Practice Problems

1. In Fig. 9 , find the location of the number(s) "c" which Rolle's Theorem promises (guarantees).

Fig. 9

For problem 3, verify that the hypotheses of Rolle's Theorem are satisfied for each of the functions on the given intervals, and find the value of the number(s) " $c$ " which Rolle's Theorem promises exists.

3. (a) $\mathrm{f}(\mathrm{x})=\sin (\mathrm{x})$ on $[0, \pi]$
(b) $f(x)=\sin (x)$ on $[\pi, 5 \pi]$

5. Suppose you toss a ball straight up and catch it when it comes down. If $\mathrm{h}(\mathrm{t})$ is the height of the ball at time $t$, then what does Rolle's Theorem say about the velocity of the ball? Why is it easier to catch a ball which someone on the ground tosses up to you on a balcony, than for you to be on the ground and catch a ball which someone on a balcony tosses down to you?

7. If $\mathrm{f}(\mathrm{x})=|\mathrm{x}|$, then $\mathrm{f}(-1)=1$ and $\mathrm{f}(1)=1$ but $\mathrm{f}^{\prime}(\mathrm{x})$ is never equal to $0$. Why doesn't this function violate Rolle's Theorem?

9. If I take off in an airplane, fly around for awhile and land at the same place I took off from, then my starting and stopping heights are the same but the airplane is always moving. Doesn't this violate Rolle's theorem which says there is an instant when my velocity is $0$?

11. Use the corollary in problem 10 to justify the conclusion that the only root of $f(x)=x^{3}+5 x-18$ is $2$. (Suggestion: What could you conclude about $\mathrm{f}'$ if $\mathrm{f}$ had another root?)

In problems 13-15, verify that the hypotheses of the Mean Value Theorem are satisfied for each of the functions on the given intervals, and find the value of a number(s) " $c$ " which Mean Value Theorem guarantees.

13. (a) $f(x)=x^{2}$ on $[0,2]$
(b) $f(x)=x^{2}-5 x+8$ on $[1,5]$

15. (a) $f(x)=5-\sqrt{x}$ on $[1,9]$
(b) $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+1$ on $[1,7]$

17. If $f(x)=|x|$, then $f(-1)=1$ and $f(3)=3$ but $f^{\prime}(x)$ is never equal to $\frac{f(3)-f(-1)}{3-(-1)}=\frac{1}{2}$. Why doesn't this function violate the Mean Value Theorem?

In problem 19, you are a traffic court judge. In each case, a speeding ticket has been given and you need to decide if the ticket is appropriate.

19. The driver in the next case heard the toll taker and says, "Your Honor, my average velocity on that portion of the toll road was only 17 miles per hour, so I could not have been speeding. I don't deserve a ticket".

21. Find a function $f$ so that $f^{\prime}(x)=3 x^{2}+2 x+5$ and $f(1)=10$.

23. Find values for $\mathrm{A}$ and $\mathrm{B}$ so that the graph of the parabola $\mathrm{f}(\mathrm{x})=\mathrm{Ax}^{2}+\mathrm{B}$ is
(a) tangent to the line $\mathrm{y}=4 \mathrm{x}+5$ at the point $(1,9)$
(b) tangent to the line $\mathrm{y}=7-2 \mathrm{x}$ at the point $(2,3)$
(c) tangent to the parabola $\mathrm{y}=\mathrm{x}^{2}+3 \mathrm{x}-2$ at the point $(0,2)$

25. Sketch the graphs of several members of the "family" of functions whose derivatives always equal $3 \mathrm{x}^{2}$. Give a formula which defines every function in this family.

27. Assume that a rocket is fired from the ground and has the upward velocity shown in Fig. 11. Estimate the height of the rocket when $t=1, 2$, and $5$ seconds.

Fig. 11

29. Use the following information to determine an equation for $f(x): f "(x)=6, f^{\prime}(0)=4$, and $f(0)=-5$.

31. Define $\mathrm{A}(\mathrm{x})$ to be the area bounded by the $\mathrm{x}$-axis, the line $\mathrm{y}=3$, and a vertical line at $\mathrm{x}$ (Fig. 13).
(a) Find a formula for $\mathrm{A}(\mathrm{x})$?
(b) Determine $A '(x)$

Fig. 13

33. Define $\mathrm{A}(\mathrm{x})$ to be the area bounded by the $\mathrm{x}$-axis, the line $\mathrm{y}=2 \mathrm{x}+1$, and a vertical line at $x$ (Fig. 15).
(a) Find a formula for $\mathrm{A}(\mathrm{x})$?
(b) Determine $\mathrm{A}^{\prime}(\mathrm{x})$

Fig. 15

In problem 35, we have a list of numbers $a_{1}, a_{2}, a_{3}, a_{4}, \ldots$, and the consecutive differences between numbers in the list are $a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3}, \ldots$

35. If $a_{1}=5$ and the difference between consecutive numbers in the list is always 3, find a formula for $a_{n}$?

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.2-Mean-Value-Theorem.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. $\mathrm{c} \approx 3,10$, and $13$.

