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Calculus with Applications

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

Chapter 10

Differential Equations

Educators


Problem 1

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=-2 x+9 x^{2}$$

Adam D.
Numerade Educator

Problem 2

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=-4 x+12 x^{2}$$

Sid W.
University of Louisville

Problem 3

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$4 x^{3}-2 \frac{d y}{d x}=0$$

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Problem 4

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$3 x^{2}-3 \frac{d y}{d x}=2$$

Sid W.
University of Louisville

Problem 5

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$7 y \frac{d y}{d x}=5 x^{2}$$

Adam D.
Numerade Educator

Problem 6

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$y \frac{d y}{d x}=x^{2}-x$$

Sid W.
University of Louisville

Problem 7

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=15 x y$$

Adam D.
Numerade Educator

Problem 8

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=x^{2} y$$

Sid W.
University of Louisville

Problem 9

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=8 x^{2} y-9 x y$$

Adam D.
Numerade Educator

Problem 10

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\left(y^{2}-y\right) \frac{d y}{d x}=x$$

Sid W.
University of Louisville

Problem 11

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{y}{x}, x>0$$

Adam D.
Numerade Educator

Problem 12

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{y}{x^{2}}$$

Sid W.
University of Louisville

Problem 13

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{y^{2}+6}{2 y}$$

Adam D.
Numerade Educator

Problem 14

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{4 e^{x}}{e^{y}}$$

Sid W.
University of Louisville

Problem 15

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{11 e^{x}}{e^{y}}$$

Adam D.
Numerade Educator

Problem 16

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{8 e^{x}}{e^{y}}$$

Sid W.
University of Louisville

Problem 17

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}+4 x^{3}=9 x^{2} ; \quad y(0)=5$$

Adam D.
Numerade Educator

Problem 18

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=8 x^{3}-9 x^{2}+x ; \quad y(1)=0$$

Sid W.
University of Louisville

Problem 19

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=8 x^{3}-3 x^{2}+7 x ; \quad y(3)=0$$

Adam D.
Numerade Educator

Problem 20

Find the particular solution for each initial value problem.
$$x \frac{d y}{d x}=x^{2} e^{3 x} ; \quad y(0)=\frac{8}{9}$$

Sid W.
University of Louisville

Problem 21

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=\frac{5 x^{3}}{4 y} ; \quad y(0)=2$$

Adam D.
Numerade Educator

Problem 22

Find the particular solution for each initial value problem.
$$x^{2} \frac{d y}{d x}-y \sqrt{x}=0 ; \quad y(1)=e^{-2}$$

Sid W.
University of Louisville

Problem 23

Find the particular solution for each initial value problem.
$$(2 x+3) y=\frac{d y}{d x} ; \quad y(0)=1$$

Adam D.
Numerade Educator

Problem 24

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=\frac{x^{2}+8}{6 y-1} ; y(0)=13$$

Sid W.
University of Louisville

Problem 25

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=\frac{5 x+1}{y-5} ; \quad y(0)=2$$

Adam D.
Numerade Educator

Problem 26

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=\frac{5 x+8}{y-3} ; \quad y(0)=4$$

Sid W.
University of Louisville

Problem 27

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=\frac{y^{2}}{x} ; \quad y(e)=3$$

Adam D.
Numerade Educator

Problem 28

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=x^{1 / 2} y^{2} ; \quad y(4)=9$$

Sid W.
University of Louisville

Problem 29

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=(y-2)^{2} e^{x-7} ; \quad y(7)=3$$

Adam D.
Numerade Educator

Problem 30

Find the particular solution for each initial value problem.
$$\frac{d y}{d x}=(y-6)^{2} e^{x-7} ; \quad y(7)=7$$

Sid W.
University of Louisville

Problem 31

Find all equilibrium points and determine their stability.
$$\frac{d y}{d x}=y\left(y^{2}-1\right)$$

Adam D.
Numerade Educator

Problem 32

Find all equilibrium points and determine their stability.
$$\frac{d y}{d x}=\left(4-y^{2}\right)(y+1)$$

Sid W.
University of Louisville

Problem 33

Find all equilibrium points and determine their stability.
$$\frac{d y}{d x}=\left(e^{y}-1\right)(y-3)$$

Adam D.
Numerade Educator

Problem 34

Find all equilibrium points and determine their stability.
$$\frac{d y}{d x}=(\ln y-1)(5-y)$$

