ap calculus ab-bc course description, may 2002 - may 2003 (sample exams)

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C A L C U L U SCalculus AB, Calculus BC

Course Description

M A Y 2 0 0 2 , M A Y 2 0 0 3

2002 Exam Date: Tuesday, May 7, morning session2003 Exam Date: Tuesday, May 6, morning session

2000-01 Development Committee and Chief Faculty Consultant

Thomas P. Dick, Oregon State University, Corvallis, ChairStella R. Ashford, Southern University, Baton Rouge, Louisiana

David M. Bressoud, Macalester College, St. Paul, Minnesota

Benjamin G. Klein, Davidson College, North Carolina

María Pérez Randle, Bishop Kenny High School, Jacksonville, Florida

Nancy W. Stephenson, Clements High School, Sugar Land, Texas

Michael R. White, Pennridge High School, Perkasie, Pennsylvania

Chief Faculty Consultant: Lawrence H. Riddle, Agnes Scott College, Decatur, Georgia

ETS Consultants: Gloria S. Dion, Chancey O. Jones, Craig L. Wright

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I.N. 990185

This Course Description is intended for use by AP® teachers for course and exam preparation in the classroom; permission for any other use must be soughtfrom the Program. Teachers may reproduce it, in whole or in part, in limitedquantities, for face-to-face teaching purposes but may not mass distribute thematerials, electronically or otherwise. This Course Description and any copiesmade of it may not be resold, and the copyright notices must be retained as theyappear here. This permission does not apply to any third-party copyrights contained herein.

The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity.Founded in 1900, the association is composed of more than 3,900 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, thePSAT/NMSQT™, the Advanced Placement Program® (AP®), and Pacesetter®. The College Board is committed to the principles of equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns.

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Dear Colleagues:

Last year more than three quarters of a million high school students bene-fited from the opportunity of studying in AP courses and then taking thechallenging AP Exams. These students experienced the power of learningas it comes alive in the classroom, as well as the practical benefits of earn-ing college credit and placement while still in high school. Behind each ofthese students was a talented, hardworking teacher. Teachers are the secretto the success of AP. They are the heart and soul of the Program.

The College Board is committed to supporting the work of AP teachersin as many ways as possible. AP workshops and Summer Institutes heldaround the globe provide stimulating professional development for 60,000teachers each year. The College Board Fellows stipends provide funds tosupport many teachers’ attendance at these institutes, and in 2001, stipendswere offered for the first time to teams of Pre-AP™ teachers as well.

Perhaps most exciting, the College Board continues to expand an inter-active Web site designed specifically to support AP teachers. At thisInternet site, teachers have access to a growing array of classroomresources, from textbook reviews to lesson plans, from opinion polls to cutting-edge exam information. I invite all AP teachers, particularly thosewho are new to the Program, to take advantage of these resources.

This AP Course Description provides an outline of content and descrip-tion of course goals, while still allowing teachers the flexibility to developtheir own lesson plans and syllabi, and to bring their individual creativity tothe AP classroom. Additional resources, including sample syllabi, can befound in the AP Teacher’s Guide that is available for each AP subject.

As we look to the future, the College Board’s goal is to provide accessto AP courses in every high school. Reaching this goal will require a lot ofhard work. We encourage you to help us build bridges to college andopportunity by finding ways to prepare students in your school to benefitfrom participation in AP.

Sincerely,

Gaston CapertonPresidentThe College Board

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Contents

Welcome to the AP Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1AP Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1AP Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Equity and Access . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Introduction to AP Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3The Courses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Resources for AP Calculus Teachers . . . . . . . . . . . . . . . . . . . . . . . . 6

Topic Outline for Calculus AB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Topic Outline for Calculus BC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Use of Graphing Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Graphing Calculator Capabilities for the Examinations . . . . . . . . 18Technology Restrictions on the Examinations . . . . . . . . . . . . . . . . 18Showing Work on the Free-Response Sections . . . . . . . . . . . . . . . 19List of Graphing Calculators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

The Examinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Calculus AB Subscore Grade for the Calculus

BC Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22The Grade Setting Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Calculus AB: Section I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Part A Sample Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . 24Part B Sample Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . 30

Answers to Calculus AB Multiple-Choice Questions . . . . . . . . . 35Calculus BC: Section I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Part A Sample Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . 36Part B Sample Multiple-Choice Questions . . . . . . . . . . . . . . . . . . . 41

Answers to Calculus BC Multiple-Choice Questions . . . . . . . . . 45Calculus AB and Calculus BC: Section II . . . . . . . . . . . . . . . . . . . . . . 46

Instructions for Section II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Calculus AB Sample Free-Response Questions . . . . . . . . . . . . . . . 48Calculus BC Sample Free-Response Questions . . . . . . . . . . . . . . . 60

AP Program Essentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72The AP Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

AP Grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Grade Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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AP and College Credit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Why Colleges Give Credit for AP Grades . . . . . . . . . . . . . . . . . . . . 73Guidelines on Granting Credit for AP Grades . . . . . . . . . . . . . . . . 73Finding Colleges That Accept AP Grades . . . . . . . . . . . . . . . . . . . . 74

AP Scholar Awards and the AP International Diploma . . . . . . . . . . . 74AP Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Test Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Teacher Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Pre-AP™ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76AP Publications and Other Resources . . . . . . . . . . . . . . . . . . . . . . . . . 77

Ordering Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Multimedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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www.collegeboard.com/ap 1

Welcome to the AP Program

The Advanced Placement Program is sponsored by the College Board, anon-profit membership association. AP offers 35 college-level courses andexams in 19 subject areas for highly motivated students in secondaryschools. Its reputation for excellence results from the close cooperationamong secondary schools, colleges, and the College Board. More than2,900 universities and colleges worldwide grant credit, advanced place-ment, or both to students who have performed satisfactorily on the exams,and 1,400 institutions grant sophomore standing to students who meettheir requirements. Approximately 13,000 high schools throughout theworld participate in the AP Program; in May 2000, they administered morethan 1.3 million AP Exams.

You will find more information about the AP Program at the back of thisCourse Description, and at www.collegeboard.com/ap. This Web site ismaintained for the AP Program by collegeboard.com, a destination Website for students and parents.

AP Courses

AP courses are available in the subject areas listed on the next page.(Unless noted, an AP course is equivalent to a full-year college course.)Each course is developed by a committee composed of college faculty andAP teachers. Members of these Development Committees are appointed bythe College Board and serve for overlapping terms of up to four years.

AP Exams

For each AP course, an AP Exam is administered at participating schoolsand multischool centers worldwide. Schools register to participate bycompleting the AP Participation Form and agreeing to its conditions. Formore details, see A Guide to the Advanced Placement Program; informa-tion about ordering and downloading this publication can be found at theback of this booklet.

Except for Studio Art — which consists of a portfolio assessment — allexams contain a free-response section (either essay or problem-solving)and another section consisting of multiple-choice questions. The modernlanguage exams also contain a speaking component, and the Music Theoryexam includes a sight-singing task.

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Equity and Access

The College Board and the Advanced Placement Program encourageteachers, AP Coordinators, and school administrators to make equity andaccess guiding principles for their AP programs. The College Board is com-mitted to the principle that all students deserve an opportunity to partici-pate in rigorous and academically challenging courses and programs. TheBoard encourages the elimination of barriers that restrict access to APcourses for students from ethnic and racial groups that have been tradi-tionally underrepresented in the AP Program.

For more information about equity and access in principle and practice,contact the National Office in New York.

AP Subject Areas AP Courses and Exams

Art Art History; Studio Art: DrawingPortfolio; Studio Art: 2-D Portfolio;Studio Art: 3-D Portfolio

Biology BiologyCalculus AB; BCChemistry ChemistryComputer Science A*; ABEconomics Macroeconomics*; Microeconomics*English Language and Composition; Literature

and Composition; International EnglishLanguage (APIEL™)

Environmental Science Environmental Science*French Language; LiteratureGerman LanguageGeography Human Geography*Government and Politics Comparative*; United States*History European; United States; WorldLatin Literature; VergilMusic Music TheoryPhysics B; C: Electricity and Magnetism*;

C: Mechanics*Psychology Psychology*Spanish Language; LiteratureStatistics Statistics*

** This subject is the equivalent of a half-year college course.

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Introduction to AP Calculus

Shaded text indicates important new information about this subject.An Advanced Placement (AP) course in calculus consists of a full high

school academic year of work that is comparable to calculus courses incolleges and universities. It is expected that students who take an APcourse in calculus will seek college credit, college placement, or both,from institutions of higher learning.

The AP Program includes specifications for two calculus courses andthe examination for each course. The two courses and the two corre-sponding examinations are designated as Calculus AB and Calculus BC.

Calculus AB can be offered as an AP course by any school that canorganize a curriculum for students with mathematical ability. This curricu-lum should include all the prerequisites for a year’s course in calculuslisted on page 6. Calculus AB is designed to be taught over a full highschool academic year. It is possible to spend some time on elementaryfunctions and still cover the Calculus AB curriculum within a year.However, if students are to be adequately prepared for the Calculus ABexamination, most of the year must be devoted to the topics in differentialand integral calculus described on pages 8 to 11. These topics are thefocus of the AP Examination questions.

Calculus BC can be offered by schools that are able to complete allthe prerequisites listed on page 6 before the course. Calculus BC is a full-year course in the calculus of functions of a single variable. It includesall topics covered in Calculus AB plus additional topics, but both coursesare intended to be challenging and demanding; they require a similardepth of understanding of common topics. The topics for Calculus BCare described on pages 12 to 16. A Calculus AB subscore grade isreported based on performance on the portion of the exam devoted toCalculus AB topics.

Both courses described here represent college-level mathematics forwhich most colleges grant advanced placement and credit. Most collegesand universities offer a sequence of several courses in calculus, and enter-ing students are placed within this sequence according to the extent oftheir preparation, as measured by the results of an AP Examination orother criteria. Appropriate credit and placement are granted by each insti-tution in accordance with local policies. The content of Calculus BC isdesigned to qualify the student for placement and credit in a course that isone course beyond that granted for Calculus AB. Many colleges providestatements regarding their AP policies in their catalogs.

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Secondary schools have a choice of several possible actions regardingAP Calculus. The option that is most appropriate for a particular schooldepends on local conditions and resources: school size, curriculum, thepreparation of teachers, and the interest of students, teachers, andadministrators.

The AP Development Committee in Calculus strongly supports the 1986statement of the Mathematical Association of America and the NationalCouncil of Teachers of Mathematics: “MAA and NCTM recommend that allstudents taking calculus in secondary school who are performing satisfac-torily in the course should expect to place out of the comparable collegecalculus course.”* This statement further recommends that students whoenroll in a calculus course in secondary school should have previouslydemonstrated mastery of algebra, geometry, coordinate geometry, andtrigonometry. This means that students should have at least the equivalentof four full years of mathematical preparation in these topics. Theadvanced topics in algebra, trigonometry, analytic geometry, and elemen-tary functions studied in depth during the fourth year of preparation arecritically important for students’ later course work in mathematics.

