CONTENTS Introduction .............................................................. 1 Chapter 1. Functions ..................................................... 2 Chapter 2. LimitsandContinuity ......................................... 43 Chapter 3. TheDerivative ................................................ 65 Chapter 4. LogarithmicandExponentialFunctions ......................... 99 Chapter 5. AnalysisofFunctionsandTheirGraphs ........................ 139 Chapter 6. ApplicationsoftheDerivative ................................. 177 Chapter 7. Integration ...................................................209 Chapter 8. ApplicationsoftheDefiniteIntegral inGeometry,Science,andEngineering ......................... 256 Chapter 9. PrinciplesofIntegralEvaluation ............................... 292 Chapter10. MathematicalModelingwithDifferentialEquations ..............343 Chapter11. InfiniteSeries ................................................361 Chapter12. AnalyticGeometryinCalculus .................................408 Chapter13. Three-DimensionalSpace;Vectors .............................448 Chapter14. Vector-ValuedFunctions ...................................... 490 Chapter15. PartialDerivatives ............................................524 Chapter16. MultipleIntegrals .............................................573 Chapter17. TopicsinVectorCalculus ..................................... 608 AppendixA. RealNumbers,Intervals,andInequalities .......................640 AppendixB. AbsoluteValue ...............................................647 AppendixC. CoordinatePlanesandLines ...................................650 AppendixD. Distance,Circles,andQuadraticEquations ......................658 AppendixE. TrigonometryReview .........................................668 AppendixF. SolvingPolynomialEquations .................................674 CALCULUS: A New Horizon from Ancient Roots EXERCISE SET FOR INTRODUCTION 123 41 1. (a) x=0:123123123:::; 1000x=123:123123123:::=123+x; 999x=123; x= = 999 333 115 (b) x=12:7777:::; 10x=127:7777:::, so 9x=10x¡x=115; x= 9 (c) x=38:07818181:::; 100x=3807:818181:::; 99x=3769:74; 3769:74 376974 41886 20943 x= = = = 99 9900 1100 550 4296 537 (d) 0:4296000:::=0:4296= = 10000 1250 repeats z }| { 22 2. (a) … is irrational, and thus has a nonrepeating decimal expansion, whereas =3:142857::: 7 22 (b) >… 7 ˆ p ! ˆ p ! 223 333 63 17+15 5 355 22 63 17+15 5 3. (a) < < p < < (b) p 71 106 25 7+15 5 113 7 25 7+15 5 ˆ p ! 333 63 17+15 5 (c) (d) p 106 25 7+15 5 (cid:181) ¶ (cid:181) ¶ 8 2 16 2 256 4. (a) If r is the radius, then D =2r so D = r = r2. The area of a circle of radius r is 9 9 81 …r2 so 256=81 was the approximation used for …. (b) 256=81…3:16049, 22=7…3:14268, and … …3:14159 so 256=81 is worse than 22=7. 5. The flrst series, taken to ten terms, adds to 3:0418; the second, as printed, adds to 3:1416. 1 1 1 1 1 1 1 6. (a) =0:111111:::= + + + + + +::: 9 10 100 1000 10000 100000 1000000 2 1 8 5 1 8 5 (b) =0:185185:::= + + + + + +::: 27 10 100 1000 10000 100000 1000000 14 3 1 1 1 1 1 (c) =0:311111:::= + + + + + +::: 45 10 100 1000 10000 100000 1000000 7 6 3 6 3 6 3 7. (a) =0:636363:::= + + + + + +::: 11 10 100 1000 10000 100000 1000000 8 2 4 2 4 2 4 (b) =0:242424:::= + + + + + +::: 33 10 100 1000 10000 100000 1000000 5 4 1 6 6 6 6 (c) =0:416666:::= + + + + + +::: 12 10 100 1000 10000 100000 1000000 8. (a) 1, 2, 1.75, 1.7321 (b) 1, 3, 2.33, 2.238, 2.2361 9. (a) 1, 4, 2.875, 2.6549, 2.6458 (b) 1, 25.5, 13.7, 8.69, 7.22, 7.0726, 7.0711 10. (a) Let x1 = 12(a+b), x2 = 12(a+x1), x3 = 12(a+x2), etc. Then b>x1 >x2 >¢¢¢>xn¡1 >xn >a so all the x’s are distinct, there are inflnitely many of them and they all lie between a and b. i (b) x=0:99999:::, 10x=9:99999:::, 9x=9, x=1 (c) (1:999999:::)=2=0:999999:::=1; yes it is consistent, as all three are equal. (d) 10x=9+x, so x=9=9=1. They are equal. 