AP Calc Pearson Review

PREPARTNGFOR TI{ECALCULUS

fiP"Gil.Gulu$: Gfa[nioal, ]lumerieal, flge[raicFINNEY, DEMANA, \ØAITS, KENNEDY

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*Advanced placement, Advance placement Program, AP and AP cenual are registered uademarks of the College

Board, which was not involved in the production of, and does not endorse, this product'

rsBN 0-32r -33574-0

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't¿;.: "l

Preparing for theAP* Calculus Exam

About the Authors v

About Your Pearson AP Guide viiAcknowledgments vüi

Part I: Introduction to the AP* AB andBC Calculus Exams 1

Part II: Precalculus Review of CalculusPrerequisites 13

Calculus Prerequisites

Precalculus - A Preparation for Calculus! t5Functions 20

Transformations 24

PolynomialFunctions 29

Rational Functions 33

ExponentialFunctions 39

SinusoidalFunctions 42

More Trigonometric Functions 46lnverse Trigonometric Relations and Functions 50

ParametricRelations 53

Numerical Derivatives and lntegrals 58

Part III: Review of AP* Calculus AB andCalculus BC Topics 6l

Functions, Graphs, and Limits

Analysis of Graphs 63

Limits of Functions 70

Asymptotic and Unbounded Behavior 76

Function Magnitudes and Their Rates of Change 82

Continuity 8s

lntermediate and Extreme Value Theorems 89Parametric, Polaç and Vector Functions 93

Derivatives

Concept of the Derivative 96

Differentiabilityand Continuity 99

Slope of a Curve at a Point tO2Local Linearity 106

lnstantaneous Rate of Change 109

Relationships between the Graphs of f and f, tt2The Mean Value Theorem u5Equations lnvolving Derivatives tt8Correspondences among the Graphs of

f, f', and f" l2lPoints of lnflection tzsConcavity of Functions t2gExtreme Values of Functions t3tAnalysis of Parametric, Polar, and

Vector Curves 134

0ptimization 136

Related Rates tl¡glmplicit Differentiation 143

Derivative as a Rate of Change t46Slope Fields lsoEuler's Method ts4l-Hôpital's Rule 158

Basic Derivatives r6tDerivative Rules 164

üi

Chain Rule 167

Derivatives of Parametric, Polar, and

Vector Functions l7L

lntegrals

Riemann Sums 173

Definite lntegral of a Rate of Change 176

Basic Properties of Definite lntegrals 179

Applications of lntegrals 182

FundamenklTheorem of Calculus r85

Antiderivative Basics 188

Antidifferentation by Substitution 191

Antidifferentation by Parts 195

Antidifferentation by Simple Partial Fractions r98

lmproper lntegrals 2ollnitialValueProblems 204

Separable Differential Equations 2o7

Numerical Approximations to Definite lntegrals 210

Polynomial Approximations and Series

Concept of Series 213

Geometrie, Harmonic, and Alternating Series 215

lntegral Test, Ratio Test, and Comparison Test 217

Taylor Polynomials 219

Maclaurin and Taylor Series 223

Manipulating Taylor Series 226

Power Series 230

Radius and lnterval of Convergence 233

LaGrange Error Bound 237

Part IV: Practice Examinations 241

Calculus AB Exam 1 243


Calculus BC Exam 1 283


PartV: Answers and Solutions 323

Part ll: Precalculus Review of Calculus

Prerequisites 325

Part lll: Review of AP* Calculus AB and

BC Topics 341

Functions, Graphs, and Limits 341

Derivatives 349

lntegrals 370

Polynomial Approximation 380

Practice Examinations





lv PREPARING FOR THE CALCULUS AP" EXAMINATION

Røy Børmn teaches Al* Calculus BC at Olympus High School in Salt Lake City, Utah. Hehas been an Advanced Placement Calculus exam reader and is an active instructor for TeachersTeaching with Technology. He has coauthored, Ad.vønced, Pløceru.ent Cølcøløs with the TI-ggand Dffiren'tiøl Eqwøtions with the TI-86. Ray received the Presidentiat Award for Excellencein Mathematics and Science Teaching in 1995. He is a srrong proponent of using technol-ogy ro teach mathematics.

fohn R- Brønstí'ngtaaghtAP* Calculus at Hinsdale Central High School in Hinsdale, Illinois.He has been an Advanced Placement Calculus exam reader and table leader, as well as an A?*Cfculus Test Development Committee member. IIe was a consultant for the Midwest Regionof the College Board. Now retired, ]ohn is a director of Illinois Advanced placement Institutesand Mathematics & Technology rnsdtutes, providing srunmer training for Ap* teachers.

John J' Dìehl has taught AP* Calculus and AP* Statistics at Hinsdale Central High Schoolin Hinsdale, Illinois. He has been an Advanced Placement Statistics exam reader and tableleader, as well as an AP* Statistics Test Development Committee member. ]ohn is a consultantfor the Midwest Region of the College Board and a Teachers Teaching with TechnologyInstitute instructor. John is coauthor oî Ad.pønced. Pløcem.ent cølcøløs with tbe TI-gg.

Greg IIíl'l teaches AP* Calculus BC at Hinsdale Central High School in Hinsdale, Ilinois.A nearly twenty-year veteran, he has been an Advanced Placement Calculus exam reader forthe last five years. As a consultant for the College Board, Greg has presented numerous Ap*Calculus workshops across the Midwest. An active Teachers te"cfring*ith Technology instruc-tor, Greg guides teachers in the appropriate uses of technologyin algebra, precallulus, andcalculus.

Køryl Tylerteaches AP* Calculus AB at Hinsdale Central High School in Hinsdale, Illinois.FIer twenty-eight years of teaching experience include eleven ye"rs of teaching calculus. Beyondher classroom, Karyl assists with instruction at Teachers Teaching with Technology Institutes,sharing her passion for using technology in the mathematics clirsro.r-.

Stepen L. Wòlson teaches AP" Calculus AB at Hinsdale Central High School in Hinsdale,Illinois' His twelve years of teaching experience include eight years of teaching calculus. Aversatile mathematician, Steve shares his deep underst"ndings with students t airring for rig-orous mathematics oral competitions.

About the Authors

About YoîlrPearson AP+ Guide

vu

Acknowledg-ents

We wish to thank the late Ross Finney and Bert Waits, Frank Demana, Dan Kennedy, and

Greg Foley for precalculus and calculus textbook contributions that made linking their books

ao Ap* Calculus objectives a nearly effortless task. Thanks also to the College Board for

their long-standing commirment ro educational excellence as exemplified by the AP* Calculus

curriculum and assessment standards.

We are indebted to the entire Pearson Addison-Wesley publishing team for their trust, guid-

ance, and patience with us as a writing team.

We also acknowledge the support of Dan Kennedy for his thoughtfirl input and wise counsel

in shaping this project, pre¿ Ctoff for working and checking all problems, and Hinsdale

fownsttip ffigh School District 8ó for its unwavering promotion of professional excellence

that continues to encograge teachers to learn and to share in community.

Finally, at the most personal level, our team expresses extra special thant$ to those closest to

,lr--o* families-for tJreir encouragement' undcrstanding, and patience during ¡he creating,

writing, rewriting, rewriting, and final rewriting phases of,this project'

vru PREPARING FOR THE CALCULUS AP- EXAMINATION

Introduction to the AP Cølculus-AB or BC-Examinøtion

Part I

{:,

.!r:.j ":

x,í| ._ _ .1_-i

Thís Book

So you are planning to take or have already enrolled in either AP* Calculus ABor AP* Calculus BC! Either AP* course will stretch and enrich your mathe-matics skills and knowledge and will culminate in an AP* examination.

Presenþ enrolled in a precalatlus course?This book will point out foundational calculus Objectives encountered in aprecalculus course and put those objectives into a calculus Big Picture context,provide succinct Content explanations, give Additional Practice problems, andmake Need More Help? connections to the Addison-Wesley Precalculus andCalculus textbooks.

Presenþ enrolled ín ø cølculus course?This book will point out the corresponding AP* Calculus Exam Objectives,identifr the Big Picture context, provide succinct Content explanations, giveAdditional Practice problems and solutions, and wherever possible make NeedMore Help? connections to Addison-Wesley Precalculus and Calculus tert-books.

In either case, this workbook supplement to your textbook will help youclearly identifr essential calculus concepts as well as improve both your under-standing of such concepts and your ability to communicate your thinking sothat you can be successful.

This book is not intended as a substitute for a full precalculus or calculuscourse or a comprehensive treatment of calculus topics. It is rather meant toact as a useful review for students who have previously studied the curriculumtopics and now want to solidifr their learning through recall and practice.

The Advanced Placement+ Cølculus AB or BC Currìculum

The AP* Calculus AB and AP* Calculus BC curricula are designed to providecourses in calculus that are equal in content to the best of collegiate courses.Each is developed by a team of educators from the high school and collegecommunities and continues to adapt itself so that a broad base of college pro-grams will offer credit for success in the respective examination. To find outwhether your prospective institution will give credit for your AP* CalculusExamination performance,you should email or call the admissions office.Youcan log on to AP* Central at www.apcentral.collegeboard.com for up-to-dateinformation about the Advanced Placement* Program.

A course in calculus is especially useful to those pursuing studies in math-ematics, as well as an ever-widening variety of engineering, science, economic,and business fields.

(Jnd.erstøndíng the Advanced Plncement+ Calculus AB or BC Examínøtíon

AP* Calculus Examinations began in 1956 as one of the earliest examinations

created by the College Board. Now offered as AP* Calculus AB and AP*

Calculus BC, these examinations are given in May during the same two-week

window as the other examinations. Your teacher or the AP* coordinator inyour school district can give you the exact date for this year's examination.You

can also check for this and other information on the AP* CentralWeb site. Test

schedules are determined long in advance and the dates and times are not flex-

ible. In the event of an emergency you may qualiff for an alternate examina-

tion, but the qualifring conditions are extremely rigid. Plan your calendar

around the examination date, and register with your teacher or coordinator to

reserve an examination. Standard fees as periodically adjusted by the College

Board apply.Th; AP* Examinations specifr calculus content for AP* Calculus AB and

AP* Calculus BC under the following three headings:

I Functions, GraPhs, and LimitsI DerivativesI Integrals

Additionall¡ a fourth heading for AP* Calculus BC only is

tr Poþomial Approximations and Series

You maybring to the examination two calculators with graphical capabil-

ities. A complete up-to-date list of acceptable graphing calculators can be

found on the College Board Web site. We highly recommend using one model

throughout the calculus course and bringing that same model to the exami-

nation. The exam assumes that your calculator has built-in capabilities to

l. plot the graph of a function within an arbitrary viewing window.

2. find the zeros of functions (solve equations numerically).

3. numerically calculate the derivative of a function.

4. numerically calculate the value of a definite integral.

Many students bring a second calculator as a backup. Fresh batteries are a

must! Since calculator memories need not be cleared before taking the exam-

ination, calculators may contain whatever additional programs students may

desire.The actual examination is currently formatted and has timing and grade

weights as follows:

Section l. Multiple Ghoice Section

Part A (Calculator Not Allowed)

Part B (Calculator Required)

Section ll. Free Response

Part A (Calculator Required)

Part B (Calculator Not Allowed)

Total of ¡15 Ouestions/l05 Minutes

28 questions/55 minutes

17 questionV50 minutes

Total of 6 Questions/90 Minubs

3 questionV45 minutes

3 questionV4S minutes

PART l: INTRODUCTI0N T0 THE AP CALCULUS-AB 0R BC-EXAMINATION

5{M ofTest

50% of Test

50% of the 5070

50% of the 50%

The Multiple Choice and Free Response sections of the examination are equal-ly weighted. They stand alone with a recommended break between. You shouldexpect to work the full allotted time. You should, if at all possible, allow timein each part of each section to go back and check your work. The MultipleChoice section is done first; you should not expect to go back to it after break.Indeed, that part of the examination will be sealed and collected at the end ofthe l05-minute time period.

Both the Multiple Choice and Free Response sections come with separatedirections for Calculator Not Allowed and Calculator Required portions. Ineither test section, a student cannot return to the Part A section after begin-ning Part B.

Calculus AB Subscore Grøile for the Cølculus BC Examinatíon

Since the Calculus BC curriculum and examination encompass the CalculusAB curriculum, the Calculus BC is able to report a Calculus AB Subscore. Adetailed explanation of the Calculus AB Subscore and its value can be foundon the College Board Web site.

Understandíng the Grading Procedure for the Advanced Placement* Calculus AB or BCExømínatíon

The scoring of an AP* Calculus Examination is done in part by machine(Section I-Multiple Choice) and in part by calculus educators (humanbeings!) who read the answers as you have communicated them (Section II-Free Response). The entire test is valued at 108 points with the MultipleChoice and Free Response sections each contributing 54 points.

In the Multiple Choice section you are awarded I point for each correctanswer and penalized one-quarter point for each incorrect answer. To convertto the possible 54-point total (reflecting 50olo of the grade), the number ofpoints is multipliedby 1.2. For example, suppose you answered 31 questionscorrectly but answered 10 questions incorrectly and left 4 questions blank.Here is the way your Multiple Choice section would be graded:

Number correct

Number incorrect

Number left blank

POINTS EARNED

PART l: INTRODUCTI0N T0 THE AP CALCULUS-AB 0R BC-EXAMINAT|0N

In the Free Response section,the reader assigns a score from 0-9 as indi-cated by the rubric or grading rule developed by the Chief Reader and a selectteam of exam leaders. A rubric is developed for each question to attain scoringconsistency.

MUUIIPLE CHOICE SCORE = 28.5 x 1.2 =

3tx1-3110 X-0.25 = -2.54X0-0

28.5

34.2

5

Suppose your answers tofollowing manner:

Question 1:

Question 2:

Question 3:

Question 4:

Question 5:

Question 6:

Your Total Exam score would be the sum of the Multiple Choice and Free

Response scores: 34.2 + 37 : 71.2.