The completion of this preparatory program can be accomplished in avariety of ways: for example, beginning the study of secondary schoolmathematics in grade 8; reorganizing the content of courses; establishingaccelerated sections for the more capable students; encouraging the elec-tion of more than one mathematics course in grade 9, 10, or 11; institutingprograms of summer study or guided independent study during theacademic year.


* The complete statement can be found in the Teacher’s Guide – AP Calculus, pages 28-30.

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The Courses

Philosophy

Calculus AB and Calculus BC are primarily concerned with developing thestudents’ understanding of the concepts of calculus and providing experi-ence with its methods and applications. The courses emphasize a multirep-resentational approach to calculus, with concepts, results, and problemsbeing expressed graphically, numerically, analytically, and verbally. Theconnections among these representations also are important.

Calculus BC is an extension of Calculus AB rather than an enhance-ment; common topics require a similar depth of understanding. Bothcourses are intended to be challenging and demanding.

Broad concepts and widely applicable methods are emphasized. Thefocus of the courses is neither manipulation nor memorization of an exten-sive taxonomy of functions, curves, theorems, or problem types. Thus,although facility with manipulation and computational competence areimportant outcomes, they are not the core of these courses.

Technology should be used regularly by students and teachers to rein-force the relationships among the multiple representations of functions, to confirm written work, to implement experimentation, and to assist ininterpreting results.

Through the use of the unifying themes of derivatives, integrals, limits,approximation, and applications and modeling, the course becomes acohesive whole rather than a collection of unrelated topics. These themesare developed using all the functions listed in the prerequisites.

Goals

• Students should be able to work with functions represented in a varietyof ways: graphical, numerical, analytical, or verbal. They should under-stand the connections among these representations.

• Students should understand the meaning of the derivative in terms of arate of change and local linear approximation and should be able to usederivatives to solve a variety of problems.

• Students should understand the meaning of the definite integral both asa limit of Riemann sums and as the net accumulation of a rate of changeand should be able to use integrals to solve a variety of problems.

• Students should understand the relationship between the derivative andthe definite integral as expressed in both parts of the FundamentalTheorem of Calculus.

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• Students should be able to communicate mathematics both orally and in well-written sentences and should be able to explain solutions to problems.

• Students should be able to model a written description of a physical situ-ation with a function, a differential equation, or an integral.

• Students should be able to use technology to help solve problems, exper-iment, interpret results, and verify conclusions.

• Students should be able to determine the reasonableness of solutions,including sign, size, relative accuracy, and units of measurement.

• Students should develop an appreciation of calculus as a coherent bodyof knowledge and as a human accomplishment.

Prerequisites

Before studying calculus, all students should complete four years of sec-ondary mathematics designed for college-bound students: courses inwhich they study algebra, geometry, trigonometry, analytic geometry, andelementary functions. These functions include those that are linear, poly-nomial, rational, exponential, logarithmic, trigonometric, inverse trigono-metric, and piecewise defined. In particular, before studying calculus,students must be familiar with the properties of functions, the algebra offunctions, and the graphs of functions. Students must also understand thelanguage of functions (domain and range, odd and even, periodic, symme-try, zeros, intercepts, and so on) and know the values of the trigonometric functions of numbers such as 0, , , , and .

Resources for AP Calculus Teachers

To keep up to date with changes in AP Calculus, it is strongly recom-mended that teachers participate in ongoing development opportunities.These include the many workshops and summer institutes focusing oncurriculum, pedagogy, and technology that are sponsored or coordinatedby the College Board at various locations around the country and aroundthe world. In addition, teachers seeking advice about initiating AP coursesare urged to obtain additional information from other teachers who areinvolved in the AP Program. Forums for such discussions are provided atworkshops sponsored by the College Board. Information about workshopsas well as lists of summer institutes may be obtained from the CollegeBoard regional offices listed on the inside back cover, or in the AP sectionof the College Board Web site (www.collegeboard.com/ap).

π2

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In addition, a number of publications may be of value to both new APteachers who are planning a course for the first time, and to experiencedAP teachers. One such publication is the latest edition of the Teacher’s

Guide – AP Calculus. This book provides information that is relevant toinitiating an AP program in calculus, and it goes into much greater detailabout the course descriptions for Calculus AB and Calculus BC. The pub-lication also suggests teaching strategies and resource materials and provides sample course syllabi prepared by experienced AP teachers.

Another useful resource for teachers — and students — is APCD™, theAP Calculus AB CD-ROM with multiple-choice exams, free-response tutori-als, and interactive explorations of selected calculus concepts.

The AP Publications Order Form can be used to order the Teacher’s

Guide – AP Calculus as well as several other publications helpful to AP Calculus teachers, such as the 1997 Released Exams and the 1998Released Exams. See the back of this booklet for more information on AP publications.

Up-to-date information about AP Calculus can also be found in the APsection of the College Board Web site. See Teacher Support at the back ofthis booklet for more information.

Detailed answers to free-response questions from previous AP CalculusExaminations are available from the following two sources:

Mathematical Olympiads SCA2PMTRichard Kalman, Executive Director c/o Betty Gasque2154 Bellmore Avenue Department of MathematicsBellmore, NY 11710-5645 Francis Marion University516-781-2400 P.O. Box 100547www.moems.org Florence, SC 29501-0547

Further information or assistance can be obtained by writing to the appro-priate College Board regional office (see listing on inside back cover).

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Topic Outline for Calculus AB*

This outline of topics is intended to indicate the scope of the course, but itis not necessarily the order in which the topics are to be taught. Teachersmay find that topics are best taught in different orders. (See the Teacher’s

Guide – AP Calculus for sample syllabi.) Although the examination isbased on the topics listed here, teachers may wish to enrich their courseswith additional topics.

I. Functions, Graphs, and Limits

Analysis of graphs. With the aid of technology, graphs of functionsare often easy to produce. The emphasis is on the interplay betweenthe geometric and analytic information and on the use of calculusboth to predict and to explain the observed local and global behaviorof a function.

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 of limits involving infinity.• Comparing relative magnitudes of functions and their rates of

change. (For example, contrasting exponential growth, polynomialgrowth, and logarithmic growth.)

Continuity as a property of functions.

• An intuitive understanding of continuity. (Close values of thedomain lead to close values of the range.)

• Understanding continuity in terms of limits.• Geometric understanding of graphs of continuous functions

(Intermediate Value Theorem and Extreme Value Theorem).


* There are no major changes to the Topic Outline from the May 2000, May 2001 edition ofthis Course Description. Minor editorial changes are not shaded.

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II. Derivatives

Concept of the derivative.

• Derivative presented graphically, numerically, and analytically.• Derivative interpreted as an instantaneous rate of change.• Derivative defined as the limit of the difference quotient.• Relationship between differentiability and continuity.

Derivative at a point.

• Slope of a curve at a point. Examples are emphasized, includingpoints at which there are vertical tangents and points at which thereare no tangents.

• Tangent line to a curve at a point and local linear approximation.• Instantaneous rate of change as the limit of average rate of change.• Approximate rate of change from graphs and tables of values.

Derivative as a function.

• Corresponding characteristics of graphs of ƒ and ƒ�.• Relationship between the increasing and decreasing behavior of ƒ

and the sign of ƒ�.• The Mean Value Theorem and its geometric consequences.• Equations involving derivatives. Verbal descriptions are translated

into equations involving derivatives and vice versa.

Second derivatives.

• Corresponding characteristics of the graphs of ƒ, ƒ�, and ƒ�.• Relationship between the concavity of ƒ and the sign of ƒ�.• Points of inflection as places where concavity changes.

Applications of derivatives.

• Analysis of curves, including the notions of monotonicity andconcavity.

• Optimization, both absolute (global) and relative (local) extrema.• Modeling rates of change, including related rates problems.• Use of implicit differentiation to find the derivative of an inverse

function.• Interpretation of the derivative as a rate of change in varied applied

contexts, including velocity, speed, and acceleration.

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Computation of derivatives.

• Knowledge of derivatives of basic functions, including power,exponential, logarithmic, trigonometric, and inverse trigonometricfunctions.

• Basic rules for the derivative of sums, products, and quotients offunctions.

• Chain rule and implicit differentiation.

III. Integrals

Interpretations and properties of definite integrals.

• Computation of Riemann sums using left, right, and midpoint evalu-ation points.

• Definite integral as a limit of Riemann sums over equal subdivisions.• Definite integral of the rate of change of a quantity over an interval

interpreted as the change of the quantity over the interval:

• Basic properties of definite integrals. (Examples include additivityand linearity.)

Applications of integrals. Appropriate integrals are used in a vari-ety of applications to model physical, biological, or economic situa-tions. Although only a sampling of applications can be included in anyspecific course, students should be able to adapt their knowledge andtechniques to solve other similar application problems. Whateverapplications are chosen, the emphasis is on using the integral of a rateof change to give accumulated change or using the method of settingup an approximating Riemann sum and representing its limit as a defi-nite integral. To provide a common foundation, specific applicationsshould include finding the area of a region, the volume of a solid withknown cross sections, the average value of a function, and the dis-tance traveled by a particle along a line.

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a

ƒ�(x)dx � ƒ(b) � ƒ(a)


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Fundamental Theorem of Calculus.

• Use of the Fundamental Theorem to evaluate definite integrals.• Use of the Fundamental Theorem to represent a particular anti-

derivative, and the analytical and graphical analysis of functions so defined.

Techniques of antidifferentiation.

• Antiderivatives following directly from derivatives of basic functions.• Antiderivatives by substitution of variables (including change of lim-

its for definite integrals).

Applications of antidifferentiation.

• Finding specific antiderivatives using initial conditions, includingapplications to motion along a line.

• Solving separable differential equations and using them in modeling.In particular, studying the equation and exponential growth.

Numerical approximations to definite integrals. Use of Riemannand trapezoidal sums to approximate definite integrals of functionsrepresented algebraically, graphically, and by tables of values.

Change to Course Description for Calculus AB — Effective for the2004 Examinations: Slope Fields

The following will be included in the topic outline for Calculus AB for the2003-2004 academic year for the 2004 AP Examinations.

“Geometric interpretation of differential equations via slopefields and the relationship between slope fields and solutioncurves for differential equations.” [Applications of derivatives]

This topic has been part of the topic outline for Calculus BC since the 1998AP Examinations.

y� � ky

32458W4 3/21/01 5:20 PM Page 11

Topic Outline for Calculus BC*

The topic outline for Calculus BC includes all Calculus AB topics.Additional topics are found in paragraphs that are marked with a plus sign(+) or an asterisk (*). The additional topics can be taught anywhere in thecourse that the instructor wishes. Some topics will naturally fit immedi-ately after their Calculus AB counterparts. Other topics may fit best afterthe completion of the Calculus AB topical outline. (See the Teacher’s

Guide – AP Calculus for sample syllabi.) Although the examination isbased on the topics listed here, teachers may wish to enrich their courseswith additional topics.

I. Functions, Graphs, and Limits

Analysis of graphs. With the aid of technology, graphs of functionsare often easy to produce. The emphasis is on the interplay betweenthe geometric and analytic information and on the use of calculusboth to predict and to explain the observed local and global behaviorof a function.

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 of limits involving infinity.• Comparing relative magnitudes of functions and their rates of

change. (For example, contrasting exponential growth, polynomialgrowth, and logarithmic growth.)


