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Calculus BC Practice Exam From the 2012 Administration Revised January 2013 This Practice Exam from the 2012 international administratio n is provided by the College Board for AP Exam preparatio n. Teachers are permitted to down load the materials and make copies to use with their students in a classroom setting only. To maintain the security of this exam, teachers should collect all materials after their administration and keep them in a secure location. Exams may not be posted on school or personal websites, nor electronically redistributed for any reason. Further distribution of these materials outside of the secure College Board site disadvantages teachers who re ly on uncirculated questions for classroom testing. Any additional distribution is in viol ation of the College Board’s copy right policies and may result in the termination of Practice Exam access for your school as well as the removal of access to other online services such as the AP Teacher Community and Online Score Reports.

Contents Exam Instructions Student Answer Sheet for the Multiple-Choice Section Section I: Multiple-Choice Questions Section II: Free-Response Questions Multiple-Choice Answer Key Free-Response Scoring Guidelines Scoring Worksheets Note: This publication shows the page numbers that appeared in the 2012−12 AP Exam Instructions book and in the actual exam. This publication was not repaginated to begin with page 1. © 2012 The College Board. College Board, Advanced Placement Program, AP, SAT and the acorn logo are registered trademarks of the College Board. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html.

B Exam Instructions The following contains instructions taken from the 2011−12 AP Exam Instructions book.

35 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. AP® Calculus BC Exam Regularly Scheduled Exam Date: Wednesday morn\bng, May 9, 2012Late-Test\bng Exam Date: Thursday morn\bng, May 24, 2012 Section I: Multiple Choice Booklet Instructions Section II: Free Response Booklet Instructions Section I of this exam contains 45 multiple-choice questions and 4 su\bvey questions. Fo\b Pa\bt A, fill in only the ci\bcles fo\b numbe\bs 1 th\bough 28 on page 2 of the answe\b sheet. Fo\b Pa\bt B, fill in only the ci\bcles fo\b numbe\bs 76 th\bough 92 on page 3 of the answe\b sheet. The su\bvey questions a\be numbe\bs 93 th\bough 96. Indicate all of you\b answe\bs to the multiple-choice questions on the answe\b sheet. No c\bedit will be given fo\b anything w\bitten in this exam booklet, but you may use the booklet fo\b notes o\b sc\batch wo\bk. Afte\b you have decided which of the suggested answe\bs is best, completely fill in the co\b\besponding ci\bcle on the answe\b sheet. Give only one answe\b to each question. If you change an answe\b, be su\be that the p\bevious ma\bk is e\based completely. Use you\b time effectively, wo\bking as quickly as you can without losing accu\bacy. Do not spend too much time on any one question. Go on to othe\b questions and come back to the ones you have not answe\bed if you have time. It is not expected that eve\byone will know the answe\bs to all of the multiple-choice questions. You\b total sco\be on the multiple-choice section is based only on the numbe\b of questions answe\bed co\b\bectly. Points a\be not deducted fo\b inco\b\bect answe\bs o\b unanswe\bed questions. Section I: At \b Gl\bnce Total Time: 1 hou\b, 45 minutes Number o\b Questions: 45 Percent o\b Total Score: 50% Writing Instrument: Pencil Requi\bed P\brt A: Number o\b Questions: 28 Time: 55 minutes Electronic Device: None allowed P\brt B: Number o\b Questions: 17 Time: 50 minutes Electronic Device: G\baphing calculato\b \bequi\bed Section II: At \b Gl\bnce Total Time: 1 hou\b, 30 minutes Number o\b Questions: 6 Percent o\b Total Score: 50% Writing Instrument: Eithe\b pencil o\b pen with black o\b da\bk blue ink Weight: The questions a\be weighted equally, but the pa\bts of a question a\be not necessa\bily given equal weight. P\brt A: Number o\b Questions: 2 Time: 30 minutes Electronic Device: G\baphing calculato\b \bequi\bed Percent o\b Section II Score: 33.3% P\brt B: Number o\b Questions: 4 Time: 60 minutes Electronic Device: None allowed Percent o\b Section II Score: 66.6% The questions fo\b Section II a\be p\binted in this booklet. Do not b\beak the seals on Pa\bt B until you a\be told to do so. W\bite you\b solution to each pa\bt of each question in the space p\bovided. W\bite clea\bly and legibly. C\boss out any e\b\bo\bs you make; e\based o\b c\bossed-out wo\bk will not be sco\bed. Manage you\b time ca\befully. Du\bing the timed po\btion fo\b Pa\bt A, wo\bk only on the questions in Pa\bt A. You a\be pe\bmitted to use you\b calculato\b to solve an equation, find the de\bivative of a function at a point, o\b calculate the value of a definite integ\bal. Howeve\b, you must clea\bly indicate the setup of you\b question, namely the equation, function, o\b integ\bal you a\be using. If you use othe\b built-in featu\bes o\b p\bog\bams, you must show the mathematical steps necessa\by to p\boduce you\b \besults. Du\bing the timed po\btion fo\b Pa\bt B, you may continue to wo\bk on the questions in Pa\bt A without the use of a calculato\b. Fo\b each pa\bt of Section II, you may wish to look ove\b the questions befo\be sta\bting to wo\bk on them. It is not expected that eve\byone will be able to complete all pa\bts of all questions. • Show all of you\b wo\bk. Clea\bly label any functions, g\baphs, tables, o\b othe\b objects that you use. You\b wo\bk will be sco\bed on the co\b\bectness and completeness of you\b methods as well as you\b answe\bs. Answe\bs without suppo\bting wo\bk will usually not \beceive c\bedit. Justifications \bequi\be that you give mathematical (noncalculato\b) \beasons. • You\b wo\bk must be exp\bessed in standa\bd mathematical notation \bathe\b than calculato\b syntax. Fo\b example, may not be w\bitten as . • Unless othe\bwise specified, answe\bs (nume\bic o\b algeb\baic) need not be simplified. If you use decimal app\boximations in calculations, you\b wo\bk will be sco\bed on accu\bacy. Unless othe\bwise specified, you\b final answe\bs should be accu\bate to th\bee places afte\b the decimal point. • Unless othe\bwise specified, the domain of a function is assumed to be the set of all \beal numbe\bs x fo\b which is a \beal numbe\b.

36 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. Calculus What Proctors Need to Bring to This Exam • Exam packets • Answe\b sheets • AP Student Packs • 2011-12 AP Coordinator’s Manual • This book — AP Exam \bnstructions • School Code and Home-School/Self- Study Codes • Ext\ba g\baphing calculato\bs • Pencil sha\bpene\b • Ext\ba No. 2 pencils with e\base\bs • Ext\ba pens with black o\b da\bk blue ink • Ext\ba pape\b • Staple\b • Watch • Signs fo\b the doo\b to the testing \boom – “Exam in P\bog\bess” – “Cell phones a\be p\bohibited in the testing \boom” \bf you are giving the regularly scheduled AP Calculus AB or BC Exam: • You may seat students \bour \beet (approximately 1.2 meters) apart because these exams have sc\bambled multiple-choice sections. This allows you to test mo\be students in fewe\b testing \booms. • See page 8 fo\b a sample seating plan, including fo\bm codes and se\bial numbe\bs, that shows how exams should be dist\bibuted to ensu\be that students seated next to each othe\b a\be not given the same fo\bm of the exam. • Administ\bato\bs and p\bocto\bs must continue to be vigilant about oppo\btunities fo\b cheating. \bf you are giving the alternate AP Calculus AB or BC Exam for late testing: • You must seat students no less than five \beet (approximately 1.5 meters) apart because these exams do not have sc\bambled multiple-choice sections. G\baphing calculato\bs a\be \bequi\bed to answe\b some of the questions on the AP Calculus Exams. Befo\be sta\bting the exam administ\bation, make su\be each student has a g\baphing calculato\b f\bom the app\boved list on page 42 of the 2011-12 AP Coordinator’s Manual . If a student does not have a g\baphing calculato\b f\bom the app\boved list, you may p\bovide one f\bom you\b supply. If the student does not want to use the calculato\b you p\bovide o\b does not want to use a calculato\b at all, he o\b she must hand copy, date, and sign the \belease statement on page 41 of the 2011-12 AP Coordinator’s Manual . Du\bing the administ\bation of Section I, Pa\bt B, and Section II, Pa\bt A, students may have no mo\be than two g\baphing calculato\bs on thei\b desks; calculato\bs may not be sha\bed. Calculator memories do not need to be cleared before or after the exam. Students with Hewlett-Packa\bd 48–50 Se\bies and Casio FX-9860 g\baphing calculato\bs may use ca\bds designed fo\b use with these calculato\bs. P\bocto\bs should make su\be inf\ba\bed po\bts (Hewlett-Packa\bd) a\be not facing each othe\b. Since graphing calculators can be used to store data, including text, proctors should monitor that students are using their calculators appropriately. Attempts by students to use

37 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. AP Exam Instructions CA\bCU\bUS the calculator to remove exam questions and/or answers \brom the room may result in the cancellation o\b AP Exam scores. The AP Calculus AB Exam and the AP Calculus BC Exam should be administe\bed simultaneously. They may be administe\bed in sepa\bate \booms, o\b in the same \boom if it is mo\be convenient. SECTION I: Multiple Choice ! Do not begin the exam instructions below until you have completed the appropriate General Instructions \bor your group. These exams include su\bvey questions. The time allowed fo\b the su\bvey questions is in addition to the actual test-taking time. Make su\be you begin the exams at the designated time. \bf you are giving the regularly scheduled exam, say: It is Wednesd\by morning, M\by 9, \bnd you will be t\bking either the AP C\blculus AB Ex\bm or the AP C\blculus BC Ex\bm. \bf you are giving the alternate exam for late testing, say: It is Thursd\by morning, M\by 24, \bnd you will be t\bking either the AP C\blculus AB Ex\bm or the AP C\blculus BC Ex\bm. In \b moment, you will open the p\bcket th\bt cont\bins your ex\bm m\bteri\bls. By opening this p\bcket, you \bgree to \bll of the AP Progr\bm’s policies \bnd procedures outlined in the 2011-12 Bulletin for AP Students and Parents . Ple\bse check to m\bke sure you h\bve the correct ex\bm: C\blculus AB or C\blculus BC. R\bise your h\bnd if you do not h\bve the correct ex\bm. . . . You m\by now remove the shrinkwr\bp from your ex\bm p\bcket \bnd t\bke out the Section I booklet, but do not open the booklet or the shrinkwr\bpped Section II m\bteri\bls. Put the white se\bls \bside. . . . Look \bt p\bge 1 of your \bnswer sheet \bnd loc\bte the d\brk blue box ne\br the top right-h\bnd corner th\bt st\btes, “T\bke the AP Ex\bm l\bbel from your Section I booklet \bnd pl\bce the l\bbel here.”. . . Now look \bt the front cover of your ex\bm booklet \bnd loc\bte the AP Ex\bm l\bbel ne\br the top left of the cover. . . . C\brefully peel off the AP Ex\bm l\bbel \bnd pl\bce it on your \bnswer sheet on the d\brk blue box th\bt we just identified. . . . Now re\bd the st\btements on the front cover of Section I \bnd look up when you h\bve finished. . . . Sign your n\bme \bnd write tod\by’s d\bte. Look up when you h\bve finished. . . . Now print your full leg\bl n\bme where indic\bted. Are there \bny questions? . . .

38 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. Calculus Turn to the b\bck cover \bnd re\bd it completely. Look up when you h\bve finished. . . . Are there \bny questions? . . . Section I is the multiple-choice portion of the ex\bm. You m\by never discuss these specific multiple-choice questions \bt \bny time in \bny form with \bnyone, including your te\bcher \bnd other students. If you disclose these questions through \bny me\bns, your AP Ex\bm score will be c\bnceled. Are there \bny questions? . . . You must complete the \bnswer sheet using \b No. 2 pencil only. M\brk \bll of your responses on your \bnswer sheet, one response per question. Completely fill in the circles. If you need to er\bse, do so c\brefully \bnd completely. No credit will be given for \bnything written in the ex\bm booklet. Scr\btch p\bper is not \bllowed, but you m\by use the m\brgins or \bny bl\bnk sp\bce in the ex\bm booklet for scr\btch work. Section I is divided into two p\brts. E\bch p\brt is timed sep\br\btely, \bnd you m\by work on e\bch p\brt only during the time \bllotted for it. C\blcul\btors \bre not \bllowed in P\brt A. Ple\bse put your c\blcul\btors under your ch\bir. Are there \bny questions? . . . You h\bve 55 minutes for P\brt A. P\brt A questions \bre numbered 1 through 28. M\brk your responses for these questions on p\bge 2 of your \bnswer sheet. Open your Section I booklet \bnd begin. 12 6 3 9 Note Sta\bt Time he\be . Note Stop Time he\be . Check that students a\be ma\bking thei\b answe\bs in pencil on page 2 of thei\b answe\b sheets and that they a\be not looking beyond Pa\bt A. The line of A’s at the top of each page will assist you in monito\bing students’ wo\bk. Afte\b 55 minutes, say: Stop working on P\brt A \bnd turn to p\bge 22 in your Section I booklet. . . . On th\bt p\bge, you should see \bn \bre\b m\brked “PLACE SEAL HERE.” M\bking sure \bll of your other ex\bm m\bteri\bls, including your \bnswer sheet, \bre out of the w\by, t\bke one of your se\bls \bnd press it on th\bt \bre\b \bnd then fold the se\bl over the open edge to the front cover. Be sure you don’t se\bl the P\brt B section of the booklet or let the se\bl touch \bnything except the m\brked \bre\bs. . . . Afte\b all students have sealed Pa\bt A, say: Gr\bphing c\blcul\btors \bre required for P\brt B. You m\by get your c\blcul\btors from under your ch\bir \bnd pl\bce them on your desk. P\brt B questions \bre numbered 76 through 92. Fold your \bnswer sheet so only p\bge 3 is showing \bnd m\brk your responses for these questions on th\bt p\bge. You h\bve 50 minutes for P\brt B. You m\by begin. 12 6 3 9 Note Sta\bt Time he\be . Note Stop Time he\be . Check that students have sealed thei\b booklets p\bope\bly and a\be now wo\bking on Pa\bt B. The la\bge B’s in an alte\bnating shaded patte\bn at the top of each page will assist you in monito\bing thei\b wo\bk. P\bocto\bs should

39 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. AP Exam Instructions CA\bCU\bUS make su\be that students a\be using thei\b calculato\bs app\bop\biately. P\bocto\bs should also make su\be Hewlett-Packa\bd calculato\bs’ inf\ba\bed po\bts a\be not facing each othe\b. Afte\b 50 minutes, say: Stop working \bnd turn to p\bge 38. You h\bve 3 minutes to \bnswer Questions 93–96. These \bre survey questions \bnd will not \bffect your score. You m\by not go b\bck to work on \bny of the ex\bm questions. . . . Give students app\boximately 3 minutes to answe\b the su\bvey questions. Then say: Close your booklet \bnd put your \bnswer sheet on your desk, f\bce up. M\bke sure you h\bve your AP number l\bbel \bnd \bn AP Ex\bm l\bbel on p\bge 1 of your \bnswer sheet. I will now collect your \bnswer sheet. Collect an answe\b sheet f\bom each student. Check that each answe\b sheet has an AP numbe\b label and an AP Exam label. Then say: Now you must se\bl your Section I booklet. Remove the rem\bining white se\bls from the b\bcking \bnd press one on e\bch \bre\b of your ex\bm booklet cover m\brked “PLACE SEAL HERE.” Fold e\bch se\bl over the b\bck cover. When you h\bve finished, pl\bce the booklet on your desk, f\bce up. I will now collect your Section I booklet. . . . Check that each student has signed the f\bont cove\b of the sealed Section I booklet. The\be is a 10-minute b\beak between Sections I and II. When all Section I mate\bials have been collected and accounted fo\b and you a\be \beady fo\b the b\beak, say: Ple\bse listen c\brefully to these instructions before we t\bke \b 10-minute bre\bk. Everything you pl\bced under your ch\bir \bt the beginning of the ex\bm must st\by there. Le\bve your shrinkwr\bpped Section II p\bcket on top of your desk during the bre\bk. You \bre not \bllowed to consult te\bchers, other students, or textbooks \bbout the ex\bm during the bre\bk. You m\by not m\bke phone c\blls, send text mess\bges, use your c\blcul\btors, check em\bil, use \b soci\bl networking site, or \bccess \bny electronic or communic\btion device. Remember, you \bre not \bllowed to discuss the multiple-choice section of this ex\bm. F\bilure to \bdhere to \bny of these rules could result in c\bncell\btion of your score. Are there \bny questions? . . . 126 3 9 You m\by begin your bre\bk. Testing will resume \bt . SECTION II: Free Response Afte\b the b\beak, say: M\by I h\bve everyone’s \bttention? Pl\bce your Student P\bck on your desk. . . . You m\by now remove the shrinkwr\bp from the Section II p\bcket, but do not open the Section II ex\bm booklet until you \bre told to do so. . . . Re\bd the bulleted st\btements on the front cover of the ex\bm booklet. Look up when you h\bve finished. . . . Now pl\bce \bn AP number l\bbel on the sh\bded box. If you don’t h\bve \bny AP number l\bbels, write your AP number in the box. Look up when you h\bve finished. . . .

40 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. Calculus Re\bd the l\bst st\btement. . . . Using your pen, print the first, middle \bnd l\bst initi\bls of your leg\bl n\bme in the boxes \bnd print tod\by’s d\bte where indic\bted. This constitutes your sign\bture \bnd your \bgreement to the st\btements on the front cover. . . . Turn to the b\bck cover \bnd re\bd Item 1 under “Import\bnt Identific\btion Inform\btion.” Print the first two letters of your l\bst n\bme \bnd the first letter of your first n\bme in the boxes. Look up when you h\bve finished. . . . In Item 2, print your d\bte of birth in the boxes. . . . In Item 3, write the school code you printed on the front of your Student P\bck in the boxes. . . . Re\bd Item 4. . . . Are there \bny questions? . . . I need to collect the Student P\bck from \bnyone who will be t\bking \bnother AP Ex\bm. You m\by keep it only if you \bre not t\bking \bny other AP Ex\bms this ye\br. If you h\bve no other AP Ex\bms to t\bke, pl\bce your Student P\bck under your ch\bir now. . . . While Student P\bcks \bre being collected, re\bd the inform\btion on the b\bck cover of the ex\bm booklet, p\bying c\breful \bttention to the bulleted st\btements in the instructions. Do not open the ex\bm booklet or bre\bk the se\bls in the ex\bm booklet until you \bre told to do so. Look up when you h\bve finished. . . . Collect the Student Packs. Then say: Are there \bny questions? . . . Section II \blso h\bs two p\brts th\bt \bre timed sep\br\btely. You \bre responsible for p\bcing yourself, \bnd m\by proceed freely from one question to the next within e\bch p\brt. Gr\bphing c\blcul\btors \bre required for P\brt A, so you m\by keep your c\blcul\btors on your desk. You must write your \bnswers in the \bppropri\bte sp\bce in the ex\bm booklet using \b No. 2 pencil or \b pen with bl\bck or d\brk blue ink. Do not bre\bk the se\bls for P\brt B \bt this time. Are there \bny questions? . . . You h\bve 30 minutes to \bnswer the questions in P\brt A. If you need more p\bper during the ex\bm, r\bise your h\bnd. At the top of e\bch extr\b piece of p\bper you use, be sure to write only your AP number \bnd the number of the question you \bre working on. Do not write your n\bme. Open your ex\bm booklet \bnd begin. 12 6 3 9 Note Sta\bt Time he\be . Note Stop Time he\be . Check that students a\be wo\bking on Pa\bt A only and w\biting thei\b answe\bs in thei\b exam booklets using pencils o\b pens with black o\b da\bk blue ink. The pages fo\b the Pa\bt A questions a\be ma\bked with la\bge 1’s and 2’s at the top of each page to assist you in monito\bing thei\b wo\bk. Afte\b 20 minutes, say: There \bre 10 minutes rem\bining in P\brt A.

41 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. AP Exam Instructions CA\bCU\bUS Afte\b 10 minutes, say: Stop working on P\brt A. C\blcul\btors \bre not \bllowed for P\brt B. Ple\bse put \bll of your c\blcul\btors under your ch\bir. . . . Turn to p\bge 13. You h\bve 1 hour for P\brt B. During this time you m\by go b\bck to P\brt A, but you m\by not use your c\blcul\btor. Remember to write your \bnswer to e\bch p\brt of e\bch problem in the \bppropri\bte sp\bce in the ex\bm booklet. Are there \bny questions? . . . Using your finger, bre\bk open the se\bls on P\brt B. Do not peel the se\bls \bw\by from the booklet. You m\by begin P\brt B. . . . 12 6 3 9 Note Sta\bt Time he\be . Note Stop Time he\be . Afte\b 50 minutes, say: There \bre 10 minutes rem\bining in P\brt B. Afte\b 10 minutes, say: Stop working \bnd close your ex\bm booklet. Pl\bce it on your desk, f\bce up. . . . If any students used ext\ba pape\b fo\b the f\bee-\besponse section, have those students staple the ext\ba sheet/s to the fi\bst page co\b\besponding to that question in thei\b exam booklets. Then say: Rem\bin in your se\bt, without t\blking, while the ex\bm m\bteri\bls \bre collected. . . . Collect a Section II exam booklet f\bom each student. Check fo\b the following: • Exam booklet f\bont cove\b: The student placed an AP numbe\b label on the shaded box, and p\binted his o\b he\b initials and today’s date. • Exam booklet back cove\b: The student completed the “Impo\btant Identification Info\bmation” a\bea. When all exam mate\bials have been collected and accounted fo\b, \betu\bn to students any elect\bonic devices you may have collected befo\be the sta\bt of the exam. \bf you are giving the regularly scheduled exam, say: You m\by not discuss these specific free-response questions with \bnyone unless they \bre rele\bsed on the College Bo\brd website in \bbout two d\bys. You should receive your score report in the m\bil \bbout the third week of July. \bf you are giving the alternate exam for late testing, say: None of the questions in this ex\bm m\by ever be discussed or sh\bred in \bny w\by \bt \bny time. You should receive your score report in the m\bil \bbout the third week of July. If any students completed the AP numbe\b ca\bd at the beginning of this exam, say: Ple\bse remember to t\bke your AP number c\brd with you. Then say: You \bre now dismissed.

42 © 2012 The College Board. Visit the College Board on the We\b: www.college\board.org. Calculus All exam mate\bials should be put in secu\be sto\bage until they a\be \betu\bned to the AP P\bog\bam afte\b you\b school’s last administ\bation. Befo\be sto\bing mate\bials, check the “School Use Only” section on page 1 of the answe\b sheet and: • Fill in the app\bop\biate section numbe\b ci\bcle in o\bde\b to view a sepa\bate AP Inst\buctional Planning Repo\bt (fo\b \begula\bly scheduled exams only) o\b Subject Sco\be Roste\b at the class section o\b teache\b level. See “Post-Exam Activities” in the 2011-12 AP Coordinator’s Manual . • Check you\b list of students who a\be eligible fo\b fee \beductions and fill in the app\bop\biate ci\bcle on thei\b \begist\bation answe\b sheets.

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B Student Answer Sheet for the Multiple-Choice Section Use this section to capture student responses. (Note that the following answer sheet is a sample, and may differ from one used in an actual exam.)

0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 Sign your legal name a\b it will appear on your college application\b. D ate To maintain the security of the exam and the validity of my AP score, \b will allow no one else to see the multiple-choice questions. \b will seal the multiple-choice booklet when asked to do so, and \b will not discuss these questions with anyone at any time after the completion of the section. \b am aware of and agree to the AP Program’s policies and procedures as outlined in the 2011-12 Bulletin for AP Students and Parents, including using test\ving accommodations \v(e.g., extended time\v, computer, etc.) only if \b ha\vve been preapproved\v by College Board Se\vrvices for Students with \vDisabilities. NAME AND EXAM AREA — COMPLETE THIS AREA AT EVERY EXAM. USE NO. 2 PENCIL ONLY A. SIGNATURE B123456789T PAGE \b Take the AP Exam label from your Section I book\Ylet and place the label here. Take an AP Number label from your AP Student Pack and place the label here. 20\b2 Answer Sheet B. LEGAL NAME Omit apostrophes, Jr., II. Legal First Name — First \b2 Letters Legal Last Name — First \b5 Letters MI A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C D D D D D D D D D D D D D D D E E E E E E E E E E E E E E E F F F F F F F F F F F F F F F G G G G G G G G G G G G G G G H H H H H H H H H H H H H H H I I I I I I I I I I I I I I I J J J J J J J J J J J J J J J K K K K K K K K K K K K K K K L L L L L L L L L L L L L L L M M M M M M M M M M M M M M M N N N N N N N N N N N N N N N O O O O O O O O O O O O O O O P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q R R R R R R R R R R R R R R R S S S S S S S S S S S S S S S T T T T T T T T T T T T T T T U U U U U U U U U U U U U U U V V V V V V V V V V V V V V V W W W W W W W W W W W W W W W X X X X X X X X X X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z – – – – – – – – – – – – – 1 2 3 4 5 6 7 8 9 1 2 A A A A A A A A A A A A A B B B B B B B B B B B B B C C C C C C C C C C C C C D D D D D D D D D D D D D E E E E E E E E E E E E E F F F F F F F F F F F F F G G G G G G G G G G G G G H H H H H H H H H H H H H I I I I I I I I I I I I I J J J J J J J J J J J J J K K K K K K K K K K K K K L L L L L L L L L L L L L M M M M M M M M M M M M M N N N N N N N N N N N N N O O O O O O O O O O O O O P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q R R R R R R R R R R R R R S S S S S S S S S S S S S T T T T T T T T T T T T T U U U U U U U U U U U U U V V V V V V V V V V V V V W W W W W W W W W W W W W X X X X X X X X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Z Z Z Z Z Z Z Z Z Z Z Z Z – – – – – – – – – – 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 0 0 1 1 2 2 3 3 4 5 6 7 8 9 6 12 7 1 8 2 9 3 10 4 11 5 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 S D. EXAM D ATE C. Y OUR AP NUMBER Month AM PM 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 F. MULTIPLE-CHOICE BOOKL\YET SERIAL NUMBER H. AP EXAM I AM TAKING USING THIS ANSWER SHEET Print E xam N ame: Print F orm C ode: Print F orm: 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Day Year I. DATE OF BIRTH J. SEX K. CURRENT GR ADE LEVEL STUDENT INFORMATION AREA — COMPLETE THIS AREA ONLY ONCE. Female Pre-9th 9th 10th 11th 12th Po\bt-12th Male L. SOCIAL SECURITY NUMBER (Optional) M. EXPECTED D ATE OF COLLE GE ENTRANCE Fall 2012 Winter/Spring 2013 Summer 2014 Undecided 2015 N. STUDENT SEARCH SERVICE ® Ye\b No I w ant t he C ollege B oard t o \b end i nformation a bout m e t o c ollege\b, u niver\bitie\b a nd g overnment \b cholar\bhip p rogram\b i ntere\bted i n \b tudent\b l ike m e. Q. PARENTAL EDUCATION LEVEL Grade \bchool Some high \bchool High \bchool diploma or equivalent Bu\bine\b\b or trade \bchool Some college A\b\bociate or two-year degree Bachelor’\b or four-year degree Some g raduate o r p rofe\b\bional \b chool Graduate or profe\b\bional degree Exam Exam ETS USE ONLY SCHOOL USE ONLY Fee Reduction Granted Section Number Option 1 Option 2 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 O. WHICH LANGUAGE DO YOU KNOW BEST? Engli\bh Engli\bh and another language about the \bame Another language G. ONLINE PROVIDER CODE E. EXAM S TART TIME 762000 0 0 1 1 2 3 4 5 6 7 8 9 Father/ Male Guardian P. ETHNICITY/R ACE American Indian or Ala\bk a Native A\bian, A\bian Amer ican or Pacific I\blander Black or Afr ican Amer ican Mexican or Mexican Amer ican Puerto Rican Other Hi\bpanic, Latino or Latin Amer ican White Other Mother/ Female Guardian 00657-88488 • UNLPDF212 Q3428/ 1-4

S. LANGUAGE — Do not complete this section unle\Yss instructed to do so. \b 2 3 4 5 6 7 8 9 A B C D E F G H I A B C D E F G H I A B C D E F G H I A B C D E F G H I A B C D E F G H I A B C D E F G H I A B C D E F G H I A B C D E F G H I A B C D E F G H I If this answer sheet is for the French Language and Culture, German Language and Culture, Italian Language and Culture, Spanish Language, or Spanish Literature Exam, please answer the following questions. Your responses will not af\Yfect your score. 1. Have you lived or \btudied for one month or more in a country where the language of the exam you are now taking i\b \bpoken? 2. Do you regularly \bpeak or hear the language at home? Ye\b No Ye\b No Indicate your answers to the exam questions in th\Yis section. If a question has\Y only four answer options, do not mark option E. Your answer sheet will be sco\Yred by machine. Use only No. 2 pencils to mark your answers on pages 2 and 3 (one response per questi\Yon). After you have determined your response, be sure to completely fill in the corresponding circle next to the number of the question you are answering. Stray marks and smudges could be read as answers, so erase carefully and completely. Any improper gridding may affect your score. Answers written in the multiple-choice booklet will no\Yt be scored. A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E \b 2 3 4 5 6 7 8 9 \b0 \b\b \b2 \b3 \b4 \b5 \b6 \b7 \b8 \b9 20 2\b 22 23 24 25 26 27 28 29 30 3\b 32 33 34 35 36 37 38 39 40 4\b 42 43 44 45 46 47 48 49 505\b 52 53 54 55 56 57 58 59 60 6\b 62 63 64 65 66 67 68 69 70 7\b 72 73 74 75 ETS USE ONLY 48, 55, 62, 75, 87 R W O OTHER R W O PT02 TOTAL PT03 PT04 DO NOT WRITE IN THIS AREA R. SURVEY QUESTIONS — Answer the survey questions in the\Y AP Student Pack. Do not put responses to exam questions in th\Yis section. PAGE 2

PAGE 3 DO NOT WRITE IN THIS AREA A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E A B C D E 76 77 78 79 80 8\b 82 83 84 85 86 87 88 89 90 9\b 92 93 94 95 96 97 98 99 \b00 \b0\b \b02 \b03 \b04 \b05 \b06 \b07 \b08 \b09 \b\b0\b\b\b \b\b2 \b\b3 \b\b4 \b\b5 \b\b6 \b\b7 \b\b8 \b\b9 \b20 Be sure each mark is dark and completely fills the circle. If a question has\Y only four answer options, do not mark option E. QUESTIONS 76–\b20 © 2011 The College Board. College Board, AP, Student Search Service and the acorn logo are regi\btered trademark\b of the College Board.

HOME ADDRESS AND SCHOOL AREA — COMPLETE THIS AREA ONLY ONCE. U. AREA CODE AND PHONE NUMBER ZIP OR POSTAL CODE COUNTRY CODE PAGE 4 STREET ADDRESS (include \btreet number, \btreet name, apartment number, etc.) CITY A A A A A A A A A A A A A A A A A A A A B B B B B B B B B B B B B B B B B B B B C C C C C C C C C C C C C C C C C C C C D D D D D D D D D D D D D D D D D D D D E E E E E E E E E E E E E E E E E E E E F F F F F F F F F F F F F F F F F F F F G G G G G G G G G G G G G G G G G G G G H H H H H H H H H H H H H H H H H H H H I I I I I I I I I I I I I I I I I I I I J J J J J J J J J J J J J J J J J J J J K K K K K K K K K K K K K K K K K K K K L L L L L L L L L L L L L L L L L L L L M M M M M M M M M M M M M M M M M M M M N N N N N N N N N N N N N N N N N N N N O O O O O O O O O O O O O O O O O O O O P P P P P P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q R R R R R R R R R R R R R R R R R R R R S S S S S S S S S S S S S S S S S S S S T T T T T T T T T T T T T T T T T T T T U U U U U U U U U U U U U U U U U U U U V V V V V V V V V V V V V V V V V V V V W W W W W W W W W W W W W W W W W W W W X X X X X X X X X X X X X X X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 / / / / / / / / / / / / / / / / / / A A A A A A A A A A A A A B B B B B B B B B B B B B C C C C C C C C C C C C C D D D D D D D D D D D D D E E E E E E E E E E E E E F F F F F F F F F F F F F G G G G G G G G G G G G G H H H H H H H H H H H H H I I I I I I I I I I I I I J J J J J J J J J J J J J K K K K K K K K K K K K K L L L L L L L L L L L L L M M M M M M M M M M M M M N N N N N N N N N N N N N O O O O O O O O O O O O O P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q R R R R R R R R R R R R R S S S S S S S S S S S S S T T T T T T T T T T T T T U U U U U U U U U U U U U V V V V V V V V V V V V V W W W W W W W W W W W W W X X X X X X X X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Z Z Z Z Z Z Z Z Z Z Z Z Z A A A A A A A A A B B B B B B B B B C C C C C C C C C D D D D D D D D D E E E E E E E E E F F F F F F F F F G G G G G G G G G H H H H H H H H H I I I I I I I I I J J J J J J J J J K K K K K K K K K L L L L L L L L L M M M M M M M M M N N N N N N N N N O O O O O O O O O P P P P P P P P P Q Q Q Q Q Q Q Q Q R R R R R R R R R S S S S S S S S S T T T T T T T T T U U U U U U U U U V V V V V V V V V W W W W W W W W W X X X X X X X X X Y Y Y Y Y Y Y Y Y Z Z Z Z Z Z Z Z Z 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 / / / / / / / 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 T. YOUR MAILING ADDRESS U\be the addre\b\b abbreviation\b from your AP Student Pack. Fill in only one circle per column. Indicate a \bpace in your addre\b\b by leaving a blank box; do not grid that column. AK AL AR AZ CA CO CT DC DE FL GAHI IA ID IL IN KS KY LA MA MD MEMI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TXUT VA VT WA WI WV WY Puerto Rico Other STATE X. FOR STUDENTS OUTSIDE THE UNITED STATES ONLY SCHOOL CODE 0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 V. SCHOOL YOU ATTEND 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 COLLEGE CODE 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 W. COLLEGE TO RECEIVE YOUR AP SCORE REPORT College Name State Country U\bing the college code li\bted in the AP Student Pack, indicate the ONE college that you want to receive your AP \bcore report. Y. EMAIL ADDRESS INTERNATIONAL PHONE School Name State Country City City If the addre\b\b gridded above i\b not complete enough for delivery of your \bcore report, plea\be fill in thi\b circle and print your complete addre\b\b below. Addre\b\b City State or Province Country ZIP or Po\btal Code By providing your email addre\b\b, you are granting the College Board permi\b\bion to u\be your email in accordance with the policie\b in the 2011-12 Bulletin for AP Students and Parents .

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B Section I: Multiple-Choice Questions This is the multiple-choice section of the 2012 AP exam. It includes cover material and other administrative instructions to help familiarize students with the mechanics of the exam. (Note that future exams may differ in look from the following content.)

At a Glance Total Time1 hour, 45 minutes Number of Questions45 Percent of Total Score50% Writing InstrumentPencil required Part A Number of Questions28 Time55 minutes Electronic DeviceNone allowed Part B Number of Questions17 Time50 minutes Electronic DeviceGraphing calculator required Instructions Section I of this exam contains 45 multiple -choice questions and 4 survey questions .For Part A ,fill in only the circles for numbers 1 through 28 on page 2 of the answer sheet . For Part B ,fill in only the circles for numbers 76 through 92 on page 3 of the answer sheet .The survey questions are numbers 93 through 96 . Indicate all of your answers to the multiple -choice questions on the answer sheet .No credit will be given for anything written in this exam booklet ,but you may use the booklet for notes or scratch work .After you have decided which of the suggested answers is best , completely fill in the corresponding circle on the answer sheet .Give only one answer to each question .If you change an answer ,be sure that the previous mark is erased completely .Here is a sample question and answer . Use your time effectively ,working as quickly as you can without losing accuracy .Do not spend too much time on any one question .Go on to other questions and come back to the ones you have not answered if you have time .It is not expected that everyone will know the answers to all of the multiple -choice questions . Your total score on the multiple -choice section is based only on the number of questions answered correctly .Points are not deducted for incorrect answers or unanswered questions . DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. AP ® Calculus BC Exam SECTION I: Multiple Choice 2012 Form I Form Code 4IBP4-Q-S 68 Minimum 20% post-consumer waste

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3- CALCULUS BC SECTION I, Part A Time—55 minutes Number of questions—28 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the av ailable space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. In this exam: (1) Unless otherwise specified, the domain of a function fis assumed to be the set of all real numbers xfor which ()fx is a real number. (2) The inverse of a trigonometric function fmay be indicated using the inverse function notation 1f- or with the prefix “arc” (e.g., 1 sin arcsin xx- = ).

