Problem Set: The Derivative as a Function

For the following exercises (1-10), use the definition of a derivative to find $f^{\prime}(x)$ .

1. $f(x)=6$

2. $f(x)=2-3x$

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3. $f(x)=\dfrac{2x}{7}+1$

4. $f(x)=4x^2$

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$8x$

5. $f(x)=5x-x^2$

6. $f(x)=\sqrt{2x}$

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$\frac{1}{\sqrt{2x}}$

7. $f(x)=\sqrt{x-6}$

8. $f(x)=\dfrac{9}{x}$

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$\frac{-9}{x^2}$

9. $f(x)=x+\dfrac{1}{x}$

10. $f(x)=\dfrac{1}{\sqrt{x}}$

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$\frac{-1}{2x^{3/2}}$

For the following exercises (11-14), use the graph of $y=f(x)$ to sketch the graph of its derivative $f^{\prime}(x)$ .

11.

The function f(x) starts at (−2, 20) and decreases to pass through the origin and achieve a local minimum at roughly (0.5, −1). Then, it increases and passes through (1, 0) and achieves a local maximum at (2.25, 2) before decreasing again through (3, 0) to (4, −20).

12.

The function f(x) starts at (−1.5, 20) and decreases to pass through (0, 10), where it appears to have a derivative of 0. Then it further decreases, passing through (1.7, 0) and achieving a minimum at (3, −17), at which point it increases rapidly through (3.8, 0) to (4, 20).

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13.

The function f(x) starts at (−2.25, −20) and increases rapidly to pass through (−2, 0) before achieving a local maximum at (−1.4, 8). Then the function decreases to the origin. The graph is symmetric about the y-axis, so the graph increases to (1.4, 8) before decreasing through (2, 0) and heading on down to (2.25, −20).

14.

The function f(x) starts at (−3, −1) and increases to pass through (−1.5, 0) and achieve a local minimum at (1, 0). Then, it decreases and passes through (1.5, 0) and continues decreasing to (3, −1).

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For the following exercises (15-20), the given limit represents the derivative of a function $y=f(x)$ at $x=a$ . Find $f(x)$ and $a$ .

15. $\underset{h\to 0}{\lim}\dfrac{(1+h)^{\frac{2}{3}}-1}{h}$

16. $\underset{h\to 0}{\lim}\dfrac{[3(2+h)^2+2]-14}{h}$

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$f(x)=3x^2+2, \, a=2$

17. $\underset{h\to 0}{\lim}\dfrac{\cos(\pi+h)+1}{h}$

18. $\underset{h\to 0}{\lim}\dfrac{(2+h)^4-16}{h}$

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$f(x)=x^4, \, a=2$

19. $\underset{h\to 0}{\lim}\dfrac{\left[2(3+h)^2-(3+h)\right]-15}{h}$

20. $\underset{h\to 0}{\lim}\dfrac{e^h-1}{h}$

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$f(x)=e^x, \, a=0$

For the following functions (21-24),

sketch the graph and
use the definition of a derivative to show that the function is not differentiable at $x=1$ .

21. $f(x)=\begin{cases} 2\sqrt{x} & \text{ if } \, 0 \le x \le 1 \\ 3x-1 & \text{ if } \, x>1 \end{cases}$

22. $f(x)=\begin{cases} 3 & \text{ if } \, x<1 \\ 3x & \text{ if } \, x \ge 1 \end{cases}$

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b. $\underset{h\to 1^-}{\lim}\frac{3-3}{h}\ne \underset{h\to 1^+}{\lim}\frac{3h}{h}$

23. $f(x)=\begin{cases} -x^2+2 & \text{ if } \, x \le 1 \\ x & \text{ if } \, x>1 \end{cases}$

24. $f(x)=\begin{cases} 2x & \text{ if } x \le 1 \\ \dfrac{2}{x} & \text{ if } \, x>1 \end{cases}$

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b. $\underset{h\to 1^-}{\lim}\frac{2h}{h}\ne \underset{h\to 1^+}{\lim}\frac{\frac{2}{x+h}-\frac{2}{x}}{h}$ .

For the following graphs (25-27),

determine for which values of $x=a$ the $\underset{x\to a}{\lim}f(x)$ exists but $f$ is not continuous at $x=a$ , and
determine for which values of $x=a$ the function is continuous but not differentiable at $x=a$ .

25.