* There are no major changes to the Topic Outline from the May 2000, May 2001 edition ofthis Course Description. Minor editorial changes are not shaded.

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Continuity as a property of functions.

• An intuitive understanding of continuity. (Close values of thedomain lead to close values of the range.)

• Understanding continuity in terms of limits.• Geometric understanding of graphs of continuous functions

(Intermediate Value Theorem and Extreme Value Theorem).

* Parametric, polar, and vector functions. The analysis of planarcurves includes those given in parametric form, polar form, and vec-tor form.

II. Derivatives

Concept of the derivative.

• Derivative presented graphically, numerically, and analytically.• Derivative interpreted as an instantaneous rate of change.• Derivative defined as the limit of the difference quotient.• Relationship between differentiability and continuity.

Derivative at a point.

• Slope of a curve at a point. Examples are emphasized, includingpoints at which there are vertical tangents and points at which thereare no tangents.

• Tangent line to a curve at a point and local linear approximation.• Instantaneous rate of change as the limit of average rate of change.• Approximate rate of change from graphs and tables of values.

Derivative as a function.

• Corresponding characteristics of graphs of ƒ and ƒ�.• Relationship between the increasing and decreasing behavior of ƒ

and the sign of ƒ�.• The Mean Value Theorem and its geometric consequences.• Equations involving derivatives. Verbal descriptions are translated

into equations involving derivatives and vice versa.

32458W4 3/21/01 5:20 PM Page 13

Second derivatives.

• Corresponding characteristics of the graphs of ƒ, ƒ�, and ƒ�.• Relationship between the concavity of ƒ and the sign of ƒ�.• Points of inflection as places where concavity changes.

Applications of derivatives.

• Analysis of curves, including the notions of monotonicity andconcavity.

+ Analysis of planar curves given in parametric form, polar form, andvector form, including velocity and acceleration vectors.

• Optimization, both absolute (global) and relative (local) extrema.• Modeling rates of change, including related rates problems.• Use of implicit differentiation to find the derivative of an inverse

function.• Interpretation 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 fordifferential equations.

+ Numerical solution of differential equations using Euler’s method.+ L’Hôpital’s Rule, including its use in determining limits and conver-

gence of improper integrals and series.

Computation of derivatives.

• Knowledge of derivatives of basic functions, including power,exponential, logarithmic, trigonometric, and inverse trigonometricfunctions.

• Basic rules for the derivative of sums, products, and quotients offunctions.

• Chain rule and implicit differentiation.+ Derivatives of parametric, polar, and vector functions.


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III. Integrals

Interpretations and properties of definite integrals.

• Computation of Riemann sums using left, right, and midpoint evalu-ation points.

• Definite integral as a limit of Riemann sums over equal subdivisions.• Definite integral of the rate of change of a quantity over an interval

interpreted as the change of the quantity over the interval:

• Basic properties of definite integrals. (Examples include additivityand linearity.)

* Applications of integrals. Appropriate integrals are used in a vari-ety of applications to model physical, biological, or economic situa-tions. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledgeand techniques to solve other similar application problems. Whateverapplications are chosen, the emphasis is on using the integral of a rateof change to give accumulated change or using the method of settingup an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applica-tions should include finding the area of a region (including a regionbounded by polar curves), the volume of a solid with known crosssections, the average value of a function, the distance traveled by a particle along a line, and the length of a curve (including a curvegiven in parametric form).

Fundamental Theorem of Calculus.

• Use of the Fundamental Theorem to evaluate definite integrals.• Use of the Fundamental Theorem to represent a particular anti-

derivative, and the analytical and graphical analysis of functions so defined.

Techniques of antidifferentiation.

• Antiderivatives following directly from derivatives of basic functions.+ Antiderivatives by substitution of variables (including change of lim-

its for definite integrals), parts, and simple partial fractions (nonre-peating linear factors only).

+ Improper integrals (as limits of definite integrals).

� b

a

ƒ�(x)dx � ƒ(b) � ƒ(a)

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Applications of antidifferentiation.

• Finding specific antiderivatives using initial conditions, includingapplications to motion along a line.

• Solving separable differential equations and using them in modeling.In particular, studying the equation and exponential growth.

+ Solving logistic differential equations and using them in modeling.

Numerical approximations to definite integrals. Use of Riemannand trapezoidal sums to approximate definite integrals of functionsrepresented algebraically, graphically, and by tables of values.

*IV. Polynomial Approximations and Series

* Concept of series. A series is defined as a sequence of partial sums,and convergence is defined in terms of the limit of the sequence of par-tial sums. Technology can be used to explore 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 their relationship to

improper integrals, including the integral test and its use in testingthe convergence of p-series.

+ The ratio test for convergence and divergence.+ Comparing series to test for convergence or divergence.

* Taylor series.

+ Taylor polynomial approximation with graphical demonstration ofconvergence. (For example, viewing graphs of various Taylor poly-nomials of the sine function approximating the sine curve.)

+ Maclaurin series and the general Taylor series centered at .

+ Maclaurin series for the functions , sin x, cos x, and .

+ Formal manipulation of Taylor series and shortcuts to computingTaylor series, including substitution, differentiation, antidifferentia-tion, and the formation of new series from known series.

+ Functions defined by power series.+ Radius and interval of convergence of power series.+ Lagrange error bound for Taylor polynomials.

11 � x

ex

x � a

y� � ky


32458W4 3/21/01 5:20 PM Page 16


Use of Graphing Calculators

Professional mathematics organizations such as the National Council ofTeachers of Mathematics, the Mathematical Association of America, andthe Mathematical Sciences Education Board of the National Academy ofSciences have strongly endorsed the use of calculators in mathematicsinstruction and testing.

The use of a graphing calculator in AP Calculus is considered an integralpart of the course. Students should be using this technology on a regularbasis so that they become adept at using their graphing calculators.Students should also have experience with the basic paper-and-pencil tech-niques of calculus and be able to apply them when technological tools areunavailable or inappropriate.

The AP Calculus Development Committee understands that new calcu-lators and computers, capable of enhancing the teaching of calculus, con-tinue to be developed. There are two main concerns that the committeeconsiders when deciding what level of technology should be required forthe examinations: equity issues and teacher development.

Over time, the range of capabilities of graphing calculators hasincreased significantly. Some calculators are much more powerful thanfirst-generation graphing calculators and may include symbolic algebrafeatures. Other graphing calculators are, by design, intended for studentsstudying mathematics at lower levels than calculus. The committee candevelop examinations that are appropriate for any given level of technol-ogy, but it cannot develop examinations that are fair to all students if thespread in the capabilities of the technology is too wide. Therefore, thecommittee has found it necessary to make certain requirements of thetechnology that will help ensure that all students have sufficient computa-tional tools for the AP Calculus Examinations. Examination restrictionsshould not be interpreted as restrictions on classroom activities. The com-mittee will continue to monitor the developments of technology and willreassess the testing policy regularly.

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Graphing Calculator Capabilities for the Examinations

The committee develops examinations based on the assumption that allstudents have access to four basic calculator capabilities used extensivelyin calculus. A graphing calculator appropriate for use on the examinationsis expected to have the built-in capability to:

1) 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, and4) numerically calculate the value of a definite integral.

One or more of these capabilities should provide the sufficient compu-tational tools for successful development of a solution to any examinationquestion that requires the use of a calculator. Care is taken to ensure thatthe examination questions do not favor students who use graphing calcula-tors with more extensive built-in features.

Technology Restrictions on the Examinations

Nongraphing scientific calculators, computers, devices with a QWERTYkeyboard, pen-input devices, and electronic writing pads are not permittedfor use on the AP Calculus Examinations.

Test administrators are required to check calculators before the exami-nation. Therefore, it is important for each student to have an approved cal-culator. The student should be thoroughly familiar with the operation ofthe calculator he or she plans to use on the examination. Calculators maynot be shared, and communication between calculators is prohibited dur-ing the examination. Students may bring to the examination one or two(but no more than two) graphing calculators from the list on page 20.

Calculator memories will not be cleared. Students are allowed to bringto the examination calculators containing whatever programs they want.

Students must not use calculator memories to take test materials out ofthe room. Students should be warned that their grades will be invalidatedif they attempt to remove test materials from the room by any method.


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Showing Work on the Free-Response Sections

Students are expected to show enough of their work for readers to followtheir line of reasoning. To obtain full credit for the solution to a free-response problem, students must communicate their methods and conclu-sions clearly. Answers should show enough work so that the reasoningprocess can be followed throughout the solution. This is particularlyimportant for assessing partial credit. Students may also be asked to usecomplete sentences to explain their methods or the reasonableness oftheir answers, or to interpret their results.

For results obtained using one of the four required calculator capabili-ties listed on page 18, students are required to write the setup (e.g., theequation being solved, or the derivative or definite integral being evalu-ated) that leads to the solution, along with the result produced by the cal-culator. For example, if the student is asked to find the area of a region,the student is expected to show a definite integral (i.e., the setup) and theanswer. The student need not compute the antiderivative; the calculatormay be used to calculate the value of the definite integral without furtherexplanation. For solutions obtained using a calculator capability otherthan one of the four required ones, students must also show the mathemat-ical steps necessary to produce their results; a calculator result alone isnot sufficient. For example, if the student is asked to find a relative mini-mum value of a function, the student is expected to use calculus and showthe mathematical steps that lead to the answer. It is not sufficient to graphthe function or use a built-in minimum finder.

When a student is asked to justify an answer, the justification mustinclude mathematical (noncalculator) reasons, not merely calculatorresults. Functions, graphs, tables, or other objects that are used in a justifi-cation should be clearly labeled.

A graphing calculator is a powerful tool for exploration, but studentsmust be cautioned that exploration is not a mathematical solution.Exploration with a graphing calculator can lead a student toward an ana-lytical solution, and after a solution is found, a graphing calculator canoften be used to check the reasonableness of the solution.

As on previous AP Examinations, if a calculation is given as a decimalapproximation, it should be correct to three places after the decimal pointunless otherwise indicated. Students should be cautioned against roundingvalues in intermediate steps before a final calculation is made. Studentsshould also be aware that there are limitations inherent in graphing calcula-tor technology; for example, answers obtained by tracing along a graph tofind roots or points of intersection might not produce the required accuracy.

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List of Graphing Calculators

Students are expected to bring a calculator with the capabilities listed onpage 18 to the examinations. AP teachers should check their own students’calculators to ensure that the required conditions are met. If a student wishesto use a calculator that is not on the list, the AP teacher must contact ETS(609-771-7300) before April 1 of the testing year to request written permissionfor the student to use the calculator on the AP Examinations.

Casio Hewlett-Packard Texas InstrumentsFX-6000 series *FX-9700 series *HP-28 series TI-73FX-6200 series *FX-9750 series *HP-38G TI-80FX-6300 series *CFX-9800 series *HP-39G TI-81FX-6500 series *CFX-9850 series *HP-40G *TI-82FX-7000 series *CFX-9950 series *HP-48 series *TI-83/TI-83 PlusFX-7300 series *CFX-9970 series *HP-49 series *TI-85FX-7400 series *Algebra FX 2.0 series *TI-86FX-7500 series Radio Shack *TI-89FX-7700 series EC-4033FX-7800 series EC-4034 OtherFX-8000 series EC-4037 MicrontaFX-8500 series Smart2

FX-8700 series SharpFX-8800 series EL-5200

*EL-9200 series*EL-9300 series*EL-9600 series

* These graphing calculators have the built-in capabilities listed on page 18.

