Calculus an Applied Approach

Ron Larson, David C, Falvo

Chapter 4

Exponential and Logarithmic Functions - all with Video Answers

Educators

Section 1

Exponential Functions

Problem 1

Evaluate each expression.
(a) $5\left(5^{3}\right)$
(b) $27^{2 / 3}$
(c) $64^{3 / 4}$
(d) $81^{1 / 2}$
(e) $25^{3 / 2}$
(f) $32^{2 / 5}$

Dwijendra Rao

Dwijendra Rao

Numerade Educator

Problem 2

Evaluate each expression.
(a) $\left(\frac{1}{5}\right)^{3}$
(b) $\left(\frac{1}{8}\right)^{1 / 3}$
(c) $64^{2 / 3}$
(d) $\left(\frac{5}{8}\right)^{2}$
(e) $100^{3 / 2}$
(f) $4^{5 / 2}$

Dwijendra Rao

Numerade Educator

Problem 3

Use the properties of exponents to simplify the expression.
(a) $\left(5^{2}\right)\left(5^{3}\right)$
(b) $\left(5^{2}\right)\left(5^{-3}\right)$
(c) $\left(5^{2}\right)^{2}$
(d) $5^{-3}$

Dwijendra Rao

Numerade Educator

Problem 4

Use the properties of exponents to simplify the expression.
(a) $\frac{5^{3}}{5^{6}}$
(b) $\left(\frac{1}{5}\right)^{-2}$
(c) $\left(8^{1 / 2}\right)\left(2^{1 / 2}\right)$
(d) $\left(32^{3 / 2}\right)\left(\frac{1}{2}\right)^{3 / 2}$

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Problem 5

Use the properties of exponents to simplify the expression.
(a) $\frac{5^{3}}{25^{2}}$
(b) $\left(9^{2 / 3}\right)(3)\left(3^{2 / 3}\right)$
(c) $\left[\left(25^{1 / 2}\right)\left(5^{2}\right)\right]^{1 / 3}$
(d(d) $\left(8^{2}\right)\left(4^{3}\right)$

Dwijendra Rao

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Problem 6

Use the properties of exponents to simplify the expression.
(a) $\left(4^{3}\right)\left(4^{2}\right)$
(b) $\left(\frac{1}{4}\right)^{2}\left(4^{2}\right)$
(c) $\left(4^{6}\right)^{1 / 2}$
(d) $\left[\left(8^{-1}\right)\left(8^{2 / 3}\right)\right]^{3}$

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Problem 7

Evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places.
$f(x)=2^{x-1}$
$\begin{array}{llll}{\text { (a) } f(3)} & {\text { (b) } f\left(\frac{1}{2}\right)} & {\text { (c) } f(-2)} & {\text { (d) } f\left(-\frac{3}{2}\right)}\end{array}$

Dwijendra Rao

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Problem 8

Evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places.
$f(x)=3^{x+2}$
$\begin{array}{llll}{\text { (a) } f(-4)} & {\text { (b) } f\left(-\frac{1}{2}\right)} & {\text { (c) } f(2)} & {\text { (d) } f\left(-\frac{5}{2}\right)}\end{array}$

Dwijendra Rao

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Problem 9

Evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places.
$g(x)=1.05^{x}$
$\begin{array}{llll}{\text { (a) } g(-2)} & {\text { (b) } g(120)} & {\text { (c) } g(12)} & {\text { (d) } g(5.5)}\end{array}$

Dwijendra Rao

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Problem 10

Evaluate the function. If necessary, use a graphing utility, rounding your answers to three decimal places.
$g(x)=1.075^{x}$
$\begin{array}{llll}{\text { (a) } g(1.2)} & {\text { (b) } g(180)} & {\text { (c) } g(60)} & {\text { (d) } g(12.5)}\end{array}$

Dwijendra Rao

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Problem 11

After $t$ years, the remaining mass $y$ (in grams) of 16 grams of a radioactive element whose half-life is 30 years is given by
$y=16\left(\frac{1}{2}\right)^{t / 30}, \quad t \geq 0$
How much of the initial mass remains after 90 years?

Dwijendra Rao

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Problem 12

After $t$ years, the remaining mass $y$ (in grams) of 23 grams of a radioactive element whose half life is 45 years is given by
$y=23\left(\frac{1}{2}\right)^{t / 45}, \quad t \geq 0$
How much of the initial mass remains after 150 years?

