Solution Manual for Calculus and Its Applications, 14th Edition

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SOLUTIONSMANUALBEVERLYFUSFIELDCALCULUS&ITSAPPLICATIONSFOURTEENTHEDITIONCALCULUS&ITSAPPLICATIONS,BRIEFVERSIONFOURTEENTHEDITIONLarry J. GoldsteinGoldstein Educational TechnologiesDavid C. LayUniversity of MarylandDavid I. SchneiderUniversity of MarylandNakhlé H. AsmarUniversity of Missouri

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CONTENTSChapter 0Functions.............................................................................................1Chapter 1The Derivative ..................................................................................27Chapter 2Applications of the Derivative ..........................................................74Chapter 3Techniques of Differentiation .........................................................122Chapter 4The Exponential and Natural Logarithmic Functions.....................148Chapter 5Applications of the Exponential and Natural LogarithmFunctions.........................................................................................178Chapter 6The Definite Integral.......................................................................194Chapter 7Functions of Several Variables .......................................................227Chapter 8The Trigonometric Functions .........................................................264Chapter 9Techniques of Integration ...............................................................283Chapter 10Differential Equations.....................................................................323Chapter 11Taylor Polynomials and Infinite Series...........................................355Chapter 12Probability and Calculus .................................................................377

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1Chapter 0 Functions0.1Functions and Their Graphs1.2.3.4.5.6.7.[2, 3)8.31, 29.[–1, 0)10.[–1, 8)11., 312.2,13.2( )3fxxx2(0)03(0)0f2(5)53(5)251510f2(3)33(3)990f2( 7)( 7)3( 7)492170f14.32( )1fxxxx32(1)11110f32( 1)( 1)( 1)( 1)10f  3211119122228f 32( )1f aaaa15.2( )2fxxx222(1)(1)2(1)(21)221faaaaaaa222(2)(2)2(2)(44)242faaaaaaaa16.( )(1)sh ss1122312211231h 332213223321h 11(1)1(1)2aah aaa17.( )32,0fxxh  33 3293231133 3211fhhhhf333111133fhfhhhhh18. 2,0fxxh 2221112111fhhhhf 221211122hhfhfhhhhhh19.a.27359332735660kxxxxThe boiling point of tungsten is 5660°C.b.  93259 566032102205fxxfxThe boiling point of tungsten is 10220°F.20.a.f(0) represents the number of laptops soldin 2015.b.2(5)1502(5)51501025185fIn 2020, the company will sell 185laptops.21.8( )(1)(2)xfxxxall real numbers such thatx≠1, 2 or,11, 22, 

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2Chapter 0 Functions22.1( )f ttall real numbers such thatt> 0 or0,23.1( )3g xxall real numbers such thatx< 3 or,3 24.4( )(2)g xx xall real numbers such thatx≠0, –2 or,22, 00, 25.26.27.28.29.function30.not a function31.not a function32.not a function33.not a function34.function35.01;71ff 36.23;10ff37.positive38.negative39.[−1, 3]40.−1, 5, 941.,15, 9 42.1, 59,43.1.03;5.037ff44.6.03f45.0, .0546.t≈347.1( )22fxxx125(3)3(32)22fNo, (3, 12) is not on the graph.48.f(x) =x(5 +x)(4 –x)f(–2) = –2(5 + (–2))(4 – (–2)) = –36No, (–2, 12) is not on the graph.49.231( )1xg xx   23 11211211gYes,1, 1is on the graph.50.24( )2xg xx  244201044263gNo,14, 4is not on the graph.51.3( )fxx3(1)(1)f aa52.5( )fxxx225(2)(2)(2)5(2)14(2)2fhhhhhhhh53.for 02( )1for 25xxfxxx  (1)11(2)123(3)134fff

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Section 0.1 Functions and Their Graphs354.21for 12( )for 2xxfxxx 1(1)11f1(2)2f2(3)39f55.2for2( )1for 22.54for 2.5xxfxxxxx2(1)(1)ff(2) = 1 + 2 = 3f(3) = 4(3) = 1256.23for24( )2for 235for 3xxfxxxxx3(1)141ff(2) = 2(2) = 42(3)3542f57.a.0.06for 503000( )0.0215for 3000xxfxxx b.30000.06 300018045000.02 450015105ff58.59.60. 100,0xR xxbxa.100 303000200301530453Rb.100 505000305050305003505033bbb61.EnteringY1= 1/X + 1will graph the function1( )1fxx. In order to graph the function1( )1fxx, you need to include parenthesesin the denominator:Y1= 1/(X + 1).62.EnteringY1= X ^ 3 / 4will graph the function( )4xfx. In order to graph the function3 4yx, you need to include parentheses inthe exponent:Y1= X ^ (3/4).63.222fxxx 64. 211fxx

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4Chapter 0 Functions0.2Some Important Functions1.21yxxy110–1–1–32.3.31yxxy1401–1–24.142yx xy2–50–4–2–35.23yx xy−1503116.0y7.0xyxy1100–1–18.321xy xy3−51–2−34( )21fxx3y( )31fxx

