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H. FREEMAN AND COMP<span class="_0 blank"></span>ANY</div><div class="t m0 x5 h11 y11 ff7 fs7 fc0 sc0 ls0 ws8">NEW YORK</div><div class="c xb y1 w2 h2"><div class="t m0 x0 h3 y2 ff1 fs0 fc0 sc0 ls0">www.elsolucionario.net</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div> <div id="pf5" class="pf w7 h9" data-page-no="5"><div class="pc pc5 w7 h9"><img fetchpriority="low" loading="lazy" class="bi x0 y7 w7 ha" alt="" src="https://files.passeidireto.com/6bf7eeaa-f34a-4479-8a10-d1e67ea494fa/bg5.png"><div class="t m0 xc h12 y12 ff8 fs8 fc0 sc0 ls0 ws9">© 2012 by W<span class="_2 blank"></span>. H. Freeman and Company</div><div class="t m0 xc h12 y13 ff8 fs8 fc0 sc0 ls0 ws9">ISBN-13: 978-1-4292-4290-5</div><div class="t m0 xc h12 y14 ff8 fs8 fc0 sc0 ls0 ws9">ISBN-10: 1-4292-4290-6</div><div class="t m0 xc h12 y15 ff8 fs8 fc0 sc0 ls0 ws9">All rights reserved</div><div class="t m0 xc h12 y16 ff8 fs8 fc0 sc0 ls0 ws9">Printed in the United States of<span class="_3 blank"> </span>America</div><div class="t m0 xc h13 y17 ff3 fs7 fc0 sc0 ls0 wsa">First Printing</div><div class="t m0 xc h12 y18 ff8 fs8 fc0 sc0 ls0 ws9">W<span class="_2 blank"></span>. H. Freeman and Company<span class="_0 blank"></span>, 41 Madison<span class="_3 blank"> </span>A<span class="_4 blank"></span>venue, New<span class="_5 blank"> </span>Y<span class="_2 blank"></span>ork, NY<span class="_5 blank"> </span>10010</div><div class="t m0 xc h12 y19 ff8 fs8 fc0 sc0 ls0 ws9">Houndmills, Basingstoke RG21 6XS, England</div><div class="t m0 xc h12 y1a ff8 fs8 fc0 sc0 ls0 wsb">www<span class="_4 blank"></span>.whfreeman.com</div><div class="c xb y1 w2 h2"><div class="t m0 x0 h3 y2 ff1 fs0 fc0 sc0 ls0">www.elsolucionario.net</div></div><a class="l"><div class="d m1" style="border-style:none;position:absolute;left:151.272000px;bottom:70.241300px;width:97.843000px;height:11.118000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div> <div id="pf6" class="pf w7 h9" data-page-no="6"><div class="pc pc6 w7 h9"><img fetchpriority="low" loading="lazy" class="bi x0 y7 w7 ha" alt="" src="https://files.passeidireto.com/6bf7eeaa-f34a-4479-8a10-d1e67ea494fa/bg6.png"><div class="t m0 xa h14 y1b ff9 fs9 fc0 sc0 ls0 wsc">C<span class="_4 blank"></span>ONTENTS</div><div class="t m0 xd h15 y1c ff3 fs6 fc0 sc0 ls0 wsd">Chapter 1<span class="_6 blank"> </span>PRECALCULUS REVIEW<span class="_7 blank"> </span>1</div><div class="t m0 xd h16 y1d ffa fs8 fc0 sc0 ls0 wse">1.1 <span class="ffb fsa wsf">Real Numbers, Functions, and Graphs<span class="_8 blank"> </span>1</span></div><div class="t m0 xd h16 y1e ffa fs8 fc0 sc0 ls0 wse">1.2 <span class="ffb fsa wsf">Linear and Quadratic Functions<span class="_9 blank"> </span>8</span></div><div class="t m0 xd h16 y1f ffa fs8 fc0 sc0 ls0 wse">1.3 <span class="ffb fsa wsf">The Basic Classes of Functions<span class="_a blank"> </span>13</span></div><div class="t m0 xd h16 y20 ffa fs8 fc0 sc0 ls0 wse">1.4 <span class="ffb fsa wsf">T<span class="_4 blank"></span>rigonometric Functions<span class="_b blank"> </span>16</span></div><div class="t m0 xd h16 y21 ffa fs8 fc0 sc0 ls0 wse">1.5 <span class="ffb fsa wsf">T<span class="_4 blank"></span>echnology: Calculators and Computers<span class="_c blank"> </span>23</span></div><div class="t m0 xe h17 y22 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_d blank"> </span>27</div><div class="t m0 xd h15 y23 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 2<span class="_6 blank"> </span>LIMITS<span class="_e blank"> </span>31</div><div class="t m0 xd h16 y24 ffa fs8 fc0 sc0 ls0 wse">2.1 <span class="ffb fsa wsf">Limits, Rates of Change, and T<span class="_4 blank"></span>angent Lines<span class="_f blank"> </span>31</span></div><div class="t m0 xd h16 y25 ffa fs8 fc0 sc0 ls0 wse">2.2 <span class="ffb fsa wsf">Limits: A Numerical and Graphical Approach<span class="_10 blank"> </span>37</span></div><div class="t m0 xd h16 y26 ffa fs8 fc0 sc0 ls0 wse">2.3 <span class="ffb fsa wsf v0">Basic Limit Laws<span class="_11 blank"> </span>46</span></div><div class="t m0 xd h16 y27 ffa fs8 fc0 sc0 ls0 wse">2.4 <span class="ffb fsa wsf">Limits and Continuity<span class="_12 blank"> </span>49</span></div><div class="t m0 xd h16 y28 ffa fs8 fc0 sc0 ls0 wse">2.5 <span class="ffb fsa wsf v0">Evaluating Limits Algebraically<span class="_13 blank"> </span>57</span></div><div class="t m0 xd h16 y29 ffa fs8 fc0 sc0 ls0 wse">2.6 <span class="ffb fsa wsf">T<span class="_4 blank"></span>rigonometric Limits<span class="_14 blank"> </span>61</span></div><div class="t m0 xd h16 y2a ffa fs8 fc0 sc0 ls0 wse">2.7 <span class="ffb fsa wsf v0">Limits at In\ufb01nity<span class="_15 blank"> </span>66</span></div><div class="t m0 xd h16 y2b ffa fs8 fc0 sc0 ls0 wse">2.8 <span class="ffb fsa wsf">Intermediate V<span class="_0 blank"></span>alue Theorem<span class="_16 blank"> </span>73</span></div><div class="t m0 xd h16 y2c ffa fs8 fc0 sc0 ls0 wse">2.9 <span class="ffb fsa wsf">The Formal De\ufb01nition of a Limit<span class="_17 blank"> </span>76</span></div><div class="t m0 xe h17 y2d ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_d blank"> </span>82</div><div class="t m0 xd h15 y2e ff3 fs6 fc0 sc0 ls0 wsd">Chapter 3<span class="_6 blank"> </span>DIFFERENTIA<span class="_4 blank"></span>TION<span class="_18 blank"> </span>91</div><div class="t m0 xd h16 y2f ffa fs8 fc0 sc0 ls0 wse">3.1 <span class="ffb fsa wsf">Definition of the Derivative<span class="_19 blank"> </span>91</span></div><div class="t m0 xd h16 y30 ffa fs8 fc0 sc0 ls0 wse">3.2 <span class="ffb fsa wsf">The Derivative as a Function<span class="_1a blank"> </span>101</span></div><div class="t m0 xd h16 y31 ffa fs8 fc0 sc0 ls0 wse">3.3 <span class="ffb fsa wsf">Product and Quotient Rules<span class="_1b blank"> </span>112</span></div><div class="t m0 xd h16 y32 ffa fs8 fc0 sc0 ls0 wse">3.4 <span class="ffb fsa wsf v0">Rates of Change<span class="_1c blank"> </span>119</span></div><div class="t m0 xd h16 y33 ffa fs8 fc0 sc0 ls0 wse">3.5 <span class="ffb fsa wsf">Higher Derivatives<span class="_1d blank"> </span>126</span></div><div class="t m0 xd h16 y34 ffa fs8 fc0 sc0 ls0 wse">3.6 <span class="ffb fsa wsf">T<span class="_4 blank"></span>rigonometric Functions<span class="_d blank"> </span>132</span></div><div class="t m0 xd h16 y35 ffa fs8 fc0 sc0 ls0 wse">3.7 <span class="ffb fsa wsf">The Chain Rule<span class="_1e blank"> </span>138</span></div><div class="t m0 xd h16 y36 ffa fs8 fc0 sc0 ls0 wse">3.8 <span class="ffb fsa wsf v0">Implicit Differentiation<span class="_1f blank"> </span>147</span></div><div class="t m0 xd h16 y37 ffa fs8 fc0 sc0 ls0 wse">3.9 <span class="ffb fsa wsf">Related Rates<span class="_20 blank"> </span>157</span></div><div class="t m0 xe h17 y38 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_19 blank"> </span>165</div><div class="t m0 xd h15 y39 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 4<span class="_6 blank"> </span>APPLICA<span class="_4 blank"></span>TIONS OF THE DERIV<span class="_0 blank"></span>A<span class="_4 blank"></span>TIVE 174</div><div class="t m0 xd h16 y3a ffa fs8 fc0 sc0 ls0 wse">4.1 <span class="ffb fsa wsf v0">Linear Approximation and Applications<span class="_21 blank"> </span>174</span></div><div class="t m0 xd h16 y3b ffa fs8 fc0 sc0 ls0 wse">4.2 <span class="ffb fsa wsf">Extreme V<span class="_0 blank"></span>alues<span class="_22 blank"> </span>181</span></div><div class="t m0 xd h16 y3c ffa fs8 fc0 sc0 ls0 wse">4.3 <span class="ffb fsa wsf">The Mean V<span class="_0 blank"></span>alue Theorem and Monotonicity<span class="_23 blank"> </span>191</span></div><div class="t m0 xd h16 y3d ffa fs8 fc0 sc0 ls0 wse">4.4 <span class="ffb fsa wsf">The Shape of a Graph<span class="_24 blank"> </span>198</span></div><div class="t m0 xd h16 y3e ffa fs8 fc0 sc0 ls0 wse">4.5 <span class="ffb fsa wsf v0">Graph Sketching and Asymptotes<span class="_25 blank"> </span>206</span></div><div class="t m0 xd h16 y3f ffa fs8 fc0 sc0 ls0 wse">4.6 <span class="ffb fsa wsf">Applied Optimization<span class="_26 blank"> </span>220</span></div><div class="t m0 xd h16 y40 ffa fs8 fc0 sc0 ls0 wse">4.7 <span class="ffb fsa wsf">Newton\u2019<span class="_4 blank"></span>s Method<span class="_27 blank"> </span>236</span></div><div class="t m0 xd h16 y41 ffa fs8 fc0 sc0 ls0 wse">4.8 <span class="ffb fsa ws10">Antiderivatives 242</span></div><div class="t m0 xe h17 y42 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_19 blank"> </span>250</div><div class="t m0 xd h15 y43 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 5<span class="_6 blank"> </span>THE INTEGRAL<span class="_28 blank"> </span>260</div><div class="t m0 xd h16 y44 ffa fs8 fc0 sc0 ls0 wse">5.1 <span class="ffb fsa wsf">Approximating and Computing Area<span class="_29 blank"> </span>260</span></div><div class="t m0 xd h16 y45 ffa fs8 fc0 sc0 ls0 wse">5.2 <span class="ffb fsa wsf v0">The De\ufb01nite Integral<span class="_2a blank"> </span>274</span></div><div class="t m0 xd h16 y46 ffa fs8 fc0 sc0 ls0 wse">5.3 <span class="ffb fsa wsf">The Fundamental Theorem of Calculus, Part I<span class="_2b blank"> </span>284</span></div><div class="t m0 xf h16 y47 ffa fs8 fc0 sc0 ls0 wse">5.4 <span class="ffb fsa wsf">The Fundamental Theorem of Calculus, Part II<span class="_2c blank"> </span>290</span></div><div class="t m0 xf h16 y48 ffa fs8 fc0 sc0 ls0 wse">5.5 <span class="ffb fsa wsf">Net Change as the Integral of a Rate<span class="_2d blank"> </span>296</span></div><div class="t m0 xf h16 y49 ffa fs8 fc0 sc0 ls0 wse">5.6 <span class="ffb fsa wsf">Substitution Method<span class="_2e blank"> </span>300</span></div><div class="t m0 x10 h17 y4a ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_19 blank"> </span>307</div><div class="t m0 xf h15 y4b ff3 fs6 fc0 sc0 ls0 wsd">Chapter 6<span class="_6 blank"> </span>APPLICA<span class="_4 blank"></span>TIONS OF THE INTEGRAL<span class="_2f blank"> </span>317</div><div class="t m0 xf h16 y4c ffa fs8 fc0 sc0 ls0 ws11">6.1 <span class="ffb fsa wsf">Area Between T<span class="_4 blank"></span>wo Curves<span class="_30 blank"> </span>317</span></div><div class="t m0 xf h16 y4d ffa fs8 fc0 sc0 ls0 ws11">6.2 <span class="ffb fsa wsf">Setting Up Integrals: V<span class="_0 blank"></span>olume, Density<span class="_4 blank"></span>, Average V<span class="_4 blank"></span>alue<span class="_31 blank"> </span>328</span></div><div class="t m0 xf h16 y4e ffa fs8 fc0 sc0 ls0 ws11">6.3 <span class="ffb fsa wsf">V<span class="_0 blank"></span>olumes of Revolution<span class="_32 blank"> </span>336</span></div><div class="t m0 xf h16 y4f ffa fs8 fc0 sc0 ls0 ws11">6.4 <span class="ffb fsa wsf">The Method of Cylindrical Shells<span class="_33 blank"> </span>346</span></div><div class="t m0 xf h16 y50 ffa fs8 fc0 sc0 ls0 ws11">6.5 <span class="ffb fsa wsf">W<span class="_0 blank"></span>ork and Energy<span class="_e