MA005: Calculus I

1. (a) $\mathrm{f}(\mathrm{t})=$ number of workers unemployed at time $t$. $\mathrm{f}^{\prime}(\mathrm{t}) > 0$ and $\mathrm{f}^{\prime \prime}(\mathrm{t}) < 0$.
(b) $\mathrm{f}(\mathrm{t})=$ profit at time $t$. $\mathrm{f}^{\prime}(\mathrm{t}) < 0$ and $\mathrm{f}^{\prime \prime}(\mathrm{t}) > 0$.
(c) $f(t)=$ population at time $t$. $(f^{\prime}(t) > 0$ and $f^{\prime \prime}(t) > 0$.

3. See Fig. 3.4P3.

Fig. 3.4P3.

5. (a) Concave up on $(0,2),(2,3+),(6,9)$. Concave down on $(3+, 6)$. (A small technical note: we have defined concavity only at points where the function is differentiable, so we exclude the endpoints and points where the function is not differentiable from the intervals of concave up and concave down.)

7. $\mathrm{g}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-9 \mathrm{x}+7$. $\mathrm{g}^{\prime \prime} (\mathrm{x})=6 \mathrm{x}-6$. $\mathrm{g}^{\prime \prime} (-1) < 0$ so $(-1,12)$ is a local maximum. $\mathrm{g}^{\prime \prime} (3) > 0$ so $(3,-20)$ is a local minimum.

9. $\mathrm{f}(\mathrm{x})=\sin ^{5}(\mathrm{x}) \cdot \mathrm{f}^{\prime \prime} (\mathrm{x})=5\left\{-\sin ^{5}(\mathrm{x})+4 \sin ^{3}(\mathrm{x}) \cdot \cos ^{2}(\mathrm{x})\right\}$. $\mathrm{f}^{\prime \prime} (\pi / 2) < 0$ so $(\pi / 2,1)$ is a local maximum. $f^{\prime \prime} (3 \pi / 2) > 0$ so $(3 \pi / 2,-1)$ is a local minimum.
$f^{\prime \prime} (\pi)=0$ and $f$ changes concavity at $x=\pi$ so $(\pi, 0)$ is an inflection point.

11. $\mathrm{d}$ and $\mathrm{e}$.

13. (a) $0$
(b) at most $1$
(c) at most $\mathrm{n}-2$.

15.

$\begin{array}{l|l|l|l}x & g(x) & g^{\prime}(x) & g^{\prime \prime}(x) \\\hline 0 & - & + & + \\1 & + & 0 & - \\2 & - & - & + \\3 & 0 & + & +\end{array}$

17. See Fig. 3.4P17.

Fig. 3.4P17.