# Thomas Calculus

## Educators

DI ### Problem 1

In Exercises $1-6$ , 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 .

DI
Doruk I.

### Problem 2

In Exercises $1-6$ , 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 . Matt J.

### Problem 3

In Exercises $1-6$ , 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 .

DI
Doruk I.

### Problem 4

In Exercises $1-6$ , 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 . Matt J.

### Problem 5

In Exercises $1-6$ , 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 .

DI
Doruk I.

### Problem 6

In Exercises $1-6$ , 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 . Matt J.

### Problem 7

In Exercises $7-10,$ find the absolute extreme values and where they
occur.

DI
Doruk I.

### Problem 8

In Exercises $7-10,$ find the absolute extreme values and where they
occur. Matt J.

### Problem 9

In Exercises $7-10,$ find the absolute extreme values and where they
occur.

DI
Doruk I.

### Problem 10

In Exercises $7-10,$ find the absolute extreme values and where they
occur. Matt J.

### Problem 11

In Exercises $11-14,$ match the table with a graph.
$$\begin{array}{l}{x} && {f^{\prime}(x)} \\ {a} && {0} \\ {b} && {0} \\ {c} && {5}\end{array}$$

DI
Doruk I.

### Problem 12

In Exercises $11-14,$ match the table with a graph.
$$\begin{array}{cc}{x} & {f^{\prime}(x)} \\ {a} & {0} \\ {b} & {0} \\ {c} & {-5}\end{array}$$ Matt J.

### Problem 13

In Exercises $11-14,$ match the table with a graph.
$$\begin{array}{cc}{x} & {f^{\prime}(x)} \\ {a} & \text{does not exist}\\ {b} & {0} \\ {c} & {-2}\end{array}$$

DI
Doruk I.

### Problem 14

In Exercises $11-14,$ match the table with a graph.
$$\begin{array}{cc}{x} & {f^{\prime}(x)} \\ {a} & \text{does not exist}\\ {b} & \text{does not exist} \\ {c} & {-1.7}\end{array}$$ Matt J.

### Problem 15

In Exercises $15-20,$ 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 $1 .$
$$f(x)=|x|, \quad-1 < x < 2$$

DI
Doruk I.

### Problem 16

In Exercises $15-20,$ 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 $1 .$
$$y=\frac{6}{x^{2}+2}, \quad-1 < x <1$$ Matt J.

### Problem 17

In Exercises $15-20,$ 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 $1 .$
$$g(x)=\left\{\begin{array}{ll}{-x,} & {0 \leq x<1} \\ {x-1,} & {1 \leq x \leq 2}\end{array}\right.$$

DI
Doruk I.

### Problem 18

In Exercises $15-20,$ 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 $1 .$
$$h(x)=\left\{\begin{array}{ll}{\frac{1}{x},} & {-1 \leq x<0} \\ {\sqrt{x},} & {0 \leq x \leq 4}\end{array}\right.$$ Matt J.

### Problem 19

In Exercises $15-20,$ 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 $1 .$
$$y=3 \sin x, \quad 0 < x < 2 \pi$$

DI
Doruk I.

### Problem 20

In Exercises $15-20,$ 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 $1 .$
$$f(x)=\left\{\begin{array}{ll}{x+1,} & {-1 \leq x<0} \\ {\cos x,} & {0 < x \leq \frac{\pi}{2}}\end{array}\right.$$ Matt J.

### Problem 21

In Exercises $21-36,$ 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.
$$f(x)=\frac{2}{3} x-5, \quad-2 \leq x \leq 3$$

DI
Doruk I.

### Problem 22

In Exercises $21-36,$ 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.
$$f(x)=-x-4, \quad-4 \leq x \leq 1$$ Matt J.

### Problem 23

In Exercises $21-36,$ 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.
$$f(x)=x^{2}-1, \quad-1 \leq x \leq 2$$

DI
Doruk I.

### Problem 24

In Exercises $21-36,$ 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.
$$f(x)=4-x^{3}, \quad-2 \leq x \leq 1$$ Matt J.

### Problem 25

In Exercises $21-36,$ 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.
$$F(x)=-\frac{1}{x^{2}}, \quad 0.5 \leq x \leq 2$$

DI
Doruk I.

### Problem 26

In Exercises $21-36,$ 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.
$$F(x)=-\frac{1}{x}, \quad-2 \leq x \leq-1$$ Matt J.

### Problem 27

In Exercises $21-36,$ 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.
$$h(x)=\sqrt{x}, \quad-1 \leq x \leq 8$$

DI
Doruk I.

### Problem 28

In Exercises $21-36,$ 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.
$$h(x)=-3 x^{2 / 3}, \quad-1 \leq x \leq 1$$ Matt J.

