MA005: Calculus I

1. (a) $\mathrm{h}$ has a root at $\mathrm{x}=1$.
(b) limits of $\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{x}) / \mathrm{g}(\mathrm{x})$: as $\mathrm{x} \rightarrow 1^{+}$ is $0$: as $\mathrm{x} \rightarrow 1^{-}$ is $0$: as $\mathrm{x} \rightarrow 3^{+}$ is $-\infty$: as $\mathrm{x} \rightarrow 3^{-}$ is $+\infty$
(c) $\mathrm{h}$ has a vertical asymptote at $\mathrm{x}=3$

3. limits of $\mathrm{h}(\mathrm{x})=\mathrm{f}(\mathrm{x}) / \mathrm{g}(\mathrm{x})$: as $\mathrm{x} \rightarrow 2^{+}$ is $+\infty$: as $\mathrm{x} \rightarrow 2^{-}$ is $-\infty$: as $\mathrm{x} \rightarrow 4^{+}$ is $0$: as $\mathrm{x} \rightarrow 4^{-}$ is $0$

5. $0$

7. $-3$

9. $0$

11. $DNE$

13. $2 / 3$

15. $0$

17. $-7$

19. $0$

21. $\cos (0)=1$

23. $\ln (1)=0$

25. (a) $\mathrm{V}(\mathrm{t})=50+4 \mathrm{t}$ gallons, and $\mathrm{A}(\mathrm{t})=0.8 \mathrm{t}$ pounds of salt
(b) $\mathrm{C}(\mathrm{t})=\frac{\text { amount of salt }}{\text { total amount of liquid }}=\frac{\mathrm{A}(\mathrm{t})}{\mathrm{V}(\mathrm{t})}=\frac{0.8 \mathrm{t}}{50+4 \mathrm{t}}$
(c) "after a long time" (as $\mathrm{t} \rightarrow \infty$), $\mathrm{C}(\mathrm{t}) \rightarrow 0.8 / 4=0.2$ pounds of salt per gallon.
(d) $\mathrm{V}(\mathrm{t})=200+4 \mathrm{t}, \mathrm{A}(\mathrm{t})=0.8 \mathrm{t}, \mathrm{C}(\mathrm{t})=\frac{0.8 \mathrm{t}}{200+4 \mathrm{t}} \rightarrow 0.8 / 4=0.2$ pounds of salt per gallon.

27. $+\infty$

29. $-\infty$

31. $-\infty$

33. $-\infty$

35. $+\infty$

37. $-\infty$

39. $1$

41. $-\infty$

43. Horizontal: $\mathrm{y}=0$. Vertical: $\mathrm{x}=0$.

45. Horizontal: $\mathrm{y}=0$. Vertical: $\mathrm{x}=3$ and $\mathrm{x}=1$

47. Horizontal: $\mathrm{y}=1$.

49. Horizontal: $\mathrm{y}=1$. Vertical: $\mathrm{x}=1$.

51. $\mathrm{y}=2 \mathrm{x}+1$. $\mathrm{x}=0$

53. $y=\sin (x)$. $x=2$

55. $\mathrm{y}=\mathrm{x}^{2}$

57. $\mathrm{y}=\cos (\mathrm{x})$. $\mathrm{x}=3$.

59. $\mathrm{y}=\sqrt{\mathrm{x}}$. $\mathrm{x}=-3$