2003calculusab-bcexams

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For fuz-tiler infonnatioll, visit apcentral.collegeboEtJ:d.c01n.

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The 2003 AP@ Calculus AB andAP Calculus BC Released Exams

• Multiple-Choice Questions, Answer Keys, and Diagnostic Guides

• Free-Response Questi.ons with:

• Scoring Commentary

• Scoring Guidelines

• Sample Student Responses

• Statistical Information About Student Performance on the 2003 Exams

Materials included in this Released Exam may not reflect the current AP Course Description and exam in

this subject, and teachers are advised to take this into account as they use these materials to supporttheir instruction of students. For up-to-date information about this AP course and exam, please downloadthe official AP Course Description from the AP Central"" Web site at apcentral.collegeboard.com.

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The College Board intends this publication for noncommercial use by teachers forcourse and exam preparation; permission for any other use must be sought from theCollege Board. Teachers may reproduce this publication, in whole or in part, in limitedprint quantities for noncommercial, face-to-face teaching purposes and distribute upto 50 print copies from a teacher to a class of middle or high school students, with eachstudent receiving no more than one copy.

This permission does not apply to any third-party copyrights contained within thispublication.

When educators reproduce this publication for noncommercial, face-to-face teachingpurposes, the following source line must be included:

2003 AP@ Calculus AB and AP Calculus BC Released Exams. Copyright © 2005by College Entrance Examination Board. Reprinted with permission. All rightsreserved. www.collegeboard.com.This material may not be mass distributedt

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Chapter I: The Ap® Process• "Vhat Is the Purpose of the AP® Calculus AB and

Calculus BC Exams?

• Who Develops the Exams?

• How Are the Exams Developed?• Section I-Multiple Choice• Section II-Free Response

• Question Types• Multiple Choice• Free Response

• Scoring the Exams• "\Tho Scores the AP Calculus AB and

Calculus BC Exams?

• Ensuring Accuracy• How the Scoring Guidelines Are Created• Training Readers to Apply the Scoring Guidelines• Maintaining the Scoring Guidelines

• Preparing Students for the Exams

This chapter will give you a brief overview of thedevelopment and scoring processes for the AP Calculus ABand Calculus BC Exams. You can find more detailedinformation at AP Central® (apcentral.collegeboard.com).

What Is the Purpose of the APCalculus AB and Calculus Be Exams?The AP Calculus AB and Calculus BC Exams are designedto assess how well a student has mastered the concepts andtechniques of calculus. The Calculus AB exam covers topicstypically included in about two-thirds of a yearlong collegelevel calculus sequence; the Calculus BC exam covers topicsincluded in a full-year. college-level calculus sequence. Themultiple-choice section of the exams is designed to testproficiency in a wide variety of topics. The free-responsesection requires students to demonstrate the ability to solveproblems involving a more extended chain of reasoning.Both Calculus AB and Calculus BC require a similar depthof understanding of common topics. and graphingcalculator use is an integral part of the courses. Both themultiple-choice section and the free-response section of theexams contain parts where a graphing calculator is requiredand parts where calculator use is prohibited. Qualifyinggrades on the AP Calculus AB Exam can allow studentsto begin their college careers with credit for a Calculus 1course. Qualifying grades on the AP Calculus BC Exam canallow students to begin their college careers with credit for afull year of calculus (Calculus I and Calculus 2). Students in

both cases also have the opportunity to register for coursesfor which calculus is a prerequisite.

Who Develops the Exams?The AP Calculus Development Committee. working withmathematics Assessment Specialists at ETS. develops theexams. This Committee is appointed by the College Boardand is composed of six teachers from secondary schools.colleges. and universities in the United States. The membersprovide different perspectives: high school teachers offervaluable advice regarding realistic expectations whenmatters of content coverage. skills required. levels ofsophistication. and clarity of phrasing are addressed.College and university faculty members ensure that thequestions are at the appropriate level of difficulty forstudents planning to continue their studies at colleges anduniversities. Both high school teachers and college facultybring technology expertise to the Committee. Committeemembers typically serve for four years.

The Chief Reader also aids in the development process.The Chief Reader attends every Committee meeting toensure that the free-response questions selected for theexams can be scored reliably. The expertise of the ChiefReader and the Committee members who have scoredexams in past y~ars is notable: they bring to bear theirvaluable experience from past AP Readings and suggestchanges to improve the quality and the performance of thequestions. In addition. the ETS mathematics AssessmentSpecialists offer their advice and guidance.

How Are the Exams Developed?The Development Committee sets the specifications for theexams. determining what will be tested and how it will betested. It also determines the appropriate level of difficultyfor the exams. based on its understanding of the level ofcompetence required for introductory calculus courses incolleges and universities. Each AP Calculus AB and CalculusBC Exam is the result of several stages of development thattogether span two or more years.

Section I-Multiple Choice

1. Development Committee members and outside itemwriters write and submit multiple-choice questionsdirected to the topic outlines in the Course Descriptionfor AP Calculus: Calculus AB. Calculus Be.

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2. ETS mathematics Assessment Specialists performpreliminary reviews to ensure that the multiple-choicequestions are worded clearly and concisely.

3. At the Committee meetings, which are held three timesa year, the Committee members review. revise, andapprove the draft questions for use on future exams.They determine whether or not a question is appropriatefor testing a student's understanding of the concepts,methods, and applications of calculus. They make surethat the questions are clear and unambiguous. that eachquestion has only one correct answer, and that thedifficulty level of the questions is appropriate. Forquestions on the calculator-required portion of theexams, care is taken to ensure fairness regardless of thegraphing calculator used.

4. From the pool of approved questions, ETS mathematicsAssessment Specialists assemble draft exams accordingto the specifications set by the Development Committee.Many of the multiple-choice questions are pretested inhigh school or college calculus classes to gather dataregarding the difficulty of the questions.

5. The Committee thoroughly reviews the draft exams invarious stages of their development, revising the individual questions and the mix of questions until it issatisfied with the result.

The Committee controls the difficulty level of the multiplechoice section by selecting a wide range of questions,subsets of which have been used in an earlier form ofthe exam.

Section II-Free Response

1. Well in advance of the administration of the exams, themembers of the Development Committee write freeresponse questions for the exams based on the examspecifications. Appropriate combinations of questionsare assembled into draft free-response sections for theexams at a Committee meeting. Questions may also beselected from a free-response pool of questions that hasbeen developed over the years by the DevelopmentCommittee members.

2. The Committee reviews and revises these questions at allstages of the development of the exams to ensure thatthey are of the highest possible quality. Most of thetime at Committee meetings is devoted to work on thefree-response sections. The Committee considers, forexample, whether the questions will offer an appropriatelevel of difficulty and whether they will elicit answers

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that allow Readers to discriminate among the responsesalong an analytic scoring scale of 0 to 9 points. An idealquestion enables the stronger students to demonstratetheir accomplishments while revealing the limitations ofless proficient students. Each free-response question,however, is designed to have a part that is accessible forthe student who is prepared to take an AP Calculus Exam.

Question TypesThe 2003 AP Calculus AB Exam and Calculus BC Exameach contain a lOS-minute multiple-choice sectionconsisting of45 questions-Part A: 55 minutes, 28 questions,no calculator allowed; Part B: 50 minutes, 17 questions,graphing calculator required-and a 90-minute free-responsesection consisting of 6 questions-Part A: 45 minutes,3 questions, graphing calculator required; Part B: 45 minutes,3 questions, no calculator allowed. During the second timedportion for Part B, students are permitted to continueworking on the questions in Part A without the use of acalculator. The two sections are designed to complementeach other and to measure a wide range of calculusconcepts and skills.

Multiple-choice questions are useful for measuring astudent's level of competence in a variety of contexts. Inaddition, they have three other strengths:

1. They are highly reliable. Reliability. or the likelihood thatstudents of similar ability levels taking a different formof the exam will receive the same scores, is controlledmore effectively with multiple-choice questions thanwith free-response questions.

2. They allow the Development Committee to include aselection of questions at various levels of difficulty,thereby ensuring that the measurement of differences instudents' achievement is optimized. For AP Exams, themost important distinctions are between studentsearning grades of 2 and 3, and 3 and 4. These distinctions are usually best accomplished by using manyquestions of middle difficulty.

3. They allow comparison of the ability level of the currentstudents with those from another year. A number ofquestions from an earlier exam are included in thecurrent one, thereby allowing comparisons to be madebetween the scores of the earlier group of students andthose of the current group. This information, along withother data, is used by the Chief Reader to establish APgrades that reflect the competence demanded by theAdvanced Placement Program®, and that can be legitimately compared with grades from earlier years.

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Free-response questions on the AP Calculus AB andCalculus BC Exams require students to solve problemsinvolving a more extended chain of reasoning. The formatallows students to use their analytical, reasoning, andwriting skills to solve problems and present cogent answersand explanations in writing their responses. Students maybe required to explain a particular concept or their methodsfor solving a problem, interpret a problem in context, justifytheir answers mathematically, or examine the reasonableness of their solutions. The two-part format for the freeresponse sections provides greater flexibility in the types ofquestions that can be given while ensuring fairness to allstudents taking the exams, regardless of the graphingcalculator used. The free-response format allows for thepresentation of uncommon yet correct responses .andpermits students to demonstrate their mastery of calculusby a show of creativity.

Free-response and multiple-choice questions areanalyzed both individually and collectively after eachadministration, and the conclusions are used to improvethe following year's exams.

Scoring the ExamsWho Scores the AP Calculus AB and CalculusBCExams?

The multiple-choice answer sheets are machine scored. Theteachers who score the free-response section of the APCalculus Exams are known as "Readers." The majority ofthese Readers are experienced faculty members who teachcalculus at a college, university, or high school in the UnitedStates or Canada, with a few coming from schools abroad.Great care is taken to obtain a broad and balanced group ofReaders. Among the factors considered before appointingsomeone to the role are school location and setting (urban,rural, etc.), type of institution (public, private), and years ofteaching experience. Every effort is made to obtain Readersof both genders and varied ethnicity. University and highschool calculus teachers in the United States and Canadawho are interested in applying to be a Reader at a futureAP Reading can complete and submit an online applicationvia AP Central (apcentral.collegeboard.com/readers), orrequest more information bye-mailing [email protected].

In June of 2003,626 calculus teachers and mathematicsprofessors gathered at Colorado State University in FortCollins, Colorado, to participate in the scoring session forthe AP Calculus Exams. Some of the most experiencedmembers of this group were asked to serve as Exam Leaders,Question Leaders, and Table Leaders, and they arrived at theReading early to help prepare for the scoring session. Theremaining Readers were divided into 42 groups of 12-16,

with each group advised and supervised by two TableLeaders. Under the guidance of the Chief Reader and ChiefReader Designate, the 4 Exam Leaders, 9 Question Leaders(who also served as Table Leaders), and 75 Table Leadersassisted in establishing scoring guidelines, selecting samplestudent responses that exemplified the guidelines, andpreparing for the training of the Readers. All of the freeresponse questions on the 2003 AP Calculus AB andCalculus BC Exams were evaluated by the Readers at thissingle, central scoring session under the supervision of theChief Reader and Chief Reader Designate.

Ensuring Accuracy

The scoring process is designed so that all Readers assignscores using the scoring guidelines in a consistent and fairmanner. The creation of detailed scoring guidelines, thethorough training of all Readers, and the various "checksand balances" applied throughout the AP Reading allcontribute to achieving this goal.

How the Scoring Guidelines Are Created

1. As the questions are being developed and reviewedbefore the Reading, the Development Committee andthe Chief Reader discuss the scoring of the free-responsequestions to ensure that they can be scored validly andreliably. The Committee provides preliminary guidanceregarding the philosophy to be used in scoring thevarious free-response questions. The Chief Readerproduces a rough draft of the scoring guidelines for eachfree-response question for Committee input.

2. During the pre-Reading period, several important tasksare completed:• First, the Chief Reader produces a draft of the scoring

guideline for each free-response problem. ExamLeaders and Question Leaders receive copies of thescoring guidelines and a set of actual studentresponses selected by ETS Assessment Specialists. TheExam Leaders and Question Leaders come to theReading site prepared to discuss revisions to thescoring guidelines based on the review of studentresponses.

• The Chief Reader, Exam Leaders, Question Leaders,Table Leaders, and ETS Assessment Specialists meet atthe Reading site to discuss, review, and revise the draftscoring guidelines. The scoring guidelines are testedby applying them to actual student responses to thequestions, and then revised and adjusted, if necessary,to reflect not only the Committee's original intent butalso the full range of actual responses that will beencountered by the Readers.

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3. Once the scoring of student responses begins. nochanges are made to the scoring guidelines. Given theexpertise of the Chief Reader and the analysis of manystudent responses by Exam Leaders. Question Leaders.and Table Leaders in the pre-Reading period. theseguidelines can be used to cover the whole range ofstudent responses. During the first day of the scoring ofa particular question. Exam Leaders. Question Leaders.and Table Leaders ensure that everyone evaluatingresponses for that question understands the scoringguideline and can apply it reliably and fairly.

Training Readers to Applythe Scoring Guidelines

Because Reader training is so vital in ensuring that studentsreceive an AP grade that accurately reflects their performance. the process is thorough:

1. On the first day of the Reading, the Chief Reader provides an overview of the exams and the scoring processto the entire group of Readers. At the Calculus Reading.Readers typically score four different free-responsequestions during the course of the Reading. These areusually two questions that are common to the CalculusAB and Calculus BC exams, plus two other questionsfrom either the Calculus AB exam or the Calculus BCexam. All Readers typically start with one of thecommon questions and receive training for scoringthat question.

2. Each Question Leader directs a discussion of the question for which he or she is responsible, commenting onthe requirements of the question and the expectationsfor student performance. The scoring guideline for theproblem is explained and discussed.

3. The Readers are trained to apply the scoring guidelinesby reading and evaluating samples of student responsesthat were selected at the pre-Reading session as clearexamples of the various score points and as the kinds ofresponses that Readers are likely to encounter. Questionand Table Leaders explain why the responses receivedparticular scores and discuss the issues encountered invarious sample responses.

4. Once the Readers understand the scoring guidelines andcan apply them uniformly, the scoring of studentresponses begins. Experienced Readers are paired as"Table Partners" with less experienced Readers.

5. Table Leaders evaluate the responses scored by eachReader early in the process to help ensure that Readersare applying the scoring guidelines correctly.

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6. Throughout the course of the Reading, Readers candiscuss with their Table Partner or Table Leader anystudent response that presents a scoring difficulty. Whennecessary. the Chief Reader. the Question Leaders. or theExam Leaders are consulted. A student response that isproblematic receives multiple readings and evaluations.

Maintaining the Scoring Guidelines

Each Reader scores a single question. not an entire exam.Steps are taken to prevent the so-called halo effect-giving aresponse a higher or lower score than it deserves because ofhow well or poorly the student performed on otherquestions on the exam:

• Each question is read by a different Reader.

• No marks of any kind are made on the students' papers.The Readers record the scores on a scannable form,which is identified only by the student's AP number.Readers are unable to see the scores given to otherresponses in the exam booklet.

• The student's identification information is concealed.Thus. each Reader can evaluate student responseswithout being prejudiced by knowledge about individualstudents. .

Here are some other methods that help ensure that Readersare adhering closely to the scoring guidelines:

• Table Leaders discuss prescored responses with Readerseach morning and during the day as necessary.

• Readers are paired; every Reader has a partner to checkconsistency and to discuss problem cases. Table Leadersare also paired to help each other on questionable calls.

• Table Leaders backread (reread) a portion of the studentpapers from each of the Readers in that Leader's group.This approach allows Leaders to guide their Readerstoward appropriate and consistent interpretations of thescoring guidelines.

• The Chief Reader and the Exam Leaders can monitoruse of the full range of the scoring scale for the groupand for each Reader by checking daily graphs of scoredistributions.

• Reliability data are periodically collected by havingReaders unknowingly rescore booklets and comparingthe scores of the first and second readings.