The function starts at (−6, 2) and increases to a maximum at (−5.3, 4) before stopping at (−4, 3) inclusive. Then it starts again at (−4, −2) before increasing slowly through (−2.25, 0), passing through (−1, 4), hitting a local maximum at (−0.1, 5.3) and decreasing to (2, −1) inclusive. Then it starts again at (2, 5), increases to (2.6, 6), and then decreases to (4.5, −3), with a discontinuity at (4, 2).

26.

The function starts at (−3, −1) and increases to and stops at a local maximum at (−1, 3) inclusive. Then it starts again at (−1, 1) before increasing quickly to and stopping at a local maximum (0, 4) inclusive. Then it starts again at (0, 3) and decreases linearly to (1, 1), at which point there is a discontinuity and the value of this function at x = 1 is 2. The function continues from (1, 1) and increases linearly to (2, 3.5) before decreasing linearly to (3, 2).

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$x=1$ , b.

$x=2$

27. Use the graph to evaluate a. $f^{\prime}(-0.5)$ , b. $f^{\prime}(0)$ , c. $f^{\prime}(1)$ , d. $f^{\prime}(2)$ , and e. $f^{\prime}(3)$ , if they exist.

The function starts at (−3, 0) and increases linearly to a local maximum at (0, 3). Then it decreases linearly to (2, 1), at which point it increases linearly to (4, 5).

For the following functions (28-30), use $f''(x)=\underset{h\to 0}{\lim}\dfrac{f^{\prime}(x+h)-f^{\prime}(x)}{h}$ to find $f''(x)$ .

28. $f(x)=2-3x$

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29. $f(x)=4x^2$

30. $f(x)=x+\dfrac{1}{x}$

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$\frac{2}{x^3}$

For the following exercises (31-36), use a calculator to graph $f(x)$ . Determine the function $f^{\prime}(x)$ , then use a calculator to graph $f^{\prime}(x)$ .

31. [T] $f(x)=-\dfrac{5}{x}$

32. [T] $f(x)=3x^2+2x+4$

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$f^{\prime}(x)=6x+2$

The function f(x) is graphed as an upward facing parabola with y intercept 4. The function f’(x) is graphed as a straight line with y intercept 2 and slope 6.

33. [T] $f(x)=\sqrt{x}+3x$

34. [T] $f(x)=\dfrac{1}{\sqrt{2x}}$

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$f^{\prime}(x)=-\frac{1}{(2x)^{3/2}}$

The function f(x) is in the first quadrant and has asymptotes at x = 0 and y = 0. The function f’(x) is in the fourth quadrant and has asymptotes at x = 0 and y = 0.

35. [T] $f(x)=1+x+\dfrac{1}{x}$

36. [T] $f(x)=x^3+1$

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$f^{\prime}(x)=3x^2$

The function f(x) starts is the graph of the cubic function shifted up by 1. The function f’(x) is the graph of a parabola that is slightly steeper than the normal squared function.

For the following exercises (37-42), describe what the two expressions represent in terms of each of the given situations. Be sure to include units.

$\dfrac{f(x+h)-f(x)}{h}$
$f^{\prime}(x)=\underset{h\to 0}{\lim}\dfrac{f(x+h)-f(x)}{h}$

37. $P(x)$ denotes the population of a city at time $x$ in years.

38. $C(x)$ denotes the total amount of money (in thousands of dollars) spent on concessions by $x$ customers at an amusement park.

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a. Average rate at which customers spent on concessions in thousands per customer.
b. Rate (in thousands per customer) at which $x$ customers spent money on concessions in thousands per customer.

39. $R(x)$ denotes the total cost (in thousands of dollars) of manufacturing $x$ clock radios.

40. $g(x)$ denotes the grade (in percentage points) received on a test, given $x$ hours of studying.

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a. Average grade received on the test with an average study time between two amounts.
b. Rate (in percentage points per hour) at which the grade on the test increased or decreased for a given average study time of $x$ hours.

41. $B(x)$ denotes the cost (in dollars) of a sociology textbook at university bookstores in the United States in $x$ years since 1990.

42. $p(x)$ denotes atmospheric pressure at an altitude of $x$ feet.

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a. Average change of atmospheric pressure between two different altitudes.
b. Rate (torr per foot) at which atmospheric pressure is increasing or decreasing at $x$ feet.

