Calculus: Early Transcendentals

Michael Sullivan, Kathleen Miranda

Chapter 0

Functions and Precalculus - all with Video Answers

Educators

Section 1

Functions and Graphs

Problem 1

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Functions are the same as equations.
(b) True or False: The domain of every function is a subset of $\mathbb{R}$.
(c) True or False: The function that for each $x$ has output $f(x)=1$ is a one-to-one function.
(d) True or False: Every global maximum of a function is also a local maximum.
(e) True or False: Every local minimum of a function is also a global minimum.
(f) True or False: The graph of a function can never cross one of its asymptotes.
(g) True or False: Average rates of change can be thought of as slopes.
(h) True or False: A function can have different average rates of change on different intervals.

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Problem 2

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A function that is defined with a formula.
(b) A function that is not defined with a formula.
(c) A formula that does not define a function.

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Problem 3

State the mathematical definition of a function, and describe its meaning in your own words. Support your answer with an example of something that is a function and an example of something that is not.

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Problem 4

Suppose $P$ is the set of people alive today and $C$ is the set of possible eye colors. Let $f: P \rightarrow C$ be the rule that assigns to each person his or her eye color. Is $f$ a function? Why or why not?

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Problem 5

Use set notation to define the domain of a function. Then use the same notation to express the domain of the function $f(x)=\sqrt{x}$.

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Problem 6

Use set notation to define the range of a function. Then use the same notation to express the range of the function $f(x)=x^{2}$.

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Problem 7

Determine whether the points (a) $(3,2)$, (b) $(1,1)$, and $(\mathrm{c})$ $(-5,2)$ lie on the graph of $f(x)=\sqrt{x+1}$, without referring to a picture of the graph of $f$.

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Problem 8

Describe the graph of the function $f(x)=3 x+2$ as a set of ordered pairs.

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Problem 9

Consider the function $f(x)=x^{2}+1$
(a) Explain why $y=5$ is in the range of $f$.
(b) Explain why $y=0$ is not in the range of $f$.
(c) Argue that the range of $f(x)=x^{2}+1$ is $[1, \infty)$.

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Problem 10

Determine whether or not each diagram that follows represents a function. If it does, find its domain and range, and determine whether it is one-to-one. If it does not, explain what goes wrong.

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Problem 11

Construct a rule $f:\{2,4,6,8,10\} \rightarrow\{1,2,3,4\}$ that is a function. Express this function three ways: as a list, as a table, and as a diagram. Is your function one-to-one? What is its range?

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Problem 12

Construct a rule $f:\{2,4,6,8,10\} \rightarrow\{1,2,3,4\}$ that is not a function. Justify your answer.

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Problem 13

If the graph of a rule $y=f(x)$ passes through $(-2,1)$ and $(2,1)$, could that rule be a function? Why or why not?

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Problem 14

A constant function is a function $f: A \rightarrow B$ with the property that there is some $b \in B$ for which $f(x)=b$ for all $x \in A$. (The output of the function is constantly the same.) Describe
(a) a constant function $f: \mathbb{R} \rightarrow \mathbb{R}$
(b) a constant function $g: \mathbb{R} \rightarrow[-5,-2]$
(c) a constant function $h: \mathbb{R} \rightarrow \mathbb{R}^{2}$

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Problem 15

The identity function for a set $A$ is the function $f: A \rightarrow A$ defined by $f(x)=x$ (so called because the output is identical to the input). For which of the following domains and ranges is there a well-defined identity function? Why or why not?
(a) $f: \mathbb{R} \rightarrow \mathbb{R}$
(b) $g: \mathbb{R}^{2} \rightarrow \mathbb{R}$
(c) $h: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$

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Problem 16

Let $P$ be the set of all people living in the United States. Give examples of each of the following functions and state their ranges:
(a) the identity function $f: P \rightarrow P$
(b) two different constant functions $g: P \rightarrow P$
(c) a non-constant, non-identity function $g: P \rightarrow P$
(d) a constant function $h: P \rightarrow \mathbb{R}$
(e) a non-constant function $h: P \rightarrow \mathbb{R}$

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Problem 17

Explain in your own words why the vertical line test determines whether a graph is a function.

