Thomas’ Calculus꞉ Early Transcendentals

Joel R. Hass, Christopher E. Heil, Maurice D. Weir, Przemyslaw Bogacki

Chapter 4

Applications of Derivatives - all with Video Answers

Educators

Section 1

Extreme Values of Functions on Closed Intervals

Problem 1

Determine from the graph whether the function has any absolute extreme values on $[a, b]$. Then explain how your answer is consistent with Theorem 1.
(GRAPH CAN'T COPY)

Katelyn Chen

Katelyn Chen

Numerade Educator

Problem 2

Determine from the graph whether the function has any absolute extreme values on $[a, b]$. Then explain how your answer is consistent with Theorem 1.
(GRAPH CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 3

Determine from the graph whether the function has any absolute extreme values on $[a, b]$. Then explain how your answer is consistent with Theorem 1.
(GRAPH CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 4

Determine from the graph whether the function has any absolute extreme values on $[a, b]$. Then explain how your answer is consistent with Theorem 1.
(GRAPH CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 5

Determine from the graph whether the function has any absolute extreme values on $[a, b]$. Then explain how your answer is consistent with Theorem 1.
(GRAPH CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 6

Determine from the graph whether the function has any absolute extreme values on $[a, b]$. Then explain how your answer is consistent with Theorem 1.
(GRAPH CAN'T COPY)

Katelyn Chen

Numerade Educator

Problem 7

Find the absolute extreme values and where they occur.
(GRAPH CAN'T COPY)

AG

Ankit Gupta

Numerade Educator

Problem 8

Find the absolute extreme values and where they occur.
(GRAPH CAN'T COPY)

AG

Ankit Gupta

Numerade Educator

Problem 9

Find the absolute extreme values and where they occur.
(GRAPH CAN'T COPY)

AG

Ankit Gupta

Numerade Educator

Problem 10

Find the absolute extreme values and where they occur.
(GRAPH CAN'T COPY)

AG

Ankit Gupta

Numerade Educator

Problem 11

Match the table with a graph.
$$
\begin{array}{cc}
\hline \boldsymbol{x} & \boldsymbol{f}^{\prime}(\boldsymbol{x}) \\
\hline a & 0 \\
b & 0 \\
c & 5 \\
\hline
\end{array}
$$

Katelyn Chen

Numerade Educator

Problem 12

Match the table with a graph.
$$
\begin{array}{cc}
\hline \boldsymbol{x} & \boldsymbol{f}^{\prime}(\boldsymbol{x}) \\
\hline a & 0 \\
b & 0 \\
c & -5 \\
\hline
\end{array}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 13

Match the table with a graph.
$$
\begin{array}{lc}
\hline \boldsymbol{x} & \boldsymbol{f}^{\prime}(\boldsymbol{x}) \\
\hline a & \text { does not exist } \\
b & 0 \\
c & -2 \\
\hline
\end{array}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 14

Match the table with a graph.
$$
\begin{array}{lc}
\hline \boldsymbol{x} & \boldsymbol{f}^{\prime}(\boldsymbol{x}) \\
\hline a & \text { does not exist } \\
b & \text { does not exist } \\
c & -1.7 \\
\hline
\end{array}
$$

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 15

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$f(x)=|x|, \quad-1<x<2$

Katelyn Chen

Numerade Educator

Problem 16

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$y=2-x^2, \quad-1<x<1$

Katelyn Chen

Numerade Educator

Problem 17

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$g(x)= \begin{cases}-x, & 0 \leq x<1 \\ x-1, & 1 \leq x \leq 2\end{cases}$

Doruk Isik

Numerade Educator

Problem 18

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$h(x)= \begin{cases}\frac{1}{x}, & -1 \leq x<0 \\ \sqrt{x}, & 0 \leq x \leq 4\end{cases}$

Katelyn Chen

Numerade Educator

Problem 19

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$y=3 \sin x, \quad 0<x<2 \pi$

Doruk Isik

Numerade Educator

Problem 20

Sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
$f(x)= \begin{cases}x+1, & -1 \leq x<0 \\ \cos x, & 0<x \leq \frac{\pi}{2}\end{cases}$

Katelyn Chen

Numerade Educator

Problem 21

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(x)=\frac{2}{3} x-5, \quad-2 \leq x \leq 3$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 22

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(x)=-x-4, \quad-4 \leq x \leq 1$

Katelyn Chen

Numerade Educator

Problem 23

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(x)=x^2-1, \quad-1 \leq x \leq 2$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 24

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(x)=4-x^3, \quad-2 \leq x \leq 1$

Katelyn Chen

Numerade Educator

Problem 25

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$F(x)=-\frac{1}{x^2}, \quad 0.5 \leq x \leq 2$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 26