1 CHAPTER 1 Functions EXERCISE SET 1.1 1. (a) around 1943 (b) 1960; 4200 (c) no; you need the year’s population (d) war; marketing techniques (e) news of health risk; social pressure, antismoking campaigns, increased taxation 2. (a) 1989; $35,600 (b) 1983; $32,000 (c) the flrst two years; the curve is steeper (downhill) 3. (a) ¡2:9;¡2:0;2:35;2:9 (b) none (c) y =0 (d) ¡1:75•x•2:15 (e) y =2:8 at x=¡2:6; y =¡2:2 at x=1:2 max min 4. (a) x=¡1;4 (b) none (c) y =¡1 (d) x=0;3;5 (e) y =9 at x=6; y =¡2 at x=0 max min 5. (a) x=2;4 (b) none (c) x•2; 4•x (d) y =¡1; no maximum value min 6. (a) x=9 (b) none (c) x‚25 (d) y =1; no maximum value min 7. (a) Breaks could be caused by war, pestilence, (cid:176)ood, earthquakes, for example. (b) C decreases for eight hours, takes a jump upwards, and then repeats. 8. (a) Yes, if the thermometer is not near a window or door or other source of sudden temperature change. (b) The number is always an integer, so the changes are in movements (jumps) of at least one unit. 9. (a) If the side adjacent to the building has length x then L = x + 2y. Since A = xy = 1000, L=x+2000=x. (b) x>0 and x must be smaller than the width of the building, which was not given. (c) 120 (d) Lmin …89:44 20 80 80 10. (a) V =lwh=(6¡2x)(6¡2x)x (b) From the flgure it is clear that 01 and f(3t)=6t for t•1. 3+1 ¡1+1 …+1 ¡1:1+1 ¡0:1 1 2. (a) g(3) = = 2; g(¡1) = = 0;g(…) = ; g(¡1:1) = = = ; 3¡1 ¡1¡1 …¡1 ¡1:1¡1 ¡2:1 21 t2¡1+1 t2 g(t2¡1)= = t2¡1¡1 t2¡2 p p (b) g(3) = 3+1 = 2;g(¡1) = 3;g(…) = …+1;g(¡1:1) = 3; g(t2 ¡ 1) = 3 if t2 < 2 and p g(t2¡1)= t2¡1+1=jtj if t2 ‚2. p p 3. (a) x6=3 (b) x•¡ 3 or x‚ 3 ExerciseSet1.2 4 (c) x2 ¡2x+5 = 0 has no real solutions so x2 ¡2x+5 is always positive or always negative. If x=0, then x2¡2x+5=5>0; domain: (¡1;+1). (d) x6=0 (e) sinx6=1, so x6=(2n+ 1)…, 2 n=0;§1;§2;::: 7 4. (a) x6=¡ 5 1 (b) x¡3x2 must be nonnegative; y =x¡3x2 is a parabola that crosses the x-axis at x=0; and 3 1 opens downward, thus 0•x• 3 x2¡4 (c) > 0, so x2 ¡4 > 0 and x¡4 > 0, thus x > 4; or x2 ¡4 < 0 and x¡4 < 0, thus x¡4 ¡20, all x 5. (a) x•3 (b) ¡2•x•2 (c) x‚0 (d) all x (e) all x 2 3 3 6. (a) x‚ (b) ¡ •x• (c) x‚0 (d) x6=0 (e) x‚0 3 2 2 7. (a) yes (b) yes (c) no (vertical line test fails) (d) no (vertical line test fails) 8. The sine of (cid:181)=2 is (L=2)=10 (side opposite over hypotenuse), so that L=20sin((cid:181)=2). 9. The cosine of (cid:181) is (L¡h)=L (side adjacent over hypotenuse), so h=L(1¡cos(cid:181)). 10. T 11. h 12. w t t t 5 10 15 13. (a) If x < 0, then jxj = ¡x so f(x) = ¡x + 3x + 1 = 2x + 1. If x ‚ 0, then jxj = x so f(x)=x+3x+1=4x+1; ‰ 2x+1; x<0 f(x)= 4x+1; x‚0 (b) If x<0, then jxj=¡x and jx¡1j=1¡x so g(x)=¡x+1¡x=1¡2x. If 0•x<1, then jxj=x and jx¡1j=1¡x so g(x)=x+1¡x=1. If x‚1, then jxj=x and jx¡1j=x¡1 so g(x)=x+x¡1=2x¡1; 8 < 1¡2x; x<0 g(x)= 1; 0•x<1 : 2x¡1; x‚1 14. (a) Ifx<5=2,thenj2x¡5j=5¡2xsof(x)=3+(5¡2x)=8¡2x. Ifx‚5=2,thenj2x¡5j=2x¡5 so f(x)=3+(2x¡5)=2x¡2; ‰ 8¡2x; x<5=2 f(x)= 2x¡2; x‚5=2 5 Chapter1 (b) If x<¡1, then jx¡2j=2¡x and jx+1j=¡x¡1 so g(x)=3(2¡x)¡(¡x¡1)=7¡2x. If ¡1•x<2, then jx¡2j=2¡x and jx+1j=x+1 so g(x)=3(2¡x)¡(x+1)=5¡4x. If x‚2, then jx¡2j=x¡2 and jx+1j=x+1 so g(x)=3(x¡2)¡(x+1)=2x¡7; 8 < 7¡2x; x<¡1 g(x)= 5¡4x; ¡1•x<2 : 2x¡7; x‚2 15. (a) V =(8¡2x)(15¡2x)x (b) ¡1

Student Solutions Manual Set for Calculus Early Transcendentals Single Variable 8th Edition

Similar Student Solutions Manual Set for Calculus Early Transcendentals Single Variable 8th Edition