Chief Reader Report on Student Responses:

Transcription

1 Chief Reader Report o Studet Resposes: 17 AP Calculus AB ad Calculus BC Free-Respose Questios Number of Readers (Calculus AB/Calculus BC): 971 Calculus AB Number of Studets Scored 316,99 Score Distributio Exam Score N %At 5 59, , , , ,59.4 Global Mea.93 Calculus BC Number of Studets Scored 13,514 Score Distributio Exam Score N %At 5 56, , , , ,7 5.3 Global Mea 3.78 Calculus BC Calculus AB Subscore Number of Studets Scored 13,55 Score Distributio Exam Score N %At 5 64, , , , , Global Mea The College Board.

2 The followig commets o the 17 free-respose questios for AP Calculus AB ad Calculus BC were writte by the Chief Reader, Stephe Davis of Davidso College. They give a overview of each freerespose questio ad of how studets performed o the questio, icludig typical studet errors. Geeral commets regardig the skills ad cotet that studets frequetly have the most problems with are icluded. Some suggestios for improvig studet preparatio i these areas are also provided. Teachers are ecouraged to atted a College Board workshop to lear strategies for improvig studet performace i specific areas. 17 The College Board.

3 Questio AB1/BC1 Topic: Modelig Volume/Related Rates-Tabular/Aalytic Max. Poits: 9 Mea Score: AB1: 3.3; BC1: 5. What were resposes expected to demostrate i their respose to this questio? I this problem studets were preseted with a tak that has a height of 1 feet. The area of the horizotal cross sectio of the tak at height h feet is give by a cotiuous ad decreasig fuctio A, where Ah is measured i square feet. Values of Ah for heights h,, 5, ad 1 are supplied i a table. I part (a) studets were asked to approximate the volume of the tak usig a left Riema sum ad idicate the uits of measure. Studets eeded to respod by 1 icorporatig data from the table i a left Riema sum expressio approximatig A h dh usig the subitervals,,, 5, ad 5, 1. [LO 3.B/EK 3.B] I part (b) studets eeded to explai that a left Riema sum approximatio for the defiite itegral of a cotiuous, decreasig fuctio overestimates the value of the itegral. [LO B/EK 3.B] I part (c) the fuctio f give by fh is preseted as a model for the area, i square feet, of.h e h the horizotal cross sectio at height h feet. Studets were asked to fid the volume of the tak usig this model, agai idicatig uits of measure. Usig the model f for cross-sectioal areas of the tak, studets eeded to express the 1 volume of the tak as fh dh ad use the graphig calculator to produce a umeric value for this itegral. [LO 3.4D/EK 3.4D] I part (d) water is pumped ito the tak so that the water s height is icreasig at the rate of.6 foot per miute at the istat whe the height of the water is 5 feet. Studets were asked to use the model from part (c) to fid the rate at which the volume of water is chagig with respect to time whe the height of the water is 5 feet, agai idicatig uits of measure. Studets eeded to realize that the volume of water i the tak, as a fuctio of its height h, is h give by Vh fx dx ad the use the Fudametal Theorem of Calculus to fid that the rate of chage of the volume of water with respect to its height is give by Vh fh. The, usig the chai rule for derivatives, studets eeded to relate the rates of chage of volume with respect to time ad height ad the rate of chage of the water s height with respect to time. Iformatio i the problem suffices to be able to fid these rates whe the water s height is 5 feet. [LO.3C/EK.3C, LO 3.3A/EK 3.3A] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, coectig multiple represetatios, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? Studets did reasoably well at supplyig uits for parts (a) ad (c); both are volume measuremets. Studets who had correct uits i parts (a) ad (c) sometimes supplied o uits or icorrect uits for part (d). I part (a) most studets were able to costruct a left Riema sum, although some studets cofused left with right. Most studets were able to recogize volume as the itegral of cross-sectioal area ad correctly completed part (c). I part (d) may studets were ot able to get the correct start o this part, which required recogizig that the volume V of the water i the tak at h dv dh height h is give by Vh fx dx or by startig with the correct relatioship for rates of chage, fh. dt dt 17 The College Board.

4 What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) some studets cofused left with right, while some other studets costructed their ow subitervals so that, 1 was divided ito three subitervals of equal legths. Volume 1 Ah dh ( ) A() (5 ) A() (1 5) A(5) I part (b) some studets used vague laguage or isufficiet or speculative reasoig to explai why the approximatio i part (a) is a overestimate. Studets eeded to focus upo the give iformatio that the fuctio A is decreasig. Because A is decreasig, a left Riema sum overestimates the value of 1 Ah dh. I part (c) some studets used a icorrect itegrad (e.g., f h or f h ) i a itegral to compute the volume of the tak. Usig the model that f h describes the area, i square feet, of a horizotal cross sectio at height h feet, the 1 volume of the tak is f h dh cubic feet. I part (d) studets struggled to fid a correct expressio for the volume of water whe the height of the water is h feet, e.g., statig that the volume is h fh, or f h, or the costat 1 f h dh. Vh h f( x) dx I part (d) studets made errors i differetiatig their water volume expressio with respect to time. Usig the Fudametal Theorem of Calculus ad the dv dh chai rule, fh. dt dt Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? Studets should experiece a variety of problems i which uits are required: area, volume, ad various rates of chage. Teachers ca work with studets to be wary of makig uwarrated assumptios. For example, i part (b) some studets treated A is decreasig to be equivalet to A is egative. The latter is ot supported by the iformatio give i the problem (see the MPAC reasoig with defiitios ad theorems) ad was ot a applicable reaso that the left Riema sum is a overestimate for the volume of the tak. Teachers should work with studets to set up their ow fuctios that describe the behavior of some object. This is particularly true whe those fuctios are give by a itegral with a variable limit of itegratio as i part (d). Teachers should the ask studets to verify particular thigs about those fuctios, which iclude aalyzig first ad secod derivatives of the fuctios that the studets have set up. 17 The College Board.

