chap5-rev-problems.xml

<?xml version="1.0"?>


<!-- This file was originally part of the book     -->
<!--   Modeling, Functions, and Graphs   -->
<!--           4th                                     -->
<!-- Copyright (C) Katherine Yoshiwara      -->

<exercises xml:id="chap5-rev-problems"  xmlns:xi="http://www.w3.org/2001/XInclude">
    <title>Review Problems</title>

<exercisegroup cols="2"><introduction><p>
    Make a table of values for the inverse function.
</p></introduction>

<exercise number="1">
    <statement><p><m>f (x) = x^3 + x + 1</m></p></statement>
    <answer><p><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center"><col halign="left"/>
        <row>
            <cell><m>y</m></cell>
            <cell><m>-1</m></cell>
            <cell><m>1</m></cell>
            <cell><m>3</m></cell>
            <cell><m>11</m></cell>
        </row>
        <row>
            <cell><m>x=f^{-1}(y)</m></cell>
            <cell><m>-1</m></cell>
            <cell><m>0</m></cell>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
        </row>
    </tabular></p></answer>
</exercise>

<exercise number="2"><statement><p><m>g(x)=x+6\sqrt[3]{x} </m></p></statement></exercise>

<exercise number="3"><statement><p><m>g(w)=\dfrac{1+w}{w-3} </m></p></statement>
<answer><p><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center"><col halign="left"/>
        <row>
            <cell><m>y</m></cell>
            <cell><m>0</m></cell>
            <cell><m>\frac{-1}{3} </m></cell>
            <cell><m>-1</m></cell>
            <cell><m>-3</m></cell>
        </row>
        <row>
            <cell><m>w=g^{-1}(y)</m></cell>
            <cell><m>-1</m></cell>
            <cell><m>0</m></cell>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
        </row>
    </tabular></p></answer>
</exercise>

<exercise number="4"><statement><p><m>f(n)=\dfrac{n}{1+n} </m></p></statement></exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>
    Use the graph to find the function values.
</p></introduction>

<exercise number="5"><statement><p><image source="images/fig-chap5-rev-5"  width="70%"><description>increasing sigmoid</description></image> 
<ol cols="2">
    <li><p><m>P^{-1}(350) </m></p></li>
    <li><p><m>P^{-1}(100)  </m></p></li>
</ol>
</p></statement>
<answer><ol cols="2">
    <li><p><m>P^{-1}(350)=40</m></p></li>
    <li><p><m>P^{-1}(100)=0  </m></p></li>
</ol></answer>
</exercise>

<exercise number="6"><statement><p><image source="images/fig-chap5-rev-6"  width="78%"><description>decay</description></image> 
<ol cols="2">
    <li><p><m>H^{-1}(200) </m></p></li>
    <li><p><m>H^{-1}(75)  </m></p></li>
</ol>
</p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="2"><introduction><p>
    For Problems 7<ndash/>12,
    <ol label="a">
        <li><p>Find a formula for the inverse <m>f^{-1} </m> of each function.</p></li>
        <li><p>Graph the function and its inverse on the same set of axes, along with the graph of <m>y=x</m>.</p></li>
    </ol>
</p></introduction>

<exercise number="7"><statement><p><m>f(x)=x+4 </m></p>
<answer><p><ol cols="2">
    <li><p><m>f^{-1} (x) = x - 4</m></p></li>
    <li><p><image source="images/fig-ans-chap5-rev-7"  width="90%"><description>line and inverse</description></image> </p></li>
</ol></p></answer>
</statement></exercise>

<exercise number="8"><statement><p><m>f(x)=\dfrac{x-2}{4} </m></p>
</statement></exercise>

<exercise number="9"><statement><p><m>f(x)=x^3-1 </m></p>
<answer><p><ol cols="2">
    <li><p><m>f^{-1} (x) =\sqrt[3]{x+1} </m></p></li>
    <li><p><image source="images/fig-ans-chap5-rev-9"  width="90%"><description>cubic and inverse</description></image> </p></li>
</ol></p></answer>
</statement></exercise>