blank"> </span>355</span></div><div class="t m0 x10 h17 y51 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_30 blank"> </span>362</div><div class="t m0 xf h15 y52 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 7<span class="_6 blank"> </span>EXPONENTIAL FUNCTIONS<span class="_34 blank"> </span>370</div><div class="t m0 xf h18 y53 ffa fs8 fc0 sc0 ls0 ws11">7.1 <span class="ffb fsa wsf">Derivative of <span class="ffc lsb ws12">f(<span class="_35 blank"></span>x<span class="_2 blank"></span>) <span class="ffd ls4">=</span><span class="ls5">b<span class="fsb ls6 v1">x</span><span class="ffb ls0 wsf">and the Number </span><span class="ls7">e<span class="ffb ls0">370</span></span></span></span></span></div><div class="t m0 xf h16 y54 ffa fs8 fc0 sc0 ls0 ws11">7.2 <span class="ffb fsa wsf">Inverse Functions<span class="_36 blank"> </span>378</span></div><div class="t m0 xf h16 y55 ffa fs8 fc0 sc0 ls0 ws11">7.3 <span class="ffb fsa wsf">Logarithms and Their Derivatives<span class="_37 blank"> </span>383</span></div><div class="t m0 xf h16 y56 ffa fs8 fc0 sc0 ls0 ws11">7.4 <span class="ffb fsa wsf">Exponential Growth and Decay<span class="_38 blank"> </span>393</span></div><div class="t m0 xf h16 y57 ffa fs8 fc0 sc0 ls0 ws11">7.5 <span class="ffb fsa wsf">Compound Interest and Present V<span class="_0 blank"></span>alue<span class="_39 blank"> </span>398</span></div><div class="t m0 xf h19 y58 ffa fs8 fc0 sc0 ls0 ws11">7.6 <span class="ffb fsa wsf">Models Involving<span class="_3a blank"> </span><span class="ffc ls8">y<span class="ffd fsb ls9 v1">\ue002</span><span class="ffd ls4">=</span><span class="lsc ws13">k(<span class="_0 blank"></span>y <span class="ffd lsa">\u2212</span><span class="lsd ws14">b) <span class="ffb ls0">401</span></span></span></span></span></div><div class="t m0 xf h16 y59 ffa fs8 fc0 sc0 ls0 ws11">7.7 <span class="ffb fsa wsf">L<span class="_2 blank"></span>\u2019Hôpital\u2019<span class="_0 blank"></span>s Rule<span class="_3b blank"> </span>407</span></div><div class="t m0 xf h16 y5a ffa fs8 fc0 sc0 ls0 ws11">7.8 <span class="ffb fsa wsf">Inverse T<span class="_4 blank"></span>rigonometric Functions<span class="_3c blank"> </span>415</span></div><div class="t m0 xf h16 y5b ffa fs8 fc0 sc0 ls0 ws11">7.9 <span class="ffb fsa wsf">Hyperbolic Functions<span class="_3d blank"> </span>424</span></div><div class="t m0 x10 h17 y5c ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_30 blank"> </span>431</div><div class="t m0 xf h15 y5d ff3 fs6 fc0 sc0 ls0 wsd">Chapter 8<span class="_6 blank"> </span>TECHNIQUES OF INTEGRA<span class="_4 blank"></span>TION<span class="_3e blank"> </span>446</div><div class="t m0 xf h16 y5e ffa fs8 fc0 sc0 ls0 ws11">8.1 <span class="ffb fsa wsf">Integration by Parts<span class="_3f blank"> </span>446</span></div><div class="t m0 xf h16 y5f ffa fs8 fc0 sc0 ls0 ws11">8.2 <span class="ffb fsa wsf">T<span class="_4 blank"></span>rigonometric Integrals<span class="_40 blank"> </span>457</span></div><div class="t m0 xf h16 y60 ffa fs8 fc0 sc0 ls0 ws11">8.3 <span class="ffb fsa wsf">T<span class="_4 blank"></span>rigonometric Substitution<span class="_41 blank"> </span>467</span></div><div class="t m0 xf h16 y61 ffa fs8 fc0 sc0 ls0 ws11">8.4 <span class="ffb fsa wsf v0">Integrals Involving Hyperbolic and Inverse Hyperbolic</span></div><div class="t m0 x10 h17 y62 ffb fsa fc0 sc0 ls0 ws15">Functions 481</div><div class="t m0 xf h16 y63 ffa fs8 fc0 sc0 ls0 ws11">8.5 <span class="ffb fsa wsf">The Method of Partial Fractions<span class="_42 blank"> </span>485</span></div><div class="t m0 xf h16 y64 ffa fs8 fc0 sc0 ls0 ws11">8.6 <span class="ffb fsa wsf">Improper Integrals<span class="_43 blank"> </span>503</span></div><div class="t m0 xf h16 y65 ffa fs8 fc0 sc0 ls0 ws11">8.7 <span class="ffb fsa wsf">Probability and Integration<span class="_44 blank"> </span>520</span></div><div class="t m0 xf h16 y66 ffa fs8 fc0 sc0 ls0 ws11">8.8 <span class="ffb fsa wsf">Numerical Integration<span class="_24 blank"> </span>525</span></div><div class="t m0 x10 h17 y67 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_30 blank"> </span>537</div><div class="t m0 xf h15 y68 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 9<span class="_6 blank"> </span>FURTHER APPLICA<span class="_4 blank"></span>TIONS OF THE</div><div class="t m0 x11 h15 y69 ff3 fs6 fc0 sc0 ls0 wsd">INTEGRAL AND T<span class="_0 blank"></span>A<span class="_4 blank"></span>YLOR</div><div class="t m0 x11 h15 y6a ff3 fs6 fc0 sc0 ls0 ws16">POL<span class="_2 blank"></span>YNOMIALS 555</div><div class="t m0 xf h16 y6b ffa fs8 fc0 sc0 ls0 ws11">9.1 <span class="ffb fsa wsf">Arc Length and Surface Area<span class="_9 blank"> </span>555</span></div><div class="t m0 xf h16 y6c ffa fs8 fc0 sc0 ls0 ws11">9.2 <span class="ffb fsa wsf">Fluid Pressure and Force<span class="_45 blank"> </span>564</span></div><div class="t m0 xf h16 y44 ffa fs8 fc0 sc0 ls0 ws11">9.3 <span class="ffb fsa wsf">Center of Mass<span class="_46 blank"> </span>569</span></div><div class="t m0 xf h16 y45 ffa fs8 fc0 sc0 ls0 ws11">9.4 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style="border-style:none;position:absolute;left:336.373000px;bottom:57.679700px;width:276.725000px;height:11.733600px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[584,"Fit"]'><div class="d m1" style="border-style:none;position:absolute;left:335.395000px;bottom:46.923600px;width:288.459000px;height:13.689600px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[600,"Fit"]'><div class="d m1" style="border-style:none;position:absolute;left:352.018000px;bottom:31.278400px;width:251.302000px;height:14.667400px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div> <div id="pf7" class="pf w7 h9" data-page-no="7"><div class="pc pc7 w7 h9"><img fetchpriority="low" loading="lazy" class="bi x0 y7 w7 ha" alt="" src="https://files.passeidireto.com/6bf7eeaa-f34a-4479-8a10-d1e67ea494fa/bg7.png"><div class="t m0 x13 h1a y6f ffa fsc fc0 sc0 ls0 ws17">iv <span class="ff3 fsd lse ws18">CALCULUS <span class="ffb lsf">CONTENTS</span></span></div><div class="t m0 x13 h15 y70 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 10<span class="_6 blank"> </span>INTRODUCTION TO DIFFERENTIAL</div><div class="t m0 x14 h15 y71 ff3 fs6 fc0 sc0 ls0 ws19">EQUA<span class="_4 blank"></span>TIONS 601</div><div class="t m0 x13 h16 y72 ffa fs8 fc0 sc0 ls0 ws1a">10.1 <span class="ffb fsa wsf">Solving Differential Equations<span class="_18 blank"> </span>601</span></div><div class="t m0 x13 h16 y73 ffa fs8 fc0 sc0 ls0 ws1a">10.2 <span class="ffb fsa wsf">Graphical and Numerical Methods<span class="_48 blank"> </span>614</span></div><div class="t m0 x13 h16 y74 ffa fs8 fc0 sc0 ls0 ws1a">10.3 <span class="ffb fsa wsf">The Logistic Equation<span class="_24 blank"> </span>621</span></div><div class="t m0 x13 h16 y75 ffa fs8 fc0 sc0 ls0 ws1a">10.4 <span class="ffb fsa wsf">First-Order Linear Equations<span class="_49 blank"> </span>626</span></div><div class="t m0 x5 h17 y76 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_30 blank"> </span>637</div><div class="t m0 x13 h15 y77 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 11<span class="_6 blank"> </span>INFINITE SERIES<span class="_4a blank"> </span>646</div><div class="t m0 x13 h16 y78 ffa fs8 fc0 sc0 ls0 ws1a">11.1 <span class="ffb fsa ws1b">Sequences 646</span></div><div class="t m0 x13 h16 y79 ffa fs8 fc0 sc0 ls0 ws1a">11.2 <span class="ffb fsa wsf">Summing an In\ufb01nite Series<span class="_4b blank"> </span>658</span></div><div class="t m0 x13 h16 y7a ffa fs8 fc0 sc0 ls0 ws1a">11.3 <span class="ffb fsa wsf">Convergence of Series with Positive T<span class="_4 blank"></span>erms<span class="_4c blank"> </span>669</span></div><div class="t m0 x13 h16 y7b ffa fs8 fc0 sc0 ls0 ws1a">11.4 <span class="ffb fsa wsf">Absolute and Conditional Convergence<span class="_4d blank"> </span>683</span></div><div class="t m0 x15 h16 y7c ffa fs8 fc0 sc0 ls0 ws1a">11.5 <span class="ffb fsa wsf">The Ratio and Root T<span class="_4 blank"></span>ests<span class="_4e blank"> </span>690</span></div><div class="t m0 x15 h16 y7d ffa fs8 fc0 sc0 ls0 ws1a">11.6 <span class="ffb fsa wsf">Power Series<span class="_4f blank"> </span>697</span></div><div class="t m0 x15 h16 y7e ffa fs8 fc0 sc0 ls0 ws1a">11.7 <span class="ffb fsa wsf">T<span class="_4 blank"></span>aylor Series<span class="_50 blank"> </span>710</span></div><div class="t m0 x16 h17 y7f ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_30 blank"> </span>727</div><div class="t m0 x15 h15 y80 ff3 fs6 fc0 sc0 ls0 wsd">Chapter 12<span class="_6 blank"> </span>P<span class="_0 blank"></span>ARAMETRIC EQUA<span class="_4 blank"></span>TIONS, POLAR</div><div class="t m0 x17 h15 y81 ff3 fs6 fc0 sc0 ls0 wsd">COORDINA<span class="_4 blank"></span>TES, AND CONIC</div><div class="t m0 x17 h15 y82 ff3 fs6 fc0 sc0 ls0 ws1c">SECTIONS 742</div><div class="t m0 x15 h16 y83 ffa fs8 fc0 sc0 ls0 ws1a">12.1 <span class="ffb fsa wsf">Parametric Equations<span class="_51 blank"> </span>742</span></div><div class="t m0 x15 h16 y84 ffa fs8 fc0 sc0 ls0 ws1a">12.2 <span class="ffb fsa wsf">Arc Length and Speed<span class="_32 blank"> </span>759</span></div><div class="t m0 x15 h16 y85 ffa fs8 fc0 sc0 ls0 ws1a">12.3 <span class="ffb fsa wsf">Polar Coordinates<span class="_52 blank"> </span>766</span></div><div class="t m0 x15 h16 y86 ffa fs8 fc0 sc0 ls0 ws1a">12.4 <span class="ffb fsa wsf">Area and Arc Length in Polar Coordinates<span class="_53 blank"> </span>780</span></div><div class="t m0 x15 h16 y7a ffa fs8 fc0 sc0 ls0 ws1a">12.5 <span class="ffb fsa wsf">Conic Sections<span class="_46 blank"> </span>789</span></div><div class="t m0 x16 h17 y87 ffb fsa fc0 sc0 ls0 wsf">Chapter Review Exercises<span class="_30 blank"> </span>801</div><div class="c xb y1 w2 h2"><div class="t m0 x0 h3 y2 ff1 fs0 fc0 sc0 ls0">www.elsolucionario.net</div></div><a class="l" 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style="border-style:none;position:absolute;left:328.550000px;bottom:558.327300px;width:258.147000px;height:9.779000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[808,"Fit"]'><div class="d m1" style="border-style:none;position:absolute;left:342.240000px;bottom:543.660300px;width:265.969000px;height:14.667000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div> <div id="pf8" class="pf w7 h9" data-page-no="8"><div class="pc pc8 w7 h9"><img fetchpriority="low" loading="lazy" class="bi x0 y7 w7 ha" alt="" src="https://files.passeidireto.com/6bf7eeaa-f34a-4479-8a10-d1e67ea494fa/bg8.png"><div class="t m0 x18 h1b y88 ffe fsa fc2 sc0 ls0 ws1d">June 7, 2011<span class="_54 blank"> </span><span class="fs2 ws1e">L<span class="_35 blank"></span>TSV SSM Second Pass</span></div><div class="t m0 x19 h1c y89 fff fse fc0 sc0 ls10">1<span class="ff10 fs9 ls0 ws1f">PRECAL<span class="_2 blank"></span>CUL<span class="_4 blank"></span>US REVIEW</span></div><div class="t m0 x19 h1d y8a ff11 fsf fc0 sc0 ls0 ws20">1.1 <span class="ff12 ws21">Real Numbers, Functions, and Graphs</span></div><div class="t m0 x19 h1e y8b ff13 fs6 fc0 sc0 ls0 wsd">Preliminary Questions</div><div class="t m0 x1a h1f y8c ff14 fsa fc0 sc0 ls0 ws22">1. <span class="ff15 ws0">Give an example of numbers <span class="ff16 ls11">a</span>and <span class="ff16 ls12">b</span>such that <span class="ff16 ls4a ws23">a<<span class="_0 blank"></span>b<span class="_4 blank"></span><span class="ff15 ls0 ws0">and <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls4">|<span class="ff16">></span></span>|<span class="ff16 lsd">b</span>|</span>.