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$F(x)=-\frac{1}{x}, \quad-2 \leq x \leq-1$

Katelyn Chen

Numerade Educator

Problem 27

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$h(x)=\sqrt[3]{x}, \quad-1 \leq x \leq 8$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 28

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$h(x)=-3 x^{2 / 3}, \quad-1 \leq x \leq 1$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 29

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=\sqrt{4-x^2}, \quad-2 \leq x \leq 1$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 30

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=-\sqrt{5-x^2}, \quad-\sqrt{5} \leq x \leq 0$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 31

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(\theta)=\sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 32

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 33

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=\csc x, \quad \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3}$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 34

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=\sec x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{6}$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 35

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(t)=2-|t|, \quad-1 \leq t \leq 3$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 36

Find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(t)=|t-5|, \quad 4 \leq t \leq 7$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 37

a. Find the absolute maximum and minimum values of each function on the given interval.
b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=x e^{-x}, \quad-1 \leq x \leq 1$

Katelyn Chen

Numerade Educator

Problem 38

a. Find the absolute maximum and minimum values of each function on the given interval.
b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.
$h(x)=\ln (x+1)-\frac{x}{2}, \quad 0 \leq x \leq 3$

Katelyn Chen

Numerade Educator

Problem 39

a. Find the absolute maximum and minimum values of each function on the given interval.
b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.
$f(x)=\frac{1}{x}+\ln x, \quad 0.5 \leq x \leq 4$

Katelyn Chen

Numerade Educator

Problem 40

a. Find the absolute maximum and minimum values of each function on the given interval.
b. Graph the function, identify the points on the graph where the absolute extrema occur, and include their coordinates.
$g(x)=e^{-x^2}, \quad-2 \leq x \leq 1$

Katelyn Chen

Numerade Educator

Problem 41

Find the function's absolute maximum and minimum values and say where they occur.
$f(x)=x^{4 / 3}, \quad-1 \leq x \leq 8$

Katelyn Chen

Numerade Educator

Problem 42

Find the function's absolute maximum and minimum values and say where they occur.
$f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8$

Katelyn Chen

Numerade Educator

Problem 43

Find the function's absolute maximum and minimum values and say where they occur.
$g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1$

Katelyn Chen

Numerade Educator

Problem 44

Find the function's absolute maximum and minimum values and say where they occur.
$h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8$

Katelyn Chen

Numerade Educator

Problem 45

Determine all critical points and all domain endpoints for each function.
$y=x^2-6 x+7$

Katelyn Chen

Numerade Educator

Problem 46

Determine all critical points and all domain endpoints for each function.
$f(x)=6 x^2-x^3$

Katelyn Chen

Numerade Educator

Problem 47

Determine all critical points and all domain endpoints for each function.
$f(x)=x(4-x)^3$

Katelyn Chen

Numerade Educator

Problem 48

Determine all critical points and all domain endpoints for each function.
$g(x)=(x-1)^2(x-3)^2$

Katelyn Chen

Numerade Educator

Problem 49

Determine all critical points and all domain endpoints for each function.
$y=x^2+\frac{2}{x}$

Katelyn Chen

Numerade Educator

Problem 50

Determine all critical points and all domain endpoints for each function.
$f(x)=\frac{x^2}{x-2}$

Katelyn Chen

Numerade Educator

Problem 51

Determine all critical points and all domain endpoints for each function.
$y=x^2-32 \sqrt{x}$

Katelyn Chen

Numerade Educator

Problem 52

Determine all critical points and all domain endpoints for each function.
$g(x)=\sqrt{2 x-x^2}$

Katelyn Chen

Numerade Educator

Problem 53

Determine all critical points and all domain endpoints for each function.
$y=\ln (x+1)-\tan ^{-1} x$

Katelyn Chen

Numerade Educator

Problem 54

Determine all critical points and all domain endpoints for each function.
$y=2 \sqrt{1-x^2}+\arcsin x$

Katelyn Chen

Numerade Educator

Problem 55

Determine all critical points and all domain endpoints for each function.
$y=x^3+3 x^2-24 x+7$

Katelyn Chen

Numerade Educator

Problem 56

Determine all critical points and all domain endpoints for each function.
$y=x-3 x^{2 / 3}$

Katelyn Chen

Numerade Educator

Problem 57

Let $f(x)=(x-2)^{2 / 3}$.
a. Does $f^{\prime}(2)$ exist?
b. Show that the only local extreme value of $f$ occurs at $x=2$.
c. Does the result in part (b) contradict the Extreme Value Theorem?
d. Repeat parts (a) and (b) for $f(x)=(x-a)^{2 / 3}$, replacing 2 by $a$.

