precalculus - mcdougal littell · graphical and algebraic approach to example solutions and...

IndianaAcademic Standards for

Precalculus

correlated to the

6/20032001

CC2

PRECALCULUSWITH LIMITS

A GRAPHING APPROACH

i

Introduction

to

Precalculus with Limits: A Graphing Approach © 2001

by Roland E. Larson, Robert P. Hostetler, and Bruce H. Edwards

Precalculus with Limits: A Graphing Approach strengthens students’ conceptual understandingand problem-solving skills by the frequent use of graphic calculators and computers and isintended for courses in which graphic calculators are readily available to all students. The textincorporates technology, problem-solving strategies, real-life applications, and conceptreinforcement to help students develop strong precalculus skills. The text adds a side-by-sidegraphical and algebraic approach to example solutions and features study guides for each chapter.Students are also given section objectives that outline the main concepts, examples that show thereal-life relevance of mathematical concepts, and concise chapter summaries.

Special Features

• Explorations exercises provide an opportunity for active participation.• Writing About Mathematics activities can be used for individual or group work, and offer

students an opportunity to express their ideas pertaining to mathematical concepts throughwritten explanations.

• Chapter Summaries are found at the end of every chapter and reinforce the section objectives.These objectives are correlated to the chapter Review Exercises.

• Synthesis exercises consist of conceptual and critical thinking exercises that require students toexpand upon and synthesize the section concepts.

A complete listing of program components is provided on the following page.

ii

Precalculus with Limits: A Graphing Approach © 2001Components

Pupil’s EditionInstructor’s Annotated Edition

AncillariesStudy and Solutions GuideTest Item FileComplete Solutions GuideGraphing Technology Guide

Technology HM Testing v6.0 (Test Generator)HM ClassPrep CD-ROM with HM Testing v6.0• includes test items, Student Success Organizer worksheets, and solutions to accompany the

material in each chapterGraphing Technology VideoVideo ProgramLearning Tools Student CD-ROM• includes animations, simulations, review exercises, and other tools to support the material in

the textInteractive CD-ROM (entire book on CD-ROM)Textbook web site

PE = Pupil’s Edition, IAE = Instructor’s Annotated Edition

Selected exercises are referenced in parentheses, otherwise entire page is applicable.1

Precalculus With Limits: A Graphing Approach © 2001correlated to

The Indiana Academic Standards for Precalculus

INSTRUCTION APPLICATIONPupil’s Edition andTeacher’s Edition

Print Ancillaries,Transparencies andTechnology

STANDARD 1 Relations and Functions Students use polynomial, rational, and algebraic functions to writefunctions and draw graphs to solve word problems, to find composite and inverse functions, and to analyzefunctions and graphs. They analyze and graph circles, ellipses, parabolas, and hyperbolas.PC.1.1 Recognize andgraph various types offunctions, includingpolynomial, rational,algebraic, and absolutevalue functions. Use paperand pencil methods andgraphing calculators.Example: Draw the graphsof the functionsy = x5 – 2x3 – 5x2,y =(2x-1) / (3x+2), andy = √ ((x+2) (x-5))

PE/IAE88-95, 100-105, 109-115, 147-155, 189-193, 199-203

Ancillaries Study and Solutions Guide86, 123

PE/IAE97, 106-107, 116-117, 156-157,192, 195-196, 204-205

Learning Tools CD ROMSections 1.1, 1.5, 1.6, 1.7, 2.2,2.6Teaching Math UsingTechnologyF23, N2

PC.1.2 Find domain,range, intercepts, zeros,asymptotes, and points ofdiscontinuity of functions.Use paper and pencilmethods and graphingcalculators.Example:Let R(x) = 1/√(x-2). Findthe domain of R(x) — i.e.,the values of x for whichR(x) is defined. Also findthe range, zeros, andasymptotes of R(x).