</span></span></span></div><div class="t m0 x19 h20 y8d ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">T<span class="_4 blank"></span>ake <span class="ff16 ls14">a<span class="ff17 ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff15 ls0 ws0">3 and <span class="ff16 ls15">b<span class="ff17 ls4">=</span></span><span class="ws26">1. Then <span class="ff16 ls4a ws23">a<b<span class="_4 blank"></span><span class="ff15 ls0 ws0">but <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls18">|=</span></span><span class="ls16">3<span class="ff16 ls4">><span class="ff15">1<span class="ff17 ls18 ws25">=|<span class="_55 blank"></span><span class="ff16 lsd">b<span class="ff17 ls0">|<span class="ff15">.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></div><div class="t m0 x1a h1f y8e ff14 fsa fc0 sc0 ls0 ws27">2. <span class="ff15 ws0">Which numbers satisfy <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls18">|=<span class="ff16 ls17">a</span></span></span>? Which satisfy <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls18 ws25">|=\u2212<span class="_55 blank"></span><span class="ff16 ls13">a<span class="ff15 ls0 ws0">?<span class="_3 blank"> </span>What about <span class="ff17">|\u2212</span></span>a<span class="ff17 ls18">|=</span><span class="ls17">a<span class="ff15 ls0">?</span></span></span></span></span></span></div><div class="t m0 x19 h20 y8f ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">The numbers <span class="ff16 ls14">a<span class="ff17 ls4">\u2265</span></span>0 satisfy <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls18">|=<span class="ff16 ls19">a</span></span></span>and <span class="ff17 ls1c">|\u2212<span class="ff16 ls1a">a</span><span class="ls18">|=<span class="ff16 ls17">a</span></span></span>. The numbers <span class="ff16 ls14">a<span class="ff17 ls4">\u2264</span></span>0 satisfy <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls18 ws25">|=\u2212<span class="_55 blank"></span><span class="ff16 ls13">a<span class="ff15 ls0">.</span></span></span></span></span></div><div class="t m0 x1a h1f y90 ff14 fsa fc0 sc0 ls0 ws22">3. <span class="ff15 ws0">Give an example of numbers <span class="ff16 ls11">a</span>and <span class="ff16 ls12">b</span>such that <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span><span class="ls4">|<span class="ff16"><</span></span></span>|<span class="ff16 ls13">a</span><span class="ls1c ws28">|+|<span class="_56 blank"></span><span class="ff16 lsd">b<span class="ff17 ls0">|<span class="ff15">.</span></span></span></span></span></span></div><div class="t m0 x19 h20 y91 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">T<span class="_4 blank"></span>ake <span class="ff16 ls14">a<span class="ff17 ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff15 ls0 ws0">3 and <span class="ff16 ls15">b<span class="ff17 ls1d">=</span></span><span class="ws26">1. Then</span></span></span></span></span></div><div class="t m0 x1b h21 y92 ff17 fsa fc0 sc0 ls0">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span><span class="ls18">|=|</span></span></div><div class="t m0 x3 h20 y93 ff17 fsa fc0 sc0 lsa">\u2212<span class="ff15">3</span>+<span class="ff15 ls0">1</span><span class="ls18 ws25">|=|<span class="_4 blank"></span>\u2212<span class="_0 blank"></span><span class="ff15 ls0">2<span class="ff17 ls18">|=</span>2<span class="ff16 ls1e">,</span><span class="ws29">but <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls1c ws28">|+|<span class="_56 blank"></span><span class="ff16 lsd">b<span class="ff17 ls18 ws25">|=|<span class="_4 blank"></span>\u2212<span class="_0 blank"></span><span class="ff15 ls0">3<span class="ff17 ls1c ws28">|+|<span class="_56 blank"></span><span class="ff15 ls0">1<span class="ff17 ls18">|=</span><span class="ls1f">3<span class="ff17 lsa">+</span><span class="ls4">1<span class="ff17">=</span></span></span>4<span class="ff16">.</span></span></span></span></span></span></span></span></span></span></span></div><div class="t m0 x19 h20 y94 ff15 fsa fc0 sc0 ls0 ws0">Thus, <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span><span class="ls4">|<span class="ff16"><</span></span></span>|<span class="ff16 ls13">a</span><span class="ls1c ws28">|+|<span class="_56 blank"></span><span class="ff16 lsd">b<span class="ff17 ls0">|<span class="ff15">.</span></span></span></span></span></div><div class="t m0 x1a h1f y95 ff14 fsa fc0 sc0 ls0 ws27">4. <span class="ff15 ws0">What are the coordinates of the point lying at the intersection of the lines <span class="ff16 ls20">x<span class="ff17 ls4">=</span></span>9 and <span class="ff16 ls20">y<span class="ff17 ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff15 ls0">4?</span></span></span></span></div><div class="t m0 x19 h20 y96 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">The point <span class="ff16">(</span>9<span class="ff16 ls21">,</span><span class="ff17">\u2212</span>4<span class="ff16 ls22">)</span>lies at the intersection of the lines <span class="ff16 ls20">x<span class="ff17 ls4">=</span></span>9 and <span class="ff16 ls20">y<span class="ff17 ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff15 ls0">4.</span></span></span></span></div><div class="t m0 x1a h1f y97 ff14 fsa fc0 sc0 ls0 ws22">5. <span class="ff15 ws0">In which quadrant do the following points lie?</span></div><div class="t m0 x19 h1f y98 ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff16">(<span class="ff15">1</span><span class="ls21">,</span><span class="ff15">4</span><span class="ls23">)</span></span>(b) <span class="ff16">(<span class="ff17">\u2212<span class="ff15">3</span></span><span class="ls21">,</span><span class="ff15">2</span><span class="ls24">)</span></span>(c) <span class="ff16">(<span class="ff15">4</span><span class="ls21">,</span><span class="ff17">\u2212<span class="ff15">3</span></span><span class="ls25">)</span></span>(d) <span class="ff16">(<span class="ff17">\u2212<span class="ff15">4</span></span><span class="ls21">,</span><span class="ff17">\u2212<span class="ff15">1</span></span>)</span></div><div class="t m0 x19 h22 y99 ff18 fsa fc0 sc0 ls0">solution</div><div class="t m0 x19 h1f y9a ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff15 ws0">Because both the <span class="ff16 ls26">x</span><span class="ws2a">- and <span class="ff16 ls26">y</span><span class="ws2b">-coordinates </span></span><span class="v0">of the point <span class="ff16">(</span>1<span class="ff16 ls21">,</span>4<span class="ff16 ls22">)</span>are positive, the point <span class="ff16">(</span>1<span class="ff16 ls21">,</span>4<span class="ff16 ls22">)</span>lies in the \ufb01rst quadrant.</span></span></div><div class="t m0 x19 h1f y9b ff14 fsa fc0 sc0 ls0 ws22">(b) <span class="ff15 ws2c">Because the <span class="ff16 ls26">x</span>-coordinate of the point <span class="ff16">(<span class="ff17">\u2212</span></span>3<span class="ff16 ls21">,</span>2<span class="ff16 ls27">)</span>is negative but the <span class="ff16 ls26">y</span>-coordinate is positive, the point <span class="ff16">(<span class="ff17">\u2212</span></span>3<span class="ff16 ls21">,</span>2<span class="ff16 ls27">)</span>lies in</span></div><div class="t m0 x19 h20 y9c ff15 fsa fc0 sc0 ls0 ws0">the second quadrant.</div><div class="t m0 x1c h1f y9d ff14 fsa fc0 sc0 ls0 ws22">(c) <span class="ff15 ws2c">Because the <span class="ff16 ls26">x</span>-coordinate of the point <span class="ff16">(</span>4<span class="ff16 ls21">,</span><span class="ff17">\u2212</span>3<span class="ff16 ls27">)</span>is positive but the <span class="ff16 ls26">y</span>-coordinate is negative, the point <span class="ff16">(</span>4<span class="ff16 ls21">,</span><span class="ff17">\u2212</span>3<span class="ff16 ls27">)</span>lies in</span></div><div class="t m0 x19 h20 y9e ff15 fsa fc0 sc0 ls0 ws0">the fourth quadrant.</div><div class="t m0 x19 h1f y9f ff14 fsa fc0 sc0 ls0 ws22">(d) <span class="ff15 ws2d">Because both the <span class="ff16 ls26">x</span><span class="ws2e">- and <span class="ff16 ls26">y</span></span>-coordinates of the point <span class="ff16">(<span class="ff17">\u2212</span></span>4<span class="ff16 ls21">,</span><span class="ff17">\u2212</span>1<span class="ff16 lsb">)</span>are negative, the point <span class="ff16">(<span class="ff17">\u2212</span></span>4<span class="ff16 ls21">,</span><span class="ff17">\u2212</span>1<span class="ff16 lsb">)</span>lies in the third quadrant.</span></div><div class="t m0 x1a h23 ya0 ff14 fsa fc0 sc0 ls0 ws27">6. <span class="ff15 ws0">What is the radius of the circle with equation <span class="ff16 ws2f">(x <span class="ff17 lsa">\u2212</span></span>9<span class="ff16 ws30">)</span><span class="fsb ls28 v1">2</span><span class="ff17 lsa v0">+<span class="ff16 ls0 ws2f">(y </span>\u2212</span><span class="v0">9<span class="ff16 ls29">)</span><span class="fsb ls2a v1">2</span><span class="ff17 ls4">=</span>9?</span></span></div><div class="t m0 x19 h24 ya1 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">The circle with equation <span class="ff16 ws2f">(x <span class="ff17 lsa">\u2212</span></span>9<span class="ff16 ws30">)</span><span class="fsb ls2b v2">2</span><span class="ff17 lsa v0">+<span class="ff16 ls0 ws2f">(y </span>\u2212</span><span class="v0">9<span class="ff16 ls2c">)</span><span class="fsb ls2a v2">2</span><span class="ff17 ls4">=</span>9 has radius 3.</span></span></div><div class="t m0 x1a h1f ya2 ff14 fsa fc0 sc0 ls0 ws27">7. <span class="ff15 ws0">The equation <span class="ff16 lsb ws31">f(<span class="_35 blank"></span>x<span class="_2 blank"></span>) <span class="ff17 ls4">=<span class="ff15 ls0 ws0">5 has a solution if (choose one):</span></span></span></span></div><div class="t m0 x19 h1f ya3 ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff15 ws0">5 belongs to the domain of <span class="ff16 lsb">f</span>.</span></div><div class="t m0 x19 h1f ya4 ff14 fsa fc0 sc0 ls0 ws22">(b) <span class="ff15 ws0">5 belongs to the range of <span class="ff16 lsb">f</span>.</span></div><div class="t m0 x19 h1f ya5 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">The correct response is <span class="ff14">(b)</span>: the equation <span class="ff16 lsb ws31">f(<span class="_35 blank"></span>x<span class="_2 blank"></span>) <span class="ff17 ls4">=<span class="ff15 ls0 ws0">5 has a solution if 5 belongs to the range of </span></span>f<span class="ff15 ls0">.