5 What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad participatio o the Olie Teacher Commuity are of high value for practice ad for advice. 17 The College Board.

6 Questio AB Topic: Modelig Rates Max. Poits: 9 Mea Score: 4.1 What were resposes expected to demostrate i their respose to this questio? The cotext for this problem is the removal ad restockig of baaas o a display table i a grocery store durig a 1- hour period. Iitially, there are 5 pouds of baaas o the display table. The rate at which customers remove baaas from the table is modeled by t f t 1 (.8 t)si 1 for t 1, where f t is measured i pouds per hour ad t is the umber of hours after the store opeed. Three hours after the store opes, store employees add baaas to the display table at a rate modeled by 3.4l 3 g t t t for 3 t 1, where g t is measured i pouds per hour ad t is the umber of hours after the store opeed. I part (a) studets were asked how may pouds of baaas are removed from the display table durig the first hours the store is ope. Studets eeded to realize that the amout of baaas removed from the table durig a time iterval is foud by itegratig the rate at which baaas are removed across the time iterval. Thus, studets eeded to express this amout as f t dt ad use the graphig calculator to produce a umeric value for this itegral. [LO 3.4E/EK 3.4E1] I part (b) studets were asked to fid f 7 ad, usig correct uits, explai the meaig of f 7 i the cotext of the problem. Studets were expected to use the graphig calculator to evaluate the derivative, ad explai that the rate at which baaas are beig removed from the display table 7 hours after the store has bee ope is decreasig by 8.1 pouds per hour per hour. [LO.3A/EK.3A1, LO.3D/EK.3D1] I part (c) studets were asked to determie, with reaso, whether the umber of pouds of baaas o the display table is icreasig or decreasig at time t 5. This ca be determied from the sig of the differece betwee the rate at which baaas are added to the table ad the rate at which they are removed from the table. Thus, studets eeded to evaluate the differece g5 f 5 o the graphig calculator ad report that the umber of pouds of baaas o the display table is decreasig because this value is egative. [LO.A/EK.A1] I part (d) studets were asked how may pouds of baaas are o the display table at time t 8. The umber of pouds of 8 baaas added to the table by time t 8 is give by g t dt, ad the umber of pouds of baaas removed from the 3 8 table by that time is give by f t dt. Thus, usig that there were iitially 5 pouds of baaas o the table, the gives the umber of pouds of baaas o the table at time 8. expressio 5 g t dt f t dt t Studets eeded to evaluate this expressio usig the umeric itegratio capability of the graphig calculator. [LO 3.4E/EK 3.4E1] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I part (a) some studets misiterpreted the fuctio f t, leadig them to respod with a aswer of f or some similar variat. I part (b) some studets had difficulty with the specific cotext of the problem, iappropriately ivokig the vocabulary ( velocity, acceleratio ) of rectiliear motio, or writig about slope i attemptig to explai the meaig of f 7. Studets also had difficulty determiig the correct uits for f (7), ofte givig the uits for f, t pouds per hour. 17 The College Board.

7 I part (c) some studets read the questio superficially, focusig either o the phrases umber of pouds of baaas or icreasig or decreasig, ad attempted to aswer the questio by cosiderig defiite itegrals or derivatives, respectively, istead of, or i additio to, the fuctio values f 5 ad g 5. Geerally, studets did well i part (d), although those studets that were icorrectly iterpretig the origial fuctios i part (a) ofte carried that icorrect iterpretatio ito part (d). What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) studets had difficulty dealig with f t as a rate at which pouds of baaas were removed so respoded with a value for f (). Others erroeously icorporated the umber of pouds of baaas o the display table whe the store opeed. The umber of pouds of baaas removed from the table durig the first two hours after the store opeed is give by f t dt. I part (b) studets made errors calculatig a symbolic derivative of f, t whereas they could use the graphig calculator to evaluate f (7) umerically. f For the explaatio i part (b), studets agai had difficulty with the meaig of f, t or failed to tie the meaig to the time t 7. After the store has bee ope 7 hours, the rate at which baaas are beig removed from the display table is decreasig by 8.1 pouds per hour per hour. I part (c) may studets assumed that a questio about icreasig or decreasig must be aswered usig a derivative, either igorig or uable to deal with f t ad gt give as rates. Whether the umber of pouds of baaas o the display table is icreasig or decreasig at time t 5 ca be determied by comparig the rate of removal of the baaas, f (5) pouds per hour, agaist the rate at which baaas are added to the table, g (5) pouds per hour. Because f(5) g(5), the umber of pouds of baaas o the display table is decreasig at time t 5. I part (d) a commo error was to overlook that baaas are ot added to the table util time t 3, icorrectly computig the et chage i the pouds of baaas o the table as 8 gt f t dt(a coverget improper itegral). Also, some studets failed to accout for the 5 pouds of baaas o the display table whe the store opeed. The umber of pouds of baaas o the table at time t 8 is give by: The umber of pouds of baaas o the table iitially, plus the umber of pouds added, mius the umber of pouds removed: g t dt f t dt 17 The College Board.