<exercise number="10"><statement><p><m>f(x)=\dfrac{1}{x+2} </m></p>
</statement></exercise>

<exercise number="11"><statement><p><m>f(x)=\dfrac{1}{x}+2 </m></p>
<answer><p><ol cols="2">
    <li><p><m>f^{-1} (x) =\dfrac{1}{x-2} </m></p></li>
    <li><p><image source="images/fig-ans-chap5-rev-11"  width="90%"><description>translated reciprocal and inverse</description></image> </p></li>
</ol></p></answer>
</statement></exercise>

<exercise number="12"><statement><p><m>f(x)=\sqrt[3]{x}-2 </m></p>
</statement></exercise>
</exercisegroup>



<exercise number="13"><statement><p>If <m>F(t) = \dfrac{3}{4}t + 2</m>, find <m>F^{-1}(2)</m>.</p></statement>
<answer><p><m>0</m></p></answer>
</exercise>

<exercise number="14"><statement><p>If <m>G(x) = \dfrac{1}{x}-4</m>, find <m>G^{-1}(3)</m>.</p></statement>
</exercise>

<exercise number="15"><statement><p>
    The table shows the revenue, <m>R</m>, from sales of the Miracle Mop as a function of the number of dollars spent on advertising, <m>A</m>. Let <m>f</m> be the name of the function defined by the table, so <m>R = f (A)</m>.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center"><col halign="left"/>
        <row>
            <cell><line><m>A</m> (thousands</line><line>of dollars)</line></cell>
            <cell><m>100</m></cell>
            <cell><m>150</m></cell>
            <cell><m>200</m></cell>
            <cell><m>250</m></cell>
            <cell><m>300</m></cell>
        </row>
        <row>
            <cell><line><m>R</m> (thousands</line><line>of dollars)</line></cell>
            <cell><m>250</m></cell>
            <cell><m>280</m></cell>
            <cell><m>300</m></cell>
            <cell><m>310</m></cell>
            <cell><m>315</m></cell>
        </row>
    </tabular></sidebyside>
    <ol>
        <li><p>Evaluate <m>f^{ -1}(300)</m>. Explain its meaning in this context.</p></li>
        <li><p>Write two equations to answer the following question, one using <m>f</m> and one using <m>f^{ -1}</m>: How much should we spend on advertising to generate revenue of <m>\$250,000</m>?</p></li>
    </ol>
</p></statement>
<answer><p><ol>
    <li><p><m>f^{-1} (300) = 200</m>: <m>\$200,000</m> in advertising results in <m>\$300,000</m> in revenue.</p></li>
    <li><p><m>f (A) = 250</m> or <m>A = f^{-1} (250)</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="16"><statement><p>
    The table shows the systolic blood pressure, <m>S</m>, of a patient as a function of the dosage, <m>d</m>, of medication he receives. Let <m>g</m> be the name of the function defined by the table, so <m>S = g(d)</m>.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor" halign="center"><col halign="left"/>
        <row>
            <cell><m>d</m> (mg)</cell>
            <cell><m>190</m></cell>
            <cell><m>195</m></cell>
            <cell><m>200</m></cell>
            <cell><m>210</m></cell>
            <cell><m>220</m></cell>
        </row>
        <row>
            <cell><m>S</m> (mm Hg)</cell>
            <cell><m>220</m></cell>
            <cell><m>200</m></cell>
            <cell><m>190</m></cell>
            <cell><m>185</m></cell>
            <cell><m>183</m></cell>
        </row>
    </tabular></sidebyside>
    <ol>
        <li><p>Evaluate <m>g^{ -1}(200)</m>. Explain its meaning in this context.</p></li>
        <li><p>Write two equations to answer the following question, one using <m>g</m> and one using <m>g^{ -1}</m>: What dosage results in systolic blood pressure of <m>220</m>?</p></li>
    </ol>
</p></statement>
</exercise>



<exercisegroup cols="2"><introduction><p>Write in exponential form.</p></introduction>

<exercise number="17"><statement><p><m>\log_{10} 0.001 = z</m></p></statement>
<answer><p><m>10^z = 0.001</m></p></answer>
</exercise>