</span></span></span></div><div class="t m0 x1a h1f ya6 ff14 fsa fc0 sc0 ls0 ws27">8. <span class="ff15 ws0">What kind of symmetry does the graph have if <span class="ff16 lsb ws32">f(<span class="_35 blank"></span><span class="ff17 ls0">\u2212<span class="ff16 ls26 ws33">x) </span><span class="ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff16 lsb ws32">f(<span class="_35 blank"></span>x<span class="_2 blank"></span>)<span class="_35 blank"></span><span class="ff15 ls0">?</span></span></span></span></span></span></div><div class="t m0 x19 h20 ya7 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff15 ws0">If <span class="ff16 lsb ws32">f(<span class="_35 blank"></span><span class="ff17 ls0">\u2212<span class="ff16 ls26 ws33">x) </span><span class="ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff16 lsb ws32">f(<span class="_35 blank"></span>x<span class="_2 blank"></span>)<span class="_35 blank"></span><span class="ff15 ls0 ws0">, then the graph of <span class="ff16 ls2d">f</span>is symmetric with respect to the origin.</span></span></span></span></span></span></div><div class="t m0 x19 h1e ya8 ff13 fs6 fc0 sc0 ls0">Exercises</div><div class="t m0 x1a h25 ya9 ff14 fsa fc0 sc0 ls0 ws22">1. <span class="ff15 ws0">Use a calculator to \ufb01nd a rational number <span class="ff16 ls2e">r</span>such that <span class="ff17">|<span class="ff16 ls2f">r</span><span class="lsa">\u2212<span class="ff16 ls30">\u03c0</span></span></span><span class="fsb ls31 v1">2</span><span class="ff17 ls4">|<span class="ff16"><</span></span><span class="ws35">10<span class="ff17 fsb v1">\u2212<span class="ff15 ls32">4</span></span>.</span></span></div><div class="t m0 x19 h26 yaa ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff16 ls33">r</span><span class="ff15 ws36">must satisfy <span class="ff16 ls34">\u03c0</span><span class="fsb ls35 v1">2</span><span class="ff17 ls36 v0">\u2212</span><span class="ws35 v0">10<span class="ff17 fsb v1">\u2212<span class="ff15 ls37">4</span></span><span class="ff16 ls38 ws37"><r <\u03c0</span></span></span></div><div class="t m0 x1d h27 yab ff15 fsb fc0 sc0 ls35">2<span class="ff17 fsa ls36 v3">+<span class="ff15 ls0 ws35">10</span></span><span class="ff17 ls0">\u2212</span><span class="ls32">4<span class="fsa ls4b ws38 v3">,o<span class="_55 blank"></span>r9<span class="_55 blank"></span><span class="ff16 ls0">.<span class="ff15 ws39">869504 </span><span class="ls38 ws37"><r <</span><span class="ff15">9</span>.<span class="ff15 ws3a">869705. </span><span class="ls39">r<span class="ff17 ls3a">=</span></span><span class="ff15">9</span>.<span class="ff15 ws39">8696 <span class="ff17 ls3b">=</span><span class="fsb v2">12337</span></span></span></span></span></div><div class="t m0 x1e h28 yac ff15 fsb fc0 sc0 ls0 ws3b">1250 <span class="fsa v2">would</span></div><div class="t m0 x19 h20 yad ff15 fsa fc0 sc0 ls0 ws0">be one such number<span class="_4 blank"></span>.</div><div class="t m0 x1f h20 yae ff15 fsa fc3 sc0 ls0 ws0">Which of (a)\u2013(f) are true for <span class="ff16 ls14">a<span class="ff17 ls18 ws25">=\u2212<span class="_55 blank"></span><span class="ff15 ls0 ws0">3 and <span class="ff16 ls15">b<span class="ff17 ls4">=</span></span>2?</span></span></span></div><div class="t m0 x20 h1f yaf ff14 fsa fc3 sc0 ls0 ws22">(a) <span class="ff16 ls4a ws3c">a<<span class="_0 blank"></span>b <span class="ff14 ls0 ws22">(b) <span class="ff17">|</span></span><span class="ls13">a<span class="ff17 ls4">|<span class="ff16"><</span><span class="ls0">|</span></span><span class="lsd">b<span class="ff17 ls3c">|<span class="ff14 ls0 ws22">(c) </span></span></span><span class="ws3d">ab > <span class="ff15 ls0">0</span></span></span></span></div><div class="t m0 x20 h29 yb0 ff14 fsa fc3 sc0 ls0 ws22">(d) <span class="ff15">3<span class="ff16 ls4a ws23">a<<span class="_0 blank"></span><span class="ff15 ls0">3<span class="ff16 ls3d">b</span><span class="ff14 ws22">(e) <span class="ff17">\u2212</span></span>4<span class="ff16 ls4a">a<<span class="_4 blank"></span><span class="ff17 ls0">\u2212<span class="ff15">4<span class="ff16 ls3e">b</span><span class="ff14 ws3e">(f) </span><span class="v4">1</span></span></span></span></span></span></span></div><div class="t m0 x21 h2a yb1 ff16 fsa fc3 sc0 ls3f">a<span class="ls40 v4"><</span><span class="ff15 ls0 v5">1</span></div><div class="t m0 x22 h2b yb1 ff16 fsa fc3 sc0 ls0">b</div><div class="t m0 x19 h2c yb2 ff19 fsa fc0 sc0 ls0 ws0">In Exer<span class="_0 blank"></span>cises 3\u20138, expr<span class="_0 blank"></span>ess the interval in terms of an inequality involving absolute value.</div><div class="t m0 x1a h1f yb3 ff14 fsa fc0 sc0 ls0 ws22">3. <span class="ff17">[\u2212<span class="ff15">2<span class="ff16 ls21">,</span>2</span>]</span></div><div class="t m0 x19 h20 yb4 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff17">|<span class="ff16 ls26">x</span><span class="ls18">|\u2264</span><span class="ff15">2</span></span></div><div class="t m0 x1f h20 yb5 ff16 fsa fc3 sc0 ls0">(<span class="ff17">\u2212<span class="ff15">4</span></span><span class="ls21">,</span><span class="ff15">4</span>)</div><div class="t m0 x1a h1f yb6 ff14 fsa fc0 sc0 ls0 ws22">5. <span class="ff16">(<span class="ff15">0</span><span class="ls21">,</span><span class="ff15">4</span>)</span></div><div class="t m0 x19 h20 yb7 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws3f">The midpoint of the interval is <span class="ff16 ls41">c<span class="ff17 ls42">=</span><span class="ls0">(</span></span><span class="ls43">0<span class="ff17">+</span></span>4<span class="ff16">)/</span><span class="ls42">2<span class="ff17">=</span></span>2, and the radius is <span class="ff16 ls44">r<span class="ff17 ls42">=</span><span class="ls0">(</span></span><span class="ls43">4<span class="ff17">\u2212</span></span>0<span class="ff16">)/</span><span class="ls42">2<span class="ff17">=</span></span>2; therefore, <span class="ff16">(</span>0<span class="ff16 ls21">,</span>4<span class="ff16">)</span></span></div><div class="t m0 x19 h20 yb8 ff15 fsa fc0 sc0 ls0 ws0">can be expressed as <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span></span>2<span class="ff17 ls4">|<span class="ff16"><</span></span>2.</div><div class="t m0 x1f h20 yb9 ff17 fsa fc3 sc0 ls0">[\u2212<span class="ff15">4<span class="ff16 ls21">,</span>0</span>]</div><div class="t m0 x1a h1f yba ff14 fsa fc0 sc0 ls0 ws22">7. <span class="ff17">[<span class="ff15">1<span class="ff16 ls21">,</span>5</span>]</span></div><div class="t m0 x19 h20 ybb ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws40">The midpoint of the interval is <span class="ff16 ls44">c<span class="ff17 ls46">=</span><span class="ls0">(</span></span><span class="ls47">1<span class="ff17">+</span></span>5<span class="ff16">)/</span><span class="ls46">2<span class="ff17">=</span></span>3, and the radius is <span class="ff16 ls48">r<span class="ff17 ls46">=</span><span class="ls0">(</span></span><span class="ls47">5<span class="ff17">\u2212</span></span>1<span class="ff16">)/</span><span class="ls46">2<span class="ff17">=</span></span>2; therefore, the</span></div><div class="t m0 x19 h20 ybc ff15 fsa fc0 sc0 ls0 ws0">interval <span class="ff17">[</span>1<span class="ff16 ls21">,</span>5<span class="ff17 ls22">]</span>can be expressed as <span class="ff17 ws30">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span></span>3<span class="ff17 ls18">|\u2264</span>2.</div><div class="t m0 x1f h20 ybd ff16 fsa fc3 sc0 ls0">(<span class="ff17">\u2212<span class="ff15">2</span></span><span class="ls21">,</span><span class="ff15">8</span><span class="ls49">)</span><span class="ff12 fsc fc0 v6">1</span></div><div class="c xb y1 w2 h2"><div class="t m0 x0 h3 y2 ff1 fs0 fc0 sc0 ls0">www.elsolucionario.net</div></div><a class="l" data-dest-detail='[6,"Fit"]'><div class="d m1" style="border-style:none;position:absolute;left:148.630000px;bottom:693.268300px;width:379.397000px;height:42.046000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[6,"Fit"]'><div class="d m1" style="border-style:none;position:absolute;left:144.719000px;bottom:647.310300px;width:251.301000px;height:17.601000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div> <div id="pf9" class="pf w7 h9" data-page-no="9"><div class="pc pc9 w7 h9"><img fetchpriority="low" loading="lazy" class="bi x0 y7 w7 ha" alt="" src="https://files.passeidireto.com/6bf7eeaa-f34a-4479-8a10-d1e67ea494fa/bg9.png"><div class="t m0 x23 h1b y88 ffe fsa fc2 sc0 ls0 ws1d">June 7, 2011<span class="_54 blank"> </span><span class="fs2 ws1e">L<span class="_35 blank"></span>TSV SSM Second Pass</span></div><div class="t m0 x13 h1a y6f ff12 fsc fc0 sc0 ls4c">2<span class="ff1a fsd lse ws41">CHAPTER 1<span class="_57 blank"> </span></span><span class="ff1b ls0 ws42">PRECALCULUS REVIEW</span></div><div class="t m0 xc h2c y7c ff19 fsa fc0 sc0 ls0 ws0">In Exer<span class="_0 blank"></span>cises 9\u201312, write the inequality in the form <span class="ff16 ls4a ws43">a<<span class="_0 blank"></span>x<<span class="_0 blank"></span>b<span class="_58 blank"></span><span class="ff19 ls0">.</span></span></div><div class="t m0 x24 h1f ybe ff14 fsa fc0 sc0 ls0 ws22">9. <span class="ff17">|<span class="ff16 ls26">x</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">8</span></span></div><div class="t m0 xc h20 ybf ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff17">\u2212<span class="ff15 ls4">8<span class="ff16 ls18 ws44"><x <</span><span class="ls0">8</span></span></span></div><div class="t m0 x20 h20 yc0 ff17 fsa fc3 sc0 ls0">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span><span class="ff15">12</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">8</span></div><div class="t m0 xc h1f yc1 ff14 fsa fc0 sc0 ls8c ws45">11 .<span class="_31 blank"> </span><span class="ff17 ls0">|<span class="ff15">2<span class="ff16 ls45">x</span></span><span class="lsa">+</span><span class="ff15">1</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">5</span></span></div><div class="t m0 xc h20 yc2 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff17">\u2212<span class="ff15 ls4">5<span class="ff16"><</span><span class="ls0">2<span class="ff16 ls45">x</span></span></span><span class="lsa">+<span class="ff15 ls4">1<span class="ff16"><</span><span class="ls22 ws46">5s<span class="_55 blank"></span>o<span class="ff17 ls0">\u2212</span><span class="ls4">6<span class="ff16"><</span><span class="ls0">2<span class="ff16 ls8d ws47">x<<span class="_0 blank"></span><span class="ff15 ls0 ws0">4 and <span class="ff17">\u2212</span><span class="ls4">3<span class="ff16 ls18 ws44"><x <</span></span>2</span></span></span></span></span></span></span></span></div><div class="t m0 x20 h20 yc3 ff17 fsa fc3 sc0 ls0">|<span class="ff15">3<span class="ff16 ls45">x</span></span><span class="lsa">\u2212</span><span class="ff15">4</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">2</span></div><div class="t m0 xc h2c yc4 ff19 fsa fc0 sc0 ls0 ws0">In Exer<span class="_0 blank"></span>cises 13\u201318, expr<span class="_0 blank"></span>ess the set of numbers <span class="ff16 ls4d">x</span>satisfying the given condition as an interval.