Calculus of a Single Variable: Early Transcendental Functions

Ron Larson, Bruce H. Edwards

Chapter 1

Preparation for Calculus - all with Video Answers

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).]
GRAPH CANT COPY
$y=-\frac{3}{2} x+3$

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
GRAPH CANT COPY
$y=\sqrt{9-x^2}$

JC

James Choi

Numerade Educator

Problem 3

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
GRAPH CANT COPY
$y=3-x^2$

Amy Jiang

Numerade Educator

Problem 4

Match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).]
GRAPH CANT COPY
$y=x^3-x$

Gregory Higby

Numerade Educator

Problem 5

In Exercises 5-14, sketch the graph of the equation by point plotting.
$y=\frac{1}{2} x+2$

William Semus

Numerade Educator

Problem 6

Sketch the graph of the equation by point plotting.
$y=5-2 x$

Carson Merrill

Numerade Educator

Problem 7

Sketch the graph of the equation by point plotting.
$y=4-x^2$

Yujie Wang

College of San Mateo

Problem 8

Sketch the graph of the equation by point plotting.
$y=(x-3)^2$

Carson Merrill

Numerade Educator

Problem 9

Sketch the graph of the equation by point plotting.
$y=|x+2|$

Carson Merrill

Numerade Educator

Problem 10

Sketch the graph of the equation by point plotting.
$y=|x|-1$

Carson Merrill

Numerade Educator

Problem 11

Sketch the graph of the equation by point plotting.
$y=\sqrt{x}-6$

Carson Merrill

Numerade Educator

Problem 12

Sketch the graph of the equation by point plotting.
$y=\sqrt{x+2}$

Carson Merrill

Numerade Educator

Problem 13

Sketch the graph of the equation by point plotting.
$y=\frac{3}{x}$

Carson Merrill

Numerade Educator

Problem 14

Sketch the graph of the equation by point plotting.
$y=\frac{1}{x+2}$

Carson Merrill

Numerade Educator

Problem 15

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

Yujie Wang

College of San Mateo

Problem 16

Describe the viewing window that yields the figure.
$y=|x|+|x-16|$
FIGURE CANT COPY

Yujie Wang

College of San Mateo

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.
$y=\sqrt{5-x}$
(a) $(2, y)$
(b) $(x, 3)$

Linh Vu

Numerade Educator

Problem 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.
$y=x^5-5 x$
(a) $(-0.5, y)$
(b) $(x,-4)$

Carson Merrill

Numerade Educator

Problem 19

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

Diego Alcala

Numerade Educator

Problem 20

Find any intercepts.
$y=4 x^2+3$

Carson Merrill

Numerade Educator

Problem 21

Find any intercepts.
$y=x^2+x-2$

James Kiss

Numerade Educator

Problem 22

Find any intercepts.
$y^2=x^3-4 x$

Carson Merrill

Numerade Educator

Problem 23

Find any intercepts.
$y=x \sqrt{16-x^2}$

Carson Merrill

Numerade Educator

Problem 24

Find any intercepts.
$y=(x-1) \sqrt{x^2+1}$

Carson Merrill

Numerade Educator

Problem 25

Find any intercepts.
$y=\frac{2-\sqrt{x}}{5 x}$

Carson Merrill

Numerade Educator

Problem 26

Find any intercepts.
$y=\frac{x^2+3 x}{(3 x+1)^2}$

Carson Merrill

Numerade Educator

Problem 27

Find any intercepts.
$x^2 y-x^2+4 y=0$

Carson Merrill

Numerade Educator

Problem 28

Find any intercepts.
$y=2 x-\sqrt{x^2+1}$

Carson Merrill

Numerade Educator

Problem 29

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

JC

James Choi

Numerade Educator

Problem 30

Test for symmetry with respect to each axis and to the origin.
$y=x^2-x$

Carson Merrill

Numerade Educator

Problem 31

Test for symmetry with respect to each axis and to the origin.
$y^2=x^3-8 x$

Carson Merrill

Numerade Educator

Problem 32

Test for symmetry with respect to each axis and to the origin.
$y=x^3+x$

Carson Merrill

Numerade Educator

Problem 33

Test for symmetry with respect to each axis and to the origin.
$x y=4$

Carson Merrill

Numerade Educator

Problem 34

Test for symmetry with respect to each axis and to the origin.
$x y^2=-10$

Carson Merrill

Numerade Educator

Problem 35

Test for symmetry with respect to each axis and to the origin.
$y=4-\sqrt{x+3}$

Carson Merrill

Numerade Educator

Problem 36

Test for symmetry with respect to each axis and to the origin.
$x y-\sqrt{4-x^2}=0$

Carson Merrill

Numerade Educator

Problem 37

Test for symmetry with respect to each axis and to the origin.
$y=\frac{x}{x^2+1}$

Carson Merrill

Numerade Educator

Problem 38

Test for symmetry with respect to each axis and to the origin.
$y=\frac{x^2}{x^2+1}$

Carson Merrill

Numerade Educator

Problem 39

Test for symmetry with respect to each axis and to the origin.
$y=\left|x^3+x\right|$