Sid W.
University of Louisville

Problem 35

(a) Solve the logistic Equation (4) in this section by observing
that
$$\frac{1}{y}+\frac{1}{N-y}=\frac{N}{(N-y) y}$$
(b) Assume 0 < y < N. Verify that $b=\left(N-y_{0}\right) / y_{0}$ in Equation (5), where y0 is the initial population size.
(c) Assume 0 < N < y for all y. Verify that $b=\left(y_{0}-N\right) / y_{0}$

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Problem 36

Suppose that 0 < $z$ < 1 for all $z$ . Solve the logistic Equation
(6) as in Exercise $35 .$ Verify that $b=e^{k x_{0}},$ where $x_{0}$ is the time
at which $z=1 / 2$ .

Sid W.
University of Louisville

Problem 37

Suppose that $0< y_{0} < N .$ Let $b=\left(N-y_{0}\right) / y_{0},$ and let
$y(x)=N /\left(1+b e^{-k x}\right)$ for all $x$ . Show the following.
$$\begin{array}{l}{\text { (a) } 0 < y(x) < N \text { for all } x} \\ {\text { (b) The lines } y=0 \text { and } y=N \text { are horizontal asymptotes of }} \\ {\text { the graph. }} \\ {\text { (c) } y(x) \text { is an increasing function. }} \\ {\text { (d) }((\ln b) / k, N / 2) \text { is a point of inflection of the graph. }} \\ {\text { (e) } d y / d x \text { is a maximum at } x_{0}=(\ln b) / k}\end{array}$$

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Problem 38

$$\begin{array}{c}{\text { Suppose that } 0 < N < y_{0} . \text { Let } b=\left(y_{0}-N\right) / y_{0} \text { and let }} \\ {y(x)=\frac{N}{1-b e^{-k x}} \text { for all } x \neq \frac{\ln b}{k}}\end{array}$$
$$\begin{array}{l}{\text { (a) } 0 < b < 1} \\ {\text { (b) The lines } y=0 \text { and } y=N \text { are horizontal asymptotes of the graph. }}\end{array}$$
$$\begin{array}{l}{\text { (c) The line } x=(\ln b) / k \text { is a vertical asymptote of the graph. }} \\ {\text { (d) } y(x) \text { is decreasing on }((\ln b) / k, \infty) \text { and on }(-\infty,(\ln b) / k)} \\ {\text { (e) } y(x) \text { is concave upward on }((\ln b) / k, \infty) \text { and concave downward on }(-\infty,(\ln b) / k)}\end{array}$$

Sid W.
University of Louisville

Problem 39

Profit The marginal profit of a certain restaurant is given by
$$\frac{d y}{d x}=\frac{200}{20-3 x}$$
where x represents the amount of money (in thousands of dollars) the restaurant spends on quality assurance. Find the profit for each money expenditure on quality assurance if the profit is $500 when the restaurant spends no money on quality assurance.
(a) $\$ 1000$
(b) $2000
(c) According to this model, can the expenditure reach $7000?

Adam D.
Numerade Educator

Problem 40

Sales Decline Sales (in thousands) of a certain product are declining at a rate proportional to the amount of sales, with a decay constant of 15% per year.
(a) Write a differential equation to express the rate of sales decline.
(b) Find a general solution to the equation in part (a).
(c) How much time will pass before sales become 25% of
their original value?

Sid W.
University of Louisville

Problem 41

Inflation If inflation grows annually at a rate of 10%, how long will it take for 1 EGP (Egyptian Pound) to lose half its value?
Elasticity of Demand Elasticity of demand was discussed in Chapter 6 on Applications of the Derivative, where it was defined as
$$E=-\frac{p}{q} \cdot \frac{d q}{d p}$$
for demand $q$ and price $p .$ Find the general demand equation $q=f(p)$ for each elasticity function. (Hint: Set each elasticity function equal to $-\frac{p}{q} \cdot \frac{d q}{d p},$ then solve for $q .$ Write the constant of integration as $\ln C$ in Exercise $43 . )$

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Problem 42

$E=\frac{4 p^{2}}{q^{2}}$

Sid W.
University of Louisville

Problem 43

$E=2$

Tani I.
Numerade Educator

Problem 44

Internet Usage During the early days of the Internet, growth in the number of users worldwide could be approximated by an exponential function. The following table gives the number of worldwide users of the Internet. Source: Internet World Stats.
Use a calculator with exponential and logistic regression capabilities to complete the following.
(a) Letting t represent the years since 1990, plot the number of worldwide users of the Internet on the y-axis against the year on the t-axis. Discuss the shape of the graph.