</div><div class="t m0 xc h1f yc5 ff14 fsa fc0 sc0 ls0 ws22">13. <span class="ff17">|<span class="ff16 ls26">x</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">4</span></span></div><div class="t m0 xc h20 yc6 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff16">(<span class="ff17">\u2212<span class="ff15">4</span></span><span class="ls21">,</span><span class="ff15">4</span>)</span></div><div class="t m0 x20 h20 yc7 ff17 fsa fc3 sc0 ls0">|<span class="ff16 ls26">x</span><span class="ls18">|\u2264</span><span class="ff15">9</span></div><div class="t m0 xc h1f yc8 ff14 fsa fc0 sc0 ls0 ws22">15. <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span><span class="ff15">4</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">2</span></span></div><div class="t m0 xc h20 yc9 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws48">The expression <span class="ff17">|<span class="ff16 ls4e">x</span><span class="ls4f">\u2212</span></span>4<span class="ff17 ls50">|<span class="ff16"><</span></span><span class="ws49">2 is equivalent to <span class="ff17">\u2212</span><span class="ls50">2<span class="ff16 ls8e ws4a"><x<span class="ff17 ls4f">\u2212</span></span>4<span class="ff16 ls51"><</span></span><span class="ws4b">2. Therefore, 2<span class="_59 blank"> </span><span class="ff16 ls8e ws4c"><x <span class="v0"><</span></span></span><span class="v0">6, which represents the</span></span></span></div><div class="t m0 xc h20 yca ff15 fsa fc0 sc0 ls0 ws0">interval <span class="ff16">(</span>2<span class="ff16 ls21">,</span>6<span class="ff16">)</span>.</div><div class="t m0 x20 h20 ycb ff17 fsa fc3 sc0 ls0">|<span class="ff16 ls45">x</span><span class="lsa">+</span><span class="ff15">7</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">2</span></div><div class="t m0 xc h1f ycc ff14 fsa fc0 sc0 ls0 ws22">17. <span class="ff17">|<span class="ff15">4<span class="ff16 ls45">x</span></span><span class="lsa">\u2212</span><span class="ff15">1</span><span class="ls18">|\u2264</span><span class="ff15">8</span></span></div><div class="t m0 xc h2d ycd ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws4d">The expression <span class="ff17">|</span>4<span class="ff16 ls52">x<span class="ff17 ls53">\u2212</span></span>1<span class="ff17 ls8f">|\u2264</span><span class="ws4e">8 is equivalent to <span class="ff17">\u2212</span><span class="ls54">8<span class="ff17">\u2264</span></span>4<span class="ff16 ls52">x<span class="ff17 ls53">\u2212</span></span><span class="ls54">1<span class="ff17">\u2264</span><span class="ls55 ws4f">8o<span class="_55 blank"></span>r<span class="ff17 ls0">\u2212</span><span class="ls56">7<span class="ff17 ls54">\u2264</span><span class="ls0">4<span class="ff16 ls57">x<span class="ff17 ls54">\u2264</span></span><span class="ws50">9. Therefore,<span class="_5a blank"> </span><span class="ff17 ls58">\u2212</span></span><span class="fsb v2">7</span></span></span></span></span></span></span></div><div class="t m0 x25 h2e yce ff15 fsb fc0 sc0 ls59">4<span class="ff17 fsa ls54 v2">\u2264<span class="ff16 ls57">x<span class="ff17 ls5a">\u2264</span></span></span><span class="ls0 v7">9</span></div><div class="t m0 x26 h2f yce ff15 fsb fc0 sc0 ls5b">4<span class="fsa ls0 v2">,</span></div><div class="t m0 xc h30 ycf ff15 fsa fc0 sc0 ls0 ws0">which represents the interval <span class="ff17 ws51">[\u2212 </span><span class="fsb v2">7</span></div><div class="t m0 x27 h2e yd0 ff15 fsb fc0 sc0 ls5b">4<span class="ff16 fsa ls5c v2">,</span><span class="ls0 v7">9</span></div><div class="t m0 x28 h2f yd0 ff15 fsb fc0 sc0 ls5b">4<span class="ff17 fsa ls0 v2">]<span class="ff15">.</span></span></div><div class="t m0 x20 h20 yd1 ff17 fsa fc3 sc0 ls0">|<span class="ff15">3<span class="ff16 ls45">x</span></span><span class="lsa">+</span><span class="ff15">5</span><span class="ls4">|<span class="ff16"><</span></span><span class="ff15">1</span></div><div class="t m0 xc h2c yd2 ff19 fsa fc0 sc0 ls0 ws0">In Exer<span class="_0 blank"></span>cises 19\u201322, describe the set as a union of \ufb01nite or in\ufb01nite intervals.</div><div class="t m0 xc h1f yd3 ff14 fsa fc0 sc0 ls0 ws22">19. <span class="ff17">{<span class="ff16 ls20">x</span><span class="ls18 ws25">:|<span class="_55 blank"></span><span class="ff16 ls45">x<span class="ff17 lsa">\u2212<span class="ff15 ls0">4</span><span class="ls4">|<span class="ff16">><span class="ff15 ls0">2<span class="ff17">}</span></span></span></span></span></span></span></span></div><div class="t m0 xc h20 yd4 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff16 ls45">x<span class="ff17 lsa">\u2212<span class="ff15 ls4">4<span class="ff16">></span><span class="ls22 ws46">2o<span class="_55 blank"></span>r<span class="ff16 ls45">x<span class="ff17 lsa">\u2212</span></span><span class="ls4">4<span class="ff16"><<span class="ff17 ls0">\u2212</span></span>2<span class="ff17">\u21d2<span class="ff16 ls8d ws47">x><span class="_0 blank"></span><span class="ff15 ls22 ws46">6o<span class="_55 blank"></span>r<span class="ff16 ls8d ws47">x<<span class="_0 blank"></span><span class="ff15 ls4">2<span class="ff17">\u21d2<span class="ff16 ls0">(<span class="ff17">\u2212\u221e</span><span class="ls21">,</span><span class="ff15">2</span><span class="lsa">)<span class="ff17">\u222a</span></span>(<span class="ff15">6</span><span class="ls21">,</span><span class="ff17">\u221e</span>)</span></span></span></span></span></span></span></span></span></span></span></span></div><div class="t m0 x20 h20 yd5 ff17 fsa fc3 sc0 ls0">{<span class="ff16 ls20">x</span><span class="ls18 ws25">:|<span class="_55 blank"></span><span class="ff15 ls0">2<span class="ff16 ls45">x<span class="ff17 lsa">+</span></span>4<span class="ff17 ls4">|<span class="ff16">></span></span>3<span class="ff17">}</span></span></span></div><div class="t m0 xc h31 yd6 ff14 fsa fc0 sc0 ls0 ws22">21. <span class="ff17">{<span class="ff16 ls20">x</span><span class="ls18 ws25">:|<span class="_55 blank"></span><span class="ff16 ls5d">x<span class="ff15 fsb ls2b v1">2</span><span class="ff17 lsa v0">\u2212<span class="ff15 ls0">1</span><span class="ls4">|<span class="ff16">><span class="ff15 ls0">2<span class="ff17">}</span></span></span></span></span></span></span></span></div><div class="t m0 xc h32 yd7 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff16 ls5e">x<span class="ff15 fsb ls5f v1">2</span><span class="ff17 ls60 v0">\u2212<span class="ff15 ls61">1<span class="ff16 ls62">></span><span class="ls63 ws52">2o<span class="_55 blank"></span>r<span class="ff16 ls64">x</span><span class="fsb ls5f v1">2</span><span class="ff17 ls60">\u2212</span><span class="ls61">1<span class="ff16 ls62"><<span class="ff17 ls0">\u2212</span><span class="ff15">2<span class="ff17">\u21d2</span></span><span class="ls65">x</span></span><span class="fsb ls66 v1">2</span><span class="ff16 ls62">></span></span>3o<span class="_55 blank"></span>r<span class="ff16 ls67">x</span><span class="fsb ls68 v1">2</span><span class="ff16 ls62"><<span class="ff17 ls0">\u2212<span class="ff15 ws49">1 (this will never happen) </span><span class="ls62">\u21d2</span></span><span class="ls90">x></span></span></span></span></span></span></div><div class="t m0 x29 h21 yd8 ff17 fsa fc0 sc0 ls69">\u221a<span class="ff15 ls63 ws52 v8">3o<span class="_55 blank"></span>r<span class="ff16 ls90 ws53">x<<span class="_4 blank"></span><span class="ff17 ls0 ws30">\u2212<span class="ls6a v9">\u221a</span><span class="ff15 ls61">3</span>\u21d2</span></span></span></div><div class="t m0 xc h33 yd9 ff16 fsa fc0 sc0 ls0">(<span class="ff17">\u2212\u221e</span><span class="ls21">,</span><span class="ff17 ws30">\u2212<span class="v9">\u221a</span><span class="ff15 v0">3<span class="ff16 lsa">)<span class="ff17">\u222a</span><span class="ls0">(</span></span></span><span class="ls6b v9">\u221a</span><span class="ff15 v0">3<span class="ff16 ls21">,</span><span class="ff17">\u221e<span class="ff16">)</span></span>.</span></span></div><div class="t m0 x20 h34 yda ff17 fsa fc3 sc0 ls0">{<span class="ff16 ls20">x</span><span class="ls18 ws25">:|<span class="_55 blank"></span><span class="ff16 ls6c">x<span class="ff15 fsb ls28 v1">2</span><span class="ff17 lsa v0">+<span class="ff15 ls0">2<span class="ff16 ls26">x</span></span><span class="ls4">|<span class="ff16">><span class="ff15 ls0">2<span class="ff17">}</span></span></span></span></span></span></span></div><div class="t m0 xc h1f ydb ff14 fsa fc0 sc0 ls0 ws22">23. <span class="ff15 ws0">Match (a)\u2013(f) with (i)\u2013(vi).</span></div><div class="t m0 xc h29 ydc ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff16 ls4a ws23">a><span class="_0 blank"></span><span class="ff15 ls6d">3<span class="ff14 ls0 ws22">(b) <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">\u2212</span><span class="ff15">5</span><span class="ls4">|<span class="ff16 ls6e"><</span></span><span class="ff15 v4">1</span></span></span></span></span></div><div class="t m0 x2a h20 ydd ff15 fsa fc0 sc0 ls0">3</div><div class="t m0 x2b h35 yde ff14 fsa fc0 sc0 ls0 ws22">(c) <span class="ff1c va">\ue002</span></div><div class="t m0 x1a h36 ydf ff1c fsa fc0 sc0 ls0">\ue002</div><div class="t m0 x1a h36 ye0 ff1c fsa fc0 sc0 ls0">\ue002</div><div class="t m0 x1a h37 ye1 ff1c fsa fc0 sc0 ls0 ws2a">\ue002<span class="ff16 ls1b v1">a<span class="ff17 ls41">\u2212</span></span><span class="ff15 vb">1</span></div><div class="t m0 x2c h38 ye2 ff15 fsa fc0 sc0 ls6f">3<span class="ff1c ls0 vc">\ue002</span></div><div class="t m0 x2d h36 ye3 ff1c fsa fc0 sc0 ls0">\ue002</div><div class="t m0 x2d h36 ye4 ff1c fsa fc0 sc0 ls0">\ue002</div><div class="t m0 x2d h39 ye5 ff1c fsa fc0 sc0 ls4">\ue002<span class="ff16 v1"><<span class="ff15 ls70">5<span class="ff14 ls0 ws22">(d) <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls4">|</span></span></span></span>><span class="ff15 ls0">5</span></span></div><div class="t m0 x2b h1f ye6 ff14 fsa fc0 sc0 ls0 ws22">(e) <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">\u2212</span><span class="ff15">4</span><span class="ls4">|<span class="ff16"><<span class="ff15 ls71">3</span></span></span></span>(f) <span class="ff15 ls4">1<span class="ff17">\u2264<span class="ff16 ls14">a</span>\u2264</span><span class="ls0">5</span></span></div><div class="t m0 x24 h1f ye7 ff14 fsa fc0 sc0 ls0 ws22">(i) <span class="ff16 ls11">a</span><span class="ff15 ws0">lies to the right of 3.</span></div><div class="t m0 x2e h1f ye8 ff14 fsa fc0 sc0 ls0 ws22">(ii) <span class="ff16 ls11">a</span><span class="ff15 ws0">lies between 1 and 7.