Solution Manual for University Calculus: Early Transcendentals , 4th Edition

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SOLUTIONSMANUALDUANEKOUBAJENNIFERA.BLUEUniversity of California, DavisSUNY Empire State CollegeUNIVERSITYCALCULUSEARLYTRANSCENDENTALSFOURTHEDITIONJoel HassUniversity of California, DavisChristopher HeilGeorgia Institute of TechnologyPrzemyslaw BogackiOld Dominion UniversityMaurice D. WeirNaval Postgraduate SchoolGeorge B. Thomas, Jr.Massachusetts Institute of Technology

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iiiTABLE OF CONTENTS1Functions 11.1 Functions and Their Graphs 11.2 Combining Functions; Shifting and Scaling Graphs 91.3 Trigonometric Functions 191.4 Graphing with Software 271.5 Exponential Functions 321.6 Inverse Functions and Logarithms 352Limits and Continuity 472.1 Rates of Change and Tangent Lines to Curves 472.2 Limit of a Function and Limit Laws 512.3 The Precise Definition of a Limit 612.4 One-Sided Limits 702.5 Continuity 752.6 Limits Involving Infinity; Asymptotes of Graphs 81Practice Exercises 92Additional and Advanced Exercises 983Derivatives 1073.1 Tangent Lines and the Derivative at a Point 1073.2 The Derivative as a Function 1133.3 Differentiation Rules 1243.4 The Derivative as a Rate of Change 1303.5 Derivatives of Trigonometric Functions 1363.6 The Chain Rule 1433.7 Implicit Differentiation 1553.8 Derivatives of Inverse Functions and Logarithms 1633.9 Inverse Trigonometric Functions 1733.10 Related Rates 1803.11 Linearization and Differentials 185Practice Exercises 193Additional and Advanced Exercises 207

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iv4Applications of Derivatives 2134.1 Extreme Values of Functions on Closed Intervals 2134.2 The Mean Value Theorem 2224.3 Monotonic Functions and the First Derivative Test 2294.4 Concavity and Curve Sketching 2434.5 Indeterminate Forms and L’Hôpital’s Rule 2674.6 Applied Optimization 2764.7 Newton's Method 2914.8 Antiderivatives 296Practice Exercises 306Additional and Advanced Exercises 3275Integrals 3355.1 Area and Estimating with Finite Sums 3355.2 Sigma Notation and Limits of Finite Sums 3405.3 The Definite Integral 3465.4 The Fundamental Theorem of Calculus 3615.5 Indefinite Integrals and the Substitution Method 3725.6 Definite Integral Substitutions and the Area Between Curves 380Practice Exercises 399Additional and Advanced Exercises 4156Applications of Definite Integrals 4216.1 Volumes Using Cross-Sections 4216.2 Volumes Using Cylindrical Shells 4336.3 Arc Length 4456.4 Areas of Surfaces of Revolution 4546.5 Work and Fluid Forces 4596.6 Moments and Centers of Mass 466Practice Exercises 479Additional and Advanced Exercises 4897Integrals and Transcendental Functions 4957.1 The Logarithm Defined as an Integral 4957.2 Exponential Change and Separable Differential Equations 5027.3 Hyperbolic Functions 507Practice Exercises 516Additional and Advanced Exercises 520

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v8Techniques of Integration 5238.1 Integration by Parts 5238.2 Trigonometric Integrals 5378.3 Trigonometric Substitutions 5468.4 Integration of Rational Functions by Partial Fractions 5568.5 Integral Tables and Computer Algebra Systems 5668.6 Numerical Integration 5788.7 Improper Integrals 588Practice Exercises 601Additional and Advanced Exercises 6159Infinite Sequences and Series 6239.1 Sequences 6239.2 Infinite Series 6359.3 The Integral Test 6449.4 Comparison Tests 6529.5 Absolute Convergence; The Ratio and Root Tests 6639.6 Alternating Series and Conditional Convergence 6699.7 Power Series 6799.8 Taylor and Maclaurin Series 6919.9 Convergence of Taylor Series 6979.10 The Binomial Series and Applications of Taylor Series 705Practice Exercises 714Additional and Advanced Exercises 72510Parametric Equations and Polar Coordinates 73110.1 Parametrizations of Plane Curves 73110.2 Calculus with Parametric Curves 73710.3 Polar Coordinates 74710.4 Graphing Polar Coordinate Equations 75310.5 Areas and Lengths in Polar Coordinates 761Practice Exercises 767Additional and Advanced Exercises 774