Note: This list will be updated, when necessary, to include new allowablecalculators. An up-to-date list is available in the AP section of the CollegeBoard Web site (www.collegeboard.com/ap).

Unacceptable machines include the following:Nongraphing scientific calculatorsPowerbooks and portable computersPocket organizers (Wizard, etc.)Electronic writing pads or pen-input devices (Newton, etc.)Devices with QWERTY keyboards (HP-95, TI-92, etc.)


32458W4 3/21/01 5:20 PM Page 20


The Examinations

The Calculus AB and BC Examinations seek to assess how well a studenthas mastered the concepts and techniques of the subject matter of the corresponding courses. Each examination consists of two sections, asdescribed below.

Section I: a multiple-choice section testing proficiency in a wide varietyof topics

Section II: a free-response section requiring the student to demonstratethe ability to solve problems involving a more extendedchain of reasoning

The time allotted for each AP Calculus Examination is 3 hours and 15minutes. The multiple-choice section of each examination consists of 45questions in 105 minutes. Part A of the multiple-choice section (28 ques-tions in 55 minutes) does not allow the use of a calculator. Part B of themultiple-choice section (17 questions in 50 minutes) contains some ques-tions for which a graphing calculator is required.

The free-response section of each examination has two parts: one partrequiring graphing calculators and a second part not allowing graphing cal-culators. The AP Examinations are designed to accurately assess studentmastery of both the concepts and techniques of calculus. The two-part for-mat for the free-response section provides greater flexibility in the types ofproblems that can be given while ensuring fairness to all students takingthe exam, regardless of the graphing calculator used. See page 20 for thelist of graphing calculators for the AP Calculus Examinations.

The free-response section of each examination consists of 6 problemsin 90 minutes. Part A of the free-response section (3 problems in 45 min-utes) contains some problems or parts of problems for which a graphingcalculator is required. Part B of the free-response section (3 problems in45 minutes) does not allow the use of a calculator. During the secondtimed portion of the free-response section (Part B), students are permittedto continue work on problems in Part A, but they are not permitted to usea calculator during this time.

In determining the grade for each examination, the scores for Section Iand Section II are given equal weight. Since the examinations are designedfor full coverage of the subject matter, it is not expected that all studentswill be able to answer all the questions.

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Calculus AB Subscore Grade for the Calculus BC Examination

A Calculus AB subscore grade is reported based on performance on theportion of the examination devoted to Calculus AB topics (approximately60% of the examination). The Calculus AB subscore grade is designed to give colleges and universities more information about the student.Although each college and university sets its own policy for awardingcredit and/or placement for AP Exam grades, it is recommended that insti-tutions apply the same policy to the Calculus AB subscore grade that theyapply to the Calculus AB grade. Use of the subscore grade in this manneris consistent with the philosophy of the courses, since common topics aretested at the same conceptual level in both Calculus AB and Calculus BC.

The Calculus AB subscore grade was first reported for the 1998Calculus BC examination. The annual percentage increase in students tak-ing the AP Calculus AB Examination has remained constant at 7% from1993 to 2000. The annual percentage increase for students taking the APCalculus BC Examination increased from 8% from 1993 to 1997 to 15%from 1997 to 2000. Based on these annual increases, it appears that morestudents who take a Calculus BC course have chosen to take the CalculusBC examination as a result of knowing that they will receive this CalculusAB subscore grade to assist with college credit and/or placement. The reli-ability of the Calculus AB subscore grade is nearly equal to the reliabilitiesof the AP Calculus AB and Calculus BC Examinations.

The following tables compare the AB subscore grades with the BCgrades for students taking the AP Calculus BC Examination in 1998, 1999,and 2000. For 2000, these data show that 99.5% of the candidates earned anAB subscore grade equal to or higher than their grade on the Calculus BCexamination. The remaining 0.5% of the candidates had an AB subscoregrade that was one grade lower than their Calculus BC grade and almostall of these candidates earned a 5 on the Calculus BC exam and a 4 for theCalculus AB subscore grade. No candidates earned an AB subscore gradethat was two or more grades lower than their Calculus BC grade.

Comparison of AB Subscore Grades

with Calculus BC Grades — May 1998

AB Subscore GradeBC Grade 1 2 3 4 5

1 28.3% 48.3% 23.3% <0.1%2 9.8% 82.7% 7.5%3 0.1% 33.2% 64.8% 1.8% 4 1.0% 68.5% 30.6% 5 7.3% 92.7%


32458W4 3/21/01 5:20 PM Page 22


Comparison of AB Subscore Grades with

Calculus BC Grades — May 1999


1 28.2% 58.6% 13.2% <0.1% 2 <0.1% 18.3% 77.4% 4.3% 3 0.8% 39.4% 57.9% 1.9% 4 1.1% 65.8% 33.1% 5 4.6% 95.4%

Comparison of AB Subscore Grades

with Calculus BC Grades — May 2000


1 24.8% 52.8% 22.3% 0.1% 2 7.0% 77.1% 15.8% 3 <0.1% 16.9% 73.2% 9.9% 4 <0.1% 35.8% 64.1% 5 1.2% 98.8%

Summary of Differences between AB Subscore Grades

and BC Grades (AB Subscore Grade Minus BC Grade)

–1 0 1 2 3 1998 2.9% 59.1% 34.2% 3.8% <0.1% 1999 2.1% 62.5% 33.0% 2.3% <0.1% 2000 0.5% 51.6% 41.5% 6.4% <0.1%

The Grade Setting Process

The Chief Faculty Consultant for AP Calculus works in conjunction withstatistical analysis and mathematics test development staff of EducationalTesting Service to establish the AP grade ranges (i.e., the range of rawscores that determine a particular AP grade). Direct comparisons are madebetween the performance of the current year’s candidates and that of for-mer candidates on a set of previously administered multiple-choice ques-tions. A statistical procedure called “equating” uses this information toprovide grade ranges for the current exam that best represent perfor-mance equivalent to the grade ranges in previous years. The Chief FacultyConsultant also considers other pertinent data, including college validitystudies and reports of table leaders from the AP Calculus Reading, toarrive at decisions on grades. The Chief Faculty Consultant then deter-

32458W4 3/21/01 5:20 PM Page 23

mines the grade ranges that convert the total raw scores (the multiple-choice score plus the free-response score) to the AP Program 5-point scaleof grades.

The AP grade ranges for the Calculus AB and Calculus BC examinationsare set first. This process takes into account only the performance of thestudents who took each exam, and does not consider the performance ofthose students who took the other exam. The grade ranges for the AB sub-score grade are set last. The process takes into account both the perfor-mance of the BC students on the AB-level material of the BC exam and theperformance of the students who took the AB exam. Scores on the AB por-tion of the BC exam are scaled to scores on the AB exam. Equating deter-mines the grade ranges for the AB subscore grade that best representperformance equivalent to the grade ranges for the AB exam.

Additional information on “AP Grades” can be found on page 72 and theCollege Board Web site.

Calculus AB: Section I

Section I consists of 45 multiple-choice questions. Part A contains 28questions and does not allow the use of a calculator. Part B contains 17questions and requires a graphing calculator for some questions. Twenty-four sample multiple-choice questions for Calculus AB are included in thefollowing sections. Answers to the sample questions are given on page 35.

Part A Sample Multiple-Choice Questions

A calculator may not be used on this part of the examination.

Part A consists of 28 questions. In this section of the examination, as acorrection for guessing, one-fourth of the number of questions answeredincorrectly will be subtracted from the number of questions answered cor-rectly. Following are the directions for Section I Part A and a representa-tive set of 14 questions.

Directions: Solve each of the following problems, using the availablespace for scratchwork. After examining the form of the choices, decidewhich is the best of the choices given and fill in the corresponding oval onthe answer sheet. No credit will be given for anything written in the testbook. Do not spend too much time on any one problem.

In this test: Unless otherwise specified, the domain of a function ƒ isassumed to be the set of all real numbers x for which ƒ(x) is a real number.


Calculus AB: Section 1

32458W4 3/21/01 5:20 PM Page 24



1.

The function ƒ, whose graph consists of two line segments, is shownabove. Which of the following are true for ƒ on the open interval(a, b)?

I. The domain of the derivative of ƒ is the open interval (a, b).II. ƒ is continuous on the open interval (a, b).

III. The derivative of ƒ is positive on the open interval (a, c).

(A) I only(B) II only(C) III only(D) II and III only(E) I, II, and III

2. What is ?

(A) �1

(B)

(C) 0(D) 1(E) The limit does not exist.

��22

cos �π2 � h� � cos �π2�h

limh → 0

x

y

a b

cO

f

32458W4 3/21/01 5:20 PM Page 25

3. At which of the five points on the graph in the figure

at the right are and

both negative?

(A) A

(B) B

(C) C

(D) D

(E) E

4. The slope of the tangent to the curve at (2, 1) is

(A)

(B) �1

(C)

(D)

(E) 0

5. Which of the following statements about the function given byis true?

(A) The function has no relative extremum.(B) The graph of the function has one point of inflection and the

function has two relative extrema.(C) The graph of the function has two points of inflection and the

function has one relative extremum.(D) The graph of the function has two points of inflection and the

function has two relative extrema.(E) The graph of the function has two points of inflection and the

function has three relative extrema.

6. If , then

(A) �2 cos 3(B) �2 sin 3 cos 3(C) 6 cos 3(D) 2 sin 3 cos 3(E) 6 sin 3 cos 3

ƒ�(0) �ƒ(x) � sin2(3 � x)

ƒ(x) � x4 � 2x 3

�314

�514

�32

y3x � y2x2 � 6

d 2y

dx2

dy

dx



x

y

O

AB

C

DE

32458W4 3/21/01 5:20 PM Page 26



7. The solution to the differential equation , where , is

(A)

(B)

(C)

(D)

(E)

8.

(A)

(B)

(C)

(D)

(E) 12

x 2 � 2x 2�3

� x � C

25

x 2�5

�23

x 2�3

� C

12

x 2 � x � C

23

x 2�3

�12

x 2�1

� C

32

�x �1

�x� C

�(x � 1)�x dx �

y � 3�34

x 4 � 15

y � 3�34

x 4 � 5

y � 3�34

x 4 � 15

y � 3�34

x 4 � 3�15

y � 3�34

x 4

y(2) � 3dy

dx�

x 3

y 2

32458W4 3/21/01 5:20 PM Page 27



tO 1

�1

1

2

3

�2

�3

�4

2 3 4 5 6 7 8 9 10 11

v(t)

Questions 9 and 10 refer to the following graph and information.

A bug is crawling along a straight wire. The velocity, v(t), of the bugat time t, , is given in the graph above.

