Calculus, Early Transcendentals

Michael Sullivan, Kathleen Miranda

Chapter 0

Preparing for Calculus - all with Video Answers

Educators

Section 1

Functions and Their Graphs

Problem 1

If $f$ is a function defined by $y=f(x),$ then $x$ is called the variable and $y$ is the $\longrightarrow$ variable.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 2

True or False The independent variable is sometimes referred to as the argument of the function.

Alexandra Ali

Numerade Educator

Problem 3

True or False If no domain is specified for a function $f,$ then the domain of $f$ is taken to be the set of all real numbers.

AG

Ankit Gupta

Numerade Educator

Problem 4

True or False The domain of the function $f(x)=\frac{3\left(x^{2}-1\right)}{x-1}$ is $\{x \mid x \neq\pm 1\}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 5

True or False A function can have more than one $y$ -intercept.

Aditya Sood

Numerade Educator

Problem 6

A set of points in the $x y$ -plane is the graph of a function if and only if every _____ line intersects the graph in at most one point.

Darian Kaulahao

Darian Kaulahao

Numerade Educator

Problem 7

If the point (5,-3) is on the graph of $f,$ then $f(_____)=$______.

AG

Ankit Gupta

Numerade Educator

Problem 8

Find $a$ so that the point (-1,2) is on the graph of $f(x)=a x^{2}+4$.

Darian Kaulahao

Darian Kaulahao

Numerade Educator

Problem 9

Multiple Choice A function $f$ is [(a) increasing, (b) decreasing,
(c) nonincreasing, (d) nondecreasing, (e) constant] on an interval $I$ if, for any choice of $x_{1}$ and $x_{2}$ in $I,$ with $x_{1}<x_{2},$ then $f\left(x_{1}\right)<f\left(x_{2}\right)$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 10

Multiple Choice A function $f$ is $[(\mathbf{a})$ even, $(\mathbf{b})$ odd $,(\mathbf{c})$ neither even nor odd $]$ if for every number $x$ in its domain, the number $-x$ is also in the domain and $f(-x)=f(x) .$ A function $f$ is $[$ (a) even, (b) odd, (c) neither even nor odd] if for every number $x$ in its domain, the number $-x$ is also in the domain and $f(-x)=-f(x)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 11

True or False Even functions have graphs that are symmetric with respect to the origin.

Cheyenne Whinham

Cheyenne Whinham

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Problem 12

The average rate of change of $f(x)=2 x^{3}-3$ from 0 to 2 is _____.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 13

For each function find:
(a) $f(0)$
(b) $f(-x)$
$(c)-f(x)$
(d) $f(x+1)$
(e) $f(x+h)$
$$
f(x)=3 x^{2}+2 x-4
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

For each function find:
(a) $f(0)$
(b) $f(-x)$
$(c)-f(x)$
(d) $f(x+1)$
(e) $f(x+h)$
$$
f(x)=\frac{x}{x^{2}+1}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 15

For each function find:
(a) $f(0)$
(b) $f(-x)$
$(c)-f(x)$
(d) $f(x+1)$
(e) $f(x+h)$
$$
f(x)=|x|+4
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 16

For each function find:
(a) $f(0)$
(b) $f(-x)$
$(c)-f(x)$
(d) $f(x+1)$
(e) $f(x+h)$
$$
\text { 16. } f(x)=\sqrt{3-x}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 17

Find the domain of each function.
$$
f(x)=x^{3}-1
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 18

Find the domain of each function.
$$
f(x)=\frac{x}{x^{2}+1}
$$

AG

Ankit Gupta

Numerade Educator

Problem 19

Find the domain of each function.
$$
v(t)=\sqrt{t^{2}-9}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 20

Find the domain of each function.
$$
g(x)=\sqrt{\frac{2}{x-1}}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 21

Find the domain of each function.
$$
h(x)=\frac{x+2}{x^{3}-4 x}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 22

Find the domain of each function.
$$
s(t)=\frac{\sqrt{t+1}}{t-5}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 23

Find the difference quotient of $f$. That is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$.
$$
f(x)=-3 x+1
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 24

Find the difference quotient of $f$. That is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$.
$$
f(x)=\frac{1}{x+3}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 25