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Preparing Students for the ExamsThe AP Calculus courses (Calculus AB and Calculus BC)consist of a full high school academic year of work that iscomparable to calculus courses in colleges and universities.The Calculus AB and Calculus BC courses have significantoverlap, with Calculus BC including additional topicsin differential and integral calculus plus the topics ofpolynomial approximations and series. Calculus BCis an extension of Calculus AB rather than an enhancement; common topics require a similar depth of understanding. Both courses are intended to be challengingand demanding.

The Course Description for AP Calculus: Calculus AB,Calculus BC includes a philosophy statement and goals forthe AP Calculus courses and mathematical prerequisites forstudents studying calculus. As teachers focus on these goals,not only will they be preparing their students for the APCalculus Exams; they will be preparing them for futurestudy and applications of mathematics. It is assumed thatmost students preparing to take an AP Calculus Exam havecompleted a course in calculus that included instruction inthe content areas outlined in the Course Description for APCalculus: Calculus AB, Calculus Be. Students who take anAP Calculus Exam are expected to demonstrate competencein calculus concepts and techniques.

Because the AP Calculus courses emphasize a multirepresentational approach, students should be able to workwith functions represented in a variety of ways: graphical,numerical, analytical, or verbal, and understand theconnections between them. Students need to practicemathematical writing skills to help communicate theirreasoning and explanations in the free-response portions ofthe exams. Students should have experience with justifyingconclusions using calculus arguments. The exams havemoved away from rote manipulation and towardquestions that probe understanding of fundamentalcalculus concepts.

Students should be comfortable with the use of a graphing calculator, particularly the four required capabilitiesgraphing a function within an arbitrary viewing window,finding zeros of functions (solving equations numerically),calculating derivatives numerically, and calculating definiteintegrals numerically. Because the exams have noncalculator parts, students are expected to know how tocompute derivatives and antiderivatives of basic functions"by hand."

To do their best on the exams, students with the requisitecalculus skills should become familiar with the format,timing, and particularly the directions for the free-responsesection of the exams so that they will know what to expectwhen they take an AP Calculus Exam.

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Chapter II: The 2003 AP CalculusAB and Calculus BC Exams• Exam Content and Format

• Giving a Practice Exam

• Instructions for Administering the Exam

• Blank Answer Sheet

• The Exam

Exam Content and FormatThe 2003 AP Calculus AB and Calculus BC Exams areeach three hours and fifteen minutes in length. There aretwo sections:

• A lOS-minute multiple-choice section consisting of45 questions, accounting for 50 percent of the finalgrade. Part A consists of28 questions in 55 minutes.

Students are not allowed to use calculators for this part.Part B consists of 17 questions in 50 minutes. Studentsare required to have a graphing calculator for this partbecause some of the questions require the use of acalculator.

• A 90-minute free-response section consisting of6 questions, accounting for 50 percent of the final grade.Part A consists of 3 questions in 45 minutes. Students arerequired to have a graphing calculator for this partbecause some of the problems require the use of acalculator. Part Bconsists of 3 questions in 45 minutes.Students are not allowed to use calculators for this part.During the administration of Part B, students can workon the questions in Part A without the use of a calculator.

.. ,. " " ,. " " " - .

PartS

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No calculator allowed.

Graphing c31culator required.

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Multiple ChOice (Section I) .

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2003 AP :C~Cu1usAB and,Calculus Be :Ei:ams Format' ,: ~::;'; ,"">':<,:, ' '/::, , ,'; ::f:)('< -.<?'":\

3 questiO~c~?~'" .....•••N~~;';;~~~.. ;:!"~i1;~~~r 45 minutes . ,,:, ,.

-'D~g the timed Porti~ri:for'Nrt B, studen~ ~a~~~rk on'.. '. ," Part A questions without the use of a calculator.'

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Giving a Practice ExamThe following pages contain the instructions as theyappeared in the 2003 AP Coordinator's Manual foradministering the AP Calculus AB and Calculus BC Exams.Following these instructions are a blank 2003 answer sheetand the 2003 AP Calculus AB and Calculus BC Exams. Ifyou plan to use these released exams to test your students,you may wish to use the instructions to create an examsituation that closely resembles an actual administration. Ifso, read only the directions in the boxes to the students; allother instructions are for the person administering theexam and need not be read aloud. Some instructions, suchas those referring to the date, the time, page numbers, andsurvey questions, are no longer relevant and should beignored.

Another publication that you might find useful is thePacket of ID-ten copies of the 2003 AP Calculus AB andCalculus BC Exams, each with a blank answer sheet. Forordering information, see the Appendix.

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Instructions for Administering the Exam(from the 2003 AP Coordinator's Manual)

IMPORTANTFor the regularly scheduled administrations,read ALL of the boxed instructions belowexcept for the boxes marked for the latetesting administrations that use an alternateform of the exam.

For the late testing administrations, readALL of the boxed instructions below exceptfor those marked specifically for the May 8thadministrations.

Graphing calculators are required to answersome of the questions on the Calculusexaminations. Before starting the examadministration, make sure each student has agraphing calculator from the approved list onpage 42 of this manual. If a student does nothave a graphing calculator from the approvedlist, you may supply one. If a student does notwant to use a calculator, have him or her write,date, and sign the release statementon page 44.During the administration of Section I, Part Band Section II. Part A only, students may haveone or two graphing calculators on their desks.Calculatormemories do not needto be clearedbefore oraftertheexam. Students with HewlettPackard 48 Series graphing calculators mayuse cards designed for use with thesecalculators. In addition, proctors should ensurethat calculator infrared ports (Hewlett Pa~kard)are not facing each other.

The Calculus AB exam and the Calculus BCexam should be administered simUltaneously.

The administration of these exams includes four surveyquestions. The time allowed for these survey questions is inaddition to the actual test-taking time.

To help you and your proctors make sure students areworking on the correct parts during Section I of the exam,the parts are identified as follows: Part A has a line ofA'sacross the top ofeach page; Pan B has a line oflarge B's inan alternating shaded pattern across the top of each page.

e Do not begin the exam instructions until you havecompleted the appropriate general instructions foryour group. •

When you have completed the general instructions, say: '.

It is Thursday afternoon, May 22, and you will betaking one of the AP Calculus Examinations.Make sure you have the correct examination:Calculus AB or Calculus BC. If you do not havethe correct exam, please raise your hand••••

Print your full name, last name first, on tbefront cover of the Section I booklet and read thedirections on the back cover. When you havefinished, look up••••

Section I is the multiple-choice portion of theexam. It is divided into two parts, PartA and PartB. Each part is timed separately and you maywork on each part only during the time allottedfor it. Part A questions are numbered 1 through28. You will mark your responses for thesequestions on page 2 of your answer sheet. Part Bquestions are numbered 76 through 92. You willmark your responses for PartB questions on page3 of your answer sheet.

Before we begin, I'd like to point out that thereare more answer ovals on your answer sheet thanthere are questions. When you reach the end ofeach part, there will be unused answer ovals.Scratch paper is not allowed, but you may use themargins or any blank space in the Section Ibooklet for scratch work.

Part B of Section I and Part A of Section IIrequire a graphing calculator. For Part A ofSection I, no calculators are allowed. Please placeall of your calculators under your chair. • • •Are there any questions?

Answer all questions regarding procedure.When you are ready to begin the exam, note the time here

____'. Then say:

AT THE MAY 8TH, ADMINISTRATIONS ONLY, SAY:

It is Thursday morning, May 8, and you will betaking one of the AP Calculus Examinations.Make sure you have the correct examination:Calculus AB or Calculus BC. Ifyou do not havethe correct exam, please raise your hand••••

Read the statements thatare on the front coverof your Section I booklet. •••

Are there any questions? •.••Now sign your name, rdl in today's date, and

print your full name, last name first, as indicated. •.•Now read the directions on the back cover.

When you have finished, look up••••

You have 55 minutes for PartA. Fold your answersheet so that only page2 is showing•••• Open yourSection I booklet to Part A and begin.

•

Allow 55 minutes. Note the time you will stop here____,. While the students are working on Part A, youand your proctors should make sure they are all markinganswers on page 2 of the answer sheet in pencil. and thatthey are not looking beyond Part A of the test. The line ofA's at the top ofeach page will assist you in monitoring thestudents' work.

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We're now going to have a short break before webegin Section II. You may get up, talk, go to therest room, or get a drink. Everything you placedunder your chair at the beginning of the exammust remain there. You are not allowed to consulttextbooks, teachers, or other students about theexam materials during the break. You may notmake phone calls, check e-mail, or access a PDA,calculator, or computer. You are not allowed todiscuss the multiple-choice section of this examwith anyoneat any time. Failure to adhere to theserules could result in invalidation of grades. Arethere any questions? •••

OK, let's begin our break. Testing will resume at:

Close your exam booklet. DO NOT put youranswer sheet inside the booklet. Put your answersheet on your desk, face up, with the fold to yourleft. I will now collect your answer sheets.

Take your Section I exam booklet and seal it withthe white seals you have set aside on your desk.Peel each seal from the backing sheet and press iton the front coverso itjustcovers the area marked"PLACE SEAL HERE." Fold it over the openedgeand press it to the back cover. Use oneseal foreach open edge. De careful not to let the sealstouch anything except the marked areas. Thenput your exam booklet on your desk with thecover face up and the fold to your left. •••

I will now coDed your Section I exam booklets. •••

Give students approximately two minutes 10 answcr thesurvey questions. Then say:

After you have collectcd an answer sheet from each student.say:

Answer all questions regarding procedure. Then say:

Collect the sealed Section I exam booklets. Be sure youreceive one from every student. Between Section I andSection II thcre is a scheduled IO-minute break. Duringthe break,calculators must remain turned ofTand mustNOT be taken from the exam room.

When you have collected and accounted for all Section1materials and are ready for the break, say:

You have 50 minutes for Part B. Now tum to PartB in your booklet and begin.

Stop working on PartA.You will now seal the PartA section of the exam booklet in the followingway: Make sure all other exam materials,including your answer sheet, are out of the wayand turn to page 20 [page 22 for the Calculus BClate testing administration] in your Section Ibooklet•••• You should see an area marked"PLACE SEAL HERE." Take one of the whiteseals you have set aside on your desk and place itso it just covers the area marked "PLACE SEALHERE." Fold it over the section of the bookletcontaining Part A only and press it to the areaindicated on the front cover. Be sure you do notseal the Part Bsection ofthe booklet,orlet the sealtouch anything except the marked areas••••

Graphing calculators may be used for Part B ofthis section. You may now place your calculatorson your desk.

Part B questions are numbered 76 through 92.Tum your answer sheet so that only page 3 isshowing and record your responses to thesequestions on that page.

Ifyou complete work on Part B before time iscalled you may check your work on Part B. YouCANNOT return to work on Part A.

After 55 minutes, say:

After alt students have properly sealed Part A of the exambooklet say:

•Stop working and tum ofT your calculator. I l- --J

------------------..... After the break, say:

•When you are ready to begin, note the time here

. . Then say:

•

Allow 50 minutes. Note the time you will stop here. . .While students are working on Part B, you and

your proctors should walk around and make sure studenlshave sealed theirbooklets properly and are now working onPart B. The large 8's in an alternating shaded pattern at thetop of each pagc will assist you in monitoring their work.

Calculators may not be shared. Communication betwccncalculators is prohibited during the exam. Proctors shouldwalk around and make sure Hewlett Packard calculators'infmred ports are not facing each other.

After 50 minutcs, say:

AT THE MAY 8TH

ADMINISTRATIONS ONLY, SAY:AT THE MAY 8TH

ADMINISTRATIONS ONLY, SAY:

10

Turn to page 34 in your exam booklet. ••• Answerthe questions numbered 93 through 96. These aresurvey questions and will not affect yourexamination grade. You may not go back at thistime to work on any of the previous questions.

Open the package containing your Section IIbooklet and read the statements on the frontcover. Do not break the seal on the blue insert.When you have finished, look up••••

Now tum to the back cover and read theinstructions at the upper left. ••• Using a pen with

2003 EXAM

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black or dark-blue ink, print your identificationinformation in the boxes and then sign your nameandwrite in today'sdate,agnleingtotheconditiomonthe front cover. •••

After you have finished, detach the perforationatthetopand fold theflap down. Moistenand pressthe glue strip firmly along the lower edge. Thiscoversyour identification informationso it will notbe known by those scoring your answers••••

Now read the instructions on the upper rightsideofthe back cover. ••• When you have rmished,put an AP number label in the indicated area. Ifyou run outofAP number labels, you may write inyour AP number where the label should go••••

Open the package containing your Section IIbooklet. Do not break the seal on the peach insert.Tum to the back coverof the booklet and read theinstructions at the upper left•••• Using a pen withblack or dark-blue ink, print your identificationinformation in the boxes•••• Now, taking care notto tear the sheet beneath the cover, detach theperforation at the top and fold the Rap down.Moisten and press the glue strip firmly along thelower edge. This. covers your identificationinformation so it will not be known by thosescoring your answers••••

Read the instructions at the upper right of theback cover••••

Now, as instructed, print your initials in the threeboxes to the left and put oneAP number label in thearea below the instructionsand one in the area to theleft. If you run out of AP number labels, you maywrite in yourAP number where the label should go. •••

Now read Item 5 [Item 6 for the late testingadministrations]. Unless you mark the box belowit, you grant Educational Testing Service and theCollege Board permission to use your freeresponse materials for educational research andinstructional purposes••••

Are there any questions? .


I need to collect Student Packs from anyone whowill be taking anotherAP Exam. Ifyou are takinganother AP Exam, please put your Student Packon your desk. You may keep it only ifyou are nottaking any more AP Exams this year.

Collect the Student Packs, then say:

Read the directions for Section II on the backofyourbooklet,payingcarefulattentiontothebulletedstatements. Look up when you have rmished. •••

AT THE MAY 8TH


The problems for Section II, PartA are printed inthe green insert. Problems for Section II, Part Bare in the blue insert.

You may use the green and blue inserts forscratch work, but be sure to show your work andwrite your answers to each part ofeach problemin the space provided for that part in the pinkSection II exam booklet. No credit will be givenfor work shown in the inserts.

The problems for Section II, PartA are printed inthe beige exam booklet. The problems for SectionII, Part B are in the peach insert. You may use thepeach insert for scratch work, but besure to showyour work and write youranswers to each part ofeach problem in the space provided for that partin the beigeSection II exam booklet. No credit willbe given for work shown in the insert.

Only No. 2 pencils may be used for Section II.Are there any questions?


Put the blue insert [peach for the late testingadministrations] aside. You may use a graphingcalculator for Part A of Section II only.

AT THE MAY 8TH


Tear out the green insert in the center of thebooklet and print your name, teacher, and schoolin the upper left-hand comer••••This insert willbe collected at the end of the administration andreturned toyou ata laterdate byyour teacher••••

•

When you are ready to begin the exam, note the time here. . Then say:

AT THE MAY 8TH


You have 45 minutes for this part of the exam.Write your answer to each part of each problemin the appropriate space in the pink Section IIexam booklet. You may open the green insert andbegin working on Section II, Part A.

11

2003 EXAM

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12

You have 45 minutes for this part of the exam.Write your answer to each part of each problemin the appropriate space in the beige Section IIexam booklet. You may open your exam bookletand begin working on Section II, Part A.

•

Allow 45 minutes. Note the time you will stop here. . While stiJdents are working on Part A.

. you and your proctors should make sure they arcwriting their answers on the appropriate pages of the pink[beige for the late testing administrations] Section II exambooklet. The pages for PartAare marked with large I's, 2's,and 3's at the top of each page to assist you in monitoringstudents' work.

You will not collect the green inserts until the end of theadministration. Students must keep the green inserts sothey may work on these problemsduring Part B without theuse of a calculator.