43. Sketch the graph of a function $y=f(x)$ with all of the following properties:

$f^{\prime}(x)>0$ for $-2\le x<1$
$f^{\prime}(2)=0$
$f^{\prime}(x)>0$ for $x>2$
$f(2)=2$ and $f(0)=1$
$\underset{x\to −\infty}{\lim}f(x)=0$ and $\underset{x\to \infty}{\lim}f(x)=\infty$
$f^{\prime}(1)$ does not exist.

44. Suppose temperature $T$ in degrees Fahrenheit at a height $x$ in feet above the ground is given by $y=T(x)$ .

Give a physical interpretation, with units, of $T^{\prime}(x)$ .
If we know that ${T}^{\prime }(1000)=-0.1,$ explain the physical meaning.

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a. The rate (in degrees per foot) at which temperature is increasing or decreasing for a given height $x$ .
b. The rate of change of temperature as altitude changes at 1000 feet is -0.1 degrees per foot.

45. Suppose the total profit of a company is $y=P(x)$ thousand dollars when $x$ units of an item are sold.

What does $\dfrac{P(b)-P(a)}{b-a}$ for [latex]0
What does $P^{\prime}(x)$ measure, and what are the units?
Suppose that $P^{\prime}(30)=5$ . What is the approximate change in profit if the number of items sold increases from 30 to 31?

46. The graph in the following figure models the number of people $N(t)$ who have come down with the flu $t$ weeks after its initial outbreak in a town with a population of 50,000 citizens.

Describe what $N^{\prime}(t)$ represents and how it behaves as $t$ increases.
What does the derivative tell us about how this town is affected by the flu outbreak?

The function starts at (0, 3000) and increases quickly to an asymptote at y = 50000.

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a. The rate at which the number of people who have come down with the flu is changing $t$ weeks after the initial outbreak.
b. The rate is increasing sharply up to the third week, at which point it slows down and then becomes constant.

For the following exercises, use the following table, which shows the height $h$ of the Saturn V rocket for the Apollo 11 mission $t$ seconds after launch.

Time (seconds)	Height (meters)
0	0
1	2
2	4
3	13
4	25
5	32

47. What is the physical meaning of $h^{\prime}(t)$ ? What are the units?

48. [T] Construct a table of values for $h^{\prime}(t)$ and graph both $h(t)$ and $h^{\prime}(t)$ on the same graph. (Hint: for interior points, estimate both the left limit and right limit and average them. An interior point of an interval $I$ is an element of $I$ which is not an endpoint of $I$ .)

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Time (seconds)	$h^{\prime}(t)$ (m/s)
0	2
1	2
2	5.5
3	10.5
4	9.5
5	7

49. [T] The best linear fit to the data is given by $H(t)=7.229t-4.905$ , where $H$ is the height of the rocket (in meters) and $t$ is the time elapsed since takeoff. From this equation, determine $H^{\prime}(t)$ . Graph $H(t)$ with the given data and, on a separate coordinate plane, graph $H^{\prime}(t)$ .

50. [T] The best quadratic fit to the data is given by $G(t)=1.429t^2+0.0857t-0.1429$ , where $G$ is the height of the rocket (in meters) and $t$ is the time elapsed since takeoff. From this equation, determine $G^{\prime}(t)$ . Graph $G(t)$ with the given data and, on a separate coordinate plane, graph $G^{\prime}(t)$ .

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$G^{\prime}(t)=2.858t+0.0857$

This graph has the points (0, 0), (1, 2), (2, 4), (3, 13), (4, 25), and (5, 32). There is a quadratic line fit to the points with y intercept near 0.
This graph has a straight line with y intercept near 0 and slope slightly less than 3.

51. [T] The best cubic fit to the data is given by $F(t)=0.2037t^3+2.956t^2-2.705t+0.4683$ , where $F$ is the height of the rocket (in m) and $t$ is the time elapsed since take off. From this equation, determine $F^{\prime}(t)$ . Graph $F(t)$ with the given data and, on a separate coordinate plane, graph $F^{\prime}(t)$ . Does the linear, quadratic, or cubic function fit the data best?

52. Using the best linear, quadratic, and cubic fits to the data, determine what $H''(t), \, G''(t)$ , and $F''(t)$ are. What are the physical meanings of $H''(t), \, G''(t)$ , and $F''(t)$ , and what are their units?

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$H''(t)=0, \, G''(t)=2.858$ , and $F''(t)=1.222t+5.912$ represent the acceleration of the rocket, with units of meters per second squared ( $\text{m/s}^2$ ).

Derivatives: Supplemental Content

Problem Set: The Derivative as a Function

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