# Precalculus with Limits

## Educators

Problem 1

Polynomial and rational functions are examples of______________functions.

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Problem 2

Exponential and logarithmic functions are examples of nonalgebraic functions, also called____________functions.

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Problem 3

You can use the___________ Property to solve simple exponential equations.

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Problem 4

The exponential function $f(x)=e^{x}$ is called the____________ _____________function, and the base $e$ is called the____________base.

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Problem 5

To find the amount $A$ in an account after $t$ years with principal $P$ and an annual interest rate $r$ compounded $n$ times per year, you can use the formula________________.

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Problem 6

To find the amount $A$ in an account after $t$ years with principal $P$ and an annual interest rate $r$ compounded continuously, you can use the formula______________.

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Problem 7

Evaluating an Exponential Function In Exercises
$7-12$ , evaluate the function at the indicated value of $x .$
Round your result to three decimal places.
Function Value
$f(x)=0.9^{x}$ $x=1.4$

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Problem 8

Evaluating an Exponential Function In Exercises
$7-12$ , evaluate the function at the indicated value of $x .$
Round your result to three decimal places.
$$Function$$
$$f(x)=0.9^{x}$$

$$Value$$
$$x=1.4$$

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Problem 9

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$
$$f(x)=5^{x}$$

$$Value$$
$$x=-\pi$$

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Problem 10

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$
$$f(x)=\left(\frac{2}{3}\right)^{5 x}$$

$$Value$$
$$x=\frac{3}{10}$$

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Problem 11

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$
$$g(x)=5000\left(2^{x}\right)$$

$$Value$$
$$x=-1.5$$

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Problem 12

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$
$$f(x)=200(1.2)^{12 x}$$

$$Value$$
$$x=24$$

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Problem 13

Matching an Exponential Function with Its
Graph In Exercises $13-16,$ match the exponential
function with its graph. [The graphs are labeled (a), (b),
(c), and (d).]

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Problem 13

Matching an Exponential Function with Its
Graph In Exercises $13-16,$ match the exponential
function with its graph. [The graphs are labeled (a), (b),
(c), and (d).]
$$f(x)=2^{x}$$

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Problem 14

Matching an Exponential Function with Its
Graph In Exercises $13-16,$ match the exponential
function with its graph. [The graphs are labeled (a), (b),
(c), and (d).]
$$f(x)=2^{x}+1$$

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Problem 15

Matching an Exponential Function with Its
Graph In Exercises $13-16,$ match the exponential
function with its graph. [The graphs are labeled (a), (b),
(c), and (d).]
$$f(x)=2^{-x}$$

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Problem 16

Matching an Exponential Function with Its
Graph In Exercises $13-16,$ match the exponential
function with its graph. [The graphs are labeled (a), (b),
(c), and (d).]
$$f(x)=2^{x-2}$$

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Problem 17

Graphing an Exponential Function In Exercises
$17-22,$ use a graphing utility to construct a table of values
for the function. Then sketch the graph of the function.
$$f(x)=\left(\frac{1}{2}\right)^{x}$$

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Problem 18

Graphing an Exponential Function In Exercises
$17-22,$ use a graphing utility to construct a table of values
for the function. Then sketch the graph of the function.
$$f(x)=\left(\frac{1}{2}\right)^{-x}$$

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Problem 19

Graphing an Exponential Function In Exercises
$17-22,$ use a graphing utility to construct a table of values
for the function. Then sketch the graph of the function.
$$f(x)=6^{-x}$$

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Problem 20

Graphing an Exponential Function In Exercises
$17-22,$ use a graphing utility to construct a table of values
for the function. Then sketch the graph of the function.
$$f(x)=6^{x}$$

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Problem 21

Graphing an Exponential Function In Exercises
$17-22,$ use a graphing utility to construct a table of values
for the function. Then sketch the graph of the function.
$$f(x)=2^{x-1}$$