(b) Use the exponential regression function on your calculator to determine the exponential equation that best fits the data. Plot the exponential equation on the same graph as the data points. Discuss the appropriateness of fitting an exponential function to these data.

(c) Use the logistic regression function on your calculator to determine the logistic equation that best fits the data. Plot the logistic equation on the same graph. Discuss the appropriateness of fitting a logistic function to these data. Which graph better fits the data?

(d) Assuming that the logistic function found in part (c) continues to be accurate, what seems to be the limiting size of the number of worldwide Internet users?

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Problem 45

Life Insurance A life insurance company invests $5000 in a bank account in order to fund a death benefit of $20,000. Growth in the investment over time can be modeled by the differential equation
$$\frac{d A}{d t}=A i$$
where $i$ is the interest rate and $A(t)$ is the amount invested at time $t($ in years). Calculate the interest rate that the investment must earn in order for the company to fund the death benefit in 24 years. Choose one of the following. Source: Society of Actuaries.
(a) $\frac{-\ln 2}{12}$ (b) $\frac{-\ln 2}{24} \quad$ (c) $\frac{\ln 2}{24} \quad$ (d) $\frac{\ln 2}{12} \quad$ (e) $\frac{\ln 2}{6}$

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Problem 46

Tracer Dye The amount of a tracer dye injected into the bloodstream decreases exponentially, with a decay constant of 4% per minute. If 5 cc are present initially, how many cubic centimeters are present after 8 minutes? (Here k will be negative.)

Sid W.
University of Louisville

Problem 47

Soil Moisture The evapotranspiration index I is a measure of soil moisture. An article on 10- to 14-year-old heath vegetation described the rate of change of I with respect to W, the amount of water available, by the equation
$$\frac{d I}{d W}=0.088(2.4-I)$$
Source: Australian Journal of Botany.
$$\begin{array}{l}{\text { (a) According to the article, } I \text { has a value of } 1 \text { when Solve the initial value problem. }} \\ {\text { (b) What happens to } I \text { as } W \text { becomes larger and larger? }}\end{array}$$

Adam D.
Numerade Educator

Problem 48

Fish Population An isolated fish population is limited to 4000 by the amount of food available. If there are now 320 fish and the population is growing with a growth constant of 2% a year, find the expected population at the end of 10 years.
Dieting A person's weight depends both on the daily rate of energy intake, say, $C$ calories per day, and on the daily rate of energy consumption, typically between 15 and 20 calories per pound per day. Using an average value of 17.5 calories per pound per day. Using an average value of 17.5 calories per pound per day, a person weighing $w$ pounds uses 17.5$w$ calories per day. If $C=17.5 w,$ then weight remains constant, and weight gain or loss occurs according to whether $C$ is greater or less than 17.5$w .$ Source: The College Mathematics Journal.

Sid W.
University of Louisville

Problem 49

To determine how fast a change in weight will occur, the most plausible assumption is that $d w / d t$ is proportional to the net excess (or deficit) $C-17.5 w$ in the number of calories per day.
(a) Assume $C$ is constant and write a differential equation to express this relationship. Use $k$ to represent the constant of proportionality. What does $C$ being constant imply?
(b) The units of $d w / d t$ are pounds per day, and the units of $C-17.5 w$ are calories per day. What units must $k$ have?
(c) Use the fact that 3500 calories is equivalent to 1 lb to rewrite the differential equation in part (a).
(d) Solve the differential equation.
(e) Let $w_{0}$ represent the initial weight and use it to express the coefficient of $e^{-0.005 t}$ in terms of $w_{0}$ and $C .$

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Problem 50

(Refer to Exercise $49 . )$ Suppose someone initially weighing 180 lb adopts a diet of 2500 calories per day.
(a) Write the weight function for this individual.
(b) Graph the weight function on the window $[0,300]$ by $[120,200] .$ What is the asymptote? This value of $w$ is the equilibrium weight $w_{e q}$ . According to the model, can a person ever achieve this weight?
(c) How long will it take a dieter to reach a weight just 2 lb more than $w_{e q} ?$