PE/IAE38-49, 74-81, 147-155, 189-194

Ancillaries Study and Solutions Guide62, 101, 119Teaching Math UsingTechnologyF1-F4, F12-F15

PE/IAE50-52, 84, 96-97, 108, 119,156-157, 195-196, 207, 210-212, 302,

Learning Tools CD ROMSections 1.3, 2.2, 2.6Teaching Math UsingTechnologyF15, F17

Precalculus with Limits: A Graphing Approach © 2001 correlated toThe Indiana Academic Standards for Precalculus





PC.1.3 Model and solveword problems usingfunctions and equations.Example: You are on thecommittee for planning theProm and need to decidewhat to charge for tickets.Last year you charged $5.00and 400 people boughttickets. Earlier experiencessuggest that for every 10¢decrease in price you willsell 50 extra tickets. Use aspreadsheet and write afunction to show how theamount of money in ticketsales depends on thenumber of 10¢ decreases inprice. Construct a graph thatshows the price and grossreceipts. What is theoptimum price you shouldset for the tickets?

PE/IAE79, 92, 115, 141-142, 193-194,203, 708, 718

PE/IAE85-87, 97, 118-119, 129, 145-146, 158-159, 172-173, 196-197, 206-207, 211-212, 702-703, 711-712, 721, 738

Learning Tools CD-ROMSections 1.1, 1.3, 1.7, 1.8, 1.9,2.1, 2.2 2.5, 2.6, 10.2, 10.3,10.4Teaching Math UsingTechnologyN1, N2, N4, N5

PC.1.4 Define, find, andcheck inverse functions.Example: Find the inversefunction ofh(x) = (x – 2)3.

PE/IAE120-126

Ancillaries Study and Solutions Guide72Teaching Math UsingTechnologyF1, F6-F9, F21

PE/IAE127-129, 192

Learning Tools CD-ROMSections 1.8, 2.6Teaching Math UsingTechnologyF5, F10, F17, F23

PC.1.5 Describe thesymmetry of the graph of afunction.Example: Describe thesymmetries of the functionsx, x2, x3, and x4.

PE/IAE94-95, 147-150

Ancillaries Study and Solutions Guide59

PE/IAE96-97, 156-157, 209

Learning Tools CD-ROMSections 1.1, 2.2

PC.1.6 Decide if functionsare even or odd.Example: Is the function tanx even, odd, or neither?Explain your answer.

PE/IAE94-95, 299


PE/IAE96-97, 300-301







PC.1.7 Applytransformations tofunctions.Example: Explain how youcan obtain the graph ofg(x) = -|2(x + 3)2 – 2|. fromthe graph of f(x) = x2.

PE/IAE100-105


PE/IAE106-107, 209

Learning Tools CD-ROMSection 1.6

PC.1.8 Understand curvesdefined parametrically anddraw their graphs.Example: Draw the graph ofthe function y = f(x), wherex = 3t + 1 andy = 2t2 – 5 for a parameter t.

PE/IAE731-735


PE/IAE736-737, 762-763 (#43-58)


PC.1.9 Compare relativemagnitudes of functionsand their rates of change.Example: Contrast thegrowth of y = x2

and y = 2x.

PE/IAE91, 150-151, 216-219, 445, 748

Ancillaries Study and Solutions Guide143Teaching Math UsingTechnologyF13-F14

PE/IAE158-159, 225-226, 275,453-454, 752

Learning Tools CD-ROMSections 1.1, 1.4Teaching Math UsingTechnologyF17

PC.1.10 Write the equationsof conic sections instandard form (completingthe square and usingtranslations as necessary),in order to find the type ofconic section and to find itsgeometric properties (foci,asymptotes, eccentricity,etc.).Example: Write theequationx2 + y2 – 10x – 6y – 25 = 0in standard form. Decidewhat kind of conic it is andfind its foci, asymptotes,and eccentricity asappropriate.

PE/IAE695-700, 704-709, 713-719,722-727, 754-757

Ancillaries Study and Solutions Guide474, 478, 484, 490

PE/IAE701-703, 710-711, 720-721,729-730, 761-763

Learning Tools CD-ROMSections 10.2, 10.3, 10.4, 10.5,10.9






STANDARD 2 Logarithmic and Exponential Functions Students solve word problems involving logarithmicand exponential functions. They draw and analyze graphs, and find inverse functions.PC.2.1 Solve wordproblems involvingapplications of logarithmicand exponential functions.Example: The amount A gmof a radioactive elementafter t years is given by theformulaA(t) = 100 e-0.02t. Find twhen the amount is 50 gm,25 gm, and 12.5 gm. Whatdo you notice about thesetime periods?