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Problem 22

Graphing an Exponential Function In Exercises
$17-22,$ use a graphing utility to construct a table of values
for the function. Then sketch the graph of the function.
$$f(x)=4^{x-3}+3$$

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Problem 23

Using the One-to-One Property In Exercises $23-26$
use the One-to-One Property to solve the equation for $x .$
$$3^{x+1}=27$$

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Problem 24

Using the One-to-One Property In Exercises $23-26$
use the One-to-One Property to solve the equation for $x .$
$$2^{x-3}=16$$

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Problem 25

Using the One-to-One Property In Exercises $23-26$
use the One-to-One Property to solve the equation for $x .$
$$\left(\frac{1}{2}\right)^{x}=32$$

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Problem 26

Using the One-to-One Property In Exercises $23-26$
use the One-to-One Property to solve the equation for $x .$
$$5^{x-2}=\frac{1}{125}$$

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Problem 27

Transforming the Graph of an Exponential
Function In Exercises $27-30,$ use the graph of $f$ to
describe the transformation that yields the graph of $g$ .
$$f(x)=3^{x}, \quad g(x)=3^{x}+1$$

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Problem 28

Transforming the Graph of an Exponential
Function In Exercises $27-30,$ use the graph of $f$ to
describe the transformation that yields the graph of $g$ .
$$f(x)=10^{x}, \quad g(x)=10^{-x+3}$$

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Problem 29

Transforming the Graph of an Exponential
Function In Exercises $27-30,$ use the graph of $f$ to
describe the transformation that yields the graph of $g$ .
$$f(x)=\left(\frac{7}{2}\right)^{x}, \quad g(x)=-\left(\frac{7}{2}\right)^{-x}$$

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Problem 30

Transforming the Graph of an Exponential
Function In Exercises $27-30,$ use the graph of $f$ to
describe the transformation that yields the graph of $g$ .
$$f(x)=0.3^{x}, \quad g(x)=-0.3^{x}+5$$

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Problem 31

Graphing an Exponential Function In Exercises
$31-34,$ use a graphing utility to graph the exponential
function.
$$y=2^{-x^{2}}$$

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Problem 32

Graphing an Exponential Function In Exercises
$31-34,$ use a graphing utility to graph the exponential
function.
$$y=3^{-|x|}$$

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Problem 33

Graphing an Exponential Function In Exercises
$31-34,$ use a graphing utility to graph the exponential
function.
$$y=3^{x-2}+1$$

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Problem 34

Graphing an Exponential Function In Exercises
$31-34,$ use a graphing utility to graph the exponential
function.
$$y=4^{x+1}-2$$

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Problem 35

Evaluating a Natural Exponential Function In
Exercises $35-38$ , evaluate the function at the indicated
value of $x .$ Round your result to three decimal places.
$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ {f(x)=e^{x}} & {x=3.2}\end{array}$

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Problem 36

Evaluating a Natural Exponential Function In
Exercises $35-38$ , evaluate the function at the indicated
value of $x .$ Round your result to three decimal places.

$$functions$$
$$f(x)=1.5 e^{x / 2}$$

$$values$$
$$x=240$$

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Problem 37

Evaluating a Natural Exponential Function In
Exercises $35-38$ , evaluate the function at the indicated
value of $x .$ Round your result to three decimal places.

$$functions$$
$$f(x)=5000 e^{0.06 x}$$

$$values$$
$$x=6$$

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Problem 38

Evaluating a Natural Exponential Function In
Exercises $35-38$ , evaluate the function at the indicated
value of $x .$ Round your result to three decimal places.

$$functions$$
$$f(x)=250 e^{0.05 x}$$

$$values$$
$$x=20$$

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Problem 39

Graphing a Natural Exponential Function In
Exercises $39-44$ , use a graphing utility to construct a
table of values for the function. Then sketch the graph of
the function.
$$f(x)=e^{x}$$

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Problem 40

Graphing a Natural Exponential Function In
Exercises $39-44$ , use a graphing utility to construct a
table of values for the function. Then sketch the graph of
the function.
$$f(x)=e^{-x}$$

