Solution Manual for Precalculus, 11th Edition

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SOLUTIONSMANUALTIMBRITTJackson State Community CollegePRECALCULUSELEVENTHEDITIONMichael SullivanChicago State University

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Table of ContentsChapter 1Graphs1.1 The Distance and Midpoint Formulas......................................................................................... 11.2 Graphs of Equations in Two Variables; Intercepts; Symmetry ................................................ 131.3 Lines ......................................................................................................................................... 261.4 Circles ....................................................................................................................................... 43Chapter Review................................................................................................................................ 56Chapter Test..................................................................................................................................... 62Chapter Projects............................................................................................................................... 64Chapter 2Functions and Their Graphs2.1 Functions................................................................................................................................... 652.2 The Graph of a Function........................................................................................................... 832.3 Properties of Functions ............................................................................................................. 922.4 Library of Functions; Piecewise-defined Functions ............................................................... 1092.5 Graphing Techniques: Transformations ................................................................................. 1212.6 Mathematical Models: Building Functions............................................................................. 139Chapter Review.............................................................................................................................. 147Chapter Test................................................................................................................................... 154Cumulative Review ....................................................................................................................... 157Chapter Projects............................................................................................................................. 161Chapter 3Linear and Quadratic Functions3.1 Properties of Linear Functions and Linear Models................................................................. 1633.2 Building Linear Functions from Data ..................................................................................... 1743.3 Quadratic Functions and Their Properties .............................................................................. 1803.4 Build Quadratic Models from Verbal Descriptions and from Data ........................................ 2033.5 Inequalities Involving Quadratic Functions............................................................................ 211Chapter Review.............................................................................................................................. 231Chapter Test................................................................................................................................... 239Cumulative Review........................................................................................................................ 241Chapter Projects............................................................................................................................. 244Chapter 4Polynomial and Rational Functions4.1 Polynomial Functions ............................................................................................................. 2474.2 Graphing Polynomial Functions; Models ............................................................................... 2574.3 Properties of Rational Functions............................................................................................. 2734.4 The Graph of a Rational Function .......................................................................................... 2834.5 Polynomial and Rational Inequalities ..................................................................................... 3394.6 The Real Zeros of a Polynomial Function .............................................................................. 3604.7 Complex Zeros; Fundamental Theorem of Algebra ............................................................... 391Chapter Review.............................................................................................................................. 400Chapter Test................................................................................................................................... 415Cumulative Review........................................................................................................................ 419Chapter Projects............................................................................................................................. 424

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Chapter 5Exponential and Logarithmic Functions5.1 Composite Functions .............................................................................................................. 4265.2 One-to-One Functions; Inverse Functions .............................................................................. 4445.3 Exponential Functions ............................................................................................................ 4665.4 Logarithmic Functions............................................................................................................ 4875.5 Properties of Logarithms ........................................................................................................ 5095.6 Logarithmic and Exponential Equations................................................................................. 5185.7 Financial Models .................................................................................................................... 5385.8 Exponential Growth and Decay Models; Newton’s Law; Logistic Growthand Decay Models .................................................................................................................... 5465.9 Building Exponential, Logarithmic, and Logistic Models from Data..................................... 555Chapter Review.............................................................................................................................. 561Chapter Test................................................................................................................................... 573Cumulative Review........................................................................................................................ 577Chapter Projects............................................................................................................................. 580Chapter 6Trigonometric Functions6.1 Angles, Arc Length, and Circular Motion .............................................................................. 5836.2 Trigonometric Functions: Unit Circle Approach .................................................................... 5926.3 Properties of the Trigonometric Functions ............................................................................. 6106.4 Graphs of the Sine and Cosine Functions ............................................................................... 6246.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions...................................... 6456.6 Phase Shift; Sinusoidal Curve Fitting ..................................................................................... 655Chapter Review.............................................................................................................................. 666Chapter Test................................................................................................................................... 674Cumulative Review........................................................................................................................ 678Chapter Projects............................................................................................................................. 682Chapter 7Analytic Trigonometry7.1 The Inverse Sine, Cosine, and Tangent Functions.................................................................. 6857.2 The Inverse Trigonometric Functions (Continued) ................................................................ 6987.3 Trigonometric Equations ........................................................................................................ 7107.4 Trigonometric Identities ......................................................................................................... 7317.5 Sum and Difference Formulas ................................................................................................ 7447.6 Double-angle and Half-angle Formulas.................................................................................. 7697.7 Product-to-Sum and Sum-to-Product Formulas...................................................................... 795Chapter Review.............................................................................................................................. 808Chapter Test................................................................................................................................... 823Cumulative Review........................................................................................................................ 828Chapter Projects............................................................................................................................. 834Chapter 8Applications of Trigonometric Functions8.1 Right Triangle Trigonometry; Applications ........................................................................... 8388.2 The Law of Sines .................................................................................................................... 8528.3 The Law of Cosines ................................................................................................................ 8678.4 Area of a Triangle ................................................................................................................... 8798.5 Simple Harmonic Motion; Damped Motion; Combining Waves ........................................... 889Chapter Review.............................................................................................................................. 899Chapter Test................................................................................................................................... 905Cumulative Review........................................................................................................................ 909Chapter Projects............................................................................................................................. 915