PE/IAE222, 235, 253, 259-265


PE/IAE226-227, 238-239, 256, 266-271, 275-277

Learning Tools CD-ROMSections 3.1, 3.2, 3.4, 3.5Teaching Math UsingTechnologyN28

PC.2.2 Find the domain,range, intercepts, andasymptotes of logarithmicand exponential functions.Example: For the functionL(x) = log10 (x – 4), find itsdomain, range, x-intercept,and asymptote.

PE/IAE229-235


PE/IAE236-239, 276


PC.2.3 Draw and analyzegraphs of logarithmic andexponential functions.Example: In the lastexample, draw the graph ofL(x).

PE/IAE217-220, 231-234

Ancillaries Study and Solutions Guide143, 149, 160, 167Teaching Math UsingTechnologyF24-F25

PE/IAE225-226, 236, 255, 275-276

Learning Tools CD-ROMSections 3.1, 3.2, 3.4, 3.5Teaching Math UsingTechnologyF28

PC.2.4 Define, find, andcheck inverse functions oflogarithmic andexponential functions.Example: Find the inverseof f(x) = 3e2x.

Opportunities to address thisstandard are found on thefollowing pages:PE/IAE120-126

Ancillaries Study and Solutions Guide72Teaching Math UsingTechnologyF24-F26

Opportunities to address thisstandard are found on thefollowing pages:PE/IAE :127-129, 134

Opportunities to address thisstandard are found in thefollowing sections:Learning Tools CD-ROMSections 1.8Teaching Math UsingTechnologyF28-F29






STANDARD 3 Trigonometry in Triangles Students define trigonometric functions using right triangles. Theysolve word problems and apply the laws of sines and cosines.PC.3.1 Solve wordproblems involving rightand oblique triangles.Example: You want to findthe width of a river that youcannot cross. You decide touse a tall tree on the otherbank as a landmark. From aposition directly oppositethe tree, you measure 50 malong the bank. From thatpoint, the tree is in adirection at 37º to your 50m line. How wide is theriver?

PE/IAE303-309, 314-319, 355-359

PE/IAE310-313, 320-322, 361-366,369-370, 374, 484

Learning Tools CD-ROMSections 4.3, 4.4, 4.8

PC.3.2 Apply the laws ofsines and cosines to solvingproblems.Example: You want to fixthe location of a mountainby taking measurementsfrom two positions 3 milesapart. From the firstposition, the angle betweenthe mountain and thesecond position is 78º. Fromthe second position, theangle between the mountainand the first position is 53º.How far is the mountainfrom each position?

PE/IAE428-433, 437-440

Ancillaries Study and Solutions Guide290-295

PE/IAE434-436, 441-443, 479-480,484, 486


PC.3.3 Find the area of atriangle given two sides andthe angle between them.Example: Calculate thearea of a triangle with sidesof length 8 cm and 6 cmenclosing an angle of 60º.

PE/IAE432-433, 440


PE/IAE434, 441, 486







STANDARD 4 Trigonometric Functions Students define trigonometric functions using the unit circle and usedegrees and radians. They draw and analyze graphs, find inverse functions, and solve word problems.

PC.4.1 Define sine andcosine using the unit circle.Example: Find the acuteangle A for whichsin 150º = sin A.

PE/IAE314-319, 323-329


PE/IAE320-322, 330-331, 368, 370


PC.4.2 Convert betweendegree and radianmeasures.Example: Convert 90º, 45º,and 30º to radians.

PE/IAE285-290


PE/IAE291-293, 368, 404


PC.4.3 Learn exact sine,cosine, and tangent valuesfor 0, π/2, π/3, π/4, π/6, andmultiples of π.Use those values to findother trigonometric values.Example: Find the values ofcos (π/2), tan (3π/4),csc (2π/3), sin-1 (-√3/2) andsin 3π

PE/IAE295-299, 314-319, 345-349

Ancillaries Study and Solutions Guide191, 197, 218

PE/IAE300-301, 320-322, 351-352, 370


PC.4.4 Solve wordproblems involvingapplications oftrigonometric functions.Example: In Indiana, theday length in hours variesthrough the year in a sinewave. The longest day of 14hours is on Day 175 and theshortest day of 10 hours ison Day 355. Sketch a graphof this function and find itsformula. Which other dayhas the same length as July4?