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Problem 41

Graphing a Natural Exponential Function In
Exercises $39-44$ , use a graphing utility to construct a
table of values for the function. Then sketch the graph of
the function.
$$f(x)=3 e^{x+4}$$

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Problem 42

Graphing a Natural Exponential Function In
Exercises $39-44$ , use a graphing utility to construct a
table of values for the function. Then sketch the graph of
the function.
$$f(x)=2 e^{-0.5 x}$$

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Problem 43

Graphing a Natural Exponential Function In
Exercises $39-44$ , use a graphing utility to construct a
table of values for the function. Then sketch the graph of
the function.
$$f(x)=2 e^{x-2}+4$$

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Problem 44

Graphing a Natural Exponential Function In
Exercises $39-44$ , use a graphing utility to construct a
table of values for the function. Then sketch the graph of
the function.
$$f(x)=2+e^{x-5}$$

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Problem 45

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$y=1.08 e^{-5 x}$$

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Problem 46

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$y=1.08 e^{5 x}$$

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Problem 47

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$s(t)=2 e^{0.12 t}$$

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Problem 48

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$s(t)=3 e^{-0.2 t}$$

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Problem 49

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$g(x)=1+e^{-x}$$

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Problem 49

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$g(x)=1+e^{-x}$$

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Problem 50

Graphing a Natural Exponential Function In
Exercises $45-50$ , use a graphing utility to graph the
exponential function.
$$h(x)=e^{x-2}$$

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Problem 51

Using the One-to-One Property In Exercises
$51-54,$ use the One-to-One Property to solve the equation
for $x .$
$$e^{3 x+2}=e^{3}$$

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Problem 52

Using the One-to-One Property In Exercises
$51-54,$ use the One-to-One Property to solve the equation
for $x .$
$$e^{2 x-1}=e^{4}$$

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Problem 53

Using the One-to-One Property In Exercises
$51-54,$ use the One-to-One Property to solve the equation
for $x .$
$$e^{x^{2}-3}=e^{2 x}$$

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Problem 54

Using the One-to-One Property In Exercises
$51-54,$ use the One-to-One Property to solve the equation
for $x .$
$$e^{x^{2}+6}=e^{5 x}$$

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Problem 55

Compound Interest In Exercises $55-58$ , complete
the table to determine the balance $A$ for $P$ dollars invested
at rate $r$ for $t$ years and compounded $n$ times per year.
$$P=\ 1500, r=2 \%, t=10$$

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Problem 56

Compound Interest In Exercises $55-58$ , complete
the table to determine the balance $A$ for $P$ dollars invested
at rate $r$ for $t$ years and compounded $n$ times per year.
$$P=\ 2500, r=3.5 \%, t=10$$

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Problem 57

Compound Interest In Exercises $55-58$ , complete
the table to determine the balance $A$ for $P$ dollars invested
at rate $r$ for $t$ years and compounded $n$ times per year.
$$P=\ 2500, r=4 \%, t=20$$

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Problem 58

Compound Interest In Exercises $55-58$ , complete
the table to determine the balance $A$ for $P$ dollars invested
at rate $r$ for $t$ years and compounded $n$ times per year.
$$P=\ 1000, r=6 \%, t=40$$

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Problem 59

Compound Interest In Exercises $59-62,$ complete
the table to determine the balance $A$ for $\$ 12,000$invested at rate$r$for$t$years, compounded continuously. $$r=4 \%$$ Check back soon! Problem 60 Compound Interest In Exercises$59-62,$complete the table to determine the balance$A$for$\$12,000$ invested
at rate $r$ for $t$ years, compounded continuously.
$$r=6 \%$$

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Problem 61

Compound Interest In Exercises $59-62,$ complete
the table to determine the balance $A$ for $\$ 12,000$invested at rate$r$for$t$years, compounded continuously. $$r=6.5 \%$$ Check back soon! Problem 62 Compound Interest In Exercises$59-62,$complete the table to determine the balance$A$for$\$12,000$ invested
at rate $r$ for $t$ years, compounded continuously.
$$r=3.5 \%$$

