Calculus of a Single Variable

Ron Larson, Bruce Edwards

Chapter 0

Preparation for Calculus - all with Video Answers

Educators

+ 1 more educators

Section 1

Graphs and Models

Problem 1

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=-\frac{3}{2} x+3
$$

Linh Vu

Linh Vu

Numerade Educator

Problem 2

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=\sqrt{9-x^{2}}
$$

Linh Vu

Numerade Educator

Problem 3

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=3-x^{2}
$$

Linh Vu

Numerade Educator

Problem 4

In Exercises $1-4,$ match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
$$
y=x^{3}-x
$$

Linh Vu

Numerade Educator

Problem 5

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\frac{1}{2} x+2
$$

William Semus

Numerade Educator

Problem 6

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=5-2 x
$$

Linh Vu

Numerade Educator

Problem 7

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=4-x^{2}
$$

Linh Vu

Numerade Educator

Problem 8

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=(x-3)^{2}
$$

Linh Vu

Numerade Educator

Problem 9

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=|x|-1
$$

Marcella Sippey

Marcella Sippey

Numerade Educator

Problem 10

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=|x|-1
$$

Linh Vu

Numerade Educator

Problem 11

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\sqrt{x}-6
$$

Linh Vu

Numerade Educator

Problem 12

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\sqrt{x+2}
$$

Linh Vu

Numerade Educator

Problem 13

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\frac{3}{x}
$$

Linh Vu

Numerade Educator

Problem 14

In Exercises $9-14,$ sketch the graph of the equation by point plotting.
$$
y=\frac{1}{x+2}
$$

Linh Vu

Numerade Educator

Problem 15

In Exercises 15 and $16,$ describe the viewing window that yields the figure.
$$
y=x^{3}+4 x^{2}-3
$$

Linh Vu

Numerade Educator

Problem 16

In Exercises 15 and $16,$ describe the viewing window that yields the figure.
$$
y=|x|+|x-16|
$$

Linh Vu

Numerade Educator

Problem 17

In Exercises 17 and $18,$ use a graphing utility to graph the equation. Move the cursor along the curve to approximate the unknown coordinate of each solution point accurate to two decimal places.
$$
y=\sqrt{5-x} \quad(\text { a) }(2, y) \quad \text { (b) }(x, 3)
$$

Linh Vu

Numerade Educator

Problem 18

In Exercises 17 and $18,$ use a graphing utility to graph the equation. Move the cursor along the curve to approximate the unknown coordinate of each solution point accurate to two decimal places.
$$
y=x^{5}-5 x \quad \text { (a) }(-0.5, y) \quad \text { (b) }(x,-4)
$$

Linh Vu

Numerade Educator

Problem 19

In Exercises $19-28$ , find any intercepts.
$$
y=2 x-5
$$

Thomas Harris

Numerade Educator

Problem 20

In Exercises $19-28$ , find any intercepts.
$$
y=4 x^{2}+3
$$

JC

Jacob Caudell

Numerade Educator

Problem 21

In Exercises $19-28$ , find any intercepts.
$$
y=x^{2}+x-2
$$

Shaurya Tathgir

Shaurya Tathgir

Numerade Educator

Problem 22

In Exercises $19-28$ , find any intercepts.
$$
y^{2}=x^{3}-4 x
$$

JC

Jacob Caudell

Numerade Educator

Problem 23

In Exercises $19-28$ , find any intercepts.
$$
y=x \sqrt{16-x^{2}}
$$

JR

Jack Ryan

Numerade Educator

Problem 24

In Exercises $19-28$ , find any intercepts.
$$
y=(x-1) \sqrt{x^{2}+1}
$$

JC

Jacob Caudell

Numerade Educator

Problem 25

In Exercises $19-28$ , find any intercepts.
$$
y=\frac{2-\sqrt{x}}{5 x}
$$

Linh Vu

Numerade Educator

Problem 26

In Exercises $19-28$ , find any intercepts.
$$
y=\frac{x^{2}+3 x}{(3 x+1)^{2}}
$$

Sheryl Ezze

Numerade Educator

Problem 27

In Exercises $19-28$ , find any intercepts.
$$
x^{2} y-x^{2}+4 y=0
$$

JC

James Choi

Numerade Educator

Problem 28

In Exercises $19-28$ , find any intercepts.
$$
y=2 x-\sqrt{x^{2}+1}
$$

JC

Jacob Caudell

Numerade Educator

Problem 29

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=x^{2}-6
$$

Linh Vu

Numerade Educator

Problem 30

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=x^{2}-x
$$

Linh Vu

Numerade Educator

Problem 31

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y^{2}=x^{3}-8 x
$$

Linh Vu

Numerade Educator

Problem 32

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=x^{3}+x
$$

Linh Vu

Numerade Educator

Problem 33

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.