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -4- 1. If 3 sin ,=y x then = dy dx (A) 3 cos x (B) 2 3cos x (C) 2 3sin x (D) 2 3sin cos- xx (E) 2 3sin cos xx 2. The position of a particle moving in the xy-plane is given by the parametric equations () 32 3 xt t t =- and () 2 12 3 . yttt =- At which of the following points (),xy is the particle at rest? (A) () 4, 12- (B) () 3, 6- (C) () 2, 9- (D) ()0, 0 (E) ()3, 4

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -5- 3. The graph of f is shown above for 04. x ££ What is the value of () 4 0 ? fx dx Ú (A) 1- (B) 0 (C) 2 (D) 6 (E) 12

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -6- 4. Which of the following integrals gives the length of the curve ln y x = from 1 x= to 2? x= (A) 2 2 1 1 1 dx x + Û Ù ı (B) 2 2 1 1 1 dx x ʈ + ˯ Û Ù ı (C) 2 2 1 1 xedx+ Ú (D) () 2 2 1 1ln xdx + Ú (E) () () 2 2 1 1ln xdx + Ú 5. The Maclaurin series for the function f is given by () ()0 . 4 n n x fx = • =-  What is the value of ()3?f (A) 3- (B) 3 7- (C) 4 7 (D) 13 16 (E) 4

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -7- 6. Using the substitution 2 3, ux=- () 4 5 2 1 3 xx dx- - Ú is equal to which of the following? (A) 13 5 2 2 udu-Ú (B) 13 5 2udu-Ú (C) 13 5 2 1 2 udu-Ú (D) 4 5 1udu-Ú (E) 4 5 1 1 2 udu-Ú 7. If arcsin ln , xy= then dy dx = (A) 2 1 y x- (B) 2 1 xy x- (C) 2 1 y x+ (D) arcsin x e (E) arcsin 2 1 x e x+

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -8- t (hours) 4 7 12 15 ()R t (liters/hour) 6.5 6.2 5.9 5.6 8. A tank contains 50 liters of oil at time 4 t= hours. Oil is being pumped into the tank at a rate () , R t where ()R t is measured in liters per hour, and t is measured in hours. Selected values of ()R t are given in the table above. Using a right Riemann sum with three subintervals and da ta from the table, what is the approximation of the number of liters of oil that are in the tank at time 15 t= hours? (A) 64.9 (B) 68.2 (C) 114.9 (D) 116.6 (E) 118.2 9. Which of the following series converge? I. 1 8 ! n n n = • Â II. 100 1 ! n n n = • Â III. ()( )( )1 1 23 n n nn n = • + ++ Â (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -9- 10. 4 32 1 - = Ú tdt (A) 1- (B) 7 8- (C) 1 2- (D) 1 2 (E) 1 11. Let f be the function defined by () 2 fx x =- for all x. Which of the following statements is true? (A) f is continuous but not differentiable at 2. x= (B) f is differentiable at 2. x= (C) f is not continuous at 2. x= (D) () 2 lim 0x fx Æ π (E) 2 x= is a vertical asymptote of the graph of f.

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -10- 12. The points () 1, 1-- and ()1, 5 - are on the graph of a function () y fx = that satisfies the differential equation 2 . dy xy dx =+ Which of the following must be true? (A) ()1, 5 - is a local maximum of f. (B) ()1, 5 - is a point of inflection of the graph of f. (C) () 1, 1-- is a local maximum of f. (D) () 1, 1-- is a local minimum of f. (E) () 1, 1-- is a point of inflection of the graph of f. 13. What is the radius of convergence of the series () 2 0 4 ? 3 n n n x = • - Â (A) 23 (B) 3 (C) 3 (D) 3 2 (E) 0

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11- 14. Let k be a positive constant. Which of the follo wing is a logistic differential equation? (A) = dy kt dt (B) = dy ky dt (C) ()1 =- dy kt t dt (D) ()1 =- dy ky t dt (E) ()1 =- dy ky y dt

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12- 15. The graph of a differentiable function f is shown above. If () () 0 , x hx f t dt = Ú which of the following is true? (A) () () ()66 6hh h

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -13- 16. Let () =y fx be the solution to the differential equation =- dy xy dx with initial condition ()13. = f What is the approximation for ()2f obtained by using Euler’s method with two steps of equal length starting at 1?=x (A) 5 4- (B) 1 (C) 7 4 (D) 2 (E) 21 4 17. For 0, x> the power series () () 246 2 11 3! 5! 7! 21! n n xxx x n -+-++- + + "" converges to which of the following? (A) cos x (B) sin x (C) sin x x (D) 2 xxee - (E) 2 1 xxee+-

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -14- 18. The graph of ,f¢ the derivative of a function f, consists of two line segments a nd a semicircle, as shown in the figure above. If ()21,f = then () 5 f-= (A) 22p- (B) 23p- (C) 25p- (D) 62 p - (E) 42 p -

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -15- 19. The function f is defined by () .2 = + x fx x What points (),xy on the graph of f have the property that the line tangent to f at (),xy has slope 1?2 (A) ()0, 0 only (B) ()11,25 only (C) ()0, 0 and () 4, 2- (D) ()0, 0 and () 2 4, 3 (E) There are no such points. 20. 1 2 0 58 32 x dx xx + ++ Û Ù ı is (A) ()ln 8 (B) ()27 ln 2 (C) ()ln 18 (D) ()ln 288 (E) divergent

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16- 21. The line 5 =y is a horizontal asymptote to the graph of which of the following functions? (A) () sin 5 = x y x (B) 5 =y x (C) 1 5 = - y x (D) 5 1 = - x y x (E) 2 2 20 14 - = + xx y x 22. The power series () 0 3n n n ax = • - Â converges at 5. x= Which of the following must be true? (A) The series diverges at 0. x= (B) The series diverges at 1. x= (C) The series converges at 1. x= (D) The series converges at 2. x= (E) The series converges at 6. x=

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -17- 23. If ()Pt is the size of a population at time t, which of the following differential equations describes linear growth in the size of the population? (A) 200 = dP dt (B) 200 = dP t dt (C) 2 100= dP t dt (D) 200 = dP P dt (E) 2 100= dP P dt 24. Let f be a differentiable function such that () () 3 sin cos 4 cos . fx xdx fx x x xdx =- + ÚÚ Which of the following could be () ? fx (A) cos x (B) sin x (C) 3 4x (D) 4x- (E) 4x

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -18- 25. 2 1 x xe dx• - Ú is (A) 1 e- (B) 1 2e (C) 1 e (D) 2 e (E) divergent

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -19- 26. What is the slope of the line tangent to the polar curve 12sin r q =+ at 0? q = (A) 2 (B) 1 2 (C) 0 (D) 1 2- (E) 2-

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -20- 27. For what values of p will both series 2 1 1 p n n = •  and 1 2 n n p = • ʈ ˯  converge? (A) 22 p -< < only (B) 11 22 p -< < only (C) 1 2 2 p (E) There are no such values of p.

AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. -21- 28. Let g be a continuously differentiable function with ()16g = and ()13. g =¢ What is () () 1 1 lim ? 6 x x gt dt gx Æ - Ú (A) 0 (B) 1 2 (C) 1 (D) 2 (E) The limit does not exist. END OF PART A OF SECTION I IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART A ONLY. DO NOT GO ON TO PART B UNTIL YOU ARE TOLD TO DO SO.

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24- CALCULUS BC SECTION I, Part B Time — 50 minutes Number of questions — 17 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Directions: Solve each of the following problems, using the available space for scratch work. After examining the form of the choices, decide which is the best of the choices given and fill in the corresponding circle on the answer sheet. No credit will be given for anything written in the exam book. Do not spend too much time on any one problem. BE SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YOU MAY NOT RETURN TO PAGE 2 OF THE ANSWER SHEET. In this exam: (1) The exact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among the choices the number th at best approximates the exact numerical value. (2) Unless otherwise specified, the domain of a function fis assumed to be the set of all real numbers xfor which ()fx is a real number. (3) The inverse of a trigonometric function fmay be indicated using the inverse function notation 1f- or with the prefix “arc” (e.g., 1 sin arcsin xx- = ).

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -25- 76. The function f, whose graph is shown above, is defined on the interval 22. x -£ £ Which of the following statements about f is false? (A) f is continuous at 0. x= (B) f is differentiable at 0. x= (C) f has a critical point at 0. x= (D) f has an absolute minimum at 0. x= (E) The concavity of the graph of f changes at 0. x= 77. Let f and g be the functions given by () = x fx e and () 4. = gx x On what intervals is the rate of change of ()fx greater than the rate of change of () ? gx (A) ()0.831, 7.384 only (B) () ,0.831 -• and ()7.384, • (C) () ,0.816 --• and ()1.430, 8.613 (D) () 0.816, 1.430- and ()8.613, • (E) () , -••

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -26- 78. The graph of the piecewise linear function f is shown above. What is the value of () ()9 132? - + Ú fx dx (A) 7.5 (B) 9.5 (C) 27.5 (D) 47 (E) 48.5 79. Let f be a function having deri vatives of all orders for 0 x> such that ()32,f = ()31, f =- ¢ ()36, f = ¢¢ and ()312. f = ¢¢¢ Which of the following is the third-degree Taylor polynomial for f about 3? x= (A) 23 2612 xx x-+ + (B) 23 232 xx x-+ + (C) ()() () 23 2363123 xx x --+ - + - (D) ()()() 23 233343 xx x --+ - + - (E) ()()() 23 233323 xx x --+ - + -

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -27- 80. The graph of ,¢f the derivative of the function f, is shown above. Which of the following statements must be true? I. f has a relative minimum at 3. =-x II. The graph of f has a point of inflection at 2. =-x III. The graph of f is concave down for 04.

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -28- 01 x

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -29- 83. If the function f is continuous at 3, x= which of the following must be true? (A) () ( ) 3 3lim x ffx Æ < (B) () () 33lim limxx fx fx-+ ÆÆ π (C) () () () 33 3lim lim xx ffxfx -+ ÆÆ == (D) The derivative of f at 3 x= exists. (E) The derivative of f is positive for 3 x< and negative for 3. x> 84. For 1.5 1.5,-

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -30- 85. The fuel consumption of a car, in miles per gallon (mpg), is modeled by () 2 20 2400 6, ʈ - Á˜Ë¯ = ss Fs e where s is the speed of the car, in miles per hour. If the car is traveling at 50 miles per hour and its speed is changing at the rate of 2 20 miles / hour , what is the rate at which its fuel consumption is changing? (A) 0.215 mpg per hour (B) 4.299 mpg per hour (C) 19.793 mpg per hour (D) 25.793 mpg per hour (E) 515.855 mpg per hour

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -31- 86. If () 0>¢ fx for all real numbers x and () 7 4 0,= Ú ftdt which of the following could be a table of values for the function ?f (A) x ()fx 4 4- 5 3- 7 0 (B) x ()fx 4 4- 5 2- 7 5 (C) x ()fx 4 4- 5 6 7 3 (D) x ()fx 4 0 5 0 7 0 (E) x ()fx 4 0 5 4 7 6

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32- 87. Let R be the region in the first quadrant bounded above by the graph of ()ln 3 , =-y x for 02.££ x R is the base of a solid for which each cross section perpendicular to the x-axis is a square. What is the volume of the solid? (A) 0.442 (B) 1.029 (C) 1.296 (D) 3.233 (E) 4.071

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -33- 88. The derivative of a function f is increasing for 0x Which of the following could be the graph of ?f (A) (B) (C) (D) (E)

BB B BB B BB B Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -34- 89. A particle moves along a line so that its acceleration for 0 ≥t is given by () 3 3 . 1 + = + t at t If the particle’s velocity at 0=t is 5, what is the velocity of the particle at 3?=t (A) 0.713 (B) 1.134 (C) 6.134 (D) 6.710 (E) 11.710 90. If the series 1 n n a = • Â converges and 0 na > for all n, which of the following must be true? (A) 1 lim 0 n n n a a + Æ• = (B) 1 na < for all n (C) 1 0 n n a = • = Â (D) 1 n n na = • Â diverges. (E) 1 n n a n = • Â converges.

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