instantaneous speed was exactly 0.2 km/min.

Question 11

Which of the following functions does NOT satisfy the hypotheses of the Mean Value Theorem on $$[1, 3]$$?

Accepted Answer

$$f(x) = \frac{1}{x-2}$$

Question 12

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f'(x) is $$always positive on $$(a, b)$$, what can you conclude about $$f(b)$$ and $$f(a)$$?

Accepted Answer

$$f(b) \u003e f(a)$$

Question 13

A pharmaceutical company tests a drug's concentration $$C(t) in $$the bloodstream, measured every minute for 30 minutes. If $$C(0) = 0 mg/L $$and $$C(30) = 60 mg/L$$, what does the Mean Value Theorem guarantee?

Accepted Answer

There is a time $$c$$ where $$C'(c) = 2 mg/L $$per minute.

Question 14

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, and $$f(a) 
eq f(b)$$, which theorem can be applied?

Accepted Answer

Mean Value Theorem

Question 15

Let $$f(x) = x^2$ on $$[0, 2]$$. Find the value c$ guaranteed by the Mean Value Theorem.

Accepted Answer

$$c = 1$$

Question 16

A patient’s heart rate $$H(t) is $$monitored continuously from $$t = 0 to$$ $$t = 60$$ minutes. If $$H(0) = 70$$ bpm and $$H(60) = 100$$ bpm, what does the Mean Value Theorem guarantee?

Accepted Answer

There is a time $$c$$ where $$H'(c) = 0.5$$ bpm/min.

Question 17

Which of the following is NOT a hypothesis of Rolle's Theorem?

Accepted Answer

$$f is $$increasing on $$[a, b]$$

Question 18

Let $$f(x) = x^3$ - 3x on$$ $$[-2, 2]. $$Does Rolle's Theorem apply? If so, how many points $$c in$$ $$(-2, 2)$$ satisfy $$f'(c) = 0$$?

Accepted Answer

Yes, and there are two such points.

Mean Value Theorem and Rolle's Theorem: 4.2

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