Dwijendra Rao

Numerade Educator

Problem 13

Match the function with its graph. [The graphs are labeled (a)-(f).]
$f(x)=3^{x}$

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Problem 14

Match the function with its graph. [The graphs are labeled (a)-(f).]
$f(x)=3^{-x / 2}$

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Problem 15

Match the function with its graph. [The graphs are labeled (a)-(f).]
$f(x)=-3^{x}$

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Problem 16

Match the function with its graph. [The graphs are labeled (a)-(f).]
$f(x)=3^{x-2}$

Dwijendra Rao

Numerade Educator

Problem 17

Match the function with its graph. [The graphs are labeled (a)-(f).]
$f(x)=3^{-x}-1$

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Problem 18

Match the function with its graph. [The graphs are labeled (a)-(f).]
$f(x)=3^{-x}-1$

Dwijendra Rao

Numerade Educator

Problem 19

Use a graphing utility to graph the function.
$f(x)=6^{x}$

Kim Matthews

Numerade Educator

Problem 20

Use a graphing utility to graph the function.
$f(x)=4^{x}$

Kim Matthews

Numerade Educator

Problem 21

Use a graphing utility to graph the function.
$f(x)=\left(\frac{1}{5}\right)^{x}=5^{-x}$

Kim Matthews

Numerade Educator

Problem 22

Use a graphing utility to graph the function.
$f(x)=\left(\frac{1}{4}\right)^{x}=4^{-x}$

Kim Matthews

Numerade Educator

Problem 23

Use a graphing utility to graph the function.
$y=2^{x-1}$

Kim Matthews

Numerade Educator

Problem 24

Use a graphing utility to graph the function.
$y=4^{x}+3$

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Numerade Educator

Problem 25

Use a graphing utility to graph the function.
$y=-2^{x}$

Kim Matthews

Numerade Educator

Problem 26

Use a graphing utility to graph the function.
$y=-5^{x}$

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Problem 27

Use a graphing utility to graph the function.
$y=3^{-x^{2}}$

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Numerade Educator

Problem 28

Use a graphing utility to graph the function.
$y=2^{-x^{2}}$

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Numerade Educator

Problem 29

Use a graphing utility to graph the function.
$s(t)=\frac{1}{4}\left(3^{-t}\right)$

Kim Matthews

Numerade Educator

Problem 30

Use a graphing utility to graph the function.
$s(t)=2^{-t}+3$

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Numerade Educator

Problem 31

The population $P$ (in millions) of the United States from 1992 through 2005 can be modeled by the exponential function $P(t)=252.12(1.011)^{t}$, where $t$ is the time in years, with $t=2$ corresponding to $1992 .$ Use the model to estimate the population in the years (a) 2008 and (b) $2012 .$

Dwijendra Rao

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Problem 32

The sales $S$ (in millions of dollars) for Starbucks from 1996 through 2005 can be modeled by the exponential function $S(t)=182.34(1.272)^{t},$ where $t$ is the time in years, with $t=6$ corresponding to $1996 .$ Use the model to estimate the sales in the years (a) 2008 and (b) 2014.

Dwijendra Rao

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Problem 33

Suppose that the value of a piece ofproperty doubles every 15 years. If you buy the property for $\$ 64,000,$ its value $t$ years after the date of purchase should be $V(t)=64,000(2)^{t / 15} .$ Use the model to approximate the value of the property (a) 5 years and (b) 20 years after it is purchased.

Dwijendra Rao

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Problem 34

After $t$ years, the value of a car that originally cost $\$ 16,000$ depreciates so that each year it is worth $\frac{3}{4}$ of its value for the previous year. Find a model for $V(t),$ the value of the car after $t$ years. Sketch a graph of the model and determine the value of the car 4 years after it was purchased.

Dwijendra Rao

Numerade Educator

Problem 35

Suppose that the annual rate of inflation averages $4 \%$ over the next 10 years. With this rate of inflation, the approximate cost $C$ of goods or services during any year in that decade will be given by
$C(t)=P(1.04)^{t}, \quad 0 \leq t \leq 10$
where $t$ is time in years and $P$ is the present cost. If the price of an oil change for your car is presently $\$ 24.95,$ estimate the price 10 years from now.

Dwijendra Rao

Numerade Educator

Problem 36

Repeat Exercise 35 assuming that the annual rate of inflation is $10 \%$ over the next 10 years and the approximate cost $C$ of goods or services will be given by $C(t)=P(1.10)^{t}, \quad 0 \leq t \leq 10$.

Dwijendra Rao

Numerade Educator

Problem 37

For the years 1998 through $2005,$ the median sales prices $y$ (in dollars) of one-family homes in the United States are shown in the table. (Source: U.S. Census Bureau and U.S. Department of Housing and Urban Development)
$\begin{array}{|c|c|c|c|c|}\hline \text { Year } & {1998} & {1999} & {2000} & {2001} \\ \hline \text { Price } & {152,500} & {161,000} & {169,000} & {175,200} \\ \hline\end{array}$
$\begin{array}{|c|c|c|c|c|}\hline \text { Year } & {2002} & {2003} & {2004} & {2005} \\ \hline \text { Price } & {187,600} & {195,000} & {221,000} & {240,900} \\ \hline\end{array}$
A model for this data is given by $y=90,120(1.0649)^{t},$ where $t$ represents the year, with $t=8$ corresponding to 1998 .
(a) Compare the actual prices with those given by the model. Does the model fit the data? Explain your reasoning.
(b) Use a graphing utility to graph the model.
(c) Use the zoom and trace features of a graphing utility to predict during which year the median sales price of one-family homes will reach $\$ 300,000$.

Kim Matthews

Numerade Educator