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AP ® Calculus AB Practice Exam From the 2 016 Administration This exam ma y no t be posted on school or persona l websites , no r electronically redistribute d fo r an y reason . Thi s Released Exa m is provided by th e Colleg e Boar d for AP Exam preparation . Teacher s ar e permitted to download th e material s an d mak e copies to use wit h thei r student s in a classroo m settin g onl y. To maintai n th e securit y of this exam , teacher s shoul d collec t all material s afte r thei r administratio n an d keep them in a secure location . Further distribution o f thes e material s outsid e of the secur e Colleg e Board sit e disadvantages teachers wh o rel y on uncirculated questions fo r classroo m testing . An y additional distributio n is in violation of the Colleg e Board ’s copyrigh t policies and ma y resul t in the terminatio n of Practice Exam acces s fo r you r school as well as the removal of access to othe r onlin e service s suc h as the A P T eacher 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 Question Descriptors and Performance Data Note: This publication shows the page numbers that appeared in the 201 5−16 AP Exam Instructions book and in th e actual exam. This publication was not repaginated to begin with page 1. © 2016 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.

Exam Instructions The following contains instructions taken from the 2015−16 AP Exam Instructions book.

36 AP® Calculus AB/BC Exam Regularly Scheduled Exam Date: Thursday morning, May 5, 2016 Late-Testing Exam Date: Thursday morning, May 19, 2016 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 I Total 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 What Proctors Need to Bring to This Exam Exam packets Answer sheets AP Student Packs 2015 - 16 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. ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.org.

37 AP Exam Instructions 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 47 of the 2015 -1 6 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 45 of the 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 305–306 for a seating chart template and instructions. See the 2015-16 AP Coordinator’s Manual for exam seating requirements (pages 49–52). If you are giving the regularly scheduled exam, say: It is Thursday 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 19, 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 2015 - 16 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. . . . ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.org.

38 Calculus 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 can still 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 of your exam booklet 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. Open your answer sheet to page 2. 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 ma y 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 ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.org.

39 AP Exam Instructions CALCULUS of the way, take one of your seals and press it on that area and then fold 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, notes, 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 ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.org.

40 Calculus electronic or communication device. Remember, you may never discuss the 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 take an AP number label from your Student Pack and place it 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 a pen with black or dark blue ink, 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, using your pen, 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. . . . 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. . . . ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.org.

Collect the Student Packs. Then say: Are there any questions? . . . 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 paper 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 question number you are working on. Do not write your name. Open your exam booklet and begin. 126 3 9 Note Start Time here . Note Stop Time here . Check that students are working 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 a question in 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. A single Incident Report may be completed for multiple students per exam subject per administration (regular or late testing) as long as all of the required information is provided. Include all exam booklets with extra sheets of paper in an Incident Report return envelope (see page 60 of the 2015-16 AP Coordinator’s Manual for complete details). Then say: Remain in your seat, without talking, while the exam materials are collected. . . . 41 AP Exam Instructions CALCULUS ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.org.

42 Calculus 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 a nd printed his or her initials and today’s date. • Ex am booklet back cover: The student completed the “Important Identification Information” area. 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 I nstructional Planning Report (for regularly scheduled exams only) or subject score roster at the class section or teacher level. See “Post-Exam Activities” in the 2015-16 AP Coordinator’s Manual . • Ch eck 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. ? 2016 The College Board. Visit the College Board on the Web: www.col\ legeboard.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 2015-16 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 2 016 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 O. 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. Yes No 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) 108603-00657 • TF216E4757.5 • Printed in U.S.A. Q4068/1- 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 B123456789T 785208 11th 12th No longer in high school

Q. 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 P. 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 © 2015 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

COMPLETE THIS AREA ONLY ONCE. ZIP OR POSTAL CODECOUNTRY 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 R. 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Section I: Multiple -Choice Questions This is the multiple -choice section of the 201 6 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 Exam2016 SECTION I: Multiple Choice DO NOT OPEN THIS BOOKLET UNTIL YOU ARE TOLD TO DO SO. At a Glance Total Time 1 hour, 45 minutes Number of Questions45 Percent of Total Score 50% Writing Instrument Pencil required Part A Number of Questions28 Time55 minutes Electronic Device None allowed Part B Number of Questions17 Time50 minutes Electronic Device Graphing 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. Form I Form Code 4LBP6-S 66

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -3- CALCULUS AB SECTION I, Part A Time — 55 minutes Number of questions — 28 A CALCULATOR MA Y 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 x - = x ).