9. According to the graph, at what time t does the bug change direction?

(A) 2(B) 5(C) 6(D) 8(E) 10

10. According to the graph, at what time t is the speed of the buggreatest?

(A) 2(B) 5(C) 6(D) 8(E) 10

0 � t � 11

32458W4 3/21/01 5:20 PM Page 28



11. What is ?

(A) �2

(B)

(C)

(D) 1(E) The limit does not exist.

12. The area of the region in the first quadrant between the graph ofand the x-axis is

(A)

(B)

(C)(D)

(E)

13. Which of the following are antiderivatives of ?

I.

II.

III.

(A) I only(B) III only(C) I and II only(D) I and III only(E) II and III only

2 ln x � ln2x

x2

ln3x

3� 6

ln3x

3

ln2x

x

163

2�32�2

83

23

�2

y � x�4 � x 2

12

�14

x 2 � 42 � x � 4x

2limx → ∞

32458W4 3/21/01 5:20 PM Page 29



14. If r is positive and increasing, for what value of r is the rate ofincrease of twelve times that of r?

(A)(B) 2(C)(D)(E) 6

Part B Sample Multiple-Choice Questions

A graphing calculator is required for some questions on this part of

the examination.

Part B consists of 17 questions. In this section of the examination, as acorrection for guessing, one-fourth of the number of questions answeredincorrectly will be subtracted from the number of questions answeredcorrectly. Following are the directions for Section I Part B and a represen-tative set of 10 questions.


In this test:

(1) The exact numerical value of the correct answer does not alwaysappear among the choices given. When this happens, select from amongthe choices the number that best approximates the exact numerical value.(2) Unless otherwise specified, the domain of a function ƒ is assumed tobe the set of all real numbers x for which ƒ(x) is a real number.

15. The average value of the function on the closed interval[�1, 1] is

(A) 0(B) 0.368(C) 0.747(D) 1(E) 1.494

ƒ(x) � e�x 2

2�3�123

3�4

r 3

32458W4 3/21/01 5:20 PM Page 30



16. The volume generated by revolving about the x-axis the regionenclosed by the graphs of and , for , is

(A)

(B)

(C)

(D)

(E)

17. Let ƒ be defined as follows, where .

Which of the following are true about ƒ?

I. ƒ(x) exists.

II. ƒ(a) exists.III. ƒ(x) is continuous at .

(A) None(B) I only(C) II only(D) I and II only(E) I, II, and III

x � a

limx → a

ƒ(x) � �x 2 � a2

x � a, for x � a,

0, for x � a.

a � 0

π� 2

0�y2 �

y2

2 � dy

π� 2

0��y

2�

y

2�2

dy

x(2x � 2x 2) dx2π� 1

0

(4 x 2 � 4 x 4) dxπ� 1

0

(2x � 2x 2)2 dxπ� 1

0

0 � x � 1y � 2 x 2y � 2 x

32458W4 3/21/01 5:20 PM Page 31

18.

Let ƒ be a function such that for all x in the closed interval[1, 2], with selected values shown in the table above. Which of thefollowing must be true about ƒ�(1.2)?

(A) ƒ�(1.2) 0(B) 0 ƒ�(1.2) 1.6(C) 1.6 ƒ�(1.2) 1.8(D) 1.8 ƒ�(1.2) 2.0(E) ƒ�(1.2) 2.0

19. Two particles start at the origin and move along the x-axis. For, their respective position functions are given by

and . For how many values of t do the particles have thesame velocity?

(A) None(B) One(C) Two(D) Three(E) Four

20. If the function g is defined by on the closed

interval , then g has a local minimum at

(A) 0(B) 1.084(C) 1.772(D) 2.171(E) 2.507

x ��1 � x � 3

g(x) � �x

0 sin(t2)dt

x2 � e�2t � 1x1 � sin t0 � t � 10

ƒ�(x) 0



x 1.1 1.2 1.3 1.4

ƒ(x) 4.18 4.38 4.56 4.73

32458W4 3/21/01 5:20 PM Page 32



(A) (B)

(C) (D)

(E)

21. The graphs of five functions are shown below. Which function has anonzero average value over the closed interval [�π, π]?

32458W4 3/21/01 5:20 PM Page 33

22. The region in the first quadrant enclosed by the y-axis and the graphsof and is rotated about the x-axis. The volume of thesolid generated is

(A) 0.484(B) 0.877(C) 1.520(D) 1.831(E) 3.040

23. Oil is leaking from a tanker at the rate of gallonsper hour, where t is measured in hours. How much oil has leaked outof the tanker after 10 hours?

(A) 54 gallons(B) 271 gallons(C) 865 gallons(D) 8,647 gallons(E) 14,778 gallons

24. If ƒ is continuous for all x, which of the following integralsnecessarily have the same value?

I.

II.

III.

(A) I and II only(B) I and III only(C) II and III only(D) I, II, and III(E) No two necessarily have the same value.

� b�c

a�c

ƒ(x � c)dx

� b�a

0 ƒ(x � a)dx

� b

a

ƒ(x)dx

R(t) � 2,000e�0.2t

y � xy � cos x



32458W4 3/21/01 5:20 PM Page 34



* Indicates a graphing calculator-active question.

Answers to Calculus AB Multiple-Choice Questions

Part A

1. D 6. B 11. B2. A 7. E 12. B3. B 8. D 13. C4. C 9. D 14. B5. C 10. E

Part B

15.* C 19.* D 22.* C16. B 20.* E 23.* D17. D 21. E 24. A18. D

32458W4 3/21/01 5:20 PM Page 35

Calculus BC: Section I

Section I consists of 45 multiple-choice questions. Part A contains 28questions and does not allow the use of a calculator. Part B contains 17questions and requires a graphing calculator for some questions. Twenty-four sample multiple-choice questions for Calculus BC are included in thefollowing sections. Answers to the sample questions are given on page 45.

Part A Sample Multiple-Choice Questions

A calculator may not be used on this part of the examination.

Part A consists of 28 questions. In this section of the examination, as acorrection for guessing, one-fourth of the number of questions answeredincorrectly will be subtracted from the number of questions answered cor-rectly. Following are the directions for Section I Part A and a representa-tive set of 14 questions.


In this test: Unless otherwise specified, the domain of a function ƒ isassumed to be the set of all real numbers x for which ƒ(x) is a real number.

1. The line perpendicular to the tangent of the curve represented by theequation at the point (�2, �4) also intersects thecurve at

(A) �6

(B)

(C)

(D) �3

(E) �12

�72

�92

x �y � x 2 � 6x � 4



32458W4 3/21/01 5:20 PM Page 36



2. If , then

(A) 1 cos (xy)(B) 1 y cos (xy)

(C)

(D)

(E)

3. Let be the solution to the differential equation

with the initial condition . What is the

approximation for ƒ(1) if Euler’s method is used, starting at with a step size of 0.5?

(A) 2

(B)

(C)

(D)

(E) 3

4. What are all values of x for which the series converges?

(A) All x except .(B)(C)(D)(E) The series diverges for all x.

5. If and if , then

(A)(B)(C)(D)(E) x 2g(x2)

2xg(x 2)g�(x)2xg(x)g(x 2)

d

dx ƒ(h(x)) �h(x) � x 2d

dx ƒ(x) � g(x)

�x� 3�3 � x � 3�x� � 3

x � 0

�

n�1 n3n

xn

2 �π2

2 �π4

2 �π6

x � 0

ƒ(0) � 2dy

dx� arcsin(xy)

y � ƒ(x)

1 � y cos(xy)1 � x cos(xy)

11 � x cos(xy)

11 � cos(xy)

��

dy

dx�y � x � sin(xy)

32458W4 3/21/01 5:20 PM Page 37

6. If F� is a continuous function for all real x, then is

(A) 0(B) F(0)(C) F(a)(D) F�(0)(E) F�(a)

7.

The slope field for a certain differential equation is shown above.Which of the following could be a specific solution to that differentialequation?

(A)(B)(C)(D)(E)

8. is

(A) �1

(B)

(C)

(D) 1(E) divergent

12

�12

�∞

0 e�2tdt

y � ln xy � cos xy � e�x

y � ex

y � x2

1h

�a�h

a

F�(x)dxlimh → 0



32458W4 3/21/01 5:20 PM Page 38



9. Which of the following series converge to 2?

I.

II.

III.

(A) I only(B) II only(C) III only(D) I and III only(E) II and III only

10. What is the area of the closed region bounded by the curve and the lines and ?

(A)

(B)

(C)

(D)

(E)

11. Which of the following integrals gives the length of the graph ofbetween and , where ?

(A)

(B)

(C)

(D)

(E) � b

a

�1 �1

4 x dx

� b

a

�1 �1

2�x dx

� b

a

�x �1

2�x dx

� b

a

�x � �x dx

� b

a

�x2 � x dx

0 a bx � bx � ay � �x

e2 � 12

e2 � 22

3 � e2

2

e2 � 32

2 � e2

2

y � 1x � 1y � e2x

12n

�

n�0

�8(�3)n

�

n�1

2n

n � 3�

n�1

32458W4 3/21/01 5:20 PM Page 39

12. A point (x, y) is moving along a curve . At the instant when

the slope of the curve is , the x-coordinate of the point is

increasing at the rate of 5 units per second. The rate of change, inunits per second, of the y-coordinate of the point is

(A)

(B)

(C)

(D)

(E)

13. The third-degree Taylor polynomial about of is

(A)

(B)

(C)

(D)

(E)

14. If and when , then

(A)(B)(C)(D)(E) tan x � 5ex

tan x � 55etan xetan x � 5etan x � 4

y �x � 0y � 5dy

dx� y sec2x

�x �x2

2�

x 3

3

�1 � x �x 2

2

x �x 2

2�

x 3

3

1 � x �x2

2

�x �x2

2�

x 3

3

ln(1 � x)x � 0

53

35

13

�13

�53

�13

y � ƒ(x)



32458W4 3/21/01 5:20 PM Page 40



Part B Sample Multiple-Choice Questions

A graphing calculator is required for some questions on this part of

the examination.

Part B consists of 17 questions. In this section of the examination, as acorrection for guessing, one-fourth of the number of questions answeredincorrectly will be subtracted from the number of questions answeredcorrectly. Following are the directions for Section I Part B and a represen-tative set of 10 questions.


In this test:

(1) The exact numerical value of the correct answer does not alwaysappear among the choices given. When this happens, select from amongthe choices the number that best approximates the exact numerical value.(2) Unless otherwise specified, the domain of a function ƒ is assumed tobe the set of all real numbers x for which ƒ(x) is a real number.

15.

Graph of f

The graph of the function ƒ above consists of four semicircles. If

, where is g(x) nonnegative?

(A) [�3, 3](B) only(C) [0, 3] only(D) [0, 2] only(E) only[�3, �2] � [0, 3]

[�3, �2] � [0, 2]

g(x) � �x

0ƒ(t)dt

O 1

1

32458W4 3/21/01 5:20 PM Page 41

16. If ƒ is differentiable at , which of the following could be false?

(A) ƒ is continuous at .(B) ƒ(x) exists.

(C) exists.

(D) ƒ�(a) is defined.(E) ƒ�(a) is defined.

17.