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Problem 51

$\mathrm{H} 1 \mathrm{N} 1$ Virus The cumulative number of deaths worldwide due to the HIN1 virus, or swine flu, at various days into the epidemic are listed below, where April $21,2009,$ was day $1 .$ Source: $B B C$ .
Use a calculator with logistic regression capability to complete the following.
(a) Plot the number of deaths $y$ against the number of days $t$ Discuss the appropriateness of fitting a logistic function to these data.
(b) Use the logistic regression function on your calculator to determine the logistic equation that best fits the data.
c) Plot the logistic regression function from part (b) on the same graph as the data points. Discuss how well the logistic equation fits the data.
(d) Assuming the logistic equation found in part (b) continues to be accurate, what seems to be the limiting size of the number of deaths due to this outbreak of the HIN virus?
(e) Discuss whether a logistic model is more appropriate than an exponential model for estimating the number of deaths due to the HIN1 virus.

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Problem 52

Population Growth The following table gives the historic and projected populations (in millions) of China and India. Source: United Nations.
Use a calculator with logistic regression capability to complete the following.

(a) Letting t represent the years since 1950, plot the Chinese population on the y-axis against the year on the t-axis. Discuss the appropriateness of fitting a logistic function to these data.
(b) Use the logistic regression function on your calculator to determine the logistic equation that best fits the data. Plot the logistic function on the same graph as the data points. Discuss how well the logistic function fits the data.
(c) Assuming the logistic equation found in part (b) continues to be accurate, what seems to be the limiting size of the Chinese population?
(d) Repeat parts (a)-(c) using the population for India.

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Problem 53

U.S. Hispanic Population A recent report by the U.S. Census Bureau predicts that the U.S. Hispanic population will increase from 35.6 million in 2000 to 102.6 million in $2050 .$ Assuming the unlimited growth model $d y / d t=k y$ fits this population growth, express the population y as a function of the year t. Let 2000 correspond to $t=0 .$ Source: U.S. Census Bureau.

Adam D.
Numerade Educator

Problem 54

U.S. Asian Population (Refer to Exercise $53 . )$ The report also predicted that the U.S. Asian population would increase from 10.7 million in 2000 to 33.4 million in $2050 .$ Repeat Exercise 53 using these data. Source: $U . S .$ Census Bureau.

Sid W.
University of Louisville

Problem 55

Plant Growth Researchers have found that the probability $P$ that a plant will grow to radius $R$ can be described by the differential equation
$$\frac{d P}{d R}=-4 \pi D R P^{2}$$
where $D$ is the density of the plants in an area. Given the initial condition $P(0)=1,$ find a formula for $P$ in terms of $R .$ Source: Ecology.

Adam D.
Numerade Educator

Problem 56

World Population The following table gives the population of the world at various times over the past two centuries, plus projections for this century. Source: The New York Times.
Use a calculator with logistic regression capability to complete the following.
(a) Use the logistic regression function on your calculator to determine the logistic equation that best fits the data.

(b) Plot the logistic function found in part (a) and the original data in the same window. Does the logistic function seem to fit the data from 1927 on? Before 1927$?$

(c) To get a better fit, subtract 0.99 from each value of the population in the table. (This makes the population in 1804 small, but not 0 or negative.) Find a logistic function that its the new data.

(d) Plot the logistic function found in part (c) and the modified data in the same window. Does the logistic function now seem to be a better fit than in part (b)?

(e) Based on the results from parts $(\mathrm{c})$ and $(\mathrm{d})$ , predict the limiting value of the world's population as time increases. For comparison, the New York Times article predicts a value of 10.73 billion. (Hint: After taking the limit, remember to add the 0.99 that was removed earlier.)

(f) Based on the results from parts $(\mathrm{c})$ and (d), predict the limiting value of the world population as you go further and further back in time. Does that seem reasonable? Explain.

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Problem 57

57. Worker Productivity A company has found that the rate at which a person new to the assembly line produces items is
$$\frac{d y}{d x}=7.5 e^{-0.3 y}$$

where $x$ is the number of days the person has worked on the line. How many items can a new worker be expected to produce on the eighth day if he produces none when $x=0 ?$

Adam D.
Numerade Educator

Problem 58

Spread of a Rumor Suppose the rate at which a rumor spreads-that is, the number of people who have heard the rumor over a period of time-increases with the number of people who have heard it. If $y$ is the number of people who have heard the rumor, then
$$\frac{d y}{d t}=k y$$
$$\begin{array}{l}{\text { where } t \text { is the time in days. }} \\ {\text { (a) If } y \text { is } 1 \text { when } t=0, \text { and } y \text { is } 5 \text { when } t=2, \text { find } k} \\ {\text { Using the value of } k \text { from part (a), find } y \text { for each time. }} \\ {\text { (b) } t=3 \quad \text { (c) } t=5 \quad \text { (d) } t=10}\end{array}
$$