<exercise number="18"><statement><p><m>\log_{3} 20 = t</m></p></statement>
</exercise>

<exercise number="19"><statement><p><m>\log_{2} 3 = x-2</m></p></statement>
<answer><p><m>2^{x-2} = 3</m></p></answer>
</exercise>

<exercise number="20"><statement><p><m>\log_{5} 3 = 6-2p</m></p></statement>
</exercise>

<exercise number="21"><statement><p><m>\log_{b} (3x+1) = 3</m></p></statement>
<answer><p><m>b^{3} = 3x+1</m></p></answer>
</exercise>

<exercise number="22"><statement><p><m>\log_{m} 8 = 4t</m></p></statement>
</exercise>

<exercise number="23"><statement><p><m>\log_{n} q = p-1</m></p></statement>
<answer><p><m>n^{p-1} = q</m></p></answer>
</exercise>

<exercise number="24"><statement><p><m>\log_{q} (p+2) = w</m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="4"><introduction><p>Simplify.</p></introduction>

<exercise number="25"><statement><p><m>10^{\log 6n} </m></p></statement>
<answer><p><m>6n</m></p></answer>
</exercise>

<exercise number="26"><statement><p><m>\log 100^x </m></p></statement>
</exercise>

<exercise number="27"><statement><p><m>\log_2 4^{x+3} </m></p></statement>
<answer><p><m>2x+6</m></p></answer>
</exercise>

<exercise number="26"><statement><p><m>3^{2\log_3 t} </m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="3"><introduction><p>Solve.</p></introduction>

<exercise number="29"><statement><p><m>\log_{3}\dfrac{1}{3}=y </m></p></statement>
<answer><p><m>-1</m></p></answer>
</exercise>

<exercise number="30"><statement><p><m>\log_{3}x=4 </m></p></statement>
</exercise>

<exercise number="31"><statement><p><m>\log_{2}y=-1 </m></p></statement>
<answer><p><m>\dfrac{1}{2} </m></p></answer>
</exercise>

<exercise number="32"><statement><p><m>\log_{5}y=-2 </m></p></statement>
</exercise>

<exercise number="33"><statement><p><m>\log_{b} 16=2 </m></p></statement>
<answer><p><m>4 </m></p></answer>
</exercise>

<exercise number="34"><statement><p><m>\log_{b}9=\dfrac{1}{2} </m></p></statement>
</exercise>

<exercise number="35"><statement><p><m>\log_{4}\left(\dfrac{1}{2}t+1\right)=-2 </m></p></statement>
<answer><p><m>\dfrac{-15}{8} </m></p></answer>
</exercise>

<exercise number="36"><statement><p><m>\log_{2}(3x-1)=3 </m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup ><introduction><p>Solve.</p></introduction>

<exercise number="37"><statement><p><m>\log_3 x + \log_3 4 = 2</m></p></statement>
<answer><p><m>\dfrac{9}{4} </m></p></answer>
</exercise>

<exercise number="38"><statement><p><m>\log_2(x + 2) - \log_2 3 = 6</m></p></statement>
</exercise>

<exercise number="39"><statement><p><m>\log_{10}(x-1) + \log_{10} (x+2) = 1</m></p></statement>
<answer><p><m>3 </m></p></answer>
</exercise>

<exercise number="40"><statement><p><m>\log_{10}(x + 2) - \log_{10} (x-3) = 1</m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="3"><introduction><p>Solve.</p></introduction>

<exercise number="41"><statement><p><m>e^x=4.7 </m></p></statement>
<answer><p><m>x\approx 1.548 </m></p></answer>
</exercise>

<exercise number="42"><statement><p><m>e^x=0.5 </m></p></statement>
</exercise>

<exercise number="43"><statement><p><m>\ln x =6.02 </m></p></statement>
<answer><p><m>x\approx 411.58 </m></p></answer>
</exercise>

<exercise number="44"><statement><p><m>\ln x=-1.4 </m></p></statement>
</exercise>

