## Problem Set 73: Derivatives

1. How is the slope of a linear function similar to the derivative?

2. What is the difference between the average rate of change of a function on the interval $\left[x,x+h\right]$ and the derivative of the function at $x?$

3. A car traveled 110 miles during the time period from 2:00 P.M. to 4:00 P.M. What was the car’s average velocity? At exactly 2:30 P.M., the speed of the car registered exactly 62 miles per hour. What is another name for the speed of the car at 2:30 P.M.? Why does this speed differ from the average velocity?

4. Explain the concept of the slope of a curve at point $x$.

5. Suppose water is flowing into a tank at an average rate of 45 gallons per minute. Translate this statement into the language of mathematics.

For the following exercises, use the definition of derivative $\underset{h\to 0}{\mathrm{lim}}\dfrac{f\left(x+h\right)-f\left(x\right)}{h}$ to calculate the derivative of each function.

6. $f\left(x\right)=3x - 4$

7. $f\left(x\right)=-2x+1$

8. $f\left(x\right)={x}^{2}-2x+1$

9. $f\left(x\right)=2{x}^{2}+x - 3$

10. $f\left(x\right)=2{x}^{2}+5$

11. $f\left(x\right)=\frac{-1}{x - 2}$

12. $f\left(x\right)=\frac{2+x}{1-x}$

13. $f\left(x\right)=\frac{5 - 2x}{3+2x}$

14. $f\left(x\right)=\sqrt{1+3x}$

15. $f\left(x\right)=3{x}^{3}-{x}^{2}+2x+5$

16. $f\left(x\right)=5$

17. $f\left(x\right)=5\pi$

For the following exercises, find the average rate of change between the two points.

18. $\left(-2,0\right)$ and $\left(-4,5\right)$

19. $\left(4,-3\right)$ and $\left(-2,-1\right)$

20. $\left(0,5\right)$ and $\left(6,5\right)$

21. $\left(7,-2\right)$ and $\left(7,10\right)$

For the following polynomial functions, find the derivatives.

22. $f\left(x\right)={x}^{3}+1$

23. $f\left(x\right)=-3{x}^{2}-7x=6$

24. $f\left(x\right)=7{x}^{2}$

25. $f\left(x\right)=3{x}^{3}+2{x}^{2}+x - 26$

For the following functions, find the equation of the tangent line to the curve at the given point $x$ on the curve.

26. $\begin{array}{ll}f\left(x\right)=2{x}^{2}-3x\hfill & x=3\hfill \end{array}$

27. $\begin{array}{ll}f\left(x\right)={x}^{3}+1\hfill & x=2\hfill \end{array}$

28. $\begin{array}{ll}f\left(x\right)=\sqrt{x}\hfill & x=9\hfill \end{array}$

For the following exercise, find $k$ such that the given line is tangent to the graph of the function.

29. $\begin{array}{ll}f\left(x\right)={x}^{2}-kx,\hfill & y=4x - 9\hfill \end{array}$

For the following exercises, consider the graph of the function $f$ and determine where the function is continuous/discontinuous and differentiable/not differentiable.

30.

31.

32.

33.

For the following exercises, use Figure 20 to estimate either the function at a given value of $x$ or the derivative at a given value of $x$, as indicated.

Figure 20

34. $f\left(-1\right)$

35. $f\left(0\right)$

36. $f\left(1\right)$

37. $f\left(2\right)$

38. $f\left(3\right)$

39. \begin{align}{f}^{\prime }\left(-1\right)\end{align}

40. \begin{align}{f}^{\prime }\left(0\right)\end{align}

41. \begin{align}{f}^{\prime }\left(1\right)\end{align}

42. \begin{align}{f}^{\prime }\left(2\right)\end{align}

43. \begin{align}{f}^{\prime }\left(3\right)\end{align}

44. Sketch the function based on the information below:

\begin{align}{f}^{\prime }\left(x\right)=2x, f\left(2\right)=4\end{align}

45. Numerically evaluate the derivative. Explore the behavior of the graph of $f\left(x\right)={x}^{2}$ around $x=1$ by graphing the function on the following domains: $\left[0.9,1.1\right]$ , $\left[0.99,1.01\right]$ , $\left[0.999,1.001\right]$, and $\left[0.9999,1.0001\right]$. We can use the feature on our calculator that automatically sets Ymin and Ymax to the Xmin and Xmax values we preset. (On some of the commonly used graphing calculators, this feature may be called ZOOM FIT or ZOOM AUTO). By examining the corresponding range values for this viewing window, approximate how the curve changes at $x=1$, that is, approximate the derivative at $x=1$.

For the following exercises, explain the notation in words. The volume $f\left(t\right)$ of a tank of gasoline, in gallons, $t$ minutes after noon.

46. $f\left(0\right)=600$

47. \begin{align}f^{\prime}\left(30\right)=-20\end{align}

48. $f\left(30\right)=0$

49. \begin{align}f^{\prime}\left(200\right)=30\end{align}

50. $f\left(240\right)=500$

For the following exercises, explain the functions in words. The height, $s$, of a projectile after $t$ seconds is given by $s\left(t\right)=-16{t}^{2}+80t$.

51. $s\left(2\right)=96$

52. \begin{align}s^{\prime}\left(2\right)=16\end{align}

53. $s\left(3\right)=96$

54. \begin{align}s^{\prime}\left(3\right)=-16\end{align}

55. $s\left(0\right)=0,s\left(5\right)=0$.

For the following exercises, the volume $V$ of a sphere with respect to its radius $r$ is given by $V=\frac{4}{3}\pi {r}^{3}$.

56. Find the average rate of change of $V$ as $r$ changes from 1 cm to 2 cm.

57. Find the instantaneous rate of change of $V$ when $r=3\text{ cm}\text{.}$

For the following exercises, the revenue generated by selling $x$ items is given by $R\left(x\right)=2{x}^{2}+10x$.

58. Find the average change of the revenue function as $x$ changes from $x=10$ to $x=20$.

59. Find \begin{align}R^{\prime}\left(10\right)\end{align} and interpret.

60. Find \begin{align}R^{\prime}\left(15\right)\end{align} and interpret. Compare \begin{align}R^{\prime}\left(15\right)\end{align} to \begin{align}]R^{\prime}\left(10\right)\end{align}, and explain the difference.

For the following exercises, the cost of producing $x$ cellphones is described by the function $C\left(x\right)={x}^{2}-4x+1000$.

61. Find the average rate of change in the total cost as $x$ changes from $x=10\text{ to }x=15$.

62. Find the approximate marginal cost, when 15 cellphones have been produced, of producing the 16th cellphone.

63. Find the approximate marginal cost, when 20 cellphones have been produced, of producing the 21st cellphone.

For the following exercises, use the definition for the derivative at a point $x=a$, $\underset{x\to a}{\mathrm{lim}}\dfrac{f\left(x\right)-f\left(a\right)}{x-a}$, to find the derivative of the functions.

64. $f\left(x\right)=\frac{1}{{x}^{2}}$

65. $f\left(x\right)=5{x}^{2}-x+4$

66. $f\left(x\right)=-{x}^{2}+4x+7$

67. $f\left(x\right)=\frac{-4}{3-{x}^{2}}$