22 TYPE PROBLEMS The AP clculus exms contin fresh crefully thought out often clever questions. This is especilly true for the free-response questions. The topics nd style of the questions re similr from yer to yer. By style I men whether the stem presents the given informtion s n eqution, grph or tble of vlues. Most of the topics cn be nd hve been presented in ech of the styles. With tht in mind this chpter will discuss the common types of questions on the exms. Some will be discussed bsed on their style nd other by the clculus topic being tested. Some of the questions test topics tht re usully found in one or severl contiguous sections of most textbook. These include the re-volume questions, differentil equtions, nd in BC the prmetric eqution-vector question, nd the power series question. The others tend to drw questions from diverse prts of the book tught t different times of the yer, yet bsed on the sme stem. Students need to be wre of this nd be redy to shift gers in the middle of the question. They my not be used to this since textbook questions tend to be bout the topic in tht section nd not drwn from previous sections. Using relesed exm questions during the yer nd especilly s prt of the review t the end of the yer will help students drw their knowledge together nd do their best on the exm. Another wy of developing this skill in students is to mke ll tests nd quizzes cumultive from the beginning of the yer. Alwys include few from previous units on ech test. 246
TYPE PROBLEMS 247 Keep in mind tht ech prt of free-response question cn be re-written s multiple-choice question. You cn lso use them s shorter open end questions on your tests. Wht follows is discussion of the 10 types of questions. Type 1: Rte & Accumultion These questions re often in rel context with lot of words describing sitution in which some things re chnging. There re usully two rtes given cting in opposite wys. These re lso known s in-out questions.students re sked bout the chnge tht the rtes produce over some time intervl either seprtely or together. The rtes my be given in n eqution, grph or tble. The question lmost lwys ppers on the clcultor llowed prt of the freeresponse exm nd the rtes re often firly complicted functions. If they re on the clcultor llowed section, students should store the equtions in the eqution editor of their clcultor nd use their clcultor to do ny eqution solving, integrtion, or differentition tht my be necessry. Shorter questions on this concept pper in the multiple-choice sections. As lwys, look over s mny questions of this kind from pst exms. Be redy to red nd pply; often these problems contin lot of writing which needs to be crefully red nd interepreted. Recognize tht the word rte mens tht derivtive is given even though the word derivtive my not pper. Recognize rte from the units given without the words rte or derivtive. b 8 f l^th dt = f^bh- f^h. nitil mount to the mount t f^th = f^t0h + 7 f l^xh dx, where t = t 0 is the initil time, nd f^t 0 h is the t0 initil mount. Understnd the question. It is often not necessry to s much computtion integrnd is the derivtive of the integrl. Explin the mening of derivtive or its vlue in terms of the context of the problem. Explin the mening of context of the problem. Justify your nswer. (FR) 2007 AB 2 (FR) 2009 AB 3 (FR) 2010 AB 1 BC 1 (MC) 2012 AB 81 BC 81 (FR) 2013 AB 1 BC 1
248 CHAPTER 22 Store functions in their clcultor recll them to do computtions on their clcultor. If the rtes re given in tble, be redy to pproximte n integrl using Riemnn sum or by trpezoids. Do mx/min or incresing/decresing nlysis. These topics re discussed in chpters 7, 14 nd 20 of this book. These questions my give the position, the velocity, or the ccelertion long with n initil condition. Students my be sked bout the motion of the prticle: its direction, when it chnges direction, when it is frthest left or right, when it turns round, how fr it trvels its position t certin time, etc. Speed, the bsolute vlue of velocity, is lso common topic. The prticle my be prticle, person, cr, etc. The position, velocity, or ccelertion my be given s n eqution, grph or tble. There re lot of different question is verstile wy to test vriety of clculus concepts. (FR) 2003 AB 5 (FR) 2006 AB 4 (FR) 2008 AB 4, BC 4 (FR) 2009 AB 1, BC 1 (MC) 2012 AB 16, 28, 79, 83, 89 BC 2, 89 (FR) 2013 AB 2 Solve n initil vlue differentil eqution problems: given the velocity (or Distinguish between position t some time, nd the totl distnce trveled during the time (displcement). The totl distnce trveled is the b vlue of velocity): 7 v^h t dt The net distnce (displ b 7 v ^h t dt nl position is the initil position plus the net chnge in distnce t from x = to x = t: s^th= s^h+7 v^xh dx Notice tht this is n ccumultion function eqution. verge vlue of function) Determine the speed nd whether it is incresing or decresing. If t some time, the velocity nd ccelertion hve the sme sign then the speed is incresing. If they hve different signs the speed is decresing. If the velocity grph is moving wy from (towrds) the t-xis the speed is incresing (decresing). Use difference quotient to pproximte derivtive from grph or tble. (Show quotient even if the denomintor is 1.) Approximte velocity or ccelertion from grph of tble.
TYPE PROBLEMS 249 Approximte n integrl using Riemnn sum or trpezoid sum with vlues from tble. Determine units of mesure. Interpret mening of derivtive or problem. Justify your nswer. These topics re discussed in chpter 9 with ides from chpters 11, 14 nd 20 of this book. Type 3: Interpreting Grphs The long nme is Here s the grph of the derivtive, tell me things bout the function. There is no eqution given nd it is not expected tht students will write the eqution; rther, students re expected to determine importnt fetures of the function or its grph directly from the grph of the derivtive. They my be sked for the loction of extreme vlues, intervls where the function is incresing or decresing, concvity, etc. They my be sked for function vlues t points, found by using the res of the regions on the derivtive s grph. The grph my be given in context nd student will be sked bout tht context. sked bout the motion nd position. Less often the function s grph my be given nd students will be sked bout its derivtives. Red informtion bout the function from the grph of the derivtive. This my be pproched using derivtive techniques or techniques using the intervl test, etc.) Write n eqution of tngent line. Evlute Riemnn or trpezoidl sums from geometry of the grph or from tble Evlute integrls from res of regions on the grph. Understnd tht the function, g^xh given grph is the grph of the integrnd, f^h. t So students should know x tht if g^xh g^h f^th dt, then gl^xh= f^xh nd gll^xh= f l^xh = +7 Justify your nswer. (FR) 2005 AB 5 (FR) 2010 AB 5 (FR) 2011 AB 4, BC 4 (MC) 2012 AB 15, 17, 76, 80, 85, 87 BC 15, 18, 76, 78, 80, 88 (FR) 2012 AB 3, BC 3 (FR) 2013 AB 4, BC 4
250 CHAPTER 22 The ides nd concepts tht cn be tested with this type question re numerous. The type ppers on the multiple-choice exms s well s the free-response. They hve ccounted for bout 20% of the points vilble on recent tests. It is very importnt tht students re fmilir with ll of the ins nd outs of this sitution. These topics re discussed minly in chpters 9 12, 14 nd 20 of this book. Type 4: Are Volume the volume of the solid formed when the region is revolved round line, or used s bse of solid with regulr cross-sections. These stndrd pplictions of the integrl the BC exm. If this questions ppers on the clcultor ctive section: Students should write the its vlue fter it. It is not required to give the ntiderivtive nd if students is (somehow) correct. Students should enter the equtions in the equtions editor nd then recll them for computtions. use them s the limits of integrtion. Students should write the coordintes of the intersection(s) on their pper nd ssign letter to ech. They my them use the letter s limit of integrtion from then on. Students should store nd recll them on the clcultor for computtions. This voids copy errors nd round off errors. Don t round. If rounded or truncted limits of integrtion result in (FR) 2008 AB 1, BC 1 (FR) 2009 AB 4 (FR) 2010 AB 5 (MC) 2012 AB 10, 92, BC 87 (FR) 2012 AB 2 re of the region between the grph nd the xis or between two grphs. n xis, by the disk/wsher method. (Shell method is never necessry, but is eligible for full credit if properly used.) or volume of the region in hlf. This involves setting up nd solving n
TYPE PROBLEMS 251 integrl eqution where the limit is the vrible for which the eqution is re of region bounded by polr curves. These topics re discussed minly in chpters 11, 12 nd 15 of this book. Type 5: Tble Questions Tbles my be used to test vriety of ides in clculus including nlysis of functions, ccumultion, position-velocity-ccelertion, theory, nd theorems mong problems re how this is tested. derivtive using difference quotient. Use the two vlues closest to the Use Riemnn sums (left, right, midpoint) or trpezoidl pproximtion to pproximte the vlue of (typiclly with subintervls of uneven length). The Trpezoidl Rule, per se, is not tested; it is expected tht students will dd the res of severl trpezoids without reference to formul. verge vlue, verge rte of chnge, Rolle s Theorem, the Men Vlue AB 3 which hs men score of less thn 1out of 9 points) Use tble vlues to evlute derivtives by the product, quotient nd chin rules. If the question is presented in some context nswers should be in tht context. Do s nd Don ts Do: Remember tht you do not know wht hppens between the vlues in the tble unless some other informtion is given. Don t ssume tht the lrgest number in the tble is the mximum vlue of the function, or the smllest vlue is the minimum. Do: Show wht you re doing even if you cn do it in your hed. If you re becuse you re required to show your work. A bld nswer, one with no work, even if correct my not receive credit. Don t do rithmetic. A long expression consisting entirely of numbers such ny wy. If you simplify correct nswer incorrectly, you will lose credit. It is oky to leve things like cos ^h 2 or even 1+ 1. Do not: Use to nswer prts of the question. While regression is good mthemtics, regression equtions re not one of the four things students my do with (FR) 2007 AB 3 (FR) 2010 AB 2, BC 2 (FR) 2011 AB 2, BC 2 (MC) 2012 AB 8, 86, 91, BC 8, 81, 86 (FR) 2013 AB 3, BC 3 (FR) 2014 AB 4, BC 4
252 CHAPTER 22 their clcultor. Regression gives only n pproximtion of our function. The exm is testing whether students cn work with numbers. Shorter questions on this concept pper in the multiple-choice sections. As lwys, look over s mny questions of this kind from pst exms. These topics re discussed in chpters 7 15 of this book. Type 6: Differentil Equtions The ctul solving of the differentil eqution is usully the min prt of the pproximtion. BC students my lso be sked to pproximte using Euler s Method. (FR) 2002 BC 5 (FR) 2007 AB 4b (FR) 2008 AB 5, BC 6 (MC) 2008 AB 22, 27 BC 24, 27 (FR) 2010 AB 6 (MC) 2012 AB 23, 25 BC 12, 16, 23 (FR) 2012 AB 5, BC 5 (FR) 2013 AB 6, BC 5 generl solution of differentil eqution using the method of seprtion of vribles (this is the only method tested). prticulr solution using the initil condition to evlute the constnt of integrtion initil vlue problem (IVP). Understnd tht proposed solution of differentil eqution is function (not number) nd if it nd its derivtive re substituted into the given differentil eqution the resulting eqution is true. This my be prt of doing the problem even if solving the differentil eqution is not required Growth-decy problems. extreme vlues fter seprting the vribles. mening, etc. The exms hve never sked students to ctully solve logistic eqution IVP. Lrge prts of the BC questions re often suitble for AB students nd contribute to the AB subscore of the BC exm. AB techers my dpt these for their students. Shorter questions on this concept pper in the multiple-choice sections. These topics re discussed minly in chpter 16 of this book.
TYPE PROBLEMS 253 Type 7: Occsionl Topics These two topics pper now nd then on the free-response exms. Sometimes they pper s full questions, other times s prt of question. Implicit Differentition to show tht it is correct. (This is becuse without the correct derivtive the rest of the question cnnot be done.) The follow-up is to nswer some question bout the function horizontl or verticl. Other times implicit differentition my be only prt of question. Implicit differentition questions lso pper on the multiple-choice sections reltion using the product rule, quotient rule, the Chin rule, etc. derivtive. coordintes of the point nd the vlue of the derivtive there. (Note: If ll tht is needed is the numericl vlue of the derivtive then the substitution 2 is often esier if done before solving for dy dx or d y dx 2 ) Anlyze the derivtive to determine where the reltion hs horizontl nd/ or verticl tngents. extreme vlues. It my lso be necessry to show tht the point where the derivtive is zero is ctully on the grph. (FR) 2004 AB 4 (FR) 2008 AB 6 (form B) (MC) 2012 AB 27 These topics re discussed minly in chpter 8 of this book. Relted Rtes Derivtives re rtes nd when more thn one vrible is involved the reltionships mong the rtes cn be found by differentiting with respect to time. The time vrible my not pper in the equtions. These questions pper occsionlly on the freeresponse sections often s prt of longer question. A simple question my pper in the multiple-choice sections. Set up nd solve relted rte problems. Know how to differentite with respect to time Interpret the nswer including units. These topics re discussed minly in chpter 10 of this book. (FR) 2001 AB 5 (FR) 2002 AB 6 (FR) 2003 AB 5, BC 5 (FR) 2008 AB 3 (MC) 2012 AB 88 BC 88