</span></div><div class="t m0 xc h3a ye9 ff14 fsa fc0 sc0 ls0 ws27">(iii) <span class="ff15 ws0">The distance from <span class="ff16 ls11">a</span>to 5 is less than<span class="_5b blank"> </span><span class="fsb v2">1</span></span></div><div class="t m0 x2f h2f yea ff15 fsb fc0 sc0 ls5b">3<span class="fsa ls0 v2">.</span></div><div class="t m0 xc h1f yeb ff14 fsa fc0 sc0 ls0 ws27">(iv) <span class="ff15 ws0">The distance from <span class="ff16 ls11">a</span>to 3 is at most 2.</span></div><div class="t m0 x2e h3b yec ff14 fsa fc0 sc0 ls0 ws22">(v) <span class="ff16 ls11">a</span><span class="ff15 ws0">is less than 5 units from<span class="_5b blank"> </span><span class="fsb v2">1</span></span></div><div class="t m0 x3 h2f yed ff15 fsb fc0 sc0 ls5b">3<span class="fsa ls0 v2">.</span></div><div class="t m0 xc h1f yee ff14 fsa fc0 sc0 ls0 ws22">(vi) <span class="ff16 ls11">a</span><span class="ff15 ws0">lies either to the left of <span class="ff17">\u2212</span>5 or to the right of 5.</span></div><div class="t m0 xc h22 yef ff18 fsa fc0 sc0 ls0">solution</div><div class="t m0 xc h1f yf0 ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff15 ws54">On the number line, numbers greater than 3 appear to the right; hence, <span class="ff16 ls4a ws23">a><span class="_0 blank"></span><span class="ff15 ls0 ws54">3 is equivalent to the numbers to the right</span></span></span></div><div class="t m0 xc h1f yf1 ff15 fsa fc0 sc0 ls0 ws0">of 3: <span class="ff14">(i)</span>.</div><div class="t m0 xc h3c yf2 ff14 fsa fc0 sc0 ls0 ws22">(b) <span class="ff17">|<span class="ff16 ls72">a</span><span class="ls73">\u2212</span><span class="ff15">5</span><span class="ls74">|</span><span class="ff15 ws55">measures the distance from <span class="ff16 ls75">a</span>to 5; hence, </span>|<span class="ff16 ls72">a</span><span class="ls73">\u2212</span><span class="ff15">5</span><span class="ls4">|<span class="ff16 ls76"><</span></span><span class="ff15 fsb v2">1</span></span></div><div class="t m0 x30 h3d yf3 ff15 fsb fc0 sc0 ls77">3<span class="fsa ls0 ws55 v2">is satis\ufb01ed by those numbers less than<span class="_5c blank"> </span></span><span class="ls0 v7">1</span></div><div class="t m0 x31 h3e yf3 ff15 fsb fc0 sc0 ls77">3<span class="fsa ls0 ws55 v2">of a unit from</span></div><div class="t m0 xc h1f yf4 ff15 fsa fc0 sc0 ls0 ws0">5: <span class="ff14">(iii)</span>.</div><div class="t m0 xc h3f yf5 ff14 fsa fc0 sc0 ls0 ws22">(c) <span class="ff17">|<span class="ff16 ls78">a</span><span class="ls79">\u2212</span><span class="ff15 fsb v2">1</span></span></div><div class="t m0 x2c h3d yf6 ff15 fsb fc0 sc0 ls5b">3<span class="ff17 fsa ls47 v2">|<span class="ff15 ls0 ws56">measures the distance from <span class="ff16 ls7a">a</span><span class="ws57">to <span class="fsb v2">1</span></span></span></span></div><div class="t m0 x32 h3d yf6 ff15 fsb fc0 sc0 ls5b">3<span class="fsa ls0 ws56 v2">; hence, <span class="ff17">|<span class="ff16 ls78">a</span><span class="ls7b">\u2212</span></span></span><span class="ls0 v7">1</span></div><div class="t m0 x33 h3e yf6 ff15 fsb fc0 sc0 ls5b">3<span class="ff17 fsa ls7c v2">|<span class="ff16"><<span class="ff15 ls0 ws56">5 is satis\ufb01ed by those numbers less than 5 units from</span></span></span></div><div class="t m0 x2b h40 yf7 ff15 fsb fc0 sc0 ls0">1</div><div class="t m0 x2b h41 yf8 ff15 fsb fc0 sc0 ls5b">3<span class="fsa ls22 v2">:<span class="ff14 ls0">(v)<span class="ff15">.</span></span></span></div><div class="t m0 xc h1f yf9 ff14 fsa fc0 sc0 ls0 ws27">(d) <span class="ff15 ws58">The inequality <span class="ff17">|<span class="ff16 ls13">a</span><span class="ls4">|<span class="ff16">></span></span></span>5 is equivalent to <span class="ff16 ls4a ws23">a><span class="_0 blank"></span><span class="ff15 ls7d ws59">5o<span class="_56 blank"></span>r<span class="ff16 ls4a ws23">a<<span class="_0 blank"></span><span class="ff17 ls0">\u2212<span class="ff15 ws58">5; that is, either <span class="ff16 ls7e">a</span>lies to the right of 5 or to the left of </span>\u2212<span class="ff15 ws58">5: <span class="ff14">(vi)</span>.</span></span></span></span></span></span></div><div class="t m0 x2b h1f yfa ff14 fsa fc0 sc0 ls0 ws27">(e) <span class="ff15 ws5a">The interval described by the inequality <span class="ff17">|<span class="ff16 ls7f">a</span><span class="ls60">\u2212</span></span>4<span class="ff17 ls80">|<span class="ff16"><</span></span>3 has a center at 4 and a radius of 3; that is, the interval consists</span></div><div class="t m0 xc h1f yfb ff15 fsa fc0 sc0 ls0 ws0">of those numbers between 1 and 7: <span class="ff14">(ii)</span>.</div><div class="t m0 x2e h1f yfc ff14 fsa fc0 sc0 ls0 ws27">(f) <span class="ff15 ws5b">The interval described by the inequality 1<span class="_5a blank"> </span><span class="ff16 ls18 ws44"><x <</span>5 has a center at 3 and a radius of 2; that is, the interval consists of</span></div><div class="t m0 xc h1f yfd ff15 fsa fc0 sc0 ls0 ws0">those numbers less than 2 units from 3: <span class="ff14">(iv)</span>.</div><div class="t m0 x20 h42 yfe ff15 fsa fc3 sc0 ls0 ws0">Describe <span class="ff1c ws2a v5">\ue003</span><span class="ff16 ls20">x<span class="ff17 ls81">:</span><span class="ls0 v4">x</span></span></div><div class="t m0 x34 h43 yff ff16 fsa fc3 sc0 ls45">x<span class="ff17 lsa">+<span class="ff15 ls6e">1</span></span><span class="ls4 v4"><<span class="ff15 ls0 ws35">0<span class="ff1c ls22 v5">\ue004</span><span class="ws0">as an interval.</span></span></span></div><div class="t m0 xc h44 y100 ff14 fsa fc0 sc0 ls0 ws22">25. <span class="ff15 ws0">Describe <span class="ff17">{<span class="ff16 ls20">x</span><span class="ls4">:<span class="ff16 ls82">x</span></span></span><span class="fsb ls2b v1">2</span><span class="ff17 lsa v0">+</span><span class="v0">2<span class="ff16 ls8d ws47">x<<span class="_4 blank"></span><span class="ff15 ls0">3<span class="ff17 ls22">}</span><span class="ws0">as an interval. <span class="ff19">Hint: </span>Plot <span class="ff16 ls20">y<span class="ff17 ls4">=</span><span class="ls83">x</span></span><span class="fsb ls2b v1">2</span><span class="ff17 lsa">+</span>2<span class="ff16 ls45">x<span class="ff17 lsa">\u2212</span></span>3.</span></span></span></span></span></div><div class="t m0 xc h45 y101 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws2a">The inequality <span class="ff16 ls84">x</span><span class="fsb ls85 v1">2</span><span class="ff17 lsa v0">+</span><span class="v0">2<span class="ff16 ls8d ws47">x<<span class="_4 blank"></span><span class="ff15 ls0 ws2a">3 is equivalent to <span class="ff16 ls86">x</span><span class="fsb ls85 v1">2</span><span class="ff17 lsa">+</span>2<span class="ff16 ls45">x<span class="ff17 lsa">\u2212</span></span><span class="ls4">3<span class="ff16"><</span></span>0. In the \ufb01gure below<span class="_0 blank"></span>, we see that the graph of</span></span></span></span></div><div class="t m0 xc h46 y102 ff16 fsa fc0 sc0 ls6e">y<span class="ff17 ls87">=</span><span class="ls88">x<span class="ff15 fsb ls89 v1">2</span><span class="ff17 ls72 v0">+<span class="ff15 ls0">2<span class="ff16 ls8a">x</span></span>\u2212<span class="ff15 ls0 ws2c">3 falls below the <span class="ff16 ls26">x</span>-axis for <span class="ff17">\u2212</span><span class="ls87">3<span class="ff16 ws5c"><x <</span></span>1. Thus, the set <span class="ff17">{</span></span></span></span><span class="v0">x<span class="ff17 ls87">:</span><span class="ls8b">x<span class="ff15 fsb ls89 v1">2</span><span class="ff17 ls72">+<span class="ff15 ls0">2</span></span></span><span class="ws5d">x<<span class="_4 blank"></span><span class="ff15 ls0">3<span class="ff17 ls27">}</span><span class="ws2c">corresponds to the interval</span></span></span></span></div><div class="t m0 xc h20 y103 ff17 fsa fc0 sc0 ls0">\u2212<span class="ff15 ls4">3<span class="ff16 ls18 ws44"><x <</span><span class="ls0">1.</span></span></div><div class="c xb y1 w2 h2"><div class="t m0 x0 h3 y2 ff1 fs0 fc0 sc0 ls0">www.elsolucionario.net</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div> <div id="pfa" class="pf w7 h9" data-page-no="a"><div class="pc pca w7 h9"><img fetchpriority="low" loading="lazy" class="bi x0 y7 w7 ha" alt="" src="https://files.passeidireto.com/6bf7eeaa-f34a-4479-8a10-d1e67ea494fa/bga.png"><div class="t m0 x18 h1b y88 ffe fsa fc2 sc0 ls0 ws1d">June 7, 2011<span class="_54 blank"> </span><span class="fs2 ws1e">L<span class="_35 blank"></span>TSV SSM Second Pass</span></div><div class="t m0 x35 h47 y6f ff1a fsd fc0 sc0 lse ws41">SECTION <span class="ffe ls0 ws5e">1.1 <span class="fsc ws5f">Real Numbers, Functions, and Graphs<span class="_5d blank"> </span><span class="ff12">3</span></span></span></div><div class="t m0 x36 h48 y104 ff1d fs10 fc4 sc0 ls0 ws60">\u2212<span class="ff15 fs11 ls91 v0">4</span><span class="ls92">\u2212<span class="ff15 fs11 ls93 v0">3</span></span>\u2212<span class="ff15 fs11 ls94 v0">2</span><span class="vd">\u2212</span><span class="ff15 fs11 vd">2</span></div><div class="t m0 x37 h49 y105 ff15 fs11 fc4 sc0 ls0">2</div><div class="t m0 x37 h49 y106 ff15 fs11 fc4 sc0 ls0">4</div><div class="t m0 x37 h49 y107 ff15 fs11 fc4 sc0 ls0">6</div><div class="t m0 x37 h49 y108 ff15 fs11 fc4 sc0 ls0">8</div><div class="t m0 x38 h49 y109 ff15 fs11 fc4 sc0 ls0">10</div><div class="t m0 x39 h4a y10a ff1e fs11 fc4 sc0 ls0">y</div><div class="t m0 x3a h4a y10b ff19 fs11 fc4 sc0 ls0">x</div><div class="t m0 x17 h49 y10c ff15 fs11 fc4 sc0 lsd2">12</div><div class="t m0 x36 h4b y10d ff1e fs11 fc4 sc0 ls0">y<span class="ff15 ls95 ws0"> <span class="ff1d fs10 ls92 ve">=</span><span class="ls0 v0"> <span class="ff19 ls96">x</span><span class="fs12 ws61 vf">2</span> <span class="ff1d fs10 ls92 v0">+</span> 2<span class="ff19">x</span><span class="ls97"> </span></span></span><span class="ff1d fs10 ws60 v0">\u2212</span><span class="ff15 ws0 v0"> 3</span></div><div class="t m0 x1f h20 y10e ff15 fsa fc3 sc0 ls0 ws0">Describe the set of real numbers satisfying <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span></span>3<span class="ff17 ls18 ws25">|=|<span class="_55 blank"></span><span class="ff16 ls45">x<span class="ff17 lsa">\u2212<span class="ff15 ls0">2</span><span class="ls1c">|+<span class="ff15 ls0 ws0">1 as a half-in\ufb01nite interval.</span></span></span></span></span></div><div class="t m0 x19 h4c y10f ff14 fsa fc0 sc0 ls0 ws22">27. <span class="ff15 ws0">Show that if <span class="ff16 ls4a ws23">a><span class="_0 blank"></span>b<span class="_58 blank"></span><span class="ff15 ls0 ws0">, then <span class="ff16 ls98">b</span><span class="ff17 fsb v1">\u2212<span class="ff15 ls99">1</span></span><span class="ff16 ls18 v0">>a</span></span></span></span></div><div class="t m0 x3b h40 y110 ff17 fsb fc0 sc0 ls0">\u2212<span class="ff15 ls32">1<span class="fsa ls0 ws0 v3">, provided that <span class="ff16 ls11">a</span>and <span class="ff16 ls12">b</span>have the same sign. What happens if <span class="ff16 ls4a ws23">a><span class="_4 blank"></span><span class="ff15 ls0 ws0">0 and <span class="ff16 lsd3 ws62">b<</span>0?