Practice Problems

In problems 1 and 3, sketch the graph of the derivative of each function.

1. Use Fig. 20.

Fig. 20

3. Use Fig. 22.

Fig. 22

In problem 5, the graph of the height of a helicopter is shown. Sketch the graph of the upward velocity of the helicopter.

5. Use Fig. 24,

Fig. 24

7. In Fig. 26, match the graphs of the functions with those of their derivatives.

Fig. 26

9. Use the Second Shape Theorem to show that $f(x)=\ln (x)$ is monotonic increasing on $(0, \infty)$.

11. A student is working with a complicated function $\mathrm{f}$ and has shown that the derivative of $\mathrm{f}$ is always positive. A minute later the student also claims that $\mathrm{f}(\mathrm{x})=2$ when $\mathrm{x}=1$ and when $\mathrm{x}=\pi$. Without checking the student's work, how can you be certain that it contains an error?

13. Fig. 29 shows the graph of the derivative of a continuous function $\mathrm{g}$.
(a) List the critical numbers of $g$.
(b) For what values of $\mathrm{x}$ does $\mathrm{g}$ have a local maximum?
(c) For what values of $\mathrm{x}$ does $\mathrm{g}$ have a local minimum?

Fig. 29

In problem 15, the graph of the upward velocities of several helicopters are shown. Use each graph to determine when each helicopter was at a relatively maximum and minimum height.

15. Use Fig. 31.

Fig. 31

In problems 17-21, use information from the derivative of each function to help you graph the function. Find all local maximums and minimums of each function.

17. $f(x)=x^{3}-3 x^{2}-9 x-5$

19. $h(x)=x^{4}-8 x^{2}+3$

21. $\mathrm{r}(\mathrm{t})=\frac{2}{\mathrm{t}^{2}+1}$

23. $\mathrm{f}(\mathrm{x})=2 \mathrm{x}+\cos (\mathrm{x})$ so $\mathrm{f}(0)=1$. Without graphing the function, you can be certain that $\mathrm{f}$ has how many positive roots? (zero, one, two, more than two)

25. $\mathrm{h}(\mathrm{x})=\mathrm{x}^{3}+9 \mathrm{x}-10$ has a root at $\mathrm{x}=1$. Without graphing $\mathrm{h}$, show that $\mathrm{h}$ has no other roots.

27. Each of the following statements is false. Give (or sketch) a counterexample for each statement.
(a) If $\mathrm{f}$ is increasing on an interval I, then $\mathrm{f}^{\prime}(\mathrm{x}) > 0$ for all $\mathrm{x}$ in $\mathrm{I}$.
(b) If $\mathrm{f}$ is increasing and differentiable on $\mathrm{I}$, then $\mathrm{f}^{\prime}(\mathrm{x}) > 0$ for all $\mathrm{x}$ in $\mathrm{I}$.
(c) If cars $A$ and $B$ always have the same speed, then they will always be the same distance apart.

29. (a) Give the equations of several different functions $h$ which all have the same derivative $h^{\prime}(x)=2 x$.
(b) Give the equation of the function $f$ with derivative $f^{\prime}(x)=2 x$ which also satisfies $f(3)=20$.
(c) Give the equation of the function $\mathrm{g}$ with $\mathrm{g}^{\prime}(\mathrm{x})=2 \mathrm{x}$, and the graph of $\mathrm{g}$ goes through $(2,7)$.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.3-First-Derivative.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. See Fig. 3.3P1.

Fig. 3.3P1.

3. See Fig. 3.3P3.

Fig. 3.3P3.

5. See Fig. 3.3P5.

Fig. 3.3P5.

7. $\mathrm{A}-\mathrm{Q}, \mathrm{B}-\mathrm{P}, \mathrm{C}-\mathrm{R}$

9. $f^{\prime}(x)=\frac{1}{x} > 0$ for $x > 0$ so $f(x)=\ln (x)$ is increasing on $(0, \infty)$.

11. If $f$ is increasing then $f(1) < f(\pi)$ so $f(1)$ and $f(\pi)$ cannot both equal $2$.

13. (a) $x=3, x=8$
(b) maximum at $x=8$
(c) none (or only at right endpoint)

15. Relative maximum height at $\mathrm{x}=6$. Relative minimum height at $\mathrm{x}=8$.

17. $f(x)=x^{3}-3 x^{2}-9 x-5$ has a relative minimum at $(3,-32)$ and a relative maximum at $(-1,0)$.

19. $\mathrm{h}(\mathrm{x})=\mathrm{x}^{4}-8 \mathrm{x}^{2}+3$ has a relative maximum at $(0,3)$ and relative minimums at $(2,-13)$ and $(-2,-13)$.

21. $\mathrm{r}(\mathrm{t})=2\left(\mathrm{t}^{2}+1\right)^{-1}$ has a relative maximum at $(0,2)$ and no relative minimums.

23. No positive roots. $f(x)=2 x+\cos (x)$ is continuous. $f(0)=1 > 0$. Since $\mathrm{f}^{\prime}(\mathrm{x})=2-\sin (\mathrm{x}) > 0$ for all $\mathrm{x}$, $\mathrm{f}$ is increasing and never decreases back to the $\mathrm{x}$-axis (a root).

25. $\mathrm{h}(\mathrm{x})=\mathrm{x}^{3}+9 \mathrm{x}-10$ and $\mathrm{h}(1)=0$. $\mathrm{h}^{\prime}(\mathrm{x})=3 \mathrm{x}^{2}+9=3\left(\mathrm{x}^{2}+3\right) > 0$ for all $\mathrm{x}$ so $\mathrm{h}$ is always increasing and can cross the $\mathrm{x}$-axis at most at one place. Since the graph of $\mathrm{h}$ crosses the $\mathrm{x}$-axis at $\mathrm{x}=1$, that is the only root of $\mathrm{h}$.

27.

Fig. 3.3P27

29. (a) $\mathrm{h}(\mathrm{x})=\mathrm{x}^{2}$, $\mathrm{x}^{2}+1$, $\mathrm{x}^{2}-7$, or, in general, $\mathrm{x}^{2}+\mathrm{C}$ for any constant $\mathrm{C}$.
(b) $f(x)=x^{2}+C$ for some value $C$ and $20=f(3)=3^{2}+C$ so $C=20-9=11$. $f(x)=x^{2}+11$.
(c) $$\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+\mathrm{C}$ for some value $\mathrm{C}$ and $7=\mathrm{g}(2)=2^{2}+\mathrm{C}$ so $\mathrm{C}=7-4=3$. $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+3$.