### Problem 29

In Exercises $21-36,$ 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.
$$g(x)=\sqrt{4-x^{2}}, \quad-2 \leq x \leq 1$$

DI
Doruk I.

### Problem 30

In Exercises $21-36,$ 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.
$$g(x)=-\sqrt{5-x^{2}}, \quad-\sqrt{5} \leq x \leq 0$$ Matt J.

### Problem 31

In Exercises $21-36,$ 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.
$$f(\theta)=\sin \theta, \quad-\frac{\pi}{2} \leq \theta \leq \frac{5 \pi}{6}$$

DI
Doruk I.

### Problem 32

In Exercises $21-36,$ 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.
$$f(\theta)=\tan \theta, \quad-\frac{\pi}{3} \leq \theta \leq \frac{\pi}{4}$$ Matt J.

### Problem 33

In Exercises $21-36,$ 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.
$$g(x)=\csc x, \quad \frac{\pi}{3} \leq x \leq \frac{2 \pi}{3}$$

DI
Doruk I.

### Problem 34

In Exercises $21-36,$ 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.
$$g(x)=\sec x, \quad-\frac{\pi}{3} \leq x \leq \frac{\pi}{6}$$ Matt J.

### Problem 35

In Exercises $21-36,$ 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.
$$f(t)=2-|t|, \quad-1 \leq t \leq 3$$

DI
Doruk I.

### Problem 36

In Exercises $21-36,$ 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.
$$f(t)=|t-5|, \quad 4 \leq t \leq 7$$ Matt J.

### Problem 37

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.
$$f(x)=x^{4 / 3},-1 \leq x \leq 8$$

DI
Doruk I.

### Problem 38

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.
$$f(x)=x^{5 / 3}, \quad-1 \leq x \leq 8$$ Matt J.

### Problem 39

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.
$$g(\theta)=\theta^{3 / 5}, \quad-32 \leq \theta \leq 1$$

DI
Doruk I.

### Problem 40

In Exercises $37-40$ , find the function's absolute maximum and minimum values and say where they occur.
$$h(\theta)=3 \theta^{2 / 3}, \quad-27 \leq \theta \leq 8$$ Matt J.

### Problem 41

In Exercises $41-50,$ determine all critical points for each function.
$$y=x^{2}-6 x+7$$

DI
Doruk I.

### Problem 42

In Exercises $41-50,$ determine all critical points for each function.
$$f(x)=6 x^{2}-x^{3}$$ Matt J.

### Problem 43

In Exercises $41-50,$ determine all critical points for each function.
$$f(x)=x(4-x)^{3}$$

DI
Doruk I.

### Problem 44

In Exercises $41-50,$ determine all critical points for each function.
$$g(x)=(x-1)^{2}(x-3)^{2}$$ Matt J.

### Problem 45

In Exercises $41-50,$ determine all critical points for each function.
$$y=x^{2}+\frac{2}{x}$$

DI
Doruk I.

### Problem 46

In Exercises $41-50,$ determine all critical points for each function.
$$f(x)=\frac{x^{2}}{x-2}$$ Matt J.

### Problem 47

In Exercises $41-50,$ determine all critical points for each function.
$$y=x^{2}-32 \sqrt{x}$$

DI
Doruk I.

### Problem 48

In Exercises $41-50,$ determine all critical points for each function.
$$g(x)=\sqrt{2 x-x^{2}}$$ Matt J.

### Problem 49

In Exercises $41-50,$ determine all critical points for each function.
$$y=x^{3}+3 x^{2}-24 x+7$$

DI
Doruk I.

### Problem 50

In Exercises $41-50,$ determine all critical points for each function.
$$y=x-3 x^{2 / 3}$$ Matt J.

### Problem 51

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=x^{2 / 3}(x+2)$$

DI
Doruk I.

### Problem 52

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=x^{2 / 3}\left(x^{2}-4\right)$$ Matt J.

### Problem 53

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=x \sqrt{4-x^{2}}$$

DI
Doruk I.

### Problem 54

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=x^{2} \sqrt{3-x}$$ Matt J.

### Problem 55

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=\left\{\begin{array}{ll}{4-2 x,} & {x \leq 1} \\ {x+1,} & {x>1}\end{array}\right.$$

DI
Doruk I.

### Problem 56

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=\left\{\begin{array}{ll}{3-x,} & {x<0} \\ {3+2 x-x^{2},} & {x \geq 0}\end{array}\right.$$ Matt J.

### Problem 57

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=\left\{\begin{array}{ll}{-x^{2}-2 x+4,} & {x \leq 1} \\ {-x^{2}+6 x-4,} & {x>1}\end{array}\right.$$

DI
Doruk I.