8 Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? May studets failed to make effective use of their graphig calculator o this problem, either because they lacked skill i its use, or because the time allotted for calculator use o the exam had elapsed. The former ca be addressed by fidig opportuities for studets to make appropriate use of calculators throughout the course, ad ot just durig exam preparatio time. The latter may be helped some by urturig time maagemet skills for tests ad assigmets. May studets had issues with a cotext that moves beyod geometry (area/volume/slope) or rectiliear motio (velocity/acceleratio). Further, studets are challeged by a fuctio that describes a rate. More practice is eeded with a variety of applied cotexts, especially with iterpretatio withi those cotexts. What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad participatio o the Olie Teacher Commuity are of high value for practice ad for advice. For this problem, the Teachig ad Assessig Calculus Modules 8 (Iterpretig Cotext) ad 5 (Aalyzig Problems i Cotext) would be the most help. 17 The College Board.

9 Questio AB3/BC3 Topic: Graphical Aalysis of f prime /FTC Max. Poits: 9 Mea Score: AB3: 3.63; BC3: 5.4 What were resposes expected to demostrate i their respose to this questio? I this problem studets were give that a fuctio f is differetiable o the iterval [ 6,5] ad satisfies f 7. For 6 x 5, the derivative of f is specified by a graph cosistig of a semicircle ad three lie segmets. I part (a) studets were asked to fid values of f 6 ad f 5. For each of these values, studets eeded to recogize that the et chage i f, startig from the give value f 7, ca be computed usig a defiite itegral of f x with a lower limit of itegratio ad a upper limit the desired argumet of f. These itegrals ca be computed usig properties of the defiite itegral ad the geometric coectio to areas betwee the graph of y f x ad the x-axis. Thus, studets eeded to add the iitial coditio f 7 to the values of the defiite itegrals for the desired values. [LO 3.C/EK 3.C1] I part (b) studets were asked for the itervals o which f is icreasig, with justificatio. Sice f is give o the iterval [ 6,5], f is differetiable, ad thus also cotiuous, o that iterval. Therefore, f is icreasig o closed itervals for which f x o the iterior. Studets eeded to use the give graph of f to see that f x o the itervals [ 6, ) ad, 5, so f is icreasig o the itervals [ 6, ] ad, 5, coectig their aswers to the sig of f. [LO.A/EK.A1-.A, LO.B/EK.B1] I part (c) studets were asked for the absolute miimum value of f o the closed iterval [ 6,5], ad to justify their aswers. Studets eeded to use the graph of f to idetify critical poits of f o the iterior of the iterval as x ad x. The they ca compute f ad f, similarly to the computatios i part (a), ad compare these to the values of f at the edpoits that were computed i part (a). Studets eeded to report the smallest of these values, f 7 as the aswer. Alteratively, studets could have observed that the miimum value must occur either at a poit iterior to the iterval at which f trasitios from egative to positive, at a left edpoit for which f is positive immediately to the right, or at a right edpoit for which f is egative immediately to the left. This reduces the optios to f 6 3 ad f 7. [LO.A/EK.A1-.A, LO.B/EK.B1, LO 3.3A/EK 3.3A3] I part (d) studets were asked to determie values of f 5 ad f 3, or to explai why the requested value does ot exist. Studets eeded to fid the value f 5 as the slope of the lie segmet o the graph of f through the poit correspodig to x 5. The poit o the graph of f correspodig to x 3 is the jucture of a lie segmet of slope o the left with oe of slope 1 o the right. Thus, studets eeded to report that f 3 does ot exist, ad explai why the give graph of f shows that f is ot differetiable at x 3. Studet explaatios could be doe by otig that the left-had ad right-had limits at x 3 of f x f3 the differece quotiet have differig values ( ad 1, respectively), or by a clear descriptio of the x 3 relevat features of the graph of f ear x 3. [LO 1.1A(b)/EK 1.1A3] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, coectig multiple represetatios, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I part (a) studets geerally icluded the iitial value ( ) 7 f i their aswers ad could make the geometric coectio betwee defiite itegrals ad areas betwee the itegrad ad the horizotal axis. However, some studets had difficulty with the eeded properties of the defiite itegrals, amely that reversig the limits of itegratio results i a itegral value of opposite sig, or that whe the itegrad dips below the horizotal axis, the area betwee the curve ad the axis cotributes egatively to the itegral value. 17 The College Board.

