MA005: Calculus I

1. $\ln (5 x)$

3. $\ln \left(x^{k}\right)$

5. $\ln (\cos (x))$

7. $\log _{2} 5 \mathrm{x}$

9. $\ln (\sin (x))$

11. $\log _{2}(\sin (x))$

13. $\log _{5} 5^{x}$

15. $x \cdot \ln (3 x)$

17. $\frac{\ln (x)}{x}$

19. $\ln (\sqrt{5 x-3})$

21. $\frac{\mathbf{d}}{\mathbf{d} \mathbf{w}} \cos (\ln (\mathrm{w}))$

23. $\frac{\mathbf{d}}{\mathbf{d t}} \ln (\sqrt{t+1})$

25. $\mathrm{D}\left(5^{\sin (\mathrm{x})}\right)$

27. $\frac{\mathbf{d}}{\mathbf{d x}} \ln (\sec (\mathrm{x})+\tan (\mathrm{x}))$

29. Find a point $\mathrm{P}$ on the graph of $\mathrm{f}(\mathrm{x})=\ln (\mathrm{x})$ so the tangent line to $\mathrm{f}$ at $\mathrm{P}$ goes through the origin.

31. Rumor. The percent of a population, $\mathrm{p}(\mathrm{t})$, who have heard a rumor by time $\mathrm{t}$ is often modeled $p(t)=\frac{100}{1+A e^{-t}}=100\left(1+A e^{-t}\right)^{-1}$ for some positive constant A. Calculate how fast the rumor is spreading, $\frac{\mathbf{d} \mathrm{p}(\mathrm{t})}{\mathbf{d t}}$.

In problems 33 – 41, find a function with the given derivative. 

33. $\mathrm{f}^{\prime}(\mathrm{x})=\frac{8}{\mathrm{x}}$

35. $f^{\prime}(x)=\frac{\cos (x)}{3+\sin (x)}$

37. $\mathrm{g}^{\prime}(\mathrm{x})=3 \mathrm{e}^{5 \mathrm{x}}$

39. $f^{\prime}(x)=2 x \cdot e^{\left(x^{2}\right)}$

41. $h^{\prime}(x)=\frac{\cos (x)}{\sin (x)}$

Problems 43 – 47 involve parametric equations.

43. At time $t$ minutes, robot $A$ is at $(t, 2 t+1)$ and robot $B$ is at $\left(t^{2}, 2 t^{2}+1\right)$.

(a) Where is each robot when $\mathrm{t}=0$ and $\mathrm{t}=1$ ?

(b) Sketch the path each robot follows during the first minute.

(c) Find the slope of the tangent line, $\mathrm{dy} / \mathrm{dx}$, to the path of each robot at $\mathrm{t}=1$ minute.

(d) Find the speed of each robot at $\mathrm{t}=1$ minute.

(e) Discuss the motion of a robot which follows the path $(\sin (\mathrm{t}), 2 \sin (\mathrm{t})+1)$ for 20 minutes.

45. For the parametric graph in Fig. 9, determine whether $\frac{d x}{d t}, \frac{d y}{d t}$ and $\frac{d y}{d x}$ are positive, negative or zero when $\mathrm{t}=1$ and $\mathrm{t}=3$.

47. $x(t)=R \cdot(t-\sin (t)), y(t)=R \cdot(1-\cos (t))$. (a) Graph $(x(t), y(t))$ for $0 \leq t \leq 4 \pi$.

(b) Find $\mathrm{dx} / \mathrm{dt}, \mathrm{dy} / \mathrm{dt}$, the tangent slope $\mathrm{dy} / \mathrm{dx}$, and speed when $\mathrm{t}=\pi / 2$ and $\pi$.

(The graph of $(\mathrm{x}(\mathrm{t}), \mathrm{y}(\mathrm{t}))$ is called a cycloid and is the path of a light attached to the edge of a rolling wheel with radius $\mathrm{R}$).

49. Describe the path of a robot whose location at time $\mathrm{t}$ is

(a) $(3 \cdot \cos (\mathrm{t}), 5 \cdot \sin (\mathrm{t}))$

(b) $(\mathrm{A} \cdot \cos (\mathrm{t}), \mathrm{B} \cdot \sin (\mathrm{t}))$

(c) Give the parametric equations so the robot will move along the same path as in part (a) but in the opposite direction.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.6-Some-Applications-of-the-Chain-Rule.pdf
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