A rectangle with one side on the x-axis has its upper vertices on thegraph of , as shown in the figure above. What is the mini-mum area of the shaded region?

(A) 0.799(B) 0.878(C) 1.140(D) 1.439(E) 2.000

y � cos x

Ox

y

ƒ(x) � ƒ(a)x � a

limx → a

limx → a

x � a

x � a



32458W4 3/21/01 5:20 PM Page 42



Time (sec) 0 10 25 37 46 60

Rate (gal/sec) 500 400 350 280 200 180

18.

The table above gives the values for the rate (in gal/sec) at whichwater flowed into a lake, with readings taken at specific times. Aright Riemann sum, with the five subintervals indicated by the data inthe table, is used to estimate the total amount of water that flowedinto the lake during the time period . What is this estimate?

(A) 1,910 gal(B) 14,100 gal(C) 16,930 gal(D) 18,725 gal(E) 20,520 gal

19.

Let ƒ be a function whose domain is the open interval (1, 5). The fig-ure above shows the graph of ƒ�. Which of the following describes therelative extrema of ƒ� and the points of inflection of the graph of ƒ�?

(A) 1 relative maximum, 1 relative minimum, and no point ofinflection

(B) 1 relative maximum, 2 relative minima, and no point of inflection(C) 1 relative maximum, 1 relative minimum, and 1 point of inflection(D) 1 relative maximum and 2 points of inflection(E) 1 relative minimum and 2 points of inflection

x

y

O 1 5

ƒ �

0 � t � 60

32458W4 3/21/01 5:20 PM Page 43

20. A particle moves along the x-axis so that at any time its velocityis given by . The total distance traveled bythe particle from to is

(A) 0.667(B) 0.704(C) 1.540(D) 2.667(E) 2.901

21. If the function ƒ is defined by and g is anantiderivative of ƒ such that , then

(A) �3.268(B) �1.585(C) �1.732(D) �6.585(E) 11.585

22. Let g be the function given by . Which

of the following statements about g must be true?

I. g is increasing on (1, 2).II. g is increasing on (2, 3).

III. .

(A) I only(B) II only(C) III only(D) II and III only(E) I, II, and III

g(3) 0

g(x) � �x

1100(t2 � 3t � 2)e�t2

dt

g(1) �g(3) � 5ƒ(x) � �x3 � 2

t � 2t � 0v(t) � ln(t � 1) � 2t � 1

t � 0



32458W4 3/21/01 5:20 PM Page 44



* Indicates a graphing calculator-active question.

Part A

1. B 6. E 11. E2. E 7. E 12. A3. C 8. C 13. A4. D 9. E 14. C5. D 10. B

Part B

15. A 19. E 22.* B16. E 20.* C 23. C17.* B 21.* B 24.* B18. C

23. The area of one loop of the graph of the polar equation is given by which of the following expressions?

(A)

(B)

(C)

(D)

(E)

24. Let g be the function given by .

For , g is decreasing most rapidly when

(A) 0.949(B) 2.017(C) 3.106(D) 5.965(E) 8.000

Answers to Calculus BC Multiple-Choice Questions

t �0 � t � 8

g(t) � 100 � 20 sin �πt

2 � � 10 cos �πt

6 �

2�2π3

0 sin (3q) dq

2�2π3

0 sin2 (3q) dq

2�π3

0 sin2 (3q) dq

2�π3

0 sin (3q) dq

4�π3

0 sin2 (3q) dq

r � 2 sin(3q)

32458W4 3/21/01 5:20 PM Page 45


Calculus AB and Calculus BC: Section II


Section II consists of six free-response problems. The problems do NOTappear in the Section II test booklet. Part A problems are printed in thegreen insert only; Part B problems are printed in a separate sealed blueinsert. Each part of every problem has a designated workspace in the testbooklet. ALL WORK MUST BE SHOWN IN THE TEST BOOKLET. (For stu-dents taking the examination at an alternate administration, the Part Aproblems are printed in the test booklet only; the Part B problems appearin a separate sealed insert.)

The instructions below are from the 2001 examinations. The free-response problems are from the 2000 examinations and include commentsdescribing their purpose and information on scoring. Additional samplequestions can be found in the AP section of the College Board Web site.

Instructions for Section II

PART A

(A graphing calculator is required for some problems

or parts of problems.)

Part A: 45 minutes, 3 problems

During the timed portion for Part A, you may work only on the problemsin Part A. The problems for Part A are printed in the green insert only.When you are told to begin, open your booklet, carefully tear out the greeninsert, and write your solution to each part of each problem in the spaceprovided for that part in the pink test booklet.

On Part A, you are permitted to use your calculator to solve an equa-tion, find the derivative of a function at a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of yourproblem, namely the equation, function, or integral you are using. If youuse other built-in features or programs, you must show the mathematicalsteps necessary to produce your results.

PART B

(No calculator is allowed for these problems.)

Part B: 45 minutes, 3 problems

The problems for Part B are printed in the blue insert only. When you aretold to begin, open the blue insert, and write your solution to each part ofeach problem in the space provided for that part in the pink test booklet.During the timed portion for Part B, you may continue to work on the prob-lems in Part A without the use of any calculator.

32458W4 3/21/01 5:20 PM Page 46



GENERAL INSTRUCTIONS FOR

SECTION II PART A AND PART B

For each part of Section II, you may wish to look over the problems beforestarting to work on them, since it is not expected that everyone will beable to complete all parts of all problems. All problems are given equalweight, but the parts of a particular problem are not necessarily givenequal weight.

• YOU SHOULD WRITE ALL WORK FOR EACH PART OF EACHPROBLEM IN THE SPACE PROVIDED FOR THAT PART IN THEPINK TEST BOOKLET. Be sure to write clearly and legibly. If youmake an error, you may save time by crossing it out rather than trying to erase it. Erased or crossed-out work will not be graded.

• Show all your work. You will be graded on the correctness and completeness of your methods as well as the accuracy of your final answers. Correct answers without supporting work may notreceive credit.

• Justifications require that you give mathematical (noncalculator)reasons and that you clearly label functions, graphs, tables, or otherobjects you use.

• Your work must be expressed in standard mathematical notationrather than calculator syntax.

For example, may not be written as fnInt(X2, X, 1, 5).

• Unless otherwise specified, answers (numeric or algebraic) need notbe simplified. If a calculation is given as a decimal approximation, itshould be correct to three places after the decimal point.

• Unless otherwise specified, the domain of a function f is assumed tobe the set of all real numbers x for which f(x) is a real number.

5

1x2 dx

32458W4 3/21/01 5:20 PM Page 47

Calculus AB Sample Free-Response Questions

AB/BC Question 1

This problem required the student to use definite integrals in computingthe area of a region R and the volumes of two solids with known crosssections. One solid was obtained by revolving the region R about the x-axis (a solid of revolution having circular cross sections) and the othersolid had base R with square cross sections perpendicular to the x-axis.The student needed to use a calculator equation-solver (numerical orgraphical) to determine the intersection of two curves that bound theregion. The calculator was also needed to compute all three definite integrals arising in this problem.

The mean score was 4.84 for the AB students and 6.72 for the BC stu-dents, indicating a reasonably strong overall performance, particularly forthe BC population.



32458W4 3/21/01 5:20 PM Page 48



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AB/BC Question 2

This problem presented the student with the velocity functions for tworunners, Runner A and Runner B. Runner A’s velocity was representedgraphically, while Runner B’s velocity was represented analytically as analgebraic function. The three parts of the problem asked the student toanswer parallel questions for both runners – (a) velocity at a given time,(b) acceleration at a given time, and (c) distance traveled over a given time interval – all with appropriate units of measurement. This problemrequired students to demonstrate their understanding of basic differentialand integral calculus concepts in multiple representations, a relatively newemphasis in the AP Calculus course description.

The mean score was 5.69 for the AB students and 7.06 for the BC stu-dents, indicating a very strong overall performance.



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AB Question 3

This problem presented the student with the graph of a derivative of thefunction f, along with information about the locations of the horizontaland vertical tangents to the derivative graph. The student was asked toanswer questions about the relative and absolute extrema of the “parent”function f and about values where f � is negative. Mathematically soundjustifications for the locations of the extrema were needed to earn fullcredit on this problem. Justifying that the absolute maximum of f occurs at the right endpoint of the closed interval required an especially carefulargument by the student in interpreting the area between the graph of f �and the x-axis as quantifying the accumulated change in the value of f (x).

The mean score was 3.41, the second lowest mean among the six ABproblems. Only 0.6 % of the approximately 136,000 AB students earned all9 points. Moreover, almost 28% of the AB population earned no points onthis problem, a disappointment considering the emphasis placed on ana-lyzing the behavior of a function given the graph of its derivative.



32458W4 3/21/01 5:20 PM Page 52

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AB Question 4

This problem asked the student to apply calculus in the context of quanti-fying the volume of water in an underground tank. The student was pre-sented with the following information: an initial amount of water in thetank, a constant rate of inflow, and a variable rate of outflow (leakage) ofwater from the tank relative to time. The introductory parts of the problemasked the student to find the amount of water that had leaked from thetank and the total amount of water in the tank at a specific time. The prob-lem proceeded to ask for a general model for the amount of water in thetank at time t and for the exact time at which this amount is a maximum.

The mean score was 3.64, indicating a reasonable performance by theAB students; this was pulled down by the relatively large number of gradesat the low end of the scale.



32458W4 3/21/01 5:20 PM Page 54

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AB/BC Question 5

This problem was relatively straightforward and presented the studentwith a curve described by a relation in terms of x and y. The first part ofthe problem asked for a verification of an algebraic expression for dy/dx

found by implicit differentiation. The student then had to make use of thisderivative expression and the original relation in determining the equa-tions and locations of various tangent lines to the curve. Note that thisproblem required the student to solve some simple algebraic equationswithout the use of a calculator.

The mean score was 4.17 for AB students and 5.74 for BC students, indi-cating a good performance by both populations.



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32458W4 3/21/01 5:20 PM Page 57

AB Question 6

This problem presented the student with a separable differential equation.Part (a) of the problem asked for a solution satisfying a given initial condi-tion. In solving this differential equation, the student needed to performsome simple antidifferentiations without the use of a calculator and cor-rectly handle the constants of integration that arise in order to find a solu-tion that satisfies the initial condition. The concluding part of the problemasked the student for both the domain and range of the particular solution(involving a logarithmic function) found in part (a).

The mean score was 2.84, indicating a poor performance. Despite thefact that this was the third separable differential equation problem in thepast four years, almost 29% of the AB population received a score of 0 andanother 10% left it blank.



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Calculus BC Sample Free-Response Questions

AB/BC Question 1

This problem required the student to use definite integrals in computingthe area of a region R and the volumes of two solids with known crosssections. One solid was obtained by revolving the region R about the x-axis (a solid of revolution having circular cross sections) and the othersolid had base R with square cross sections perpendicular to the x-axis.The student needed to use a calculator equation-solver (numerical orgraphical) to determine the intersection of two curves that bound theregion. The calculator was also needed to compute all three definite integrals arising in this problem.

The mean score was 4.84 for the AB students and 6.72 for the BC stu-dents, indicating a reasonably strong overall performance, particularly forthe BC population.