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vi11Vectors and the Geometry of Space 77711.1 Three-Dimensional Coordinate Systems 77711.2 Vectors 78211.3 The Dot Product 78811.4 The Cross Product 79411.5 Lines and Planes in Space 80111.6 Cylinders and Quadric Surfaces 809Practice Exercises 815Additional and Advanced Exercises 82312Vector-Valued Functions and Motion in Space 82912.1 Curves in Space and Their Tangents 82912.2 Integrals of Vector Functions; Projectile Motion 83612.3 Arc Length in Space 84212.4 Curvature and Normal Vectors of a Curve 84612.5 Tangential and Normal Components of Acceleration 85312.6 Velocity and Acceleration in Polar Coordinates 857Practice Exercises 859Additional and Advanced Exercises 86613Partial Derivatives 86913.1 Functions of Several Variables 86913.2 Limits and Continuity in Higher Dimensions 87913.3 Partial Derivatives 88713.4 The Chain Rule 89613.5 Directional Derivatives and Gradient Vectors 90613.6 Tangent Planes and Differentials 91213.7 Extreme Values and Saddle Points 92113.8 Lagrange Multipliers 937Practice Exercises 949Additional and Advanced Exercises 965

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vii14Multiple Integrals 97114.1 Double and Iterated Integrals over Rectangles 97114.2 Double Integrals over General Regions 97414.3 Area by Double Integration 98814.4 Double Integrals in Polar Form 99314.5 Triple Integrals in Rectangular Coordinates 99914.6 Applications 100514.7 Triple Integrals in Cylindrical and Spherical Coordinates 101114.8 Substitutions in Multiple Integrals 1024Practice Exercises 1031Additional and Advanced Exercises 103915Integrals and Vector Fields 104515.1 Line Integrals 104515.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 105115.3 Path Independence, Conservative Fields, and Potential Functions 106315.4 Green's Theorem in the Plane 106915.5 Surfaces and Area 107715.6 Surface Integrals 108715.7 Stokes' Theorem 109815.8 The Divergence Theorem and a Unified Theory 1105Practice Exercises 1112Additional and Advanced Exercises 112216First-Order Differential Equations 112716.1 Solutions, Slope Fields, and Euler's Method 112716.2 First-Order Linear Equations 113716.3 Applications 114116.4 Graphical Solutions of Autonomous Equations 114516.5 Systems of Equations and Phase Planes 1152Practice Exercises 1158Additional and Advanced Exercises 1166

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viii17Second-Order Differential Equations 116917.1 Second-Order Linear Equations 116917.2 Nonhomogeneous Linear Equations 117417.3 Applications 118217.4 Euler Equations 118617.5 Power-Series Solutions 1189BAppendix 1197B.1 Relative Rates of Growth 1197B.2 Probability 1202B.3 Conics in Polar Coordinates 1210B.4 Taylor’s Formula for Two Variables 1220B.5 Partial Derivatives with Constrained Variables 1223