This composite score is then subject to the Chief Reader's interpretation

of the cut points for that particular examination year's results. These cut

points are set shortly after examinations are scored and are based on several

factors, including statistical comparability with other years' examinations, the

distributions of performance on the various parts of the current year's exam-

ination, and previous years of grade distributions.Your results will be available from the College Board by phone, usually

before mid-Iuly, as they are mailed out to you and your school shortly after

that. By August your scores are also sent to any college or university that you

indicate in the general information section of your answer packet. If you do

not wish to communicate your scores immediately to your institution or if you

are unsure of what institution you will be attending, 1lou need not indicate any

schools; the results would then come only to your high school and you. You

can have them transmitted later to your chosen schools in accordance withCollege Board policies. There may be a fee for this service.

Test-Takíng Strategies for an Advønced Placement* Calculus Examinatíon

You should approach the AP* Calculus Examination the same way you would

any major test in your academic career. Just remember that it is a one-shot

deal-youmust be at your peak performance level on the day of the test. For

that reason you should do everything that your "coach" tells you to do. In most

cases your coach is your classroom teacher. It is very likely that your teacher

has some experience, based on workshop information or previous students'

performance, to share with you.You should also analyze your own test-taking abilities. At this stage in

your education, you probably know your strengths and weaknesses in test-

taking situations. You may be very good at multiple choice but weaker inessays, or perhaps it is the other way around. Whatever your particular abili-

ties are, evaluate them and respond accordingly. Spend more time on your

weaker points. In other words, rather than spending time in your comfortzone where you need less work, try to improve your soft spots. In all cases,

the Free Response questions are scored in the

8 points

6 points

8 points

5 points

6 points

4 points

FREE RESPONSE SCORE : 37 points

6

k**.

PART l: INTRoDUCTI0N T0 THE AP CALCULUS-AB 0R BC-EXAMINATI0N

concentrate on clear cornmunication of your strategies, techniques, and con-clusions.

The following table presents some ideas in a quick and easy form. It isdivided into two sections: general strategies for approaching the examinationday and specific strategies for addressing paiticular tlpes of questions on theexamination.

General Strategies for AP" Examination Preparation

Time

ThroughtheYear

DOs @

r Register with your teacher/coordinatorr Pay your fee (if applicable) on timer Take good notesr Work with others in study groupsr Review on a regular basisr Evaluate your test-taking strengths andweaknesses-keep track of how successfulyou are when guessing

The

WeekBefore

ïheNightBefore

r Combine independent and group reviewr Get típs from your teacher: Do lots of mixed review problemsr Check your exam date, time, and locationr Review the appropriate AF Calculussyllabus (AB or BC)

Exam Day

DON'Ts @

r Put new batteries in your calculator; checkfor illegal programsr Lay out your clothes and supplies so thatyou are ready to go out the doorr Do a short review: Go to bed at a reasonable hour

r Procrastinater Avoid homework and labsr Wait until the last moment to pull it together forquizzes and testsr Rely on others for your own progressr Scatter your work products (notes, labs, reviews,tests)r lgnore your weak areas-remediate as you go along

Exam Night

r Get up a little earlier than usualI Eat a good breakfast/lunchI Put some hard candy in your pocket in caseyou need an energy boost during the testr Get to your exam location 15 minutesearly

r Procrastinater ïhínk you are the only one who is stressedr Forget your priorities-this test is a one-shot deal

r Relax-you earned it I Worry-it's over

r Study all nightr Get caught without f¡:esh batteries

PART l: INTRODUCTION T0 THE AP CALCULUS-AB 0R BC-EXAMINAT|ON

II

Sleep inPanic with last-minute cramming

7

Ouestion Type

MultipleChoice

Specific Strategies to use During the AP" Examination

DOs O

r Underline key words and phrases; circleimportant information you will useI Look at the answer format so that you do

not do unnecessary stepsI Anticipate the likely errors for the type ofquestion being asked-watch out for obvious

choices: When formulas and calculations are

needed, write down what you are doing so

you can check your procedurer Eliminate as many answers as possibler Guess only when you are comfortable withthe number of possible answersn Check the units of your answers

(All Ouestions)Response

DON'Ts @

r Look over all the questions before you

start and do the ones that seem easiest firstn Read the entire question in oll of its ports

r Underline what is being asked

r Carry out your strategy by clearly indicat-ing your steps: Move on to the next part of the question

if you cannot answer one Partr Make up a reasonable answer for one part

if the next part requires the previous answer("Suppose my answer to Part A had been

20 sq. units. Using that area value and the

integral I{r!)¿x ='lr.Qo),1 will find the

number k for which the line x : k divides

the region equally.")t Write neatly, compactly, and clearly and

use calculus vocabulary correctlyr When useful, include graphs/sketches thatillustrate your answer; be sure to label axesI Mark up sketches provided to illustrateyour thinkingI COMMUNICATE CLEARLY-answer thequestion asked and place your answer in theC0NTEXI' of the question¡ Review your response to make sure that itshows good mathemoticol thi nking

r Rush-reading the question correctly is the key toanswering the question correctlyr Do unnecessary calculations (sometimes equations

can be left in any form)I Fall into the traps your teacher warned you about-aquestion that looks too easy may have one of these

traps built inr Scribble your work-if you need to review your

procedure you wind up having to repeat itr Guess haphazardlyr Spend more than 2-3 minutes on any one question

-if you don't know move onI Close the book until time is up

r Feel as

with #5; pick the one you like best to start offr Forget to answer the quesfion osked in your haste towrite an answerr Begin without a plant Move on until you have read what you have writtenfor each part to make sure you have not left outimportant words, punctuation, or numbersr Get stuck on a question-if you have no idea how toproceed after thinking about it, move onI Scribble-a human being must be able to decipheryour responser Waste time erasing unless spaee is an issue-anything crossed out will not be read as part ofyouranswert Round during computation-wait until you reach a

final answer to roundI Run on-you are likely to say somethingINCORRECTLY that will diminish your previously

correct responser fusume that the size of the space provided is

proportional to the answer desired

you to start with #1 a

I PART l: INTR0DUCTI0N T0 THE AP CALCULUS-AB 0R BC-EXAMINATI0N

Topìcs from the Advanced Placement+ Currículum for Calculus AB, Cølculus BC

The AP* Calculus Examination is based on the following Topic Outline. Foryour convenience, we have noted all Calculus AB and Calculus BC objectiveswith clear indications of topics required only by the Calculus BC Exam. Theoutline cross references each objective with our primary textbooks:Precalculus: Graphical, Numerical, Algebraic by Demana, waits, Fole¡ andKennedy and Calculus: Graphical, Numerical, Algebraic by Finne¡ Demana,Waits, and Kennedy.

Use this outline to keep track of your review. Be sure to cover every topicassociated with the exam you are taking. Check it offwhen you have reviewedthe topic from your text and then review the topic in this book.

Topic Outline for AP" Calculus AB and AP" Calculus BC

(Excerpted from the College Boord's Course Description-Colculus: Cotcutus AB, Cotcutus BC, Moy 2oo7)

l. Calculus Exam

AABBCBABBC

81 AB BC

82 AB BC

83 AB BC

C ABBCC1 AB BC

C2 AB BC

C3 AB BC

DABBCDl AB BC

D2 AB BC

D3 AB BC

EBC

ll. Calculus Exam

Analysis of graphs

Limits of functions (including one-sided limits)

An intuitive understanding of the limiting process

Calculating limits using algebra

Estimating limits from graphs or tables of data

Asymptotic and unbounded behavior

Understanding asymptotes in terms of graphical behavior

Describing asymptotic behavior in terms oflimits involving infìnity

Comparing relative magnitudes of functionsand their rates of change

Continuity as a property offunctionsAn intuitive understanding of continuityUnderstanding continuity in terms of limits

Geometric understanding of graphs of continuous functionsParametric, polar, and vector functions

Derivatives

Functions, Graphs, and Limits

A1

A2

A3

A4

B1

B2

AB BC

AB BC

AB BC

AB BC

AB BC

AB BC

AB BC

AB BC

Precalculus

1.2

10.3

r0.3

10.3

1.2,2.7

1.2,2.7

1.2,2.7

1.2

1.2

2.3

Concept of the derivative

Derivative presented graphically, numerically, and analytically 10.1

Derivative interpreted as an instantaneous rate of change 10.1

Derivative defined as the limit of the difference quotient 10.1

Relationship between differentiability and continuity 10.,l

Derivative at a point

Slope of a curve at a point 10.1

PART l: INTRODUCTION T0 THE AP CALCULUS-AB 0R BC-EXAMtNATt0N

Celculus

1.2-1.6

2.1,2.2

2.1,2.2

2.1,2.2

2.2

Tangent line to a curve at a point and locallinear approximation

2.2

2.2,2.4,8.3

2.3

2.3

2.3,4.1-4.3

10.1- 10.3

CalculusPrecalculus

2.4-4.5

2.4

2.4-3.'l

3.2

2.4

2.4,4.510.1

9

B3

B4

c1

c2

c3

C4

D1

D2

D3

E1

AB BC

AB BC

AB BC

AB BC

AB BC

lnstantaneous rate of change as the limit ofaverage rate of change 10'1

Approximate rate of change from graphs and tables of values 10'1

Derivative as a function

AB BC

AB BC

Corresponding characteristics of graphs of f and f'Relationship between the increasing and decreasing

behavior of fand the sign of f'The Mean Value Theorem and its geometric consequences

Equations involving derivatives (Verbal descriptions are

trånslated into equations involving derivatives and vice versa.)

Second derivativesAB

AB.AB

AB

AB

AB

BC

BC

BC

BC

BC

BC

E2

Corresponding characteristics of graphs of f , f' and f" 1'2

Relationship between the concavity of f and the sign of f" 1'2

Points of inflection as places where concavíty changes

Appl ications of derivatives

E3

BC

AB BC

E4 AB BC

E5 AB BC

Analysis of curves, including the notions ofmonotonicity and concavitY

E6

Analysis of planar curves given in parametric form,

polar form, and vector form, including velocityand acceleration vectors

0ptimization, both absolute (global) and relative(local)extrema 1'6

Modeling rates of change, including related rate problems

Use of implicit differentiation to find the derivative

of an inverse function

lnterpretation of the derivative as a rate of change

in varied applied contexts, including velocity, speed,

and acceleration

Geometric interpretation of differential equations via

slope fields and the relationship between slope fields

and solution curves for differential equations

Numerical solution, of differential equations using

Euler's method -LHôpital's Rule, including its use in determining limits

and convergence of improper integrals and series

Computation of derivatives

Knowledge of derivatives of basic functions, including

poweç eiponential, logarithmic, trigonometric, and inverse

trigonometric functions

AB BC

AB BC

BC

E7

E8

10.1

10.1

2.1, 10.1

2.4,3.4

2.4,3.4

3.1,4.3

4.1,4.3

4.2

3.4,3.5,4.6,6.4,6.5

4.3

4.3

4.3

4.1-4.3

10.1-10.3

4.3,4.4

4.6

3.7

3.4

6.1

6.1

8.2, 9.5E9

FABBCFl AB BC

BC

F2

F3

t4

AB BC

AB BC,, Bc

2.3

t-'1.

'tr '

Basic rules for the derivative of sums, products,

and quotients of functions

Chain rule and implicit differentiation

Derivatives of parametric, polar, and vector functions

10

I

PART I: INTRODUCTION TO THE AP CALCULUS_AB OR BC_EXAMINATION

1.4

3.3,3.5

3.3, 3.8,3.9

3.6,3.7

10.1-10.3

lll. Calculus Exam

A1

A2

AB BC

AB BC

A3

lnterpretations and properties of definite integrals

Definite integral as a limit of Riemann sums overequal subdivisions

AB BC

AB BC81a

Definite integral of the rate of change of a quantity

over an interval interpreted as the change of the quantity

overthe(interval: Ilftr¡dx: f(b) - f(o)Basic properties of definite integrals(Examples include additivity and linearity.)

Applications of integrals

Appropriate integrals are used in a variety of applicationsto model physical, biological, or economic situations.Although only a sampling of applications can be included in

any specific course, students should be able to adapt theirknowledge and techniques.

Appropriate integrals are used . . . specificapplications should include . . . fínding the areaof a region (including a region bounded by polar curves)... the distance traveled by a particle along a line,and the length of a curve (including a curve given inparametric form).

Fundamental Theorem of Calculus

Use of the Fundamental Theorem to evaluate definite integrals

Use of the Fundamental Theorem to represent a

particular antiderivative, and the analytical and graphicalanalysis of functions so derived

Tech niques of antidifferentiationAntiderivatives following directly from derivativesof basic functionsAntiderivatives by substitution of variables (including

change of limits for definite integrals)

Antiderivatives by parts, and simple partial fractions(nonrepeating linear factors only)

lmproper integrals (as limits of definite integrals)

Appl ications of a ntidifferentiation

Finding specific antiderivatives using initial conditions,including applications to motion along a line

Solving separable differential equations and using themin modeling. (ln particular, studying the equations y' : kyand exponential growth)

Solving logistic differential equations and using themin modeling

Nu merical a pproximations to definite i ntegrals

Use of Riemann and trapezoidal sums to approximatedefinite integrals of functions represented algebraically,graphically, and by tables of values

lnteqrals

B1b BC

Cl AB BC

C2 AB BC

D1

Precalculus

ù2a

AB BC

AB BC

BC

BC

AB BC

AB BC

BC

AB BC

D2b

5.1,5.2

D3

Calculus

E1

5.1, 5.4

5.2, 5.3

E2

E3

F1

5.4, 5.5,6.4,6.5,7.1-7.5

7.4,10.1,103

PART l: INTRODUCTION T0 THE AP CALCULUS-AB 0R BC-EXAMINATI0N

5.4,6.1

4.2,6.1,6.2

6.2

3.3,3.4,7.4 6.3,6.58.3

6.1, 7 .1

5.4

6.4

5.2, 5.5

6.5

1t

lV. Calculus Exam

A1

81 BC

82 BC

83 BC

84 BC

85 BC

BC

Polynomial Approximations and Series

Concept of series

A series is defined as a sequence of partial sums, andconvergence is defined in terms of the limit of thesequence of partial sums. Technology can be used toexplore convergence or divergence.