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Section 0.2 Some Important Functions59.21xyxy3211–3–110.11.  9309 933fxxfThey-intercept is (0, 3).1930933xxx  Thex-intercept is1 , 0 .312.1( )12fxx 1(0)(0)112f  They-intercept is (0, –1).11101222xxx  Thex-intercept is (–2, 0).13.f(x) = 5They-intercept is (0, 5).There is nox-intercept.14.f(x) = 14They-intercept is (0, 14).There is nox-intercept.15.50xy0500yyThex- andy-intercept is (0, 0).16.232xy23 021yyThey-intercept is (0, 1). 2232 0323xxx  Thex-intercept is2 , 0 .317.a.Cost is $(24 + 200(.45)) = $114.b.f(x) = .45x+ 2418.Letxbe the volume of gas (in thousands ofcubic feet) extracted.f(x) = 5000 + .10x19.Letxbe the number of days of hospitalconfinement.f(x) = 700x+ 190020.64035065 mphxx21.50( ), 0100105xfxxxFrom example 6, we know thatf(70) = 100.The cost to remove 75% of the pollutant is50 7575125.10575fThe cost of removing an extra 5% is$125−$100 = $25 million. To remove thefinal 5% the cost isf(100) –f(95) = 1000 – 475 = $525 million.This costs 21 times as much as the cost toremove the next 5% after the first 70% isremoved.22.a.20(85)(85)$10010285fmillionb.f(100) –f(95) = 1000 – 271.43$728.57million23. 1KfxxVVa.f(x) = .2x+ 50We have.2KVand150.VIf150,Vthen1 .50VNow,.2KVimplies150.2,Kso111.550250K

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6Chapter 0 Functionsb.111,0,KKyxVVVVVso they-intercept is10,V.Solving10,KxVVwe get11 ,K xxVVK  so thex-interceptis1 , 0K.24.From 17(b),1 , 0Kis thex-intercept. Fromthe experimental data, (–500, 0) is also thex-intercept. Thus1500K 1.500KAgain from 17(b),10,Vis they-intercept.From the experimental data, (0, 60) is also they-intercept. Thus1160.60VV25.234yxxa= 3,b= –4,c= 026.2262122333xxyxx1 ,3ab= –2,23c27.2321yxxa= –2,b= 3,c= 128.2324yxxa= 4,b= –2,c= 329.21yxa= –1,b= 0,c= 130.2132yxx 1 ,3,2abc 31. 224fxxxa= 2,b= –4,c= 0vertex:   44,1,11,22 22 2ff  xy002032.2( )43gttt a= –1,b= 4,c= –3vertex: 44,2,22, 12121ggxy0–3103033.3for2( )21for2xfxxxx< 2x≥2x3fxx21fxx13250337

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Section 0.2 Some Important Functions734.1for 04( )223for 45xxfxxx 0≤x< 44≤x≤5x 12fxxx23fxx0045215733235.4for 02( )22for 231for3xxfxxxxx0≤x< 22≤x< 3x4fxxx22fxx042213523x≥3x1fxx344536.4for 01( )84for 1224for2xxfxxxxx0≤x< 11≤x< 2x4fxxx84fxx0014122322x≥2x24fxx203237.100100( ),1111fxxxf 38.51( ),2fxxx51112232f39.2( ),10fxxx222(10)1010f40.( ),fxxxf41.( ),fxxx= –2.5(–2.5)2.52.5f 42.( ),fxx23x 222333f 

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8Chapter 0 Functions43.44.45.46.0.3The Algebra of Functions1.22( )( )(1)991fxg xxxxx2.222( )( )(1)(52)34fxh xxxx3.23( )( )(1)(9 )99fx g xxxxx4.23( ) ( )(9 )(52)4518gx h xxxxx5.222( )1111( )999999f tttttg tttttt6.2( )9( )52gtth tt7.2212(2)(3)32(3)(2)316xxxxxxxxx8.2323(2)( 2)(6)62(6)(2)6812xxxxxxxxx 9.2(4)()(8)84(8)(4)41232xxx xxxxxxxxxx 10.2()(5)(3)35(3)(5)2815xxxxx xxxxxxxx11.225(5)(10)(10)1010(10)(10)2550100xxxxx xxxxxxxx12.2266(6)(6)(6)(6)66(6)(6)27236xxxxxxxxxxxx13.225(5)(5)(2)25(2)(5)2210310xxxxxxxxxxxxxx14.221(31)(2)(1)231(2)(31)22372ttttttttttttt15.225525310xxxxxxxx16.2251455313145xxxxxxxx17.225525257105xxxxxxxxxxxx18.221122313132ssssssssssss19.2215(1)14(1)25(1)163456xxxxxxxxxxxx20.225(2)(2)25(2)237(2)(7)(3)( )1214(7)7xxxxxxxxxxxxxx xxx