</span></span></span></span></div><div class="t m0 x19 h20 y111 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff15 ws0">Case 1a: If <span class="ff16 ls11">a</span>and <span class="ff16 ls12">b</span>are both positive, then <span class="ff16 ls4a ws23">a><span class="_0 blank"></span>b<span class="_0 blank"></span><span class="ff17 ls4">\u21d2<span class="ff15">1<span class="ff16 ls9a">><span class="fsb ls0 v2">b</span></span></span></span></span></span></div><div class="t m0 x3c h4d y112 ff16 fsb fc0 sc0 ls9b">a<span class="ff17 fsa ls9c v1">\u21d2</span><span class="ff15 ls0 v10">1</span></div><div class="t m0 x3d h4e y113 ff16 fsb fc0 sc0 ls9d">b<span class="fsa ls9e v1">></span><span class="ff15 ls0 v7">1</span></div><div class="t m0 x8 h4f y112 ff16 fsb fc0 sc0 ls9f">a<span class="ff15 fsa ls0 v1">.</span></div><div class="t m0 x3e h20 y114 ff15 fsa fc0 sc0 ls0 ws2f">Case 1b: If <span class="ff16 lsa0">a</span>and <span class="ff16 ls74">b</span>are both negative, then <span class="ff16 ls4a ws23">a><span class="_0 blank"></span>b<span class="_0 blank"></span><span class="ff17 ls4">\u21d2<span class="ff15">1<span class="ff16 lsa1"><<span class="fsb ls0 v2">b</span></span></span></span></span></div><div class="t m0 x3f h50 y115 ff16 fsb fc0 sc0 lsa2">a<span class="ff15 fsa ls0 ws2f v1">(since <span class="ff16 lsa0">a</span>is negative) <span class="ff17 lsa3">\u21d2</span><span class="fsb v2">1</span></span></div><div class="t m0 x40 h51 y116 ff16 fsb fc0 sc0 ls9d">b<span class="fsa ls9e v1">></span><span class="ff15 ls0 v7">1</span></div><div class="t m0 x41 h52 y115 ff16 fsb fc0 sc0 lsa4">a<span class="ff15 fsa ls0 ws2f v1">(again, since <span class="ff16 ls74">b</span>is negative).</span></div><div class="t m0 x3e h53 y117 ff15 fsa fc0 sc0 ls0 ws0">Case 2: If <span class="ff16 ls4a ws23">a><span class="_0 blank"></span><span class="ff15 ls0 ws0">0 and <span class="ff16 lsd3 ws62">b<<span class="_0 blank"></span><span class="ff15 ls0 ws0">0, then<span class="_5b blank"> </span><span class="fsb v2">1</span></span></span></span></span></div><div class="t m0 x42 h50 y118 ff16 fsb fc0 sc0 lsa5">a<span class="fsa ls4 v1">><span class="ff15 ls0 ws0">0 and<span class="_5b blank"> </span><span class="fsb v2">1</span></span></span></div><div class="t m0 x43 h4e y119 ff16 fsb fc0 sc0 ls9d">b<span class="fsa ls4 v1"><<span class="ff15 ls22 ws63">0s<span class="_55 blank"></span>o <span class="fsb ls0 v2">1</span></span></span></div><div class="t m0 x33 h4e y119 ff16 fsb fc0 sc0 lsa6">b<span class="fsa lsa7 v1"><</span><span class="ff15 ls0 v7">1</span></div><div class="t m0 x44 h4f y118 ff16 fsb fc0 sc0 lsa8">a<span class="ff15 fsa ls0 ws0 v1">. (See Exercise 2f for an example of this).</span></div><div class="t m0 x1f h20 y11a ff15 fsa fc3 sc0 ls0 ws0">Which <span class="ff16 ls4d">x</span>satisfy both <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span></span>3<span class="ff17 ls4">|<span class="ff16"><</span></span>2 and <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span></span>5<span class="ff17 ls4">|<span class="ff16"><</span></span>1?</div><div class="t m0 x19 h54 y11b ff14 fsa fc0 sc0 ls0 ws22">29. <span class="ff15 ws0">Show that if <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">\u2212</span></span>5<span class="ff17 ls4">|<span class="ff16 lsa9"><</span></span><span class="fsb v2">1</span></span></div><div class="t m0 xb h2e y11c ff15 fsb fc0 sc0 lsaa">2<span class="fsa ls0 ws0 v2">and <span class="ff17">|<span class="ff16 lsab">b</span><span class="lsa">\u2212</span></span>8<span class="ff17 ls4">|<span class="ff16 lsac"><</span></span><span class="fsb v2">1</span></span></div><div class="t m0 x45 h2f y11c ff15 fsb fc0 sc0 ls5b">2<span class="fsa ls0 ws0 v2">, then <span class="ff17">|<span class="ff16 ws64">(a </span><span class="lsa">+<span class="ff16 lsd ws65">b) </span>\u2212</span></span>13<span class="ff17 ls4">|<span class="ff16"><</span></span>1. <span class="ff19">Hint: </span>Use the triangle inequality<span class="_4 blank"></span>.</span></div><div class="t m0 x19 h22 y11d ff18 fsa fc0 sc0 ls0">solution</div><div class="t m0 x3 h20 y11e ff17 fsa fc0 sc0 ls0">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsab">b</span>\u2212</span><span class="ff15">13</span><span class="ls18 ws25">|=|<span class="_55 blank"></span><span class="ff16 ls0 ws64">(a <span class="ff17 lsa">\u2212</span><span class="ff15">5</span><span class="lsa">)<span class="ff17">+</span></span><span class="ws66">(b <span class="ff17 lsa">\u2212</span><span class="ff15">8</span>)<span class="ff17">|</span></span></span></span></div><div class="t m0 x46 h20 y11f ff17 fsa fc0 sc0 lsd4 ws67">\u2264|<span class="_58 blank"></span><span class="ff16 ls1b">a<span class="ff17 lsa">\u2212<span class="ff15 ls0">5</span><span class="ls1c ws28">|+|<span class="_56 blank"></span><span class="ff16 lsab">b<span class="ff17 lsa">\u2212<span class="ff15 ls0">8</span><span class="lsad">|<span class="ff15 ls0 ws0">(by the triangle inequality)</span></span></span></span></span></span></span></div><div class="t m0 x46 h29 y120 ff16 fsa fc0 sc0 ls6e"><<span class="ff15 ls0 v4">1</span></div><div class="t m0 x45 h2a y121 ff15 fsa fc0 sc0 lsae">2<span class="ff17 ls41 v4">+</span><span class="ls0 v5">1</span></div><div class="t m0 xf h55 y121 ff15 fsa fc0 sc0 ls6e">2<span class="ff17 ls4 v4">=</span><span class="ls0 v4">1<span class="ff16">.</span></span></div><div class="t m0 x1f h20 y122 ff15 fsa fc3 sc0 ls0 ws0">Suppose that <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">\u2212</span></span>4<span class="ff17 ls18">|\u2264</span>1.</div><div class="t m0 x20 h1f y123 ff14 fsa fc3 sc0 ls0 ws27">(a) <span class="ff15 ws0">What is the maximum possible value of <span class="ff17">|<span class="ff16 ls45">x</span><span class="lsa">+</span></span>4<span class="ff17">|</span>?</span></div><div class="t m0 x20 h56 y124 ff14 fsa fc3 sc0 ls0 ws22">(b) <span class="ff15 ws0">Show that <span class="ff17">|<span class="ff16 lsaf">x</span></span><span class="fsb ls2b v1">2</span><span class="ff17 lsa v0">\u2212</span><span class="v0">16<span class="ff17 ls18">|\u2264</span>9.</span></span></div><div class="t m0 x19 h1f y125 ff14 fsa fc0 sc0 ls0 ws22">31. <span class="ff15 ws0">Suppose that <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">\u2212</span></span>6<span class="ff17 ls18">|\u2264</span>2 and <span class="ff17">|<span class="ff16 lsd">b</span><span class="ls18">|\u2264</span></span>3.</span></div><div class="t m0 x19 h1f y126 ff14 fsa fc0 sc0 ls0 ws27">(a) <span class="ff15 ws0">What is the largest possible value of <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span></span>|</span>?</span></div><div class="t m0 x19 h1f y127 ff14 fsa fc0 sc0 ls0 ws27">(b) <span class="ff15 ws0">What is the smallest possible value of <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span></span>|</span>?</span></div><div class="t m0 x19 h20 y128 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff17">|<span class="ff16 lsb0">a</span><span class="lsb">\u2212</span><span class="ff15">6</span><span class="ls18">|\u2264</span><span class="ff15 ws68">2 guarantees that 4<span class="_5a blank"> </span></span><span class="ls4">\u2264<span class="ff16 ls14">a</span>\u2264</span><span class="ff15 ws68">8, while </span>|<span class="ff16 lsd">b</span><span class="ls18">|\u2264</span><span class="ff15 ws68">3 guarantees that </span>\u2212<span class="ff15 ls4">3<span class="ff17">\u2264<span class="ff16 ls15">b</span>\u2264</span><span class="ls0 ws69">3. Therefore 1<span class="_59 blank"> </span></span><span class="ff17">\u2264<span class="ff16 lsb0">a</span><span class="lsb">+<span class="ff16 ls15">b</span></span>\u2264</span><span class="lsd5 ws6a">11 .</span></span></span></div><div class="t m0 x19 h20 y129 ff15 fsa fc0 sc0 ls0 ws0">It follows that</div><div class="t m0 x19 h1f y12a ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff15 ws0">the largest possible value of <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span><span class="ls22">|</span></span></span>is 1<span class="_4 blank"></span>1; and</span></div><div class="t m0 x19 h1f y12b ff14 fsa fc0 sc0 ls0 ws22">(b) <span class="ff15 ws0">the smallest possible value of <span class="ff17">|<span class="ff16 ls1b">a</span><span class="lsa">+<span class="ff16 lsd">b</span><span class="ls22">|</span></span></span>is 1.</span></div><div class="t m0 x1f h20 y12c ff15 fsa fc3 sc0 ls0 ws0">Prove that <span class="ff17">|<span class="ff16 ls26">x</span><span class="ls1c ws28">|\u2212|<span class="_56 blank"></span><span class="ff16 ls26">y<span class="ff17 ls18 ws25">|\u2264|<span class="_55 blank"></span><span class="ff16 ls45">x<span class="ff17 lsa">\u2212</span><span class="ls26">y<span class="ff17 ls0">|<span class="ff15 ls22">.</span><span class="ff19 ws6b">Hint: <span class="ff15 ws0">Apply the triangle inequality to </span></span></span><span class="ls4d">y<span class="ff15 ls0 ws0">and </span></span></span>x<span class="ff17 lsa">\u2212</span><span class="ls26">y<span class="ff15 ls0">.</span></span></span></span></span></span></span></div><div class="t m0 x19 h1f y12d ff14 fsa fc0 sc0 ls0 ws22">33. <span class="ff15 ws0">Express <span class="ff16 lsb1">r</span><span class="fsb ls2a vd">1</span><span class="ff17 ls4 v0">=</span><span class="v0">0<span class="ff16 ws30">.</span>27 as a fraction. <span class="ff19">Hint: </span>100<span class="ff16 ws30">r</span><span class="fsb ls2b vd">1</span><span class="ff17 lsa">\u2212<span class="ff16 lsb2">r</span></span><span class="fsb lsb3 vd">1</span>is an integer<span class="_4 blank"></span>. Then express <span class="ff16 ws30">r</span><span class="fsb ls2a vd">2</span><span class="ff17 ls4">=</span>0<span class="ff16">.</span><span class="ws6c">2666 <span class="ff16 lsb4 ws6d">... </span></span>as a fraction.</span></span></div><div class="t m0 x19 h57 y12e ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff15 ws0">Let <span class="ff16 ws30">r</span><span class="fsb ls2a vd">1</span><span class="ff17 ls4 v0">=</span><span class="v0">0<span class="ff16 ws30">.</span>27. W<span class="_2 blank"></span>e observe that 100<span class="ff16 ws30">r</span><span class="fsb ls2a vd">1</span><span class="ff17 ls4">=</span>27<span class="ff16 ws30">.</span><span class="ws26">27. Therefore,<span class="_5a blank"> </span>100<span class="ff16 ws30">r</span><span class="fsb ls2b vd">1</span><span class="ff17 lsa">\u2212<span class="ff16 lsb2">r</span></span><span class="fsb ls2a vd">1</span><span class="ff17 ls4">=</span>27<span class="ff16 ws30">.</span><span class="ws6e">27 <span class="ff17 lsa">\u2212</span>0<span class="ff16 lsb5">.