Series of constants

Motivating examples, including decimal expansion

Geometric series with applications

The harmonic series

Alternating series with error bound

Terms of series as areas of rectangles and theirrelationship to improper integrals, including the integral

tèst and its use in testing the convergence of p-series

The ratio test for convergence or dívergence

Comparing series to test for convergence and divergence

Taylor series

Taylor polynomial approximation with graphicaldemonstration of convergence (For example, viewinggraphs of various Taylor polynomials of the sine functionapproximating the sine curve.)

Maclaurin series and the general Taylor series centeredat''x: o

Maclaurin series for the functions ex, sin x, cos Í,and 1l( - x)

Formal manipulation of ïaylor series and shortcuts tocomputing Taylor series including substitution,differentiation, antidifferentiation, and the formation ofnew series from known series

Functions defined by power series

Radius and interval of convergence of power series

Lagrange error bound for Taylor polynomials

B6

87

c1

BC

BC

c2

BC

c3

c4

BC

Precalculus

BC

c5

c6

c7

BC

Celculus

BC

BC

BC

9.1

9.1

9.1

9.5

9.5

9.5

9.4

9.4

t2

9.2

9.2

9.2

9.1, 9.2

9.1, 9.2

9.1, 9.4, 9.5

9.3

PART l: INTRODUCTION T0 THE AP CALCULUS-AB 0R BC-EXAMINATION

Precølculus Review ofC ølculus Pr er equisit es

Part II

::,'i:

i

i

;

Precalcul ss-fi Preparationfor Galculus!

As the word implies, precalculus is a preparation for calculus. Whether yourprecalculus course ases Precalculus: Graphical, Numerical, Algebraic byDemana,Waits, Fole¡ and Kennedy (hereafter Precalculus) or some other text-book, the course should provide a good foundation for advanced mathemati-cal study. If an AP* Calculus course is in your future, you should know the spe-cific content, concepts, and skills taught in a precalculus course that will beencountered frequently throughout your calculus course.

The College Board AP* Calculus Course Description booklet, iÙ..4;ay 2007,describes the following prerequisites needed for calculus:

Before studying calculus, all students should complete four years of second-ary mathematics designed for college-bound students: courses in which theystudy algebra, geometry trigonometry anal¡ic geometry and elementaryfunctions. These functions include those that are linear, poþomial, ration-al, exponential,logarithmic, trigonometric, inverse trigonometric, and piece-

wise defined. In particular, before studying calculus, students must be famil-,iar with the properties of functions, the algebra of functions, and the graphsof functions. Students must also understand the language of functions(domain and range, odd and even, periodic, symmetry, zeros, intercepts, andso on) and know the values of the trigonometric functions of the numbers 0,

i,i,l,f, and their multiples.

[AP Calculus Course Description, May 2007]

This precalculus section is structured to identifr the important precalcu-lus Objectives, describe the importance of each in a Big Picture calculus con-text, provide succinct Content explanations, give Additional Practice prob-lems, and point to resources if you Need More Help. Ten topics-identified as

Calculus Prerequisite Knowledge-are listed below and cross referenced to thePrecølculus textbook.

l5

AP* PreparationTopic

0

1

2

3

4

5

6

7

8

I10

Calculus PrerequisiteKnowledge

Basic functions

Functions

Transformations

Polynomial functions

Rational Functions

Exponential functions

Sinusoidal functions

For those using Precalculus, there are excellent features and calculus cues

with the book. A few of them are noted below:

I Chapter P-Prerequisites. This wonderfully concise openingchapter identifies mathematical content, algebraic manipula-tion skills, and technology-related knowledge needed in both Lprecalculus and calculus courses.

I Chapter 1, Section 1 provides a problem-solving process thatincorporates the traditional algebraic methods as well as thegraphical and numerical methods associated with graphingutilities.

I Chapter 1, Section 3 highlights the twelve basic functionsthatìreusedthroughoutcalcu1usandcapturestheirrespec-tive distinctives. This display is so useful that it is reprintedon the next pages of this book. Knowledge of these functionswill be incredibly important because they are used constantlyto illustrate calculus concepts and to model real-world phe-

nomenon (e.g., linearit¡ exponential growth, or periodicity).

I Throughoat Precalculus, many examples and topics are

marked with an icon, * , to point out concepts that fore-

shadow calculus concepts such as limits, extrema, asymP-

totes, and continuity. For your convenience, the tâble that fol-

lows shows each Precalculus icon location and references the

AP* Calculus Topic Outline found in Part I. :

Other trigonometric functions

I nverse trigonometric functions

Parametric relations

Numerícal derivatives and integrals

PrecalculusTextbook

1.3

1.3

1.5

2.3

2.7

3.1,3.2

4.4

4.5

4.7

6.3

10.4

16 PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREQUISITES

Precalculuslcon Location

1.1,p.74

1.2, p. 91

1.2, p.92

1.2, p. 100

1.3, p. 110

1.4, p. 121

2.1, p. 172

2.1, p. 179

2.1, pp.180-181

2.3, p.202

2.3, p. 206

2.7, p.248

2.7, pp.251-252

3.1, p.277

3.1, p.281

3.3, p. 303

3.4, pp.312-313

3.5, p. 321

4.1, p. 354

4.7, p. 419

5.1, p.446

5.2, p.459

5.4, p.472

6.3, p. 528

7.4, p. 608

8.6, p. 690

9.4, p.747

Precalculuslcon Reference Description

Solving equations algebraically

Continuity

I ncreasing/decreasing functions

End behavior

Ana lyzing functions graphically

Decomposing functions

Rate of change

Maximum revenue

Free fall

Local extrema

lntermediate value

Rational functions

0ptimization applications

Exponential change

Exponential base e

Base e logarithms

Change of base; base b

Exponential equations

Angular/linear motion'

lnverse trig function composition

Pythagorean identities

ldentities in calculus

Power-reducing identities

Motion of objects

Partíal fraction decomposition

Ouadric surfaces

lnfinite series

AP" Calculus ObjectiveOutline Code

t.A

t.D

il. c2

t.c

l.A

ll. F3

il.AAB

ll. E3

il. E6

il. E3

t. Dl

t. c1

ll. E3

ll. E4

ll. E4

ll. E4

il. E4

il. E4

ll. E6

ll. E4

ll. Fl

ll. Fl

II. FI

ll. E6

t. c2

ll. F3

t.B

PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREQUISITES 17

Twelve Basíc Functions

The IdentityFunction

lnteresting fact This is fte only function that acts on every real number

by leaving it alone.

The Cubing Function

f(x) = x

The SquaringFunction

lnteresting fact The origin is called a "point of inflection" for this curve

because the graph changes curvature at that point.

The Square Root Function

-s-4-3-2-t,-l

lnteresting fact The graph of this function, called a parabola, hæ a

reflection property that is useful in making flashlighb and satellite

dishes.

The Recþrocal Function

f(x) = x3

12345

f@) = x2

lnteresting fact Put any positive number into your calculator. Take the

square root. Then hke the square root again. Then take the square root

again, and so on. Eventually you will always get 1.

lnteresting fact This curve, called a hyperbola, also has a reflection

property that is useful in satellite dishes.

The Exponential Function

l8

f(x) ="8

f<ù= I

t''

lnteresting fact The number e is an inational number (like ø) that

shows up in a variety of applications. The symbols e and n were both

brought into popular use by the great Swiss mathematician Leonhard

Euler (1 707-1 783).

Í(x) = e'

PART ll: PRECALCULUS-REVIEW 0F CALCULUS PREREQUISITES

j.

lr' .. '

The Natural Logarithm Function

lnteresting fact This function increases very slowly. lf the x-axis andpaxis were bofi scaled wiü unit lengfis of one inch, you would have

to travel more than two and a half miles along the curve just to get afoot above the x-axis.

The Cosine Function

23456

f@) =lnx

The Sine Function

lnteresting fact The local extrema of üe cosine function occur exactlyat the zeros of the sine function, and vice versa.

The Greatest Integer Function

lnteresting fact This function and the sinus cavities in your head derive

their names from a common root the Latin word for "bay." This is due

to a l2th-century mistake made by Robert of Cheste¡ who translated a

word incorrecüy from an Arabic manuscript.

The Absolute Value Function

,f(x) = cos ¡

"f(¡) = sin¡

lnteresting fact This function has a jump discontinuity at every integervalue of ¡. Similar-looking functions are called step functions.

lnteresting factThis function has an abrupt change of direction (a "cor-

ner") at the origin, while üe other functions are all "smooth" on theirdomains.

The Logistic Function

PART ll: PRECALCULUS-REVIEW 0F CALCULUS PREREOUISITES

12345

"f(¡) = int¡

/(-r) = lrl = ¿þs ¡

-5-4-3-2-r

lnteresting fact ïhere are two horizonhl asymptotes, the x-axis and

the line y : 1. This function provides a model for many applications in

biology and business.

12345

l9

t obiectives:

Funct¡ons

BígPìcture

. Identiff the domain and range for a given function.

. Compute the ¡-intercepts and the 7-intercept for a given function.

Content and Prøctíce

Functions are the key mathematical concept in precalculus and calculus. You

should understand the definition of a function and how functions are

described by equations and tables. Every function has a corresponding graph.

The ability to look at functions from an algebraic, numerical, and graphical

perspective will be a great aid in understanding the concepts of precalculus

and calculus.

A, reløtion is defined a set of ordered pairs, usually of real numbers. A functionis a relation for which each ordered pair (¿ l) has a unique r-coordinate; thatis, no two pairs may have the same.r-coordinate. Some functions are not writ-ten as (x, y) pairq but the definition still holds: the first coordinate, whatever

the variable, must be unique.The domain of afunction is the set of all x-coordinates (first coordinates).

It is understood that if a domain is not given for a function, we should select

the largest domain possible-that is, all possible real values of x that can be

used. Two very common reasons that restrict domains to only certain real

numbers ate zero denominators and square roots of negative numbers. The

range of a function is the set of all y-coordinates (second coordinates).

Sometimes the easiest way to confirm the range of a function is to inspect its

graph.The y-intercepr of a function is the pair that has an x-coordinate of 0; the

corresponding point on the graph intersects the 7-axis. The x-intercepts of afunction are any ordered pairs that have a 7-coordinate of 0. The correspon-

ding points on the graph intersect the x-axis.

\ 1. A tunction/is defined as f(x) : \/; + 4.

20

(A) Identifr the domain of the function.

(B) Identifr the range of the function.

(C) Compute the 7-intercept.

(D) Compute the ¡-intercept(s).

(E) Sketch a graph of the function.

\2. A function/is defined as f(x) : x2 - 2x - 3.

(A) Identifr the domain of the function.

(B) Identifr the range of the function.

(C) Compute the 7-intercept.

(D) Compute the x-intercept(s).

(E) Sketch a graph of the function.

12345

PART II: PRECALCULUS REVIEW OF CALCULUS PREREOUISITES

-5 -4 -3 -2 -1 12345

2t

Aildìtional Prøctíce

1. The domain for the function f(*) : -! it\/x-2

(A)x>0 (B)x12 (C)x=2(D)x>z (E) x>2

2. For each function below:

(A) Sketch a graph. (Can you do it without the use of a calculator?)

(B) Identi4' the domain.

(C) Identi4' the range.

1i' f(x\: i

ä. g(x) : t/i

v

1

. [ ,r

Iv

1

. l r

Iv

1

. l rI

I

I

v

1

I

v

1

I

')

t/ \ 1txL. klxl

xt-9

Domain:Range:

iv. k(x) :

22

Domain:Range:

I

-{*

v. P(x): x2

Domain:Range:

Domain:Range:

PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREOUISIÏES

Domain:Range:

vi. q(x) : sin x

vä. s(x) : tanx

3. Determine the ordered pairs of all intercepts of /(x) : x3 - gx.

4. The range of the piecewise function defined by

f(x): {':;-"r"' îil o

(A) {all real numbers} (B) {y > t} (C) {y < t}(D) {y + t} (E) {y > o}

+

" Need More Heþ rûlíth . . . See . . .

Understanding functions? Precalculus, Section 1.3

+Domain:Range:

Domain:Range:

PART ll: PRECALCULUS REVIEW 0F CALCULUS PRERE0UISITES 23

* obiectives:

Transformations

Big Picture

. Write the function rule, given a parent function and a set oftransformations.

. IdentiS' the transformations for a given function rule and parent function.

Content ønd Practíce

In addition to being familiar with a basic set of parent functions and theircorrêsponding graphs, you should be able to describe important features ofother functions in the family of functions associated with a particular parent

function.

We relate graphs using transformations, which are functions that map real

numbers to real numbers. By acting on the x.coordinates andT-coordinates ofpoints, transformations change graphs in predictable ways.

The key transformations are translations, reflections, stretches and

shrinks.

Translatíons

Let c be a positive real number. Then the following transformations result intranslations (shifts) of the graph of y : f(x)z

Horizontal translations

y : f(x - c) a translation to the right by c units

y : f(x + c) a translation to the left by c units

Vertical translations

24

y: f(x) + c

v: f(x) - c

a translation upward of c units

a translation downward of c units

Reflections

The following transformations result in reflections of the graph of y : f(x):

Across the r-axis

y : -f(x)

Stretches ønd Shrínl'r.

Across the y-axis

Let c be a positive real number. Then the following transformation result instretches or shrinks ofthe graph of y : f(x):

Horizontal stretches or shrinks

y : f(-x)

v : f(i)

Vertical stretches or shrinks

y: c'f(x)

a stretch by a factor of c, if c ) ta shrink by a factor oî c, if c ( I

When two vertical transformations are used, the order of operations pre_vails. For example, a vertical stretch would be done before

"'.r.rti.¿ ,írift.

However' when two horizontal transformations are used, it is generally easierto describe the transformation in reverse-for example, a shift before astretch.

1. A function g is defined as g(r) : 21/¡ ¡ 4 ¡ 3.

(A) Identifrthe domain of the function.