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Chapter 9Polar Coordinates; Vectors9.1 Polar Coordinates.................................................................................................................... 9199.2 Polar Equations and Graphs.................................................................................................... 9289.3 The Complex Plane; De Moivre’s Theorem ........................................................................... 9589.4 Vectors.................................................................................................................................... 9719.5 The Dot Product...................................................................................................................... 9849.6 Vectors in Space ..................................................................................................................... 9909.7 The Cross Product................................................................................................................... 996Chapter Review............................................................................................................................ 1007Chapter Test................................................................................................................................. 1016Cumulative Review...................................................................................................................... 1020Chapter Projects........................................................................................................................... 1023Chapter 10Analytic Geometry10.2 The Parabola ....................................................................................................................... 102610.3 The Ellipse .......................................................................................................................... 104110.4 The Hyperbola .................................................................................................................... 105810.5 Rotation of Axes; General Form of a Conic ....................................................................... 107810.6 Polar Equations of Conics................................................................................................... 109110.7 Plane Curves and Parametric Equations ............................................................................. 1100Chapter Review............................................................................................................................ 1115Chapter Test................................................................................................................................. 1124Cumulative Review...................................................................................................................... 1129Chapter Projects........................................................................................................................... 1131Chapter 11Systems of Equations and Inequalities11.1 Systems of Linear Equations: Substitution and Elimination............................................... 113511.2 Systems of Linear Equations: Matrices .............................................................................. 115811.3 Systems of Linear Equations: Determinants ....................................................................... 118311.4 Matrix Algebra.................................................................................................................... 119711.5 Partial Fraction Decomposition .......................................................................................... 121611.6 Systems of Nonlinear Equations ......................................................................................... 123511.7 Systems of Inequalities ....................................................................................................... 126311.8 Linear Programming ........................................................................................................... 1279Chapter Review............................................................................................................................ 1292Chapter Test................................................................................................................................. 1307Cumulative Review...................................................................................................................... 1316Chapter Projects........................................................................................................................... 1319Chapter 12Sequences; Induction; the Binomial Theorem12.1 Sequences............................................................................................................................ 132212.2 Arithmetic Sequences ......................................................................................................... 133212.3 Geometric Sequences; Geometric Series ............................................................................ 134112.4 Mathematical Induction ...................................................................................................... 135312.5 The Binomial Theorem ....................................................................................................... 1362Chapter Review............................................................................................................................ 1369Chapter Test................................................................................................................................. 1373Cumulative Review...................................................................................................................... 1376Chapter Projects........................................................................................................................... 1379

