Solution Manual for Mathematics for Elementary Teachers with Activities, 5th Edition

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SOLUTIONSMANUALDUANEKOUBAUniversity of California, DavisTHOMAS’CALCULUSEARLYTRANSCENDENTALSFOURTEENTHEDITIONBased on the original work byGeorge B. Thomas, JrMassachusetts Institute of Technologyas revised byJoel HassUniversity of California, DavisChristopher HeilGeorgia Institute of TechnologyMaurice D. WeirNaval Postgraduate School

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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 35Practice Exercises 45Additional and Advanced Exercises 552Limits and Continuity 612.1 Rates of Change and Tangents to Curves 612.2 Limit of a Function and Limit Laws 652.3 The Precise Definition of a Limit 752.4 One-Sided Limits 832.5 Continuity 882.6 Limits Involving Infinity; Asymptotes of Graphs 94Practice Exercises 105Additional and Advanced Exercises 1113Derivatives 1193.1 Tangents and the Derivative at a Point 1193.2 The Derivative as a Function 1253.3 Differentiation Rules 1363.4 The Derivative as a Rate of Change 1423.5 Derivatives of Trigonometric Functions 1483.6 The Chain Rule 1573.7 Implicit Differentiation 1683.8 Derivatives of Inverse Functions and Logarithms 1763.9 Inverse Trigonometric Functions 1863.10 Related Rates 1933.11 Linearization and Differentials 198Practice Exercises 206Additional and Advanced Exercises 220

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iv4Applications of Derivatives 2274.1 Extreme Values of Functions 2274.2 The Mean Value Theorem 2384.3 Monotonic Functions and the First Derivative Test 2454.4 Concavity and Curve Sketching 2594.5 Indeterminate Forms and L’Hôpital’s Rule 2874.6 Applied Optimization 2964.7 Newton's Method 3114.8 Antiderivatives 316Practice Exercises 326Additional and Advanced Exercises 3485Integrals 3555.1 Area and Estimating with Finite Sums 3555.2 Sigma Notation and Limits of Finite Sums 3605.3 The Definite Integral 3665.4 The Fundamental Theorem of Calculus 3815.5 Indefinite Integrals and the Substitution Method 3915.6 Definite Integral Substitutions and the Area Between Curves 398Practice Exercises 418Additional and Advanced Exercises 4346Applications of Definite Integrals 4436.1 Volumes Using Cross-Sections 4436.2 Volumes Using Cylindrical Shells 4556.3 Arc Length 4676.4 Areas of Surfaces of Revolution 4766.5 Work and Fluid Forces 4826.6 Moments and Centers of Mass 493Practice Exercises 507Additional and Advanced Exercises 5187Integrals and Transcendental Functions 5237.1 The Logarithm Defined as an Integral 5237.2 Exponential Change and Separable Differential Equations 5317.3 Hyperbolic Functions 5377.4 Relative Rates of Growth 545Practice Exercises 550Additional and Advanced Exercises 556

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v8Techniques of Integration 5598.1 Using Basic Integration Formulas 5598.2 Integration by Parts 5728.3 Trigonometric Integrals 5868.4 Trigonometric Substitutions 5958.5 Integration of Rational Functions by Partial Fractions 6048.6 Integral Tables and Computer Algebra Systems 6158.7 Numerical Integration 6268.8 Improper Integrals 6378.9 Probability 649Practice Exercises 658Additional and Advanced Exercises 6729First-Order Differential Equations 6819.1 Solutions, Slope Fields, and Euler's Method 6819.2 First-Order Linear Equations 6909.3 Applications 6949.4 Graphical Solutions of Autonomous Equations 6999.5 Systems of Equations and Phase Planes 706Practice Exercises 712Additional and Advanced Exercises 72010Infinite Sequences and Series 72310.1 Sequences 72310.2 Infinite Series 73510.3 The Integral Test 74310.4 Comparison Tests 75210.5 Absolute Convergence; The Ratio and Root Tests 76210.6 Alternating Series and Conditional Convergence 76810.7 Power Series 77810.8 Taylor and Maclaurin Series 79110.9 Convergence of Taylor Series 79710.10 The Binomial Series and Applications of Taylor Series 805Practice Exercises 814Additional and Advanced Exercises 825

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vi11Parametric Equations and Polar Coordinates 83111.1 Parametrizations of Plane Curves 83111.2 Calculus with Parametric Curves 84011.3 Polar Coordinates 85011.4 Graphing Polar Coordinate Equations 85511.5 Areas and Lengths in Polar Coordinates 86311.6 Conic Sections 86911.7 Conics in Polar Coordinates 880Practice Exercises 890Additional and Advanced Exercises 90112Vectors and the Geometry of Space 90712.1 Three-Dimensional Coordinate Systems 90712.2 Vectors 91212.3 The Dot Product 91812.4 The Cross Product 92312.5 Lines and Planes in Space 93012.6 Cylinders and Quadric Surfaces 939Practice Exercises 944Additional and Advanced Exercises 95213Vector-Valued Functions and Motion in Space 95913.1 Curves in Space and Their Tangents 95913.2 Integrals of Vector Functions; Projectile Motion 96613.3 Arc Length in Space 97513.4 Curvature and Normal Vectors of a Curve 97913.5 Tangential and Normal Components of Acceleration 98713.6 Velocity and Acceleration in Polar Coordinates 993Practice Exercises 996Additional and Advanced Exercises 1003

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vii14Partial Derivatives 100714.1 Functions of Several Variables 100714.2 Limits and Continuity in Higher Dimensions 101714.3 Partial Derivatives 102514.4 The Chain Rule 103414.5 Directional Derivatives and Gradient Vectors 104414.6 Tangent Planes and Differentials 105014.7 Extreme Values and Saddle Points 105914.8 Lagrange Multipliers 107514.9 Taylor's Formula for Two Variables 108714.10 Partial Derivatives with Constrained Variables 1090Practice Exercises 1093Additional and Advanced Exercises 111115Multiple Integrals 111715.1 Double and Iterated Integrals over Rectangles 111715.2 Double Integrals over General Regions 112015.3 Area by Double Integration 113415.4 Double Integrals in Polar Form 113915.5 Triple Integrals in Rectangular Coordinates 114515.6 Moments and Centers of Mass 115115.7 Triple Integrals in Cylindrical and Spherical Coordinates 115815.8 Substitutions in Multiple Integrals 1172Practice Exercises 1179Additional and Advanced Exercises 118616Integrals and Vector Fields 119316.1 Line Integrals 119316.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 119916.3 Path Independence, Conservative Fields, and Potential Functions 121116.4 Green's Theorem in the Plane 121716.5 Surfaces and Area 122516.6 Surface Integrals 123516.7 Stokes' Theorem 124616.8 The Divergence Theorem and a Unified Theory 1253Practice Exercises 1260Additional and Advanced Exercises 1270

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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.Case1: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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