Smple Questions PREPARING FOR THE AP (AB) CALCULUS EXAMINATION B B A B tngent line,, f '() = lim h f( + h) f() h +h f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x n] n B B A B tngent line,, f '() = lim h f( + h) f() h +h f(x) dx = lim [f(x ) x + f(x ) x + f(x ) x +...+ f(x ) x ] n n Venture Pulishing 9 Brtlet Street Andover, MA 8 Phone/Fx (58) 896-9486
Prefce This workook is intended for students prepring to tke the Advnced Plcement Clculus AB Exmintion. It contins six prctice tests tht re sed on the course description pulished y the College Bord. We hve tried to mke ech of the six tests in this workook s much like the ctul AP Exm s possile. For exmple, in the pproprite sections, there re questions tht require students to mke decisions out whether to use the grphing clcultor lot, little, or not t ll. In order to provide greter supply of this type prolem, our exms require the use of clcultor in out hlf the prolems of Section I Prt B, nd ll of Section II Prt A. Ech student is expected to hve grphing clcultor tht hs the cpility to: () produce the grph of function within n ritrry viewing window, () find the zeros of function, () compute the derivtive of function numericlly, nd (4) compute definite integrls. In the free-response sections, solutions otined using one of these four cpilities need only show the setup. Solutions using other clcultor cpilities must show the mthemticl steps tht led to the nswer. In either cse, correct nswer lone will not receive full credit. As in the AP Course Description for Mthemtics, our exmintions re in two sections of equl weight. Section I is ll multiple-choice nd Section II is ll free-response.. Section I Prt A (8 questions in 55 minutes). Clcultors my not e used in this prt of the exm.. Section I Prt B (7 questions in 5 minutes). Clcultors re llowed.. Section II Prt A ( questions in minutes). Clcultors re required 4. Section II Prt B (4 questions in 6 minutes). Clcultors my not e used nd the student my go ck to Prt A if there is time. We hve tried to crete the prolems in the spirit of clculus reform. Clculus reform implies chnge in the mode of instruction s well s incresed focus on concepts nd less ttention to symolic mnipultion; emphsis on modeling nd pplictions; use of technology to explore nd deepen understnding of concepts; projects nd coopertive lerning. We hve included questions where functions re defined grphiclly nd numericlly, s well s symoliclly, in order to give the students more prctice in this type of nlysis. We wish to thnk the memers of the Phillips Acdemy Mthemtics Deprtment for their generous contriutions of ides, prolems nd dvice. Their vlule ssistnce in testing the prolems in the clssroom hs mde us quite confident out the vlidity of the exms. Roert Clements of Phillips Exeter Acdemy provided excellent editoril ssistnce nd insightful comments. In the hope of providing future students with etter workook, the uthors welcome your suggestions, corrections, prolems of ll sorts, nd feedck in generl. Plese send your comments to: Venture Pulishing 9 Brtlet Street, Suite 55 Andover, M 8 Phone/Fx 58-896-9486 E-Mil gwest@tic.net
Student Prefce There re six exmintions in this workook. Use them s suggested y your techer, ut out two weeks prior to the AP Exm you should try to find three hour nd thirty minute lock of time to work through one entire exm. Ech prt of the exm should e crefully timed. Allow fifty-five minutes for Section I Prt A, fifty minutes for Section I Prt B, nd ninety minutes for Section II. Tke ten minute rek etween Prt A nd Prt B nd lso etween Prt B nd Section II. This will give you good mesure of the topics tht need more intensive review s well s give you feel for the energy nd enthusism needed on three hour nd fifteen minute exm. Repet the ove routine on second exm four or five dys efore the AP to check your progress. The questions on these exms re designed to e s much like the ctul AP Exms s possile. However, we hve included greter percentge of medium level nd difficult prolems nd fewer esy ones, in order to help you gin stmin nd endurnce. If you do stisfctory jo on these exms, then you should e confident of doing well on the ctul AP Exm. The nswers to the multiple-choice questions nd selected free-response questions re in the ck of the workook. A complete solution mnul for ll the prolems is ville from Venture Pulishing. No mtter how much of n exm you do t one sitting, we strongly urge you to check your nswers when you re finished, not s you go long. You will uild your confidence if you DO NOT use the "do prolem, check the nswer, do prolem" routine. The following is list of common student errors:. If f () c =, then f hs locl mximum or minimum t x = c.. If f () c =, then the grph of f hs n inflection point t x = c.. If f ( x) = g ( x), then f( x) = g( x). 4. d dx f( y) f y) 5. Volume y wshers is ( R r) dx. 6. Not expressing nswers in correct units when units re given. 7. Not providing dequte justifiction when justifiction is requested. 8. Wsting time ersing d solutions. Simply cross out d solution fter writing the correct solution. 9. Listing clcultor results without the supporting mthemtics. Recll tht clcultor is to e used primrily to: ) grph functions, ) compute numericl pproximtions of derivtive nd definite integrl, c) solve equtions.. Not nswering the question tht hs een sked. For exmple, if sked to find the mximum vlue of function, do not stop fter finding the x- vlue where the mximum vlue occurs.
4 EXAM I CALCULUS AB SECTION I PART A Time 55 minutes Numer of questions 8 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION Directions: Solve ech of the following prolems, using the ville spce for scrtchwork. After exmining the form of the choices, decide which is the est of the choices given nd fill in the ox. Do not spend too much time on ny one prolem. In this test: () Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers x for which f( x) is rel numer. () The inverse of trigonometric function f my e indicted using the inverse function nottion f or with the prefix rc (e.g., sin x = rcsin x).. If f ( x) = ln( x ), then the grph of y = f( x) is decresing if nd only if (A) < x < (B) < x (C) < x < (D) x > (E) x >. For x, the slope of the tngent to y = xcos x equls zero whenever (A) tn x = x (B) tn x = x (C) tn x (D) sin x (E) cos x = = = x x x Copyright Venture Pulishing
EXAM I Section I Prt A Multiple-Choice. The function F is defined y Fx ( ) = Gx [ + Gx ( )] 5 grph of the function G where the grph of the function G is shown t the right. 4 The pproximte vlue of F () is (A) 7 4 5 6 7 8 (B) (C) (D) (E) 4. 6 x + x dx = (A) ln 4 + (B) ln + 4 (C) ln + (D) ln 4 + 4 (E) ln + x 5. A reltive mximum of the function f( x) = ( ln ) x (A) (B) (C) (D) e (E) e occurs t Copyright Venture Pulishing
No Clcultors EXAM I Section I Prt A 6. Use right-hnd Riemnn sum with 4 equl sudivisions to pproximte the integrl - x dx. (A) (B) (C) 8.5 (D) 8 (E) 6 7. An eqution of the line tngent to the grph of y= x + x + t its point of inflection is (A) y = x + (B) y (C) y (D) y (E) y = x 7 = x + 5 = x + = x + 7 8. cos( x) dx = (A) sin( x) + C (B) sin( x) + C (C) sin( x) + C (D) sin( x) + C (E) 5 sin( x) + C Copyright Venture Pulishing
4 EXAM I Section I Prt A Multiple-Choice 9. Wht is lim x 9x + 4x +? (A) (B) 4 (C) (D) (E) The limit does not exist.. Let the first qudrnt region enclosed y the grph of y = nd the lines x = nd x x = 4 e the se of solid. If cross sections perpendiculr to the x-xis re semicircles, the volume of the solid is (A) π 64 (B) π (C) π 6 (D) π 8 (E) π 4. Let f( x) = ln x + e x. Which of the following is TRUE t x =? (A) f is incresing (B) f is decresing (C) f is discontinuous (D) f hs reltive minimum (E) f hs reltive mximum Copyright Venture Pulishing
No Clcultors EXAM I Section I Prt A 5 EXAM I CALCULUS AB SECTION I PART B Time 5 minutes Numer of questions 7 A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION Directions: Solve ech of the following prolems, using the ville spce for scrtchwork. After exmining the form of the choices, decide which is the est of the choices given nd fill in the ox. Do not spend too much time on ny one prolem. In this test: () The exct numericl vlue of the correct nswer does not lwys pper mong the choices given. When this hppens, select from mong the choices the numer tht est pproximtes the exct numericl vlue. () Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers x for which f ( x) is rel numer. () The inverse of trigonometric function f my e indicted using the inverse function nottion f or with the prefix rc (e.g., sin x = rcsin x).. The function f is defined on the intervl [ 4, 4] nd its grph is shown to the right. Which of the following sttements re true? I. lim f( x) = x f II. lim ( + h ) f ( ) = h h III. lim f( x) = f( ) x + 4 grph of f - - - 4 - - - (A) I only (B) II only (C) I nd II only (D) II nd III only (E) I, II, III. For f( x) = sin x nd gx ( ) = 5. x on the intervl [, ], the instntneous rte of chnge of f is greter thn the instntneous rte of chnge of g for which vlue of x? (A).8 (B) (C).9 (D). (E).5 Copyright Venture Pulishing
6 EXAM I Section I Prt A Multiple-Choice. Which of the following is the solution to the differentil eqution dy dx condition y( ) =? / (A) y = ( x ), x> / (B) y = ( x ), x< / (C) y = x, x< / (D) y = x, x> / (E) y = x, x = y with initil = 4. If y x = 7,then d y dx (A) 6 7 y (B) y (C) (D) y (E) 9 4 y 5. The grphs of functions f nd g grph of f 5 re shown t the right. 4 If hx ( ) = gf [ ( x)], which of the following sttements re true out the function h? I. h( ) = 4. II. h is incresing t x =. III. The grph of h hs horizontl tngent t x = 4. 4 grph of g 5 4 5 4 5 (A) I only (B) II only (C) I nd II only (D) II nd III only (E) I, II, III Copyright Venture Pulishin
Clcultors EXAM I Section II Prt A 7 9. The rte t which ice is melting in pond is given y dv t = +, where V is the volume dt of ice in cuic feet, nd t is the time in minutes. Wht mount of ice hs melted in the first 5 minutes? (A) 4.49 ft (B) 4.5 ft (C) 4.5 ft (D) 4.55 ft (E) 4.57 ft. The region shded in the figure t the right is rotted out the x-xis. Using the Trpezoid Rule with 5 equl sudivisions, the pproximte volume of the resulting solid is (A) (B) 47 (C) 7 (D) 54 (E) 4 5 4 grph of y= f( x) y = f(x) 4 5 6 7. A prticle moves long the x-xis so tht t time t, its position is given y xt () = ( t+ )( t ). For wht vlues of t is the velocity of the prticle incresing? (A) ll t (B) < t < C) < t < (D) < t < E) t < or t > Copyright Venture Pulishing
8 EXAM I Section I Prt A Multiple-Choice CALCULUS AB SECTION II, PART A Time minutes Numer of prolems A grphing clcultor is required for some prolems or prts of prolems. Before you egin Prt A of Section II, you my wish to look over the prolems efore strting to work on them. It is not expected tht everyone will e le to complete ll prts of ll prolems nd you will e le to come ck to Prt A (without clcultor), if you hve time fter Prt B. All prolems re given equl weight, ut the prts of prticulr solution re not necessrily given equl weight. You should write ll work for ech prolem in the spce provided. Be sure to write clerly nd legily. If you mke n error, you my sve time y crossing it out rther thn trying to erse it. Ersed or crossed out work will not e grded. SHOW ALL YOUR WORK. Clerly lel ny functions, grphs, tles, or other ojects you use. You will e grded on the correctness nd completeness of your methods s well s your finl nswers. wers without supporting work my not receive credit. Justifictions require tht you give mthemticl (nonclcultor) resons. You re permitted to use your clcultor in Prt A to solve n eqution, find the derivtive of function t point, or clculte the vlue of definite integrl. However, you must clerly indicte in your exm ooklet the setup of your prolem, nmely the eqution, function, or integrl you re using. If you use other uilt-in fetures or progrms, you must show the mthemticl steps necessry to produce your results. Your work must e expressed in mthemticl nottion rther thn clcultor syntx. For exmple, 5 x dx my not e written s fnint(x, X,, 5). Unless otherwise specified, nswers (numeric or lgeric) need not e simplified. If you use deciml pproximtions in your clcultions, you will e grded on ccurcy. Unless otherwise specified, your finl nswers should e ccurte to three plces fter the deciml point. Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers x for which f ( x) is rel numer. THE EXAM BEGINS ON THE NEXT PAGE PLEASE TURN OVER Copyright Venture Pulishin
Clcultors EXAM I Section II Prt A 9. Two functions, f nd g, re defined on the closed intervl 4 x 4. A grph of the function f is given in the following figure. grph of f -4 - - - 4 - - - The tle elow contins some vlues of the continuous function g. x 4 4 g(x) 9 5 6 () Find f (). () Approximte g (. ) Show your work. (c) If the function h is defined y hx ( ) = gf [ ( x)], evlute: i) h( ) nd ii) h () (d) Find 4 f( x) dx Copyright Venture Pulishing
EXAM I Section II Prt A Multiple-Choice. A wter tnk holds 5 gllons of wter t time t =. During the time intervl t, wter is pumped into the tnk t the rte of π sin( t) wt () = 4e gllons per hour. At time t = 8 hours second pump egins removing wter from the tnk t the rte of Rt ()= t + t gllons per hour. () How mny gllons of wter enter the tnk during the time intervl t 8? () At wht time during the time intervl t 8 hours is the mount of wter incresing most rpidly? (c) Wht is the totl mount of wter in the tnk t t = hours? (d) Is the mount of wter in the tnk incresing or decresing t t = hours? Justify your nswer. Copyright Venture Pulishing
Clcultors EXAM I Section II Prt A Time - 6 minutes Numer of prolems - 4 A grphing clcultor my NOT e used on this prt of the exmintion. During the timed portion for prt B, you my go ck nd continue to work on the prolems in prt A without the use of clcultor.. Cr A hs positive velocity vt () s it trvels long stright rod, where v is differentile function of t. The velocity of the cr is recorded for severl selected vlues of t over the intervl t 6 seconds, s shown in the tle elow. t ( seconds) 4 5 6 v(t) (feet per second) 5 4 7 4 44 () Use the dt from the tle to pproximte the ccelertion of Cr A t t = 5 seconds. Show the computtion tht led to your nswer. Indicte units of mesure. () Use the dt from the tle to pproximte the distnce trveled y Cr A over the time intervl t 6 seconds y using midpoint Riemnn sum with sudivisions of equl length. Show the work tht led to your nswer. (c) Cr B trvels long the sme rod with n ccelertion of t ()= ft / sec. x+ 9 At time t = seconds, the velocity of Cr B is ft/sec. Which cr is trveling fster t t = 4 seconds? Show the work tht led to your nswer Copyright Venture Pulishing
wers EXAM I SECTION I PART A EXAM II SECTION I PART A. A. A. C. C. C. D. B. D. B. A. B. A. E. C. D. B. A. C 4. C 4. B 4. D 4. C 4. D 4. B 5. E 5. B 5. B 5. D 5. A 5. B 6. D 6. E 6. B 6. C 6. C 6. E 7. A 7. C 7. C 7. E 7. C 7. A 8. D 8. C 8. E 8. A 8. E 8. E 9. B 9. B 9. B 9. B. B. A. D. D EXAM I SECTION I PART B EXAM II SECTION I PART B. D 7. B. C. E 7. E. D. C 8. B 4. C. D 8. A 4. D. C 9. C 5. B. D 9. A 5. B 4. E. C 6. D 4. C. A 6. C 5. D. E 7. E 5. C. D 7. D 6. D. E 6. A. C EXAM I SECTION II PART A. ) ) c) i) ii) 9 d) 4 8 sin( π t). ) 4e dt= 68. 6 gllons ) 6 hours c).6 d) Amount of wter is decresing since W() R() =.4 < EXAM II SECTION II PART A. ).764 ).46 c).67. ) 4 gls ) 9.54 c) 7.99 gls EXAM I SECTION II PART B. ) 4. ft / sec ) ft c) cr A is fster 4. c) y = x d) y = x + Ce x y = Ce dy dx x dx ( + Ce ) = dx x = + Ce = + Ce 5. ) y = x 5 ) x = c).75 d) x =,, e) x = 4 6. ) A = sin(arccos k) k Arccos k ) π 6 c) da dt = x x EXAM II SECTION II PART B. ) ) c) y = ( x ) d) 6 6 4, ( ) 4. ) miles / min ) t = 4 nd t = 7 c) 8 d) / 5miles / min 5. ) y = x+ 5 ) rel mx x = c) x =, 4 d) f() = 8 6. ) x < or x > ) Rel mx t, Rel min t, c) x > d) no inflection point. Copyright Venture Pulishing