Find the difference quotient of $f$. That is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$.
$$
f(x)=\sqrt{x+7}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 26

Find the difference quotient of $f$. That is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$.
$$
f(x)=\frac{2}{\sqrt{x+7}}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 27

Find the difference quotient of $f$. That is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$.
$$
f(x)=x^{2}+2 x
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 28

Find the difference quotient of $f$. That is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0$.
$$
f(x)=(2 x+3)^{2}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 29

Determine whether the graph is that of a function by using the Vertical-line Test. If it is, use the graph to find
(a) the domain and range
(b) the intercepts, if any
(c) any symmetry with respect to the $x$ -axis, $y$ -axis, or the origin.

James Kiss

Numerade Educator

Problem 30

Determine whether the graph is that of a function by using the Vertical-line Test. If it is, use the graph to find
(a) the domain and range
(b) the intercepts, if any
(c) any symmetry with respect to the $x$ -axis, $y$ -axis, or the origin.

James Kiss

Numerade Educator

Problem 31

Determine whether the graph is that of a function by using the Vertical-line Test. If it is, use the graph to find
(a) the domain and range
(b) the intercepts, if any
(c) any symmetry with respect to the $x$ -axis, $y$ -axis, or the origin.

James Kiss

Numerade Educator

Problem 32

Determine whether the graph is that of a function by using the Vertical-line Test. If it is, use the graph to find
(a) the domain and range
(b) the intercepts, if any
(c) any symmetry with respect to the $x$ -axis, $y$ -axis, or the origin.

James Kiss

Numerade Educator

Problem 33

Piecewise-defined function:
(a) Find $f(-1), f(0), f(1)$ and $f(8)$
(b) Graph $f$.
(c) Find the domain, range, and intercepts of $f$.
$$
f(x)=\left\{\begin{array}{lll}
x+3 & \text { if } & -2 \leq x<1 \\
5 & \text { if } & x=1 \\
-x+2 & \text { if } & x>1
\end{array}\right.
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 34

Piecewise-defined function:
(a) Find $f(-1), f(0), f(1)$ and $f(8)$
(b) Graph $f$.
(c) Find the domain, range, and intercepts of $f$.
$$
f(x)=\left\{\begin{array}{ll}
2 x+5 & \text { if } \quad-3 \leq x<0 \\
-3 & \text { if } \quad x=0 \\
-5 x & \text { if } \quad x>0
\end{array}\right.
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 35

Piecewise-defined function:
(a) Find $f(-1), f(0), f(1)$ and $f(8)$
(b) Graph $f$.
(c) Find the domain, range, and intercepts of $f$.
$$
f(x)=\left\{\begin{array}{ll}
1+x & \text { if } \quad x<0 \\
x^{2} & \text { if } \quad x \geq 0
\end{array}\right.
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 36

Piecewise-defined function:
(a) Find $f(-1), f(0), f(1)$ and $f(8)$
(b) Graph $f$.
(c) Find the domain, range, and intercepts of $f$.
$$
f(x)=\left\{\begin{array}{ll}
\frac{1}{x} & \text { if } x<0 \\
\sqrt[3]{x} & \text { if } x \geq 0
\end{array}\right.
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 37

Use the graph of the function $f$ to answer the following questions.
Find $f(0)$ and $f(-6)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 38

Use the graph of the function $f$ to answer the following questions.
Is $f(3)$ positive or negative?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 39

Use the graph of the function $f$ to answer the following questions.
Is $f(-4)$ positive or negative?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 40

Use the graph of the function $f$ to answer the following questions.
For what values of $x$ is $f(x)=0 ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 41

Use the graph of the function $f$ to answer the following questions.
For what values of $x$ is $f(x)>0 ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 42

Use the graph of the function $f$ to answer the following questions.
What is the domain of $f ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 43

Use the graph of the function $f$ to answer the following questions.
What is the range of $f ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 44

Use the graph of the function $f$ to answer the following questions.
What are the $x$ -intercepts?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 45

Use the graph of the function $f$ to answer the following questions.
What is the $y$ -intercept?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 46

Use the graph of the function $f$ to answer the following questions.
How often does the line $y=\frac{1}{2}$ intersect the graph?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 47