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Chapter 13Counting and Probability13.1 Counting.............................................................................................................................. 138213.2 Permutations and Combinations ......................................................................................... 138513.3 Probability........................................................................................................................... 1390Chapter Review............................................................................................................................ 1397Chapter Test................................................................................................................................. 1399Cumulative Review...................................................................................................................... 1400Chapter Projects........................................................................................................................... 1403Chapter 14A Preview of Calculus: The Limit, Derivative, and Integral of a Function14.1 Investigating Limits Using Tables and Graphs................................................................... 140614.2 Algebraic Techniques for Finding Limits ........................................................................... 141214.3 One-sided Limits; Continuity.............................................................................................. 141614.4 The Tangent Problem; The Derivative................................................................................ 142314.5 The Area Problem; The Integral ......................................................................................... 1432Chapter Review............................................................................................................................ 1446Chapter Test................................................................................................................................. 1453Chapter Projects........................................................................................................................... 1456Appendix AReviewA.1 Algebra Essentials................................................................................................................ 1462A.2 Geometry Essentials............................................................................................................. 1467A.3 Polynomials.......................................................................................................................... 1473A.4 Synthetic Division................................................................................................................ 1481A.5 Rational Expressions............................................................................................................ 1483A.6 Solving Equations ................................................................................................................ 1488A.7 Complex Numbers; Quadratic Equations in the Complex Number System ........................ 1502A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications ..... 1508A.9 Interval Notation; Solving Inequalities ................................................................................ 1515A.10nth Roots; Rational Exponents............................................................................................ 1527Appendix BGraphing UtilitiesB.1 The Viewing Rectangle........................................................................................................ 1537B.2 Using a Graphing Utility to Graph Equations ...................................................................... 1538B.3 Using a Graphing Utility to Locate Intercepts and Check for Symmetry ............................ 1543B.5 Square Screens ..................................................................................................................... 1545

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1Chapter 1GraphsSection 1.11.02.5388 3.22342554.22211601213600372161Since the sum of the squares of two of the sidesof the triangle equals the square of the third side,the triangle is a right triangle.5.12bh6.true7.x-coordinate or abscissa;y-coordinate orordinate8.quadrants9.midpoint10.False; the distance between two points is nevernegative.11.False; points that lie in Quadrant IV will have apositivex-coordinate and a negativey-coordinate.The point1, 4lies in Quadrant II.12.True;1212,22xxyyM 13.b14.a15.(a)Quadrant II(b)x-axis(c)Quadrant III(d)Quadrant I(e)y-axis(f)Quadrant IV16.(a)Quadrant I(b)Quadrant III(c)Quadrant II(d)Quadrant I(e)y-axis(f)x-axis17.The points will be on a vertical line that is twounits to the right of they-axis.

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Chapter 1:Graphs218.The points will be on a horizontal line that isthree units above thex-axis.19.221222(,)(20)(10)21415d P P20.221222(,)( 20)(10)( 2)1415d P P21.221222(,)( 21)(21)( 3)19110d P P22.221222(,)2( 1)(21)319110d P P 23. 221222(,)(53)4428464682 17d P P 24. 221222(,)214034916255d P P 25.221222(,)4( 7)(03)11(3)1219130  d P P26.221222(,)422( 3)2542529d P P 27.221222(,)(65)1( 2)131910 d P P28.221222(,)6(4)2( 3)1051002512555d P P  29.221222(,)2.3( 0.2)1.1(0.3)2.50.86.250.646.892.62 d P P30.221222(,)0.31.21.12.3( 1.5)( 1.2)2.251.443.691.92 d P P31.22122222(,)(0)(0)()()d P Pababab 32.221222222(,)(0)(0)()()22d P Paaaaaaaa 33.( 2,5),(1,3),( 1, 0)ABC  222222222222(,)1( 2)(35)3( 2)9413(,)11(03)( 2)( 3)4913(,)1( 2)(05)1( 5)12526d A Bd B Cd A C       

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Section 1.1:The Distance and Midpoint Formulas3Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)1313261313262626d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)2111313132213 square units2Ad A Bd B C34.( 2, 5),(12, 3),(10,11)ABC 222222222222(,)12( 2)(35)14( 2)1964200102(,)1012( 113)( 2)( 14)4196200102(,)10( 2)( 115)12(16)14425640020d A Bd B Cd A C       Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)10210220200200400400400d A Bd B Cd A CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21 102 10221 100 2100 square units2Ad A Bd B C35.(5,3),(6, 0),(5,5)ABC 222222222222(,)6(5)(03)11(3)1219130(,)56(50)(1)512526(,)5(5)(53)1021004104226d A Bd B Cd A C   Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:

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Chapter 1:Graphs4222222(,)(,)(,)1042613010426130130130d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 1(,)(,)211042621 2262621 2 26226 square unitsAd A Cd B C36.( 6, 3),(3,5),( 1, 5)ABC  222222222222(,)3( 6)( 53)9( 8)8164145(,)13(5( 5))( 4)1016100116229(,)1(6)(53)5225429d A Bd B Cd A C       Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)29229145294 2914529116145145145d A Cd B Cd A BThe area of a triangle is12Abh. In thisproblem, 1(,)(,)2129 22921 2 29229 square unitsAd A Cd B C37.(4,3),(0,3),(4, 2)ABC222222222222(,)(04)3( 3)(4)0160164(,)402( 3)45162541(,)(44)2( 3)05025255d A Bd B Cd A C    Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:

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Section 1.1:The Distance and Midpoint Formulas5222222(,)(,)(,)45411625414141d A Bd A Cd B CThe area of a triangle is12Abh. In thisproblem, 1(,)(,)21 4 5210 square unitsAd A Bd A C38.(4,3),(4, 1),(2, 1)ABC222222222222(,)(44)1( 3)04016164(,)2411( 2)04042(,)(24)1( 3)( 2)44162025d A Bd B Cd A C  Verifying that ∆ ABC is a right triangle by thePythagorean Theorem:222222(,)(,)(,)4225164202020d A Bd B Cd A CThe area of a triangle is12Abh. In this problem, 1(,)(,)21 4 224 square unitsAd A Bd B C39.The coordinates of the midpoint are:1212( ,),224435 ,2280,22(4, 0)xxyyx y   40.The coordinates of the midpoint are:1212( ,),222204,2204,220, 2xxyyx y   41.The coordinates of the midpoint are:1212( ,),221840,2274,227 , 22     xxyyx y

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Chapter 1:Graphs642.The coordinates of the midpoint are:1212( ,),222432,2261,2213,2xxyyx y   43.The coordinates of the midpoint are:1212( ,),227951,22416 ,22(8,2)   xxyyx y44.The coordinates of the midpoint are:1212( ,),224232,2221,2211,2xxyyx y   45.The coordinates of the midpoint are:1212( ,),2200,22,22xxyyx yabab   46.The coordinates of the midpoint are:1212( ,),2200,22,22xxyyx yaaaa   47.The x coordinate would be235and the ycoordinate would be523. Thus the newpoint would be5,3.48.The new x coordinate would be123  andthe new y coordinate would be6410. Thusthe new point would be3,1049.a.If we use a right triangle to solve theproblem, we know the hypotenuse is 13 units inlength. One of the legs of the triangle will be2+3=5. Thus the other leg will be:222225132516914412bbbbThus the coordinates will have an y value of11213  and11211 . So the pointsare3,11and3,13.b.Consider points of the form3,ythat are adistance of 13 units from the point2,1. 2221212222223( 2)1512512226dxxyyyyyyyy     22222213226132261692260214301113yyyyyyyyyy11011yyor13013yy Thus, the points3,11and3,13are adistance of 13 units from the point2,1.

Solution Manual for Precalculus, 11th Edition - Page 13 preview image

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Section 1.1:The Distance and Midpoint Formulas750.a.If we use a right triangle to solve theproblem, we know the hypotenuse is 17 units inlength. One of the legs of the triangle will be2+6=8. Thus the other leg will be:222228176428922515bbbbThus the coordinates will have an x value of11514 and11516. So the points are14,6and16,6.b.Consider points of the form,6xthat area distance of 17 units from the point1, 2. 2221212222221262182164265dxxyyxxxxxxx 22222217265172652892650222401416xxxxxxxxxx14014xx or16016xxThus, the points14,6and16,6are adistance of 13 units from the point1, 2.51.Points on thex-axis have ay-coordinate of 0. Thus,we consider points of the form, 0xthat are adistance of 6 units from the point4,3.22212122222243016831689825dxxyyxxxxxxx  222222268256825368250811( 8)( 8)4(1)( 11)2(1)864448108228634332xxxxxxxxx 433xor433xThus, the points433, 0and433, 0areon thex-axis and a distance of 6 units from thepoint4,3.52.Points on they-axis have anx-coordinate of 0.Thus, we consider points of the form0,ythatare a distance of 6 units from the point4,3.2221212222224034961696625dxxyyyyyyyyy 