### Problem 58

In Exercises $51-58$ , find the critical points and domain endpoints for
each function. Then find the value of the function at each of these
points and identify extreme values (absolute and local).
$$y=\left\{\begin{array}{ll}{-\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4},} & {x \leq 1} \\ {x^{3}-6 x^{2}+8 x,} & {x>1}\end{array}\right.$$ Matt J.

### Problem 59

In Exercises 59 and $60,$ give reasons for your answers.
Let
$$f(x)=(x-2)^{2 / 3}$$
$$\begin{array}{l}{\text { a. Does } f^{\prime}(2) \text { exist? }} \\ {\text { b. Show that the only local extreme value of } f \text { occurs at } x=2 \text { . }} \\ {\text { c. Does the result in part (b) contradict the Extreme Value }} \\ {\text { Theorem? }} \\ {\text { d. Repeat parts (a) and (b) for } f(x)=(x-a)^{2 / 3}, \text { replacing } 2 \text { by } a \text { . }}\end{array}$$

DI
Doruk I.

### Problem 60

In Exercises 59 and $60,$ give reasons for your answers.
Let
$$f(x)=\left|x^{3}-9 x\right|$$
$$\begin{array}{ll}{\text { a. Does } f^{\prime}(0) \text { exist? }} & {\text { b. Does } f^{\prime}(3) \text { exist? }} \\ {\text { c. Does } f^{\prime}(-3) \text { exist? }} & {\text { d. Determine all extrema of } f}\end{array}$$ Matt J.

### Problem 61

In Exercises $61-62,$ show that the function has neither an absolute
minimum nor an absolute maximum on its natural domain.
$$y=x^{11}+x^{3}+x-5$$

DI
Doruk I.

### Problem 62

In Exercises $61-62,$ show that the function has neither an absolute
minimum nor an absolute maximum on its natural domain.
$$y=3 x+\tan x$$ Matt J.

### 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 differen-
tiable at $x=0 .$ Is this consistent with Theorem 2$?$ Give reasons

DI
Doruk I.

### 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. Matt J.

### 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.

DI
Doruk I.

### 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 Matt J.

### Problem 67

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

DI
Doruk I.

### Problem 68

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

### 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.

DI
Doruk I.

### 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)? Matt J.

### Problem 71

Graph the functions in Exercises $71-74 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$f(x)=|x-2|+|x+3|, \quad-5 \leq x \leq 5$$

DI
Doruk I.

### Problem 72

Graph the functions in Exercises $71-74 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$g(x)=|x-1|-|x-5|, \quad-2 \leq x \leq 7$$ Matt J.

### Problem 73

Graph the functions in Exercises $71-74 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$h(x)=|x+2|-|x-3|, \quad-\infty < x < \infty$$

DI
Doruk I.

### Problem 74

Graph the functions in Exercises $71-74 .$ Then find the extreme values
of the function on the interval and say where they occur.
$$k(x)=|x+1|+|x-3|, \quad-\infty< x < \infty$$ Matt J.

### Problem 75

In Exercises $75-80,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform
the following steps.
$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$
$$f(x)=x^{4}-8 x^{2}+4 x+2, \quad[-20 / 25,64 / 25]$$

DI
Doruk I.

### Problem 76

In Exercises $75-80,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform
the following steps.
$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$
$$f(x)=-x^{4}+4 x^{3}-4 x+1, \quad[-3 / 4,3]$$ Matt J.

### Problem 77

In Exercises $75-80,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform
the following steps.
$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$
$$f(x)=x^{2 / 3}(3-x), \quad[-2,2]$$

DI
Doruk I.

### Problem 78

In Exercises $75-80,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform
the following steps.
$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$
$$f(x)=2+2 x-3 x^{2 / 3}, \quad[-1,10 / 3]$$ Matt J.

### Problem 79

In Exercises $75-80,$ you will use a CAS to help find the absolute
extrema of the given function over the specified closed interval. Perform
the following steps.
$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$
$$f(x)=\sqrt{x}+\cos x, \quad[0,2 \pi]$$

DI
Doruk I.
In Exercises $75-80,$ you will use a CAS to help find the absolute
$$\begin{array}{l}{\text { a. Plot the function over the interval to see its general behavior there. }} \\ {\text { b. Find the interior points where } f^{\prime}=0 . \text { (In some exercises, }} \\ {\text { you may have to use the numerical equation solver to approximate }} \\ {\text { a solution.) You may want to plot } f^{\prime} \text { as well. }}\\{\text { c. Find the interior points where } f^{\prime} \text { does not exist. }} \\ {\text { d. Evaluate the function at all points found in parts (b) and (c) }} \\ {\text { and at the endpoints of the interval. }} \\ {\text { e. Find the function's absolute extreme values on the interval }} \\ {\text { and identify where they occur. }}\end{array}$$
$$f(x)=x^{3 / 4}-\sin x+\frac{1}{2}, \quad[0,2 \pi]$$ 