<exercise number="45"><statement><p><m>4.73=1.2e^{0.6x} </m></p></statement>
<answer><p><m>x\approx 2.286 </m></p></answer>
</exercise>

<exercise number="46"><statement><p><m>1.75=0.3e^{-1.2x} </m></p></statement>
</exercise>
</exercisegroup>



<exercisegroup cols="4"><introduction><p>Simplify.</p></introduction>

<exercise number="47"><statement><p><m>e^{(\ln x)/2} </m></p></statement>
<answer><p><m>\sqrt{x} </m></p></answer>
</exercise>

<exercise number="48"><statement><p><m>\ln \left(\dfrac{1}{e} \right)^{2n} </m></p></statement>
</exercise>

<exercise number="49"><statement><p><m>\ln \left(\dfrac{e^k}{e^3} \right) </m></p></statement>
<answer><p><m>k-3 </m></p></answer>
</exercise>

<exercise number="50"><statement><p><m>e^{\ln(e+x)} </m></p></statement>
</exercise>
</exercisegroup>



<exercise number="51"><statement><p>
    In 1970, the population of New York City was <m>7,894,862</m>. In 1980, the population had fallen to <m>7,071,639</m>.
    <ol>
        <li><p>Write an exponential function using base <m>e</m> for the population of New York over that decade.</p></li>
        <li><p>By what percent did the population decline annually?</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="2">
    <li><p><m>P = 7,894,862e^{-0.011t}</m></p></li>
    <li><p><m>1.095\%</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="52"><statement><p>
    In 1990, the population of New York City was <m>7,322,564</m>. In 2000, the population had fallen to <m>8,008,278</m>.
    <ol>
        <li><p>Write an exponential function using base <m>e</m> for the population of New York over that decade.</p></li>
        <li><p>By what percent did the population increase annually?</p></li>
    </ol>
</p></statement>
</exercise>

<exercise number="53"><statement><p>
    You deposit <m>\$1000</m> in a savings account paying <m>5\%</m> interest compounded continuously
    <ol>
        <li><p>Find the amount in the account after <m>7</m> years.</p></li>
        <li><p>How long will it take for the original principal to double?</p></li>
        <li><p>Find a formula for the time <m>t</m> required for the amount to reach <m>A</m>.</p></li>
    </ol>
</p></statement>
<answer><p><ol cols="3">
    <li><p><m>$1419.07 </m></p></li>
    <li><p><m>13.9</m> years</p></li>
    <li><p><m>t = 20 \ln\left(\dfrac{A}{1000} \right)</m></p></li>
</ol></p></answer>
</exercise>

<exercise number="54"><statement><p>
    The voltage, <m>V</m>, across a capacitor in a certain circuit is given by the function
    <me>V(t) = 100(1-e^{-0.5t}) </me>
    where <m>t</m> is the time in seconds.
    <ol>
        <li><p>Make a table of values and graph <m>V(t)</m> for <m>t = 0</m> to <m>t = 10</m>.</p></li>
        <li><p>Describe the graph. What happens to the voltage in the long run?</p></li>
        <li><p>How much time must elapse (to the nearest hundredth of a second) for the voltage to reach <m>75</m> volts?</p></li>
    </ol>
</p></statement>
</exercise>

<exercise number="55"><statement><p>Solve for <m>t</m>: <m>y = 12 e^{-kt} + 6</m></p></statement>
<answer><p><m>t=\dfrac{-1}{k}\ln\left(\dfrac{y-6}{12} \right) </m></p></answer>
</exercise>

<exercise number="56"><statement><p>Solve for <m>k</m>: <m>N = N_0 + 4 \ln(k + 10)</m></p></statement>
</exercise>

<exercise number="57"><statement><p>Solve for <m>M</m>: <m>Q=\dfrac{1}{t}\left(\dfrac{\log M}{\log N} \right) </m></p></statement>
<answer><p><m>M=N^{Qt} </m></p></answer>
</exercise>

<exercise number="58"><statement><p>Solve for <m>t</m>: <m>C_H = C_L\cdot 10^{k}t </m></p></statement>
</exercise>