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -4- 1. If cos 2 , y x = then = dy dx (A) 2sin 2 x - (B) sin 2 x - (C) sin 2 x (D) 2sin 2 x (E) 2sinx 2. () 10 231 - Úxx dx = (A) 10 34 34 ʈ -+ Á˜ ˯ xx xC (B) () 11 31 11 - + x C (C) () 11 231 11 - + xx C (D) () 11 31 33 - + x C (E) () 11 331 33 - + xx C

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -5- 3. 4 2 91 lim 43 x x x Æ• + + is (A) 1 3 (B) 3 4 (C) 3 2 (D) 9 4 (E) infinite= 4. If= () 5 , 1 = + x y x then = dy dx (A) () 4 51 +x (B ) () 4 4 1 + x x (C) ( 4 4 51 + x x ) (D) ( 4 6 51 + x x ) (E) ( ()4 6 52 11+ + xx x )

t (minutes) 0 479 ( )rt (gallons per minute) 9 643 5. Water is flowing into a tank at the rate ( )rt , where ( )rt is measured in gallons per minute and t is measured in minutes. The tank contains 15 gallons of water at time 0 t = . Values of ( )rt for selected values of t are given in the table above. Using a trapezoidal sum with th e three intervals indicated by the table, what is the approximation of the number of gall ons of water in the tank at time 9 t = ? (A) 52 (B) 57(C) 67(D) 77 (E) 79 6. The slope of the line tangent to the graph of ()ln 1 =- y x at 1 =- x is (A) 1 - (B) 1 2 - (C) 1 2 (D) ln 2 (E) 1 AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -6-

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -7- 7. For which of the following pairs of functions f and g is ( ) ( ) limx f x gx Æ• infinite? (A) ( ) 2 2 f xx x =+ and ( ) 2 ln gx x x =+ (B) ( ) 3 3 f xx = and ( ) 4 gx x = (C) ( ) 3x fx = and ( ) 3 gx x = (D) ( ) 3 3 x fxex =+ and ( ) 2 2 x gx e x =+ (E) ( ) ( ) ln 3 f xx = and ( ) ( ) ln 2 gx x =

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -8- 8. 4 2 0 9 = + Ú x dx x (A) 2 - (B) 2 15 - (C) 1 (D) 2 (E) 5 9. Let f be the function with derivative given by ( ) ( 2 2 2 1 x fx x - = ¢ + ) . On what interval is f decreasing? (A) [ ) 0, • only (B) ( ] ,0 -• only (C) 11 , 33 È - Í Î only ˘ ˙ ˚ (D) ( ) , -•• (E) There is no such interval.

10. ()xeed+ Ú x= (A) xeC + (B) 2 xeC+ (C) xeeC++ (D) 1 x eex + ++ C (E) xeexC++ AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -9-

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -10-= 11. The graph of the function f is shown in the figure above. Which of the following could be the graph of , the f ¢ derivative of f ? (A) (B) (C) (D) (E)

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -11-= 12. If 0 < c

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -12-= 14. If ( y fx= ) is a solution to the differential equation 2x dy e dx = with the initial condition ()0f =2 , which of the following is true? (A) () 2 1 x fx e =+ (B) () 2 2 x fx xe = (C) () 2 1 x t fx e dt = Ú (D) () 2 0 2 x t fx e dt =+ Ú (E) () 2 2 2 x t fx e dt =+ Ú

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -13-= 15. A function (ft ) gives the rate of evaporation of water, in liters per hour, from a pond, where t is measured in hours since 12 noon. Which of the following gives the meaning of () 10 4 f td Ú t in the context described? (A) The total volume of water, in liters, that evaporated from the pond during the first 10 hours after 12 noon (B) The total volume of water, in liters, that evaporated from the pond between 4 P.M. and 10 P.M. (C) The net change in the rate of evapora tion, in liters per hour, fr om the pond between 4 P.M. and 10 P.M. (D) The average rate of evaporation, in liters per hour, from the pond between 4 P.M. and 10 P.M. (E) The average rate of change in the rate of evapora tion, in liters per hour per hour, from the pond between 4 P.M. and 10 P.M.

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -14-= 16. The first derivative of the function f is given by () 4 3=- ¢ fx x 3 12 x. What are the x-coordinates of the points of inflection of the graph of f ? (A) 3=x only (B) 4 =x only (C) 0 =x and 2 =x (D) 0 =x and 3=x (E) 0 =x and 4 =x 17. Let f be the function defined by () 1 fx x =. What is the average value of f on the interval [ ]4, 6 ? (A) 1 24 - (B) 5 24 (C) 13 ln 22 = (D) 3 ln 2 (E) 1ln 2 2 =

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -15-= 18. The points ( )3, 0 , ( ),0 ,x 2 1 ,, x x ʈ Á˜ ˯ and 2 1 3, x Ê Á Ë ˆ ˜ ¯ are the vertices of a rectangle, where 3 x ≥ , as shown in the figure above. For what value of x does the rectangle have a maximum area? (A) 3 (B) 4 (C) 6 (D) 9 (E) There is no such value of x. 19. What are all values of x for which 2 3 Úxtdt is equal to 0 ? (A) 2 - only (B) 0 only (C) 2 only(D) 2 - and 2 only (E) 2, - 0, and 2

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -16-= 20. Let h be the function defined by ( ) 2 4 sin x hx t p = Ú dt. Which of the following is an equation for the line tangent to the graph of h at the point where 4 x p = ? (A) 1 2 y = (B) 2 y x = (C) 4 yx p =- (D) ( )1 24 yx p =- (E) ( )2 2 yx p =- 4

x (fx ) –1 –30 0 –2 3 10 5 18 21. The table above gives selected valu es for a twice-differentiable function f. Which of the following must be true? . (A) f has no critical points in the interval 1 x -< < 5 (B) () 8 fx =¢ for some value of x in the interval 1 x -< < 5 . (C) () 0 fx >¢ for all values of x in the interval 1 x -< < 5 . (D) () 0 fx < ¢¢ for all values of x in the interval 1 x -< < 5 . (E) The graph of f has no points of inflection in the interval 1 x -< < 5 . AAAAAAAAAAAAAAAAA AAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -17-=

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -18-= 22. A particle moves along the x-axis so that at time t 0 ≥ , the acceleration of the particle is 15 at = () t. The position of the particle is 10 when 0 t = , and the position of the particle is 20 when 1 t= . What is the velocity of the particle at time 0 t = ? (A) - 14 (B) 0 (C) 5(D) 6 (E) 10 23. Which of the following is the so lution to the differential equation 2 2 1 dy xy dx x = + whose graph contains the point ( )0, 1 ? (A) 2x ye = (B) 2 1 yx =+ (C) 2 ln 1 yx =+( ) (D) ( ) 2 1ln 1 yx =+ + (E) ( ) 2 12ln 1 y =+ + x

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -19-= 24. Sand is deposited into a pile with a circular base. The volume V of the pile is given by 3 3 r V =, where r is the radius of the base, in feet. The circumference of the base is increasing at a constant rate of 5p feet per hour. When the circumference of the base is 8 p feet, what is the rate of change of the volume of the pile, in cubic feet per hour? (A) 8 p (B) 16 (C) 40(D) 40p (E) 80p 25. 11 0 lim h h ee h-- - Æ - is (A) 1 - (B) 1 e - (C) 0 (D) 1 e (E) nonexistent

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. == GO ON TO THE NEXT PAGE. -20-= 26. Let f be the function given by () 3 5 = fx x + x. For what value of x in the closed interval [ ]1, 3 does the instantaneous rate of change of f equal the average rate of change of f on that interval? (A) 7 3 (B) 13 3 (C) 5 (D) 6 (E) 19 3 = 27. If= 2 4, -=- xyeye then at 1 2 = x and 2, y = dy dx = (A) 4 e (B) 2 e (C) 4 8 -e e (D) 4 4 -e e (E) 84 - e e

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA Unauthorized copying or reuse of any part of this page is illegal. = -21-= 28. Let f be the function defined by ( ) 32 fxxx =+ x + . Let ( ) ( ) 1 gx f - = x, where ( )3g =1. What is the value of ( )g¢3? (A) 1 39 (B) 1 34 (C) 1 6 (D) 1 3 (E) 39 END OF PART A OF SECTION I IF YOU FINISH BEFO RE 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.

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -24-= CALCULUS AB SECTION I, Part B Time — 50 minutes Number of questions — 17 A GRAPH ING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAM. Dire ctions: 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. B E SURE YOU ARE USING PAGE 3 OF THE ANSWER SHEET TO RECORD YOUR ANSWERS TO QUESTIONS NUMBERED 76–92. YO U 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 x - = x ). B B

BBB B B B B BB Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -25-= 76. The graph of a function f is shown above. Which of th e following limits does not exist? (A) ( 1limx fx- Æ ) (B) ( 1 limx fx Æ ) (C) ( 3limx fx- Æ ) (D) ( 3 limx fx Æ ) (E) () 5 limx fx Æ

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -26-= 77. Let f be a function that is continuous on the closed interval [ ]1, 3 with ()110 = f and ()3 18 = f . Which of the following statements must be true? (A) () 10 2 18 £f £ (B) f is increasing on the interval [ ]1, 3 . (C) () 17 = fx has at least one solution in the interval [ ]1, 3 . (D) () 8 =¢ fx has at least one solution in the interval (1, 3 ). (E) () 3 1 20 > Ú fxdx B B

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -27-= 78. Let R be the region bounded by the graphs of , =x y e 3, = y e and 0 = x . Which of the following gives the volume of the solid formed by revolving R about the line 1 =-y ? (A) () 3 2 3 0 1 p -+ Ú x ee dx (B) ( 3 2 3 0 1 p -- Ú x ee d ) x (C) () 3 2 3 0 1 p È˘ -+ Í˙ Î˚ Ú x ee dx (D) () 3 2 3 0 1 p È -- Í Î Ú x ee dx ˘ ˙ ˚ (E) ()( ) 3 22 3 0 11 p È +- + Í Î Ú x ee ˘ ˙ ˚dx 79. The number of people who have entered a mu seum on a certain day is modeled by a function ( )ft , where t is measured in hours since the museum opened that day. The number of people who have left the museum since it opened that sa me day is modeled by a function ( )gt . If ( ) ( ) 380 1.02 t ft = ¢ and ( ) ( ) 4 240 240 sin 12 t gt p - Ê =+ ¢ Á ˈ ˜ ¯ , at what time t, for 1 t ££ 11, is the number of people in the museum at a maximum? (A) 1 (B) 7.888 (C) 9.446 (D) 10.974 (E) 11 B B

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -28-= = x 0123 ( )fx 5 236 ( )fx ¢ –3 1 34 80. The derivative of the function f is continuous on the closed interval [ ]0, 4 . Values of f and f ¢ for selected values of x are given in the table above. If ( ) 4 0 8, ftdt = ¢ Ú then ( )4 f = (A) 0 (B) 3(C) 5(D) 10 (E) 13 BB

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -29-= 81. A slope field for a differential equa tion is shown in the figure above. If ( ) y fx = is the particular solution to the differential equation through the point ( ) 1, 2 - and ( ) ( ) 3, hx x f x = then ( ) 1 h -= ¢ (A) 6 - (B) 2 - (C) 0 (D) 1 (E) 12 82. If f is a continuous function such that ( )2 f =6, which of the following statements must be true? (A) ( ) 1 lim 2 3x fx Æ = (B) ( ) 2 lim 2 12x fx Æ = (C) ( ) ( ) 2 2 lim 6 2 x fx f x Æ - = - (D) ( ) 2 2 lim 36x fx Æ = (E) ( ) ( ) 2 2 lim 36x fx Æ = BB

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -30-= 83. A particle moves along a straight line with velocity given by ( ) 3 5 t vt e =+ for time 0 t ≥. What is the acceleration of the particle at time 4 t = ? 84. A home uses fuel oil at the rate (A) 0.422 (B) 0.698 (C) 1.265 (D) 8.794 (E) 28.381 () () 10 8 sin 60t rt =+ gallons per day , where t is the number of days from the beginning of the heating season. To the nearest ga llon, what is the total amount of fuel oil used from 0 t = to 60 t = days? (A) 7 gal (B) 14 gal (C) 600 gal (D) 821 gal (E) 1004 gal BB

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -31-= 85. The function f is defined on the open interval 0.4 2.4

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -32-= 86. The graph of the function f, which has a domain of [ ]0, 7 , is shown in the figure above. The graph consists of a quarter circle of radius 3 and a segment with slope 1 - . Let b be a positive number such that () 0 0 bfx dx = Ú . What is the value of b ? (A) 3.760 (B) 5.548 (C) 5.65 9 (D) 6.76 0 (E) There is no such value of b. BB

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -33-= 87. The first derivative of the function g is given by () () 2 cos p =¢ gx x for 0.5 1. -< x< 5. On which of the following intervals is g decreasing? (A) 0.5 0 -

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -34-= 88. The height above the ground of a passenger on a Ferris wheel t minutes after the ride begins is modeled by the differentiable function H, where ( )H t is measured in meters. Which of th e following is an interpretation of the statement ()7.5 15.708 H =¢ ? (A) The Ferris wheel is turning at a rate of 15.708 meters per minute when the passenge r is 7.5 meters above the ground. (B) The Ferris wheel is turning at a rate of 15.708 meters per minute 7.5 minutes after the ride begins. (C) The passenger’s height above the ground is increasing by 15.708 meters per minute when the passenger is 7.5 meters above the ground. (D) The passenger’s height above the ground is increas ing by 15.708 meters per minute 7.5 minutes after the ride begins. (E) The passenger is 15.708 meters above the ground 7.5 minutes after the ride begins. = 89. A particle moves along a straight line for 6 seconds so that its velocity, in centimeters per second, is modeled by the graph shown. During the time interval 0 t £ 6 , what is the total distance the particle travels? £ (A) 2 cm (B) 3.5 cm (C) 4 cm (D) 6.5 cm (E) 8.5 cm BB

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -35-= 90. Let f be a twice-differentiable function on the open interval (, ab ). If () 0 >¢ fx on (, ab ) and () 0 < ¢¢ fx on (, ab ), which of the following could be the graph of f ? (A) (B) (C) (D) (E) BB

BBB B BB B Unauthorized copying or reuse of any part of this page is illegal. = = GO ON TO THE NEXT PAGE. -36-= 91. The graphs of f and g are shown above. If () () (hx f x g = ), x then ()6 h =¢ (A) 9 - (B) 7 - (C) 1 (D) 7 (E) 9 92. In the xy-plane, the graph o f the twice-differentiable function ( = y fx ) is concave up on the open interval (0, 2 ) and is tangent to the line 32 =- yx at 1 = x . Which of the following statements must be true about the derivative of f ? (A) () 3 £¢ fx on the interval ( )0.9, 1 . (B) () 3 ≥¢ fx on the interval ( )0.9, 1 . (C) () 0 ¢ fx on the interval (0.9, 1.1 ). (E) (¢ f x ) is constant on the interval (0.9, 1.1 ). B B

BBB B BB B -37- END OF SECTION I IF YOU FINISH BEF ORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON PART B ONLY. DO NOT GO ON TO S ECTION II UNTIL YOU ARE TOLD TO DO SO. ____________________ ____________________________ MAKE SU RE YOU HAVE DONE THE FOLLOWING. • PLACED YOUR AP NUMBER LABEL ON YOUR ANSWER SHEET • WRITTEN AND GRIDDED YOUR AP NUMBER CORRECTLY ON YOUR ANSWER SHEET • TAKEN THE AP EXAM LABEL FROM THE FRONT OF THIS BOOKLET AND PLACED IT ON YOUR ANSWER SHEET AFTER TI ME HAS BEEN CALLED, TURN TO PAGE 38 AND ANSWER QUESTIONS 93–96. B B

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