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

$f(b) > f(a)$

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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