$$
x y=4
$$

Amy Jiang

Numerade Educator

Problem 34

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
x y^{2}=-10
$$

Linh Vu

Numerade Educator

Problem 35

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=4-\sqrt{x+3}
$$

JC

James Choi

Numerade Educator

Problem 36

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
x y-\sqrt{4-x^{2}}=0
$$

Linh Vu

Numerade Educator

Problem 37

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=\frac{x}{x^{2}+1}
$$

Linh Vu

Numerade Educator

Problem 38

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=\frac{x^{2}}{x^{2}+1}
$$

Linh Vu

Numerade Educator

Problem 39

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
y=\left|x^{3}+x\right|
$$

Jacob Denson

Numerade Educator

Problem 40

In Exercises $29-40,$ test for symmetry with respect to each axis and to the origin.
$$
|y|-x=3
$$

Jacob Denson

Numerade Educator

Problem 41

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=2-3 x
$$

Linh Vu

Numerade Educator

Problem 42

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=-\frac{3}{2} x+6
$$

Linh Vu

Numerade Educator

Problem 43

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{1}{2} x-4
$$

Linh Vu

Numerade Educator

Problem 44

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{2}{3} x+1
$$

Linh Vu

Numerade Educator

Problem 45

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=9-x^{2}
$$

Linh Vu

Numerade Educator

Problem 46

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x^{2}+3
$$

Linh Vu

Numerade Educator

Problem 47

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=(x+3)^{2}
$$

Linh Vu

Numerade Educator

Problem 48

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=2 x^{2}+x
$$

Linh Vu

Numerade Educator

Problem 49

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x^{3}+2
$$

Linh Vu

Numerade Educator

Problem 50

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x^{3}-4 x
$$

Linh Vu

Numerade Educator

Problem 51

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=x \sqrt{x+5}
$$

Linh Vu

Numerade Educator

Problem 52

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\sqrt{25-x^{2}}
$$

Linh Vu

Numerade Educator

Problem 53

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
x=y^{3}
$$

Linh Vu

Numerade Educator

Problem 54

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
x=y^{2}-4
$$

Linh Vu

Numerade Educator

Problem 55

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{8}{x}
$$

Linh Vu

Numerade Educator

Problem 56

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=\frac{10}{x^{2}+1}
$$

Linh Vu

Numerade Educator

Problem 57

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=6-|x|
$$

Linh Vu

Numerade Educator

Problem 58

In Exercises $41-58,$ sketch the graph of the equation. Identify any intercepts and test for symetry.
$$
y=|6-x|
$$

Linh Vu

Numerade Educator

Problem 59

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
y^{2}-x=9
$$

Linh Vu

Numerade Educator

Problem 60

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
x^{2}+4 y^{2}=4
$$

Linh Vu

Numerade Educator

Problem 61

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
x+3 y^{2}=6
$$

Himanshu Gupta

Numerade Educator

Problem 62

In Exercises $59-62$ , use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$$
3 x-4 y^{2}=8
$$

Linh Vu

Numerade Educator

Problem 63

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{c}{x+y=8} \\ {4 x-y=7}\end{array}
$$

Linh Vu

Numerade Educator

Problem 64

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{l}{3 x-2 y=-4} \\ {4 x+2 y=-10}\end{array}
$$

Linh Vu

Numerade Educator

Problem 65

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{aligned} x^{2}+y &=6 \\ x+y &=4 \end{aligned}
$$

Linh Vu

Numerade Educator

Problem 66

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{aligned} x^{2}+y &=6 \\ x+y &=4 \end{aligned}
$$

Linh Vu

Numerade Educator

Problem 67

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{aligned} x^{2}+y^{2} &=5 \\ x-y &=1 \end{aligned}
$$

Linh Vu

Numerade Educator

Problem 68

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{c}{x^{2}+y^{2}=25} \\ {-3 x+y=15}\end{array}
$$

Linh Vu

Numerade Educator

Problem 69

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{l}{y=x^{3}} \\ {y=x}\end{array}
$$

Linh Vu

Numerade Educator

Problem 70

In Exercises $63-70,$ find the points of intersection of the graphs of the equations.
$$
\begin{array}{l}{y=x^{3}-4 x} \\ {y=-(x+2)}\end{array}
$$

Linh Vu

Numerade Educator

Problem 71

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=x^{3}-2 x^{2}+x-1} \\ {y=-x^{2}+3 x-1}\end{array}
$$

Linh Vu

Numerade Educator

Problem 72

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=x^{4}-2 x^{2}+1} \\ {y=1-x^{2}}\end{array}
$$

Linh Vu

Numerade Educator

Problem 73

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=\sqrt{x+6}} \\ {y=\sqrt{-x^{2}-4 x}}\end{array}
$$

Linh Vu

Numerade Educator

Problem 74

In Exercises $71-74,$ use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{array}{l}{y=-|2 x-3|+6} \\ {y=6-x}\end{array}
$$

Linh Vu

Numerade Educator

Problem 75

Modeling Data The table shows the Consumer Price Index (CPI) for selected years. (Source: Bureau of Labor Statistics)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1975} & {1980} & {1985} & {1990} & {1995} & {2000} & {2005} \\ \hline \mathrm{CPI} & {53.8} & {82.4} & {107.6} & {130.7} & {152.4} & {172.2} & {195.3} \\ \hline\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find
a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the CPI and $t$ represents the year, with $t=5$ corresponding to $1975 .$
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.
(c) Use the model to predict the CPI for the year 2010.