(B) Identi$'the range of the function.

a stretch by a factor of c, if c ) Ia shrink by a factor of c, if c 1 I

PART II: PRECALCULUS REVIEW OF 0ALOULUS PREREQUISITES

(c) Describe the transformations that show how the graph of thisfunction is obtained from the graph of the parent"fuictionf(x) : t/i.

25

(D) Sketch a graPh of the function'

9

I7

6

5

4

3

2

I

2. The eraph of a function g(x) is obtained from the graph of the parent

fu*f# f@) : x2 by añ x-axis reflection, a vertical stretch by 3, a

vertical túift dorvtt 4,anda horizontal shift right 1'

(A) Write the equation that describes the rule for the function'

(B) Sketch a graPh of the function'

5

4

3

2

I

26

(C) Identify the range of the function'

l

i

i

-1-2-3-4-5

PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREQUISITES

i.:

i,

3. The equation of the graph of y : sin x reflected

(A) 7 : sin(-x)

(B) y: -sinx

(C) /: sin(r - l)

(D) y: -sin(-x)

(E) r : sinT

Adilìtìonal Practìce

in the ¡-axis is

1. The graph of f(x) : lxlis shown in the figure.

PART ll: PRECALCULUS REVTEW 0F CALCULUS PRERE0U|S|TES 27

Sketch a graPh of each function'

(A) y: -f(* - L) + 2

-5 -4 -3 -2 -1 12345

(B) y:zf(!+r)

Which of the following represents a vertical shift up 3 and a horizontal

shift left a of f(x) = lrl?(A) s(¡) : lx - al- 3

(C) s(x\:lx-+l+t(E) s(¡) : lx + 3l- a

-5 -4 -3 -2 -1

3. Which of the following represents the graph of y : /(*) first shifted

down I and then reflected in the x-axis?

12345

Need More HeIPWíth

(A) y:f(-x)-L(c) ,: -f(x) * 1

(E) y:f(-x)+1

(B) s(x):lx+al-3(D) g(x) : lx * 4l+ 3

Tiansformations?

28

(B) y:-(f(x)-1)(D) y:lf@ - 1)l

See ...Precalculus, Section 1.5

PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREOUISITES

* objectives:

Polynomial Functions

Bíg Picture

' Determine the possible number of real zeros for a poþomial function.' Determine the possible number of extrema for a pãþomial function.. Determine the end behavior of a poþomial funition.'Determine the intervals where a poþomial function is increasing or

decreasing.. Compute zeros and extrema of a poþomial function.

You should be familiar with the family of functions known as polynomialfunc-tions. These functions are continuous. The degree and the teuai"g coedcientdescribe specific patterns of increasing/decreasìng and end behavior as well asthe possible number of zeros and extreme pointsl

For this section we are assuming poþomial functions with real coefficientsand unrestricted domains.

A polynomial function of odd degree always has at least one real zero.(since function "zeros" equate to graphical x-iniercepts, that also guaranteesthat the graph of a poþomial of odd degree must iniersect the,-*i, at leastonce.) Because nonreal zeros occur in conjugate pairs, the possible number ofreal zeros increases by two, up to the degree ofthe functìon. Thus, a third-degree poþomial function has one o, thr.. real zeros, and a ñfth-degreepoþomial.function has one, three, or five real zeros. similarl¡ a fouith-degree poþomial function has zero, two, or four real zeros.

l. use a graphing^ calculator to graph a variety of cubic equations:y: )c3,y: x3 - 3,y: x3 - x2,/: x3 - x2 - 2,andothersofyour choice.

(A) Note the number of zeros of each function.


N

(B) Note the number of x-intercepts of the corresponding graph.

29

(C) Note the number of extrema of each function'

(D) Note the end behaviors of each function'

Nz. Useagraphingcalculatortograp!avarietyoffourth-degreeequations:y: J,i: in - x,l: x4 - x2,l: x4 - x3 + 2x2 - )c - l,and

others ofyour choice.

(A) Note the number of zeros of each function'

(B)Notethenumberof¡-interceptsofthecorrespondinggraph.

(C) Note the number of extrema of each functio¡r'

(D) Note the end behaviors of each function'

When afunction such as y : (x- 5)3 ot y : xs has a single factor that

is repeated, in general form (r - c)^,we say that the respective zero has mul-

tiplicity m. So for y : (x - 5)3 in which (x - 5) is a repeated factor,5 is a

z.ero of.multiplicity 3. For y : x5,0 is a zero of multipliclty 5'

The end behavior of a poþomial function of even degree is the same as

x approaches both positive and negative infinity. when the leading coefficient

is positive, the end behavior is: as ¡ Ð Ñ, l+ 6' and as ¡-> -oo' y -è æ'

When the leading coefficient is negative' the end behavior is: as

¡+oo,y-è-Ø,andas¡-à-oo,y--æ'Becauseofthisbehaviorapoþomial function of even degree has an odd number of extrema' For exam-

ple, a fourth-degree function has three or one extrema'

The end behavior of a poþomial function of odd degree is different as x

approaches both positive and negative infinit¡ when the leading coefficient is

pori irr., the end Lehavior is: as ¡ -+ oo' / è @' and as t,-t - *' y --è - æ'

When the leading coefficient is negative' the end behavior is: as

¡+oo, lè-6, and as ¡Ð-oo, ytæ' Because of this behavior' a

potyrrolniut function of odd degree has an even number of extrema' For exam-

ple, a fifih-degree function has four two, or no extrema'

30 PARTII:PRE0ALOULUSREVIEW0F0AL0ULUSPREREQUISITES

;_- ìr

ìl

i. "ir,.

\¡. Afunction/isdefined asf(x):2x3 - 9x2 - Z4rc + 31.

(A) Identif' the possible number of real zeros.

(B) Identift the possible number of exrreme points.

(C) Predict the end behavior off

(D) Compute all real zeros.

(E) Compute the coordinates of all extrema.

(F) Describe the rising/falling behavior of the graph.

(G) sketch the graph off confirm wirh a graphing calculator.

\ ¿. fff(x):(:',c+tinct real zeros?

(A) I (B)

PART II: PRECALCULUS REVIEW OF CALCULUS PREREQUISITES

lX¡ - Ð3þc2 + 4), then /(x) has how many dis-

2 (c)3 (D)4 (E) 6

3t

Aililítionøl Prøctíce

\1. Afunction/is defined as /(¡) : x4 - 8x2 + 7'

(A) Identiff the possible number of real zeros'

(B) Identiff the possible number of ertreme points'

(C) Predict the end behavior of f

(D) Compute all real zeros.

(E) Compute the coordinates of all extrema'

(F) Describe the rising/falling behavior of the graph'

(G) Sketch the graph off Confirm with a graphing calculator'

\ z. Which of the following describes the possible number of real zeros

andextremaforafifth-degreepoþomialfunction?

(A) 5 real zeros;4 extrema

(B) 4,2,or 0 real zeros;3 or I extrema

(C) 5 real zeros; 4,2, at 0 extrema

(D) 5,3, or 1 real zeros;4 extrema

(E) 5, 3, or 1 real zeros;4,2, or 0 extrema

NeeilMoreHelPtüith . . .

Poþomial functions?

32

See ,..

PART ll: PRECALCULUS REVIEW 0F cAlqq!!!-llE!!g!l!l]Eq

Precalculus, Section 2.3ì

t obiectives:

Rational Functions

Big Pícture

. Determine the domain for a rational function.' Determine the vertical asymptotes and removable discontinuities.'Determine the end behavior for a rational function, including horizontal

asymptotes.

You should be familiar with the family of functions known as rational func-¡ions. These functions are usually continuous over their domain, but have val-ues that are not in the domain. Thus, they are often not continuous every-where. You should be able to determine the behavior of the function at thesepoints, In addition, you should be able to describe the end behavior. All of thisinformation should be used to sketch a graph. The ability to look at thesefunctions from an algebraic, numerical, and graphical peispective will be agreat aid in understanding the calculus concept of limits.

A rational function is a ratio of two poþomial functions.

Content øndPractìce

f(*):# s@): x2-4

Domains: If the function in the denominator has any realzeros, these values are not in the domain.

. The domain of f(x) does not include -3.. The domain of g(x) does not include 2.

. The domain of h(x) does not include 0 or 6.

. The domain of k(x) does not include -1.Removable discontinuities:Yol should determine if the func-tion has a removable discontinuity. you could factor thenumerator and denominator separately. If a linear factor inthe denominator appears at least as often in the numerator,then the function will have a removable discontinuity at thevalue that makes that factor zero.

x-2 h(x) : x2-3x-lgx2-6* k(x):d+#

33

g(x) has a removable discontinuity at 2 because it has a

common factor of (x - 2) in its numerator and

denominator. Since lim g(x) : 4, (2,4) is identified as

the ordered pair ursJcilted with the removable discon-

tinuity.

h(r) has a removable discontinuity at 6 fot a similar

reason. Since lim g(x) : tlr, (6,'lr) ßidentified asx'-->6'

the ordered pår ãssociated with the removable discon-

tinuity.

I Verticøl øsymptotes: You should determine if the function has

a vertical áry-ptot.. Again, considering the linear factors in

the numeraiorãnd denominator, if the factor is in the

denominator onl¡ or is a factor of the denominator more

times than in the numerator, then the function has a vertical

"îï*ilïiii:iifiliHli:'í,-

In the language of calculus, we wish to determine the values that are not

in the domain. Then, we investigate the limit as x approaches that value' If the

limit is finite, then we have a Ãmovable discontinuity; if it is infÌnite, then

there is an asYmPtote.

I Endbehavior:You should also investigate the end behavior' Ifthedegreeofthenumeratorislessthanthedegreeofthe.denoriinator, then the end behavior is zero and the function

has a horizontal asYmPtote, Y : 0'

. k(¡) has a horizontal asymptote, f : 0'

J Horizontal asymptotes: lf the degree of the mrmerator is the

same as the áegiee of the denominator the function has a

horizontal asymptote but not atzeto.The value is deter-

mined by the ratio of the leading coefficients'

. l(¡) has a horizontal asymptote, y : 5'

. h(x) has a horizontal asymptote, y : l'

I Slant asymptotes: If the degree of the numerator is one higher

than thá dågree of the den-ominator, the function has a linear

asymptote tut it is not horizontal; rather, it is a slant (or

oblique) asYmPtote.

34 PARTII:PRECALCULUSREVIEW0FCALCULUSPREREQUISITES

'T

I Parabolic asymptotes: If the degree of the numerator is morethan one higher than the degree of the denominator, thefunction has an end behavior asymptote but it is not linear.For example, if the degree of the numerator is two higher,expect a parabolic asymptote.

In the case of either slant and parabolic asymptotes, the end behavior asymp-tote is determined by the quotient (without remainder) of the ratio of the twopoþomials.

' g(x) hasaslantasymptote, y: x * 2.

. None of the functions f g, h, or k, has a parabolicasymptote.

\ 1. A function/is defined as /(x) : fff,(A) Write the domain.

(B) Write the ordered pair for any removable discontinuities.

(C) Write the equation for any vertical asymptotes.

(D) Write the equation for any horizontal asymptotes

(E) Describe the end behavior off

(F) Compute the 7-intercept.

PART II: PRECALCULUS REVIEW OF CALCULUS PREREQUISITES 35

(G) Sketch the graph ofl

ru2. /(x) is defined as /(x) : *.(A) Complete the table of values for f(x).

36

te

0

0.9

0.99

0.999

1.001

1.01

1.1

2

(B) Referring to the table of values, describe the graph of f near

x : I. ..

'',:,

f(x)

PART'll: PRECALCULUS REVIEW 0F CALCULUS PREREQUISITES

Aildìtionøl Practíce

1. A function/is defined as /(x) :

(A) Write the domain

(B) Write the ordered pair for any removable discontinuities.

(C) Write the equation for any vertical asymptotes.

(D) Write the equation for any other asymptotes.


(F) Compute the 7-intercept.

x2+x+3x-l

(G) Sketch the graph off Confirm with a graphing calcularor.

l0

9

8

7

6

5

4

3

2

I


-5 -4 -3 -2 -r-l_)

-3

-4

-5

-6

t2345

37

2. Which of the following best describes the behavior of the function

f(*) : #at the values not in its domain?

(A) One vertical asymptote, no removable discontinuities

(B) Two vertical asymPtotes

(C) Two removable discontinuities

(D) One removable discontinuity, one vertical asymptote, x : 2

(E) One removable discontinuity, one vertical asymptote, x : -2

NeedMoreffielPWith .

Rational functions?

I

See ...Pr ecalculus, Section 2. 7

38 PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREOUISITES

* obiectives:

Exponential Functions

Bìg Picture

. Determine the range for an exponential function.

. Determine the end behavior for an exponential function, includinghorizontal asymptotes.

Content andPractíce

You should be familiar with the family of functions known as exponentialfunctions. These functions usually have a domain of all real numberi and arecontinuous over their domain. You should be able to describe the end behav-ior, which often includes a horizontal asymptote in one direction. All of thisinformation should be used to sketch a graph.

A basic exponential function has an equation of the form /(x) : a. b*.

Tiansformed functions may have additional coefficients, such as

f(x) : a.b'-h * k. For a: L,b : 3,h: O,andft - -4,thefunction is

f(x) : ll. - 4 and,its graph is shown in the figure.

The end behavior of such functions will always be infinite as r-+ oo (ifb > I) or Í+ -oo (if 0 < b < l) and have a horizontal asymptote for theother direction. Related behavior will mean that the function is either increas-ing or decreasing and that the graph of the function is rising or falling.

-5 -4 -3 -2 -1

39

Exponential functions model many real phenomena' such as population

growth, half-life deca¡ and Newton's law of cooling. For many of these Tod-ã1r ttt" y-interceptrepresents an initial value (at time 0) for the variable that y

represents.- F*ample: A townt population is 3000 and projècted to double every l0

years. An exponential model for the population, P, expected in d decades,

wouldbeP: 3000'2d.

\ 1. A tunctionfis defined as /(x) : z+(l). + +.

(A) Compute the y-intercePt.

(B) Compute f(x) for xe {-2, -1,1,2}-

(C) Is/increasing or decreasing? Explain.

(D) Write the equation for the horizontal asymptote.


