MA005: Calculus I

1. (a) $2$    (b) $0$   (c) DNE (does not exist)   (d) $1.5$

3. (a) $1$    (b) $3$    (c) $1$    (d) $\approx 0.8$

5. See Fig. 1.2P5

7. (a) $2$     (b) $1$   (c) DNE     (d) $2$        (e) $2$

    (f) $2$     (g) $1$    (h) $2$         (i) DNE

9. (a) When $\mathrm{v}=0, \mathrm{~L}=\mathrm{A}$.

(b) $\lim\limits_{v \rightarrow c^{-}} \mathrm{A} \sqrt{1-\frac{\mathrm{v}^{2}}{\mathrm{c}^{2}}}=0$

11. (a) $4$    (b) $1$    (c) $2$    (d) $0$    (e) $1$    (f) $1$

13. (a) Slope of the line tangent to the graph of $\mathrm{y}=\cos (\mathrm{x})$ at the point $(0,1)$. (b) Slope $=0$.

15. (a) $approx 1$    (b) $\approx 3.43$     (c) $\approx 4$

17. $\begin{array}{ll}\text { at } x=-1: \text { a } & \text { at } x=0: b \quad \text { at } x=1: c \quad d=2: d \\ \text { at } x=3: c \text { at } x=4: b & \text { at } x=5: a\end{array}$

19. Verify each step.

21. Several different lists will work. Here is one example.

Put $a_{n}=1 /(n \pi)$ for $n=1,2,3, \ldots$ so $a_{n}$ approaches 0 and $\sin \left(a_{n}\right)=\sin \left(\frac{1}{1 /(n \pi)}\right)=\sin (n \pi)=0$ for all $n$.

Put $\mathrm{bn}$ $=\frac{1}{2 \mathrm{n} \pi+\pi / 2}$ for $\mathrm{n}=1,2,3, \ldots$ so $\mathrm{bn}$ approaches 0 and $\sin (b n)=\sin (2 n \pi+\pi / 2)=\sin (\pi / 2)=1$ for all $n$.

Put $\mathrm{bn}$ $=\frac{1}{2 \mathrm{n} \pi+\pi / 2}$ for $\mathrm{n}=1,2,3, \ldots$ so $\mathrm{bn}$ approaches 0 and $\sin (b n)=\sin (2 n \pi+\pi / 2)=\sin (\pi / 2)=1$ for all $n$.