207 AP Calculus BC Sample Studet Resposes ad Scorig Commetary Iside: RR Free Respose Questio 6 RR Scorig Guidelie RR Studet Samples RR Scorig Commetary 207 The College Board. College Board, Advaced Placemet Program, AP, AP Cetral, ad the acor logo are registered trademarks of the College Board. Visit the College Board o the Web: www.collegeboard.org. AP Cetral is the official olie home for the AP Program: apcetral.collegeboard.org
AP CALCULUS BC 207 SCORING GUIDELINES Questio 6 (a) f ( 0) = 0 f ( 0) = f ( 0) = ( ) = f ( 0) = 2( ) = 2 ( 4 f ) ( 0) = 3( 2) = 6 ( 4 : f ( 0 ), f ( 0 ), ad f ) ( 0) 3 : : verify terms : geeral term The first four ozero terms are 2 3 4 2 2 3 6 4 x x x 0 + x + x + x + x = x +. 2! 3! 4! 2 3 4 The geeral term is + ( ) x. (b) For x =, the Maclauri series becomes = ( ) +. 2 : coverges coditioally with reaso The series does ot coverge absolutely because the harmoic series diverges. The series alterates with terms that decrease i magitude to 0, ad therefore the series coverges coditioally. x + x 2 3 4 ( ) (c) t t t t f ( t) dt = t dt 0 + + + + 2 3 4 0 + + 2 3 4 5 t t t t ( ) t = + + + + 2 3 2 4 3 5 4 ( + ) + + 2 3 4 5 x x x x ( ) x = + + + + 2 6 2 20 ( + ) t= x t = 0 : two terms 3 : : remaiig terms : geeral term (d) The terms alterate i sig ad decrease i magitude to 0. By the alteratig series error boud, the error P4 ( ) g 2 ( 2) is bouded ( 2) 5 by the magitude of the first uused term,. 20 ( 2) Thus, ( ) ( ) 5 P g 4 = <. 2 2 20 32 20 500 : error boud 207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
AP CALCULUS BC 207 SCORING COMMENTARY Questio 6 Overview I this problem studets were preseted with a fuctio f that has derivatives of all orders for < x < such ( + ) ( ) that f ( 0) = 0, f ( 0) =, ad f ( 0) = f ( 0) for all. It is also stated that the Maclauri series for f coverges to f( x ) for x <. I part (a) studets were asked to verify that the first four ozero terms of 2 3 4 x x x the Maclauri series for f are x + ad to write the geeral term of this Maclauri series. The thdegree term of the Taylor polyomial for f about x = 0 is x. f ( 0) = 0 ad f ( 0) = are give, ad 2 3 4 ( ) f ( 0)! ( + ) the give recurrece relatio for f ( 0) ca be readily applied to see that f ( 0) =, f ( 0) = 2, ( 4 ) ( ) + f ( 0) = 6, ad f ( 0) = ( ) (! ). Usig these derivative values, studets eeded to cofirm that the ) + x ( first four ozero terms of the Maclauri series for f are as give, ad that the geeral term is. [LO 4.2A/EK 4.2A] I part (b) studets were asked to determie, with explaatio, whether the Maclauri series for f coverges absolutely, coverges coditioally, or diverges at x =. Substitutig x =, studets should ( + ) have obtaied that the Maclauri series for f evaluated at x = is. Studets eeded to coclude that this series coverges coditioally, otig that the series coverges by the alteratig series test ad that + ( ) is the diverget harmoic series. [LO 4.A/EK 4.A4-4.A6] I part (c) studets were asked to fid = the first four ozero terms ad the geeral term of the Maclauri series for g( x) = f ( t) dt. Studets eeded to fid these terms by itegratig the Maclauri series for f term-by-term. [LO 4.2B/EK 4.2B5] I part (d) usig the fuctio g defied i part (c), the expressio P ( 2 ) represets the th-degree Taylor polyomial for g about x = 0 evaluated at 4( ) g( ) P x =. Studets were directed to use the alteratig series error boud to show that 2 <. Studets may have observed that the terms of the Taylor polyomial for g about x = 0, 2 2 500 evaluated at x =, alterate i sig ad decrease i magitude to 0. Thus, the alteratig series error boud ca 2 ( 2) be applied to see that ( ) ( ) 5 P g 4 = =, showig that the error i the approximatio is less 2 2 20 32 20 tha. [LO 4.B/EK 4.B2] This problem icorporates the followig Mathematical Practices for AP Calculus 500 (MPACs): reasoig with defiitios ad theorems, coectig cocepts, implemetig algebraic/computatioal processes, buildig otatioal fluecy, ad commuicatig. 0 x 207 The College Board. Visit the College Board o the Web: www.collegeboard.org.
AP CALCULUS BC 207 SCORING COMMENTARY Questio 6 (cotiued) Sample: 6A Score: 9 The respose eared all 9 poits: 3 poits i part (a), 2 poits i part (b), 3 poits i part (c), ad poit i part (d). I part (a) the studet computes the umerical values of the derivatives i lie 3 ad eared the first poit. The studet uses the derivatives i lie 4 to verify the give expressio ad eared the secod poit. The studet produces a correct geeral term i lie 5 usig with a expoet of rather tha +, which is still correct. The studet eared the third poit. I part (b) the studet correctly uses the alteratig series test to draw a coclusio of coverges, idetifies the harmoic series as diverget, ad draws the correct coclusio of coverges coditioally. The studet eared both poits. I part (c) the studet produces the first four terms ad eared the first 2 poits. The studet presets a correct geeral term usig idices of rather tha + ad eared the third poit. I part (d) the studet correctly computes the alteratig series error ad idetifies it as the boud o the error. Sample: 6B Score: 6 The respose eared 6 poits: poit i part (a), 2 poits i part (b), 3 poits i part (c), ad o poit i part (d). I part (a) the studet does ot preset the umerical values of the derivatives or iclude a proper verificatio. The studet did ot ear the first 2 poits. The studet produces a correct geeral term ad eared the third poit. The studet is ot pealized for usig both lower ad upper cases for i this questio. I part (b) the studet correctly idetifies the series as the alteratig harmoic ad the absolute value series as the harmoic. The studet draws the correct coclusio of coverges coditioally. The studet eared both poits. I part (c) the studet produces the first four terms ad the geeral term. All 3 poits were eared. I part (d) the studet does ot produce the umerical error value to verify that the error is less tha. The studet did ot ear the poit. 500 Sample: 6C Score: 3 The respose eared 3 poits: 2 poits i part (a), poit i part (b), o poits i part (c), ad o poit i part (d). I part (a) the studet computes the umerical values of the derivatives embedded i lie 2 ad eared the first poit. The studet s use of derivatives i the verificatio lie is correct ad eared the secod poit. The studet does ot preset a geeral term, so the third poit was ot eared. I part (b) the studet cocludes that the alteratig series coverges coditioally, but the studet does ot explicitly address the series of absolute values of the terms. The studet eared of the 2 poits. I part (c) the studet produces four terms, but oly the first term is correct. The studet did ot ear either of the first 2 poits. The studet does ot produce a sufficiet geeral term because lie is formulaic. The studet did ot ear the third poit. I part (d) the studet correctly computes the error boud based o the work i part (c) but does ot coect P 4( 2 ) = to the error. The 32 5 studet did ot ear the poit. 207 The College Board. Visit the College Board o the Web: www.collegeboard.org.