(F) Sketch the graph ofl

I

I

40 PART ll: PRECALCULUS BEVIEW 0F CALCULUS PREREQUISITES

Aililitionøl Prøctíce

1. A function/is defined as f(x) : 3.2*.(A) Compute the 7-intercept.

(B) Compute f(x) for xe {-2, -1,1,21.

(C) Is/increasing or decreasing?.

(D) Write the equation for the horizontal asymptote.


(F) Sketch the graph off,

2. A radioactive substance decays so that half of the substance decaysevery 2 minutes.If 100 g of the substance are present initiall¡ howmany grams will be present after 4 minutes and I minutes, respectively?

(A) 6.2s,0.39062s

(B) 25,6.25

(c) 2s, r2.5

(D) 50, t2.5

(E) 50,25

NeedMoreHelpWith . . .

Exponential functions?


See ...Precalculus, Sections 3.I-3.3

4t

* obiectives:

Sinusoidal Functions

BígPícture

. Determine the equations for a sinusoidal function for a particular graph.

. Determine the transformations for a sinusoidal function, given an equation.

Functions are the key mathematical concept in precalculus and calculus. You

should be familiar with the famity of functions known as trigonometric func-tions. The first type we investigate are sinusoidøI functions, defined as a sine otcosine function.You should be able to sketch a graph for these types of func-

tions, given the equation. You should also be able to describe the transforma-

tions that show how the graph compares to the basic sine or cosine graph. The

ability to look at these functions from an algebraic, numerical, and graphical

perspective will be a great aid in understanding the concepts of precalculus

and calculus.

Content anil Prøctíce

The two basic sinusoidal functions are y: sin (x) and 7 : cos (l).Transformed functions will have additional coefficients, such as

y : a. sin [b(x - h)] + k. The coefficient a determines the amplitude, the

same transformation as a vertical stretch. The coefficient k determines the ver-

tical shift. The coefficient h determines the horizontal shift, which in

trigonometry is often referred to as a phase shift. The coefficient b determines

the horizontal stretch or shrink. Because sinusoidal graphs are periodic,^this

coefficient also determines the period of the graph, found by computi"yffi.

t. A function/is defined as f(x): 3 cos lå (" - i)] * t.

(A) Determine the amplitude.

(B) Determine the vertical shift.

42

(C) Determine the range of the function.

.J.

.t.

(D) Determine the horizontal shift.

(E) Determine the period.

(F) Write the coordinates of two local maximum and two local mini-mum points.

(G) Sketch the graph.

The graph of a sinusoidal function is shown.

(A) Write the coordinates of two local maximum and two local mini-mum points.

(B) Determine the amplitude.

(C) Determine the vertical shift.

(D) Determine the period.

PART II: PRECALCULUS REVIEW OF CALCULUS PREREQUISITES 43

(E) Write the equation of a cosine function that has this graph.Identifr the horizontal shift for this function.

Aililìtíonøl Practìce

(F) Write the equation of a sine function that has this graph. Identifythe horizontal shift for this function.

l. A sinusoidal function has a local maximum at (2,8) and the next local

minimum at(6, -2).(A) Determine the amplitude.

(B) Determine the vertical shift.

(C) Determine the range of the function.

(D) Determine the period.

(E) Write the equation of a cosine function that has this graph.Identi$rthe horizontal shift for this function.

(F) Write the equation of a sine function that has this graph. Identiffthe horizontal shift for this function.

(G) Sketch the graph.

44 PART II: PRECALCULUS REVIEW OF CALCULUS PREREQUISITES

',,- 1

2. A sinusoidal function has a local maximum at (0,2) and the next min-imum at(nf 4, -Z).Acorrect equation for the function is

(A) y:2cos(ax)(B) y:zsin(¿r)(C) y:4cos(x)(D) y:4cos(4x)(E) y: sin(+x)

3. The functioÍr y : -3 cos (x - nla) has a local maximum at whichpoint?

(A) (nl+,r)(B) (zr/a,3)

(C) (tn/+,3)

(D) (snl+,r\(E) (snl+,t)

Need More Heþ tMith .

Sinusoidal functions?


See ...Pr ecalculu s, Section 4.4

45

W obiectives:

More Trigonometric Functions

BígPìcture

. Sketch the graph of a trigonometric function.

. Determine the transformations for a trigonometric function, given an

equation.

You should be familiar with the family of functions known as ffigonometric

functions.In the previous section, we explored sine and cosine functions. This

section explores the other trigonometric functions, the tangent, cotangent,

secant, and cosecant functions. You should be able to sketch a graph for these

t1ryes of functions, given the equation. You should also be able to describe the

transformations that show how the graph compares to the basic trigonomet-

ric graph. The ability to look at these functions from an algebraic, numerical,

and grãphical perspective will be a great aid in understanding the concepts ofprecalculus and calculus.


I

l

l

The first new trigonometric function is y : tan (¡). One definition states that

tan (x) : ä8.Therefore, the domain of this function does not include all

real numbers; it excludes values where cos (x) : 0, namely the odd multiples

of nf Z.The graph of y : tan (x) will have vertical asymptotes at these values.

Tiansformed functions may have additional coefficients. However, the main

transformations are those that transform these vertical asymptotes-the hor-

izontal stretch, shrink, or shift. The basic function y : tan(x) has period ø,

and the function y : tan(bx) has neriod ff.

46

.1.

Each of the sine, cosine, and tangent functions has a reciprocal function.They are the cosecant, secant, and cotangent functions, respectiveþ All ofthese reciprocal functions also have vertical asymptotes. The cotangent func-tion has period

ff, U.tt the secant and cosecant functions have neriod ffi.

1. A tunction/is defined as f(x): t"" (j").

(A) Determine the period.

(B) Determine the domain.

(C) Determine the equations of the vertical asymptotes.

(D) Determine the x-intercepts.

(E) Sketch the graph, showing several periods.

2. Write the transformations that describe how the graph ofg(x) : 2 csc(3(x - ø)) * I compares to the graph off(*) : csc (x).

PART ll: PRECALCULUS REVIEW 0F CALCULUS PRERE0UISITES 47

3. Sketch each pair of functions on the same set of axes; show at least twoperiods and label the axes.

(A) /: cosxand y: sec)c

(B) 7 : sin xandY : ,t"G - î)

(C) y: tanxand y : cotx

Aililítíonøl Practìce

+2

l. A function/is defined as /(x) : 3 sec (2x) + l.(A) Determine the period.

(B) Determine the domain.

48

(C) Determine the equations of the vertical asymptotes.

(D) Determine the vertical shift.

PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREQUISITES

")

(E) Write the coordinates of two local maximum and two local mini-mum points.

(F) Sketch the graph, showing several periods.

2. If the graph of f(x) : cot (x) is transformed by a horizontal shrink of

å *d a horizontal shift left n,the result is the graph of

(A) s(x) : cot ff(r - r)](B) s(x) : cot l|(x * ')](C) g(x) : cotlfi(x - T))

(D) g(¡) : cot !a(x + n)l

(E) g(x) : cot(bc + n)

3. The function y : tan(xl3\ has an x-intercept at

(A) îrl3(B) 7r

(C) 2n

(D) 3n

(E) 6r

NeedMoreHelpWíth ...tigonometric functions?

PARï ll: PRECALCULUS REVIEW 0F CALCULUS PREREOUISITES

See . ".Precalculus, Section 2.3

49

* obiectives:

lnverse TrigonometricRelations and Functions

BìgPícture

. Identifr the domain and range of an inverse trigonometric function.

. Compute values for inverse trigonometric relations and functions.

Cantent andPractìce

Each of the trigonometric functions has aninverserelntion. These relations are

not functiots becaot. the original functions are not one-to-one. You should

be familiarwith the infinite set of values for an inverse relation. However, if the

range of the inverse relation is properly restricted, then the inverse is a func-

tion with only one specific value. You should be familiar with the domain and

range of each of these functions.

The equation sin (x) : j ttus an infinite number of solutions. In the

Precalculustextbook, the symbol sin-t(|) denotes a single value, f. The func-

tion /(x) : sin-l(x) is the inverse sine function and it has domain [-1, 1]

and range l-n,:rl. The restricted range is selected so that the inverse will

have the same domain, will actually be a function, and will pass the vertical

line test. We wish to accomplish the same thing with the inverse cosine but

cannot use the same range because it would not pass the vertical line test.

Thus, each of the trigonometric functions has an inverse. The sine, cosine, and

tangent are summarized in the table below. There are certain values, such as

our sin-t(|) example, that you should be able to compute without a calcula-

tor; others require the use of a calculator.

50

Function

sin -r(¡)cos -r(x)

tan -1(x)

Domain

[-1, 1]

[-1, 1]

(-oo, oo)

l-o nlL 2,2J

[0, ø]

?t,i)

Sometimes you are asked to solve an equation such as sin (x) : -You can use the inverse trig function to find a solution. However, the equation

actually h?r an infinite number of solutions. For example,

,i"-t (-+) : -i,but sin is also negative in Quadrant III so f is also a

solution. The complete set of solutions could be written as

{-t* 2ntr,!+2nrr}.

1. Compute exact values.

/ \Æ\(A) .or-t { -\--l \ 2)

(B) tan-r(-t)

(c),,"-'(+)

ru 2. Evaluate. (3 decimal places)

(A) cos-1(0.823)

(B) tan-t(2.+)

(C) sin-t(-o.ozt)

(D) sec-r(-0.511)

ú2

(B) sin (x) - -1

(C) tan (x) : -+PART ll: PRECALCULUS REVIEW 0F CALCULUS PREREOUISITES

3. Write all solutions.

(A) cos (x) : j

51

Addítìonal Prøctíce

ru1. Evaluate. (3 decimal places)

(A) cos-1(-o.z+o)

(B) tan -t(-t.s)

(C) sin -r(o.szs)

Solve each equation.

(A) 3*tan(x):2

(B) 4cosz(x):3

(C) zsin2(x) : sin(¡)

(D) cos2(r') : 4

3. sin -

(A)

(B)

(c)

(D)

(E)

,( 1\ -\-z) -7T-6T-5

7f6

5r6

7n6

NeedMoreHelpWíth . . .

Inverse trigonometric functions?Solving trigonometric equations?

52

Þr:¡**

See...

Precalculus, Section 4.7Precalculus, Sections 5.1-5.4

PART II: PRECALCULUS REVIEW OF CALCULUS PREREQUISITES

* objectives:

Parametric Relations

Big Pícture

. Given parametric equations, plot relations by hand or calculator.' control the speed and direction of the plot by varying t and its increments

or by varying the equations.. Produce parametric equations for Cartesian equations.'convert parametric equations to cartesian equations (eliminate the

parameter).. Model motion problems.

Content and Practíce

Parametrics offer a powerful method to plot many relations whether or notthey are functions. ihey also allow us to model motion, since we have morecontrol over how points are plotted. Beyond this course, you will work withthe calculus of parametrics, so gaining a high level of comfort with them nowwill assure future success.

when you first learned to plot lines, you probably used a chart where youchose x-values and plugged them into an equation to produce 7-values. withparametrics, the x- and 7-values are produced independently by substitutingfor a third variable, ú, called the parameter. In modeling motion, r usually rep-resents time.

1. Given the following parametric equations, produce a table of valuesand plot the relation. The table has been started for you.

t-2-10

1

2

3

x6

v-4

xt: t2 - t

lt: 2t

53

2. Using substitution, convert the parametric equations in Problem 1 to

CartÃian form. This is called eliminating the parameter. (Hinr: Solve

for x as a function of 7.)

When parametric equations contain trigonometric. functions, we often

rely on a trigonometric^identity rather thai substitution to eliminate the

paíameter. Cõnsider the parameiric equations below.

If we use the trigonometric identity L * tanz 0 : sec2 g, the equation

becomesl+2y:)c2.

3. Convert to Cartesian coordinates: f : 3 sin (r), y : 4 cos (r)'(Hint: Divide each by the constant first.)

x¿ : sec (r)

Any function canbe converted to parametric form simply by letting the

independent variable be ú. So, for instanc ê, I : \/7 + ,-can be converted

to ,c : t and y : \/t2 + z;.We must rcalize,though, that due to limitations

on the values of f we will not always produce a complete graph. For instance,

y:2)c - l canbedefinedparametricallyas ,c: tandy:2t - l,butif r

goes from - 10 to 10, we would only see a plot of a segment from (- 10' -21)to (10, 19).

Yr: ttan2(t)

4, Determine parametric equations to plot the right half of the parabola

y : (rc - t)'. Graph it on your calculator to see if you have achieved

your goal.

5. What is the effect of changing the increments of t, or ústep on the cal-

culator? Find out by exploring. Try the following examples, comparing

AtoBandAtoC.

Xt:

It:

54

(A) tq: 2t - |

ft: t + I

,mln

,max

+:&step

fmin : 0

t*u*: 3

ts¡¿p : 0.2

PART ll; PRECALCULUS REVIEW 0t CALCULUS PREREQUISITES

(B) xÍ: 2t - | fmin : 0

/t: t+I f-*:3úr¡"n : 0.02

(C) &:2t-l fmin: 3

/t: t+l f**: 0

h¡sp : -0.2

(A) A compared to B: What was the effect of making the frr.n smaller?Explain why it caused that effect.

(B) A compared to C: What was the effect of a negative rrl.o? Explainwhy it caused that effect.

6. compare the next two plots, where just the functions were changedslightly. (Make sure you plot in radians.) Explain the similarities anddifferences in the plots.

(D) xr: 3 cos (r) úr.,in : 0

ft : 3 sin (r) t^*: 6.3

f51sp : 0.1

(E) x, : sin (r) fmi,, : 0

y, : cos (t) t^*: 6.3

úr,.n : 0. 1

Parametric equations also allow us to model motion problems. we canmodel vertical motion, projectiles launched at an angle, circular motion, andmany other kinds of motion.

To model vertical motion parametricall¡ let r equal any constant, e.g.,x : l-Let y : -l6f * vst f fts where z6 is the initial velocity in ft/sec aidh6 is the initial height in feet.