Af~er 45 minutes say:

AT THE MAY 8THI ADMINISTRATIONS ONLY, SAY:

Stop working on Part A. Because a graphingcalculator cannot be used for Part B of Section II,place your calculators under your chair. You maykeep the green insertand return to problems in PartAofSection II during this time, but you may not usea calculator. Print your name, teacher, and school inthe upper left-hand corner of the blue insert. Thisinsertwillbecollectedattheend oftheadministrationand returned toyouatalaterdatebyyourteacher. •••

StopworkingonPartA.Becauscagraphingcalculatorcannot be used for Part B of Section II, place yourcalculators under your chair. You may return toproblems in PartAofSection II during this time, butyou may not use a calculator.The peach insert will becollected at theend ofthe adJninmration and will notbe returned toyou ••••

•

When you are ready to begin, note the time here. . Then say:

Open the blue [peach for the late testingadministrations] insert,and begin work onSectionII, Part B. You have 45 minutes for this part oftheexam. Write your answer to each part of eachproblem in the appropriate place in the pink[beige for the late testing administrations] SectionII exam booklet. H you finish your work on PartB, you may return to the problems in Part A, butyou may not use a calculator. Remember that nocredit will be given for work shown in the insert.

•Allow 45 minutes. Note the time yo~ will stop here

. .While the students are workmg on Part B, youand your proctors should make sure they are writing theiranswers in the pink [beige for the late testingadministrations] Section II test booklets. Students shouldnot be using calculators for this portion of the exam.Students may return to the problems in PartA, but they maynot use calculators.

After 45 minutes, say:

AT THE MAY 8THADMINISTRATIONS ONLY, SAY:

Stop working and close your exam booklet. Put iton your desk, face up, with the fold to your left.Put your green and blue inserts next to it. •••

I will now collect your Section II booklets andthe green and blue inserts. Remain in your seats,without talking, while the exam materials arebeing collected••••

You should receive your grade reports by midJuly and grades will be available by phonebeginning July 1st.

Stop working and close your exam booklet. Put iton your desk, face up, with the fold to your left. Putyour peach insert next to it. •••

I will now collect your Section II booklets andthe peach inserts. Remain in your seats, withouttalking, while they are being collected. •••

You should receive your grade reports by midJulyand gradeswill beavailableby phonebeginningJuly 1st.

Collect the Section II booklets and the inserts. Be sure youhave a Section II booklet, a green insert, and a blue insert[peach insert only for the late testing administrations] fromevery student. Check the back ofeach Section II booklet tomake sure the student'sAP number appears in the box [twoboxes for the late testing administrations). The green andblue inserts must be stored securely for no less than 48hours (2 school days) after they are collected. After the 48hour holding time, the blue and green inserts may be givento the appropriate AP teacher(s) for return to the students.Peach inserts for alternate forms of the exam used for latetesting must be returned to ETS.

When all examination materials have been collected,students may be dismissed.

Separate the CalculusAB exam materials from those forCalculus BC. Fill in the necessary information for theCalculus examinations on the S&R form.

Alternate exams should be recorded on the pink ply ofthe S&R form. Put the exam materials in locked storageuntil they are returned to ETS after your school's lastadministration. Sec "ActivitiesAftertheExam"in this manual.

2003 EXAM

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P. STUDENT SEARCH SERVICE' OFTHE COLLEGE BOARD(ccmprete ON1.YIt JOU ant. BOPHOMOfiE onJUNIOR)

IEfSUSE !

ONLY Fj

@20036)2004@2005@2006

oNo

••

aYes

•

e> Fan8 WmterlSpring@ Summer9 Undecided

•

Report toTnchtNiShtdent Glide Roster

@ @@.~ @) @@@@)Fee Recluctlcln GnntocI(tee AP CocntkIlItet'f ManrmJfor d9ta1led 1rtB1rI:dIons)

oOptton 1 0 Optton 2 0 0JItf0n 3

.>,

a Yes, I want the College Board to send lnformalicn about me tocoOoges, universities, and govarnment scholarshlp programsInterested In students lileu me.

o No. I do not wan' the CoIlago Board to send fnfnrmalkm about meto colIogos, universities, and government scholarship programsthrough the Studen' Search 5ervic:e.

•

SCHOOLUSE

ONLY

.....

Ii HoSEX"

<D Male '0

@ Female if

•• ••

Essay ChoicesFill In the ovals that correspond to thenumbers of the essay questions youanswered In this oxamlnatlon.1 2 3 4 5 e 1 8 9 10 11 12

o 0 a 0 0 a 0 0 0 a 0 a

COll'fllghl 0 2003 CollOlJo EnlrollCO Examlnatlcn Board.All rtgtlto merYOd. College BonnI. Adv3ncccI PlacorncntPrognun. AP, Student Seard1 ScrvIco. and \he acorn logo

B10 roglstaIod~ ollho Co!Iogo En1roncoExominntIon BcxtRi.

• •

Prirttemmlnatlclnname~' _

RlIln the appropriate oval below fer examinatfon aama and !WIIlIm.07 0 U.S. History 53 0 Geography: Human13 0 Art History 55 0 German Language14 0 Art: Studio Drawing 57 0 Gov. & Pol.: U.S.15 0 Art: StudIo 2·0 Design 58 0 Gov. & Pol.: Camp.16 0 Art: StudIo 3·0 Design 60 0 latin: Vergll20 0 Biology 61 0 latin Utoraturo25 0 Chemistry 66 0 Calculus AS31 0 Computer Sclence A 68 0 CBlculus Be33 0 Computer ScIence AS 75 0 Music Theory34 0 Economics: MIcro 78 0 Physics B35 0 Economics: Macro 80 Physics C: Mach. a E & M 036 0 Eng. Language & Compo 85 0 Psychology37 0 Eng. L1torature & Compo 87 0 SpanIsh Language40 0 Environmental Science 89 0 Spanish Literature43 0 European History 90 0 Statistlcs48 0 French Langooge 93 0 Wortcl History51 <5 French Utoroture• • •

• •• •• •NAME AND EXAM AREA - COMPLETE THIS AREA AT EVERY EXAMINATION

A. S1GNATURE ITo~ I!lo~ fA !!to (IIlIm and !It W!ll:ly fA my AP•• I will a::low no ClIIll Olhar \tIllIl myI(!ll to SllO lhG1---------'~ quoslion$ andwlll5CnJ lho~llt Il!CllDn wtlen askud to do 10.1 will nat cl:salIs 1I10s0 Cl\IllSl!oIls

wi1h Il1YOftO III MI tiM aflCflho ~0lI0n 01 1M~ sediDn.1 am llwnrG of Mld "V'Cll1o lho Program"policies CIld Ilrocedunls as~ In 1M 2003 Bu1ImtIIbrAPSbtcnlJ tIfId P.rnln!s.

2003 EXAM

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Q. THIS SECTION IS FORTHE SURVEY QUESTIONS INTHE AP STUDENT PACK. (DO NOT PUT RESPONSES TOEXAM QUESTIONS IN THIS SECTION.) BE SURE EACH MARK IS DARK AND COMPu:t:ELY FILLSTHE OVAL

~a;lIege AP·:&~~~~lBoard nOG.... rl

1(3)([)@@(D<D@2(3)([)@@>(D<D@3(3)([)@@(D<D<ID

PAGE 2

4(3)@@@><D<D@5(3)@@@>CD<D@6(3)@@@>CD<D@

7(3)@@@><D<D@8(3)@@@>CD<D@9(!)@@@(D<D@

•••

DO NOT COMPLETETHIS SECTION UNLESS INSTRUCTEDTO DO SO.

R. If thiS answer sheet is for the French Language, French Uterature, German Language, Spanish Language. or SpanishUterature Examination. please answer the following questions. (Your responses will not affect your grade.)

1. Have you lived or studied for one month or more In a country where thelanguage of the exam you are now taking Is spoken?

2. Do you regularly speak or hear the language at home?

o Yes

o Yes

o No

o No

•

•

14

INDICATE YOUR ANSWERS TO THE EXAM QUESTIONS IN THIS SECTION. IF A QUESTION HAS ONLY FOUR ANSWEROPTIONS, DO NOT MARK OPTION (E). YOUR ANSWER SHEET WILL BE SCORED BY MACHINE. USE ONLY NO.2PENCILS TO MARK YOUR ANSWERS ON PAGES 2 AND 3 (ONE RESPONSE PER QUESTION). AFTER YOU HAVEDETERMINEDYOUR RESPONSE, BE SURETO COMPLEtELY FILL INTHE OVAL CORRESPONDINGTOTHE NUMBER OFTHE QUESTION YOU ARE ANSWERING. STRAY MARKS AND SMUDGES COULD BE READ AS ANSWERS. SO ERASECAREFULLY AND COMPLETELY. ANY IMPROPER GRIDDING MAY AFFECTYOUR GRADE.

1(3)([)@@(D 26(3)([)@@(D 51 (!) @ @ @ CD2(!)([)@@>(D 27 (!) ([) @ @> CD 52 (!) @ @ @ (D3(!)([)@@(D 28 (!) ([) @ @ (D 53(!)@@@(D4(3)([)@)@(D 29 (!) ([) @ @ (D 54(!)@@@CD5(3)@@@(D 30@@@@(D 55 (!)@@@>CD6(!)([)@@CD 31 (!) @ @ @ (D 56 (!) ([) @.@ •• (!)

7 (!) @ @ @.CD 32(!)@@@(D 57(!)([)@@CD8(!)C!)@@CD 33(!)C!)@@>CD 58 @ C!) @@CD9@C!)@@>CD 34@C!)@@>(D 59@C!)@@@

10 (!) C!) @ @ CD 35@([)@@CD 6O@C!)@@CD11 (!) ([) @ @> @ 36@@@@>(D 61 (!) @ @ @ (D12 (!) ([) @ @ CD 37 (!) ([) @ @ CD 62@@@@CD13 (!) <!> @ @ (D 38(!)([)@@(D 63@@@@>(D14 (3) <!> @ @ CD 39 (3) ([) @ @> (D 64@@@@>(D15 (3) @ @ @> CD 40 (!) <!> @ @> (D 65 (!) <!> @ @ @16 (!) ([) @ @> CD 41 @ <!> @ @ (D 66 (!) @ @@).CD17 @ ([) @ @> CD 42@@@@>CD 67@C!)@@(!)18 @ C!) @ @> (D 43@C!)@@CD 68 @ C!) @ .. @ ••..•• (!)19@@@@CD 44@C!)@@>(D 69 (!) C!) @@>@2O(!)C!)@@CD 45@C!)@@(!) 70@@@@(!)21 (!) <!> @ @ CD 46 (!) <!> @ @ (!) 71 (!) <!> @ @> (!)22@([)@@>(!) 47 (3) <!> @ @ CD 72@<!>@@(!)23(!)([)@@>(!) 48(!)@@@>CD 73 (!)@@@(D24 (!) ([) @ @> CD 49 (!) ([) @ @ (D 74 (!) @ @ @ (D25(!)@@@>(D 50(!)([)@@CD 75 (3)@@@CD

FOR QUESTIONS 76-151, SEE PAGE 3.

DO NOTWRITE INTHIS AREA.

en: ·:L..:.i n~0.; t"'ili.:' ~::-: e.g !t= :£:1 i7-2~i i.{:~.~ r?,-~~

f~ -r;;r.; 2::<': Ll:; 2i1 frill r;m-·ss· l.?:.:.L: 11:::1 ~;OCj :::r::;

••

•••

••••

2003 EXAM

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BE SURE EACH MARK IS DARK AND COMPLETELY F1LLSTHE OVAL IF A QUESTION HAS ONLY FOURANSWER OPTIONS, DO NOT MARK OPTION E.

76 @@@@CD 101 @ ® @ @ CD 126 @ @ @ @ CDn@@@@CD 102 @ @ @ @ CD 127 @ @ @ @ CD78 @@@@CD 103 (]) @ @ @ CD 128 @ @ @ @ CD79@@@@CD 104 (]) ® @ @ CD 129 @ @ @ @ CD80(])@@@CD 105 @ ® @ @ CD 130 @ @ @ @ CD81 @ @@ @ CD 106 @@ @ @ CD 131 @ @ @) @ CD82 @@@ @ <D 107 @® @> @ (D 132 (]) @ @)(§)<CD83@@@@<D 108 (]) ® @> @ <D 133 (]) @ @)@ ®84@@@@)<D 109 @ ® @> @ CD 134 (]) @ @ @) CD85(])@@@(D 110 @ ® @> @ CD 135 <!> @ @>@ <D86@@@@)CD 111 @ @ @) @ CD 136 (]) @ @> @ (D87 @@@@CD 112 @ @ @ @) CD 137 (]) @ @) @ <D88 @@@@CD 113 @ @ @ @ <D 138 (]) @ @> @ CD89 @@@@CD 114 @ ® @) @ CD 139 (]) ® @> @ <D9O@@@@CD 115 @ ® @> @ CD 140 (]) @ @) @ CD91 @ @ @ @) CD 116 (]) ® @> @ <D 141 (]) @ @) @) ®92@@@@)CD 117 @ @ @> @) CD 142 (]) @ @> @ ®93@@@@)CD 118 (]) @ @) @) CD 143 (]) @ @> @ CD94 @ @@ @ CD 119 @@ @> @ CD 144 @ @ @>@®95 @ @<@ @CD 120 @ @ @> @ CD 145 0 @ @) @eD96 @ @ @ @) CD 121 (]) ® @> @ CD 146 (]) @ @> @ ®97 @@@@CD 122 @ @ @> @ CD 147 (]) @ @> @ ®98 @@@@CD 123 @ @ @> @ CD 148 (]) @ @> @ CD99 @@@@)CD 124 @ @ @> @ CD 149 @ @ @> @ ®

100 @ @ @ @ <D 125 @ @ @> @ CD 150 0 @ @> @ CD151 (]) @ @> @ CD

•

•

•

••

•

~1!l8 APeADVANCED

~ college PLACEMENTBoard ,.DCu.·PAGE 3

MULTlPLE·CHOICE BOOKLETSERIAL NUMBER

••

•

ETS USE ONLY

R W FS

PT1

PT2

PT3

PT4

TOT

EQ

TA1

TA2

DO NOTWRlTE IN THIS AREA.t1":t L:·:.;': G~:~: h1tJ~; L:~ ..' r;~J !WTI r:':J EiJ

•L= ~:s m:: L:..:; ~.::-_-~

'"""=~ ~<:>", l.. ,t~.·:A.~ "-<..>. >. ~ •• ~~; ~:. ~'.. ~ : ~,'.~1

15

2003 EXAM

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~

Q)

• • • • • • • • • • ••S. YOUR MAfUHG ADDRESS I ADDRE$S AND ScHOOLAREA~ COMPLETETHIS AREA ONLYQNCE. PAGE 4

• 'IIlUA lIlWlE REPOIlTWlLL BE • USING THE _TlOHS OIVENIN~ N> ST1IllENT IW:l<. PRINT ADDllESS • INIllCA1E A SlW:EIN~_89 BY l.EAVINO A Ill.ANJ( BOX.IIAI1S)TO rnt9 AllllRESS INMY. lHTO lIOnS PRO\I1llEIl. 1F'IIlUA_89 DOl:S NOT m, SEE ITEU U. BELOW. ClllLY ONE OYAL ISTO BE FlLLED IN FOR EACH COUBIN.

Slrool t;llY STATE ZIP OR POSTAL CODE T.

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ICOUtmri

'0 AL a 0 NE I I I I I I I coca : '.'i'

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--L(j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) 2lOMO "OWA (j) (j) (j) (j) (j) (j) (j) (j) (j)9 9 9 @ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 @ 9 9 9 9 9 9 9 9 9 99 9 nOMA (DOWV 999 9 @ 9 @ 9 9 @@

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@@@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ v. @ @ @ @ @ @ @ @ @ o ~(j) (j) (j) @ (j) @ (j) (j) 0 @ @ 6) @ @ @ @ (j) @ @ @ @ (j) @ @ @ @ @ @@ @ @ @ @ @ @ @ TELEPHONE @ @ 0 6) (j) (j) @ 6) @ @ @@ @ @ @ @ , @ @ @@ @, @ @ @ 8 @ @ @ @ @ @ 0 @ @ @ T @ @ 0 (j) 6) 0 (j) 6) @ @ @ @ @ @ @ @@ @ @ !..!

~NII your international

@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ telephone number Is @ @ @ @ @ @ @ @ @•• @@ @@ @@ @ @ @@ @@ @@ @ @ @ @@@ @@ @ @ @@ EU @ @ @ @ @ @ @ @ @ @ longer lIlan 10 digits, @@ @ @ @ @@ @ @

@ @ @@ @ @ @ @ @ @ @ @ @ @ @ @ @@ @@ @@@ @ @ @ pM @@@ @@ @@ @ @ @ write In tho enllre @@@ @ @ @@@ @@@@ @ @ @ @ @@I @@ @ @ @ @ @ @@I@ @ @ @@ @ HB @ @ @ @ @ @ @ @ @ @ number below. @@ @ @ @ @ @ @@ ..,:

o 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 000 0 0 000 OE @ @ @ @ @ @ @ @ @ @ 00 0 0 0 o 0 0 0@I@ @ @ @@ @ @@@@ @ @ @ @@ @@I@ @ @ @ @ @ NR o 0 0 o 0 0 o 0 0 0 @@ @ @ @ @@@ @

y @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @:. E @ @ @ @ @ @ @ @@ @ @ @ @ @ @ @ @ @ @00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 000 0 0 0 0 0 @ @ @ @ @ @ @ @ @ @ e e e e e e e ;

'. ' ..

;. U. If the address grldded above Is not complete enough W. SCHOOLYOU ATTEND X. COLLEGETO RECSVEYOUR AP ORADEfor delivery or your grade report, please mlln thle ClVIIl -. 0

SChooICodo CollegoCodoand print your complete address below. SChool Name, City, and Slate College Name and Address

I I I I I +- Mallo sure you have correctly entered your I I I Using tile Cellege Code list In the AP Student Pack. Indicate

-.. @ @ @ @ @ @SChool Code. filled In the appropriate owls,

@ @ @@the ene Cllliege thai yeu want te receive your AP Grade

1:. and completed the lnlormation below. Repert by writing In the Cllllelle code number and griddlng

I~'L:(j) @ (j) @ (j) (j) @ (j) @6) tlte ovals 10 lite lelt. Also complete the In'ermallen belew,

-.. @ @ @ @ @ @ . @ @ @@@ @ @ @ @ @ School Name @ @ @@

IE @ @ @ @@ @ @ @@ @ClJy Staro Of PrcMnclJ , @ @ @ @ @@ @ @ @ @ CotIogo N4II1O

@@@@@@ City @ @ @ @c:c....y Z1por_e- ~~(j)~00 (j) (1) (j) (1) City

@@@@@@ @ @ @ @. , @@@@@@ SIlI!8 , @ @ @ @ SIII!8• • • - -

2003 EXAM

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The Calculus AB Exam

CALCULUS AB

A CALCULATOR CANNOT BE USED ON PART A OF SECTION I. A GRAPHING CALCULATOR FROM THEAPPROVED LIST IS REQUIRED FOR PART B OF SECTION I AND FOR PART A OF SECTION II OF THEEXAMINATION. CALCULATOR MEMORIES NEED NOT BE CLEARED. COMPUTERS. NONGRAPHINGSCIENTIFIC CALCULATORS. CALCULATORS WITH QWERTY KEYBOARDS. AND ELECTRONIC WRITINGPADS ARE NOT ALLOWED. CALCULATORS MAY NOT BE SHARED AND COMMUNICATION BETWEENCALCULATORS IS PROHIBITED DURING THE EXAMINATION. A'ITEMPTS TO REMOVE TEST MATERIALSFROM THE ROOM BY ANY METHOD WILL RESULT IN THE INVALIDATION OF TEST SCORES.

SECTION I

Time- 1 hour and 45 minutes

All questions are given equal weight.

Percent of total grade-50

Part A: 55 minutes. 28 multiple-choice questionsA calculator'is NOT allowed.

Part B: 50 minutes. 17 multiple-choice questionsA graphing calculator is required.

Parts A and B of Section I are in this examination booklet; Parts A and B of Section II. which consist of longer problems.are in a separate. sealed package.

General Instructions

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE INSTRUCTED TO DO SO.

INDICATE YOUR ANSWERS TO QUESTIONS IN PART A ON PAGE 2 OF THE SEPARATE ANSWER SHEET. THEANSWERS TO QUESTIONS IN PART B SHOULD BE INDICATED ON PAGE 3 OF THE ANSWER SHEET. No creditwill be given for anything written in this examination booklet. but you may use the booklet for notes or scratchwork. Afteryou have decided which of the suggested answers is best. COMPLETELY fill in the corresponding oval on the answersheet. Give only one answer to each question. If you change an answer. be sure that the previous mark is erased completely.

Example:

What is the arithmetic mean of the numbers 1. 3. and 6 ?

(A)

7(B) 3"

(C) 3

(D) 130

7(E) 2

Sample Answer

Many candidates wonder whether or not to guess the answers to questions about which they are not certain. In this sectionof the examination. as a correction for haphazard guessing. one-fourth of the number of questions you answer incorrectlywill be subtracted from the number of questions you answer correctly. It is improbable. therefore. that mere guessing willimprove your score significantly; it may even lower your score. and it does take time. If. however. you are not sure ofthe best answer but have some knowledge of the question and are able to eliminate one or more of the answer choicesas wrong. your chance of answering correctly is improved. and it may be to your advantage to answer such a question.

Use your time effectively. working as rapidly as you can without losing accuracy. Do not spendtoo much time on questions that are too difficult. Go on to other questions and come back to thedifficult ones later if you have time. It is not expected that everyone will be able to answer allthe multiple-choice questions.

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CALCULUSABSECTION I, Part A

Time-55 minutes

Number of questions-28

Calculus AB

A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining theform of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answersheet. No credit will be given for anything written in the test book. Do not spend too much time on anyone problem.

In this test:

(1) Unless otherwise specified, the domain of a function! is assumed to be the set of all real numbers x for which!(x) is a real number.

(2) The inverse of a trigono~etric function! may be indicated using the inverse function notation ! -lor with the

prefix "arc" (e.g., sin -1 x = arcsin x).

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GO ON TO THE NEXT PAGE.

2003 EXAM

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Calculus AB

( 3)2 dy1. If y = x + 1 , then dx =

~artA

(B) 2(x3 + 1)

-4(A)~

4(C) e-4 -1

1 e-4(0) --4 4 (E) 4 - 4e-4

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~artA ICalculus AB

3. For x ;;:: 0, the horizontal line y = 2 is an asymptote for the graph of the function f Which of the followingstatements must be true?

(A) f(O) = 2

(B) f(x) * 2 for all x;;:: 0

(C) f(2) is undefined.

(D) lim f(x) = 00x-t2

(E) lim f(x) = 2x-too

(E) 13

-5(D) (3x + 2)2(C) (3x ~ 2)2

2x+3 dy4. If y = 3x + 2 ' then dx =

(A) 12x + 13 (B) 12x - 13(3x + 2)2 (3x + 2)2

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CalculusAB

1r

5. f sinxdx =

EIIII!fIIart A I

(B) .J22

J2(C) -2-1 (D) _.J2 + 12

(E) J2 - 12

6. . x 3- 2x2 + 3x - 4

hm =X~OO 4x3 - 3x2 + 2x - 1

(A) 4 (B) 1 (C) !4

(D) 0 (E) -1

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y

-+----+---~--_+--___........x-2 -1

Graph off'

Calculus AB .

7. The graph of f'. the derivative of the function f, is shown above. Which of the following statements is trueabout f ?

(A) f is decreasing for -1 S x S 1.

(B) f is increasing ft?r -2 S x S O.

(C) f is increasing for 1 S x S 2.

(D) f has a local minimum at x = O.

(E) f is not differentiable at x =-1 and x = 1.

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(C) .!.4

Calculus AB

8. Jx 2 cos(x3 )dx =

(A) - jsin(x3) + C

(B) tSin(x3) + C

x3

)(C) - TSin(x3 + C

x3

)(D) TSin(x3 + C

(E) X; Sin( X44) + C

9. If/(x) = In(x + 4 + e-3X). then /'(0) is

2(A) -5 (D) ~

5

~artA I

(E) nonexistent

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Calculus AB

10. The function f has the property that f(x), f'(X), and f"(x) are negative for all real values x. Which of the

following could be the graph of f ?

(A)y

-------,o,-l------X

(B)y

-------,,-I------X

(C)

(E)

y

-----+-----_,x

y

-----+------xo

(D)y

-----4------X

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CalculusAB

11. Using the substitution u = 2x + I, J: .J2x + I dx is equivalent to

I 11/2(A) -2 ..[U du-1/2

1 1.2(B) 2 o..[U duI rS

(C) '2 JI ..[U du (D) J: JU du

12. The rate of change of the volume, V, of water in a tank with respect to time, I, is directly proportional to thesquare root of the volume. Which of the following is a differential equation that describes this relationship?

(A) Vet) = k.Jt

(B) V(I) = k.JV

(C) ~~ = k.Jt

(D) dV =...!...dl .JV

(E) ez = k.JV

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y

--~L..---+---+--T--l--+-+--~7I'-_X

Graph off

Calculus AB

13. The graph of a function f is shown above. At which value of x is f continuous, but not differentiable?

(A) a (B) b (C) c (D) d (E) e

26

14. If y = x2 sin 2x, then : =

(A) 2xcos2x

(B) 4xcos 2x

(C) 2x(sin 2x + cos 2x)

(D) 2x(sin 2x - x cos 2x)

(E) 2x(sin 2x + x cos 2x)

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Calculus AB

15. Let f be the function with derivative given by /,(x) = x 2- ~. On which of the following intervals

xis f decreasing?

(A) (-00, -1] only

(B) (-00,0)

(C) [-1.0) only

(D) (0. t'2]

(E) [t'2, 00)

16. If the line tangent to the graph of the function f at the point (1. 7) passes through the point (-2. -2).then /,(1) is

(A) -5 (B) 1 (C) 3 (D) 7 (E) undefined

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17. Let I be the function given by I(x) = 2xex• The graph of I is concave down when

Calculus AB

(A) x < -2 (B) x > -2 (C) x < -1 (D) x > -1 (E) x < 0

x -4 -3 -2 -1 0 1 2 3 4

g'(x) 2 3 0 -3 -2 -1 0 3 2

18. The derivative g' of a function g is continuous and has exactly two zeros. Selected values of g' are given inthe table above. If the domain of g is the set of all r:eal numbers, then g is decreasing on which of the followingintervals?

(A) -2 S x S 2 only

(B) -1 S x S 1 only

(C) x ~ -2

(D) x ~ 2 only

(E) x S -2 or x ~ 2

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Calculus AB~artA

19. A curve has slope 2x + 3 at each point (x. y) on the curve. Which ofthe following is an equation for this curveif it passes through the point (1. 2) ?

(A) y = 5x - 3

(B) y = x2 + 1

(C) Y = x 2 + 3x

(D) y =x 2 + 3x - 2

(E) y = 2x2 + 3x - 3

{

X + 2 if x S 3f(x) = 4x - 7 if x > 3

20. Let f be the function given above. Which of the following statements are true about f ?

I. lim f(x) exists.x~3

II. f is continuous at x = 3.

m. f is differentiable at x = 3.

(A) None

(B) I only

(C) II only

(0) I and II only

(E) I. II, and ill

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21. The second derivative of the function I is given by I"(x) = x(x - a)(x - b)2. The graph of I" is shownabove. For what values of x does the graph of I have a point of inflection?

(A) 0 and a only (B) 0 and m only (C) b and j only (D) 0, a, and b (E) b, j, and k

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Calculus AB

y

-~----~--...x

~artA I

22. The graph of f', the derivative of f, is the line shown in the figure above. If f(O) = 5, then f(1) =

(A) 0 (B) 3 (C) 6 (D) 8 (E) 11

23. ~ (J:2

sin(t 3) dt) =

(A) - COs(x6) (B) sin(x3

) (C) sin(x6) (D) 2x sin(x 3

)

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24. Let f be the function defined by f (x) = 4x3 - 5x + 3. Which of the following is an equation of the line tangentto the graph of f at the point where x = -1 ?

(A) y =7x - 3

(B) y = 7x + 7

(C) y =7x + 11

(D) y = -5x - 1

(E) y = -5x - 5

25. A particle moves along the x-axis so that at time t ~ 0 its position is given by x(t) = 2t3- 21t2 + 72t - 53.

At what time t is the particle at rest?

(A) t =1 only

(B) t = 3 only

7(C) t ="2 only

7(D) t =3 and t ="2

(E) t =3 and t =4

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CalculusAB~artA I

26. What is the slope of the line tangent to the curve 3i - 2x2 = 6 - 2X)' at the point (3.2)?

(A) 0 (D) 67

(E) ~3

27. Let / be the function defined by lex) = x 3 + x. If g(x) =/-1 (x) and g(2) =1. what is the value of g'(2) ?

117(A) 13 (B) 4 (C) 4 (D) 4 (E) 13

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Calculus AB

28. Let g be a twice-differentiable function with g'(x) > 0 and g"(x) > 0 for all real numbers x, such thatg(4) = 12 and g(5) = 18. Of the following. which is a possible value for g(6)?

(A) 15 (B) 18 (C) 21 (D) 24 (E) 27

Unauthorized copying or reuse of34 any part of this page Is Illegal.

END OF PART A OF SECTION I

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Calculus AB~artB ;

CALCULUSABSECTION I, Part BTime-SO minutes

Number of questions-I7

A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ONTIllS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining thefonn of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answersheet. No credit will be given for anything written in the test book. Do not spend too much time on anyone problem.

BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TOQUESTIONS NUMBERED 76-92.

YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.

In this test:

(1) The exact numerical value of the correct answer does not always appear among the choices given. When thishappens, select from among the choices the number that best approximates the exact numerical value.

(2) Unless otherwise specified, the domain of a function I is assumed to be the set ofall real numbers x for whichI(x) is a real number.

(3) The inverse of a trigonometric function I may be indicated using the inverse function notation I -lor with theprefix "arc" (e.g., sin -I x = arcsin x).

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~artB ICalculus AB

76. A particle moves along the x-axis so that at any time t ~ 0, its velocity is given by v(t) = 3 + 4.1 cos(0.9t).What is the acceleration of the particle at time t = 4?

(A) -2.016 (B) -0.677 (e) 1.633

y

(D) 1.814 (E) 2.978

77. The regions A, B, and C in the figure above are bounded by the graph of the function f and the x-axis. If the

area of each region is 2, what is the value of J3 (I(x) + l)dx?-3

(A) -2 (B) -1 (C) 4 (D) 7 (E) 12

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CalculusAB~artB I

78. The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in thearea of the circle at the instant when the circumference of the circle is 201r meters?

(A) 0.041r m2/sec

(B) 0.41r m2/sec

(C) 41r m 2/sec

(D) 201r m 2/sec

(E) W01r m2/sec

79. For which of the following does lim j<x) exist?x-.4

II. y m. y

:/1

x x1 4 0 1 4

Graphofj Graphofj1 4

Graph ofj

I. Y

--=+~f--+--+--+---+--"'x

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and ill only

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Calculus AB

80. The function f is continuous for -2 :::; x :::; 1 and differentiable for -2 < x < 1. If f(-2) =-5 and f{l) =4,which of the following statements could be false?

(A) There exists e, where -2 < e < 1, such that fee) = O.

(B) There exists e, where -2 < e < 1, such that f'(e) =o.(C) There exists e, where -2 < e < 1, such that fee) = 3.

(D) There exists e, where -2 < e < 1, such that f'(e) =3.

(E) There exists e, where -2 ::; e :::; I, such that fee) ~ f(x) for all x on the closed interval -2 :::; x :::; 1.

81. Let f be the function with derivative given by f'(x) = sin(x2 + 1). How many relative extrema does f have

on the interval 2 < x < 4 ?

(A) One (B) Two (C) Three (D) Four (E) Five

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Calculus AB~artB I

78. The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in thearea of the circle at the instant when the circumference of the circle is 20n- meters?

(A) 0.04n- m2/sec

(B) OAn- m 2/sec

(C) 4n- m 2/sec

(D) 20n- m 2/sec

(E) lOOn- m2/sec

79. For which ofthe following does lim f(x) exist?x-.4

I. y

--=fo---ll-+---l--+---+---X4

Graph off

II. Y

--=fo---l-+--+-+--+--"X4

Graph off

ill. y

I-~O:f-+--+--+--41--i--X

Graph off

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and ill only

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~artB ICalculus AB

80. The function f is continuous for -2 S x S 1 and differentiable for -2 < x < 1. If f(-2) = -5 and f{l) =4,which of the following statements could be false?

(A) There exists e, where -2 < e < 1, such that fee) = O.

(B) There exists e, where -2 < e < 1, such that f '(e) = O.

(C) There exists e, where -2 < e < 1. such that fee) = 3.

(D) There exists e, where -2 < e < 1, such that f'(e) = 3.

(E) There exists e, where -2 S e S 1, such that fee) ~ f(x) for all x on the closed interval -2 S x S 1.

81. Let f be the function with derivative given by f'ex) = sin(x 2 + 1). How many relative extrema does f have

on the interval 2 < x < 4 ?

(A) One (B) Two (C) Three (0) Four (E) Five



2003 EXAM

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Calculus AB B!BI_~artB

82. The rate of change ofthe altitude of a hot-air balloon is given by ret) = t 3- 4t2 + 6 for 0 ~ t ~ 8. Which of

the following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?

13.514

(A) r(t)dt1.572

(B) J: ret) dt

r2.667(C) Jo ret) dt

13.514

(D) r'(t) dt\.572

r2.667(E) Jo r'(t) dt

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Calculus AB

83. The velocity, in ftlsec, of a particle moving along the x-axis is given by the function V(/) = e' + Ie'. What is theaverage velocity of the particle from time I = 0 to time I = 3 ?

(A) 20.086 ftlsec

(B) 26.447 ftlsec

(C) 32.809 ftlsec

(D) 40.671 ftlsec

(E) 79.342 ftlsec

84. A pizza, heated to a temperature of 350 degrees Fahrenheit (OF), is taken out of an oven and placed in a 75°F

room at time I = 0 minutes. The temperature of the pizza is changing at a rate of -11Oe-0.4r degrees Fahrenheit

per minute. To the nearest degree, what is the temperature of the pizza at time I = 5 minutes?

(A) 112°F (D) 238°F

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Calculus AB

85. If a trapezoidal sum ovempproximates J:f(X) dx, and a right Riemann sum underapproximates J:f(x) dx,

which of the following could be the graph of Y = f (x) ?

(A) Y (B) Y

4

3

2

1

x1 2 3 4

(0) Y

4

3

2

1x0 1 2 3 4

0 x1 2 3 4

(C) Y

4

3

2

1

x0 1 2 3 4

(E) Y

4

3

2

1x

2 3 4

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2003 EXAM

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~artB ICalculus AB

86. The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = tan -I x, thehorizontal line y = 3, and the vertical line x = 1. For this solid, each cross section perpendicular to the x-axis isa square. What is the volume of the solid?

(A) 2.561 (B) 6.612 (C) 8.046 (D) 8.755 (E) 20.773

87. The function f has first derivative given by f'(x) = ~ 3' What is the x-coordinate ofthe inflectionl+x+x

point of the graph of f ?

(A) 1.008 (B) 0.473 (C) 0 (D) -0.278 (E) The graph of f has no inflection point.

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CalculusAB~artB ;

88. On the closed interval [2. 4]. which of the following could be the graph of a function / with the property that

4 ~ 2 J24/(/) dl = 1?

(A) Y

4

3

2

1

(B) Y

4

3

2

OL-...---l--~--4--4-_X

123 4 OL..----l--4---4-~_x

I 234

(C) Y (D) Y4 4

3 3

/2 2

1 1

x x0 1 2 3 4 0 1 2 3 4

(E) Y

4

3

2

1

OL-...--+-----+---x

123 4

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~artB ICalculusAB

89. Let / be a differentiable function with /(2) =3 and /'(2) = -5, and let g be the function defined byg(x) = x/(x). Which of the following is an equation of the line tangent to the graph of g at the point where

x = 2?

(A) Y =3x

(B) Y - 3 = -Sex - 2)

(C) Y - 6 = -sex - 2)

(D) Y - 6 = -7(x - 2)

(E) Y - 6 = -10(x - 2)

90. For all x in the closed interval [2. 5]. the function / has a positive first derivative and a negative secondderivative. Which of the following could be a table of values for f?

(A) x lex)2 73 94 125 16

(B) x lex)2 73 114 145 16

(C) x I(x)

2 163 124 95 7

(D) x I(x)

2 163 144 115 7

(E) x I(x)

2 163 134 105 7

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Calculus AB~artB I

91. A particle moves along the x-axis so that at any time 1 > 0. its acceleration is given by a(l) = In(1 + 2t). If the

velocity of the particle is 2 at time 1 = I, then the velocity of the particle at time 1 = 2 is

•

(A) 0.462 (B) 1.609 (e) 2.555 (D) 2.886 (E) 3.346

92. Let g be the function given by g(x) =J: sin(t 2)dt for -1 S x :s; 3. On which of the following intervals is g

decreasing?

(A) -1 S x :s; 0

(B) 0 S x :s; 1.772

(C) 1.253 S x :s; 2.171

(D) 1.772 S x S 2.507

(E) 2.802 S x S 3

END OF SECTION I

AFTER TIME HAS BEEN CALLED, TURN TO THE NEXT PAGE ANDANSWER QUESTIONS 93·96.

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~artB ICalculusAB

93. Which graphing calculator did you use during the examination?

(A) Casio 6300, Casio 7300, Casio 7400, Casio 7700, Tl-73, Tl-80, or TI-81(B) Casio 9700, Casio 9800, Sharp 9200, Sharp 9300, TI-82, or TI-85(C) Casio 9850, Casio FX 1.0, Sharp 9600, Sharp 9900, TI-83m-83 Plus, or TI-86(D) Casio 9970, Casio Algebra FX 2.0, lIP 38G, lIP 39G, lIP 40G, lIP 48 series, lIP 49 series, or TI-89(E) Some other graphing calculator

94. During your Calculus AB course, which of the following best describes your calculator use?

(A) I used my own graphing calculator.(B) I used a graphing calculator furnished by my school, both in class and at home.(C) I used a graphing calculator furnished by my school only in class.(D) I used a graphing calculator furnished by my school mostly in class, but occasionally at home.(E) I did not use a graphing calculator.

95. During your Calculus AB course, which of the following describes approximately how oftena graphing calculator was used by you or your teacher in classroom learning activities?

(A) Almost every class(B) About three-quarters of the classes(C) About ~me-halfof the classes(D) About one-quarter of the classes(E) Seldom or never

96. During your Calculus AB course, which of the following describes the portion of testing time youwere allowed to use a graphing calculator?

(A) All or almost all of the time(B) About three-quarters of the time(C) About one-half of the time(D) About one-quarter of the time(E) Seldom or never

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•

2003 EXAM

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Calculus AB

CALCULUSABSECTION II

Time - 1 hour and 30 minutesPercent of total grade - SO

Part A: 45 minutes, 3 problemsPart B: 45 minutes, 3 problems

PARTA(A graphing calculator is required for some problems or parts of problems.)During the timed portion for PartA. you may work only on the problems in Part A. The problems for Part A are printed in the greeninsert only. When you are told to begin. open your booklet, carefully tear out the green insert. and write your solution to each partof each problem in the space provided for that part in the pink test booklet.

On Part A. you are permitted to use your calculator to solve an equation. find the derivative of a function at a point, or calculatethe value of a definite integral. However. you must clearly indicate the setup of your problem, namely the equation, function. orintegral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produceyour results.

PART B (No calculator is allowed for these problems.)The problems for Part B are printed in the blue insert only. When you are told to begin, open the blue insert, and writeyour solution to each part of each problem in the space provided for that part in the pink test booklet. During the timed portionfor Part B, you may keep the green insert and continue to work on the problems in Part A without the use of any calculator.

GENERAL INSTRUCTIONS FORSECTION (( PARTAAND PARTBFor each part of Section II. you may wish to look over the problems before starting to work on them, since it is not expected thateveryone will be able to complete all parts of all problems. All problems are given equal weight. but the parts of a particularproblem are not necessarily given equal weight.

• YOU SHOULDWRITEALLWORKFOREACH PARTOFEACH PROBLEM INTHE SPACEPROVIDED FORTHATPART IN THE PINK TEST BOOKLET. Be sure to write clearly and legibly. Ifyou make an error, you may save time bycrossing it out rather than trying to erase it. Erased or crossed-out work will not be graded.

• Show all your work. Clearly label any functions. graphs. tables. or other objects that you use. You will be graded on thecorrectness and completeness of your methods as well as your answers. Answers without supporting work may notreceive credit.

• Justifications require that you give mathematical (noncalculator) reasons.• Your work must be expressed in standard mathematical notation rather than calculator syntax.

For example. rx 2dx may not be written as fnlnt(X2

• X, I, 5).

• Unless otherwise specified. answers (numeric or algebraic) need not be simplified.• Ifyou use decimal approximations in calculations. you will be graded on accuracy. Unless otherwise specified, your final

answers should be accurate to three places after the decimal point.• Unless otherwise specified. the domain of a function f is assumed to be the set of all real numbers x for which

f(x) is a real number.

47

2003 EXAM

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~artA ICALCULUSAB

SECTION II, Part ATime-45 minutes

Number of problems-3

A graphing calculator is required for some problems or parts of problems.

y

Calculus AB

1

1

1. Let R be the shaded region bounded by the graphs of y =JX and y = e-3x and the vertical line x = 1,

as shown in the figure above.

(a) Find the area of R.

(b) Find the volume of the solid generated when R is revolved about the horizontal line y = 1.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis isa rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.

48UlUluthorized copying or reuse ofany part of this page I. illegal.


2003 EXAM

of 107

CalculusAB mllllfl?art A I2. A particle moves along the x-axis so that its velocity at time 1 is given by

V(I) =-(1 + 1) sin(1; )-At time 1 = 0, the particle is at position x = 1.

(a) Find the acceleration of the particle at time 1 =2. Is the speed of the particle increasing at I =2 ? Why orwhy not?

(b) Find all times 1 in the open interval 0 < 1 < 3 when the particle changes direction. Justify your answer.

(c) Find the total distance traveled by the particle from time I =0 until time I = 3.

(d) During the time interval 0 SIS 3, what is the greatest distance between the particle and the origin? Showthe work that leads to your answer.

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2003 EXAM

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~artA IR(t)

c 70o'g, 60

§ 50U)c8 40-~ 30

~ 20

~ 10~

o'--+--+--+----il--+--+--+---l-+__.- t10 20 30 40 50 60 70 80 90

TIme

CalculusAB

I R(I)(minutes) (gallons per minute)

0 20

30 30

40 40

50 55

70 65

90 70

3. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twicedifferentiable and strictly increasing function R of time I. The graph of R and a table of selected values ofR(I), for the time interval 0 SIS 90 minutes, are shown above.

(a) Use data from the table to find an approximation for R'(45). Show the computations that lead to your

answer. Indicate units of measure.

(b) The rate of fuel consumption is increasing fastest at time I =45 minutes. What is the value of R"(45) ?

Explain your reasoning.

(c) Approximate the value of f:R(t) dl using a left Riemann sum with the five subintervals indicated by the

data in the table. Is this numerical approximation less than the value of f:R(I) dt ? Explain your reasoning.

(d) For 0 < b S 90 minutes, explain the meaning of f:R(I) dt in terms of fuel consumption for the plane.

Explain the meaning of tf:R(t) dl in tenns of fuel consumption for the plane. Indicate units of measure in

both answers.

END OF PART A OF SECTION II

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2003 EXAM

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CalculusAB

(-3, 1)

CALCULUSABSECTION II, Part B

Time-4S minutesNumber of problems-3

No calculator is allowed for these problems.

y

2

~artB I

(4,-2)

Graph of f'

4. Let f be a function defined on the closed interval -3 S x S 4 with 1(0) = 3. The graph of I', the derivative

of f, consists of one line segment and a semicircle, as shown above.

(a) On what intervals, if any, is I increasing? Justify your answer.

(b) Find the x-coordinate of each point of inflection of the graph of I on the open interval -3 < x < 4. Justifyyour answer.

(c) Find an equation for the line tangent to the graph of I at the point (0,3).

(d) Find 1(-3) and1(4). Show the work that leads to your answers.

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2003 EXAM

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~artB ICalculus AB

for 0 S x S 3

for 3 < x S 5,

52

5. A coffeepot has the shape of a cylinder with radius 5 inches, as shown in the figure above. Let h be the depth ofthe coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume Vof coffee in the pot is changing at the rate of -51r.JJi cubic inches per second. (The volume V of a cylinder with

radius r and height h is V =1rr2h.)

dh .ffi(a) Show that di =- T .

(b) Given that h = 17 at time t = 0, solve the differential equation ~; = -.JJ for h as a function of t.

(c) At what time t is the coffeepot empty?

6. Let f be the function defined by

f(x) ={..rx:+T for 0 S x S 35 - x for 3 < x S 5.

(a) Is f continuous at x = 3? Explain why or why not.

(b) Find the average value of f(x) on the closed interval 0 S x S 5.

(c) Suppose the function g is defined by

g(x) = {k.JX+Tm.x + 2

where k and m are constants. If g is differentiable at x = 3, what are the values of k and m ?

END OF EXAM

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2003 EXAM

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The Calculus BC Exam

CALCULUSBC

A CALCULATOR CANNOT BE USED ON PART A OF SECTION I. A GRAPHING CALCULATOR FROM THEAPPROVED LIST IS REQUIRED FOR PART B OF SECTION I AND FOR PART A OF SECTION II OF THEEXAMINATION. CALCULATOR MEMORIES NEED NOT BE CLEARED. COMPUTERS, NONGRAPHINGSCIENTIFIC CALCULATORS, CALCULATORS WITH QWERTY KEYBOARDS, AND ELECTRONIC WRITINGPADS ARE NOT ALLOWED. CALCULATORS MAY NOT BE SHARED AND COMMUNICATION BETWEENCALCULATORS IS PROHIBITED DURING THE EXAMINATION. ATTEMPTS TO REMOVE TEST MATERIALSFROM THE ROOM BY ANY METHOD WILL RESULT IN THE INVALIDATION OF TEST SCORES.

SECTION I

Time- 1 hour and 45 minutesAll questions are given equal weight.

Percent of total grade-50

Part A: 55 minutes, 28 multiple-choice questionsA calculator is NOT allowed.

Part B: 50 minutes, 17 multiple-choice questionsA graphing calculator is required.

Parts A and B of Section I are in this examination booklet; Parts A and B of Section II, which consist of longer problems,are in a separate, sealed package.

General Instructions

DO NOT OPEN THIS BOOKLET UNTIL YOU ARE INSTRUCTED TO DO SO.

INDICATE YOUR ANSWERS TO QUESTIONS IN PART A ON PAGE 2 OF THE SEPARATE ANSWER SHEET. THEANSWERS TO QUESTIONS IN PART B SHOULD BE INDICATED ON PAGE 3 OF THE ANSWER SHEET. No creditwill be given for anything written in this examination booklet, but you may use the booklet for notes or scratchwork. Afteryou have decided which of the suggested answers is best, COMPLETELY fill in the corresponding oval on the answersheet. Give only one answer to each question. If you change an answer, be sure that the previous mark is erased completely.

Example:

What is the arithmetic mean of the numbers 1, 3, and 6 ?

(A) 1

7(B) 3

(C) 3

(D) ~O

7(E) 2

Sample Answer

Many candidates wonder whether or not to guess the answers to questions about which they are not certain. In this sectionof the examination, as a correction for haphazard guessing, one-fourth of the number of questions you answer incorrectlywill be subtracted from the number of questions you answer correctly. It is improbable, therefore, that mere guessing willimprove your score significantly; it may even lower your score, and it does take time. If, however, you are not sure ofthe best answer but have some knowledge of the question and are able to eliminate one or more of the answer choicesas wrong, your chance of answering correctly is improved, and it may be to your advantage to answer such a question.

Use your time effectively, working as rapidly as you can without losing accuracy. Do not spendtoo much time on questions that are too difficult. Go on to other questions and come back to thedifficult ones later if you have time. It is not expected that everyone will be able to answer allthe multiple-choice questions. .

53

2003 EXAM

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~artA I

CALCULUSBCSECTION I, Part A

Time-55 minutesNumber of questions-28

Calculus Be

A CALCULATOR MAY NOT BE USED ON TIllS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining thefonn of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answersheet. No credit will be given for anything written in the test book. Do not spend too much time on anyone problem.

In this test:

(1) Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for whichf(x) is a real number.

(2) The inverse of a trigonometric function f may be indicated using the inverse function notation f -lor with the

preftx "are" (e.g., sin -I x = arcsin x).

54Unauthorized copying or reuse 0'any part 0' this page Is Illegal.


2003 EXAM

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Calculus BC

1. If y = sin(3x), then :z =

~artA ;

(A) -3 cos(3x) (B) - cos(3x) (C) - j cos(3x) (D) cos(3x) (E) 3 cos(3x)

2. •• eX - cos x - 2x .1m IS

x-+o x 2 - 2x

1(A) -

2(B) 0 (D) 1 (E) nonexistent

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2003 EXAM

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~artA I

3. J(3x + 1)5dx =

(A) (3x 1~ 1)6 + C

(3x + 1)6 C(B) 6 +

(3x + 1)6 C(C) 2 +

(3X2 )6-+x

(D) 2 + C2

(E) (3;' + X)' + C

Calculus BC

3(E) - 4 tan 13

4. For 0 S t S 13, an object travels along an elliptical path given by the parametric equations x = 3 cos t andy = 4 sin t. At the point where t = 13. the object leaves the path and travels along the line tangent to the pathat that point. What is the slope of the line on which the object travels?

(A) - i (B) _1 (C) _ 4 tan 13 (0) _ 43 4 3 3 tan 13



2003 EXAM

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Calculus BC~artA I

5. Let y = f(x) be the solution to the differential equation : = x + y with the initial condition f(l) = 2. What

is the approximation for f(2) ifEuler's method is used, starting at x = 1 with a step size of 0.5 ?

(A) 3 (B) 5 (C) 6 (D) 10 (E) 12

foo 1

6. What are all values of p for which -2- dx converges?1 x P

(A) P < -1

(B) p>O

(C) 1p>-

2

(D) p>l

(E) There are no values of p for which this integral converges.

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2003 EXAM

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~artA ICalculus BC

7. The position of a particle moving in the xy-plane is given by the parametric equations x = t3- 3t2 and

y = 2t3 - 3t2 - 12t. For what values of I is the particle at rest?

(A) -1 only (B) 0 only (C) 2 only (D) -1 and 2 only (E) -1, 0, and 2

58

8. Jx 2 cos(x3) dx =

(A) - tSin(x3) + C

(B) tSin(x3

) + C

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2003 EXAM

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Calculus BC

9. Iff(x) = In(x + 4 + e-3.t), then /,(0) is

2(A) -

5

00 211+110. What is the value of~ --?

£.J 3"11=1

(D) ~5

~artA I

(E) nonexistent

(A) I (B) 2 (C) 4 (D) 6 (E) The series diverges.

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2003 EXAM

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~artA ICalculus BC

~ 211. The Maclaurin series for~ is L x fl

• Which of the following is a power series expansion for x 2?x '1=0 1 - x

(A) 1 + x 2 + x 4 + x 6 + x 8 + ...

(B) x 2 + x 3 + x 4 + x S + ...

(C) x 2 + 2x3 + 3x4 + 4xs + ...

(D) xl + x 4 + x 6 + x 8 + .

(E) x 2 _ x 4 + x 6 _ x 8 + .

12. The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to thesquare root of the volume. Which of the following is a differential equation that describes this relationship?

(A) V(t) = k../i

(B) V(t) = k.JV

(C) ~~ = k../i

dV k(D) dt = .JV

(E) ": = k.JV



2003 EXAM

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CalculusBC

y

Graphofj

ErIIIIfI!artA

13. The graph of a function j is shown above. At which value of x is j continuous, but not differentiable?

(A) a (B) b (C) c (D) d (E) e

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2003 EXAM

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~artA ICalculus Be

y

---/11--/ / I /--/1/1-/11 "- / I , I II , , I I II , , I I 13 x-I I I , I-/ I " ,--/1/1--//1/---/11

3\\,---\\, ..... -\ \ ,,--I \ \ \'-1 I I \ ,I I I I I ,

-31 I I I I \I I I \ ,, \ \ \ ,\ \ , ,--\ \ ' ..... -7\\,---

-3

14. Shown above is a slope field for which of the following differential equations?

(A) dy =x (B) dy =~ (C) dy =.i. (0) dy =.£.dx y dx y2 dx y dx y

62Unauthorized copying or rouse ofony part of this page Is lIIegol.


2003 EXAM

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Calculus BC

15. The length of a curve from x = 1 to x = 4 is given by f .J1 + 9x4 dx. If the curve contains the point (1,6),

which of the following could be an equation for this curve?

(A) y = 3 + 3x2

(B) y = 5 + x 3

(C) Y = 6 + x 3

(D) y = 6 - x 3

16 9 s(E) y =T + x + "5 x

16. If the line tangent to the graph of the function / at the point (1,7) passes through the point (-2, -2),then /'(1) is

(A) -5 (B) 1 (C) 3 (D) 7 (E) undefined

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2003 EXAM

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~artA ICalculus BC

17. A curve C is defined by the parametric equations x = ,2- 41 + 1 and y = 1

3• Which of the following is an

equation of the line tangent to the graph of C at the point (-3, 8) ?

(A) x = -3

(B) x = 2

(C) y = 8

27(D) y =- 1O(x + 3) + 8

(E) y = 12(x + 3) + 8



2003 EXAM

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Calculus BC

7

6 1\

1\5\4\

3\2 \

11\

0 I \ 2 3 4 5 6 7I

\ V ~/'"2~

V3 ./

Graph off

18. The graph of the function f shown in the figure above has horizontal tangents at x = 3 and x = 6.

r2x ,If g(x) = Jo f(t)dt, what is the value of g (3) ?

(A) 0 (B) -1 (C) -2 (D) -3 (E) -6

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2003 EXAM

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~artA ICalculus BC

19. A curve has slope 2x + 3 at each point (x, y) on the curve. Which of the following is an equation for this curveif it passes through the point (1, 2) ?

(A) Y = 5x - 3

(B) y = x 2 + 1

(C) Y = x 2 + 3x

(D) y = x2 + 3x - 2

(E) y = 2x2 + 3x - 3

4 S 6 n+3

20. A function I has Macla.urin series given by ~! + ~! + ~! + ... + (:+ 1)! + .... Which of the following is an

expression for I(x) ?

(A) -3x sin x + 3x2

(B) - cos(x2) + 1

(C) _x2 cos x + x 2

(D) x 2ex - x 3 _ x 2

(E) ex2

- x 2- 1

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2003 EXAM

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Calculus Be~artA I

21. The number of moose in a national park is modeled by the function M that satisfies the logistic differential

equation ~~ = 0.6M(1 - :0). where t is the time in years and M(O) = 50. What is I~~ M(t)?

(A) 50 (B) 200 (C) 500 (D) 1000 (E) 2000

co

22. What are all values of p for which the infinite series L n converges?n=l n P + 1

(A) p> 0

Unauthortzod copying or rouse ofany part 0' this page Is Illegal.

(B) p ~ 1 (C) p> 1 (D) p ~ 2 (E) p > 2

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2003 EXAM

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68

~artA I

23. Jx sin(6x)dx =

(A) -x cos(6x) + sin(6x) + C

(B) - ~ cos(6x) + 3~ sin(6x) + C

(C) - ~ cos(6x) + ~ sin(6x) + C

(0) ~ cos(6x) + 316 sin(6x) + C

(E) 6x cos(6x) - sin(6x) + C

24. Which of the following series diverge?

I. f (Si~2rn=O

00 1II. L 'Vn

n=1

III. f(-+.-)n=1 e + 1

(A) III only

(B) I and II only

(C) I and III only

(0) II and III only

(E) I, II, and III

Unauthorized copying or muse 0'any part of this page 'S Illegal.

Calculus BC


2003 EXAM

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Calculus BC

x 2 5 10 14

I(x) 12 28 34 30

25. The function I is continuous on the closed interval [2,14] and has values as shown in the table above. Using

J14

the subintervals [2,5], [5,10], and [10, 14], what is the approximation of 2 I(x) dx found by using a right

Riemann sum?

(A) 296 (B) 312 (C) 343 (D) 374 (E) 390

26. f 2x dx-(x + 2)(x + 1) -

(A) Inlx + 2/ + lnlx + 11 + c(B) lnlx + 21 + Inlx + 11 - 3x + C

(C) -4lnlx + 21 + 2lnlx + 11 + c(D) 4 Inlx + 21 - 2 lnlx + 11 + c

2 1 2(E) 2 Inl xl + "3 x + '2x + C

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2003 EXAM

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~artA ICalculus BC

(C) In(x6 + 1)

28. What is the coefficient of x 2 in the Taylor series for 1 2 about x = O?(l + x)

(A) .!.6

(C) 1 (D) 3 (E) 6

70Unauthorized copying or reuse ofany part of this page Is "'ogul.

END OF PART A OF SECTION I

2003 EXAM

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Calculus BC~artB

CALCULUSBCSECTION I, Part BTime-SO minutes

Number of questions-I7

A GRAPlllNG CALCULATOR IS REQUIRED FOR SOME QUESTIONS ONTHIS PART OF THE EXAMINATION.

Directions: Solve each of the following problems, using the available space for scratchwork. After examining theform of the choices, decide which is the best of the choices given and fill in the corresponding oval on the answersheet. No credit will be given for anything written in the test book. Do not spend too much time on anyone problem.

BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TOQUESTIONS NUMBERED 76-92.

YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET.

In this test:

(1) The exact numerical value of the correct answer does not always appear among the choices given. When thishappens, select from among the choices the number that best approximates the exact numerical value.

(2) Unless otherwise specified, the domain of a function I is assumed to be the set of all real numbers x for whichI(x) is a real number.

(3) The inverse of a trigonometric function f may be indicated using the inverse function notation I-lor with the

prefix "arc" (e.g., sin-I x = arcsin x).

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2003 EXAM

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~artB I

y

--_~.J----4-----.Jtr--_x

Graph ofl

76. The graph of the function I is shown above. Which of the following statements must be false?

(A) I(a) exists.

(B) I(x) is defined for 0 < x < a.

(C) I is not continuous at x = a.

(D) lim I(x) exists.x-+a

(E) lim I'(x) exists.x-+a

Calculus BC

77. Let P(x) = 3x2- 5x3 + 7x4 + 3xs be the fifth-degree Taylor polynomial for the function I about x = o.

What is the value of 1"'(0) ?

(A) -30 (B) -15 (C) -5 5(D) -

61

(E) -6

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2003 EXAM

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Calculus BC~artB I

78. The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase in thearea of the circle at the instant when the circumference of the circle is 20tr meters?

.,(A) O.04tr m-/sec

(B) OAtr m2/scc.,

(C) 4tr m-/sec

(D) 20tr m 2/scc

(E) 100tr m2/scc

x f(x) f'(x) g(x) g'(x)

-1 6 5 3 -2

I 3 -3 -I 2

3 I -2 2 3

79. The table above gives values of f. 1'. g. and g' at selected values of x. If hex) = f(g(x»). then 11'(1) =(A) 5 (B) 6 (C) 9 (D) 10 (E) 12

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2003 EXAM

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~artB ICalculus BC

100e-OJ'

80. Insects destroyed a crop at the rate of -3r tons per day. where time t is measured in days. To the nearest2-e

ton, how many tons did the insects destroy during the time interval 7 :::;; t :::;; 14 ?

(A) 125 (B) 100 (C) 88 (D) 50 (E) 12

y

3-1--+---:::---1--------.x

Graph of f

81. The graph of the function f is shown in the figure above. The value of lim sin(f (x» isx~1

(A) 0.909 (B) 0.841 (C) 0.141 (D) -0.416 (E) nonexistent

74Unouthorlzed copying or reuse ofany port of this pogo Is lIIegol.


2003 EXAM

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Calculus Be --r!artB I

82. The rate of change of the altitude of a hot-air balloon is given by r(t) = t 3 - 4t2 + 6 for 0 S t S 8. Which ofthe following expressions gives the change in altitude of the balloon during the time the altitude is decreasing?

13.514

(A) r(t)dt1.572

(B) J: r(t) dt

r2.667(C) Jo r(t) dt

13.514

(D) r'{t) dt1.572

r2.667(E) Jo r'(t) dt

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2003 EXAM

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x 0 I 2 3 4

I(x) 2 3 4 3 2

Calculus Be

83. The function I is continuous and differentiable on the closed interval [0, 4]. The table above gives selectedvalues of f on this interval. Which of the following statements must be true?

(A) The minimum value of Ion [0,4] is 2.

(B) The maximum value of Ion [0, 4] is 4.

(C) I(x) > 0 for 0 < x < 4

(D) I'(X) < 0 for 2 < x < 4

(E) There exists c, with 0 < c < 4, for which f'(c) = O.

84. A particle moves in thexy-plane so that its position at any time I is given by X(I) = 12 and yet) = sin(41).What is the speed of the particle when I = 3 ?

(A) 2.909 (B) 3.062 (C) 6.884 (D) 9.016 (E) 47.393

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2003 EXAM

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Calculus BC EIIIIlI!art B I

85. If a trapezoidal sum overapproximates J:f(X) dx, and a right Riemann sum underapproximates J:f(X) dx,

which of the following could be the graph of Y = f(x) ?

(A) Y (B) Y

4

3

2

1

x x1 2 3 4 1 2 3 4

(C) Y (D) Y

4 4

3 32 2

1 1x x0 1 2 3 4 0 1 2 3 4

(E) Y

4

3

2

1x

1 2 3 4

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2003 EXAM

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Calculus Be

86. Let f be the function with derivative defmed byf '(x) = sin(x 3 ) on the interval -1.8 < x < 1.8. How many

points of inflection does the graph of f have on this interval?

(A) Two (B) Three (C) Four (D) Five (E) Six

87. A particle moves along the x-axis so that at any time I ~ 0, its velocity is given by ve,) = cos(2 - 12

).

The position of the particle is 3 at time t = O. What is the position of the particle when its velocity is firstequal to O?

(A) 0.411 (B) 1.310 (C) 2.816 (D) 3.091 (E) 3.411

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CalculusBC~artB I

88. On the closed interval [2. 4]. which of the following could be the graph of a function f with the property that

4 ~ 2 J:/(t) dt = 1?

(A) y

4

3

2

1

x0 1 2 3 4

(B) Y

4

3

2

OL--+--+---lo--+--X

234

(e) Y (D) Y4 4

3 3

/2 2

1 1

x x0 1 2 3 4 0 1 2 3 4

(E) Y

4

3

2

1

O'-----+---+--+--X

1 2 3 4

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~artB ICalculus Be

89. The region bounded by the graph of y = 2x - x 2 and the x-axis is the base of a solid. For this solid, each crosssection perpendicular to the x-axis is an equilateral triangle. What is the volume of the solid?

(A) 1.333 (B) 1.067 (e) 0.577

y

(D) 0.462 (E) 0.267

90. The graph of f'. the derivative of the function f. is shown above. If f(O) =O. which of the following must betrue?

I. f(O) > f(1)

II. f(2) > f(1)

III. f(l) > f(3)

(A) I only

(B) II only

(C) III only

(D) I and IT only

(E) II and III only



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CalculusBC~artB

91. The height h, in meters, of an object at time 1 is given by h(l) = 241 + 2413/2 - 161 2• What is the height of the

object at the instant when it reaches its maximum upward velocity?

(A) 2.545 meters

(B) 10.263 meters

(C) 34.125 meters

(D) 54.889 meters

(E) 89.005 meters

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~artB ICalculus BC

92. Let f be the function defined by f(x) = x + In x. What is the value of c for which the instantaneous rate ofchange of f at x = c is the same as the average rate of change of f over [1,4] ?

(A) 0.456 (B) 1.244 (C) 2.164 (D) 2.342 (E) 2.452

END OF SECTION I

AFTER TIME HAS BEEN CALLED, TURN TO THE NEXT PAGE ANDANSWER QUESTIONS 93-96.

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Calculus Be~artB I

93. Which graphing calculator did you use during the examination?

(A) Casio 6300, Casio 7300, Casio 7400, Casio 7700, TI-73, TI-80, or TI-81(B) Casio 9700, Casio 9800, Sharp 9200, Sharp 9300, TI-82, or TI-85(C) Casio 9850, Casio FX 1.0, Sharp 9600, Sharp 9900, TI-83ffI-83 Plus, or TI-86(0) Casio 9970, Casio Algebra FX 2.0, lIP 38G, HP 39G, lIP 40G, lIP 48 series, lIP 49 series, or TI-89(E) Some other graphing calculator

94. During your Calculus BC course, which of the following best describes your calculator use?

(A) I used my own graphing calculator.(B) I used a graphing calculator furnished by my school, both in class and at home.(C) I used a graphing calculator furnished by my school only in class.(0) I used a graphing calculator furnished by my school mostly in class, but occasionally at home.(E) I did not use a graphing calculator.

95. During your Calculus BC course, which of the following describes approximately how oftena graphing calculator was used by you or your teacher in classroom learning activities?

(A) Almost every class(B) About three-quarters of the classes(C) About one-half of the classes(0) About one-quarter of the classes(E) Seldom or never .

96. During your Calculus BC course, which of the following describes the portion of testing time youwere allowed to use a graphing calculator?

(A) Allor almost all of the time(B) About three-quarters of the time(C) About one-half of the time(D) About one-quarter of the time(E) Seldom or never

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Calculus Be

CALCULUSBCSECTION II

Time - I hour and 30 minutesPercent of total grade - 50

Part A: 45 minutes, 3 problemsPart B: 45 minutes, 3 problems

PARTA (A graphing calculator is required for some problems or parts ofproblems.)

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

On Part A, you arc permitted to use your calculator to solve an equation. find the derivative of a function at a point, or calculatethe value of a definite integral. However, you must clearly indicate the setup of your problem, namely the equation, function. orintegral you are using. If you use other built-in features or programs, you must show the mathematical steps necessary to produceyour results.

PART B (No calculator is allowed for these problems.)

The problems for Part B are printed in the blue insert only. When you arc told to begin, open the blue insert, and writeyour solution to each part of each problem in the space provided for that part in the pink test booklet. During the timed portionfor Part B, you may keep the green insert and continue to work on the problems in Part A without the use of any calculator.

GENERAL INSTRUCTIONS FOR SECTION II PARTAAND PARTBFor each part of Section II, you may wish to look over the problems before starting to work on them. since it is not expected thateveryone will be able to complete all parts of all problems. All problems are given equal weight. but the parts of a particularproblem are not necessarily given equal weight.

• YOU SHOULDWRITEALL WORK FOREACH PARTOFEACH PROBLEMINTIlESPACEPROVIDED FORTIIATPART IN THE PINK TEST BOOKLET. Be sure to write clearly and legibly. Ifyou make an error. you may save time bycrossing it out rather than trying to erase it. Erased or crossed-out work will not be graded.

• Show all your work. Clearly label any functions, graphs. tables, or other objects that you use. You will be graded on thecorrectness and completeness of your methods as well as your answers. Answers without supporting work may notreceive credit.

• Justifications require that you give mathematical (noncalculator) reasons.• Your work must be expressed in standard mathematical notation rather than calculator syntax.

For example, JIS

x 2dx may not be written as fnInt(X2, X, 1.5).

• Unless otherwise specified, answers (numeric or algebraic) need not be simplified.• Ifyou use decimal approximations in calculations, you will be graded on accuracy. Unless otherwise specified. your final

answers should be accurate to three places after the decimal point.• Unless otherwise specified, the domain of a function! is assumed to be the set of all real numbers x for which

!(x) is a real number.

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Roberto

Typewritten Text

p. 84 was blank

~artA ICALCULUSBC

SECTION II, Part ATime-45 minutes


Calculus Be

A graphing calculator is required for some problems or parts of problems.

y

I

-=f---------+...;;;...--xI

1. Let R be the shaded region bounded by.the graphs of y = JX and y = e-3x and the vertical line x = I,

as shown in the figure above.

(a) Find the area of R.

(b) Find the volume of the solid generated when R is revolved about the horizontal line y = 1.

(c) The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis isa rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.



2003 EXAM

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Calculus BC

yB

~artA I

- ......~------------ .......xD A

2. A particle starts at point A on the positive x-axis at time I =0 and travels along the curve from A to B to C

to D, as shown above. The coordinates of the particle's position (X(I), y(I») are differentiable functions of I,

where X'(I) =~ = -9COs( r;:) sin(H~) and y'(I) =~ is not explicitly given. At time I = 9, the

particle reaches its final position at point D on the positive x-axis.

(a) At point C, is : positive? At point C, is :: positive? Give a reason for each answer.

(b) The slope ofthe curve is undefined at point B. At what time I is the particle at point B?

(c) The line tangent to the curve at the point (x(S), yeS») has equation y = ~ x - 2. Find the velocity vector

and the speed of the particle at this point.

(d) How far apart are points A and D, the initial and final positions, respectively, of the particle?

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2003 EXAM

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~artA Iy

""o~-------------f-------x

Calculus Be

3. The figure above shows the graphs of the line x =j y and the curve C given by x =~1 + l. Let S be the

shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P.

(a) Find the coordinates of point P and the value of : for curve C at point P.

(b) Set up and evaluate an integral expression with respect to y that gives the area of S.

(c) Curve C is a part ofthe curve x 2 - y2 = 1. Show that x 2 - y2 =1 can be written as the polar equation

r 2 = 1 .cos2e ,.... sin 2e

(d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar angle e thatrepresents the area of S.

END OF PART A OF SECTION II

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Calculus BC

CALCULUSBCSECTION II, Part B

Time-45 minutes


No calculator is allowed for these problems.

y

2

(-3. 1)

(4. -2)

Graph of f'

~artB I

4. Let j be a function defmed on the closed interval -3 S x S 4 with j(O) = 3. The graph of j'. the derivative

of f, consists of one line segment and a semicircle. as shown above.

(a) On what intervals. if any. is j increasing? Justify your answer.

(b) Find the x-coordinate of each point of inflection of the graph of j on the open interval -3 < x < 4. Justifyyour answer.

(c) Find an equation for the line tangent to the graph of j at the point (0.3).

(d) Find j(-3) andj(4). Show the work that leads to your answers.

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~artB ICalculus Be

5. A coffeepot has the shape of a cylinder with radius 5 inches. as shown in the figure above. Let h be the depth ofthe coffee in the pot, measured in inches, where h is a function of time t, measured in seconds. The volume Vof coffee in the pot is changing at the rate of -5rrJli cubic inches per second. (The volume V of a cylinder withradius r and height h is V = rrr 2h.)

dh Jli(a) Show that dt = - T .

(b) Given that h = 17 at time t = 0, solve the differential equation ~~ = -1 for h as a function of t.


6. The function f is defined by the power series

00 (_l)n x 2n x2 x 4 x6 (_I)n x 2n

f(x) =Io (2n + I)! =1 - 3T + Sf -7T + ... + (2n + I)! + '"

for all real numbers x.

(a) Find f '(0) and f "(0). Determine whether f has a local maximum, a local minimum, or neither at x = O.

Give a reason for your answer.

(b) Show that 1 - ~! approximates f(l) with errorless than 160 .(c) Show that y = f(x) is a solution to the differential equation xy' + Y = cos x.

END OF EXAM

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Chapter III: Answers tothe 2003 AP Calculus AB andCalculus BC Exams• Section I: Multiple Choice

• Section I Answer Key and Percent AnsweringCorrectly

• Analyzing Your Students' Performance on theMultiple-Choice Section

• Diagnostic Guide for the 2003 AP Calculus AB andCalculus BC Exams

• Section II: Free Response• Comments from the Chief Reader• Commentary, Scoring Guidelines, and Sample

Student Responses

• AB/BC Question 1• AB Question 2• AB Question 3• AB/BC Question 4• AB/BC Question 5• AB Question 6• BC Question 2• BC Question 3• BC Question 6

Section I: Multiple ChoiceOn the following page are the correct answers to themultiple-choice questions, the percent ofAP students whoanswered each question correctly by AP grade, and the totalpercent answering correctly.

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Section I Answer Key and Percent Answering Correctly

Calculus AB Part ATotal

Item Correct Percent Correct by Grade PercentNo. Answer 5 4 3 2 1 Correct

1 E 99 98 97 94 75 932 D 84 62 40 23 12 453 E 65 47 37 30 22 414 D 96 92 86 79 55 835 D 86 72 57 43 23 586 C 97 92 85 76 55 827 B 95 87 69 42 18 64

8 B 95 84 67 47 22 659 A 82 60 40 23 9 4410 8 98 93 79 61 35 7511 C 62 46 35 25 14 3712 E 94 86 76 67 49 7613 A 96 89 78 65 41 7514 E 97 92 83 69 39 77

15 D 90 74 55 36 16 5616 C 86 68 52 40 30 5617 A 83 58 34 18 11 4218 A 75 61 52 41 27 5219 D 69 45 31 23 18 3820 D 78 53 37 29 26 4521 A 79 54 31 16 9 3922 D 80 52 34 22 13 4123 E 55 35 24 17 12 2924 C 95 87 68 46 24 6625 E 94 84 70 51 24 6726 8 90 76 57 38 20 5827 8 46 18 9 8 13 1828 E 72 43 23 11 6 32

Calculus AB Part BTotal

Item Correct Percent Correct by Grade PercentNo. Answer 5 4 3 2 1 Correct76 C 97 94 87 73 38 7977 C 55 26 13 8 10 2378 C 94 81 63 45 33 6579 D 86 69 52 37 23 5580 8 63 38 25 18 11 3181 D 83 68 53 37 21 5482 A 73 52 33 19 11 3983 A 79 66 51 33 15 5084 A 65 34 17 9 6 2685 A 77 57 43 30 19 4686 8 79 50 26 14 12 37fJ7 B 79 61 44 30 19 4788 C 82 62 42 24 12 4689 D 84 56 30 13 7 3990 8 92 80 60 37 17 5991 E 54 26 17 12 9 2492 D 88 70 49 32 18 53

92

Calculus BC Part ATotal

Item Correct Percent Correct by Grade PercentNo. Answer 5 4 3 2 1 Correct

1 E 99 98 97 94 84 962 C 82 70 60 49 33 673 A 98 94 88 76 51 884 D 94 86 76 59 34 795 C 88 69 51 35 23 656 C 87 73 62 46 30 697 C 43 25 20 19 14 298 B 94 87 77 64 39 819 A 91 80 68 54 29 7410 C 61 42 32 25 15 4311 D 64 38 31 24 19 4412 E 95 90 85 79 66 fJ7

13 A 96 90 84 73 55 86

14 E 66 45 33 22 13 4515 B 84 61 43 25 13 5816 C 94 82 72 59 45 7817 A 63 32 16 6 5 3618 C 44 18 10 5 4 2419 D 79 60 47 36 24 5920 D 43 19 13 11 10 2621 8 53 28 20 14 14 3322 E 48 24 14 8 4 2823 B 88 75 64 49 28 7024 D 63 38 28 20 18 4225 D 86 65 49 33 19 6226 D 90 79 67 57 41 7527 E 71 46 32 22 14 4728 D 62 36 23 15 9 39

Calculus BC Part BTotal

Item Correct Percent Correct by Grade PercentNo. Answer 5 4 3 2 1 Correct76 E 88 78 71 61 44 7577 A 88 74 64 57 54 7478 C 95 87 75 61 41 8079 D 88 71 55 36 20 6680 A 98 94 87 78 58 8981 A 79 63 52 41 28 6282 A 76 57 42 28 17 5483 E 96 87 76 59 37 8084 C 86 63 47 34 23 6285 A 88 73 60 45 29 6986 C 60 41 32 27 24 44fJ7 C 77 42 25 13 9 4688 C 84 68 56 41 22 6589 D 74 44 26 12 7 4590 8 80 64 49 32 18 5991 B 61 41 30 20 13 4292 C 81 54 35 18 13 53

2003 EXAM

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Scoring Guideline for AB/BC Question 1

Let R be the shaded region bounded by the graphs of 11 = JX nnd 11 = c-:l r and

the "crticnl line x = I. Ill' shown in thc figure Ilbm·c.

(a) Find the arcn of R.

(b) Find the volume of the solid gencrated when R is re\'Oh'ed about thc horizontal

line 11 = 1.

(c) The region R is the base of a solid. For this solid, each cross section

perpendicular to the x-axis is a rectangle whose height is 5 times the length of its

base in region R. Find the "olul1lc of this solid.

Point of intersection

e-;!.r = .JX at (T, S) = (0.238734, 0.488604)

(a) Area = h: (5 - e-3.r )d:z:

= 0.442 or 0.443

1: Correct limits in an integral in

(a). (b), or (c)

11 : integrand

.J .- . 1: answer

(b) Volume = irh~((1-e-;!r):.! -(I-,JX):.!)d:Z~

= 0.453 ir or 1.42:~ or 1.424

(c) Length = ...IX - e--3,,.

Height = ;)(5 - c-3.r )

fl r- 'J:1Volume - J7'5(vx _c-..r) dx = 1.554

3:

3:

2 : integrand

< -1 > reversal

< -1 > error with constant

< -1 > omits 1 in one radius

< - 2 > other errors

1 : answer

2 : integrand

<: - 1 > incorrect but has

...IX - e-:h

as a factor

1 : answer

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Scoring Guideline for AB Question 2

A particle mo\'es along the :v-a.'i:is so that. its velocity at time t is gh'en by

tI (t) = - (t + 1) sin ( t; ).At time t = 0, the particle is at position x = l.

(a) Find the acceleration of the particle at time t = 2. Is the speed of the pnrticle increasing at. t = 2'1

"~hy or why not'?

(b) Find /\11 times t in the open interval 0 < t < 3 when the particle changes direction. .Justify your

allSwer.

(c) Find the total distance trnveled by the particle from time t = 0 until time t = 3.

(d) During the time interval 0 :::; t :::; 3. what is the greatest distance between the particle and the

origin'? Show the work that leads to your answer.

(a) a(2) = 11'(2) = 1.587 or 1.588

v(2) = -3sin(2) < 0

Speed is decreasing since a(2) > 0 and 0(2) < o.

2

(b) v(t) = 0 when t~ = ir

t = .J2T. or 2.506 or 2.507

Sincev(t) < 0 for 0 < t < ...rEi and 'v(t) > 0 for

.../2i < t < 3, the particle changes directions at

t = .J'Fi,

13

(c) Distance = Iv(t)ldt = 4.333 or 4,334II

(d) fu-lEr,1(t)dt = -3,265

.IEx(~) = x(O) +1 .. v(t)dt = -2.265

II

Since the total distance from t = 0 to t = 3 is

4.334, the particle is still to the left of the origin

at t = 3. Hence the greatest distance from the

origin is 2.265.

I1 : a(2)

2: 1 : speed decrCllSing

with reason

I1 : t = .J2ii only2:

1 : justification

j1 : limits

:3: 1 : integrand

1 : answer

j 1:± (distance particle travels

2: while velocity is negative)

1: answer

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2003 EXAM

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I Rill(minules) (g3l10ns peT minute)

0 20

30 30

40 40

50 55

70 65

90 70

(a) Use data from the table to find an approximation

for R'(45). Show the computations that lead to

The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a

twice-differentiable and strictly increasing function R Rill

of time t. The graph of Rand n table of selected 1:values of R (t ), for the time interval 0 $ t :5 90 ~ 50

c8 -10

~ 30minutes, are shown above.

o 10 20 30 .w 50 60 70 80 90your answer. Indicate units of measure. 'lime

(b) The rate of fuel consumption is increasing fastest at time t = 45 minutes. What is the value of

R"(45)? Explain your rensoning.

l!lll

(c) Approximate the value of R(t)dt using a left Riemann sum with the five subintervals indicatedII

by the data in the table. Is this numericnl approximation less than the value of 1!J() R(t)dt ?(I

(d)

Explain your reasoning.

For 0 < b :5 90 minutes, explain the meaning of Ib R ( t )dt in terms of fuel consumption for the()

plane. Explain the ~eaning of ifob R (t )dt in terms of fuel consumption for the plane. Indicate

units of measure in both answers.

(a) R'(45) :::::: R(50) - R(40) = 55 - 40 1 : a differcnce quoticnt using50 - 40 10 numbcrs from tablc and

= 1.5 gal/min:? 2:interval that contains 45

1 : 1.5 gal/min2

(b) R"(45) = 0 since R'(t) has a ma.ximum at !1 : R"(45) = 02:

t = 45. 1 : rea<;on

190 !I : value of left Riemann sum(c) o R(t) dt ::::: (30)(20) + (10)(30) + (10)(40) 2:

I : "less" with reason+(20)(55) + (20)(65) = 3700

Yes, this approximation is less because the

graph of R is increasing 011 the interval.

(d) fob R(t) dt is the total amount of fuel in2 : mcanings

.f'gallons consumed for the first b minutes. I : meaning of R(t) dt()

llb 3: Ill'- R(t)dt is the average value of the rate of I : meaning of b () R(t) dtb 0

fuel consumption in gallons/min during the < - 1 > if no reference to time b

first b minutes. I : units in both answers

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Lct I be a function defined 011 the closed interval -3 ::; 3: ::; .. with

I (0) = 3. The graph of I'. the derivative of I. consists of one line

segmcnt and a semicircle. as shown abovc.

(a) On what inten·als. if any. is I increasing? Justify your answer.

(b) Find the vcoordinate of each point of inflection of the graph of Ion the open interval -3 < x < 4. Justify your answer.

(c) Find an equation for the line tangent to the graph of I at the

point (0.3).(d) Find I (-3) and I (4). Show the work that leads to your answers.

(-3. II

--~--~f--"":"+"~-~~--.\

(-1.-2)

Graph off'

(a) The function lis increasing on [-3.-2J since

f' > 0 for -3 ~ x < -2.

(b) :1: = 0 and x = 2

I' changes from decreasing to increasing at

x = 0 and frOln increasing to decreasing at

x=2

1 : interval

1 : reason

1 : x = 0 and ;,; = 2 only

1 : justification

(c) 1'(0) = -2

Tangent line is y. = -2:1: + 3.

Ju ,

(d) 1(0) - 1(-3) = _:/ (t)dt.

113= 2(1)(1) - 2(2)(2) = -2

3 91(-3) = 1(0) + 2" = 2

1(4) - 1(0) = J~I I'(t)dt

=-(8 - ~(2)21i") = -8 + 21i"

1(4) = 1(0) - 8 + 21i" = -5 + 21i"

1 : equation

1 · + (! - ?). - 2 -

(difference of areas

of triangles)

1 : answer for J(-3) using FTC

4 :

(1 ,.)1: ± 8 - 2(2)-1i"

(area of rectangle

- area of scmicircle)

1 : answer for 1(4) using FTC

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A coffeepot has the shape of a cylinder with radius 5 inches. as shown in the figure

above. Let It be the depth of the coffee in the pot, measured in inches. where It is a

function of time t. measured in seconds. The volume F of coffee in the pot is

changing at the rate of -51r.Jii cubic inches per second. (The volume 11 of a cylinder

with radius l' and height h is F = 1r1'2h. )

dh .Jii(a) Show that - = --dt 5 .

(b) Given that h = 17 at time t = 0, solve the differential equation ddh

= -..r: fort <>

h as a fWlction of t.


,,:0.,,:';;:<+:11II in

:.':, _.....-_.... ·:·>;:;··1

(a) V = 257rh

dV dh .- = 257r- = -57r.J7idt dt

dh -57r.JTi .J7idt = 257r = --5-

(b) dh = _ .Jfidt 5

3:

5:

dV1: dt = -57r./Ji

dV1 : computes dt1 : shows result

1 : separates variables

1 : antiderivatives

1 : constant of integration

1 : uses initial condition h = 17

when t = 0

12.J7i = -st + C

2m = 0 + C

t = 10m

1 : solves for II

Note: max 2/5 11-1-0-0-0] if no constant

of integration

Note: 0/5 if no separation of variables

1 : answer

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Let f be the function defined by

j..rx+T for 0 ~ x ~ 3f(x) =

5 - x for 3 < x ~ 5.

(a) Is f continuous at x = 3'? Explain why or why not.

(b) Find t.he average value of f(x) on t.he closed interval 0 ~ x ~ 5.

(c) Suppose the function 9 is defined by

jk..rx+T for 0 ~ x ~ 3g(x) =. mx + 2 for 3 < x ~ 5.

where k and m are constants. If 9 is differentiable at x = 3, what are the values of k and TTl?

(a) f is continuous at x = 3 because

lim f(3.:) = lim f(3.:) = 2.r-3- r-3+

Therefore, lim f(x) = 2 = f(3) ..r-3

2:

1 : answers "yes" and equates the

\'alues of the left- and right-hand

limits

1 : explanation involving limits

(b) fa f(x)dx = f3 f(x) dx + fa f(x)d3.:Jo JIJ J 3

= ~ (3.: + 1)% 13

+ (5X _ ~ x2) I

r

,3 u 2:\

=(16_~) (25_21)=203 3 + 2 2 3

1 J:5 4Avcrage value: ~ f(x)dx = :-i) IJ 3

1 : k f3 f(x)dx + k f'j f(x)dxJu J3

(where k ~ 0)

4: 1: antiderivative of .Jx + 1

1 : antiderivative of 5 - x

1 : evaluation and answer

(c) Sinee 9 is continuous at x = 3, 2k = 3m + 2., /.)k for 0 < x < 39 (x) = - x + 1

m for 3 < x < 5

I· '() k I I' '()r_n.~Il.- 9 X = -4 an( nil 9 x = m

.r-3+

Sincc these two limits exist and 9 is

differentiable at x = 3. the two limits are

I 1'} kequa . lUS 4 = m.

•J 88m = 3m + 2; m = .:. and k = -

5 5

3:

1 : 2k = 3m + 2

kl:-=m

41 : values for k and m

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J~.

Scoring Guideline for Be Question 2

A particle starts at point A on the positive :I;-axis at time t = 0 and travels

along the curve from A to B to C to D, as shown above. The coordinates of

the particle's position (x (t). y (t ») are differentiable functions of t. where

, dx 9 ( Ilt) . (rr..['[+T) d' dy . I' . I .x (0 = dt = - cos 6"" sm 2 an y (t) = dt IS not. exp IClt y given.

At time t = 9, the particle reaches its final position at. point D on the positive x-axis.

() ' . C· dy ..? A . C' dx ..? G' t' ha .'1.t pomt . IS -i' positive. t pomt ,IS -d poslttve. Hoe a reason lor eac answer.ct t

(b) The slope of the curve is undefined at point B. At what time t is the particle at point B '!

(c) The line tangent to the curve at the point (x(8), y(8») has equation y = ~x - 2. Find the

velocit.y vect.or and the speed of the particle at this point.

(d) How far apart are points A and D, the initial and final positions, respectively. of the particle?

( ) A 0 C ely . .. b ( ) .a t pomt 'dl IS not positive eeause y l IS

decreasing along the arc BD a.'; t increases.

A . C elx . .. b ( ) .t pomt , -d IS not positive ecause x t IS.[

decreasing along the arc BD as {increases.

/1

dy . . . 1: -1- not pOSItive Wit I reason

2: d

1d:r.: 0 0 • 1

: -el not pOSItIve Wit 1 reasont

(b) dx (7it ) . (rrJT+I)dt = 0; cos 6" = 0 or sm 2 = 0

rrt rr rr..rt+I- = ? or 2 = rr; t = 3 for both.6 _

Particle is at point B at I = 3.

elx1 : sets - = 0

dt1 : t = 3

(c) x'(8) = -9cos(~)Sin(3;) =-~

y'(8) dy 5x'(8) = d:r.: = 9

y'(8) = ~x'(8) = -~9 2

The velocity vector is < -4.5,-2.5 >.

Speed = ../4.52 + 2.52 = 5.147 or 5.148

1 : :1:'(8)

1 : y'(8)

1 : speed

(d) x(9) - .r.(O) = fo9 x'(l)dt

= -39.255The initial and final positions are 39.255 apart.

11 : integral

2 :1 : answer

147

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5The figure ahove shows the graphs of the line x = '3 y and the curve C given by y

x = J1 + !J"!. Let S be the shaded region bounded by the two graphs ami the

va-xis. The line and the curve intersect at point P.

elx .(a) Find the coordinates of point P and the value of -/ for curve Cat. pomt P.

l!l

(b) Set up and evaluate an integral expression with respect to y that gives the

area of S. I

(c) Curve C is a part of the curve x2 - y2 = 1. Show that x2 - y2 = 1 can he "Titten as the polar

.., 1equation r- = ., . .) .

cos- 0 - SIn- 0

(d) Use the polar equation given in part (c) to set up lUI integral expression with respect to the polar

angle 0 that represents the area of S.

(a)5 .)

At P, 3y = .,Jl + y~, sO

S. 5 5lllce x = - y, x = -.

3· 4

3y = 4'

1 : coordinates of P

1 t!;,; P: -J. at

(.1/

(b) Area f%( 5 )= Jo ../1 + y2 - 31/ dy

= 0.346 or 0.347

1 : limits

1 : integrand

1 : answer

(c)

(d)

x = l' cos 0; y = l' sin 0

x2 _ y2 = 1 => 1'2 cos2 () - 1.2 sin2 0 = 1

., 11'~ = 'J • .,

cos- 0 - sur 0

Let {3 be the angle that segment OP makes with

1/ % 3the x-axis. Then tan (3 = ~ = %= '5'

Itan- 1(%) 1 .,

Area = - 1'~ dO(J 2

11Ian-I(%) 1= - .). dO

2 0 cos~ () - sin2 ()

2 :

2 : I

1: substitutes x = rcosO and

1/ = l' sin () into x2- 1/ = 1

1 : isolates 1'2

1 : limits

1 : integrand and constant

155

2003 EXAM

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The function 1 is defined 1>y the power series

::lO (-1)" :1:2/1 x'.! xl x 6 ( _1)/1 X'.!II

I(x) = ~(211 +1)! = 1- 3T + 5f - 7! + ... + (27/ +1)! + ...

for all real numbers a:.

(a) Find /,(0) and 1"(0). Determine whether 1 has a local maximum, a local minimum, or neither at

x = O. Give a reason for your answer.

(1)) Show that 1 - ;! approximates 1(1) with error less than 1~0'

(c) Show that y = 1(3:) is a solution to the differential equation xy' + y = cosx.

(a) f'(O) = coefficient of x term = 0

f"(O) = 2(coefficient of x2 term) = 2(- ;!) '-}f has a local ma....imum at 3: = 0 because 1'(0) = 0 and

1"(0) < o.

4:

1 : f'(O)

1 : f"(O)

1 : critical point answer

1 : reason

1 1 1 (-1)"(b) f(l) = 1 - - + - - - + ... + + ...

3! 5! 7! (211 + I)!

This is an alternating series whose terms decrease in

absolute value with limit O. Thus, the error is less than the

first omitted term. so If(l) - (1 - ;!)I ::; ;! = 1~0 < 1~0'

11 : error bound < 100

(c), 2x 4x:1 6x5 ( -1)"2nx2,,-1

y = - 3! + 5! -7T + ... + (211 + I)! + ..., 2:/:2 4x4 6xlJ (-1 )1I2nx211

xy = - 3! + 5f -7T + ... + (21/ + I)! + ...

xy' + Y = 1 - (~ + .!.) x2-l- (i. + .!.) :r,1 - (.Q. + '!')xG + ...

3! 3! '5! 5!' 7! 7!

( 1)11 (21/ 1) 211+ - + x + ...(2n + I)! (21/ + I)! '

1 ') 1 , 1 lJ ( -I)" ?/,= 1 - -:r- + - x" - - 'r) + ... +-- x- + ...2!" 4!' 61" (2n)!

= cos:/;

OR

1 : series for y'

1 : series for :ry'

4:

1 : series for xy' + Y

1 : identifies series as cos x

OR

X3 1

f( ) . ( I)" '),,-1xy = X X = X - - + ... + - x- . + ...3! (2n + I)!

= sin:r

xy' + 11 = (xy)' = (sin x)' = cos :/:

4 :

1 : series for xf(x)

1 : identifies series as sin:1.'

1 : handles xy' + y

1 : makes connection

163

2003 EXAM

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Table 4.2AB-Calculus AB Scoring Worksheet

Section I: Multiple Choice

I - (114 x --- )1 x 1.2000 = _Number Correct Number Wrong Weighted

(out of 45) Section I Score(If less than zero, enter

zero; do not round)

Section II: Free Response

Question 1 ______ X 1.0000 = _ ....... _(out of 9)

Question 2 ______ X 1.0000 = _'-- _(out of 9) (Do not round)

Question 3 ______ X 1.0000 = _~...;..".;..........,....;.;_

(out of 9) (DOnot round)

Question 4 ______ X 1.0000 = _'-- _

AP Grade Conversion Chart

Calculus AB

(out of 9) (1)ono~.tound) CompositeScore Range" APGrade

54321

66-10847-6529-4616-28

0-15

• The students' scores are weighted according toformulas determined in advance each year by theDevelopment Commlttee to yield raw compositescores; the Chief Reader IS responsible for converting composite scores to the 5-point AP scale

(Do not round)

Sum = _

(out of 9)

(out of 9)

______ X 1.0000 = _..;..;.;,;. ...;......,--

______ X 1.0000 = _

Question 6

Question 5

WeightedSection II Score(Do not round)

Composite Score

-------+-------=-------Weighted

Section I ScoreWeighted

Section II ScoreComposite Score(Round to nearest

whole number)

173

2003 EXAM

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Table 4.2BC-Calculus BC Scoring Worksheet

Section I: Multiple Choice

[ - (1/4 x )] x 1.2000 = _

Number Correct Number Wrong Weighted(out of 45) Section I Score

(If less than zero, enterzero; do not round)

Section II: Free Response

Question 1

Question 2

______ X 1.0000 = __........................_(out of 9)

______ X 1.0000 = _

(out of 9) (Do nottoUno)

'-''";'''';::<:::::':/I:

".'--'.,-,,-~'

Question 3 ______ X 1.0000 = .........._

(out of 9) (Do not round)

Question 4 ______ X 1.0000 = ' '..."....'''.....~''._'_

AP Grade Conversion Chart

Calculus BC

(out of 9) (Do not ro~rid) CompositeScore Range* AP Grade

______ X 1.0000 = -

______ X 1.0000 = ............_• The students' scores are weighted according to

formulas determined in advance each year by theDevelopment Committee to Yleld raw compositescores: the Cluef Reader is responsible for converting composite scores to the 5,pomt AP scale,

Question 5

Question 6

(out of 9)

(out of 9)

(Do not round)

(Do not round)

Sum = _

WeightedSection II Score(Do not round)

64-10853-6336-5226-350-25

54

321

Composite Score

-------+-------=-------

176

WeightedSection I Score

WeightedSection II Score

Composite Score(Round to nearest

whole number)

2003 EXAM

of 107

2003calculusab-bcexams

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