<exercise number="59"><statement><p>Express <m>P(t) = 750e^{0.32t}</m> in the form <m>P(t) = P_0b^t</m>.</p></statement>
<answer><p><m>P (t) = 750 (1.3771)^t</m></p></answer>
</exercise>

<exercise number="60"><statement><p>Express <m>P(t) = 80e^{-0.6t}</m> in the form <m>P(t) = P_0 b^t</m>.</p></statement>
</exercise>

<exercise number="61"><statement><p>Express <m>N(t) =600(0.4)^{t}</m> in the form <m>N(t) = N_0 e^{kt}</m>.</p></statement>
<answer><p><m>N(t) = 600 e^{-0.9163t}</m></p></answer>
</exercise>

<exercise number="62"><statement><p>Express <m>N(t) =100(1.06)^{t}</m> in the form <m>N(t) = N_0 e^{kt}</m>.</p></statement>
</exercise>

<exercise number="63"><statement><p>
    Plot the values on a log scale.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor"><col right="major"/>
        <row>
            <cell><m>x</m></cell>
            <cell><m>0.04</m></cell>
            <cell><m>45</m></cell>
            <cell><m>1200</m></cell>
            <cell><m>560,000</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement>
<answer><p><image source="images/fig-ans-chap5-rev-63"  width="80%"><description>log scale</description></image> </p></answer>
</exercise>


<exercise number="64"><statement><p>
    Plot the values on a log scale.</p><p>
    <sidebyside><tabular left="minor" right="minor" top="minor" bottom="minor"><col right="major"/>
        <row>
            <cell><m>x</m></cell>
            <cell><m>0.0007</m></cell>
            <cell><m>0.8</m></cell>
            <cell><m>3.2</m></cell>
            <cell><m>2500</m></cell>
        </row>
    </tabular></sidebyside>
</p></statement></exercise>

<exercise number="65"><statement>
    <sidebyside widths="50% 40%">
    <p>The graph describes a network of streams near Santa Fe, New Mexico. It shows the number of streams of a given order, which is a measure of their size. Use the graph to estimate the number of streams of orders <m>3</m>, <m>4</m>, <m>8</m>, and <m>9</m>. (Source: Leopold, Wolman, and Miller)</p>
    <p><image source="images/fig-chap5-rev-65"><description>stream order on semi-log scale</description></image></p>
</sidebyside></statement>
<answer><p>Order <m>3</m>: <m>17,000</m>; Order <m>4</m>: <m>5000</m>; Order <m>8</m>: <m>40</m>; Order <m>9</m>: <m>11</m></p></answer>
</exercise>

<exercise number="66"><statement><p>
    Large animals use oxygen more efficiently when running than small animals do. The graph shows the amount of oxygen various animals use, per gram of their body weight, to run <m>1</m> kilometer. Estimate the body mass and oxygen use for a kangaroo rat, a dog, and a horse. (Source: Schmidt-Neilsen, 1972)</p><p>
    <sidebyside width="85%"><image source="images/fig-chap5-rev-66"/></sidebyside>
</p></statement></exercise>

<exercise number="67"><statement><p>The pH of an unknown substance is <m>6.3</m>. What is its hydrogen ion concentration?</p></statement>
<answer><p><m>5\times 10^{-7}</m></p></answer>
</exercise>

<exercise number="68"><statement><p>The noise of a leaf blower was measured at <m>110</m> decibels. What was the intensity of the sound waves?</p></statement>
</exercise>

<exercise number="69"><statement><p>A refrigerator produces <m>50</m> decibels of noise, and a vacuum cleaner produces <m>85</m> decibels. How much more intense are the sound waves from a vacuum cleaner than those from a refrigerator?</p></statement>
<answer><p><m>3160</m></p></answer>
</exercise>

<exercise number="70"><statement><p>In 2004, a magnitude <m>9.0</m> earthquake struck Sumatra in Indonesia. How much more powerful was this quake than the 1906 San Francisco earthquake of magnitude <m>8.3</m>?</p></statement>
</exercise>

</exercises>