Sid W.
University of Louisville

Problem 59

Snowplow One morning snow began to fall at a heavy and constant rate. A snowplow started out at $8 : 00$ A.M. At $9 : 00$ A.M. it had traveled 2 miles. By $10 : 00$ A.M. it had traveled 3 miles. Assuming that the snowplow removes a constant volume of snow per hour, determine the time at which it started snowing. (Hint: Let $t$ denote the time since the snow started to fall, and let $T$ be the time when the snowplow started out. Let $x,$ the distance the snowplow has traveled, and $h,$ the height of the snow, be functions of $t .$ The assumption that a constant volume of snow per hour is removed implies that the speed of the snowplow times the height of the snow is a constant. Set up and solve differential equations involving $d x / d t$ and $d h / d t .$ ) Source: The American Mathematical Monthly.

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Problem 60

Radioactive Decay The amount of a radioactive substance decreases exponentially, with a decay constant of 3$\%$ per month.
(a) Write a differential equation to express the rate of change.
(b) Find a general solution to the differential equation from part (a).
(c) If there are 75 $\mathrm{g}$ at the start of the decay process, find a particular solution for the differential equation from part (a).
(d) Find the amount left after 10 months.

Newton's Law of Cooling Newton's law of cooling states that the rate of change of temperature of an object is proportional to the difference in temperature between the object and the surrounding medium. Thus, if $T$ is the temperature of the object after $t$ hours and $T_{M}$ is the (constant) temperature of the surrounding medium, then
$$\frac{d T}{d t}=-k\left(T-T_{M}\right) $$
where $k$ is a constant. Use this equation in Exercises $61-64$

Sid W.
University of Louisville

Problem 61

Show that the solution of this differential equation is
$$T=C e^{-k t}+T_{M}$$
where $C$ is a constant.

Adam D.
Numerade Educator

Problem 62

According to the solution of the differential equation for Newton's law of cooling, what happens to the temperature of an object after it has been in a surrounding medium with constant temperature for a long period of time? How well does this agree with reality?

Newton's Law of Cooling When a dead body is discovered, one of the first steps in the ensuing investigation is for a medical examiner to determine the time of death as closely as possible. Have you ever wondered how this is done? If the temperature of the medium (air, water, or whatever) has been fairly constant and less than 48 hours have passed since the death, Newton's law
of cooling can be used. The medical examiner does not actuallysolve the equation for each case. Instead, a table based on the formula is consulted. Use Newton's law of cooling to work the following exercises. Source: The College Mathematics Journal.

Sid W.
University of Louisville

Problem 63

Assume the temperature of a body at death is $98.6^{\circ} \mathrm{F}$ , the temperature of the surrounding air is $68^{\circ} \mathrm{F},$ and at the end of one hour the body temperature is $90^{\circ} \mathrm{F}$ .
$$\begin{array}{l}{\text { (a) What is the temperature of the body after } 2 \text { hours? }} \\ {\text { (b) When will the temperature of the body be } 75^{\circ} \mathrm{F} ?} \\ {\text { (c) Approximately when will the temperature of the body be }} \\ {\quad \text { within } 0.01^{\circ} \text { of the surrounding air? }}\end{array}$$

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Problem 64

Suppose the air temperature surrounding a body remains at a constant $10^{\circ} \mathrm{F}, C=88.6,$ and $k=0.24$
$$\begin{array}{l}{\text { (a) Determine a formula for the temperature at any time } t .} \\ {\text { (b) Use a graphing calculator to graph the temperature } T \text { as a }} \\ {\text { function of time } t \text { on the window }[0,30] \text { by }[0,100] \text { . }}\end{array}$$
$$\begin{array}{l}{\text { (c) When does the temperature of the body decrease more }} \\ {\text { rapidly: just after death, or much later? How do you }} \\ {\text { know? }} \\ {\text { (d) What will the temperature of the body be after } 4 \text { hours? }} \\ {\text { (e) How long will it take for the body temperature to reach }} \\ {40^{\circ} \mathrm{F} ? \text { Use your calculator graph to verify your answer. }}\end{array}$$

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