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Finite Mathematics and Calculus with Applications

Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Chapter 14

Applications of the Derivative - all with Video Answers

Educators

Section 1

Absolute Extrema

00:42

Problem 1

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:34

Problem 2

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:28

Problem 3

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:20

Problem 4

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:37

Problem 5

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:33

Problem 6

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:41

Problem 7

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:34

Problem 8

Find the locations of any absolute extrema for the functions with graphs as follows.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:47

Problem 9

What is the difference between a relative extremum and an absolute extremum?

Amy Jiang
Amy Jiang
Numerade Educator
01:11

Problem 10

Can a relative extremum be an absolute extremum? Is a relative extremum necessarily an absolute extremum?

Amy Jiang
Amy Jiang
Numerade Educator
02:22

Problem 11

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x^{3}-6 x^{2}+9 x-8 ; [0,5]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:58

Problem 12

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x^{3}-3 x^{2}-24 x+5 ; [-3,6]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:01

Problem 13

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{1}{3} x^{3}+\frac{3}{2} x^{2}-4 x+1 ; [-4,2]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:54

Problem 14

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{1}{3} x^{3}-\frac{1}{2} x^{2}-6 x+3 ; [-4,4]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:24

Problem 15

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x^{4}-18 x^{2}+1 ; [-4,4]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:09

Problem 16

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x^{4}-32 x^{2}-7 ; [-5,6]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:40

Problem 17

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{1-x}{3+x} ; [0,3]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:38

Problem 18

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{8+x}{8-x} ;[4,6]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:57

Problem 19

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{x-1}{x^{2}+1} ; [1,5]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:49

Problem 20

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{x}{x^{2}+2} ; [0,4]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:35

Problem 21

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\left(x^{2}-4\right)^{1 / 3} ; [-2,3]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:38

Problem 22

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\left(x^{2}-16\right)^{2 / 3} ; [-5,8]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:18

Problem 23

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=5 x^{2 / 3}+2 x^{5 / 3} ; d[-2,1]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:46

Problem 24

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x+3 x^{2 / 3} ; [-10,1]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:58

Problem 25

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x^{2}-8 \ln x ;[1,4]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:18

Problem 26

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=\frac{\ln x}{x^{2}} ;[1,4]$$

Amy Jiang
Amy Jiang
Numerade Educator
02:36

Problem 27

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x+e^{-3 x} ;[-1,3]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:42

Problem 28

Find the absolute extrema if they exist, as well as all values of $x$ where they occur, for each function, and specified domain. If you have one, use a graphing calculator to verify your answers.
$$f(x)=x^{2} e^{-0.5 x} ; [2,5]$$

Amy Jiang
Amy Jiang
Numerade Educator
00:37

Problem 29

Graph each function on the indicated domain, and use the capabilities of your calculator to find the location and value of the absolute extrema.
$$f(x)=\frac{-5 x^{4}+2 x^{3}+3 x^{2}+9}{x^{4}-x^{3}+x^{2}+7} ;[-1,1]$$

Amy Jiang
Amy Jiang
Numerade Educator
00:30

Problem 30

Graph each function on the indicated domain, and use the capabilities of your calculator to find the location and value of the absolute extrema.
$$f(x)=\frac{x^{3}+2 x+5}{x^{4}+3 x^{3}+10} ;[-3,0]$$

Amy Jiang
Amy Jiang
Numerade Educator
01:52

Problem 31

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=2 x+\frac{8}{x^{2}}+1, x>0$$

Amy Jiang
Amy Jiang
Numerade Educator
01:17

Problem 32

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=12-x-\frac{9}{x}, x>0$$

Amy Jiang
Amy Jiang
Numerade Educator
01:26

Problem 33

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=-3 x^{4}+8 x^{3}+18 x^{2}+2$$

Amy Jiang
Amy Jiang
Numerade Educator
01:21

Problem 34

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=x^{4}-4 x^{3}+4 x^{2}+1$$

Amy Jiang
Amy Jiang
Numerade Educator
01:27

Problem 35

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=\frac{x-1}{x^{2}+2 x+6}$$

Amy Jiang
Amy Jiang
Numerade Educator
01:04

Problem 36

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=\frac{x}{x^{2}+1}$$

Amy Jiang
Amy Jiang
Numerade Educator
01:30

Problem 37

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=\frac{\ln x}{x^{3}}$$

Amy Jiang
Amy Jiang
Numerade Educator
00:43

Problem 38

Find the absolute extrema if they exist, as well as all values of $x$ where they occur.
$$f(x)=x \ln x$$