Carson Merrill

Numerade Educator

Problem 40

Test for symmetry with respect to each axis and to the origin.
$|y|-x=3$

Carson Merrill

Numerade Educator

Problem 41

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

Linh Vu

Numerade Educator

Problem 42

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=-\frac{3}{2} x+6$

Linh Vu

Numerade Educator

Problem 43

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=\frac{1}{2} x-4$

Yujie Wang

College of San Mateo

Problem 44

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=\frac{2}{3} x+1$

Yujie Wang

College of San Mateo

Problem 45

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=9-x^2$

Carson Merrill

Numerade Educator

Problem 46

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=x^2+3$

Yujie Wang

College of San Mateo

Problem 47

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=(x+3)^2$

Yujie Wang

College of San Mateo

Problem 48

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=2 x^2+x$

Yujie Wang

College of San Mateo

Problem 49

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=x^3+2$

Yujie Wang

College of San Mateo

Problem 50

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=x^3-4 x$

Yujie Wang

College of San Mateo

Problem 51

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=x \sqrt{x+5}$

Carson Merrill

Numerade Educator

Problem 52

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=\sqrt{25-x^2}$

Carson Merrill

Numerade Educator

Problem 53

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$x=y^3$

Yujie Wang

College of San Mateo

Problem 54

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$x=y^2-4$

Yujie Wang

College of San Mateo

Problem 55

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=\frac{8}{x}$

Carson Merrill

Numerade Educator

Problem 56

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=\frac{10}{x^2+1}$

Carson Merrill

Numerade Educator

Problem 57

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=6-|x|$

Carson Merrill

Numerade Educator

Problem 58

Sketch the graph of the equation. Identify any intercepts and test for symmetry.
$y=|6-x|$

Carson Merrill

Numerade Educator

Problem 59

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

Yujie Wang

College of San Mateo

Problem 60

Use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$x^2+4 y^2=4$

Yujie Wang

College of San Mateo

Problem 61

Use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$x+3 y^2=6$

Yujie Wang

College of San Mateo

Problem 62

Use a graphing utility to graph the equation. Identify any intercepts and test for symmetry.
$3 x-4 y^2=8$

Yujie Wang

College of San Mateo

Problem 63

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

Georgiann Andersen

Georgiann Andersen

Numerade Educator

Problem 64

Find the points of intersection of the graphs of the equations.
$$
\begin{aligned}
& 3 x-2 y=-4 \\
& 4 x+2 y=-10
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 65

Find the points of intersection of the graphs of the equations.
$\begin{aligned} x^2+y & =6 \\ x+y & =4\end{aligned}$

Carson Merrill

Numerade Educator

Problem 66

Find the points of intersection of the graphs of the equations.
$\begin{aligned} x & =3-y^2 \\ y & =x-1\end{aligned}$

Carson Merrill

Numerade Educator

Problem 67

Find the points of intersection of the graphs of the equations.
$$
\begin{aligned}
x^2+y^2 & =5 \\
x-y & =1
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 68

Find the points of intersection of the graphs of the equations.
$$
\begin{aligned}
x^2+y^2 & =25 \\
-3 x+y & =15
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 69

Find the points of intersection of the graphs of the equations.
$$
\begin{aligned}
& y=x^3 \\
& y=x
\end{aligned}
$$

Yujie Wang

College of San Mateo

Problem 70

Find the points of intersection of the graphs of the equations.
$$
\begin{aligned}
& y=x^3-4 x \\
& y=-(x+2)
\end{aligned}
$$

Yujie Wang

College of San Mateo

Problem 71

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

Carson Merrill

Numerade Educator

Problem 72

Use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{aligned}
& y=x^4-2 x^2+1 \\
& y=1-x^2
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 73

Use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{aligned}
& y=\sqrt{x+6} \\
& y=\sqrt{-x^2-4 x}
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 74

Use a graphing utility to find the points of intersection of the graphs. Check your results analytically.
$$
\begin{aligned}
& y=-|2 x-3|+6 \\
& y=6-x
\end{aligned}
$$

Carson Merrill

Numerade Educator

Problem 75

Modeling Data The table shows the Consumer Price Index (CPI) for selected years.
$$
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline \text { Year } & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 \\
\hline \text { CPI } & 53.8 & 82.4 & 107.6 & 130.7 & 152.4 & 172.2 & 195.3 \\
\hline
\end{array}
$$
(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.

Carson Merrill

Numerade Educator

Problem 76

Modeling Data The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years.
$$
\begin{array}{|l|c|c|c|c|c|c|}
\hline \text { Year } & 1990 & 1993 & 1996 & 1999 & 2002 & 2005 \\
\hline \text { Number } & 5 & 16 & 44 & 86 & 141 & 208 \\
\hline
\end{array}
$$
(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

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

Yujie Wang

College of San Mateo

Problem 78

The resistance $y$ in ohms of 1000 feet of solid copper wire at $77^{\circ} \mathrm{F}$ can be approximated by the model
$$
y=\frac{10,770}{x^2}-0.37, \quad 5 \leq x \leq 100
$$
where $x$ is the diameter of the wire in mils ( 0.001 in.). Use a graphing utility to graph the model. If the diameter of the wire is doubled, the resistance is changed by about what factor?

Diego Alcala

Numerade Educator

Problem 79

In Exercises 79 and 80, write an equation whose graph has the given 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

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

Yujie Wang

College of San Mateo

Problem 81

Write an equation whose graph has the given property. (There may be more than one correct answer.)
(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.

Charles Carter

Numerade Educator

Problem 82

Match the equation or equations with the given characteristic.
(i) $y=3 x^3-3 x$
(ii) $y=(x+3)^2$
(iii) $y=3 x-3$
(iv) $y=\sqrt[3]{x}$
(v) $y=3 x^2+3$
(vi) $y=\sqrt{x+3}$
(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

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

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.

Carson Merrill

Numerade Educator

Problem 85

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.

Carson Merrill

Numerade Educator

Problem 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.

Carson Merrill

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

Linh Vu

Numerade Educator

Problem 88

Find an equation of the graph that consists of all points $(x, y)$ having the given distance from the origin.
The distance from the origin is $K(K \neq 1)$ times the distance from $(2,0)$.

Yujie Wang

College of San Mateo