Norman Atentar

Numerade Educator

Problem 58

Let $f(x)=\left|x^3-9 x\right|$.
a. Does $f^{\prime}(0)$ exist?
b. Does $f^{\prime}(3)$ exist?
c. Does $f^{\prime}(-3)$ exist?
d. Determine all extrema of $f$.

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 59

Show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
$y=x^{11}+x^3+x-5$

Katelyn Chen

Numerade Educator

Problem 60

Show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
$y=3 x+\tan x$

Katelyn Chen

Numerade Educator

Problem 61

Show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
$y=\frac{1-e^x}{e^x+1}$

Katelyn Chen

Numerade Educator

Problem 62

Show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
$y=2 x-\sin 2 x$

Katelyn Chen

Numerade Educator

Problem 63

A minimum with no derivative The function $f(x)=|x|$ has an absolute minimum value at $x=0$ even though $f$ is not differentiable at $x=0$. Is this consistent with Theorem 2 ? Give reasons for your answer.

Carson Merrill

Numerade Educator

Problem 64

Even functions If an even function $f(x)$ has a local maximum value at $x=c$, can anything be said about the value of $f$ at $x=-c$ ? Give reasons for your answer.

Vincenzo Zaccaro

Vincenzo Zaccaro

Numerade Educator

Problem 65

Odd functions If an odd function $g(x)$ has a local minimum value at $x=c$, can anything be said about the value of $g$ at $x=-c$ ? Give reasons for your answer.

Lucas Finney

Numerade Educator

Problem 66

No critical points or endpoints exist We know how to find the extreme values of a continuous function $f(x)$ by investigating its values at critical points and endpoints. But what if there are no critical points or endpoints? What happens then? Do such functions really exist? Give reasons for your answers.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 67

The function
$$
V(x)=x(10-2 x)(16-2 x), \quad 0<x<5,
$$
models the volume of a box.
a. Find the extreme values of $V$.
b. Interpret any values found in part (a) in terms of the volume of the box.

Katelyn Chen

Numerade Educator

Problem 68

Cubic functions Consider the cubic function
$$
f(x)=a x^3+b x^2+c x+d .
$$
a. Show that $f$ can have 0,1 , or 2 critical points. Give examples and graphs to support your argument.
b. How many local extreme values can $f$ have?

Matt Just

Numerade Educator

Problem 69

Maximum height of a vertically moving body The height of a body moving vertically is given by
$$
s=-\frac{1}{2} g t^2+v_0 t+s_0, \quad g>0,
$$
with $s$ in meters and $t$ in seconds. Find the body's maximum height.

Mutahar Mehkri

Numerade Educator

Problem 70

Peak alternating current Suppose that at any given time $t$ (in seconds) the current $i$ (in amperes) in an alternating current circuit is $i=2 \cos t+2 \sin t$. What is the peak current for this circuit (largest magnitude)?

Khushbu Rani

Numerade Educator

Problem 71

Then find the extreme values of the function on the interval and say where they occur.
$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 72

Then find the extreme values of the function on the interval and say where they occur.
$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$

Carson Merrill

Numerade Educator

Problem 73

Then find the extreme values of the function on the interval and say where they occur.
$h(x)=|x+2|-|x-3|,-\infty<x<\infty$

Bobby Barnes

University of North Texas

Problem 74

Then find the extreme values of the function on the interval and say where they occur.
$k(x)=|x+1|+|x-3|,-\infty<x<\infty$

Jonathon Brumley

Jonathon Brumley

Numerade Educator

Problem 75

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=x^4-8 x^2+4 x+2, \quad[-20 / 25,64 / 25]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 76

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=-x^4+4 x^3-4 x+1, \quad[-3 / 4,3]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 77

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=x^{2 / 3}(3-x),[-2,2]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 78

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=2+2 x-3 x^{2 / 3}, \quad[-1,10 / 3]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 79

$f(x)=\sqrt{x}+\cos x,[0,2 \pi]$

Aayush Gupta

Numerade Educator

Problem 80

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 81

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=\pi x^2 e^{-3 x / 2},[0,5]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 82

You will use a CAS to help find the absolute extrema of the given function over the specified closed interval. Perform the following steps.
a. Plot the function over the interval to see its general behavior there.
b. Find the interior points where $f^{\prime}=0$. (In some exercises, you may have to use the numerical equation solver to approximate a solution.) You may want to plot $f^{\prime}$ as well.
c. Find the interior points where $f^{\prime}$ does not exist.
d. Evaluate the function at all points found in parts (b) and (c) and at the endpoints of the interval.
e. Find the function's absolute extreme values on the interval and identify where they occur.
$f(x)=\ln (2 x+x \sin x)$,$[1,15]$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator