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Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 3

Applications of Differentiation

Educators


Problem 1

In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum.
$$
f(x)=\frac{x^{2}}{x^{2}+4}
$$

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Problem 2

In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum.
$$
f(x)=\cos \frac{\pi x}{2}
$$

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Problem 3

In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum.
$$
g(x)=x+\frac{4}{x^{2}}
$$

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Problem 4

In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum.
$$
f(x)=-3 x \sqrt{x+1}
$$

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Problem 5

In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum.
$$
f(x)=(x+2)^{2 / 3}
$$

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Problem 6

In Exercises $1-6,$ find the value of the derivative (if it exists) at each indicated extremum.
$$
f(x)=4-|x|
$$

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Problem 7

In Exercises $7-10$ , approximate the critical numbers of the function shown in the graph. 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.
(GRAPH NOT COPY)

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Problem 8

In Exercises $7-10$ , approximate the critical numbers of the function shown in the graph. 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.
(GRAPH NOT COPY)

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Problem 9

In Exercises $7-10$ , approximate the critical numbers of the function shown in the graph. 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.
(GRAPH NOT COPY)

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Problem 10

In Exercises $7-10$ , approximate the critical numbers of the function shown in the graph. 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.
(GRAPH NOT COPY)

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Problem 11

In Exercises $11-16,$ find any critical numbers of the function.
$$
f(x)=x^{3}-3 x^{2}
$$

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Problem 12

In Exercises $11-16,$ find any critical numbers of the function.
$$
g(x)=x^{4}-4 x^{2}
$$

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Problem 13

In Exercises $11-16,$ find any critical numbers of the function.
$$
g(t)=t \sqrt{4-t}, t<3
$$

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Problem 14

In Exercises $11-16,$ find any critical numbers of the function.
$$
f(x)=\frac{4 x}{x^{2}+1}
$$

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Problem 15

In Exercises $11-16,$ find any critical numbers of the function.
$$
\begin{array}{l}{h(x)=\sin ^{2} x+\cos x} \\ {0 < x < 2 \pi}\end{array}
$$

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Problem 16

In Exercises $11-16,$ find any critical numbers of the function.
$$
\begin{array}{l}{f(\theta)=2 \sec \theta+\tan \theta} \\ {0 < \theta < 2 \pi}\end{array}
$$

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Problem 17

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=3-x,[-1,2]
$$

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Problem 18

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=\frac{2 x+5}{3},[0,5]
$$

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Problem 19

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
g(x)=x^{2}-2 x,[0,4]
$$

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Problem 20

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
h(x)=-x^{2}+3 x-5,[-2,1]
$$

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Problem 21

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=x^{3}-\frac{3}{2} x^{2},[-1,2]
$$

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Problem 22

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=x^{3}-12 x,[0,4]
$$

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Problem 23

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
y=3 x^{2 / 3}-2 x,[-1,1]
$$

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Problem 24

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
g(x)=\sqrt[3]{x},[-1,1]
$$

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Problem 25

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
g(t)=\frac{t^{2}}{t^{2}+3},[-1,1]
$$

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Problem 26

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=\frac{2 x}{x^{2}+1},[-2,2]
$$

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Problem 27

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
h(s)=\frac{1}{s-2},[0,1]
$$

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Problem 28

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
h(t)=\frac{t}{t-2},[3,5]
$$

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Problem 29

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
y=3-|t-3|,[-1,5]
$$

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Problem 30

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
g(x)=\frac{1}{1+|x+1|},[-3,3]
$$

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Problem 31

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=[x],[-2,2]
$$

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Problem 32

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
h(x)=[[2-x]],[-2,2]
$$

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Problem 33

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
f(x)=\cos \pi x,\left[0, \frac{1}{6}\right]
$$

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Problem 34

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
g(x)=\sec x,\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]
$$

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Problem 35

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
y=3 \cos x,[0,2 \pi]
$$

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Problem 36

In Exercises $17-36,$ locate the absolute extrema of the function on the closed interval.
$$
y=\tan \left(\frac{\pi x}{8}\right),[0,2]
$$

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Problem 37

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{ll}{f(x)=2 x-3} \\ {\text { (a) }[0,2]} & {\text { (b) }[0,2)} \\ {\text { (c) }(0,2]} & {\text { (d) }(0,2)}\end{array}
$$

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Problem 38

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{l}{f(x)=5-x} \\ {\text { (a) }[1,4]} & {\text { (b) }[1,4)} \\ {\text { (c) }(1,4]} & {\text { (d) }(1,4)}\end{array}
$$

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Problem 39

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{l}{f(x)=x^{2}-2 x} \\ {\text { (a) }[-1,2] \quad \text { (b) }(1,3]} \\ {\text { (c) }(0,2) \quad \text { (d) }[1,4)}\end{array}
$$

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Problem 40

In Exercises $37-40$ , locate the absolute extrema of the function (if any exist) over each interval.
$$
\begin{array}{l}{f(x)=\sqrt{4-x^{2}}} \\ {\text { (a) }[-2,2] \quad \text { (b) }[-2,0)} \\ {\text { (c) }(-2,2) \quad \text { (d) }[1,2)}\end{array}
$$

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Problem 41

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=\left\{\begin{array}{ll}{2 x+2,} & {0 \leq x \leq 1} \\ {4 x^{2},} & {1<x \leq 3}\end{array},[0,3]\right.
$$

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Problem 42

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=\left\{\begin{array}{ll}{2-x^{2},} & {1 \leq x<3} \\ {2-3 x,} & {3 \leq x \leq 5}\end{array} \quad[1,5]\right.
$$

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Problem 43

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=\frac{3}{x-1}, \quad(1,4]
$$

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Problem 44

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=\frac{2}{2-x}, \quad[0,2)
$$

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Problem 45

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=x^{4}-2 x^{3}+x+1, \quad[-1,3]
$$

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Problem 46

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=\sqrt{x}+\cos \frac{x}{2}, \quad[0,2 \pi]
$$

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Problem 47

In Exercises 47 and $48,$ (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).
$$
f(x)=3.2 x^{5}+5 x^{3}-3.5 x, \quad[0,1]
$$

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Problem 48

In Exercises 47 and $48,$ (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).
$$
f(x)=\frac{4}{3} x \sqrt{3-x}, \quad[0,3]
$$

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Problem 49

In Exercises 49 and $50,$ 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 4.6.)
$$
f(x)=\sqrt{1+x^{3}}, \quad[0,2]
$$

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Problem 50

In Exercises 49 and $50,$ 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 4.6.)
$$
f(x)=\frac{1}{x^{2}+1}, \quad\left[\frac{1}{2}, 3\right]
$$

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Problem 51

In Exercises 51 and $52,$ 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 4.6.)
$$
f(x)=(x+1)^{2 / 3}, \quad[0,2]
$$

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Problem 52

In Exercises 51 and $52,$ 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 4.6.)
$$
f(x)=\frac{1}{x^{2}+1}, \quad[-1,1]
$$

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Problem 53

Writing Write a short paragraph explaining why a continuous function on an open interval may not have a maximum or minimum. Illustrate your explanation with a sketch of the graph of such a function.

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Problem 54

Decide whether each labeled point is an absolute maximum or minimum, a relative maximum or minimum, or neither.

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Problem 55

In Exercises 55 and $56,$ 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 56

In Exercises 55 and $56,$ 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 $x=5$

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Problem 57

In Exercises $57-60$ , determine from the graph whether $f$ has a minimum in the open interval $(a, b) .$
A.(GRAPH NOT COPY)
B.(GRAPH NOT COPY)

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Problem 58

In Exercises $57-60$ , determine from the graph whether $f$ has a minimum in the open interval $(a, b) .$
A.(GRAPH NOT COPY)
B.(GRAPH NOT COPY)

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Problem 59

In Exercises $57-60$ , determine from the graph whether $f$ has a minimum in the open interval $(a, b) .$
A.(GRAPH NOT COPY)
B.(GRAPH NOT COPY)

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Problem 60

In Exercises $57-60$ , determine from the graph whether $f$ has a minimum in the open interval $(a, b) .$
A.(GRAPH NOT COPY)
B.(GRAPH NOT COPY)

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Problem 61

Power 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 62

Lawn Sprinkler 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?

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Problem 63

Honeycomb 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 $S$ .

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Problem 64

Highway Design 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 $y=a x^{2}+b x+c,-500 \leq$ $x \leq 500$ , 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?

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Problem 65

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
The maximum of a function that is continuous on a closed interval can occur at two different values in the interval.

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Problem 66

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If a function is continuous on a closed interval, then it must have a minimum on the interval.

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Problem 67

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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 68

True or False? In Exercises $65-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
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 69

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 70

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 71

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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