Use the graph of the function $f$ to answer the following questions.
How often does the line $x=5$ intersect the graph?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 48

Use the graph of the function $f$ to answer the following questions.
For what values of $x$ does $f(x)=3 ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 49

Use the graph of the function $f$ to answer the following questions.
For what values of $x$ does $f(x)=-2 ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 50

Use the graph of the function $f$ to answer the following questions.
On what interval(s) is the function $f$ increasing?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 51

Use the graph of the function $f$ to answer the following questions.
On what interval(s) is the function $f$ decreasing?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 52

Use the graph of the function $f$ to answer the following questions.
On what interval(s) is the function $f$ constant?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 53

Use the graph of the function $f$ to answer the following questions.
On what interval(s) is the function $f$ nonincreasing?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 54

On what interval(s) is the function $f$ non increasing?
On what interval(s) is the function $f$ non decreasing?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 55

Answer the questions about the function
$$g(x)=\frac{x+2}{x-6}$$.
What is the domain of $g$ ?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 56

Answer the questions about the function
$$g(x)=\frac{x+2}{x-6}$$.
Is the point (3,14) on the graph of $g ?$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 57

Answer the questions about the function
$$g(x)=\frac{x+2}{x-6}$$.
If $x=4,$ what is $g(x) ?$ What is the corresponding point on the graph of $g$ ?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 58

Answer the questions about the function
$$g(x)=\frac{x+2}{x-6}$$.
If $g(x)=2$, what is $x$ ? What is(are) the corresponding point(s) on the graph of $g$ ?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 59

Answer the questions about the function
$$g(x)=\frac{x+2}{x-6}$$.
List the $x$ -intercepts, if any, of the graph of $g$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 60

Answer the questions about the function
$$g(x)=\frac{x+2}{x-6}$$.
What is the $y$ -intercept, if there is one, of the graph of $g$ ?

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 61

Determine whether the function is even, odd, or neither. Then determine whether its graph is symmetric with respect to the $y$ -axis, the origin, or neither.
$$
h(x)=\frac{x}{x^{2}-1}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 62

Determine whether the function is even, odd, or neither. Then determine whether its graph is symmetric with respect to the $y$ -axis, the origin, or neither.
$$
f(x)=\sqrt[3]{3 x^{2}+1}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 63

Determine whether the function is even, odd, or neither. Then determine whether its graph is symmetric with respect to the $y$ -axis, the origin, or neither.
$$
G(x)=\sqrt{x}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 64

Determine whether the function is even, odd, or neither. Then determine whether its graph is symmetric with respect to the $y$ -axis, the origin, or neither.
$$
F(x)=\frac{2 x}{|x|}
$$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 65

Find the average rate of change of $f(x)=-2 x^{2}+4$ :
(a) From 1 to 2
(b) From 1 to 3
(c) From 1 to 4
(d) From 1 to $x, x \neq 1$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 66

Find the average rate of change of $s(t)=20-0.8 t^{2}$ :
(a) From 1 to 4
(b) From 1 to 3
(c) From 1 to 2
(d) From 1 to $t, t \neq 1$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 67

The graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 68

The graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 69

The graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 70

The graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 71

The graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 72

The graph of a piecewise-defined function is given. Write a definition for each piecewise-defined function. Then state the domain and the range of the function.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 73

The monthly $\operatorname{cost} C,$ in dollars, of manufacturing $x$ road bikes is given by the function
$$
C(x)=0.004 x^{3}-0.6 x^{2}+250 x+100,500
$$
(a) Find the average rate of change of the $\operatorname{cost} C$ of manufacturing from 100 to 101 road bikes.
(b) Find the average rate of change of the $\operatorname{cost} C$ of manufacturing from 500 to 501 road bikes.
(c) Interpret the results from parts (a) and (b).

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 74

The weekly cost in dollars to produce $x$ tons of steel is given by the function
$$
C(x)=\frac{1}{10} x^{2}+5 x+1500
$$
(a) Find the average rate of change of the cost $C$ of producing from 500 to 501 tons of steel.
(b) Find the average rate of change of the cost $C$ of producing from 1000 to 1001 tons of steel.
(c) Interpret the results from parts (a) and (b).

Eric Mockensturm

Eric Mockensturm

Numerade Educator