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AB/BC Question 2

This problem presented the student with the velocity functions for tworunners, Runner A and Runner B. Runner A’s velocity was representedgraphically, while Runner B’s velocity was represented analytically as analgebraic function. The three parts of the problem asked the student toanswer parallel questions for both runners – (a) velocity at a given time,(b) acceleration at a given time, and (c) distance traveled over a given time interval – all with appropriate units of measurement. This problemrequired students to demonstrate their understanding of basic differentialand integral calculus concepts in multiple representations, a relatively newemphasis in the AP Calculus course description.

The mean score was 5.69 for the AB students and 7.06 for the BC stu-dents, indicating a very strong overall performance.



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BC Question 3

This problem presented the student with the values of a function f and allof its derivatives at x � 5. From this information, the student needed tofind the third-degree Taylor polynomial for f and determine the radius ofconvergence for the Taylor series for f about x � 5. The problem con-cluded by asking the student to show that the sixth-degree Taylor polyno-mial for f about x � 5 could be used to approximate f (6) with error lessthan 1/1000. Since the resulting Taylor series is a convergent alternatingseries when x � 6, the seventh-degree term can be used to find a suitablebound on this error.

All BC examinations in recent years have had a problem on series, andstudent performances have generally been weak. The mean score on thisquestion was 3.29, again relatively weak, but similar to the result on the1999 BC exam.



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BC Question 4

This problem presented the student with a velocity vector for a particlemoving in the plane, along with its position at the specific time t � 1. Thefirst parts of the problem asked the student to find the acceleration vectorand the position of the particle at time t � 3. These questions required thestudent to perform both some simple differentiation and antidifferentiationwithout the use of a calculator. The remaining parts of the problem turnedto the geometry of the particle’s path. The student had to find the time atwhich the slope of the path attains a given value, as well as the limitingvalue of the slope as t approaches infinity.

The mean score was 5.57, indicating a reasonably strong student performance.



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AB/BC Question 5

This problem was relatively straightforward and presented the studentwith a curve described by a relation in terms of x and y. The first part ofthe problem asked for a verification of an algebraic expression for dy/dx

found by implicit differentiation. The student then had to make use of thisderivative expression and the original relation in determining the equa-tions and locations of various tangent lines to the curve. Note that thisproblem required the student to solve some simple algebraic equationswithout the use of a calculator.

The mean score was 4.17 for AB students and 5.74 for BC students, indicating a good performance by both populations.



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BC Question 6

This problem presented the student with a separable differential equa-tion. The use of slope fields, a relatively new topic in the AP Calculus BCcourse description, is prominent in the first two parts of this problem. Thestudent had to sketch a slope field for the differential equation using agiven lattice of points in the coordinate plane. The student was then askedto use this slope field to mathematically “disqualify” a particular curve as a solution to the differential equation. The next part of the problem askedfor a solution satisfying a given initial condition. In solving this differentialequation, the student needed to perform some simple antidifferentiationswithout the use of a calculator and correctly handle the constants of inte-gration that arise in order to find a solution that satisfies the initial con-dition. The concluding part of the problem asked for the range of thisparticular solution. Note that the slope field found earlier in the problemcould be used to help a student check the reasonableness of a solution and its range.

This question was split graded with parts (a) and (b) being BC materialonly (slope fields) and parts (c) and (d) contributing to the Calculus ABsubscore grade (solution and range for a separable differential equation).The mean score on parts (a) and (b) was 2.25 (out of 3), with about 54%earning all 3 points. The mean score on the last two parts was 2.91 (out of 6) for a total mean of 5.16. This indicates a reasonably strong studentperformance, although even the BC students continue to have difficultysolving a separable differential equation. Almost a quarter of the BC population earned no points on parts (c) and (d).



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AP Program Essentials

The AP Reading

In June, the free-response sections of the exams, as well as the portfoliosin Studio Art, are scored by college and secondary school teachers at theAP Reading. Thousands of these faculty consultants participate, under thedirection of a Chief Faculty Consultant in each field. The experience offersboth significant professional development and the opportunity to networkwith like-minded educators; if you are an AP teacher or a member of a col-lege faculty and would like to serve as a faculty consultant, you can applyonline in the AP section of the College Board’s Web site. Alternatively,send an e-mail message to [email protected], or call Performance ScoringServices at 609 406-5383.

AP Grades

The faculty consultants’ judgments on the essay and problem-solving ques-tions are combined with the results of the computer-scored multiple-choicequestions, and the total raw scores are converted to AP’s 5-point scale:

AP GRADE QUALIFICATION

5 Extremely Well Qualified4 Well Qualified3 Qualified2 Possibly Qualified1 No Recommendation

Grade Distributions

Many teachers want to compare their students’ grades with the nationalpercentiles. Grade distribution charts are available in the subject pages ofthe AP Web site, as is information on how the cut-off points for each APgrade are calculated.

AP and College Credit

Advanced placement and/or credit is awarded by the college or university,not the College Board or the AP Program. The best source of specific andup-to-date information about an individual institution’s policy is its catalogor Web site.

x5-2554-15 CEEB WRAP BACK 3/21/01 1:17 AM Page 72

Why Colleges Give Credit for AP Grades

Colleges need to know that the AP grades they receive for their incomingstudents represent a level of achievement equivalent to that of studentswho take the same course in the colleges’ own classrooms. That equiva-lency is assured through several Advanced Placement Program processes:

• College faculty serve on the committees that develop the coursedescriptions and examinations in each AP subject.

• College faculty are responsible for standard setting and are involvedin the evaluation of student responses at the AP Reading.

• AP courses and exams are updated regularly, based on both theresults of curriculum surveys at up to 200 colleges and universitiesand the interactions of committee members with professional orga-nizations in their discipline.

• College comparability studies are undertaken in which the perfor-mance of college students on AP Exams is compared with that ofAP students to confirm that the AP grade scale of 1–5 is properlyaligned with current college standards.

In addition, the College Board has commissioned studies that use a “bottom-line” approach to validating AP Exam grades by comparing theachievement of AP versus non-AP students in higher-level college courses.For example, in the 1998 Morgan and Ramist “21-College” study, AP stu-dents who were exempted from introductory courses and who completeda higher-level course in college are compared, on the basis of their collegegrades, with students who completed the prerequisite first course in col-lege, then took the second, higher-level course in the subject area. Suchstudies answer the question of greatest concern to colleges — are their APstudents who are exempted from introductory courses as well prepared tocontinue in a subject area as students who took their first course in col-lege? To see the results of several college validity studies, go to the APpages of the College Board’s Web site. (The aforementioned Morgan andRamist study can be downloaded from the site in its entirety.)

Guidelines on Granting Credit for AP Grades

If you are an admission administrator and need guidance on setting a policy for your college, you will find the College and University Guide to

the Advanced Placement Program useful; see the back of this booklet forordering information. Alternatively, contact your local College BoardRegional Office, as noted on the inside back cover of this booklet.




Finding Colleges That Accept AP Grades

In addition to contacting colleges directly for their AP policies, studentsand teachers can use College Search, an online resource maintained by theCollege Board through its Annual Survey of Colleges. College Search canbe accessed via the College Board’s Web site (www.collegeboard.com). Itis worth remembering, though, that policies are subject to change. Contactthe college directly to get the most up-to-date information.

AP Scholar Awards and the AP International Diploma

The AP Program offers a number of awards to recognize high school students who have demonstrated college-level achievement through APcourses and exams. In addition, the AP International Diploma (APID) cer-tifies the achievement of successful AP candidates who plan to apply to auniversity outside the United States.

For detailed information on AP Scholar Awards and the APID, includingqualification criteria, visit the AP Web site or contact the College Board’sNational Office. Students’ questions are also answered in the AP Bulletin

for Students and Parents; information about ordering and downloadingthe Bulletin can be found at the back of this booklet.

AP Calendar

To get an idea of the various events associated with running an AP pro-gram and administering the AP Exams, please refer to this year’s edition ofA Guide to the Advanced Placement Program; information about orderingand downloading the Guide can be found at the back of this booklet.

Test Security

The entire AP Exam must be kept secure until the scheduled administrationdate. Except during the actual exam administration, exam materials mustbe placed in locked storage. Forty-eight hours after the exam has beenadministered, the green and blue inserts from the free-response section(Section II) are available for teacher and student review.* However, the

multiple-choice section (Section I) must remain secure both before

and after the exam administration. No one other than candidates taking

*The alternate (make-up) form of the free-response section is NOT released.



the exam can ever have access to or see the questions contained in this sec-tion — this includes AP Coordinators and AP teachers. The multiple-choicesection must never be shared or copied in any manner.

Various combinations of selected multiple-choice questions are reusedfrom year to year to provide an essential method of establishing high examreliability, controlled levels of difficulty, and comparability with earlierexams. These goals can only be attained when the multiple-choice ques-tions remain secure. This is why teachers cannot view the questions andstudents cannot share information about these questions with anyone fol-lowing the exam administration.

To ensure that all students have an equal chance to perform on theexam, AP Exams must be administered in a uniform manner. It isextremely important to follow the administration schedule and all

procedures outlined in detail in the most recent AP Coordinator’s

Manual. The manual also includes directions on how to deal with mis-conduct and other security problems. Any breach of security should bereported immediately through the test security hot line (call 800 353-8570,e-mail [email protected], or fax 609 406-9709).

Teacher Support

Look for these enhanced Web resources at www.collegeboard.com/ap

• Information about AP Exam development, administration, scoring andgrading, fees, and scheduling.

• Program news, such as exam format changes, opinion polls (teachersurveys, ad hoc polls), and profiles of successful teachers and AP programs.

• A searchable catalog of teaching resources, including: course topicoutlines, sample syllabi and lesson plans, strategies and tips, topicbriefs, links, and textbook reviews.

• A searchable catalog of professional development opportunities (e.g.,workshops, summer institutes, conferences). New and experiencedAP teachers are invited to attend workshops and institutes to learnthe fundamentals of teaching an AP course, as well as the latestexpectations for each course and exam. Sessions ranging from oneday to three weeks in length are held year-round. Dates, locations,topics, and fee information are also available through the CollegeBoard’s Regional Offices.



• Online forums for exchanging ideas with AP teachers.

• Sample multiple-choice and free-response questions.

To supplement these online resources, there are a number of AP publica-tions, CD-ROMs, and videos that can assist AP teachers. Please see the following pages for an overview and for ordering information.

Pre-AP™

Preparing Students for Challenging Courses;Preparing Teachers for Student Success

Pre-AP has two objectives: (1) to promote access to AP for all students;(2) to provide professional development through content-specific strate-gies to build a rigorous curriculum. Teachers employ Pre-AP strategies andmaterials to introduce skills, concepts, and assessment methods that pre-pare students for success when they take AP and other challenging aca-demic courses. Schools use Pre-AP strategies to strengthen and align thecurriculum across grade levels, and to increase the academic challenge forall students.

Pre-AP professional development is available to teachers throughBuilding Success workshops and through AP Vertical Teams™ confer-ences and workshops.