Amy Jiang
Amy Jiang
Numerade Educator
03:33

Problem 39

Find the absolute maximum and minimum of $f(x)=$ $2 x-3 x^{2 / 3}(a)$ on the interval $[-1,0.5] ;(b)$ on the interval $[0.5,2] .$

Amy Jiang
Amy Jiang
Numerade Educator
03:14

Problem 40

Let $f(x)=e^{-2 x} .$ For $x>0,$ let $P(x)$ be the perimeter of the rectangle with vertices $(0,0),(x, 0),(x, f(x))$ and $(0, f(x)) .$ Which of the following statements is true? Society of Actuaries.
a. The function $P$ has an absolute minimum but not an absolute maximum on the interval $(0, \infty) .$
b. The function $P$ has an absolute maximum but not an absolute minimum on the interval (0, $\infty$ ).
c. The function $P$ has both an absolute minimum and an absolute maximum on the interval $(0, \infty)$ .
d. The function $P$ has neither an absolute maximum nor an absolute minimum on the interval $(0, \infty),$ but the graph of the function $P$ does have an inflection point with positive $x$ -coordinate.
e. The function $P$ has neither an absolute maximum nor an absolute minimum on the interval $(0, \infty),$ and the graph of the function $P$ does not have an inflection point with positive $x$ -coordinate.

Jessica Wellington
Jessica Wellington
University of Missouri - Columbia
01:12

Problem 41

Bank Robberies The number of bank robberies in the United States for the years 2000–2009 is given in the following figure. Consider the closed interval [2000, 2009]. Source: FBI.
a. Give all relative maxima and minima and when they occur on the interval.
b. Give the absolute maxima and minima and when they occur on the interval. Interpret your results.

Amy Jiang
Amy Jiang
Numerade Educator
01:27

Problem 42

Bank Burglaries The number of bank burglaries (entry into or theft from a bank during nonbusiness hours) in the United States for the years 2000–2009 is given in the figure on the following page. Consider the closed interval [2000, 2009]. Source: FBI.
a. Give all relative maxima and minima and when they occur on the interval.
b. Give the absolute maxima and minima and when they occur on the interval. Interpret your results.

Amy Jiang
Amy Jiang
Numerade Educator
01:44

Problem 43

Profit The total profit $P(x)$ (in thousands of dollars) from the sale of $x$ hundred thousand automobile tires is approximated by
$$P(x)=-x^{3}+9 x^{2}+120 x-400, \quad x \geq 5$$
Find the number of hundred thousands of tires that must be sold to maximize profit. Find the maximum profit.

Amy Jiang
Amy Jiang
Numerade Educator
02:53

Problem 44

Profit A company has found that its weekly profit from the sale of $x$ units of an auto part is given by
$$P(x)=-0.02 x^{3}+600 x-20,000. $$
Production bottlenecks limit the number of units that can be made per week to no more than $150,$ while a long-term contract requires that at least 50 units be made each week. Find the maximum possible weekly profit that the firm can make.

Amy Jiang
Amy Jiang
Numerade Educator
05:01

Problem 45

Average Cost Find the minimum value of the average cost for the given cost function on the given intervals.
$C(x)=x^{3}+37 x+250$ on the following intervals.
a. $1 \leq x \leq 10 \qquad$ b. $10 \leq x \leq 20$

Jessica Wellington
Jessica Wellington
University of Missouri - Columbia
05:01

Problem 46

Average Cost Find the minimum value of the average cost for the given cost function on the given intervals.
$C(x)=81 x^{2}+17 x+324$ on the following intervals.
a. $1 \leq x \leq 10 \qquad$ b. $10 \leq x \leq 20$

Jessica Wellington
Jessica Wellington
University of Missouri - Columbia
00:18

Problem 47

Each graph gives the cost as a function of production level. Use the method of graphical optimization to estimate the production level that results in the minimum cost per item produced.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:24

Problem 48

Each graph gives the cost as a function of production level. Use the method of graphical optimization to estimate the production level that results in the minimum cost per item produced.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:27

Problem 49

Each graph gives the profit as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
00:17

Problem 50

Each graph gives the profit as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced.
GRAPH

Amy Jiang
Amy Jiang
Numerade Educator
04:21

Problem 51

A marshy region used for agricultural drainage has become contaminated with selenium. It has been determined that flushing the area with clean water will reduce the selenium for a while, but it will then begin to build up again. A biologist has found that the percent of selenium in the soil $x$ months after the flushing begins is given by
$$f(x)=\frac{x^{2}+36}{2 x}, \quad 1 \leq x \leq 12$$
When will the selenium be reduced to a minimum? What is the minimum percent?

