Practice Problems

Work through the odd-numbered problems 1-49. Once you have completed the problem set, check your answers.


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,
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