10 I part (b) most studets did a good job of focusig o where the graph of f is above the x-axis to coclude that f x, so f is icreasig. Some studets icorrectly cosidered the figure to be the graph of f ad gave a icorrect aswer o that basis. I part (c) may studets could idetify x as a critical poit for f, but there were problems i recogizig that edpoits eed to be cosidered ad a global argumet give to justify a absolute extremum. I part (d) studets were successful fidig f ( 5) by computig a slope o the graph of f but had difficulty explaiig why f (3) does ot exist i a mathematically satisfyig way. What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) studets made errors itegratig right-to-left. 6 f ( 6) f( ) fx dx f( ) fx dx 6 I part (b) some studets pulled the icorrect iformatio from the graph of f (e.g., where f is icreasig, rather tha where f is positive). The graph of f is above the x-axis o [ 6, ) ad o (, 5), so f is icreasig o [ 6, ] ad o [, 5]. I part (c) studets explaied why f has a relative miimum at x, rather tha justifyig a absolute miimum there. Cadidates for the absolute miimum are edpoits ( x 6 ad x 5 ) ad critical poits ( x ad x ). The smallest of the values f ( 6), f ( ), f (), or f (5) is the absolute miimum value for f o [ 6,5]. I part (d) studets used imprecise laguage (e.g., the graph of f has a vertex ) i attemptig a explaatio of why f 3 does ot exist. f x f3 For x 3,, whereas for 3 x 5, x 3 f x f3 f x f3 1. Thus, the limit of as x 3 x 3 x 3 does ot exist, so f (3) does ot exist. Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? I geeral, teachers ca place more emphasis o studets commuicatio skills i covertig graphical iformatio to covey mathematically saliet reasos ad justificatios. I particular, studets ca practice usig mathematical otatio to explai, from a fuctio s graph, how a fuctio fails to be differetiable at a particular poit. What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad 17 The College Board.

11 participatio o the Olie Teacher Commuity are of high value for practice ad for advice. For this problem, the Teachig ad Assessig Calculus Module 4 (Justifyig Properties ad Behaviors of Fuctios) would be the most help. 17 The College Board.

12 Questio AB4/BC4 Topic: Modelig with Separable Differetial Equatio Max. Poits: 9 Mea Score: AB4: 1.54; BC4: 3.13 What were resposes expected to demostrate i their respose to this questio? The cotext for this problem is the iteral temperature of a boiled potato that is left to cool i a kitche. Iitially at time t, the potato s iteral temperature is 91 degrees Celsius, ad it is give that the iteral temperature of the potato exceeds 7 degrees Celsius for all times t. The iteral temperature of the potato at time t miutes is modeled by the dh 1 fuctio H that satisfies the differetial equatio H 7, where Ht is measured i degrees Celsius ad dt 4 H 91. I part (a) studets were asked for a equatio of the lie taget to the graph of H at t, ad to use this equatio to approximate the iteral temperature of the potato at time t 3. Usig the iitial value ad the differetial 1 equatio, studets eeded to fid the slope of the taget lie to be H ad report the equatio 4 of the taget lie to be y t. Studets eeded to fid the approximate temperature of the potato at t 3 to be d H degrees Celsius. [LO.3B/EK.3B] I part (b) studets were asked to use to determie whether dt the approximatio i part (a) is a uderestimate or overestimate for the potato s iteral temperature at time t 3. d H 1 dh 1 Studets eeded to use the give differetial equatio to calculate H 7. The usig the give dt 4 dt 16 d H iformatio that the temperature always exceeds 7 degrees Celsius, studets eeded to coclude that for all dt times t. Thus, the graph of H is cocave up, ad the lie taget to the graph of H at t lies below the graph of H (except at the poit of tagecy), so the approximatio foud i part (a) is a uderestimate. [LO.1D/EK.1D1, LO.A/EK.A1] I part (c) a alterate model, G, is proposed for the iteral temperature of the potato at times t 1. dg Gt is measured i degrees Celsius ad satisfies the differetial equatio G 7 3 with G 91. dt Studets were asked to fid a expressio for Gt ad to fid the iteral temperature of the potato at time t 3 based o this model. Studets eeded to employ the method of separatio of variables, usig the iitial coditio G 91 to 1 t resolve the costat of itegratio, ad arrive at the particular solutio 3 Gt 7. Studets should the have 3 reported that the model gives a iteral temperature of G3 54 degrees Celsius for the potato at time t 3. [LO 3.5A/EK 3.5A] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I geeral, studets particularly Calculus AB studets cotiue to struggle with questios that preset a differetial equatio i which the derivative is expressed i terms of oly the depedet variable. See 11 questio AB5/BC5 dw 1 db 1 ( W 3 ; AB mea score 1.63, BC mea score 3.53) ad 1 AB5/BC5 ( 1 B ; AB mea dt 5 dt 5 score.87, BC mea score 4.75). The lack of appearace of the idepedet variable, t, seems to stymie may studets. I part (a) most studets kew they had to use the differetial equatio to get the slope of the taget lie but were cofused about what value to substitute for H, some optig for H istead of H 91. As a result, may studets had a taget lie passig through the correct poit,, 91, but with a icorrect slope, The College Board.

13 I part (b) may studets failed to apply the chai rule, resultig i dh dt, d H d dh d dh dh. d dh istead of Also, may studets argued for a overestimate or uderestimate from the value of dt dt dt dh dt dt d H at a sigle test poit, failig to uderstad the eed to cosider a iterval spaig the times from t to t 3 dt (ad thus missig the sigificace of the statemet i the stem that the potato s iteral temperature is greater tha 7 C for all times t ). I part (c) may studets stumbled attemptig to separate variables, resultig i either o separatio or a icorrect separatio. Some studets omitted a costat of itegratio at the atiderivative step, thus those studets were ot eligible to ear remaiig poits. Fially, some studets wet straight from the equatio 3G t to compute the value of G whe t 3 without givig a explicit expressio for Gt. What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) computig the slope of the taget dh 1 7 lie as ( 7) dt 4 4 t dh dt t 1 (91 7) 16 4 I part (b) failig to apply the chai rule: d 1 H 7 1 dt 4 4 I part (b) failig to ascertai cocavity of the graph of H o a iterval cotaiig t ad t 3; e.g., basig a respose o just d H dt H 43. d dh H H dt 4 4 dt 16 7 d H 1 H 7 for t because H 7 for dt 16 t. Thus, the graph of H is cocave up o a iterval cotaiig t ad t 3, so usig a lie taget to the graph of H at t to approximate H (3) must result i a uderestimate. dg I part (c) from G 7 3, dt icorrectly movig to 3 Gt G 7 dt G 7 C 5 dg From 7 3 G, 7 3 dg G 1, so dt dt G 7 3 dg ( 1) dt 3 G 7 t C. ad The College Board.