PART ll: PRECALCULUS REVTEW 0F CALCULUS pREREOUtStTES 55

Projectile motion at an angle requires changing the equations slightly to

)c:1laóos(0).rand f : -76t2 * vssin (0)'t * ho,wheregistheinitialangle irom the horizontal. These equations take into account the horizontal

anã vertical components of projectile motion as described in the Precnlculus

text. They ignore air resistance.

Circular motion is often modeled using :'c: r'cos(b'Ð + c and

y : r. sin (b . t) + d, where ris the radius of motion, c is the horizontal shift,

d is a vertical shift, andb is determined by the period'

7. A projectile is fired straight up from the ground with an initial velocity

of AS feet per second. Write parametric equations to model the

motion.

8. A Ferris wheel has a diameter of 30 feet. Its lowest point is 8 feet off

the ground. If it turns clockwise one full rotation each 20 seconds,

*it p.tu*etric equations to model a passenger's motion starting

from the bottom and riding six full rotations.

Aildítìonal Practíce

1. The parametric equations x¡ : 2t + 3 and yr: {tll plot a

portion of alan

(A) Line

(D) Ellipse

56

(B) Parabola

(E) Hyperbola

: .' \

f .*+"

(C) Circle

PART ll: PRECALCULUS REVIEW OF CALCULUS PREREQUISITES

i..i,. _

A ball is thrown with an initial velocity of 4g feet per second at anangle of 35' with the ground. If the ball is released at an initial heighto{-s

!ee! offthe ground, approximately how far will it travel horizon-tally before strfüng the ground?

Need More Help With .

(A) 74 feet(D) 42 feet

Parametric equations?

(B) 68 feet

(E) 36 feet

(C) 45 feet

See ...Precalculus, Section 6.3

PART II: PRECALCULUS REVIEW OF CALOULUS PREREQUISITES 57

t obiectives:

Numerical Derivativesand lntegrals

BígPicture

. Estimate the slope of a curve at a particular point'

. Compute an average rate of chanqg'

. Write an equation äf ,n. tangentfin e at a particular point on a particular

curve.. Estimate the area under a curve'

The two most fundamental concepts in all of calculus are those of a derivatitte

;d "; integral. The derivative function tells us the slope of a curve at any

p"i",. e deñnite integral is used to compute the area under a curve.

your work with limits should have already developgf the idea that places

where a function looks "curved" may actuily belocally linear (straight over

infinitely small intervals). This atlows us to talk about the slope of nonlinear

functions.Wedefinetheslopeofasecantonafunction/(x)tobe

Content ønil Practíce

As the size of h gets smaller, the secant more and more accurately approli-

mates the slope of tfr. i""gent line (if it exists) to the function at a given point

(x,f(x)). The

f(*) : f(x+h)-f(x)

is the actual slope of the tangent to the function, when we can evaluate that

limit.When we cannot evaluate the limit, we can be satisfied with a fairþ accu-

rate numerical approximation we call the numerical derivative'Yout calculator

should have a buitt-in function to evaluate the numerical derivative' Most cal-

culators use the symmetric difference quotient to estimate the derivative' The

syrnmetric ditrerence quotienJ uses points 0.001 units to the right and left of

iír. pf".. we are trying to find the tJngent. Calculating the slope of the secant

58

limh--->0

f(x+h)-f(Ð

,.1

between those points usually provides a good approximation of the slope ofthe tangent:

The particular slmtax for using a numerical derivative function on your calcu_lator may be discussed in class or can be found in your calculator manual.

Be aware that built-in numerical derivatives work only on functions. Anumerical derivative must be calculated manually when only discrete data areavailable. Under those circumstances, we find thå slope on the smallest inter-val containing the point whose derivative we are r..kirrg.

we also use the limit concept to compute the area rinder a curve. First wepartition the domain of the function into small intervals. For each interval wedraw a rectangle that estimates the area under the curve. The actual heightused can be chosen from the left edge, the right edge, or the center of the inter_val' The sum of the area of the reciangles estimates the area under the curve.Your graphing calculator should also have a built-in function to estimate thisatea.

l|lÁn N f(*+h)-f(x-h)2h

t. Byusing f(0.999) andl(r.001), findan approximation of the slope ofthe tangent to the function f(x) : e2* at )c : l.Use the built_innumerical derivative on your calculator to verifr your answer.

, withå:0.001.

Aìldítional Practice

2. Estimate the area under f(x) : x2 + r over [0,3]. sketch a graph andshade the appropriate area.

1. which of the following is the equation of the line tangent toy : +,x2 + 2xat the point where x : 2?

(A) y:4x (B) y:4xt6 (C) y:4x*2(D) y : 4x - 2 (E) No tangent can be drawn.

PARI ll: PRECALCULUS REVTEW 0F CALCULUS PREREQU|S|TES 59

") The heisht of an obiect dropped from a 200-foot building is given by

;':-;;3'--¿:P,*ú.r. risäeasured in seconds and h is measured in

ieet. What is the velocity of the object 2 seconds after it is dropped?

(A) -16 ft/sec

(D) 64 ft/sec

3.Thenumberofgallonsofwaterinatubúminutesaftertheplugispulled is shown in the table'

(B) -32ftlsec(E) 32ftlsec

(A) Find the average rate of change in the volume in the first 4 minutes'

(Include units.)

(B) use the data to find an estimate of how fast the volume is chang-

ing at the Z.75-minute mark' Show your work'

NeeilMoreHelPWìth .

rn r02 8S 76 65 57 50

(C) -64ftlsec

Numerical derivatives?

See...

60

Precalculus, Section I 0.4


'_.Jr

Review of AP+ CølculusAB ønd BC Topics

Part III

'\'

AP* 0biective:

Big Picture

Analysis of Graphs

Predict and explain behavior of a function. Interplay between the geometric and anal¡ic information

Analyzing graphs is a critical tool in the study of calculus. with the use ofgraphs we can make conjectures, solve problems, and support our writtenwork. The graphing calculator has enabled us to quickly and easily producegraphs of functions. It is very important, howevei tttui we have an under-standing of the graphs of basic functions and their behaviors in order to deter-mine which function best models a given situation and to select an appropriateviewing window on our calculators or axis labels on our graphs. tn piecalculus,your studies included the following key function behaviors:ào-"in and range,whether a function is odd or even, symmetr¡ whether a function is periodicãrcontinuous, zeros, intercepts, asymptotes, extrema, and translations.

The Acorn AP* Course Description guide states that as a prerequisite to cal-culus, students should be familiar with the properties of the gr"phr of linear,polynomial, rational, exponential, logarithmic, trigonomeiric, inversetrigonometric, and piecewise functions.In your precalculus class, you studiedtwelve basic functions. These twelve are very useful for understunáittg graphsand their transformations.


The ldentityFunction

12345

y:x

The Squaring Function

t2345

y:x2

63

The Cubing Function

t2345

The Square Root Function

i:f

The Recþrocal Function

The Natural Logarithm Function

y: \/x

The Exponential Function

I1t:-/x

12345

64

f:lnx

/={

The Sine Function

5

43.,

I

y : sLtr)c

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS'..;*

The Cosine Function

The Greatest Integer Function

y: cosx

The Absolute Value Function

\ e¿¿;øonalpractice

y: intx

The Logistic Function

y:lxl

-5 -4-3-2-1

Many years ago, assessment of specific precalculus knowledge was explicitlycovered within annual AP* Calculus Exams. Characteristics of functiãns ortheir graphs such as domain and range, zeros, intercepts, and symmetry wereassessed in both multiple choice and open-ended questions. The followingonce-used AP* Free Response questions were originally created to be solveãwithout the use of a graphing calculator-try your hand!

1. Letfbethe real-valued function defined bV f@) : \/I + 6;. Givethe domain and range off Í197çAB l: Denotes the 1976 FreeResponse Problem l-AP* Calculus AB Exam.l

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0ptCS

v: Il*e-x

65

2. Given f(*\ : x3 - 3x2 - 4x * l},findallzerosof thefunctionf

lLe76-AB 2l

Letf(x): cosxfor0 3 xs 2r,letg(¡) : lnrforallx > 0'LetS

be the composition of gwith f; thatis, S(¡) : sU@D' 1L977-AB ll(A) Find the domain of S.

(B) Find the range of S.

(C) Find the zeros of S.

Given the function/defined by f(x) : x3 - x2 - 4x * 4, find the

zcros of f. [197S-AB 1]4.

5. The curve in the figure represents the graph off, where

f(x) : x2 - zxfor all real numbers x. [I979-AB 6)

(A) On the axes provided, sketch the graph of y : lf(") l.

66 PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS

I

(B) On the axes provided, sketch the graph of y : /( l"l).

6. Let f(x) -- lnx2 for x ) 0 and g(x) : ek for r > 0. Letíbe thecomposition oflwith g; that is, H(x) : f(g(x)), and let Kbe the com-position of gwithf, that is, K(x) : Sç@)). [19s0-AB 3]

(A) Find the domain of H and write an expression for H(x) that doesnot contain the exponential function.

(B) Find the domain of K and write an e¡pression for K(x) that doesnot contain the exponential function.

(C) Find an expression for f-T(x),where /-l denotes the inversefunction of f, and find the domain of f -r.

7. Given the function f(x) : cos r - cos2 xîor -n s x s z¡. Find thex-intercepts of the graph ofl [19S0-AB 5]

8. Given f(x) : 5 _å Írss2--Al 2l

(A) Find the zeros of f.


(B) write an equation for each vertical and each horizontal asymptoteto the graphoff.

67

(C) Describe the symmetry of the graph off

s. Le:t f(x): {*# for -Ç 1 x 1 f. stut"whether/is an even or

an odd function. )ustifr your answer. [19S4-AB 2]

ro. Let f(x) : t5+. [1es6-AB 2]

(A) Describe the symmetry of the graph of f'

(B) Write an equation for each vertical and each horizontal asymp-

tote off

\ The following five AP* Multiple Choice questions [1969-AB] were

created to beìoked without the use of a graphing calculator.

11. which of the following defines a function f for which f(-*) : -f(x)?(A) f(x) : xz (B) /(x) : sin x (C) /(r) : cos

'c(P) /(x) : log x (E) f(x) : e"

[1e6e-MC 1]

12. ln (x - 2) < 0 if and onlY if(A) x13(D) x> 2

[1e6e-MC 2]

L3. The set of all points (et, t),where t is a real number, is the graph of

v:(A) * (B) ,rtx. (c) *"tt". (D)

[1e6e-MC 10]

68

(B) 0(x(3(E) x) 3

(C) 21x13

Ilnx' (E) ln ¡.

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICSs

14. lf f(x) : *+and g(x) : 2x,then the solurion set of

fk(,*)): s(/(x)) is

(A) {+} (B) {2}. (c) {3}.

(D) {-t,z}. (E) {å,r}

[1e6e-MC 12]

15. If the function/is defined by f(x) : x5 - l, then.f -1, the inversefunction ofl is defined by f-t(x¡ :

1r(A) -;--. (B)

Yx + t' (B) #r' (c',

NeedMoreflelpWith ... See ...Functions and their properties? precalculus, Section 1.2

Calculus, Section 1.2

The twelve basic functions? precalculus, Section 1.3

Graphical transformations? precalculus, Section 1.5

Rational functions? Precalculus, Section 2.7

Exponential functions? Calculus, Section 1.3

Logarithmic functionsz. precalculus, Section 3.3


Sinusoids? precalculus, Section 4.4


(D) t/i - t. (E) í. +1.[196e-MC #t4]

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0ptCS 69

AP* 0biective:

Limits of Functions

BígPícture

Limits of functions, including one-sided limits. An intuitive understanding of the limiting process. Calculating limits using algebra. Estimating limits from graphs or tables of data

To determine the behavior of a function, we evaluate íts limits at significant

points of its domain. We can use limits to determine where a function is con-

iinoo.r, and where it has asymptotes, as well as to predict its values and end

behavior. The concept of limits ullo*r us to find instantaneous rates of change,

a study that leads to diff.r.trtial calculus. We also use limits to estimate areas

urrder corrr.s, which leads to integral calculus. You should be able to determine

limits using methods of substitution, algebra, graphing, or numerical approx-

imation.

When we are finding the limit of a function, we are determining the value that

the functio n, f(x), ãpproaches as x gets very close to some particular -u"1":.

This does not meari ìhat /(x) takes on that value at x, but rather that itøpproøchesthat value. Additionall¡ we sometimes find a limit of a function as

x "pproaches

either infìnity or negative infinity (see example 5).

For a limit to exist as.r approaches some value c, the limit of the function

must be the same as x is appióached from both the left- and right-hand sides

of c. If c is either a right- * l.ft-huttd endpoint' we may be able to find a one-

sided limit at that point.

Content andPractíce

A function f(x) has a limit, L, as r approaches c if and only if

lim-/(x) : lim /(x) : I.X--+C tC+C

we simply write lim f(x) : r (if the conditions above are met)'x-c

70

't¿

You will notice that the definition specifies that three different conditionsbe met.

ri^_f(*): )y_fl*):LThis is easily done by evaluating the three parts separately, as the followingexample illustrates.

We shall review finding two-sided and one-sided limits as well as limits thatinvolve infinity.

Two-Sììleil Límíts

One method used to find a limit of a function is to substitute that value intothe function. Find the following limits using substitution.

lim /(x) - -2x--->2-

tím f(x) - -lx--->2'

5hss lim f(*) + lim /(x),x+2- x+2+

lim f(x) does not exist.x--->2'

Y: f(x)[-6, 6] by l-4,41

1. lim(x3 - 5) :x--->2

3.(A) limx+3

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS

tx2

2 .r^(*):

Ç9

+

(B) The substitution method was not helpful in part (A). Why?

):6

7l

(C) Use algebra to solve thisproblem.

limx--'+3

*',-9 :x2-5x+6(x)

limx-">3

limx'--+3

One-sídedlímíts

ä(

3

X

(D) Use a table of function values toapproximate jfà/(r).

4.

x

Límíæ lwolvínglnfiníty

5.