</span><span class="ws6f">27 <span class="ff17 ls4">=</span></span><span class="ws0">27 and</span></span></span></span></span></div><div class="t m0 x47 h58 y12f ff16 fsa fc0 sc0 ls0 ws30">r<span class="ff15 fsb ls2a vd">1</span><span class="ff17 ls6e v0">=</span><span class="ff15 v4">27</span></div><div class="t m0 x33 h2a y130 ff15 fsa fc0 sc0 ls0 ws70">99 <span class="ff17 lsb6 v4">=</span><span class="v5">3</span></div><div class="t m0 x48 h59 y130 ff15 fsa fc0 sc0 lsd5 ws71">11 <span class="ff16 ls0 v4">.</span></div><div class="t m0 x19 h5a y131 ff15 fsa fc0 sc0 ls0 ws0">Now<span class="_4 blank"></span>, let <span class="ff16 lsb7">r</span><span class="fsb ls2a vd">2</span><span class="ff17 ls4 v0">=</span><span class="v0">0<span class="ff16">.</span><span class="ws26">2666. Then<span class="_5 blank"> </span>10<span class="ff16 ws30">r</span><span class="fsb ls2a vd">2</span><span class="ff17 ls4">=</span>2<span class="ff16 ws30">.</span><span class="ws0">666 and 100<span class="ff16 ws30">r</span><span class="fsb ls2a vd">2</span><span class="ff17 ls4">=</span>26<span class="ff16 ws30">.</span></span>666. Therefore,<span class="_5a blank"> </span>100<span class="ff16 ws30">r</span><span class="fsb ls28 vd">2</span><span class="ff17 lsa">\u2212</span>10<span class="ff16 lsb8">r</span><span class="fsb ls2a vd">2</span><span class="ff17 ls4">=</span>26<span class="ff16 ws30">.</span><span class="ws6e">666 <span class="ff17 lsa">\u2212</span>2<span class="ff16 lsb5">.</span><span class="ws6f">666 <span class="ff17 ls4">=</span></span><span class="ws0">24 and</span></span></span></span></div><div class="t m0 x47 h5b y132 ff16 fsa fc0 sc0 ls0 ws30">r<span class="ff15 fsb ls2a vd">2</span><span class="ff17 ls6e v0">=</span><span class="ff15 v4">24</span></div><div class="t m0 x33 h2a y133 ff15 fsa fc0 sc0 ls0 ws70">90 <span class="ff17 lsb9 v4">=</span><span class="v5">4</span></div><div class="t m0 x48 h59 y133 ff15 fsa fc0 sc0 ls0 ws72">15 <span class="ff16 v4">.</span></div><div class="t m0 x1f h20 y134 ff15 fsa fc3 sc0 ls0 ws0">Represent 1<span class="ff16">/</span>7 and 4<span class="ff16">/</span>27 as repeating decimals.</div><div class="t m0 x19 h5c y135 ff14 fsa fc0 sc0 ls0 ws27">35. <span class="ff15 ws73">The text states: <span class="ff19">If the decimal expansions of numbers <span class="ff16 lsba">a</span>and <span class="ff16 ls75">b</span>agr<span class="_0 blank"></span>ee to <span class="ff16 ls54">k</span>places, then <span class="ff17">|<span class="ff16 ls75">a</span><span class="ls74">\u2212<span class="ff16 lsd">b</span><span class="ls18">|\u2264</span></span><span class="ff15 ws35">10</span><span class="fsb v1">\u2212<span class="ff16 lsbb">k</span></span><span class="ff15 v0">. Show that the</span></span></span></span></div><div class="t m0 x19 h5d y136 ff15 fsa fc0 sc0 ls0 ws74">converse is false: For all <span class="ff16 ls52">k</span>there are numbers <span class="ff16 lsbc">a</span>and <span class="ff16 lsbd">b</span>whose decimal expansions <span class="ff19">do not agr<span class="_0 blank"></span>ee at all<span class="_5a blank"> </span><span class="ff15">but <span class="ff17">|<span class="ff16 lsbd">a</span><span class="lsbe">\u2212<span class="ff16 lsd">b</span><span class="ls18">|\u2264</span></span></span><span class="ws35">10<span class="ff17 fsb v1">\u2212<span class="ff16 lsbf">k</span></span><span class="v0">.</span></span></span></span></div><div class="t m0 x19 h57 y137 ff18 fsa fc0 sc0 ls0 ws34">solution <span class="ff15 ws75">Let <span class="ff16 lsc0">a<span class="ff17 lsc1">=</span></span>1 and <span class="ff16 lsc2">b<span class="ff17 lsc1">=</span></span>0<span class="ff16 lsc3">.</span><span class="v0">9 (see the discussion before Example 1). The decimal expansions of <span class="ff16 lsc4">a</span>and <span class="ff16 ls80">b</span>do not</span></span></div><div class="t m0 x19 h5e y138 ff15 fsa fc0 sc0 ls0 ws0">agree, but <span class="ff17">|</span><span class="lsa">1<span class="ff17">\u2212</span></span>0<span class="ff16 lsc5">.</span><span class="v0">9<span class="ff17 ls4">|<span class="ff16"><</span></span><span class="ws35">10<span class="ff17 fsb v1">\u2212<span class="ff16 lsc6">k</span></span></span>for all <span class="ff16 lsc7">k</span>.</span></div><div class="t m0 x1f h20 y139 ff15 fsa fc3 sc0 ls0 ws0">Plot each pair of points and compute the distance between them:</div><div class="t m0 x20 h1f y13a ff14 fsa fc3 sc0 ls0 ws22">(a) <span class="ff16">(<span class="ff15">1</span><span class="ls21">,</span><span class="ff15">4</span><span class="ls22">)</span><span class="ff15 ws0">and </span>(<span class="ff15">3</span><span class="ls21">,</span><span class="ff15">2</span><span class="lsc8">)</span></span>(b) <span class="ff16">(<span class="ff15">2</span><span class="ls21">,</span><span class="ff15">1</span><span class="ls22">)</span><span class="ff15 ws0">and </span>(<span class="ff15">2</span><span class="ls21">,</span><span class="ff15">4</span>)</span></div><div class="t m0 x20 h1f y13b ff14 fsa fc3 sc0 ls0 ws22">(c) <span class="ff16">(<span class="ff15">0</span><span class="ls21">,</span><span class="ff15">0</span><span class="ls22">)</span><span class="ff15 ws0">and </span>(<span class="ff17">\u2212<span class="ff15">2</span></span><span class="ls21">,</span><span class="ff15">3</span><span class="lsc9">)</span></span>(d) <span class="ff16">(<span class="ff17">\u2212<span class="ff15">3</span></span><span class="ls21">,</span><span class="ff17">\u2212<span class="ff15">3</span></span><span class="ls22">)</span><span class="ff15 ws0">and </span>(<span class="ff17">\u2212<span class="ff15">2</span></span><span class="ls21">,</span><span class="ff15">3</span>)</span></div><div class="t m0 x19 h1f y13c ff14 fsa fc0 sc0 ls0 ws22">37. <span class="ff15 ws0">Find the equation of the circle with center <span class="ff16">(</span>2<span class="ff16 ls21">,</span>4<span class="ff16">)</span>:</span></div><div class="t m0 x19 h1f y13d ff14 fsa fc0 sc0 ls0 ws22">(a) <span class="ff15 ws0">with radius <span class="ff16 lsca">r<span class="ff17 ls4">=</span></span>3.</span></div><div class="t m0 x19 h1f y13e ff14 fsa fc0 sc0 ls0 ws22">(b) <span class="ff15 ws0">that passes through <span class="ff16">(</span>1<span class="ff16 ls21">,</span><span class="ff17">\u2212</span>1<span class="ff16">)</span>.</span></div><div class="t m0 x19 h22 y13f ff18 fsa fc0 sc0 ls0">solution</div><div class="t m0 x19 h5f y140 ff14 fsa fc0 sc0 ls0 ws27">(a) <span class="ff15 ws0">The equation of the indicated circle is <span class="ff16 ws2f">(x <span class="ff17 lsa">\u2212</span></span>2<span class="ff16 ws30">)</span><span class="fsb ls2b v1">2</span><span class="ff17 lsa v0">+<span class="ff16 ls0 ws2f">(y </span>\u2212</span><span class="v0">4<span class="ff16 ls29">)</span><span class="fsb ls2a v1">2</span><span class="ff17 ls4">=</span><span class="ws35">3<span class="fsb ls2a v1">2</span><span class="ff17 ls4">=</span>9.</span></span></span></div><div class="t m0 x19 h1f y141 ff14 fsa fc0 sc0 ls0 ws22">(b) <span class="ff15 ws0">First determine the radius as the distance from the center to the indicated point on the circle:</span></div><div class="t m0 x49 h60 y142 ff16 fsa fc0 sc0 lsca">r<span class="ff17 ls4">=<span class="ff1c ls0 ws2a v11">\ue005</span></span><span class="ls0 v0">(<span class="ff15 lsa">2<span class="ff17">\u2212</span><span class="ls0">1</span></span><span class="lscb">)<span class="ff15 fsb ls2b vf">2</span><span class="ff17 lsa">+</span></span>(<span class="ff15 lsa">4<span class="ff17">\u2212</span></span>(<span class="ff17">\u2212<span class="ff15">1</span></span><span class="ws30">))<span class="ff15 fsb lscc vf">2</span><span class="ff17 ls4">=<span class="ls0 v12">\u221a</span></span><span class="ff15">26</span>.</span></span></div><div class="t m0 x19 h61 y143 ff15 fsa fc0 sc0 ls0 ws0">Thus, the equation of the circle is <span class="ff16 ws2f">(x <span class="ff17 lsa">\u2212</span></span>2<span class="ff16 ws30">)</span><span class="fsb ls2b v1">2</span><span class="ff17 lsa v0">+<span class="ff16 ls0 ws2f">(y </span>\u2212</span><span class="v0">4<span class="ff16 ls2c">)</span><span class="fsb lscd v1">2</span><span class="ff17 ls4">=</span>26.</span></div><div class="t m0 x1f h20 y144 ff15 fsa fc3 sc0 ls0 ws76">Find all points with integer coordinates located at a distance 5 from the origin. Then \ufb01nd all points with integer</div><div class="t m0 x20 h20 y145 ff15 fsa fc3 sc0 ls0 ws0">coordinates located at a distance 5 from <span class="ff16">(</span>2<span class="ff16 ls21">,</span>3<span class="ff16">)</span>.</div><div class="t m0 x19 h1f y146 ff14 fsa fc0 sc0 ls0 ws22">39. <span class="ff15 ws0">Determine the domain and range of the function</span></div><div class="t m0 x32 h21 y147 ff16 fsa fc0 sc0 lsce">f<span class="ff17 ls18 ws25">:{<span class="_55 blank"></span><span class="ff16 lsd6 ws77">r, s<span class="_5e blank"> </span>, t<span class="_5f blank"> </span>, u<span class="_60 blank"> </span><span class="ff17 ls18 ws25">}\u2192{<span class="_55 blank"></span><span class="ff16 lsd7 ws78">A, B, C, D<span class="_60 blank"> </span>, E<span class="_60 blank"> </span><span class="ff17 ls0">}</span></span></span></span></span></div><div class="t m0 x19 h20 y148 ff15 fsa fc0 sc0 ls0 ws0">de\ufb01ned by <span class="ff16 lsb ws31">f(<span class="_35 blank"></span>r<span class="_4 blank"></span>) <span class="ff17 ls4">=</span><span class="ls0">A<span class="ff15 ls22">,</span></span>f(<span class="_35 blank"></span>s<span class="_2 blank"></span>) <span class="ff17 ls4">=</span><span class="lscf">B<span class="ff15 ls22">,</span></span>f(<span class="_35 blank"></span>t<span class="_4 blank"></span>) <span class="ff17 ls4">=</span><span class="lscf">B<span class="ff15 ls22">,</span><span class="ls0 ws79">f (u)<span class="_5a blank"> </span><span class="ff17 ls4">=</span><span class="lsd0">E</span><span class="ff15">.</span></span></span></span></div><div class="t m0 x19 h20 y149 ff18 fsa fc0 sc0 ls0 ws24">solution <span class="ff15 ws0">The domain is the set <span class="ff16 lsd1">D<span class="ff17 ls18 ws25">={<span class="_55 blank"></span><span class="ff16 lsd6 ws77">r, s<span class="_5e blank"> </span>, t<span class="_5f blank"> </span>, u<span class="_60 blank"> </span><span class="ff17 ls0">}<span class="ff15 ws0">; the range is the set </span></span><span class="lsd1">R<span class="ff17 ls18 ws25">={<span class="_55 blank"></span><span class="ff16 lsd7 ws78">A, B, E<span class="_60 blank"> </span><span class="ff17 ls0">}<span class="ff15">.</span></span></span></span></span></span></span></span></span></div><div class="c xb y1 w2 h2"><div class="t m0 x0 h3 y2 ff1 fs0 fc0 sc0 ls0">www.elsolucionario.net</div></div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,-18.000000,-34.015700]}'></div></div>
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