PE/IAE329, 345-349, 355-359


PE/IAE293-294, 331-333, 343, 351-354, 361-365, 369, 372

Learning Tools CD-ROMSections 4.1, 4.5, 4.7, 4.8

PC.4.5 Define and graphtrigonometric functions(i.e., sine, cosine, tangent,cosecant, secant,cotangent).Example: Graph y = sin xand y = cos x, and comparetheir graphs.

PE/IAE323-327, 334-338


PE/IAE330-331, 341-342







PC.4.6 Find domain,range, intercepts, periods,amplitudes, and asymptotesof trigonometric functions.Example: Find theasymptotes of tan x and findits domain.

PE/IAE323-328, 334-339


PE/IAE330-331, 341-342, 369


PC.4.7 Draw and analyzegraphs of translations oftrigonometric functions,including period,amplitude, and phase shift.Example: Draw the graph ofy = 5 + sin (x –π/3).

PE/IAE323-328, 334-339


PE/IAE330-331, 341-342, 370-371


PC.4.8 Define and graphinverse trigonometricfunctions.Example: Graphf(x) = sin-1x.

PE/IAE345-350


PE/IAE351-353, 371


PC.4.9 Find values oftrigonometric and inversetrigonometric functions.Example: Find the values ofsin π/2 and tan-1 √3.

PE/IAE345-350


PE/IAE351-352, 354, 371


PC.4.10 Know that thetangent of the angle that aline makes with the x-axis isequal to the slope of theline.Example: Use a righttriangle to show that theslope of a line at 135º to thex-axis is -1.

Opportunities to address thisstandard are found on thefollowing pages:PE/IAE25-30, 826-830


Opportunities to address thisstandard are found on thefollowing pages:PE/IAE33-34, 833







PC.4.11 Make connectionsbetween right triangleratios, trigonometricfunctions, and circularfunctions.Example: Angle A is a 60ºangle of a right trianglewith a hypotenuse of length14 and a shortest side oflength 7. Find the exactsine, cosine, and tangent ofangle A. Find the realnumbers x, 0 < x < 2π, withexactly the same sine,cosine, and tangent values.

PE/IAE295-299, 303-308, 317-319


PE/IAE300-302, 310-311, 320-322,369-370







STANDARD 5 Trigonometric Identities and Equations Students prove trigonometric identities, solvetrigonometric equations, and solve word problems.PC.5.1 Know the basictrigonometric identitycos2x + sin2x = 1 and provethat it is equivalent to thePythagorean Theorem.Example: Use a righttriangle to show thatcos2x + sin2x = 1.

PE/IAE376-380, 384-388


PE/IAE381-382, 389-391, 422


PC.5.2 Use basictrigonometric identities toverify other identities andsimplify expressions.Example: Show that(tan2x/ 1+ tan2x)= sin2 x.

PE/IAE384-388


PE/IAE369, 389-391, 422


PC.5.3 Understand anduse the addition formulasfor sines, cosines, andtangents.Example: Prove thatsin (A + B) = sinA cosB +cosA sinB and use it to finda formula for sin 2x.

PE/IAE404-407


PE/IAE408-410, 423


PC.5.4 Understand anduse the half-angle anddouble-angle formulas forsines, cosines, and tangents.Example: Prove thatcos2x = 1/2 + 1/2cos2x.

PE/IAE411-417


PE/IAE418-419, 423-424


PC.5.5 Solve trigonometricequations.Example: Solve3 sin 2x = 1 for x between 0and 2π.

PE/IAE392-399


PE/IAE400-401


PC.5.6 Solve wordproblems involvingapplications oftrigonometric equations.Example: In the exampleabout day length inStandard 4, for how long inwinter is there less than 11hours of daylight?

PE/IAE398-399


PE/IAE401-403, 424







STANDARD 6 Polar Coordinates and Complex Numbers Students define polar coordinates and complexnumbers and understand their connection with trigonometric functions.PC.6.1 Define polarcoordinates and relatepolar coordinates toCartesian coordinates.Example: Convert the polarcoordinates (2, π/3) to(x, y) form.

PE/IAE739-741


PE/IAE742-744, 763 (#59-88)


PC.6.2 Representequations given inrectangular coordinates interms of polar coordinates.Example: Represent theequation x2 + y2 = 4 interms of polar coordinates.

PE/IAE739-741


PE/IAE743-744, 763 (#81-88)


PC.6.3 Graph equations inthe polar coordinate plane.Example: Graphy = 1 – cos θ

PE/IAE745-751, 754-757


PE/IAE752-753, 758-759,764 (#89-114)


PC.6.4 Define complexnumbers, convert complexnumbers to trigonometricform, and multiplycomplex numbers intrigonometric form.Example: Write 3 + 3i and2 – 4i in trigonometric formand then multiply theresults.

PE/IAE467-474


PE/IAE475-477, 482 (#101-126)


PC.6.5 State, prove, anduse De Moivre’s Theorem.Example: Simplify(1 – i)23.

PE/IAE471-474


PE/IAE476 (#71-88), 482 (#117-120)







STANDARD 7 Sequences and Series Students define and use arithmetic and geometric sequences andseries, understand the concept of a limit, and solve word problems.PC.7.1 Understand anduse summation notation.Example: Write the terms

of ∑

5

1

2n

PE/IAE622-624


PE/IAE626-627, 689 (#15-26)


PC.7.2 Find sums ofinfinite geometric series.Example: Find the sum of1 + 1/2 + 1/4 + 1/8 + 1/16 +…

PE/IAE622-623, 642-643


PE/IAE627 (#109-112), 645 (#69-84),689 (#77-80)


PC.7.3 Prove and use thesum formulas forarithmetic series and forfinite and infinite geometricseries.Example: Prove thata + ar + ar2 + ar3 + ar4 + …= a / (1 – r).

PE/IAE618-624, 638-643


PE/IAE625-626, 644-646, 688-689


PC.7.4 Use recursion todescribe a sequence.Example: Write the firstfive terms of the Fibonaccisequence with a1 =1, a2 =1,and an = an-1 + an-2 for n ≥ 3.

PE/IAE620, 629-634


PE/IAE626, 635-637, 688-689


PC.7.5 Understand anduse the concept of limit of asequence or function as theindependent variableapproaches infinity or anumber. Decide whethersimple sequences convergeor diverge.Example: Find the limit asn → ∞ of the sequence(2n-1)/(3n+2) and the limitas x→ ∞ of the function(x2-x5 )/(x-5).

PE/IAE806-812, 816-823, 835-840


PE/IAE813-815, 824-825, 841-842,852-854






PC.7.6 Solve wordproblems involvingapplications of sequencesand series.Example: You put $100 inyour bank account today,and then each day put halfthe amount of the previousday (always rounding to thenearest cent). Will you everhave $250 in your account?

PE/IAE624, 633-634, 643


PE/IAE627-628, 636-637, 645-647688 (#31-32), 689 (#53-54, 81),690 (#82-84)







STANDARD 8 Data Analysis Students model data with linear and non-linear functions.PC.8.1 Find linear modelsusing the median fit andleast squares regressionmethods. Decide whichmodel gives a better fit.Example: Measure the wristand neck size of eachperson in your class andmake a scatter plot. Findthe median fit line and theleast squares regressionline. Which line is a betterfit? Explain your reasoning.

PE/IAE549, A18-A19, A31-A32

Ancillaries Study and Solutions Guide2, 7Teaching Math UsingTechnologyG1-G2, G4-G6

PE/IAE549, A20-A21

Learning Tools CD-ROMSection A.8Teaching Math UsingTechnologyG2-G3, G7, N4

PC.8.2 Calculate andinterpret the correlationcoefficient. Use thecorrelation coefficient andresiduals to evaluate a“best-fit” line.Example: Calculate andinterpret the correlationcoefficient for the linearregression model in the lastexample. Graph theresiduals and evaluate thefit of the linear equation.