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Problem 63

Trust Fund On the day of a child's birth, a parest,
deposits $\$ 30,000$in a trust fund that pays 5$\%$interest, compounded continuouly. Determine the balance in this account on the child's 25 th birthday. Check back soon! Problem 64 Trust Fund A philanthropist deposits$\$5000$ in
a trust fund that pays 7.5$\%$ interest, compounded
continuously. The balance will be given to the college
from which the philance will be given to the college
money has earned interest for 50 years. How much will

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Problem 65

Inflation Assuming that the annual rate of inflation
averages 4$\%$ over the next 10 years, the approximate
costs $C$ of goods or services during any year in that
decade will be modeled by $C(t)=P(1.04)^{t},$ where $t$ is
the time in years and $P$ is the present cost. The price of
an oil change for your car is presently $\$ 23.95 .$Estimate the price 10 years from now. Check back soon! Problem 66 Computer Virus The number$V$of computers infected by a virus increases according to the model$V(t)=100 e^{4.6052 t},$where$t$is the time in hours. Find the number of computers infected after (a) 1 hour, (b) 1.5 hours, and$(\mathrm{c}) 2$hours. Check back soon! Problem 67 Population Growth The projected populations of the United States for the years 2020 through 2050 can be modeled by$P=290.323 e^{0.0083 t}$, where$P$is the population (in millions) and$t$is the time (in years), with$t=20$corresponding to$2020 . \quad$(Source: U.S. Census Bureau) (a) Use a graphing utility to graph the function for the years 2020 through 2050 . (b) Use the table feature of the graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, during what year will the population of the United States exceed 400 million? Check back soon! Problem 68 Population The populations$P$(in millions) of Italy from 2000 through 2012 can be approximated by the model$P=57.563 e^{0.0052}$, where$t$represents the year, with$t=0$corresponding to$2000 .$(Source: U.S. Census Bureau, International Data Base) (a) According to the model, is the population of Italy increasing or decreasing? Explain. (b) Find the populations of Italy in 2000 and 2012 . (c) Use the model to predict the populations of Italy in 2020 and$2025 .$Check back soon! Problem 69 Radioactive Decay Let$Q$represent a mass of radioactive plutonium$(239 \mathrm{Pu})($in grams), whose half-life is$24,100$years. The quantity of plutonium present after$t$years is$Q=16\left(\frac{1}{2}\right)^{t / 24,100}$(a) Determine the initial quantity (when$t=0 )$. (b) Determine the quantity present after$75,000$years. (c) Use a graphing utility to graph the function over the interval$t=0$to$t=150,000 .$Check back soon! Problem 70 Radioactive Decay Let$Q$represent a mass of carbon 14$(14 \mathrm{C})$(in grams), whose half-life is 5715 years. The quantity of carbon 14 present after$t$years is$Q=10\left(\frac{1}{2}\right)^{t / 5715}$. (a) Determine the initial quantity (when$t=0 )$. (b) Determine the quantity present after 2000 years. (c) Sketch the graph of this function over the interval$t=0$to$t=10,000 .$Check back soon! Problem 71 Depreciation After$t$years, the value of a wheelchair conversion van that originally cost$\$49,810$ depreciates so
that each year it is worth $\frac{7}{8}$ of its value for the previous year.
(a) Find a model for $V(t),$ the value of the van after
$t$ years.
(b) Determine the value of the van 4 years after it was
purchased.

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Problem 72

Immediately following an injection, the concentration of a drug in the
bloodstream is
300 milligrams
per milligrams
After $t$ hours, the
concentration is
75$\%$ of the level of
the previous hour.
(a) Find a model for
$C(t)$ , the concentration of the drug after $t$ hours.
(b) Determine the concentration of the drug after
8 hours.