• Building Success is a two-day workshop that assists English and his-tory teachers in designing curricula for grade 7 and above. Teacherslearn strategies to help students engage in active questioning, analysis,and constructing arguments. Workshop topics include assessment,interdisciplinary teaching and learning, and vertical planning.

• AP Vertical Teams are trained via one-day workshops, two-day con-ferences, and five-day summer institutes; they enable middle schooland high school teachers to prepare Pre-AP students for academicsuccess in AP courses and in college. Topics include organizing effec-tive teams, aligning curricula, and developing content-specific teach-ing strategies.

• Setting the Cornerstones: Building the Foundation of AP

Vertical Teams is a two-day workshop designed to provide informa-tion about the College Board and the AP Program, and to suggeststrategies for establishing coherence, commitment, collegiality, andcollaboration among the members of an AP Vertical Team.



For more information about Building Success workshops and for sched-ules of AP Vertical Teams workshops and conferences, contact yourCollege Board Regional Office. Alternatively, contact Mondy Raibon, Pre-AP Initiatives, AP Program, The College Board, 45 Columbus Avenue,New York, NY 10023-6992; 212 713-8156; [email protected].

AP Publications and Other Resources

A number of AP publications, CD-ROMs, and videos are available to helpstudents, parents, AP Coordinators, and high school and college facultylearn more about the AP Program and its courses and exams. To identifyresources that may be of particular use to you, refer to the following key.

Students and Parents SP AP Coordinators and

Administrators A

Teachers T College Faculty C

Ordering Information

You have several options for ordering publications:

• Online. Visit the College Board store to see descriptions and picturesof AP publications and to place your order.

• By mail. Send a completed order form with your payment or creditcard information to: Advanced Placement Program, Dept. E-06, P.O. Box 6670, Princeton, NJ 08541-6670. If you need a copy of the order form, you can download one from the AP Library (www.collegeboard.com/ap/library).

• By fax. Credit card orders can be faxed to AP Order Services at 609 771-7385.

• By phone. Call AP Order Services at 609 771-7243, Monday throughFriday 8:00 a.m. to 9:00 p.m., and Saturday 9:00 a.m. to 4:45 p.m. ET.Have your American Express, MasterCard, or VISA information ready.This phone number is for credit card orders only.

Payment must accompany all orders not on an institutional purchase orderor credit card, and checks should be made payable to the College Board.The College Board pays fourth-class book rate postage (or its equivalent)on all prepaid orders; you should allow two to three weeks for delivery.Postage will be charged on all orders requiring billing and/or requesting afaster method of shipment.



Publications may be returned within 15 days of receipt if postage is pre-paid and publications are in resalable condition and still in print. Unlessotherwise specified, orders will be filled with the currently available

edition; prices are subject to change without notice.

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Items marked with a computer mouse icon can be downloaded for freefrom the AP Library (www.collegeboard.com/ap/library).

AP Bulletin for Students and Parents: Free SP

This bulletin provides a general description of the AP Program, includingpolicies and procedures for preparing to take the exams, and registeringfor the AP courses. It describes each AP Exam, lists the advantages of tak-ing the exams, describes the grade reporting and award options availableto students, and includes the upcoming exam schedule.

College and University Guide to the AP Program: $10 C, A

This guide is intended to help college and university faculty and adminis-trators understand the benefits of having a coherent, equitable AP policy.Topics included are validity of AP grades; developing and maintainingscoring standards; ensuring equivalent achievement; state legislation sup-porting AP; and quantitative profiles of AP students by each AP subject.

Course Descriptions: $12 SP, T, A, C

Course Descriptions provide an outline of the AP course content, explainthe kinds of skills students are expected to demonstrate in the correspond-ing introductory college-level course, and describe the AP Exam. They alsoprovide sample multiple-choice questions with an answer key, as well assample free-response questions. A complete set of Course Descriptions isavailable for $100.

A Guide to the Advanced Placement Program: Free A

Written for both administrators and AP Coordinators, this guide is dividedinto two sections. The first section provides general information about AP,such as how to organize an AP program at your high school, the kind oftraining and support that is available for AP teachers, and a look at the AP Exams and grades. The second section contains more specific detailsabout testing procedures and policies and is intended for AP Coordinators.

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Interpreting and Using AP Grades: Free A, C, T

A booklet containing information on the development of scoring stan-dards, the AP Reading, grade-setting procedures, and suggestions on howto interpret AP grades.

Pre-AP: Achieving Equity, Emphasizing Excellence: Free A, T

An informational brochure describing the Pre-AP concept and outliningthe characteristics of a successful Pre-AP program.

Released Exams: $20

($30 for “double” subjects: Calculus, Computer Science,

Latin, Physics) T

About every four years, on a staggered schedule, the AP Program releasesa complete copy of each exam. In addition to providing the multiple-choicequestions and answers, the publication describes the process of scoringthe free-response questions and includes examples of students’ actualresponses, the scoring standards, and commentary that explains why theresponses received the scores they did.

Packets of 10: $30. For each subject with a released exam, you can pur-chase a packet of 10 copies of that year’s exam for use in your classroom(e.g., to simulate an AP Exam administration).

Secondary School Guide to the AP Program: $10 A, T

This guide is a comprehensive consideration of the AP Program. It coverstopics such as developing or expanding an AP program; gaining faculty,administration, and community support; AP Grade Reports, their use andinterpretation; AP Scholar Awards; receiving college credit for AP; APteacher training resources; descriptions of successful AP programs in nineschools around the country; and “Voices of Experience,” a collection ofideas and tips from AP teachers and administrators.

Student Guides

(available for Calculus, English, and U.S. History): $12 SP

These are course and exam preparation manuals designed for high schoolstudents who are thinking about or taking a specific AP course. Eachguide answers questions about the AP course and exam, suggests helpfulstudy resources and test-taking strategies, provides sample questions withanswers, and discusses how the free-response questions are scored.

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Teacher’s Guides: $12 T

For those about to teach an AP course for the first time, or for experi-enced AP teachers who would like to get some fresh ideas for the class-room, the Teacher’s Guide is an excellent resource. Each Teacher’s Guidecontains syllabi developed by high school teachers currently teaching the AP course and college faculty who teach the equivalent course at colleges and universities. Along with detailed course outlines and inno-vative teaching tips, you’ll also find extensive lists of recommended teaching resources.

AP Vertical Team Guides T, A

An AP Vertical Team (APVT) is made up of teachers from different gradelevels who work together to develop and implement a sequential curricu-lum in a given discipline. The team’s goal is to help students acquire theskills necessary for success in AP. To help teachers and administratorswho are interested in establishing an APVT at their school, the CollegeBoard has published three guides: AP Vertical Teams in Science,

Social Studies, Foreign Language, Studio Art, and Music Theory: An

Introduction ($12); A Guide for Advanced Placement English Vertical

Teams ($10); and Advanced Placement Program Mathematics Vertical

Teams Toolkit ($35). A discussion of the English Vertical Teams guide, andthe APVT concept, is also available on a 15-minute VHS videotape ($10).

Multimedia

EssayPrep® SP, T

EssayPrep is available through the AP subject pages of the College Board’sWeb site. Students can select an essay topic, type a response, and get anevaluation from an experienced reader. The service is offered for the free-response portions of the AP Biology, English Language and Composition,English Literature and Composition, and U.S. History Exams. The fee is$15 per response for each evaluation. SAT® II: Writing Subject Test topicsare also offered for a fee of $10. Multiple evaluations can be purchased ata 10–20% discount.



APCD®: $49 (home version),

$450 (multi-network site license) SP, T

These CD-ROMs are available for Calculus AB, English Language, EnglishLiterature, European History, Spanish Language, and U.S. History. Theyeach include actual AP Exams, interactive tutorials, and other featuresincluding exam descriptions, answers to frequently asked questions, study-skill suggestions, and test-taking strategies. There is also a listing ofresources for further study and a planner to help students schedule andorganize their study time.

Videoconference Tapes: $15 SP, T, C

AP has conducted live, interactive videoconferences for various subjects,enabling AP teachers and students to talk directly with the DevelopmentCommittees that design and develop the AP courses and exams. Tapes of these events are available in VHS format and are approximately 90 minutes long.

AP: Pathway to Success

(video — available in English and Spanish): $15 SP, T, A, C

This 25-minute video takes a look at the AP Program through the eyes ofpeople who know AP: students, parents, teachers, and college admissionstaff. They answer such questions as: “Why do it?” “Who teaches APcourses?” and “Is AP for you?” College students discuss the advantagesthey gained through taking AP courses, such as academic self-confidence,improved writing skills, and college credit. AP teachers explain what thechallenge of teaching AP courses means to them and their school, andadmission staff explain how they view students who have stretched them-selves by taking AP Exams. There is also a discussion of the impact thatan AP program has on an entire school and its community, and a look atresources available to assist AP teachers, such as regional workshops,teacher conferences, and summer institutes.


This Course Description is intended for use by AP® teachers for course and exam preparation in the classroom; permission for any other use must be soughtfrom the Program. Teachers may reproduce it, in whole or in part, in limitedquantities, for face-to-face teaching purposes but may not mass distribute thematerials, electronically or otherwise. This Course Description and any copiesmade of it may not be resold, and the copyright notices must be retained as theyappear here. This permission does not apply to any third-party copyrights contained herein.

The College Board is a national nonprofit membership association dedicated to preparing, inspiring, and connecting students to college and opportunity.Founded in 1900, the association is composed of more than 3,900 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 22,000 high schools, and 3,500 colleges, through major programs and services in college admission, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, thePSAT/NMSQT™, the Advanced Placement Program® (AP®), and Pacesetter®. The College Board is committed to the principles of equity and excellence, and that commitment is embodied in all of its programs, services, activities, and concerns.

Copyright © 2001 by College Entrance Examination Board. All rights reserved.College Board, Advanced Placement Program, AP, APCD, EssayPrep, and the acorn logo are registered trademarks of the College Entrance ExaminationBoard. AP Vertical Teams, APIEL, and Pre-AP are trademarks owned by theCollege Entrance Examination Board. Other products and services may betrademarks of their respective owners.

Visit College Board on the Web: www.collegeboard.com/ap.

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C A L C U L U SCalculus AB, Calculus BC

Course Description

M A Y 2 0 0 2 , M A Y 2 0 0 3

2002 Exam Date: Tuesday, May 7, morning session2003 Exam Date: Tuesday, May 6, morning session

2000-01 Development Committee and Chief Faculty Consultant

Thomas P. Dick, Oregon State University, Corvallis, ChairStella R. Ashford, Southern University, Baton Rouge, Louisiana

David M. Bressoud, Macalester College, St. Paul, Minnesota

Benjamin G. Klein, Davidson College, North Carolina

María Pérez Randle, Bishop Kenny High School, Jacksonville, Florida

Nancy W. Stephenson, Clements High School, Sugar Land, Texas

Michael R. White, Pennridge High School, Perkasie, Pennsylvania

Chief Faculty Consultant: Lawrence H. Riddle, Agnes Scott College, Decatur, Georgia

ETS Consultants: Gloria S. Dion, Chancey O. Jones, Craig L. Wright

www.collegeboard.com/ap

I.N. 990185

ap calculus ab-bc course description, may 2002 - may 2003 (sample exams)

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