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Problem 18

Explain in your own words why the horizontal line test determines whether a function is one-to-one.

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Problem 19

Show that $f(x)=x^{2}+1$ is not one-to-one, using values of $f$ (not the horizontal line test).

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Problem 20

Define what it means for a function $f$ with domain $\mathbb{R}$ to have (a) a global minimum at $x=c$ and (b) a local minimum at $x=c$.

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Problem 21

Define what it means for a function $f$ with domain $\mathbb{R}$ to be (a) negative on an interval $I$ and (b) decreasing on an interval $L$.

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Problem 22

Make a labeled graph that illustrates why it makes sense that a function is increasing on an interval $I$ if, for all $b>a$ in $I$, we have $f(b)>f(a)$. Include labels for $a, b, f(a)$, and $f(b)$, and for the interval $I$.

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Problem 23

How is the formula for average rate of change related to the formula for computing slope?

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Problem 24

Illustrate on a graph of $f(x)=1-x^{2}$ that the average rate of change of $f$ on $[-1,3]$ is $-2$.

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Problem 25

For each local maximum $x=c$ in the following graph, approximate the largest possible $\delta>0$ so that $f(c) \geq f(x)$ for all $x \in(c-\delta, c+\delta)$. Similarly, for the one local minimum $x=b$, find the largest $\delta$ so that $f(b) \leq f(x)$ for all $x \in(b-\delta, b+\delta) .$

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Problem 26

Use the definition of a local maximum to explicitly argue why the function graphed in Exercise 25 does not have a local maximum at $x=0$.

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Problem 27

Find the domain and range of each function. Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$$
f(x)=\sqrt{x-1}
$$

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Problem 28

Find the domain and range of each function. Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$$
f(x)=\sqrt{x}-1
$$

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Problem 29

Find the domain and range of each function. Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$$
f(x)=\frac{1}{x+2}
$$

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Problem 30

Find the domain and range of each function. Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$$
f(x)=\frac{1}{\sqrt{5-x}}
$$

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Problem 31

Find the domain and range of each function. Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$$
f(x)=\frac{1}{x^{2}+1}
$$

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Problem 32

Find the domain and range of each function. Use a graphing utility or plot points to sketch a graph of the function, and illustrate the domain and range on the graph.
$$
f(x)=\frac{1}{x^{2}-1}
$$

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Problem 33

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

Harsh Gadhiya

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Problem 34

Find the domain of each function.
$$
f(x)=\frac{3 x+1}{2 x-1}
$$

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Problem 35

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

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Problem 36

Find the domain of each function.
$$
f(x)=\frac{1}{\sqrt{x^{2}-4}}
$$

Ahmad Reda

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Problem 37

Find the domain of each function.
$$
f(x)=\frac{1}{\sqrt{(x-1)(x+3)}}
$$

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Problem 38

Find the domain of each function.
$$
f(x)=\sqrt{\frac{1}{x^{2}}-4}
$$

Ahmad Reda

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Problem 39

Find the domain of each function.
$$
f(x)=\frac{\sqrt{x^{2}-1}}{x^{2}-9}
$$

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Problem 40

Find the domain of each function.
$$
f(x)=\frac{x^{3 / 4}}{3 x-5}
$$

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Problem 41

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

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Problem 42

Find the domain of each function.
$$
f(x)=\frac{\sqrt{x^{2}-1}}{\sqrt{x^{2}-9}}
$$

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Problem 43

Evaluate each function at the values indicated. Simplify your answers if possible.
If $f(x)=x^{2}+1$, find
(a) $f(-4)$
(b) $f\left(a^{3}\right)$
(c) $f(f(x))$

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Problem 44

Evaluate each function at the values indicated. Simplify your answers if possible.
If $k(x)=\frac{x^{2}}{x+1}$, find
(a) $k(5)$
(b) $k(x+h)$
(c) $k(k(x))$