2.7

2.8

f(x)

2.9

8.1429

3

7.25

3.1

6.55s6

3.2

At the endpoints, we can find onlY

the right-hand limit as xapproaches -3 and the left-handlimit as r approaches 8.

(A) lim */(x) :tc-'->- 3

(B) lim-f(r) :r-+E

5.5455

5.t667

72

[-8,8] by [-8,8]

f(*):

(A)

(B)

(c)

(D)

(E)

(F)

x2-x-2x-3

lim f(x):x"'+-l-lim f(x) :

x--->-l'

lim /(x) :x-+2-

lim. /(¡) :x-+2-

lim l(x) :f-+-OO

lim /(x) :f++OO


'!

Problem 5 illustrates the following properties.

If either lim_/(x) : too or lim /(x) : *oo,thentheline x: aisax,4a x+avertical øsyffiptote of the graph of the function y : f(x).

Addìtìonal Practíce

If either lim /(x) : bor lim /(x) : b,thentheline /: bisaf++OO f+-OO

horizontal asymptote of the graph of the function y : f(x).

1. f(x\: {':- t' x = 2r' L¡'-1, x)2

(A) lim /(x) : (B) lim./(x) :x-">2- x-+2-

(C) What does this imply about the liryf(xX Explain.

, [2x-3, x=2Let f(x) : {r\' lx"*a, x)2.Use one-sided limits to find the value of ø so that -lin1/(¡) : l.

3.

[-15, 15] by [-8,8]

f(x):#::PART lll: REVIEW 0F AP. CALCULUS AB AND BC TOPICS

(A)

(B)

(c)

lim f(x) :f--à-OO

lim /(x) :.f--++OO

Conclusion

73

4.

l-tz,Izlby [-10, 10]

(A) lim /(x) :f--+-OO

(B) lim f(x\ :f--++OO-

(c) lim /(x) :x+-2-

(D) lim /(x) :x'-->2

(E) Conclusions

5.

f(*) : 2x2+3x-5x2-4

[-4,4] by [-1,3]

f(x):+

(A)

(B)

6.

lim f(¡) :¡--+0-

lim f(x\:f--++OO-

Use the values in the table to approximate lim f(x).¡--+-1.8

x

- 1.83

-r.82

74

f(x)

- 1.81

-22.5r

- 1.8

-22.54

-t.79

-22.57

-t.78- r.77

-22.63

þ-"I

-22.66

-22.69


7. The graph of which of the following equations has 7 : 1 as anasymptote?

(A) y : cosx

*2(D) y : ., -L (E) /: -lnxx--5

If lim f(x) : I, where I is a real number, which of the following mustxèa

be true?

I' f(a): til. lim /(x) :

x--->a-

m. lim /(x) :x--->a'

(A) I only(D) II and III

(B) y:ex

g. lim 4t2+x-7 _,+-oo X. - 5X - 3

(A) o (B)

(D) 1 (E)

L

L

(c) y: x3

x2+l

(B)

(E)

I and II[,II, and III

10. If the graph of y : ry++has a horizontal asym ptote y : -2, a

verticalasymptote x:4,andanx-interceptof 1.5,thena - b * c:

(A) -3. (B) 1. (c) s. (D) -e. (E) -1.


Limits?

(C) I and III

Z3

Nonexistent


(c) 4

See...

Precalculus, Section 10.3

Calculus, Sections 2.1 and 2.2

75

AP* 0biective:

Asymptotic and UnboundedBehavior

Bìg Pìcture

Asymptotic and unbounded behavior. Understanding asymptotes in terms of graphical behavior. Describing asymptotic behavior in terms of limits involving infinity


Graphical models help us visualize the relationships between the variablequantities of the numerical or algebraic models. By understanding the behav-

iors of the graphs, we can make predictions in their real-world applications.Graphs of past and present business data can help predict future growth.Calculus studies include many different types of optimization and related rate

problems, which analyze practical situations. A good knowledge of basic

graphs and their behavior is invaluable in solving these problems.

When we are looking for restrictions on a function's domain, we need to deter-mine if there are any vertical asymptotes. Similarþ finding horizontal asymp-

toteswill,help us determine a function's range. The concept of boundedness ofa graph also gives us insight as to the range. If we know a function's domainand range, we know on what intervals we can expect to evaluate andanalyze a

function.

76

Vertícøl Asymptotes

Consider the

lim -f(x) : oo andx--->-6

lim,/(x) : -ooJú+-6'

Horízontøl Asymptotes

f(*) :[-15, l0] by [-8,8]

We may conclude that the line r : -6 is

a vertical asymptote of the graph of f(x).lim_/(x) - -oo and lim./(x) : oo

tc+l Í+l'

We may conclude that the line x : I is avertical asymptote of the graph of f(x).

If either lim_/(x) : too or lim-/(x) : *oo,thentheline x: aisax+a - x--->a-'

vertical asymptote of the graph of the function y : f(x).

x2+5x-6-x-l 3

Looking at the graph of f(x) above, we see

,tg5/(x) : 0. We may conclude that the line 7 =

ofthe graph of f(x).

If either"ll;/(") : b or"IT*í") : b, then the line f : b is a

horizontal ûsymptote of the graph of the function y : f(x).

PART lll: REVIEW 0F AP* CALCULUS AB AND BC T0PICS

that lim /(x) : 0 andJr--+-OO

0 is a horizontal asymptote

77

Consider the following functions.

il.

[-10, 10] by [-8,8]

f(x) : *3-4x2*x-rx2-3x+s

[-30, 30] by l-2, al

You may remember the following short cuts for determining the horizontalasymptotes of rational functions.

1. If the degree of the numerator is less than the degree of the denomina-

tor, the horizontal asymptote is 7 : g.

s(x) : zxz+x-3x2+zx+4

2.

Note that the graph of h@) above has a slnnt asymptote. A slant asymptote willbe found when the degree of the numerator is I greater than the degree of the

denominator. Using long division,h(x) 3a1be

rewritten as

- -*2 + *x - 9h\x):+.-2+ffi.For large values of x,h(x)approaches the line y : +x - Z,which is the equa-

tion of its slant asymPtote.

End Behøvior Model

Suppose we want to determine the behavior of a function /where lxl is very

large. We use limits to define and determine the existence of an end behavior

model.

If the degree of the numerator is equal to the degree of the denominatot

the horizontal asymptote is y :

3. If the degree of the numerator is greater than the degree of the

denominator, there is no horizontal asymptote.

h(x) :

[-10, 10]by [-10, 10]

xa-2f+5i*3r*1zf+4*-x-5

leading coefficient of numerator

leading coefficient of denominator

78

t"


i-

left end behavior model for f ifand only,f "IT-

fP : ,.

f(x\right end behavior model for f if and only if

"ITt õ : t.

Comparing function graphs or table values of ordered pairs can be useful insuggesting or confirming the end behavior model of a given function.

Consider the following functions.

A functiongis a

f(x):2xa-5x3+x2+LS(x) : zxa

l-4,4lby l-4,201

For large values of x the graph oflwill begin to look much like the graph of

g. Considering the limits in our end behavior definition above, we use divi-

f(*) f(*) 2xa - 5x3 + x2 + |s10n to exDress ^ as a sum.

S(x) *" * "*-^-'g(x)

2xa

(, * -s*t + r' + t). u auo-, easilvthat lim f9. : r and\ 2x* / t- -- r--oog(x)

fk\lim r)* : l. We therefore conclude that g(x) : 2x4 is both a left end and

¡+oo g(x)

right end behavior model for f(x) : 2x4 - 5x3 + x2 + I.

Boundedness


A function/is bounded below if there is some number b that is less than orequal to every number in the range of f. Any such number b is called a lowerbound off

79

A function /is bounded above if there is some number B that is greater

than or equal to every number in the range of f. Any such number B is called

an upper bound offA function/is bounded if it is bounded both above and below.

Not bounded above; bounded below

Determine the equations of asymptotes for the following functionsalgebraically. Confirm your answers graphically.

-Bounded above; not bounded below

r. f(x):¿#if

h(x) :

4. Which of the functions in Problems 1-3 is bounded above and below?

M3xa+l

Bounded above; bounded below

\s.

2. s(x) :

Which of the following is a left end behavior model for

f(x): x2 - 3e-*?

(A) y:x2(D) y:x2-3

2x2+3x-5

80

x-3

(B)

(E)

v:v:

-3e-xe-

(c) y : 3e-x


Addítíonøl Practíce

\r. rhe tunctio n f(x) : $-] i,2x'*II. unbounded.

III. bounded below by y : -3.m. bounded above by y : 2.

ry. bounded below by y : 2.

\ z. If f(x) : e' + 2, which of the following lines is an asymptote to the

(A) I(D) II and III

NeedMoreHelp Wíth

graph of f?

(A) y: -2(D) x: 2

(B) IV only(E) II only

Asymptotes and end behaviormodels?

Boundedness?

(B)

(E)

x:0y:0

(C) III only

(c) y:2

See ...

PARï lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS

Precalculus, Sections L.2 and2.7Calculus, Section 2.2

Precalculus, Sections 1.2

8l

AP* Obiective: Asymptotic and unbounded behavior. Comparing relative magnitudes of functions and their rates of change (for

example, contrasting exponential growth, poþomial growth, and

logarithmic growth)

BìgPìchre

There are many functions that have values that increase as their x-values

increase. To get a better understanding of the behavior of these functions, we

often compare them to the exponential function, which grows very rapidly, orto the logarithmic function, which grows very slowly.

Content andPractice

To compare exponential, poþomial, and logarithmic functions, we shall use

the following definitions.

Function Magnitudes and TheirRates of Ghange

.l

Let f(x) and g(r) be positive for sufficiently large values of rc.

/grows faster than g (and g grows slower than/) ¿5 ¡ + oo if1.

"g#:* or "ts#:o2. f andg grow at the same rate u5 ¡ + oo if

fi-9:L*0. (Iisfinite)7¿+æ g\XC)

82

!,",,*

Example.l; Let f(*) : g(x) + h(x), where g(x) : 5x3 andh(x) : -x2 + 3x - 6. Using the definition above,

rr 5x3-x2+3x-6 ,. (5*t -x2+3x-6\urTl

-:

lllft l-----;-r---------------- t,+oo 5x"

"---oo \5xr 5xt /

: u*(t*-"2+1t-o)¡-æ \ 5xt /

:1*0-1.

From this we see that f(x) andg(x) grow at the same rate. This is wh¡ for largevalues of x, we can ignore the terms of h(x) in f (x). This is also why we can say

that g(x) is a right end behavior model of f(x).

Example 2: Compare the growth rates of ¡s and e* as )c+ oo.

Æ* : Æ# useIJHôPital'sRule'

2Ox3: llfil -----;-,+oo e*

60x2:[m.f--+oo e'"

t20x:Ifmrf+OO e'"

r. 120: [ñl ---;-x4+æ ê*

-0

Therefore, xs grows slower than ex as r+ oo.

Example3; Compare the growth rates of log'e/;and ln xas x+ oo.

-t_ lnxlosVx 3ln l0

lim __i_: lim _.r+oo ln x xioo ln ¡

3ln l0

^rTherefore logYx and ln rc grow at the same rate as ,c + oo.

PART lll: REVIEW 0F AP. CALCULUS AB AND BC TOPICS 83

Addítíonal Prøctíce

1. List the functions ê*,3", and x3 in order from slowest-growing tofastest-growing as r --à oo.

(A) e',3*,x3

(D) ¡3, 3t, ê*

Which of the following functions grow at the same rate as r --à oo?

I. f(x): *3

r. g(x): \/F¡æm. h(x): VFl¡;rv. i(x\ : x5-_ 4x2 + 3r. x'*2x-9(A) I and II only(C) I,II, and IV only(E) II,III, and IV only

(B) 3*, )c3, e'

(E) e*, x3,3*

NeeilMoreHelpWîth . . .

Relative rates of growth?

(C) x3, €*,3*

(B) I and IV only(D) I,II, and III only

84

See ...Calculus, Section 8.3


AP* Objective:

Continu¡tV

Bíg Picture

Continuity as a property of functions. An intuitive understanding of continuity. Understand continuity in term of limits

Content ønd Prøctìce

In precalculus, your first experience with continuity was studying functionsand their properties. This study included finding various limits of a functionover its domain. Continuity is presented early in the Calculus text since mosttheorems in the course reþ on this property. Remember, a function can beexamined for continuity at a point, on an interval, or over its entire domain.

Intuitively we think of a function as continuous over an interval if its graphcan be sketched in one continuous motion without lifting the pencil. In study-ing functions that are discontinuous, we can learn more about the behaviorsof continuous functions.

By considering the graph, identiS' all points of discontinuity in each ofthe following.

1. f(x) : tanx

3. h(x): {i _:', :=In your own words, write a sentence to explain howyou identified the dis-

continuities of each function in Problems 1-3.

4' f(*)

5.

6.

s@)

h(x)

-l-l

2. g(x) : x2-2*-3x*l

85

The Catculøs book presents the following definition of continuity at an interior

point of a domain.

A function y : f(x) is continuous'at an interior point c of its domain if

!Y,f(*) : f('\

If c is an endpoint of its domain, only the appropriate one-sidedlimit is checked.

Many students find it helpfut to recognize that this definition asks three dis-

tinct questions:

I. Does /(c) exist? (What do I get at xc :

II. Does lim/(x) exist? (What do I expectx--->c

ru. Does lim f(x) : /(cX (Is what I gettc+c

expected to get as r aPProaches c?)

Look again at the functions in Problems 1-3. Determine which part of the def-

inition of continuity is not satisfied in each.

7' f(x)

8' s@)

e. h(x)

Aililìtional Practíce

c?)

to get as r approaches c?)

at x : c equal to what I

In Problems 1-3, use the definition of continuity to decide whether each of the

following functions is continuous at the specified value of x.If tt is not con-

tinuous, explain why the function does not meet the definition. , ,l'

1. f(x) : l*l atx: 3

86

s(x) : {î, * t'

::t, at ¡ : o


3. h(x):f-*'*t' x12

[]r*e, x>2

4. Letlbe the function defined as follows:

f(x) : {o;1i *, :::(A) If a : 3 and b : 2, is /continuous for all x? Justifr your answer.

atx:2

(B) Describe all values of a and b for which/is a continuous function.