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1CHAPTER 1 FUNCTIONS1.1FUNCTIONS AND THEIR GRAPHS1.domain(,); range[1,) 2.domain[0,); range(, 1] 3.domain[ 2,);y in range andy510x0ycan be any positive real numberrange[0,).4.domain(, 0][3,);yin range and230yxxycan be any positive real numberrange[0,).5.domain(, 3)(3,);y in range and43,tynow if433300,tttor if3t43300ttycan be any nonzero real numberrange(, 0)(0,). 6.domain(,4)(4, 4)(4,);y  in range and2216,tynow if2221641600,ttt or if222216164416160ttt, or if2221641600tttycan be any nonzeroreal number18range(,](0,).  7.(a)Not the graph of a function ofxsince it fails the vertical line test.(b)Is the graph of a function ofxsince any vertical line intersects the graph at most once.8.(a)Not the graph of a function ofxsince it fails the vertical line test.(b)Not the graph of a function ofxsince it fails the vertical line test.9.base222322; (height)height;xxxxarea is12( )a x(base)(height)2331224( );xxxperimeter is( )3 .pxxxxx10.2222side length;dsssdsand area is2212asad11.LetDdiagonal length of a face of the cube andthe length of an edge. Then222Ddand2222323.dDdThe surface area is2226362ddand the volume is233/23333.dd12.The coordinates ofPare,xxso the slope of the line joiningPto the origin is1(0).xxxmxThus,211,,.mmxx13.2222225552511124244416245;(0)(0)()xyyxLxyxxxxx  22202025202025255254416164xxxxxx14.22222222233;(4)(0)(34)(1)yxyx Lxyyyyy42242211yyyyy

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2Chapter 1 Functions15.The domain is(,). 16.The domain is(,). 17.The domain is(,). 18.The domain is(, 0].19.The domain is(, 0)(0,).20.The domain is(, 0)(0,).21.The domain is(,5)( 5,3][3, 5)(5,)  22.The range is[5,).23.Neither graph passes the vertical line test(a)(b)

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Section 1.1 Functions and Their Graphs324.Neither graph passes the vertical line test(a)(b)111oror11xyyxxyxyyx   25.012010xy26.012100xy27.224,1( )2 ,1xxF xxxx28.1,0( ), 0xxG xxx29.(a)Line through (0, 0) and (1, 1):;yxLine through (1, 1) and (2, 0):2yx ,01( )2, 12xxfxxx(b)2,010,12( )2,230,34xxfxxx30.(a)Line through (0, 2) and (2, 0):2yx Line through (2, 1) and (5, 0):01115233,m so511333(2)1yxx  51332, 02( ),25xxfxxx

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4Chapter 1 Functions(b)Line through( 1, 0)and(0,3):300( 1)3,m so33yx Line through (0, 3) and134202(2,1) :2,m so23yx 33,10( )23,02xxfxxx31.(a)Line through( 1, 1)and (0, 0):yx Line through (0, 1) and (1, 1):1yLine through (1, 1) and (3, 0):01113122,m so311222(1)1yxx  312210( )10113xxfxxxx(b)Line through( 2,1)and (0, 0):12yxLine through (0, 2) and (1, 0):22yx Line through(1,1)and(3,1):1y 32.(a)Line through2, 0Tand (T, 1):102(/2),TTTmso22201TTTyxx2220, 0( )1,TTTxfxxxT(b)223232,0,( ),,2TTTTAxAxTfxATxAxT33.(a)0 for[0, 1)xx  (b)0 for( 1, 0]xx   34.xx    only whenxis an integer.35.For any real number,1,xnxnwherenis an integer. Now:1(1).nxnnxn  By definition:and.xnxnxn      Soxx   for all realx.36.To findf(x) you delete the decimal orfractional portion ofx, leaving onlythe integer part.1220( )2201113xxfxxxx

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Section 1.1 Functions and Their Graphs537.Symmetric about the originDec:x  Inc: nowhere38.Symmetric about they-axisDec:0x Inc:0x 39.Symmetric about the originDec: nowhereInc:00xx  40.Symmetric about they-axisDec:0x Inc:0x 41.Symmetric about they-axisDec:0x Inc:0x 42.No symmetryDec:0x Inc: nowhere

