Calculus Early Transcendental Functions

Ron Larson, Bruce Edwards

Chapter 4

Applications of Differentiation - all with Video Answers

Educators

AR

Section 1

Extrema on an Interval

Problem 1

Find the value of the derivative (if it exists) at each indicated extremum.
$$f(x)=\frac{x^{2}}{x^{2}+4}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 2

Find the value of the derivative (if it exists) at each indicated extremum.
$$f(x)=\cos \frac{\pi x}{2}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 3

Find the value of the derivative (if it exists) at each indicated extremum.
$$g(x)=x+\frac{4}{x^{2}}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 4

Find the value of the derivative (if it exists) at each indicated extremum.
$$f(x)=-3 x \sqrt{x+1}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 5

Find the value of the derivative (if it exists) at each indicated extremum.
$$f(x)=(x+2)^{2 / 3}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 6

Find the value of the derivative (if it exists) at each indicated extremum.
$$f(x)=4-|x|$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 7

Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 8

Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 9

Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 10

Determine whether the function has a relative maximum, a relative minimum, an absolute maximum, an absolute minimum, or none of these at each critical number on the interval shown.

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 11

Find the critical numbers of the function.
$$f(x)=x^{3}-3 x^{2}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 12

Find the critical numbers of the function.
$$g(x)=x^{4}-8 x^{2}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 13

Find the critical numbers of the function.
$$g(t)=t \sqrt{4-t}, t<3$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 14

Find the critical numbers of the function.
$$f(x)=\frac{4 x}{x^{2}+1}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 15

Find the critical numbers of the function.
$$\begin{aligned}&h(x)=\sin ^{2} x+\cos x\\&0<x<2 \pi\end{aligned}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 16

Find the critical numbers of the function.
$$\begin{aligned}&f(\theta)=2 \sec \theta+\tan \theta\\&0<\theta<2 \pi\end{aligned}$$

Diane Koenig

Numerade Educator

Problem 17

Find the critical numbers of the function.
$$f(t)=t e^{-2 t}$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 18

Find the critical numbers of the function.
$$g(x)=4 x^{2}\left(3^{x}\right)$$

Diane Koenig

Numerade Educator

Problem 19

Find the critical numbers of the function.
$$f(x)=x^{2} \log _{2}\left(x^{2}+1\right)$$

Carson Merrill

Numerade Educator

Problem 20

Find the critical numbers of the function.
$$g(t)=2 t \ln t$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 21

Find the absolute extrema of the function on the closed interval.
$$f(x)=3-x,[-1,2]$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 22

Find the absolute extrema of the function on the closed interval.
$$f(x)=\frac{3}{4} x+2,[0,4]$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 23

Find the absolute extrema of the function on the closed interval.
$$g(x)=2 x^{2}-8 x,[0,6]$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 24

Find the absolute extrema of the function on the closed interval.
$$h(x)=5-x^{2},[-3,1]$$

Jacquelyn Trost

Jacquelyn Trost

Numerade Educator

Problem 25

Find the absolute extrema of the function on the closed interval.
$$f(x)=x^{3}-\frac{3}{2} x^{2},[-1,2]$$

Kyle Christian

Numerade Educator

Problem 26

Find the absolute extrema of the function on the closed interval.
$$f(x)=2 x^{3}-6 x,[0,3]$$

Kyle Christian

Numerade Educator

Problem 27

Find the absolute extrema of the function on the closed interval.
$$y=3 x^{2 / 3}-2 x,[-1,1]$$

Diane Koenig

Numerade Educator

Problem 28

Find the absolute extrema of the function on the closed interval.
$$g(x)=\sqrt[3]{x},[-8,8]$$

Diane Koenig

Numerade Educator

Problem 29

Find the absolute extrema of the function on the closed interval.
$$h(s)=\frac{1}{s-2},[0,1]$$

Diane Koenig

Numerade Educator

Problem 30

Find the absolute extrema of the function on the closed interval.
$$h(t)=\frac{t}{t+3},[-1,6]$$

Diane Koenig

Numerade Educator

Problem 31

Find the absolute extrema of the function on the closed interval.
$$y=3-|t-3|,[-1,5]$$

Diane Koenig

Numerade Educator

Problem 32

Find the absolute extrema of the function on the closed interval.
$$g(x)=|x+4|,[-7,1]$$

Diane Koenig

Numerade Educator

Problem 33

Find the absolute extrema of the function on the closed interval.
$$f(x)=[|x|],[-2,2]$$

Diane Koenig

Numerade Educator

Problem 34

Find the absolute extrema of the function on the closed interval.
$$h(x)=[|2-x|],[-2,2]$$

Carson Merrill

Numerade Educator

Problem 35

Find the absolute extrema of the function on the closed interval.
$$f(x)=\sin x,\left[\frac{5 \pi}{6}, \frac{11 \pi}{6}\right]$$

Kyle Christian

Numerade Educator

Problem 36

Find the absolute extrema of the function on the closed interval.
$$g(x)=\sec x,\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$$

Diane Koenig

Numerade Educator

Problem 37

Find the absolute extrema of the function on the closed interval.
$$y=3 \cos x,[0,2 \pi]$$

Kyle Christian

Numerade Educator

Problem 38

Find the absolute extrema of the function on the closed interval.
$$y=\tan \left(\frac{\pi x}{8}\right),[0,2]$$

Diane Koenig

Numerade Educator

Problem 39

Find the absolute extrema of the function on the closed interval.
$$f(x)=\arctan x^{2},[-2,1]$$

Carson Merrill

Numerade Educator

Problem 40

Find the absolute extrema of the function on the closed interval.
$$g(x)=\frac{\ln x}{x},[1,4]$$

Diane Koenig

Numerade Educator

Problem 41

Find the absolute extrema of the function on the closed interval.
$$h(x)=5 e^{x}-e^{2 x},[-1,2]$$

Diane Koenig

Numerade Educator

Problem 42

Find the absolute extrema of the function on the closed interval.
$$y=x^{2}-8 \ln x,[1,5]$$

Diane Koenig

Numerade Educator

Problem 43

Find the absolute extrema of the function on the closed interval.
$$y=e^{x} \sin x,[0, \pi]$$

Carson Merrill

Numerade Educator

Problem 44

Find the absolute extrema of the function on the closed interval.
$$y=x \ln (x+3),[0,3]$$

Carson Merrill

Numerade Educator

Problem 45

Find the absolute extrema of the function (if any exist) on each interval.
$f(x)=2 x-3$
(a) [0,2]
(b) [0,2)
(c) (0,2]
(d) (0,2)

Kyle Christian

Numerade Educator

Problem 46

Find the absolute extrema of the function (if any exist) on each interval.
$f(x)=\sqrt{4-x^{2}}$
(a) [-2,2]
(b) [-2,0)
(c) (-2,2)
(d) [1,2)

Diane Koenig

Numerade Educator

Problem 47

Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval.
$$f(x)=\frac{3}{x-1}$$

Diane Koenig

Numerade Educator

Problem 48

Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval.
$$f(x)=\frac{2}{2-x}, \quad[0,2)$$

Diane Koenig

Numerade Educator

Problem 49

Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval.
$$f(x)=\sqrt{x+4} e^{x^{2} / 10}, \quad[-2,2]$$

Diane Koenig

Numerade Educator

Problem 50

Use a graphing utility to graph the function and find the absolute extrema of the function on the given interval.
$$f(x)=\sqrt{x}+\cos \frac{x}{2}, \quad[0,2 \pi]$$

Carson Merrill

Numerade Educator

Problem 51

(A) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval.
(b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a).
$$f(x)=3.2 x^{5}+5 x^{3}-3.5 x, \quad[0,1]$$

Carson Merrill

Numerade Educator

Problem 52

(A) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval.
(b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a).
$$f(x)=\frac{4}{3} x \sqrt{3-x}, \quad[0,3]$$

Carson Merrill

Numerade Educator

Problem 53

(A) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval.
(b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a).
$$f(x)=\left(x^{2}-2 x\right) \ln (x+3), \quad[0,3]$$

Carson Merrill

Numerade Educator

Problem 54

(A) use a computer algebra system to graph the function and approximate any absolute extrema on the given interval.
(b) Use the utility to find any critical numbers, and use them to find any absolute extrema not located at the endpoints. Compare the results with those in part (a).
$$f(x)=(x-4) \arcsin \frac{x}{4}, \quad[-2,4]$$

Carson Merrill

Numerade Educator

Problem 55

Use a computer algebra system to find the maximum value of $\left|f^{\prime \prime}(x)\right|$ on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section $5.6 .)$
$$f(x)=\sqrt{1+x^{3}}, \quad[0,2]$$

Carson Merrill

Numerade Educator

Problem 56

Use a computer algebra system to find the maximum value of $\left|f^{\prime \prime}(x)\right|$ on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section $5.6 .)$
$$f(x)=\frac{1}{x^{2}+1}, \quad\left[\frac{1}{2}, 3\right]$$

Carson Merrill

Numerade Educator

Problem 57

Use a computer algebra system to find the maximum value of $\left|f^{\prime \prime}(x)\right|$ on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section $5.6 .)$
$$f(x)=e^{-x^{2} / 2}, \quad[0,1]$$

Carson Merrill

Numerade Educator

Problem 58

Use a computer algebra system to find the maximum value of $\left|f^{\prime \prime}(x)\right|$ on the closed interval. (This value is used in the error estimate for the Trapezoidal Rule, as discussed in Section $5.6 .)$
$$f(x)=x \ln (x+1), \quad[0,2]$$

Carson Merrill

Numerade Educator

Problem 59

Use a computer algebra system to find the maximum value of $\left|f^{(4)}(x)\right|$ on the closed interval. (This value is used in the error estimate for Simpson's Rule, as discussed in Section $5.6 .)$
f(x)=(x+1)^{2 / 3}, \quad[0,2]

Carson Merrill

Numerade Educator

Problem 60

Use a computer algebra system to find the maximum value of $\left|f^{(4)}(x)\right|$ on the closed interval. (This value is used in the error estimate for Simpson's Rule, as discussed in Section $5.6 .)$
$$f(x)=\frac{1}{x^{2}+1}, \quad[-1,1]$$

Carson Merrill

Numerade Educator

Problem 61

Explain why the function $f(x)=\tan x$ has a maximum on $[0, \pi / 4]$ but not on $[0, \pi]$

Lucas Finney

Numerade Educator

Problem 62

Determine whether each labeled point is an absolute maximum or minimum, a relative maximum
or minimum, or none of these.
FIGURE CANT COPY

Carson Merrill

Numerade Educator

Problem 63

Graph a function on the interval [-2,5] having the given characteristics.
Absolute maximum at $x=-2$
Absolute minimum at $x=1$
Relative maximum at $x=3$

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Problem 64

Graph a function on the interval [-2,5] having the given characteristics.
Relative minimum at $x=-1$
Critical number (but no extremum) at $x=0$
Absolute maximum at $x=2$
Absolute minimum at $x=5$

Diane Koenig

Numerade Educator

Problem 65

Determine from the graph whether $f$ has a minimum in the open interval $(a, b)$

Diane Koenig

Numerade Educator

Problem 66

Determine from the graph whether $f$ has a minimum in the open interval $(a, b)$

Diane Koenig

Numerade Educator

Problem 67

Determine from the graph whether $f$ has a minimum in the open interval $(a, b)$
GRAPHS CANT COPY

Diane Koenig

Numerade Educator

Problem 68

Determine from the graph whether $f$ has a minimum in the open interval $(a, b)$

Diane Koenig

Numerade Educator

Problem 69

The formula for the power output $P$ of a battery is
$P=V I-R I^{2}$
where $V$ is the electromotive force in volts, $R$ is the resistance in ohms, and $I$ is the current in amperes. Find the current that corresponds to a maximum value of $P$ in a battery for which $V=12$ volts and $R=0.5$ ohm. Assume that a 15 -ampere fuse bounds the output in the interval $0 \leq I \leq 15 .$ Could the power output be increased by replacing the 15 -ampere fuse with a 20-ampere fuse? Explain.

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Problem 70

A lawn sprinkler is constructed in such a way that $d \theta / d t$ is constant, where $\theta$ ranges between $45^{\circ}$ and $135^{\circ}$ (see figure). The distance the water travels horizontally is
$x=\frac{v^{2} \sin 2 \theta}{32}, \quad 45^{\circ} \leq \theta \leq 135^{\circ}$
where $v$ is the speed of the water. Find $d x / d t$ and explain why this lawn sprinkler does not water evenly. What part of the lawn receives the most water?
For more information on the "calculus of lawn sprinklers," see the article "Design of an Oscillating Sprinkler" by Bart Braden in Mathematics Magazine. To view this article, go to MathArticles.com.

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Problem 71

The surface area of a cell in a honeycomb is $S=6 h s+\frac{3 s^{2}}{2}\left(\frac{\sqrt{3}-\cos \theta}{\sin \theta}\right)$
where $h$ and $s$ are positive constants and $\theta$ is the angle at which the upper faces meet the altitude of the cell (see figure). Find the angle $\theta(\pi / 6 \leq \theta \leq \pi / 2)$ that minimizes the surface area $S$
For more information on the geometric structure of a honeycomb cell, see the article "The Design of Honeycombs" by Anthony L. Peressini in UMAP Module $502,$ published by COMAP, Inc., Suite 210,57 Bedford Street, Lexington, MA.

Carson Merrill

Numerade Educator

Problem 72

In order to build a highway, it is necessary to fill a section of a valley where the grades (slopes) of the sides are $9 \%$ and $6 \%$ (see figure). The top of the filled region will have the shape of a parabolic arc that is tangent to the two slopes at the points $A$ and $B$. The horizontal distances from $A$ to the $y$ -axis and from $B$ to the $y$ -axis are both 500 feet.
(a) Find the coordinates of $A$ and $B$
(b) Find a quadratic function $y=a x^{2}+b x+c$ for $-500 \leq x \leq 500$ that describes the top of the filled region.
(c) Construct a table giving the depths $d$ of the fill for $x=-500,-400,-300,-200,-100,0,100,200,300$
$400,$ and 500
(d) What will be the lowest point on the completed highway? Will it be directly over the point where the two hillsides come together?

Carson Merrill

Numerade Educator

Problem 73

The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

Diane Koenig

Numerade Educator

Problem 74

If a function is continuous on a closed interval, then it must have a minimum on the interval.

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Problem 75

If $x=c$ is a critical number of the function $f,$ then it is also a critical number of the function $g(x)=f(x)+k,$ where $k$ is a constant.

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Problem 76

If $x=c$ is a critical number of the function $f,$ then it is also a critical number of the function $g(x)=f(x-k),$ where $k$ is a constant.

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Problem 77

Functions Let the function $f$ be differentiable on an interval $I$ containing $c .$ If $f$ has a maximum value at $x=c$ show that $-f$ has a minimum value at $x=c$

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Problem 78

Critical Numbers Consider the cubic function $f(x)=a x^{3}+b x^{2}+c x+d,$ where $a \neq 0 .$ Show that $f$ can have zero, one, or two critical numbers and give an example of each case.

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Problem 79

Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0, a]$ with the property that the region $R=\{(x, y) ; 0 \leq x \leq a$ $0 \leq y \leq f(x)\}$ has perimeter $k$ units and area $k$ square units for some real number $k$

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