Nick Johnson
Nick Johnson
Numerade Educator
04:04

Problem 52

Salmon Spawning The number of salmon swimming upstream to spawn is approximated by
$$S(x)=-x^{3}+3 x^{2}+360 x+5000, 6 \leq x \leq 20,$$
where $x$ represents the temperature of the water in degrees Celsius. Find the water temperature that produces the maximum number of salmon swimming upstream.

Jessica Wellington
Jessica Wellington
University of Missouri - Columbia
09:38

Problem 53

Molars Researchers have determined that the crown length of first molars in fetuses is related to the postconception age of the tooth as
$$L(t)=-0.01 t^{2}+0.788 t-7.048$$
where $L(t)$ is the crown length (in millimeters) of the molar $t$ weeks after conception. Find the maximum length of the molar $t$ of first molars during weeks 22 through $28 .$ Source: American Journal of Physical Anthropology.

Regina Hays
Regina Hays
Numerade Educator
04:08

Problem 54

Fungal Growth Because of the time that many people spend indoors, there is a concern about the health risk of being exposed to harmful fungi that thrive in buildings. The risk appears to increase in damp environments. Researchers have discovered that by controlling both the temperature and the relative humidity in a building, the growth of the fungus $A$ . versicolor can be limited. The relationship between temperature and relative humidity, which limits growth, can be described by
$$R(T)=-0.00007 T^{3}+0.0401 T^{2}-1.6572 T+97.086$$
$$15 \leq T \leq 46,$$
where $R(T)$ is the relative humidity (in percent) and $T$ is the temperature (in degrees Celsius). Find the temperature at which the relative humidity is minimized. Source: Applied and Environmental Microbiology.

Jessica Wellington
Jessica Wellington
University of Missouri - Columbia
04:20

Problem 55

Gasoline Mileage From information given in a recent business publication, we constructed the mathematical model
$$M(x)=-\frac{1}{45} x^{2}+2 x-20, 30 \leq x \leq 65$$
to represent the miles per gallon used by a certain car at a speed of $x$ mph. Find the absolute maximum miles per gallon and the absolute minimum and the speeds at which they occur.

Regina Hays
Regina Hays
Numerade Educator
02:49

Problem 56

Gasoline Mileage For a certain sports utility vehicle,
$$M(x)=-0.015 x^{2}+1.31 x-7.3,30 \leq x \leq 60$$
represents the miles per gallon obtained at a speed of $x$ mph. Find the absolute maximum miles per gallon and the absolute minimum and the speeds at which they occur.

Jessica Wellington
Jessica Wellington
University of Missouri - Columbia
07:50

Problem 57

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.
Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$
Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$
Where should the cut be made in order to minimize the sum of the areas enclosed by both figures?

Charles Carter
Charles Carter
Numerade Educator
10:02

Problem 58

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.
Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$
Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$
Where should the cut be made in order to make the sum of the areas maximum? (Hint: Remember to use the endpoints of a domain when looking for absolute maxima and minima.)

Ma. Theresa Alin
Ma. Theresa Alin
Numerade Educator
08:49

Problem 59

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.
Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$
Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$
For the solution to Exercise 57, show that the side of the square equals the diameter of the circle, that is, that the circle can be inscribed in the square.*

Trinity Steen
Trinity Steen
Numerade Educator
02:50

Problem 60

Area A piece of wire 12 $\mathrm{ft}$ long is cut into two pieces. (See the figure.) One piece is made into a circle and the other piece is made into a square. Let the piece of length $x$ be formed into a circle. We allow $x$ to equal 0 or $12,$ so all the wire may be used for the square or for the circle.
Radius of circle $=\frac{x}{2 \pi} \qquad$ Area of circle $=\pi\left(\frac{x}{2 \pi}\right)^{2}$
Side of square $=\frac{12-x}{4} \quad$ Area of square $=\left(\frac{12-x}{4}\right)^{2}$
Information Content Suppose dots and dashes are transmitted over a telegraph line so that dots occur a fraction $p$ of the time (where $0 < p < 1 )$ and dashes occur a fraction $1-p$ of the time. The information content of the telegraph line is given by $I(p),$ where
$$I(p)=-p \ln p-(1-p) \ln (1-p)$$
a. Show that $I^{\prime}(p)=-\ln p+\ln (1-p)$
b. Set $I^{\prime}(p)=0$ and find the value of $p$ that maximizes the information content.
c. How might the result in part b be used?

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator