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AP ® Calculus AB Practice Exam From the 2 015 Administration This Practic e Exa m is provided by the Colleg e Boar d fo r AP Exam preparation . Teacher s ar e permitted to download the material s an d mak e copies to use wit h thei r student s in a classroo m settin g onl y. T o maintai n th e securit y of this exam , teacher s shoul d collec t al l material s afte r thei r administratio n an d keep the m in a secure location . Exam s ma y no t be poste d on schoo l or personal websites, no r electronically redistributed for an y reason. Fu rther distributio n of thes e material s outside of the secur e Colleg e Board sit e disadvantages teacher s wh o rel y on uncirculated questions for classroo m testing . An y additional distribution is in violation of the Colleg e Board ’s copyright policies an d ma y resul t in the terminatio n of Practic e Exa m acces s fo r you r schoo l as well as the removal of access to other online services such as the A P Teache r Communit y an d Onlin e Scor e Repo rts.

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 Worksheet Note: This publication shows the page numbers that appeared in the 201 4−15 AP Exam Instructions book and in th e actual exam. This publication was not repaginated to begin with page 1. © 2015 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: ww w.collegeboard.com/inquiry/cbpermit.html.

Exam Instructions The following contains instructions taken from the 2014−15 AP Exam Instructions book.

35 AP® Calculus AB/BC Exam Regularly Scheduled Exam Date: Tuesday morning, May 5, 2015 Late-Testing Exam Date: Thursday morning, May 21, 2015 Section I Total Time, Calculus AB: 1 hr. 45 min. Section II Total Time, Calculus AB: 1 hr. 30 min. Section I Total Time, Calculus BC: 1 hr. 45 min. Section II Total Time, Calculus BC: 1 hr. 30 min. Section ITotal Time: 1 hour 45 minutes Number of Questions: 45 *The number of questions may vary slightly depending on the form of the exam. * Percent of Total Score: 50% Writing Instrument: Pencil required Part A: Number of Questions: 28 Time: 55 minutes No calculator allowed Part B: Number of Questions: 17 Time: 50 minutes Graphing calculator required Section II Total Time: 1 hour 30 minutes Number of Questions: 6 Percent of Total Score: 50% Writing Instrument: Either pencil or pen with black or dark blue ink Note: For Section II, if students finish Part A before the end of the timed 30 minutes for Part A, they cannot begin working on Part B. Students must wait until the beginning of the timed 60 minutes for Part B. However, during the timed portion for Part B, students may work on the problems in Part A without the use of a calculator. Part A: Number of Questions: 2 Time: 30 minutes Percent of Section II Score: 33.3% Graphing calculator required Part B: Number of Questions: 4 Time: 60 minutes Percent of Section II Score: 66.6% No calculator allowed CALCULUS What Proctors Need to Bring to This Exam Exam packets Answer sheets AP Student Packs 2014 - 15 AP Coordinator’s Manual This book — AP Exam Instructions AP Exam Seating Chart template(s) School Code and Home-School/Self- Study Codes Extra graphing calculators Pencil sharpener Container for students’ electronic devices (if needed) Extra No . 2 pencils with erasers Extra pens with black or dark blue ink Extra paper Stapler Wa t c h Signs for the door to the testing room “Exam in Progress” • • • • • • • • • • • • • • • • – – “Cell phones are prohibited in the testing room” SEATING POLICY FOR AP CALCULUS AB AND CALCULUS BC EXAMS Testing Window Exams Administered at Schools in the United States, Canada, Puerto Rico, and the U.S. Virgin Islands Exams Administered at Schools Outside the United States, Canada, Puerto Rico, and the U.S. Virgin Islands Regularly Scheduled Exams Students must be seated no less than fourfeet apart. Students must be seated no less than fivefeet apart. Late-Testing Exams Students must be seated no less than fivefeet apart. ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

36 Calculus Graphing calculators are required to answer some of the questions on the AP Calculus Exams . Before starting the exam administration, make sure each student has a graphing calculator from the approved list on page 46 of the 2014 -1 5 AP Coordinator’s Manual . If a student does not have a graphing calculator from the approved list, you may provide one from your supply . If the student does not want to use the calculator you provide or does not want to use a calculator at all, he or she must hand copy, date, and sign the release statement on page 44 of the 2014 - 15 AP Coordinator’s Manual . During the administration of Section I, Part B, and Section II, Part A, students may have no m ore than two graphing calculators on their desks . Calculators may not be shared . Calculator memories do not need to be cleared before or after the exam. Students with Hewlett-Packard 48–50 Series and Casio FX-9860 graphing calculators may use cards designed for use with these calculators . Proctors should make sure infrared ports (Hewlett-Packard) are not facing each ot her . 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 the calculator to remove exam questions and/or answers from the room may result in the cancellation of AP Exam scores. The AP Calculus AB Exam and the AP Calculus BC Exam should be administered simultaneously . They may be administered in separate rooms, or in the same room if it is more convenient . SECTION I: Multiple Choice ! Do not begin the exam instructions below until you have completed the appropriate General Instructions for your group. These exams include survey questions . The time allowed for the survey questions is in addition to the actual test-taking time . Make sure you begin the exams at the designated time . Remember: You must complete a seating chart for this exam . See pages 279–280 for a seating chart template and instructions . See the 2014-15 AP Coordinator’s Manual for exam seating requirements (pages 48–50, 88) . If you are giving the regularly scheduled exam, say: It is Tuesday morning, May 5, and you will be taking either the AP Calculus AB Exam or the AP Calculus BC Exam. If you are giving the alternate exam for late testing, say: It is Thursday morning, May 21, and you will be taking either the AP Calculus AB Exam or the AP Calculus BC Exam. In a moment, you will open the packet that contains your exam materials. By opening this packet, you agree to all of the AP Program’s policies and procedures outlined in the 2014 - 15 Bulletin for AP Students and Parents . Please check to make sure you have the correct exam: Calculus AB or Calculus BC. Raise your hand if you do not have the correct exam. . . . You may now remove the shrinkwrap from your exam packet and take out the Section I booklet, but do not open the booklet or the shrinkwrapped Section II materials. Put the white seals aside. . . . ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

37 Carefully remove the AP Exam label found near the top left of your exam booklet cover. Now place it on page 1 of your answer sheet on the light blue box near the top right-hand corner that reads “AP Exam Label.” If students accidentally place the exam label in the space for the number label or vice versa, advise them to leave the labels in place . They should not try to remove the label; their exam will be processed correctly . Read the statements on the front cover of Section I and look up when you have finished. . . . Sign your name and write today’s date. Look up when you have finished. . . . Now print your full legal name where indicated. Are there any questions? . . . Turn to the back cover and read it completely. Look up when you have finished. . . . Are there any questions? . . . You will now take the multiple-choice portion of the exam. You should have in front of you the multiple-choice booklet and your answer sheet. You may never discuss these specific multiple-choice questions at any time in any form with anyone, including your teacher and other students. If you disclose these questions through any means, your AP Exam score will be canceled. You must complete the answer sheet using a No. 2 pencil only. Mark all of your responses beginning on page 2 of your answer sheet, one response per question. Completely fill in the circles. If you need to erase, do so carefully and completely. No credit will be given for anything written in the exam booklet. Scratch paper is not allowed, but you may use the margins or any blank space in the exam booklet for scratch work. Section I is divided into two parts. Each part is timed separately, and you may work on each part only during the time allotted for it. Calculators are not allowed in Part A. Please put your calculators under your chair. Are there any questions? . . . You have 55 minutes for Part A. Part A questions are numbered 1 through 28. Mark your responses for these questions on page 2 of your answer sheet. Open your Section I booklet and begin. 12 6 3 9 Note Start Time here . Note Stop Time here . Check that students are marking their answers in pencil on page 2 of their answer sheets and that they are not looking beyond Part A . The line of A’s at the top of each page will assist you in monitoring students’ work . After 45 minutes, say: There are 10 minutes remaining. After 10 minutes, say: Stop working on Part A and turn to page 22 in your Section I booklet. . . . On that page, you should see an area marked “PLACE SEAL HERE.” Making sure all of your other exam materials, including your answer sheet, are out of the way, take one of your seals and press it on that area and then fold AP Exam Instructions CALCULUS ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

38 Calculus the seal over the open edge to the front cover. Be sure you don’t seal the Part B section of the booklet or let the seal touch anything except the marked areas. . . . After all students have sealed Part A, say: Graphing calculators are required for Part B. You may get your calculators from under your chair and place them on your desk. Part B questions are numbered 76 through 92. Fold your answer sheet so only page 3 is showing and mark your responses for these questions on that page. You have 50 minutes for Part B. You may begin. 126 3 9 Note Start Time here . Note Stop Time here . Check that students have sealed their booklets properly and are now working on Part B . The large B’s in an alternating shaded pattern at the top of each page will assist you in monitoring their work . Proctors should make sure that students are using their calculators appropriately . Proctors should also make sure Hewlett-Packard calculators’ infrared ports are not facing each other . After 40 minutes, say: There are 10 minutes remaining. After 10 minutes, say: Stop working and turn to page 38. You have 3 minutes to answer Questions 93–96. These are survey questions and will not affect your score. You may not go back to work on any of the exam questions. . . . Give students approximately 3 minutes to answer the survey questions . Then say: Close your booklet and put your answer sheet on your desk, face up. Make sure you have your AP number label and an AP Exam label on page 1 of your answer sheet. Sit quietly while I collect your answer sheets. Collect an answer sheet from each student . Check that each answer sheet has an AP number label and an AP Exam label . After all answer sheets have been collected, say: Now you must seal your Section I booklet. Remove the remaining white seals from the backing and press one on each area of your exam booklet cover marked “PLACE SEAL HERE.” Fold each seal over the back cover. When you have finished, place the booklet on your desk, face up. I will now collect your Section I booklet. . . . Collect a Section I booklet from each student . Check that each student has signed the front cover of the sealed Section I booklet . There is a 10-minute break between Sections I and II . When all Section I materials have been collected and accounted for and you are ready for the break, say: Please listen carefully to these instructions before we take a 10-minute break. All items you placed under your chair at the beginning of this exam must stay there, and you are not permitted to open or access them in any way. Leave your shrinkwrapped Section II packet on top of your desk during the break. You are not allowed to consult teachers, other students, or textbooks during the break. You may not make phone calls, send text messages, use your calculators, check email, use a social networking site, or access any electronic or communication device. Remember, you may never discuss the ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

39 multiple-choice questions at any time in any form with anyone, including your teacher and other students. If you disclose these questions through any means, your AP Exam score will be canceled. Are there any questions? . . . 126 3 9 You may begin your break. Testing will resume at . SECTION II: Free Response After the break, say: May I have everyone’s attention? Place your Student Pack on your desk. . . . You may now remove the shrinkwrap from the Section II packet, but do not open the Section II exam booklet until you are told to do so. . . . Read the bulleted statements on the front cover of the exam booklet. Look up when you have finished. . . . Now place an AP number label on the shaded box. If you don’t have any AP number labels, write your AP number in the box. Look up when you have finished. . . . Read the last statement. . . . Using your pen, print the first, middle and last initials of your legal name in the boxes and print today’s date where indicated. This constitutes your signature and your agreement to the statements on the front cover. . . . Turn to the back cover and complete Item 1 under “Important Identification Information.” Print the first two letters of your last name and the first letter of your first name in the boxes. Look up when you have finished. . . . In Item 2, print your date of birth in the boxes. . . . In Item 3, write the school code you printed on the front of your Student Pack in the boxes. . . . Read Item 4. . . . Are there any questions? . . . I need to collect the Student Pack from anyone who will be taking another AP Exam. You may keep it only if you are not taking any other AP Exams this year. If you have no other AP Exams to take, place your Student Pack under your chair now. . . . While Student Packs are being collected, read the information on the back cover of the exam booklet, paying careful attention to the bulleted statements in the instructions. Do not open the exam booklet or break the seals in the exam booklet until you are told to do so. Look up when you have finished. . . . Collect the Student Packs . Then say: Are there any questions? . . . AP Exam Instructions CALCULUS ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

40 Calculus Section II also has two parts that are timed separately. You are responsible for pacing yourself, and may proceed freely from one question to the next within each part. Graphing calculators are required for Part A, so you may keep your calculators on your desk. You must write your answers in the appropriate space in the exam booklet using a No. 2 pencil or a pen with black or dark blue ink. Do not break the seals for Part B at this time. Are there any questions? . . . You have 30 minutes to answer the questions in Part A. If you need more p aper during the exam, raise your hand. At the top of each extra sheet of paper you use, be sure to write only your AP number and the number of the question you are working on. Do not write your name. Open your exam booklet and begin. 12 6 3 9 Note Start Time here . Note Stop Time here . Check that students are work ing on Part A only and writing their answers in their exam booklets using pencils or pens with black or dark blue ink . The pages for the Part A questions are marked with large 1s or 2s at the top of each page to assist you in monitoring their work . After 20 minutes, say: There are 10 minutes remaining in Part A. After 10 minutes, say: Stop working on Part A. Calculators are not allowed for Part B. Please put all of your calculators under your chair. . . . Turn to page 13. You have 1 hour for Part B. During this time you may go back to Part A, but you may not use your calculator. Remember to show your work, and write your answer to each part of each problem in the appropriate space in the exam booklet. Are there any questions? . . . Using your finger, break open the seals on Part B. Do not peel the seals away from the booklet. You may begin Part B. 12 6 3 9 Note Start Time here . Note Stop Time here . After 50 minutes, say: There are 10 minutes remaining in Part B. After 10 minutes, say: Stop working and close your exam booklet. Place it on your desk, face up. . . . If any students used extra paper for the free-response section, have those students staple the extra sheet(s) to the first page corresponding to that question in their exam booklets . Complete an Incident Report and include any exam booklets with extra sheets of paper in an Incident Report return envelope (see page 57 of the AP Coordinator’s Manual for details) . Then say: Remain in your seat, without talking, while the exam materials are collected. . . . Collect a Section II exam booklet from each student . Check for the following: • Exam booklet front cover: The student placed an AP number label on the shaded box, and printed his or her initials and today’s date . • Exam booklet back cover: The student completed the “Important Identification Information” area . ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

41 When all exam materials have been collected and accounted for, return to students any electronic devices you may have collected before the start of the exam . If you are giving the regularly scheduled exam, say: You may not discuss or share these specific free-response questions with anyone unless they are released on the College Board website in about two days. Your AP Exam score results will be available online in July. If you are giving the alternate exam for late testing, say: None of the questions in this exam may ever be discussed or shared in any way at any time. Your AP Exam score results will be available online in July. If any students completed the AP number card at the beginning of this exam, say: Please remember to take your AP number card with you. You will need the information on this card to view your scores and order AP score reporting services online. Then say: You are now dismissed. All exam materials must be placed in secure storage until they are returned to the AP Program after your school’s last administration . Before storing materials, check the “School Use Only” section on page 1 of the answer sheet and: • Fill in the appropriate section number circle in order to access a separate AP Instructional Planning Report (for regularly scheduled exams only) or subject score roster at the class section or teacher level . See “Post-Exam Activities” in the 2014 -1 5 AP Coordinator’s Manual . • C heck your list of students who are eligible for fee reductions and fill in the appropriate circle on their registration answer sheets . Be sure to give the completed seating chart to the AP Coordinator . Schools must retain seating charts for at least six months (unless the state or district requires that they be retained for a longer period of time) . Schools should not return any seating charts in their exam shipments unless they are required as part of an Incident Report . AP Exam Instructions CALCULUS ? 2015 The College Board. Visit the College Board on the Web: www.collegeboard.org.

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. )

Sign your legal name as it will appear on your college applications. Date To maintain the security of the exam and the validity of my AP score, I w\ ill allow no one else to see the multiple-choice questions. I will seal the multiple-choice booklet when asked to do so, and I will not dis\ cuss these questions with anyone at any time after completing the section. I am aware of and agree to the AP Program’s policies and procedures as outlined in the 2014-15 Bulletin for AP Students and Parents, including using testing accommodations (e.g., extended time, computer\ , etc.) only if I have been preapproved by College Board Services for Students with Disabilities.COMPLETE THIS AREA AT EVERY EXAM.USE NO. 2 PENCIL ONLY A. SIGNATURE PAGE 1 AP Exam Label (from Section I Booklet) AP Number Label (from Student Pack) Answer Sheet 2 015 B. LEGAL NAME Omit apostrophes, Jr., II. Legal First Name ? First 12 Letters Legal Last Name ? First 15 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 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 DATE C. YOUR AP NUMBER Month AM PM F. MULTIPLE-CHOICE BOOKLET SERIAL NUMBER H. AP EXAM I AM TAKING USING THIS ANSWER SHEET Exam Name: Form Code: Form: COMPLETE THIS AREA ONLY ONCE. SCHOOL USE ONLY Fee Reduction Granted Section Number Option 1 Option 2 E. EXAM START TIME 0 0 1 1 2 3 4 5 6 7 8 9 1 2 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 G. ONLINE PROVIDER CODE Day 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 L. SOCIAL SECURITY NUMBER (Optional) 104246-00657• UNLWEB315 Q3914/1- 4 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 M. COLLEGE TO RECEIVE YOUR AP SCORE REPORT College Name State Country Using the college code listed in the AP Student Pack, indicate the ONE college that you want to receive your AP score report. City 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 School Name State Country J. SCHOOL YOU ATTEND City 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 INTERNATIONAL PHONE I. AREA CODE AND PHONE NUMBER 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 DayYear K. DATE OF BIRTH N. CURRENT GRADE LEVEL Not yet in 9th grade 9th 10th 11th 12th No longer in high school B123456789T 779934

P. LANGUAGE — Do not complete this section unless instructed to do so. 1 2 34 5 67 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 and Culture, or Spanish Literature and Culture Exam, please answer the following questions. Your responses will not affect your score. 1. Have you lived or studied for one month or more in a country where the language of the exam you are now taking is spoken? Yes No DO NOT WRITE IN THIS AREA O. SURVEY QUESTIONS — Answer the survey questions in the AP Student Pack. Do not put responses to exam questions in this section. PAGE 2 COMPLETE THIS AREA AT EACH EXAM (IF APPLICABLE). Indicate your answers to the exam questions in this section (pages 2 and 3). Mark only one response per question for Questions 1 through 120. If a question has only four answer options, do not mark option E. Answers written in the multiple-choice booklet will not 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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 QUESTIONS 1–75 Yes No 2. Do you regularly speak or hear the language at home? A B C D A B C D You must use a No. 2 pencil and marks must be complete. Do not use a mechanical pencil. It is very important that you fill in the entire circle darkly and completely. If you change your response, erase as completely as possible. Incomplete marks or erasures may affect your score. COMPLETE MARK EXAMPLES OF INCOMPLETE MARKS SELECTED MEDIA EXAMS R W O OTHER EXAMS R W O PT02 TOTAL PT03 Subscore (if applicable) PT04 Subscore (if applicable) ETS USE ONLY Exam Exam 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

/ / / – . . . . . 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 / / / – . . . . . 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 / / / – . . . . . 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 / / / – . . . . . 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 / / / – . . . . . 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 For Students Taking AP Biology / / / – . . . . . 0 0 0 0 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 7 7 7 7 7 8 8 8 8 8 9 9 9 9 9 121123 122 124125 126 Write your answer in the boxes at the top of the griddable area and fill in the corresponding circles. Mark only one circle in any column. You will receive credit only if the circles are filled in correctly. 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 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 10 0 101 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 110111 11 2 11 3 11 4 11 5 11 6 11 7 11 8 11 9 120 Be sure each mark is dark and completely fills the circle. If a question has only four answer options, do not mark option E. QUESTIONS 76–120 © 2014 The College Board. College Board, AP, Student Search Service and the acorn logo are registered trademarks of the College Board. QUESTIONS 121–126 A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D 131 132 133 134 135 136 137 138139 140 141 142 For Students Taking AP Physics 1 or AP Physics 2 Mark two responses per question. You will receive credit only if both correct responses are selected. QUESTIONS 131–142

X. ETHNICITY/RACE COMPLETE THIS AREA ONLY ONCE. ZIP OR POSTAL CODE COUNTRY CODE PAGE 4 STREET ADDRESS (include street number, street 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 Q. YOUR MAILING ADDRESS Use the address abbreviations from your AP Student Pack. Fill in only one circle per column. Indicate a space in your address by leaving a blank box; do not grid that column. AK AL AR AZ CA CO CT DC DE FL GA HI 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 TX UT VAVT WA WI WV WY Puerto Rico AA AE AP Other S TAT E R. FOR STUDENTS OUTSIDE THE UNITED STATES ONLY T. EMAIL ADDRESS If your address does not fit in the spaces provided in Item Q, fill in as many circles as you can, then fill in the circle in Item R and print the remainder of your address in the spaces provided. Address City State or Province Country ZIP or Postal Code By providing your email address, you are granting the College Board permission to use your email in accordance with the policies in the 2014-15 Bulletin for AP Students and Parents. S. STUDENT IDENTIFIER (Student ID Number) U. STUDENT SEARCH SERVICE ® Colleges and scholarship programs may request your information to inform you of educational opportunities and financial aid. Would you like us to supply your information? If you don’t answer and previously chose to participate in this service, we will continue providing your information. Colleges and scholarship programs may request your information to inform you of educational opportunities and financial aid. Would you like us to supply your information? If you don’t answer and previously chose to participate in this service, we will continue providing your information.Y. PARENTAL EDUCATION LEVEL American Indian or Alaska Native Asian, Asian American or Pacific Islander Black or African American Mexican or Mexican American Puerto Rican Other Hispanic, Latino or Latin American White Other W. WHICH LANGUAGE DO YOU KNOW BEST? Grade school Some high school High school diploma or equivalent Vocational or trade school Some college Associate or two-year degree Bachelor’s or four-year degree Some graduate or professional school Graduate or professional degree Mother or female guardian Father or male guardian V. SEX Female Male English English and another language about the same Another language Yes No In the first column, indicate the highest level of education of your parent/guardian. If you have two parents/guardians, indicate the level of education for your other parent/ guardian in the second column. In the appropriate column for each parent/guardian, indicate whether this is your mother or female guardian or your father or male guardian.

Section I: Multiple -Choice Questions This is the multiple -choice section of the 201 5 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.)

AP ® Calculus AB Exam SECTION I: Multiple Choice 2015 DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. Instructions At a Glance Total Time 1 hour, 45 minutes Number of Questions 45 Percent of Total Score 50% Writing Instrument Pencil required Part A Number of Questions 28 Time 55 minutes Electronic Device None allowed Part B Number of Questions 17 Time 50 minutes Electronic Device Graphing calculator required 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. Form I Form Code 4KBP6-S 66

is a real number. AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA CALCULUS AB 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) Unl ess otherwise specified, the domain of a function fis assumed to be the set of all real numbers xfor which ( fx ) -f1 (2) The inverse of a trigonometric function fmay be indicated using the inverse function notation or with the prefix “arc” - (e.g., sin 1 x = arcsin x ). Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -3-

Û 2x 1 Ù 5e + dx=( x) ı 5 e 2x + 2 + C 2 2 x 5 2xe + ln x + C 2 2 x 2 5 e + + C 2x x 5 2e + ln x+ C 2x 1 10 e - + C 2x ( ) = x+ 3 fx x f¢(4)= 1 16 㔠=ㄶ 7 2 㐹=㐠 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 1. (A) (B) (C) (D) (E) 2. If , then (A) (B)(C) 1(D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -4-

Ú x2 (x3 + 5) 6 dx = 1 3 6 (x + 5) + C 3 1 3 1 㐠 6 x ( x + 5x ) + C 3 4 1 ㌠ 7 㜠(x + 5) + C 3 2 3 7x (x + 5) + C 7 1 ㌠ 7 =(x + 5 ) + C 21 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 3. (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -5-

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -6- x 025 30 50 (fx ) 46812 4. The values of a continuous function f for selected values of x are given in the table above. What is the value of the left Riemann sum approximation to Ú fxd 0 50 () x using the subintervals [0, 25 , [25, 30 , and [30, 50 ] ] ]? (A) 290 (B) 360(C) 380(D) 390 (E) 430 Ï 2 x sin px for x

3sec 2 x+2? 6sec 2 xtan x fgx( () ) ? 2 ()gx - 4 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 6. Which of the following is an antiderivative of 3 sec x+2 x (A) 3tan x (B) 3tan x+ 2 x (C) 3sec x+2 x (D) (E) 7. If fx = x - 4 () 2 and g is a differentiable function of x, what is the derivative of (A) ( 2gx ) (B) 2gx¢ () (C) 2 ¢() xg x (D) 2gxg x ¢() ( (E) ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -7-

dy=y 2 4 -y 2 dx ( ) x0 -

p dy x = , = 3 dx 1 - 2 1 - 4 1 2 2x -9 lim 2 x Æ- 3 x -2 x -15 is 3 5 3 4 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 10. If y =sin x cos x, then at 1 4 (A) (B) (C)(D)(E) 1 11. (A) 0 (B) (C) (D) 1 (E) nonexistent Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -9-

È p˘ 0, Í ˙ Î 2 ˚? 㐠 - p 2 2 4 p dy = dx x 2 2 x 3 y 2 2 x + 2 13 y AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 12. What is the average rate of change of y = cos 2 ( x)on the interval (A) (B) -1 (C) 0 (D) (E) 3 2 y +=y x 13. If , then 2 2x -3y (A) 0 (B) (C) (D) (E ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -10-

y=¢ ()fx? AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 14. The graph of y= fx() on the closed interval [0, 4 ] is shown above. Which of th e following could be the graph of (A) (B) (D) (C) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -11-

Ï3 x - 2 if x< 1 () = Ì fx Óln 3 (x - 2 ) if x ≥ 1 lim ( ) I. lim -fx() = + fx xÆ 1 xÆ 1 II. lim -fx( ) = lim + ¢ fx ¢( ) xÆ 1 xÆ 1 III. f is differentiable at x=1. AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 15. Let f be the function defined above. Which of the following statements about f are true? (A) None (B) I only (C) II only (D) II and III only (E) I, II, and III Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -12-

fc¢() = 3 - 42 3 -82 3 ㈲ 3 2 = 㐲 = = = AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 3 2 fx( ) = 2x -4x +1. The application of the Mean Value Theorem to f on the 16. The function f is defined by interval 1 £ x£3 guarantees the existence of a value c, where 1 < c< 3 , such that (A) 0 (B) 9 (C) 10 (D) 14 (E) 16 vh() =3 h 17. The velocity v, in meters per second, of a certain type of wave is given by , where h is the depth, in meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter, of the velocit y of the wave with respect to the depth of the water, when the depth is 2 meters? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -13-

18. If dy -t 2 and y() = 20 =- 10e dt 0 ,what is the value of y (6)? 2 3 2 3 4 3 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA -6 20 e -3 20 e -2 20 e -3 10 e -3 5e (A) (B) (C) ( D)(E) 19. Let f be the function with derivative defined by fx = x -4 x ¢( ) 3 . At which of the following values of x does the graph of f have a point of inflection? (A) 0 (B) (C) (D)( E) 2 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -14-

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. GO ON T O THE NEXT PAGE.= -15-= 20. Let f be the function given by () ()( () 2 42 3 1 -- = - xx fx x ). If the line = y b is a horizontal asymptote to the graph of f, then =b (A) 0 (B) 1(C) 2(D) 3 (E) 4 21. The base of a solid is the region bounded by the x-axis and the graph of 2 1 y x = . For the solid, each cross - section perpendicular to the x-axis is a square. What is the volume of the solid? (A) 2 3 (B) 4 3 (C) 2 (D) 2 3p (E) 43p

fx = 2 x +1 ( ) kx k < 0 k = 0 k > 0 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA , where k is a constant. For what values of k, if any, is f strictly decreasing on the interval ( )-1, 1 ? 22. Let f be the function given by (A) (B) (C) (D) k >1 only (E) There are no such values of k. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -16-

x=- 1? y 3 -3 ( -= e x +1 ) y3 -e( -= x+1 ) y 3 -= 0 y3 1 () + 1 -= x e y 3 3 () +1 -= e x AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA x 3-t y=-3 Ú edt-1 23. Which of the following is an equation for the line tangent to the graph of where (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -17- at the point

dy 2 = 5y dx y(0)= 3 x 9 5 y = e 1 5x y = e 9 y = e 5x + 9 3 y = 115 x - 3 y = 115 x + AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 24. Which of the following is the so lution to the differential equation with the initial condition ? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -18-

pp sin ( + h )-sin () 3 3 lim h Æ 0 h is 1 2 3 2 p 6 ㈠ 3 p 3 4 ㌠ AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA 25. (A) 0(B) (C) 1 (D) (E) nonexistent 26. An object moves along a straight line so that at any time t ≥ 0 its velocity is given by vt( ) = 2cos 3 (t). What is the distance traveled by the object from t= 0 to the first time that it stops? (A) 0 (B) (C)(D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -19-

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -20- x (fx ) ()¢ fx 0 49 0 1 2–8 2 –1 –80 27. The table above gives selected values for a differentiable and decreasing function f and its derivative. If - 1f is the inverse function of f, what is the value of - 1 ¢ (f ) 2() ? (A) - 80 (B) 1 - 8 (C) 1 - 80 (D) 1 80 (E) 1 8 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. = -21-= 28. The top of a 15-foot-long ladder rests against a vertic al wall with the bottom of the ladder on level ground, as shown above. The ladder is sliding down th e wall at a constant rate of 2 feet per second. At what rate, in radians per second, is the acute angle between the bottom of th e ladder and the ground changing at the instant the bottom of the ladder is 9 feet from the base of the wall? (A) 2 9 - (B ) 1 6 - (C) 2 25 - (D ) 2 25 (E) 1 9 END OF PA RT 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.

PART B STARTS ON PAGE 24. -23-

is a real number. - (e.g., sin 1 x = arcsin x ). B B B B B B B B B CALCULUS AB 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 ) -f1 (3) The inverse of a trigonometric function fmay be indicated using the inverse function notation or with the prefix “arc” Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -24-

P¢ (1)= 0.076 B B B B B B B B B 76. The function Pt( ) models the population of the world, in billions of people, where t is the number of years since January 1, 2010. Which of the following is the best interpretation of the statement ? (A) On February 1, 2010, the population of the world wa s increasing at a rate of 0.076 billion people per year. (B) On January 1, 2011, the populati on of the world was increasing at a ra te of 0.076 billion people per year. (C) On January 1, 2011, the population of the world was 0.076 billion people. (D) From January 1, 2010 to January 1, 2011, the populat ion of the world was increasing at an average rate of 0.076 billion people per year. (E) When the population of the world was 1 billion people, the population of the world was increasing at a rate of 0.076 billion people per year. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -25-

B B B B B B B B B x 0 24 6810 (fx ) 5 78 0–15 –20 77. Let f be a differentiable function with selected values gi ven in the table above. What is the average rate of change of f over the closed interval 0 £ x£ 10 ? (A) 6 - (B) - 5 2 (C) 2 - (D) - 2 5 (E) 2 5 78. The rate at which motor oil is leaking from an automobile is modeled by the function L defined by 2 () 1sin t for time t≥ 0 Lt =+ () . Lt() is measured in liters per hour, and t is measured in hours. How much oil leaks out of the automobile during the first half hour? (A) 1.998 liters (B) 1.247 liters (C) 0.969 liters (D) 0.541 liters (E) 0.531 liters Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -26-

Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -27- B B B B B B B B B x ( )fx ()fx ¢ ( )gx ()gx¢ 0 342p 79. The table above gives values of the differentiable functions fand g and their derivatives at x= 0. If h ( ) x =( ) fx gx ( ) , what is the value of h 0¢() ? (A) 83 p - 4 (B) 3 p -8 4 (C) 4 p (D) - p 23 2 (E) + p 83 4

BBB BB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -28-= 80. The figure above shows the graph of , the derivative of a function f, for f ¢ 0 x ££ 2 . What is the value of x at which the absolute minimum of f occurs? (A) 0 (B) 1 2 (C) 1(D) 3 2 (E) 2 81. What is the area of the region enclosed by the graphs of= 2 4 = y x- x and 2 = x y? (A) 1.707 (B) 2.829(C) 5.389(D) 8.886 (E) 21.447

x = 2. (2, 4 ). x = 2. B B B B B B B B B x = 0. (E) f has a local maximum at 82. The graph of f ¢, the derivative of f, is shown above. The line tangent to the graph of f ¢ at x = 0 is vertical, and f ¢ is not differentiable at x = 2. Which of the following statements is true? (A) f ¢ does not exist at (B) f is decreasing on the interval (C) The graph of f has a point of inflection at (D) The graph of f has a point of inflection at x = 0. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -29-

dx 4 2 =-10t +9t +8t dt t = 2 fx dx = Ú 3 ( ) 1 B B B B B B B B B 83. A particle moves along the x-axis so that its position at time t> 0 is given by xt() and . The acceleration of the particle is zero when (A) 0.387 (B) 0.831 (C) 1.243(D) 1.647 (E) 8.094 7 Ú fx dx=42 ( ) 1 3 Ú fx d() x=- 32 7 84. The function f is continuous on the closed interval [1, 7 ]. If and , then (A) -148 (B) 10 (C) 12(D) 20 (E) 148 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -30-

BBB B B B B BB Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -31-= 85. Let ( = y fx) define a twice-differentiable function and let ( = y tx) be the line tangent to the graph of f at 2 = x . If () ( ≥ tx f x ) for all real x, which of the following must be true? (A) (2) 0 ≥ f (B) (2) 0 ≥¢ f (C) (2) 0 £¢ f (D) (2) 0 ≥ ¢¢ f (E) (2) 0 £ ¢¢ f

lim fx =0 () xÆ 2 lim fx =- () • xÆ 2 lim () =• fx xÆ 2 lim () = fx 2 xÆ • lim ( ) = • fx xÆ • B B B B B B B B B 86. The vertical line x = 2 is an asymptote for the graph of the function f. Which of the following statements must be false? (A) (B) (C) (D) (E) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -32-

()hx () 1 - = Ú x f t dt . B B B B B B B B B 87. The graph of the piecewise linear function f is shown above. Let h be the function given by On which of the following intervals is hincreasing? (A) [ 1, 3 ] - (B) [0, 5 ] (C) [2, 5 ]only (D) [2, 9 ] (E) [3, 9 ]only Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -33-

x= a. x = a. fx ()() = fa . x= a. B B B B B B B B B 2 fx () ¢() = sin x 88. The first derivative of the function f is given by . At which of the following values of x does f have a local minimum? (A) 2.507 (B) 2.171 (C) 1.772(D) 1.253 (E) 0 89. If lim fx =faxa () ( Æ ), then which of the following statements about f must be true? (A) f is continuous at (B) f is differentiable at (C) For all values of x, (D) The line y ( = fa )is tangent to the graph of fat (E) The line x = a is a vertical asymptote of the graph of f. Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -34-

119 F∞ f 2() = B B B B B B B B B 90. The temperature F, in degrees Fahrenheit (∞F ), of a cup of coffee t minutes after it is poured is given by - 0.093 t Ft ( ) =72 +118 e . To the nearest degree, what is the av erage temperature of the coffee between t= 0 and t= 10 minutes? (A) 93 F ∞ (B) (C) 146 F ∞ (D) 149 F∞ (E) 154 F∞ 2 fx() =cos x ¢ () 91. If and f 3 =7() ,then (A) 0.241 (B) 5.831(C) 6.416(D) 6.759 (E) 7.241 Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -35-

[3,2- ] 0 Ú hxd() x -3 B B B B B B B B B 92. The graph of the function h is shown in the figure above. Of the following, which has the greatest value? (A) Average value of h over - [3,2] (B) Average rate of change of h over 2 () x Ú hxd- 3 (C) (D) (E) h (¢ 0 ) Unauthorized copying or reuse of any part of this page is illegal. GO ON TO THE NEXT PAGE. -36-

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