3. (a) $\mathrm{c}=\pi / 2$ (b) $\mathrm{c}=3 \pi / 2,5 \pi / 2,7 \pi / 2,9 \pi / 2$

5. Rolle's Theorem asserts that the velocity $\mathrm{h}^{\prime}(\mathrm{t})$ will equal $0$ at some point between the time the ball is tossed and the time it comes back down. The ball is not moving as fast when it reaches the balcony from below.

7. The function does not violate Rolle's Thm. because the function does not satisfy the hypotheses of the theorem: $\mathrm{f}$ is not differentiable at $0$, a point in the interval $-1 < \mathrm{x} < 1$.

9. No. The velocity is not the same as the rate of change of altitude, since altitude is only one of the components of position. Rolle's Theorem only says there was a time when my altitude was not changing.

11. Since $\mathrm{f}^{\prime}(\mathrm{x})=3 \mathrm{x}^{2}+5, \mathrm{f}^{\prime}(\mathrm{x})=0$ has no real roots. If $\mathrm{f}(\mathrm{x})=0$ for a value of $\mathrm{x}$ other than 2, then by the corollary from Problem 8, we would have an immediate contradiction.

13. (a) $\mathrm{f}(0)=0$, $\mathrm{f}(2)=4$, $\mathrm{f}^{\prime}(\mathrm{c})=2 c$. $\frac{4-0}{2-0}=2 \mathrm{c}$ implies that $\mathrm{c}=1$.
(b) $f(1)=4, f(5)=8$, $f^{\prime}(c)=2 c-5$. $\frac{8-4}{5-1}=2 c-5$ implies that $c=3$.

15. (a) $f(1)=4, f(9)=2$, $f^{\prime}(c)=\frac{-1}{2 \sqrt{c}} \cdot \frac{2-4}{1-9}=\frac{-1}{2 \sqrt{c}}$ implies $\frac{-1}{4}=\frac{-1}{2 \sqrt{c}}$ so $c=4$.
(b) $f(1)=3$, $f(7)=15$, $f^{\prime}(c)=2$. $\frac{15-3}{7-1}=2$ so any $c$ between $1$ and $7$ will do.

17. The hypotheses are not all satisfied since $\mathrm{f}^{\prime}(\mathrm{x})$ does not exist at $\mathrm{x}=0$ which is between $-1$ and $3$.

19. Guilty. All we know is that $\mathrm{f}^{\prime}(\mathrm{c})=17$ at some point, but this does not prove that the motorist "could not have been speeding".

21. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2}+5 \mathrm{x}+\mathrm{c} . \mathrm{f}(1)=7+\mathrm{c}=10$ when $\mathrm{c}=3$. Therefore, $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{x}^{2}+5 \mathrm{x}+3$.

23. (a) $\quad \mathrm{f}^{\prime}(\mathrm{x})=2 \mathrm{Ax}$. We need $\mathrm{A}(1)^{2}+\mathrm{B}=9$ and $2 \mathrm{~A}(1)=4$ so $\mathrm{A}=2$ and $\mathrm{B}=7$ and $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^{2} + 7$.
(b) $\mathrm{A}(2)^{2}+\mathrm{B}=3$ and $2 \mathrm{~A}(2)=-2$ so $\mathrm{A}=-1 / 2$ and $\mathrm{B}=5$ and $\mathrm{f}(\mathrm{x})=\frac{-1}{2} \mathrm{x}^{2}+5$.
(c) $\mathrm{A}(0)^{2}+\mathrm{B}=2$ and $\mathbf{2} \mathbf{A}(0)=3$. There is no such $\mathrm{A}$. The point $(0,2)$ is not on the parabola $y=x^{2}+3 x-2$

25. $\mathrm{f}(\mathrm{x})=\mathrm{x}^{3}+\mathrm{C}$, a family of "parallel" curves for different values of $\mathrm{C}$.

27. $\mathrm{v}(\mathrm{t})=300$. Assuming the rocket left the ground at $\mathrm{t}=0$, we have $\mathrm{y}(1)=300 \mathrm{ft}, \mathrm{y}(2)=600 \mathrm{ft}, \mathrm{y}(5)=1500 \mathrm{ft}$.

29. $\mathrm{f}^{\prime \prime}(\mathrm{x})=6, \mathrm{f}^{\prime}(0)=4, \mathrm{f}(0)=-5 . \mathrm{f}(\mathrm{x})=3 \mathrm{x}^{2}+4 \mathrm{x}-5$.

31. (a) $\mathrm{A}(\mathrm{x})=3 \mathrm{x}$
(b) $A^{\prime}(x)=3$.

33. (a) $\mathrm{A}(\mathrm{x})=\mathrm{x}^2+\mathrm{x}$
(b) $A^{\prime}(x)=2x + 1$.

35. $\mathrm{a}_{1}=5, \mathrm{a}_{2}=\mathrm{a}_{1}+3=5+3=8, \mathrm{a}_{3}=\mathrm{a}_{2}+3=(5+3)+3=11, \mathrm{a}_{4}=\mathrm{a}_{3}+3=(5+3)+3+3=14$.

In general, $a_{n}=5+3(n-1)=2+3 n$.