254 CHAPTER 22 Type 8: Prmetric nd Vector Questions ( BC topic) Rther thn the AB question bout prticle moving on line (Type 2), the BC exms hve things moving in the plne. The position or velocity is given s pir of prmetric equtions or vector. Students re sked questions bout the motion, the velocity, ccelertion, nd speed. Vectors my be written using prentheses, ( ), or pointed brckets,, or even i - j form. The pointed brckets seem to be the most populr right now, but ny nottion is llowed. (FR) 2010 BC 3 (FR) 2011 BC 1 (MC) 2012 BC 4 (FR) 2012 BC 2 (FR) 2014 BC 5c prticulr time. Given the velocity or or position. This is type of IVP differentil eqution question. Determine when the prticle is moving left or right (yl^th = 0). Determine when the prticle is moving up or down (xl^th = 0). t. Use the distnce trveled. Shorter questions on these ides pper in the multiple-choice sections. As These topics re discussed minly in chpter 19 of this book. Type 9: Polr Equtions ( BC topic) Every few yers question bout polr equtions ppers s free-response on the exm. The question concerns clculus concepts, so knowing the nmes or grphs of the vrious curves is not necessry. The grphs re usully given. If the topic is not on on the multiple-choice section. (FR) 2005 BC 2 (FR) 2007 BC 3 (MC) 2012 BC 91 (FR) 2013 BC 2 re enclosed by grph or between two grphs using the formul. Use the formuls x^ih= r^ih cos ^ih nd y^ih= r^ih sin ^ih to
TYPE PROBLEMS 255 Convert from polr to rectngulr form, Clculte the coordintes of point on the grph, nd dy Clculte d i nd dx (using the product rule). di Discuss the motion of prticle moving on the grph by discussing the mening of dr d i (motion towrds or wy from the pole), dy (motion in the di verticl direction) or dx (motion in the horizontl direction). d i dy dy/ di slope dx = t point on the grph. dx/ di These topics re discussed in chpter 19 of this book. Type 10: Sequences nd Series ( BC topic) Convergence tests for sequences pper on both sections of the BC Clculus exm. In the multiple-choice section students my be sked to sy if series converges or which of severl series converge. On the free-response section there is usully one full question devoted to power series. The Rtio test is used most often to determine the rdius of convergence nd the other tests to determine if the series convergences t the end points of the intervl of convergence. Students should be fmilir with nd ble to write severl terms nd the generl or given series. They my do this by substituting into series, differentiting it or integrting it. Use convergence tests to determine if series converges. The test to be used is rrely given so students need to know when to use ech of the common tests. This my be prt the endpoint nlysis for the intervl of convergence. Write the terms of Tylor or Mclurin series by clculting the Distinguish between the Tylor series for function nd the function itself. Do NOT sy tht the Tylor polynomil is equl to the function; sy it is pproximtely equl. Write series by substituting into known series, by differentiting or integrting known series, or by some other lgebric mnipultion of series. Know (memorize) the Mclurin series for sin ^xh, cos ^xh, e x nd 1 1 - x m. (FR) 2010 BC 6 (FR) 2011 BC 6 (FR) 2012 BC 4d, 6 (MC) 2012 BC 5, 9, 13, 17, 22, 27, 79, 90 (FR) 2014 BC 6
256 CHAPTER 22 convergence. This is usully done by using the Rtio test nd checking the endpoints. Be fmilir with geometric series, its rdius of convergence, nd the be ble S 3 = 1 1 - r. Re-writing rtionl expression s the sum of geometric series nd then writing the series. Be fmilir with the hrmonic series (diverges) nd lternting hrmonic series (converges). function t point in the intervl of convergence. Determine the error bound for convergent series (Alternting series error bound nd Lgrnge error bound). function (e.g. extreme vlues). This list is long, but only few of these items cn be sked in ny given yer. The series question on the exm is usully quite strightforwrd. As I hve suggested before, look t nd work s mny pst exm questions to get n ide of wht is sked These topics re discussed minly in chpters 17 nd 18 of this book. (FR) 2009 AB 3, BC 3 it for them, if you like. It contins hints nd suggestions to help them get redy for the exms. Much of this ws mentioned previously for you in this book; it is given here, in one plce, for you nd your students.