5. Which of the following functions are continuous for all real numbers xlI. f(*) : l*lIII. f(x) : tanx

ilI. f(x):3x2+x-7(A) I only (B) II only (C) III only(D) I and II (E) I and III

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS 87

6. Let f(x) :is true?

x3*zx2-29x-42

(A) /(¡) has a removable discontinuity at ,c : -3.

(B) f(x) has a jump discontinuity at x : 3.

(C) lf f(3) : +,then /(r) is continuous at ¡ : 3.

(n) f(x) has nonremovable discontinuities at x : -i and x : 3.

(E) lim /(r) : oox-+-3

x2-g

NeedMoreHelpWíth .. .

Continuity?

. Which of the following statements

See ...Precølculus, Section 1.2



AP* 0bjective:

lntermediate and ExtremeValue Theorems

Bíg Picture

Continuity as a property of functions

' Geometric understanding of graphs of continuous functions (IntermediateValue Theorem and Extreme Value Theorem)

Considerable time is spent in algebra II and precalculus courses learning howto find the zeros of a function and then'using this information to sketch itsgraph. fu you extended your mathematical knowledge in calculus, you learnedthat the zeros of the graph of a functionb derivative will enable us to deter-mine a function's maximum or minimum values. These are values we needwhen solving optimization problems.

In precalculus, you used synthetic division to determine if a number was azf:ro of apoþomial function. The Intermediate Value Theorem heþs us deter-mine where such zeros exist. The ExtremeValue Theorem gives us insight as towhether a function has maxima or minima. It is important to remember thatthe extreme values are the maximum or minimumivalues of the function.

Content ønd Prøctice

The Intermediate Value Theorem

A function y : f(x) that iscontinuous on a closed interval[a, b] takes on every value between

f(a) and f(b) on (a, b).

If ys is between f(a) and f(b), thenyo : fG) for some c in (a, b).

89

It is essential that f be a continuous function in order to apply the

Intermediate Value Theorem, as illustrated below'

However, looking at the interval l- 4, 2l where / is continuous, we see that

/(x) does take on everyvalue between f(-a) and f(2)'

12345

(-*-t. -.4<x32f(x): I

ll*-r, 21x<4We see that /(1) - -) and f(4) : 4'

However, /(x) does not take on all val-

ues between -2 and4 on the interval

[1,4].This is because /(x) is not a con-

tinuous function on the interval [1,4].

If /is continuous on a closed interval lø bl, then / has both a

maximum value and a minimum value on the interval'

Maxima and minima can occur at

illustrated in the figures.

Maximum and minimum

at interior points

interior points or at the endpoints, as

ab

Maximum and minimum at endPoints

90

Minimum at endpoint; maximum at

interior point

PART lll: REVIEW 0F AP. CALCULUS AB AND Bt MlCS

We can see that / must be continuous on a closed interval in order toapply the Extreme Value Theorem by analyzing the following graph.

Adilítional Practíce

I+.fl'): l_"

Use the graph of f to the right for problems

1. Explain how the Intermediate ValueTheorem is used to veri$r that/hasa zefobetween x : 2 and x : 3.

On the interval [-4,4], there is no maxi-mum value for f(x). There is also no maxi-mum on [-4, 1) since this is not a closedinterval. There is a maximum on l-4,0.91.

+3, -4<*3, l<

2. Approximate the maximum andminimum values oflon the interval[0,2].

x1l

x<4

3. The function/is continuous on the closed interval [-2,I1. Somevalues of f are shown in the table.

I and2-


The equation f(x)[-1, 1] if k:(A) 1. (B)

x

f(x)

-2-3

: I must have at least two solutions in the interval

-l7

Jt'

0

k

I

(c) 2.

3

(D) å. (E) 3.

91

4. A functionfis continuous on [-4, 1] and has its maximum at (-3, 5) )

and its minimum ut(1, -ø).Which of the following statements must

be false?

(A) The graph oflcrosses both axes.

(B) /is always decreasing on [-4, t ]. I

'

(c) f(-z) : 0

(D) /(-r) : 6

(E) f(0):2 *""

:--

\s.I-etf(x):|.o,(")_å|*nnisthemaximumvalueattainedbyf?13,-\/F\^'(A) + (B) L G) z (D) r¡ (E) 2n

NeedMoreffielPWíth,., See "'Intermediate value Theorem? Precalculus, section 2.3


Extreme Value Theorem? Calculus, Section 4'1


I

AP* 0bjective Parametric, polar, and vector functions

Big Pícture

In function mode,7 is a function of the independent variable x.

y: f(x)

In parametric mode, xandy are both functions of the independent parameter ú.

x: f(t)y : g(t)

In polar mode, r is a function of the independent variable 0.

r: f@)

Vector-valued functions utilize parametric equations.

r(r) : <f (t),g(Ð> with alrernative notationr(r): f(t)i+ g(t)j


Parametric equations are often used to describe the motion of a particle in theplane. A graphing calculator can be used to see the path and direction of theparticle.

l. During the time period from f : 0 to t : íseconds, a particle movesalong the path given by

x(t): cos¡rty(t):5sin¡rt

(A) Find the position of the particle when t : 2.5.

Parametric, Pola[ and VectorFunctions

93

(B) Sketch the graph of the path of the particle from f : 0 to t : 6'

Indicate the direction of the particle along its path.

(c) How many times does the particle pass through the point found

in part (a)?

A calculator maybe used on some problems to graph polar functions, but

take care when using a graph to solve a system of polar equations.

æ 2. Solve the following system.

From the graph there appear to be two solutions: (0, 0) and (+, n)

However, substituting the ordered pairs in the system of equations shows only

the second pair is a solution. This can be seen graphically if the calculator is

placed in simultaneous mode before the graphs are drawn.

Vector-valued functions can be handled on the calculator using paramet-

ric mode.

ru 3. The position of a moving particle is given by the vector function

r(r) : <cos(zr t),t - 1> with alternate notation

r(r) : cos(ø t)i + (t - l)i

(A) Find the position vector for the particle at t : l'l

r : 2sin9r:2cos9

94

(B) Graph the path of the particle for 0 < t < 2.


'j.

*.

Aililítíonal Practice

1. The position of a particle in a plane is described by the vector-valuedfunction r(t) : <e-t, cos f>. what is the position of the particle att : t¡12?.

(A) (r-otz, o) (B) (e-rt2, t) (c) (-s-,tz,0)

(D) (-e-rt2, t) (E) ln g)\'' z)

\ Z. Consider the following polar functions.

11 :4sin0fz:2

(A) Graph the functions.

(B) Find the points of intersection of the graphs of 11 and 12.

ru 3. A particle moves along the path specified by the following parametricequations.

x(t) : sin2ty(t) : cos2t

Sketch the path of the particle.

NeedMoreHelpWith ... See ..,Parametric equations? Precalculus, Section 6.3


Vector-valued functions? Calculus, Section 10.2

Polar functions? Calculøs, Section 10.3

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0ptCS 95

AP* Obiective:

Goncept of the Derivative

BigPicture

Concept of the derivative. Derivative presented graphically, numerically, and anal¡ically. Derivative defined as an instantaneous rate of change. Derivative defined as the limit of the difference quotient


The concept of the d.erivative is a critical part of almost everything we do in

calculus; it is important to familiarize ourselves with the derivative as seen

from different perspectives: graphicall¡ numericall¡ and anal¡icaþ This

familiarity will in turn allow us to understand and apply the derivative in a

variety of situations.

From a graphical perspective, a function's derivative at any specific point can

be thoulht of as th. tlop. of the graph of that function at that point. Shown

at right-is the graph of a quadratic function (in bold) and of its derivative.

Notiãe that high positive or negative derivative values indicate rapid growth or

decline in the function and a steeply sloping graph, whereas derivative values

close to zero'indicate little or no change in the values of the original function

and an approximately horizontal graph.

96

-5.0-4.0-3.0-2.

1. Consider the graph shown at right.

(A) Give the value of the derivativeof this function atx : -3-

(B) Give the value of the derivativeof this function at x : I.

Similarþ the derivative of a function can be thought of as the rate atwhich the function's value is changing at a specific instant. For example,if f(t)measures the position of a moving particle at time ú, then f'(5) represents thevelocity of that particle at the moment when t : 5.

2. If y : /(x) is a profit function measuring the amount of profit (in dol-lars) as a result of manufacturing and selling x basketballs, what is thesignificance of /'(SS0)? Make sure you use specific units.

Since we generally require two points to calculate slope, the task of find-

ing slope at a single point will require a new strategy. We will use the standardf(b\ - f(a\

slopeformuIa'ffifortwopointsonthecurve'andthentakethelimit

of this difference quotient as one point approaches the other.

Eventually, we will be able to develop anaþical techniques for finding thederivative as a function related to the original function.

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS 97

Aililitíonal Practíce

1. Let y : g(x) be a function that measures the water depth in a pool xminutes after the pool begins to fill. Then g'(25) represents:

I. The rate at which the depth is increasing 25 minutes after the

pool starts to fiIl.

il. The average rate at which the depth changes over the first 25

minutes.

III. The slope of the graph of gat the point where x : 25.

(A) I only (B) II only (C) III onlY

(D) I and II (E) I and III (F) I,II, and III

2. The functiofr y:/(x) measures the fish population in Blue Lake at

time r, where x is measured in years since |anuary 1, 1950. Iff'(25) : 5oo, it means that

(A) there are 500 fish in the lake in 1975.

(B) there are 500 more fish in 1975 than there were in 1950.

(C) on the aveÍage,the fish population increased by 500 per year over

the first 25 years following 1950.

(D) on lanaary 1,lg75,the fish population was growingatarate of 500

fish per year.

(E) none of the above.


Concept of derivative?

98

See...Precalculus, Section I 0. I

Calculus, Section 2.4-3.6

PART lll: REVIEW 0F AP. CALCULUS AB AND BC TOPICS :'..':"r'

AP* 0bjective:

Big Picture

Differentiab¡l¡ty and Gontinu¡ty

Content ønd Prøctice

Concept of the derivative. Relationship between differentiability and continuity

Most, but not all, of the functions we encounter in calculus will be differen-tiable over their entire domains. Before we can confidently apply the rulesregarding derivatives, we need to be able to recognize the exceptions to the rule.

A function that is differentiable at a point or over an interval will always becontinuous there, but the converse is not true: There are situations where acontinuous function may not have a derivative. To rephrase this, a functionthat is discontinuous at a point will definitely not have a derivative at thatpoint. A continuous function, on the other hand, will still fail to have a deriv-ative at any point where it has a corner, a cusp, or a vertical tangent.

l. Consider the function shownat right. At what domainvalues does the functionappear to be

(A) differentiable?

(B) continuous but notdifferentiable?

(C) neither continuousnor differentiable?

-5.0 -4.0 -3.0 -2.0 -t.0

99

Additionøl Practíce

1. Letl be a function with /'(5) : 8' Which of the following statements

is true?

(A) /must be continuous at )c : 5.

(B) /is definitely not continuous at x : 5.

(C) There is not enough information to determine whether or not flx)is continuous at x : 5.

2. Consider the function y : f(x)shown at right.

(A) At what x-values is/discontinuous?

(B) At what ¡-values would thisfunction not be differentiable?

Suppose / is a function such that /'(9) is undefined. Which of the

following statements is true?

(A) /must be continuous at x = 9.

(B) /is definitely not continuous at x : 9.

(C) There is not enough information to determine whether or not/iscontinuous atx: 9,

4. Supposethat/is afunctionthatis continuous atx : -11.Which ofthefollowing statements is true?

(A) /must be differentiable at x : -ll.(B) /is definitely not differentiable at x : - 11.

(C) There is not enough information to determine whether or not/(x)is differentiable at x : -Il.


5. Which of the following statements are always true?

I. A function that is continuous at x : c must be differentiable atx: c.

il. A function that is differentiable at x : c must be continuous atx: c.

ilI. A function that is not contiwous at x : c m:ust not be differen-tiable atx,: c.

ry. A function that is not dífferentiable at )c : c must notbe continu-ousatx:c.

(A) None of them (B) I and III (C) il and IV(D) I and IV (E) II and III (F) I,II,III, and IV

NeedMoreHelpWith... See...Differentiability? Precalculus, Section 10.1

Continuity and differentiability? Calculus, Section 3.2

PART lll: REVIEW 0F AP. CALCULUS AB AND BC T0PICS t0l

AP* 0bjective:

Slope of a Curve at a Point

BìgPìcture

Derivative at a point. Slope of a curve at a point. Examples emphasized include points at which

there are vertical tangents and points at which there are no tangents.

Content and Prac-tíce

We often use graphs to capture the relationship between variables (distance

versus time, for instance). We know from previous courses that when we dO

this for a linear function, the constant slope of that function represents the

rate of change of one variable with respect to the other. By defining slope fornonlinear functions, we can extend this same concept to a much broader range

of situations.

The value of a functiorls derivative at a specific point can be thought of as the

slope of the tangent line to the function s graph at that point. This interpreta-tion can be quite helpful, both as a means of approximating the value of aderivative and as a means for identifring points at which the derivative will be

undefined.

1. Estimate the slope of each curve at point P

y (B)(A)

to2

't

Although the conventional formula for slope requires two distinct points,the slope of the tangent line to a function at a specific point can be determinedby finding the slope between that point and a nearby point on the curve, andthen finding the limit as the nearby point approaches the original point.

Theslopeofthecurve y: f(x) atthe

point (a, f(a)) ís

limh--->(

f(a+h)-f(a)t--->O

if arnd only if it exists.

ru 2. Considerthe function f(x\ : x2 + 3.

(A) Using your calculator to view an appropriate graph of the func-tion, estimate the slope of the curve at x : 3.

PART lll: REVIEW 0F AP. CALCULUS AB AND BC TOPICS

(B) ]ustiS' your answer anaþicall¡ using the definition of the slope ofa curve.

ïhe tangent slope is

.. f(a+n-f@)ilflt:h+0 n

103

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