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Finite Mathematics and Calculus with Applications

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

Chapter 13

Graphs and the Derivative

Educators


Problem 1

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 2

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 3

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 4

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 5

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 6

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 7

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 8

Find the open intervals where the functions graphed as follows are (a) increasing, or (b) decreasing.
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Problem 9

For each of the exercises listed below, suppose that the function that is graphed is not $f(x),$ but $f^{\prime}(x) .$ Find the open intervals where $f(x)$ is (a) increasing or (b) decreasing.

Exercise 1

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Problem 10

For each of the exercises listed below, suppose that the function that is graphed is not $f(x),$ but $f^{\prime}(x) .$ Find the open intervals where $f(x)$ is (a) increasing or (b) decreasing.

Exercise 2

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Problem 11

For each of the exercises listed below, suppose that the function that is graphed is not $f(x),$ but $f^{\prime}(x) .$ Find the open intervals where $f(x)$ is (a) increasing or (b) decreasing.

Exercise 7

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Problem 12

For each of the exercises listed below, suppose that the function that is graphed is not $f(x),$ but $f^{\prime}(x) .$ Find the open intervals where $f(x)$ is (a) increasing or (b) decreasing.

Exercise 8

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Problem 13

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=2.3+3.4 x-1.2 x^{2}$$

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Problem 14

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=1.1-0.3 x-0.3 x^{2}$$

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Problem 15

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=\frac{2}{3} x^{3}-x^{2}-24 x-4$$

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Problem 16

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=\frac{2}{3} x^{3}-x^{2}-4 x+2$$

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Problem 17

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=4 x^{3}-15 x^{2}-72 x+5$$

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Problem 18

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=4 x^{3}-9 x^{2}-30 x+6$$

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Problem 19

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=x^{4}+4 x^{3}+4 x^{2}+1$$

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Problem 20

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=3 x^{4}+8 x^{3}-18 x^{2}+5$$

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Problem 21

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=-3 x+6$$

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Problem 22

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=6 x-9$$

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Problem 23

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=\frac{x+2}{x+1}$$

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Problem 24

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=\frac{x+3}{x-4}$$

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Problem 25

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=\sqrt{x^{2}+1}$$

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Problem 26

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=x \sqrt{9-x^{2}}$$

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Problem 27

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=x^{2 / 3}$$

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Problem 28

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=(x+1)^{4 / 5}$$

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Problem 29

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=x-4 \ln (3 x-9)$$

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Problem 30

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=\ln \frac{5 x^{2}+4}{x^{2}+1}$$

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Problem 31

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=x e^{-3 x}$$

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Problem 32

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=x e^{x^{2}-3 x}$$

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Problem 33

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=x^{2} 2^{-x}$$

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Problem 34

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$f(x)=x 2^{-x^{2}}$$

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Problem 35

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=x^{2 / 3}-x^{5 / 3}$$

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Problem 36

For each function, find (a) the critical numbers; (b) the open intervals where the function is increasing; and (c) the open intervals where it is decreasing.
$$y=x^{1 / 3}+x^{4 / 3}$$

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Problem 37

A friend looks at the graph of $y=x^{2}$ and observes that if you start at the origin, the graph increases whether you go to the right or the left, so the graph is increasing everywhere. Explain why this reasoning is incorrect.

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Problem 38

Use the techniques of this chapter to find the vertex and intervals where $f$ is increasing and decreasing, given
$$f(x)=a x^{2}+b x+c$$
where we assume $a>0 .$ Verify that this agrees with what we found in Chapter 10 .

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Problem 39

Repeat Exercise 38 under the assumption $a<0$

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Problem 40

Where is the function defined by $f(x)=e^{x}$ increasing? Decreasing? Where is the tangent line horizontal?

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Problem 41

Repeat Exercise 40 with the function defined by $f(x)=\ln x$

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Problem 42

a. For the function in Exercise 15, find the average of the critical numbers.
b. For the function in Exercise 15, use a graphing calculator to find the roots of the function, and then find the average of those roots.
c. Compare your answers to parts a and b. What do you notice?
d. Repeat part a for the function in Exercise 17.
e. Repeat part b for the function in Exercise 17.
f. Compare your answers to parts d and e. What do you notice? It can be shown that the average of the roots of a polynomial (including the complex roots, if there are any) and the critical numbers of a polynomial (including complex roots of $f^{\prime}(x)=0$ if there are any ) are always equal.

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Problem 43

For each of the following functions, use a graphing calculator to find the open intervals where $f(x)$ is (a) increasing, or (b) decreasing.
$$f(x)=e^{0.001 x}-\ln x$$

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Problem 44

For each of the following functions, use a graphing calculator to find the open intervals where $f(x)$ is (a) increasing, or (b) decreasing.
$$f(x)=\ln \left(x^{2}+1\right)-x^{0.3}$$

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Problem 45

A county realty group estimates that the number of housing starts per year over the next three years will be
$$H(r)=\frac{300}{1+0.03 r^{2}}$$
where $r$ is the mortgage rate (in percent).
a. Where is $H(r)$ increasing?
b. Where is $H(r)$ decreasing?

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Problem 46

Suppose the total cost $C(x)$ (in dollars) to manufacture a quantity $x$ of weed killer (in hundreds of liters) is given by
$$C(x)=x^{3}-2 x^{2}+8 x+50$$
a. Where is $C(x)$ decreasing?
b. Where is $C(x)$ increasing?

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Problem 47

A manufacturer sells video games with the following cost and revenue functions (in dollars), where $x$ is the number of games sold, for $0 \leq x \leq 3300$ .
$$\begin{array}{l}{C(x)=0.32 x^{2}-0.00004 x^{3}} \\ {R(x)=0.848 x^{2}-0.0002 x^{3}}\end{array}$$
Determine the interval(s) on which the profit function is increasing.