PE/IAEA18-A19

Ancillaries Teaching Math UsingTechnologyG8-G12

PE/IAEA20-A21

Learning Tools CD-ROMSection A.8Teaching Math UsingTechnologyG13

PC.8.3 Find a quadratic,exponential, logarithmic,power, or sinusoidalfunction to model a data setand explain the parametersof the model.Example: Drop a ball andrecord the height of eachbounce. Make a graph ofthe height (vertical axis)versus the bounce number(horizontal axis). Find anexponential function of theform y = a•bx that fits thedata and explain theimplications of theparameters a and b in thisexperiment.

PE/IAE224, 235, 243, 258-265

Ancillaries Study and Solutions Guide143, 149, 155, 167Teaching Math UsingTechnologyG14-G18, G20-G24, G26-G30,G33-G35, G37-G40

PE/IAE227, 237, 245, 256-257, 268-273, 277-278

Learning Tools CD-ROMSections 3.1, 3.2, 3.3, 3.5Teaching Math UsingTechnologyG19, G24-G25, G31-G32, G36,G41






STANDARD 9 Mathematical Reasoning and Problem Solving Students use a variety of strategies to solveproblems.PC.9.1 Use a variety ofproblem-solving strategies,such as drawing a diagram,guess-and-check, solving asimpler problem,examining simplerproblems, and workingbackwards.Example: The half-life ofcarbon-14 is 5,730 years.The original concentrationof carbon-14 in a livingorganism was 500 grams.How might you find the ageof a fossil of that livingorganism with a carbon-14concentration of 140 grams?

Found throughout the text.See, for example:

PE/IAE6, 19, 80-81, 103, 261, 350,439, 483, 632

Ancillaries Study and Solutions Guide167Teaching Math UsingTechnologyG33-G34, G37-G40


PE/IAE11, 24, 85, 118-119, 268, 270,343, 443

Learning Tools CD-ROMSection 3.5Teaching Math UsingTechnologyG35-G36, G41

PC.9.2 Decide whether asolution is reasonable in thecontext of the originalsituation.Example: John says theanswer to the problem inthe first example is about10,000 years. Is his answerreasonable? Why or whynot?


PE/IAE43, 49, 123, 125, 139, 149,221, 393, 407, 425, 657


PE/IAE21, 51, 401 (#61-64)


Students develop and evaluate mathematical arguments and proofs.

PC.9.3 Decide if a givenalgebraic statement is truealways, sometimes, ornever (statements involvingrational or radicalexpressions, trigonometric,logarithmic or exponentialfunctions).Example: Is the statementsin 2x = 2 sinx cosx truealways, sometimes, ornever? Explain your answer.

PE/IAE393-396


PE/IAE273 (#74-77), 278 (#137-142),400-401 (#1-6, 25-42)

Learning Tools CD-ROMSections 5.3, A.1, A.7






PC.9.4 Use the propertiesof number systems andorder of operations tojustify the steps ofsimplifying functions andsolving equations.Example: Simplify(( 5/ x+2)+( 2/ x+3))/(( 1/x+3)+ ( 7/ x-2)),explaining why you cantake each step.

PE/IAE38-49, 74-81

PE/IAE50-52, 68-69

Learning Tools CD-ROMSections A.1, A.5

PC.9.5 Understand that thelogic of equation solvingbegins with the assumptionthat the variable is anumber that satisfies theequation, and that the stepstaken when solvingequations create newequations that have, in mostcases, the same solution setas the original. Understandthat similar logic applies tosolving systems of equationssimultaneously.Example: A student solvingthe equationx +√x– 30 = 0 comes upwith the solution set{25, 36}. Explain why{25, 36} is not the solutionset to this equation, andwhy the “check” step isessential in solving theequation.

PE/IAE38-49


PE/IAE50-52, 68-69

Learning Tools CD-ROMSections 2.5, A.5

PC.9.6 Define and use themathematical inductionmethod of proof.Example: ProveDe Moivre’s Theorem usingmathematical induction.

PE/IAE648-653


PE/IAE654-655, 690 (#85-96)


precalculus - mcdougal littell · graphical and algebraic approach to example solutions and...

Documents