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Section 0.3 The Algebra of Functions921.2255555(5)(5)255(5)552310531550xxxxxxxxxxxxxxxx22.111112122tttttt,t≠023.155151151515uuuuuuuu,u≠024.222222221111,013331xxxxxxxxx25.611xxfxx26.3266618125151h ttttt27.3251111xxxhxxx28.6661xg xx29.323232323251(51)1(51)515ttg tttttttt30.32326(51)(51)fxxxx31.222222()22xhxxxhhxxhh32.211()xxhhxhxx xhxxh33.2222224444(2)442(42)42ththtththtthhtthhthhhthhhth34.3332233223222255335533(33)33ththtt hthhtht hthhhtthhhhtthh35.a.221( )3000802023000160040CA tttttb.2(2)30001600(2)40(2)30003200160$6040C36.a.222(( )).1(105)25(105)200.1(10010025)2501252001024077.5Cf ttttttttb.2(4)10(4)240(4)77.5$1197.50C37.11( )(81)(81)88h xfxxxh(x) converts from British to U.S. sizes.38.f(x+ 1):[−10, 10] by [0, 20]f(x– 1):[−10, 10] by [0, 20](continued on next page)

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10Chapter 0 Functions(continued)f(x+ 2):[−10, 10] by [0, 20]f(x– 2):[−10, 10] by [0, 20]The graph off(x+a) is the graph off(x)shifted to the left (ifa> 0) or to the right (ifa< 0) byaunits.39.f(x) + 1:[−5, 5] by [−5, 15]f(x) – 1:[−5, 5] by [−5, 15]f(x) + 2:[−5, 5] by [−5, 15]f(x) – 2:[−5, 5] by [−5, 15]The graph off(x) +cis the graph off(x)shifted up (ifc> 0) or down (ifc< 0) bycunits.40.This is the graph of2( )fxxshifted 1 unit tothe right and 2 units up.[−5, 5] by [−5, 15]41.This is the graph of2( )fxxshifted 2 unitsto the left and 1 unit down.[−5, 5] by [−5, 15]42.[−4, 4] by [−10, 10]They are not the same function.43.[−15, 15] by [−10, 10] 1111,11xxxffxfxxxxxxxx

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Solution Manual for Calculus and Its Applications, 14th Edition - Page 15 preview image

Section 0.4 Zeros of Functions—The Quadratic Formula and Factoring110.4Zeros of Functions—The QuadraticFormula and Factoring1.2( )276fxxx22760xxa= 2,b= –7,c= 624494(2)(6)11bac247132,242bbacxa2.2( )321fxxx23210xxa= 3,b= 2,c= –122444(3)( 1)164bac242bbacxa241 ,1633.2( )4129fttt241290tt2212( 12)4(4)(9)422(4)120382bbacta4.21( )14fxxx21104xx 21241412114(1)422102bbacxa 5.2( )234fxxx 22340xx22334(–2)(–4)422(–2)3234bbacxa23is undefined, sof(x) has no real zeros.6.2( )1171faaa211710aa227( 7)4(11)(1)422(11)757575,222222bbacaa7.25410xx224( 4)4(5)( 1)422(5)4364611,10105bbacxa8.2450xx224( 4)4(1)(5)422(1)442bbacxa4is undefined, so there is no real solution.9.2151353000xx2242135( 135)4(15)(300)2(15)135225135155, 43030bbacxa10.25204zz225442224(1)272(1)22727,22bbacza11.236502xx 2322326( 6)4(5)42266662, 2333bbacxa

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Solution Manual for Calculus and Its Applications, 14th Edition - Page 16 preview image

12Chapter 0 Functions12.291240xx2212( 12)4(9)(4)422(9)1202183bbacxa13.2815(5)(3)xxxx14.21016(2)(8)xxxx15.216(4)(4)xxx16.21(1)(1)xxx17.222312123443(2)(2)3(2)xxxxxxx18.222212182692(3)(3)2(3)xxxxxxx19.22304221522(3)(5)xxxxxx  20.22151233( 54)3(5)(1)xxxxxx  21.23(3)xxxx22.241(21)(21)xxx23.32622 (3)233xxx xx xx  24.232166166(8)(2)(8)(2)xxxxxxxxxx xx 25.32111xxxx26.321255525xxxx27.3282723469xxxx28.321118224xxxx   29.2214497xxx30.221142xxx31.222563428100xxxxx228( 8)4(2)(–10)422(2)81448125,144bbacxay= 3x+ 4 = 15 + 4 = 19y= –3 + 4 = 1Points of intersection: (5, 19), (–1, 1)32.22109911180xxxxx(x– 9)(x– 2) = 0x= 9, 2y=x– 9 = 9 – 9 = 0y= 2 – 9 = –7Points of intersection: (9, 0), (2, –7)33.2244122yxxyxx22224412226802(34)02(4)(1)04,1xxxxxxxxxxx224444(4)44yxx2( 1)4( 1)49yPoints of intersection: (4, 4), (–1, 9)34.239yx2253yxx22239253560(3)(2)03,2xxxxxxxx 22393( 3)936yx23( 2)921yPoints of intersection: (–3, 36), (–2, 21)

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