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Problem 45

Evaluate each function at the values indicated. Simplify your answers if possible.
If $l(a, b, c)=\sqrt{a^{2}+b^{2}+c^{2}}$, find:
(a) $l(5,3,2)$
(b) $l(3,0,4)$
(c) $l(x, y, z)$

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Problem 46

Evaluate each function at the values indicated. Simplify your answers if possible.
If $g(v)=\left(v-1, v, v^{2}\right)$, find
(a) $g(0)$
(b) $g(1)$
(c) $g(x+1)$

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Problem 47

Evaluate each function at the values indicated. Simplify your answers if possible.
If $F(u, v, w)=(3 u+v, u-w, v+2 w)$, find
(a) $F(2,3,5)$
(b) $F(5,2,3)$
(c) $F(a, b, 0)$

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Problem 48

For each piecewise-defined function,
(a) calculate $f(-1), f(0), f(1)$, and $f(2)$, and (b) sketch a graph of $f$.
$$
f(x)=\left\{\begin{aligned}
x^{2}+3, & \text { if } x<0 \\
3-x, & \text { if } x \geq 0
\end{aligned}\right.
$$

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Problem 49

For each piecewise-defined function,
(a) calculate $f(-1), f(0), f(1)$, and $f(2)$, and (b) sketch a graph of $f$.
$$
f(x)=\left\{\begin{aligned}
3 x+1, & \text { if } x \leq 0 \\
4, & \text { if } 0<x \leq 1 \\
x^{3}, & \text { if } x>1
\end{aligned}\right.
$$

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Problem 50

For each piecewise-defined function,
(a) calculate $f(-1), f(0), f(1)$, and $f(2)$, and (b) sketch a graph of $f$.
$$
f(x)=\left\{\begin{aligned}
4 x-1, & \text { if } x<0 \\
2, & \text { if } x=0 \\
-3 x+5, & \text { if } x>0
\end{aligned}\right.
$$

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Problem 51

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=x^{2}-0.1
$$

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Problem 52

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=\left(x^{2}-5\right)^{7}
$$

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Problem 53

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=x^{5}-3 x^{4}-7 x
$$

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Problem 54

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=x^{3}-11 x^{2}+10 x
$$

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Problem 55

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=x^{2}-17 x-18
$$

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Problem 56

Use a graphing utility to sketch a graph of each function. Use trial and error to find a graphing window so that your graph represents the local and global behavior of the function. Include the $x$ and $y$ ranges of your window in your answer.
$$
f(x)=\frac{2 x^{2}-2}{x^{2}-3 x-5}
$$

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Problem 57

Describe the key properties of each graph in Exercises $57-62$, including the following:
- domain and range;
- locations of roots, intercepts, local and global maxima and minima, and inflection points;
- intervals on which the function is positive or negative, increasing or decreasing, and concave up or down;
- any horizontal or vertical asymptotes.

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Problem 58

Describe the key properties of each graph in Exercises $57-62$, including the following:
- domain and range;
- locations of roots, intercepts, local and global maxima and minima, and inflection points;
- intervals on which the function is positive or negative, increasing or decreasing, and concave up or down;
- any horizontal or vertical asymptotes.

Carson Merrill

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Problem 59

Describe the key properties of each graph in Exercises $57-62$, including the following:
- domain and range;
- locations of roots, intercepts, local and global maxima and minima, and inflection points;
- intervals on which the function is positive or negative, increasing or decreasing, and concave up or down;
- any horizontal or vertical asymptotes.

Carson Merrill

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Problem 60

Describe the key properties of each graph in Exercises $57-62$, including the following:
- domain and range;
- locations of roots, intercepts, local and global maxima and minima, and inflection points;
- intervals on which the function is positive or negative, increasing or decreasing, and concave up or down;
- any horizontal or vertical asymptotes.

Carson Merrill

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Problem 61

Describe the key properties of each graph in Exercises $57-62$, including the following:
- domain and range;
- locations of roots, intercepts, local and global maxima and minima, and inflection points;
- intervals on which the function is positive or negative, increasing or decreasing, and concave up or down;
- any horizontal or vertical asymptotes.

Carson Merrill

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Problem 62

Describe the key properties of each graph in Exercises $57-62$, including the following:
- domain and range;
- locations of roots, intercepts, local and global maxima and minima, and inflection points;
- intervals on which the function is positive or negative, increasing or decreasing, and concave up or down;
- any horizontal or vertical asymptotes.

Carson Merrill

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Problem 63

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Domain $\mathbb{R}$, concave up everywhere, and decreasing everywhere.

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Problem 64

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Domain $\mathbb{R}$, concave down everywhere, and decreasing everywhere.

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Problem 65

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Domain $\mathbb{R}$, concave up everywhere, increasing everywhere, and negative everywhere.

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Problem 66

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Domain $\mathbb{R}$, concave down everywhere, increasing everywhere, and negative everywhere.

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Problem 67

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Always increasing, with two horizontal asymptotes, one at $y=-2$ and one at $y=2$.

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Problem 68

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Domain $(0, \infty)$, always negative, and always increasing.

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Problem 69

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Four roots but no $y$ -intercept.

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Problem 70

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Concave down on $(-\infty, 2)$, concave up on $(2, \infty)$, and always increasing.

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Problem 71

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Concave down on $(-\infty, 0)$ and concave up on $(0, \infty)$ but without an inflection point at $x=0$.

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Problem 72

Sketch the graph of functions $f$ that satisfy the lists of conditions given , if possible.
Average rate of change of 3 on $[0,2]$, average rate of change of $-1$ on $[0,1]$, and average rate of change of 0 on $[-2,2]$.

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Problem 73

Find the average rate of change of the function $f$ on the interval $[a, b]$.
$$
f(x)=-0.5+4.2 x, \quad[a, b]=[1,3.5]
$$

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Problem 74

Find the average rate of change of the function $f$ on the interval $[a, b]$.
$$
f(x)=3,[a, b]=[-100,100]
$$

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Problem 75

Find the average rate of change of the function $f$ on the interval $[a, b]$.
$$
f(x)=\sqrt{x+1},[a, b]=[1,9]
$$

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Problem 76

Find the average rate of change of the function $f$ on the interval $[a, b]$.
$$
f(x)=\frac{1-x}{1+x^{3}}, \quad[a, b]=[0,0.5]
$$

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Problem 77

Find the average rate of change of the function $f$ on the interval $[a, b]$.
$$
f(x)=\frac{1}{x}, \quad[a, b]=[0.9,1.1]
$$

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Problem 78

Find the average rate of change of the function $f$ on the interval $[a, b]$.
$$
f(x)=(x-2)^{2}+\frac{3}{x}, \quad[a, b]=[-2,2]
$$

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Problem 79

Sketch and label the graph of a function that describes the given situation.
An island warthog population initially grows quickly, but as space and food become sparse on the island, the population growth slows down. Eventually the population of the island levels off at 512 warthogs.

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Problem 80

Sketch and label the graph of a function that describes the given situation.
Susie is late for calculus class and leaves her dorm in a panic. She hurries towards the math building, but about halfway there, she realizes she has left her notebook in her room. She sprints back to her dorm and gets her notebook. Coming out of the dorm, she sprains her ankle, so the best she can do is limp as fast as she can to her classroom.

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Problem 81

Sketch and label the graph of a function that describes the given situation.
Suppose that after you drink a cup of coffee the amount of caffeine in your body rises sharply and then decreases by half every hour. You have one cup of coffee in the morning and then no more.

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Problem 82

Sketch and label the graph of a function that describes the given situation.
On a dare, you go skydiving. Gravity causes you to fall faster and faster as you plummet towards the ground. When you open your parachute, your speed is drastically reduced. After opening your parachute you approach the ground at a constant speed.
(a) Graph your distance from the ground as a function of time.
(b) Graph your velocity as a function of time.

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Problem 83

For each situation described, identify any independent and dependent variables, and express their relationship as an equation in multivariable function notation (see Example 9):
(a) $H$ is the length of the hypotenuse of the right triangle with legs of length $a$ and $b$.
(b) $V$ is the volume of a rectangular prism $\left({ }^{\prime \prime}\right.$ box $\left.^{\prime \prime}\right)$ with dimensions $x, y$, and $z$.

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Problem 84

If your rain-catching bucket starts empty and collects 3 inches of rain during a 6 -hour rainstorm, what is the average rate of change of the level of rainwater in the bucket over the 6 hours that it rained? Did the rain necessarily collect in the bucket at a constant rate?

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Problem 85

A disgruntled pet store owner abandoned an unknown number of groundhogs on a small island in 1996 . Since then it has been determined that the average rate of change of the groundhog population was 4 groundhogs per year and that the groundhog population was a linear function of time. When the abandoned groundhogs were discovered in 2001 , there were 376 groundhogs on the island. How many groundhogs did the disgruntled pet store owner originally leave on the island?

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Problem 86

The number $N$ of operating drive-in movie theatres in the state of Virginia in various years $y$ is given in the table below.
$$
\begin{array}{|r|r|r|r|r|r|r|}
\hline y & 1958 & 1967 & 1972 & 1977 & 1982 & 1999 \\
\hline N & 143 & 90 & 102 & 87 & 56 & 9 \\
\hline
\end{array}
$$
(a) Find the average rate of change in the number of drive-in movie theatres in Virginia over each time interval between table entries.
(b) Describe the units and real-world significance of these average rates of change.
(c) Over which time period was the average rate of change the most drastic? On average, assuming that no new theatres were built, how many drive-ins closed per year during that period?

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Problem 87

In your first job after graduating from college you make $\$ 36,000$ a year before taxes. After four years you get a raise of $\$ 2,500$. Two years after that you change jobs and go to work for a company that pays you $\$ 49,000$ a year.
(a) Construct a piecewise-defined function that describes your pretax income in the year that is $t$ years after you graduate from college.
(b) Write down a function that describes the total amount of money you will have earned $t$ years after graduating from college.
(c) How many years after graduating from college will you have earned a total of one million pre-tax dollars?

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Problem 88

The following table shows the year 2000 Federal Tax Rate Schedule for single filers:
$$
\begin{array}{|r|r|r|r|r|}
\hline \text { Over } & {\text { Not over Amount }} & \text { Plus \% } & \text { Of amt. over } \\
\hline \$ 0 & \$ 26,250 & \$ 0 & 15 \% & \$ 0 \\
\hline \$ 26,250 & \$ 63,550 & \$ 3,937 & 28 \% & \$ 26,250 \\
\hline \$ 63,550 & \$ 132,600 & \$ 14,381 & 31 \% & \$ 63,550 \\
\hline \$ 132,600 & \$ 288,350 & \$ 35,787 & 36 \% & \$ 132,600 \\
\hline \$ 288,350 & - & \$ 91,857 & 39.6 \% & \$ 288,350 \\
\hline
\end{array}
$$

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Problem 89

Use Definition $0.2$ to prove that the range of the function $f(x)=3 x-1$ is $\mathbb{R}$.

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Problem 90

Use Definition $0.1$ to prove that a graph represents a function if and only if it passes the vertical line test.

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Problem 91

Use Definition $0.5$ to prove that a function is one-to-one if and only if its graph passes the horizontal line test.

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Problem 92

Use the contrapositive form of Definition $0.5$ to prove that the function $f(x)=3 x+1$ is one-to-one.

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Problem 93

Use the definition of decreasing to prove that the function $f(x)=1-3 x$ is decreasing on $(-\infty, \infty)$.

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Problem 94

Use the definition of increasing to prove that the function $f(x)=\frac{1}{3-x}$ is increasing on $(-\infty, 3) .$

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Problem 95

Prove that the average rate of change of the linear function $f(x)=-2 x+4$ on any interval $I$ is always equal to $-2$.

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Problem 96

Show that the average rate of change of every linear function $f(x)=m x+b$ is constant, that is, the same over any choice of interval.

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