14 Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? Teachers ca add this problem as a practice problem to 11 AB5/BC5 ad 1 AB5/BC5 ad expad this collectio with more differetial equatios problems ivolvig separatio of variables i which the derivative is expressed oly i terms of the depedet variable. Further, teachers ca adapt may of their separatio of variable problems ivolvig variables x, y, ad/or t to equivalet problems usig other letters for the variables. Studets should have the flexibility to trasfer skills leared i x s ad y s to equivalet eviromets with other variable ames. What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad participatio o the Olie Teacher Commuity are of high value for practice ad for advice. For this problem, the Teachig ad Assessig Calculus Modules 6 (Iterpretig Notatioal Expressios) ad 7 (Applyig Procedures) would be the most help. 17 The College Board.

15 Questio AB5 Topic: Particle Motio Max. Poits: 9 Mea Score: 3.59 What were resposes expected to demostrate i their respose to this questio? I this problem, two particles, P ad Q, are movig alog the x-axis. For t 8, the positio of particle P is give by xp t l t t 1, while particle Q has positio 5 at t ad velocity vq t t 8t 15. I part (a) studets were asked for those times t, t 8, whe particle P s motio is to the left. Usig the chai rule, studets eeded to fid a expressio for the velocity of particle P at time t by differetiatig the positio xp. t By aalyzig the sig of this derivative to determie those times t with xp t, studets should have cocluded that particle P is movig left for t 1. [LO.1C/EK.1C-.1C4, LO.3C/EK.3C1] I part (b) studets were asked for all times t, t 8, durig which both particles travel i the same directio. Usig the velocity of particle P, vpt xp t, foud i part (a), ad the give velocity vq t for particle Q, studets eeded to fid those subitervals of t 8 o which both vp t ad vq t have the same sig. Studets should have respoded that for 1 t 3 ad for 5 t 8, otig that both vp t ad vq () t are positive o these itervals, both particles travel i the same directio (to the right). There is o time whe both velocities are egative. [LO.3C/EK.3C1] I part (c) studets were asked for the acceleratio of particle Q at time t, ad to determie, with explaatio, whether particle Q s speed is icreasig, decreasig, or either at time t. Studets eeded to differetiate v t to fid that the acceleratio of particle Q is give by Q aqt vq t t 8, ad report that particle Q s acceleratio at time t is aq 4. Studets should have explaied that particle Q s speed is decreasig at time t because the velocity ad acceleratio of particle Q have opposite sigs at that time. [LO.1C/EK.1C, LO.3C/EK.3C1] I part (d) studets were asked to fid the positio of particle Q the first time it chages directio. Usig the aalysis of the sig of vq () t doe i part (b), studets should have cocluded that the first chage of directio of particle Q s motio occurs at time t 3. The et chage i positio 3 of particle Q across the time iterval, 3 is give by v Q t dt. Studets eeded to evaluate this itegral usig the Fudametal Theorem of Calculus ad use the iitial positio of particle Q to fid that particle Q s positio at time 3 t 3 is 5 v Q t dt 3. [LO 3.3B(b)/EK 3.3B, LO 3.4C/EK 3.4C1] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I part (a) the vast majority of studets kew to cosider vpt x Pt to determie the directio of motio of particle P durig the give time iterval. May studets, however, were uable to correctly apply the chai rule i computig x Pt. I part (b) studets seemed to be aware that the two particles travel i the same directio whe their velocity values have the same sig but were challeged to do a sig aalysis of the velocity fuctios, or to merge two sig aalyses whe the subitervals of oe had edpoits differig from those of the other. I part (c) studets could correctly compute the acceleratio of particle Q as v t t 8 but failed to take ito accout the sig of v Q(), as well as the sig of acceleratio at time t, to determie whether the speed of particle Q is icreasig, decreasig, or either at time t. Q 17 The College Board.