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Problem 48

Profit A manufacturer of $\mathrm{CD}$ players has determined that the profit $P(x)$ (in thousands of dollars) is related to the quantity $x$ of CD players produced (in hundreds) per month by
$$P(x)=-(x-4) e^{x}-4, \quad 0<x \leq 3.9$$
a. At what production levels is the profit increasing?
b. At what levels is it decreasing?

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Problem 49

Social Security Assets The projected year-end assets in the Social Security trust funds, in trillions of dollars, where $t$ represents the number of years since 2000 , can be approximated by
$$A(t)=0.0000329 t^{3}-0.00450 t^{2}+0.0613 t+2.34$$
where $0 \leq t \leq 50 .$ Source: Social Security Administration.
a. Where is $A(t)$ increasing?
b. Where is $A(t)$ decreasing?

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Problem 50

Unemployment The annual unemployment rates of the U.S. civilian noninstitutional population for $1990-2009$ are shown in the graph. When is the function increasing? Decreasing?
Constant? Source: Bureau of Labor Statistics.
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Problem 51

The graph shows the amount of air pollution removed by trees in the Chicago urban region for each month of the year. From the graph we see, for example, that the ozone level starting in May increases up to June, and then abruptly decreases.
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a. Are these curves the graphs of functions?
b. Look at the graph for particulates. Where is the function increasing? Decreasing? Constant?
c. On what intervals do all four lower graphs indicate that the
corresponding functions are constant? Why do you think the functions are constant on those intervals?

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Problem 52

Spread of Infection The number of people $P(t)$ (in hundreds) infected $t$ days after an epidemic begins is approximated by
$$P(t)=\frac{10 \ln (0.19 t+1)}{0.19 t+1}$$
When will the number of people infected start to decline?

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Problem 53

In Exercise 55 in the section on Polynomial and Rational Functions, we gave the function defined by
$$A(x)=0.003631 x^{3}-0.03746 x^{2}+0.1012 x+0.009$$
as the approximate blood alcohol concentration in a $170-\mathrm{lb}$ woman $x$ hours after drinking 2 oz of alcohol on an empty stomach, for $x$ in the interval $[0,5] .$ Source: Medicolegal Aspects of Alcohol Determination in Biological Specimens.
a. On what time intervals is the alcohol concentration increasing?
b. On what intervals is it decreasing?

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Problem 54

Drug Concentration The percent of concentration of a drug in the bloodstream $x$ hours after the drug is administered is given by
$$K(x)=\frac{4 x}{3 x^{2}+27}$$
a. On what time intervals is the concentration of the drug increasing?
b. On what intervals is it decreasing?

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Problem 55

Suppose a certain drug is administered to a patient, with the percent of concentration of the drug in the bloodstream t hours later given by
$$K(t)=\frac{5 t}{t^{2}+1}$$
a. On what time intervals is the concentration of the drug increasing?
b. On what intervals is it decreasing?

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Problem 56

The aortic pressure-diameter relation in a particular patient who underwent cardiac catheterization can be modeled by the polynomial
$$\begin{aligned} D(p)=0.000002 p^{3}-& 0.0008 p^{2}+0.1141 p+16.683 \\ & 55 \leq p \leq 130 \end{aligned}$$
where $D(p)$ is the aortic diameter (in millimeters) and $p$ is the aortic pressure (in mmHg). Determine where this function is increasing and where it is decreasing within the interval given.

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Problem 57

The metabolic rate of a person who has just eaten a meal tends to go up and then, after some time has passed, returns to a resting metabolic rate. This phenomenon is known as the thermic effect of food. Researchers have indicated that the thermic effect of food for one particular person is
$$F(t)=-10.28+175.9 t e^{-t / 13}$
$where $F(t)$ is the thermic effect of food (in $\mathrm{kJ} / \mathrm{hr} )$ and $t$ is the number of hours that have elapsed since eating a meal.
a. Find $F^{\prime}(t)$
b. Determine where this function is increasing and where it is decreasing. Interpret your answers.

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Problem 58

Researchers have developed the following function that can be used to accurately predict the
weight of Holstein cows (females) of various ages:
$$W_{1}(t)=619\left(1-0.905 e^{-0.002 t}\right)^{1.2386}$$
where $W_{1}(t)$ is the weight of the Holstein cow (in kilograms) that is $t$ days old. Where is this function increasing?

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Problem 59

The standard normal probability function is used to describe many different populations. Its graph is the well known normal curve. This function is defined by
$$f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$$
Give the intervals where the function is increasing and decreasing.

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Problem 60

The figure shows estimated totals of nuclear weapons inventory for the United States and the Soviet Union (and its successor states) from 1945 to 2010.
a. On what intervals were the total inventories of both countries increasing?
b. On what intervals were the total inventories of both countries decreasing?
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Problem 61

The following graph shows the horsepower and torque as a function of the engine speed for a 1964 Ford Mustang.
a. On what intervals is the power increasing with engine speed?
b. On what intervals is the power decreasing with engine speed?
c. On what intervals is the torque increasing with engine speed?
d. On what intervals is the torque decreasing with engine speed?
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Problem 62

As a mathematics professor loads more weight in the back of his Subaru, the mileage goes down. Let $x$ be the amount of weight (in pounds) that he adds, and let
$y=f(x)$ be the mileage (in mpg).
a. Is $f^{\prime}(x)$ positive or negative? Explain.
b. What are the units of $f^{\prime}(x) ?$

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