Solution Manual for Precalculus, 11th Edition - Page 14 preview image

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Chapter 1:Graphs8222222266256625366250611( 6)(6)4(1)( 11)2(1)63644680226453252yyyyyyyyy 325y or325y Thus, the points0,325and0,325are on they-axis and a distance of 6 units from thepoint4,3.53.a.To shift 3 units left and 4 units down, wesubtract 3 from thex-coordinate and subtract4 from they-coordinate.23,541,1b.To shift left 2 units and up 8 units, wesubtract 2 from thex-coordinate and add 8 tothey-coordinate.22,580,1354.Let the coordinates of pointBbe,xy. Usingthe midpoint formula, we can write182,3,22xy  .This leads to two equations we can solve.122145xxx  832862yyy PointBhas coordinates5,2.55.1212,,22xxyyMx y .111,( 3, 6)Pxy and( ,)( 1, 4)x y , so122222312231xxxxxx and122222642862yyyyyyThus,2(1, 2)P.56.1212,,22xxyyMx y .222,(7,2)Pxyand( ,)(5,4)x y, so1211127521073xxxxxxand121112( 2)428( 2)6yyyyyy  Thus,1(3,6)P.57.The midpoint of AB is:0600,223, 0D The midpoint of AC is:0404,222, 2E The midpoint of BC is:6404,225, 2F 2222(,)04(34)(4)(1)16117d C D 2222(,)26(20)(4)21642025d B E2222(,)(20)(50)2542529d A F

Solution Manual for Precalculus, 11th Edition - Page 15 preview image

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Section 1.1:The Distance and Midpoint Formulas958.Let12(0, 0),(0, 4),( ,)PPPx y221222122222222222,(00)(40)164,(0)(0)416,(0)(4)(4)4(4)16dPPdPPxyxyxydPPxyxyxyTherefore,222248168162yyyyyyywhich gives2222161223xxx Two triangles are possible. The third vertex is23, 2or23, 2.59.221222(,)(42)(11)(6)0366d P P222322(,)4(4)( 31)0(4)164d PP   221322(,)(42)( 31)(6)(4)3616522 13d P P  Since222122313(,)(,)(,)d P Pd PPd P P,the triangle is a right triangle.60.221222(,)6( 1)(24)7(2)49453d P P  222322(,)46( 52)(2)(7)44953d PP  221322(,)4( 1)( 54)5(9)2581106d P P   Since222122313(,)(,)(,)d P Pd PPd P P,the triangle is a right triangle.Since1223,,dPPdPP, the triangle isisosceles.Therefore, the triangle is an isosceles righttriangle.61.221222(,)0(2)7( 1)28464682 17d P P  222322(,)30(27)3(5)92534d PP 221322(,)3( 2)2( 1)5325934d P P  Since2313(,)(,)d PPd P P, the triangle isisosceles.Since222132312(,)(,)(,)d P Pd PPd P P,the triangle is also a right triangle.Therefore, the triangle is an isosceles righttriangle.

Solution Manual for Precalculus, 11th Edition - Page 16 preview image

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Chapter 1:Graphs1062.221222(,)4702( 11)(2)121412555d P P 222322(,)4(4)(60)86643610010d PP 221322(,)4762( 3)4916255d P PSince222132312(,)(,)(,)d P Pd PPd P P,the triangle is a right triangle.63.Using the Pythagorean Theorem:222229090810081001620016200902127.28 feetdddd90909090d64.Using the Pythagorean Theorem:222226060360036007200720060284.85 feetdddd60606060d65.a.First: (90, 0), Second: (90, 90),Third: (0, 90)(0,0)(0,90)(90,0)(90,90)XYb.Using the distance formula:2222(31090)(1590)220( 75)5402552161232.43 feetd c.Using the distance formula:2222(3000)(30090)30021013410030 149366.20 feetd66.a.First: (60, 0), Second: (60, 60)Third: (0, 60)(0,0)(0,60)(60,0)(60,60)xyb.Using the distance formula:2222(18060)(2060)120(40)1600040 10126.49 feetd c.Using the distance formula:2222(2200)(22060)2201607400020 185272.03 feetd

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