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6Chapter 1 Functions43.Symmetric about the originDec: nowhereInc:x  44.No symmetryDec:0x Inc: nowhere45.No symmetryDec:0x Inc: nowhere46.Symmetric about they-axisDec:0x Inc:0x 47.Since a horizontal line not through the origin is symmetric with respect to they-axis, but not with respect to theorigin, the function is even.48.55555111()( )and()()( ).xxxfxxfxxfx   Thus the function is odd.49.Since22( )1()1().fxxxfx The function is even.50.Since22[( )][()()]fxxxfxxx and22[( )][( )( )]fxxxfxxx  the function is neithereven nor odd.51.Since333( ),()()( ).gxxx gxxxxxg x   So the function is odd.52.4242( )31()3()1(),gxxxxxgxthus the function is even.53.22111()1( )().xxg xgxThus the function is even.54.2211( );()( ).xxxxg xgxg x  So the function is odd.55.111111( );();( ).ttth thth tSince( )( ) and( )(),h th th tht the function is neither even nor odd.

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Section 1.1 Functions and Their Graphs756.Since33|||() |,( )()tth thtand the function is even.57.( )21,()21.h tthtt So( )().( )21,h thth tt so( )( ).h th t The function is neither evennor odd.58.( )2| |1 and()2||12| |1.h tthtttSo( )()h thtand the function is even.59.( )sin 2 ;()sin 2( ).gxx gxxg x  So the function is odd.60.22( )sin;()sin( ).gxxgxxg xSo the function is even.61.( )cos3 ;()cos3( ).gxx gxxg xSo the function is even.62.( )1cos;()1cos( ).gxx gxxg xSo the function is even.63.11133325(75); 60180sktkksttt64.222212960(18)4040;40(10)4000 joulesKc vccKvK65.24241245624; 10kksssrkrs66.314700147002450010003914.714700; 23.4628.2 inkkVVVPkPV67.32( )(142 )(222 )472308 ; 07.Vfxxxxxxxx68.(a)Lethheight of the triangle. Since the triangle is isosceles,22222.ABABABSo,222121hhBis at(0, 1)slope of1AB The equation ofABis( )1;[0, 1].yfxxx (b)2( )22 (1[0, 1].)22 ;A xxyxxxx x69.(a)Graphhbecause it is an even function and rises less rapidly than does Graphg.(b)Graphfbecause it is an odd function.(c)Graphgbecause it is an even function and rises more rapidly than does Graphh.70.(a)Graphfbecause it is linear.(b)Graphgbecause it contains (0, 1).(c)Graphhbecause it is a nonlinear odd function.

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8Chapter 1 Functions71.(a)From the graph,421( 2, 0)(4,)xxx (b)4422110xxxx228(4)(2)42220:1000xxxxxxxxx4xsincexis positive;228(4)(2)42220:1000xxxxxxxxx2x sincexis negative;sign of(4)(2)xxSolution interval:( 2, 0)(4,)72.(a)From the graph,3211(,5)( 1, 1)xxx   (b)Case1:x 3(1)321112xxxx33225.xxx Thus,(,5)x  solves the inequality.Case11:x3211xx3(1)12xx33225xxx whichis true if1.x Thus,( 1, 1)x solves the inequality.Case 1:x32113xxx322x5x which is never true if1,xso no solution here.In conclusion,(,5)( 1, 1).x   73.A curve symmetric about thex-axis will not pass the vertical line test because the points (x,y) and( ,)xylieon the same vertical line. The graph of the function( )0yfxis thex-axis, a horizontal line for which thereis a singley-value, 0, for anyx.74.price405 ,xquantity30025x( )Rx(405 )(30025 )xx75.222222;hhxxhxcost5(2 )10xh22( )1010hC hh522h76.(a)Note that2 mi10,560 ft,so there are22800xfeet of river cable at $180 per foot and(10,560)xfeet of land cable at $100 per foot. The cost is22( )180800C xx100(10,560-x).(b)(0)$1, 200, 000(500)$1,175,812(1000)$1,186,512(1500)$1, 212, 000(2000)$1, 243, 732(2500)$1, 278, 479(3000)$1,314,870CCCCCCCValues beyond this are all larger. It would appear that the least expensive location is less than 2000 feetfrom the pointP.

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