Bobby Barnes

University of North Texas

Problem 76

Modeling Data The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years. (Source: Cellular Telecommunications and Intemet Association)
$$
\begin{array}{|c|c|c|c|c|c|c|}\hline \text { Year } & {1990} & {1993} & {1996} & {1999} & {2002} & {2005} \\ \hline \text { Number } & {5} & {16} & {44} & {86} & {141} & {208} \\ \hline\end{array}
$$
(a) Use the regression capabilities of a graphing utility to find a mathematical model of the form $y=a t^{2}+b t+c$ for the data. In the model, $y$ represents the number of subscribers and $t$ represents the year, with $t=0$ corresponding to $1990 .$
(b) Use a graphing utility to plot the data and graph the model. Compare the data with the model.
(c) Use the model to predict the number of cellular phone subscribers in the United States in the year 2015 .

Bobby Barnes

University of North Texas

Problem 77

Break-Even Point Find the sales necessary to break even $(R=C)$ if the cost $C$ of producing $x$ units is $C=5.5 \sqrt{x}+10,000$
and the revenue $R$ from selling $x$ units is
$R=3.29 x$

Bobby Barnes

University of North Texas

Problem 78

Copper Wire The resistance $y$ in ohms of 1000 feet of solid copper wire at $77^{\circ} \mathrm{F}$ can be approximated by the model
$$y=\frac{10,770}{x^{2}}-0.37, \quad 5 \leq x \leq 100$$

Linh Vu

Numerade Educator

Problem 79

In Exercises 79 and $80,$ write an equation whose graph has the indicated property. (There may be more than one correct answer.)
The graph has intercepts at $x=-4, x=3,$ and $x=8$

Linh Vu

Numerade Educator

Problem 80

In Exercises 79 and $80,$ write an equation whose graph has the indicated property. (There may be more than one correct answer.)
The graph has intercepts at $x=-\frac{3}{2}, x=4,$ and $x=\frac{5}{2}$

Linh Vu

Numerade Educator

Problem 81

(a) Prove that if a graph is symmetric with respect to the $x$ -axis and to the $y$ -axis, then it is symmetric with respect to the origin. Give an example to show that the converse is not true.
(b) Prove that if a graph is symmetric with respect to one axis and to the origin, then it is symmetric with respect to the other axis.

Bobby Barnes

University of North Texas

Problem 82

Match the equation or equations with the given characteristic.
$\begin{array}{ll}{\text { (i) } y=3 x^{3}-3 x} & {\text { (ii) } y=(x+3)^{2} \quad \text { (iii) } y=3 x-3} \\ {\text { (iv) } y=\sqrt[3]{x}} & {\text { (v) } y=3 x^{2}+3 \quad \text { (vi) } y=\sqrt{x+3}}\end{array}$
(a) Symmetric with respect to the $y$ -axis
(b) Three $x$ -intercepts
(c) Symmetric with respect to the $x$ -axis
(d) $(-2,1)$ is a point on the graph
(e) Symmetric with respect to the origin
(f) Graph passes through the origin

Bobby Barnes

University of North Texas

Problem 83

True or False? In Exercises $83-86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $x$ -axis, then $(4,-5)$ is also a point on the graph.

Linh Vu

Numerade Educator

Problem 84

True or False? In Exercises $83-86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $y$ -axis, then $(4,-5)$ is also a point on the graph.

Linh Vu

Numerade Educator

Problem 85

True or False? In Exercises $83-86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c>0$ and $a \neq 0$ , then the graph of $y=a x^{2}+b x+c$ has two $x$ -intercepts.

Linh Vu

Numerade Educator

Problem 86

True or False? In Exercises $83-86$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $b^{2}-4 a c=0$ and $a \neq 0,$ then the graph of $y=a x^{2}+b x+c$ has only one $x$ -intercept.

Linh Vu

Numerade Educator

Problem 87

In Exercises 87 and $88,$ find an equation of the graph that consists of all points $(x, y)$ having the given distance from the origin. (For a review of the Distance Formula, see Appendix C.)

The distance from the origin is twice the distance from $(0,3)$

Linh Vu

Numerade Educator

Problem 88

In Exercises 87 and $88,$ find an equation of the graph that consists of all points $(x, y)$ having the given distance from the origin. (For a review of the Distance Formula, see Appendix C.)

The distance from the origin is $K(K \neq 1)$ times the distance from $(2,0) .$

Linh Vu

Numerade Educator