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Problem 73

True or False? In Exercises 73 and $74,$ determinewhether the statement is true or false. Justify your
The line $y=-2$ is an asymptote for the graph of
$f(x)=10^{x}-2$

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Problem 74

True or False? In Exercises 73 and $74,$ determine whether the statement is true or false. Justify your
$$e=\frac{271,801}{99,990}$$

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Problem 75

Think About It In Exercises $75-78$ , use properties of
exponents to determine which functions (if any) are the
same.
$$\begin{array}{l}{f(x)=3^{x-2}} \\ {g(x)=3^{x}-9} \\ {h(x)=\frac{1}{9}\left(3^{x}\right)}\end{array}$$

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Problem 76

Think About It In Exercises $75-78$ , use properties of
exponents to determine which functions (if any) are the
same.
$$\begin{array}{l}{f(x)=4^{x}+12} \\ {g(x)=2^{2 x+6}} \\ {h(x)=64\left(4^{x}\right)}\end{array}$$

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Problem 77

Think About It In Exercises $75-78$ , use properties of
exponents to determine which functions (if any) are the
same.
\begin{aligned} f(x) &=16\left(4^{-x}\right) \\ g(x) &=\left(\frac{1}{4}\right)^{x-2} \\ h(x) &=16\left(2^{-2 x}\right) \end{aligned}

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Problem 78

Think About It In Exercises $75-78$ , use properties of
exponents to determine which functions (if any) are the
same.
$$\begin{array}{l}{f(x)=e^{-x}+3} \\ {g(x)=e^{3-x}} \\ {h(x)=-e^{x-3}}\end{array}$$

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Problem 79

Solving Inequalities Graph the functions $y=3^{x}$
and $y=4^{x}$ and use the graphs to solve each inequality.
(a) $4^{x}<3^{x}$

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Problem 80

Graphical Analysis Use a graphing utility to graph
each function. Use the graph to find where the function
is increasing and decreasing, and approximate any
relative maximum or minimum values.
(a) $f(x)=x^{2} e^{-x} \quad$ (b) $g(x)=x 2^{3-x}$

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Problem 81

Graphical Analysis Use a graphing utility to
graph $y_{1}=(1+1 / x)^{x}$ and $y_{2}=e$ in the same viewing
window. Using the trace feature, explain what happens
to the graph of $y_{1}$ as $x$ increases.

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Problem 82

Graphical Analysis Use a graphing utility to graph
$f(x)=\left(1+\frac{0.5}{x}\right)^{x}$ and $g(x)=e^{0.5}$

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Problem 82

Graphical Analysis Use a graphing utility to graph
$f(x)=\left(1+\frac{0.5}{x}\right)^{x}$ and $g(x)=e^{0.5}$
in the same viewing window. What is the relationship
between $f$ and $g$ as $x$ increases and decreases without
bound?

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Problem 83

Graphical Analysis Use a graphing utility to graph
each pair of functions in the same viewing window.
Describe any similarities and differences in the graphs.
(a) $y_{1}=2^{x}, y_{2}=x^{2}$
(b) $y_{1}=3^{x}, y_{2}=x^{3}$

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Problem 84

HOW DO YOU SEE IT? The figure shows
the graphs of $y=2^{x}, y=e^{x}, y=10^{x}$ ,
$y=2^{-x}, y=e^{-x}$ , and $y=10^{-x}$ . Match each
function with its graph. IThe graphs are labeled
(a) through (f). Explain your reasoning.

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Problem 85

Think About It Which functions are exponential?
(a) 3$x$
(b) 3$x^{2}$
(c) $3^{x}$
(d) $2^{-x}$

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Problem 86

Compound Interest Use the formula
$A=P\left(1+\frac{r}{n}\right)^{n t}$
to calculate the balance of an investment when $P=\$ 3000$,$r=6 \%,$and$t=10\$ years, and compounding is done
(a) by the day, (b) by the hour, (c) by the minute, and
(d) by the second. Does increasing the number of
compoundings per year result in unlimited growth of
the balance? Explain.

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