MA005: Calculus I

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 $\mathrm{C}=7-4=3$. $\mathrm{g}(\mathrm{x})=\mathrm{x}^{2}+3$.