16 I part (d) studets could atidifferetiate positio at the desired time. vq, t but may failed to cosider the particle s iitial positio to fid its What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? I geeral, studets were challeged by otatio ivolvig subscripts, so that they ofte dropped subscripts P or Q, or redefied the fuctios xp t ad vq t with ew ames to avoid subscripts. This could lead to a great deal of cofusio over which particle the studet was discussig. Some studets used iappropriate iterval otatio, e.g., 1 3 istead of 1 t 3 1, 3. or Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) may studets had paretheses errors, computig 1 t xp t t. t t 1 t t 1 1 t x P t (t ). t t 1 t t 1 I part (b) ot sythesizig correct sig aalysis of vp t ad vq, t stoppig at vp t for t 1 ad v t for 1 t 8, while P vqt for t 3, vq t for 3 t 5, ad v t for 5 t 8. Q vpt for t 1 ad vp t for 1 t 8, while vq t for t 3, vq t for 3 t 5, ad vq t for 5 t 8. Thus, vp t ad vq t are both positive for 1 t 3 ad for 5 t 8. There is o time, t 8, whe both velocities are egative. (Note that a well orgaized ad labeled sig chart ca aid this sythesis.) I part (c) arguig (just) from aq () that particle Q s speed is decreasig. Because aq () ad vq (), particle Q s speed is decreasig at time t. I part (d) ot accoutig for the iitial positio: from v t t 8t 15, Q 1 3 xq t t 4t 15 t, so 3 x (3) Q xq 3 xq vq t dt 5 t 4t 15t 3 5 ( ) t3 t Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? Teachers ca emphasize correct ad clear commuicatio ad fid opportuities to icrease studets comfort with subscripts as a clarifyig otatio. 17 The College Board.

17 What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad participatio o the Olie Teacher Commuity are of high value for practice ad for advice. For this problem, the Teachig ad Assessig Calculus Module 4 (Justifyig Properties ad Behaviors of Fuctios) would be the most help. 17 The College Board.

18 Questio AB6 Topic: Aalysis of Fuctios-Tabular/Graphical Max. Poits: 9 Mea Score: 3.3 What were resposes expected to demostrate i their respose to this questio? si x This problem deals with multiple fuctios. Fuctio f is defied by fx cos x e. Fuctio g is differetiable ad values of g x ad g x correspodig to iteger values of x from x 5 to x, iclusive, are give i a table. Fuctio h is defied o [ 5,5] ad the graph of h, comprised of five lie segmets, is give. I part (a) studets were asked for the slope of the lie taget to the graph of f at x. Usig the sum ad chai rules for differetiatio ad the derivatives of trigoometric ad expoetial fuctios to differetiate f x, studets eeded to evaluate f to fid the slope of the taget lie. [LO.1C/EK.1C-.1C4, LO.3B/EK.3B1] I part (b) the fuctio k is defied by k x h f x, ad studets were asked to fid k. Studets eeded to apply the chai rule ad determie the value of h from the graph of h to arrive at the value for k. [LO.1C/EK.1C4, LO.A/EK.A] I part (c) the fuctio m is defied by mx gx hx, ad studets were asked to fid m. Studets eeded to apply the product ad chai rules for differetiatio, fid values for g 4 ad g 4 i the table for g, ad use the graph of h to determie h ad h, to fid m g4 h g4 h 3. [LO.1C/EK.1C3-.1C4, LO.A/EK.A, LO.3B/EK.3B1] I part (d) studets were asked to determie whether there is a umber c i the iterval [ 5, 3] such that gc 4, ad to justify their aswers. Usig the table for g, studets g3 g5 should have cofirmed that 4. Give that g is differetiable, studets should have cocluded that g 3 5 is cotiuous o [ 5, 3] ad, thus, recogize that the hypotheses for the Mea Value Theorem are satisfied, ad aswered i the affirmative that a umber c exists i the iterval [ 5, 3] such that gc 4. [LO 1.B/EK 1.B1, LO.4A/EK.4A1] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, coectig multiple represetatios, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I part (a) most studets kew that the slope of the lie taget to the graph of f at x is give by f ( ), but some studets made errors usig the chai rule, or i statig derivatives of sie ad cosie whe attemptig to fid f x. Some studets also misevaluated si, si, ad/or cos. I part (b) may studets were challeged by the abstract ature of the fuctio k. I geeral, studets could evaluate a derivative of h by calculatig the slope of a lie segmet i the give graph of h. However, studets ofte failed to apply the chai rule to differetiate k, ad some studets made errors i evaluatig f ( ). I part (c) studets agai were challeged with the abstract presetatio of a fuctio, i this case m. Some studets used a icorrect product rule ad/or made a chai rule error i fidig the derivative of g( x). Fially, some studets miscomputed the value of h () from the graph of h. I part (d) may studets attempted to justify a egative respose, appealig to a (false) coverse of either the Itermediate Value Theorem (with g ) or the Mea Value Theorem (with g). Studets that recogized 4 as the average rate of chage of g across the iterval [ 5, 3] ofte failed to declare the full hypotheses eeded to apply the Mea Value Theorem, usually omittig that g is cotiuous o [ 5, 3]. 17 The College Board.

19 What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) studets failed to apply the chai si x rule, computig f x to be si( x) e. Applyig the chai rule, si x f x ( si( x)) e cos x. I part (b) studets agai made chai rule errors, computig k x to be h fx. Applyig the chai rule, k x h f x f x. I part (c) studets made product ad chai rule errors, computig m x to be g( x) h x. Applyig the product ad chai rules, m x g( x) h x g( x) h x. I part (d) studets gave icomplete justificatios for a applicatio of the Mea Value Theorem, ofte omittig the cotiuity hypothesis. g is differetiable, so g is cotiuous o [ 5, 3]. The average rate of chage of g across [ 5, 3] is 1 3 ( 5) 4. Thus, by the Mea Value Theorem, there is at least oe value c, 5 c 3, for which gc 4. Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? Part of otatioal fluecy is the ability to apply rules such as the chai or product rules for differetiatio to circumstaces where fuctio ames are other tha f ad g i what the studet may cosider stadard positios. Studets ca icrease their dexterity with such fuctios through practice fidig derivatives of abstractly-defied fuctios i terms of products, quotiets, or compositios, where the costituet fuctios are preseted with a variety of ames ad i a variety of ways: umerically (table), graphically, or aalytically. Also, teachers ca reiforce that justificatios of theorem applicatios require verificatio that the coditios required by the hypotheses of the theorem are met. Studets should get i the practice of highlightig these coditios wheever they state the theorem or apply the theorem to a particular istace. What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad participatio o the Olie Teacher Commuity are of high value for practice ad for advice. For this problem, the Teachig ad Assessig Calculus Modules (Selectig Procedures) ad 3 (Establishig Coditios for Defiitios ad Theorems) would be the most help. 17 The College Board.

20 Questio BC Topic: Polar-Area/Aalysis of Curves Max. Poits: 9 Mea Score: 3.1 What were resposes expected to demostrate i their respose to this questio? I this problem a polar graph is supplied for the curves f 1 sicos ad g cos for. Regios R, bouded by the graph of r f ad the x-axis, ad regio S, bouded by the graphs of r f, r g, ad the x-axis, are idetified o the graph. I part (a) studets were asked for the area of R. Studets eeded to recogize that regio R is traced by the polar ray segmet from r to r f for ad use the graphig 1 calculator to evaluate the area of R as the umeric value of f d. [LO 3.4D/EK 3.4D1] I part (b) studets were asked to produce a equatio ivolvig oe or more itegrals that ca be solved for k, k, such that the ray k divides S ito two regios of equal areas. Studets eeded to recogize that regio S is traced by the polar ray segmet from r f to r g for. The ray k divides S ito two subregios with areas 1 k g f d 1 ad g f d. Studets should have reported a equatio equivalet k to settig these two expressios equal to each other, or settig oe of them equal to half of the area of S, which is give by 1 g f d. [LO 3.4D/EK 3.4D1] I part (c) w is defied as the distace betwee the poits with polar coordiates f, ad g,. Studets were asked to write a expressio for w ad to fid w A, the average value of w for. Studets eeded to recogize that w g f ad use the graphig w d calculator to evaluate the average value wa. [LO 3.4B/EK 3.4B1] I part (d) studets were asked to fid the value of for which w w A, ad to determie whether w is icreasig or decreasig at that value of. Importig the value of w A from part (c), studets eeded to use the graphig calculator to solve w wa to obtai Studets should have reported this value rouded or trucated to three decimal places. Studets should the have reported that w is decreasig at this value of because the calculator reports a egative value for w [LO 3.4D/EK 3.4D1] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, coectig multiple represetatios, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I parts (a) ad (b) studets kew to use defiite itegrals to determie area bouded by curves, but were challeged relatig to the polar cotext, sometimes providig itegral expressios that mixed rectagular ad polar perspectives. I part (c) studets geerally were able to provide the itegral setup for the average value of w for. Some studets had challeges i recogizig that w g f o the iterval i questio. I part (d) studets kew that, i this cotext, the sig of the derivative of w ca determie whether w is icreasig or decreasig at a particular value of. However, may studets failed to demostrate the calculator skill to solve w w for a value of at which they could test w. A 17 The College Board.

21 What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) some studets expressed the area of 1 / 1 R as f d or as 1. f d 1 The area of R is give by f d. I part (b) some studets represeted double the area of S as g f d. Double the area of S is give by g f d. I part (b) some studets did ot coect the ray k as idicatig k as a limit of itegratio, istead writig a equatio such as 1 k g f d. If the ray k divides S ito two regios of equal areas, the the area bouded by f ad g from to k must match the area bouded by f ad g from k to, i.e., 1 k 1 g f d g f k I part (c) some studets cofused the average value of w across with the average w() w of the edpoit values,. I part (d) some studets who could fid a solutio to w wa gave isufficiet reaso for their coclusios, such as w is decreasig because the graphs are gettig closer together. The average value of w across is 1 w d g f d. w wa at.518. w is decreasig at.518 because w.518. Based o your experiece at the AP Readig with studet resposes, what advice would you offer to teachers to help them improve the studet performace o the exam? Part of the richess of free-respose questios is that they ofte brig together topics from various parts of the course. I a typical course, the otio of average value is rarely ecoutered i problems ivolvig polar fuctios, or is the issue of icreasig versus decreasig. Oe should ot strive to expose studets to every possible mixture of topics, but it is importat to have a variety of surprise ecouters so that studets are ot surprised o the AP Exam. There is o magic resource for this (besides possibly previous free-respose questios), but the creativity to write problems that combie topics ca be as ivigoratig for the teacher as it is urturig for their studets. This problem added to the evidece that may studets have difficulty with the radial perspective ivolved i thikig polar. This is more a precalculus issue tha a calculus oe, but polar fuctios i calculus provide may opportuities to recall ad reiforce this perspective. While may aspects of calculus with polar fuctios ivolve the coectio to a rectagular framework (e.g., the rectagular otio of slope ), oe of the parts of this problem required a coversio to rectagular coordiates. Ideed, such a coversio was a impedimet to studet progress o these problems. Thus, time 17 The College Board.

22 spet acclimatig studets to the polar eviromet, perhaps aided by relatig it to situatios such as weather radar or a ship s soar, ca make problems such as these less itimidatig. What resources would you recommed to teachers to better prepare their studets for the cotet ad skill(s) required o this questio? I geeral, review of previous Chief Reader reports (kow previously as the Studet Performace Q&A), as well as applicable sectios of the AP Calculus AB ad AP Calculus BC Course ad Exam Descriptio (particularly the Mathematical Practices for AP Calculus) will serve teachers well. Also, previously released exam questios ad participatio o the Olie Teacher Commuity are of high value for practice ad advice. 17 The College Board.

23 Questio BC5 Topic: Aalysis of Fuctios/Improper Itegral/Series Max. Poits: 9 Mea Score: 3.54 What were resposes expected to demostrate i their respose to this questio? 3 I this problem the fuctio f is defied by fx. I part (a) studets were asked to fid the slope of the x 7x 5 lie taget to the graph of f at x 3. Studets eeded to differetiate f ad fid the slope of the lie taget to the graph of f at x 3 by evaluatig f 3. [LO.3B/EK.3B1] I part (b) studets were asked to fid the x-coordiate of each critical poit of f i the iterval 1 x.5 ad to classify each critical poit as the locatio of a relative miimum, a relative maximum, or either, justifyig these classificatios. Studets should have observed that f is differetiable o 1 x.5 ad foud that f 7 x has just oe solutio, x, i this iterval. The studets eeded to determie 4 7 that f has a relative maximum at x by otig that f chages sig from positive to egative at the critical poit 4 7 x. [LO.A/EK.A1] I part (c) studets were give the partial fractio decompositio for f x ad asked to 4 evaluate f x dx or to show that the itegral diverges. Studets should have expressed the give improper itegral as 5 a limit of proper itegrals, lim b f x dx, ad used the partial fractio decompositio for f x to fid that b 5 b b 5 5 fx dx l b 5 l b 1 l 5 l 4 l 5 l. Applyig limit theorems, studets eeded to b l. [LO 1.1C/EK 1.1C1-5 take the limit of this expressio as b to fid that the improper itegral coverges to 1.1C, LO 3.D/EK 3.D1-3.D] I part (d) studets were asked to determie whether the series f coverges or 5 diverges, statig the coditios of the test used for this determiatio. Studets eeded to combie the results of part (c) 1 with the itegral test or use a limit compariso test to the coverget p-series to fid that the series f 5 5 coverges. For either test, studets should have observed that the ecessary coditios hold, amely that f is cotiuous, positive, ad decreasig o 5,. [LO 4.1A/EK 4.1A6] This problem icorporates the followig Mathematical Practices for AP Calculus (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, buildig otatioal fluecy, ad commuicatig. How well did the respose address the course cotet related to this questio? How well did the resposes itegrate the skills required o this questio? I part (a) studets were geerally successful i kowig to differetiate ad computig f (3) to fid the slope of the idicated taget lie. 7 I part (b) most studets could fid the critical poit x but were less successful i justifyig a classificatio of the 4 critical poit as a relative miimum, relative maximum, or either. I part (c) may studets were successful i fidig the atiderivative, but did ot employ limit otatio appropriately to describe the process of evaluatig a improper itegral. I part (d) may studets were able to idetify a appropriate test (almost always the itegral test or limit compariso test) to use i determiig whether the give series coverges or diverges. However, the applicatio of the test was ofte iformal, lackig i appropriate limit otatio, usig laguage that cofused a series with the sequece of its terms, or givig a icomplete descriptio of the coditios ecessary to apply the covergece test. 17 The College Board.

24 What commo studet miscoceptios or gaps i kowledge were see i the resposes to this questio? Commo Miscoceptios/Kowledge Gaps Resposes that Demostrate Uderstadig I part (a) some studets had likage errors: usig = to coect uequal expressios, such as 3 f x (4x 7) (x 7x 5) 3 f (3) 5. (18 1 5) 3 f x (4x 7) (x 7x 5) Thus, the slope of the taget lie is 3 f (3) 5. (18 1 5) I part (b) some studets did ot costrai their aalysis to the iterval 1 x.5 ad failed to recogize x 1 ad x.5 as asymptotes for the graph of f, resultig i a faulty aalysis. For example, a studet with a correct derivative 7 could fid the oly critical poit at x ad 4 7 the explicitly cosider itervals, 4 ad 7,. Usig test poits 4 f 1 ad f (), the coclude that f has a relative f x is defied o 1 x.5 ad f x oly at x Because f chages from positive to egative at f has a relative maximum at x 7. 4 x 7, maximum at x 7. 4 I part (c) studets missed the atiderivative of, did t employ appropriate limit x 5 otatio, ad made errors i evaluatio, such as 1 5 dx x 5 x 1 l x 5 l x 1 5 (umber). Thus, the itegral diverges. [ ] b 1 5 x 5 x 1 b b lim l l l l dx lim l x 5 l x 1 b b Thus, the itegral coverges. 5 I part (d) studets gave argumets that had a superficial resemblece to correctess but were lackig i good commuicatio of the ecessary mathematics. For example, ad 1 coverges, so the series coverges by the compariso test. The studet could use part (c) ad apply the itegral test: f is cotiuous, positive, ad decreasig o [5, ). Because f x dx coverges, f must also coverge 5 1 by the itegral test. Or use the limit compariso test: f ad for The College Board.