AP Calculus BC. Sample Student Responses and Scoring Commentary. Inside: Free Response Question 6. Scoring Guideline.

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1 208 AP Calculus BC Sample Studet Resposes ad Scorig Commetary Iside: Free Respose Questio 6 RR Scorig Guidelie RR Studet Samples RR Scorig Commetary College Board, Advaced Placemet Program, AP, AP Cetral, ad the acor logo are registered trademarks of the College Board. AP Cetral is the official olie home for the AP Program: apcetral.collegeboard.org

2 AP CALCULUS BC 208 SCORING GUIDELINES Questio 6 (a) The first four ozero terms are + + The geeral term is ( ) : first four terms 2 : { : geeral term (b) ( ) ( )( ) lim + ( ) lim = = + + ( + ) < for < Therefore, the radius of covergece of the Maclauri series for f is. OR The radius of covergece of the Maclauri series for l ( + ) is, so the series for f( ) = l ( + ) coverges absolutely for <. < < Therefore, the radius of covergece of the Maclauri series for f is. + + ( ) Whe =, the series is ( ) =, = = diverges by compariso to the harmoic series. which Whe =, the series is ( ) = ( ), = = coverges by the alteratig series test. which The iterval of covergece of the Maclauri series for f is <. 5 : OR 5 : : sets up ratio : computes limit of ratio : radius of covergece : cosiders both edpoits : aalysis ad iterval of covergece : radius for l ( + ) series : substitutes : radius of covergece : cosiders both edpoits : aalysis ad iterval of covergece (c) By the alteratig series error boud, a upper boud for P 4( 2 ) f ( 2 ) is the magitude of the et term of the alteratig series. 2 8 P4 ( 2) f( 2) < = : uses fifth-degree term 2 : as error boud : aswer

3 The Maclauri series for l(l + ) is give by NO CALCULATOR ALLOWED 6 ( X (-lf O its iterval of covergece, this series coverges to lo ( + ). Let f be the fuctio defied, by 'f() = ( + } (a) Write the first four ozero terms ad the geeral term of the Maclauri series for f +-ti\ r--,/\-.;. r, \.-j ""' C\ ', "7 J.! L -' \:.{ "2... ) '\ "..,... 'f,., 'S" r- '\. "l "l, 7 ) 8' \.{ t t i-. ' +(- l\t-\ "It-\,., " -.;- -- :/\,, l "+\ ":\ + - \ I\ I\ Uauthorized copyig or,usa of ay pert of this page ls Illegal. k - - \ + l S} _-,<(t\ :> "2- ) ½ -..,_, '"As Cotiue questio 6 o page 2.

4 NO CALCULATOR-ALLOWED.', (.,() (\,.,..e..e.-') C6 4 ftcr,t., H.. " '...J I "'. \ --\ L-"; L-- -'?,Lil-) (c_) Let P4() be the fourth-degree Taylor polyomial for f a ou? = O Use the,alter_oatig_sr.ries...e:a:or boudtofidaupperboudfor!p4(2)-j(2), _ 7\ S: \ \?' l <t,\ - ff\ \ Li - o) 't-\ \ &) \.\l \,;... -::. f)j\.\ 8\ :,_ Uauthorized copyig or reuse of ay part of this page Is Illegal. -2- GO ON TO THE NEXT PAGE.

5 6,. 6 NO,CALCULAT-OR ALLOWED 6 6. The Maclauri series for I ( + ) is give by 2 4 " X (- tyi+i +..., O its iter,,! of cov«gece, this se,ios cove.ges e t f be the fuctio defioed by f () == l. ( + ). (a - te the first four o.:ero t -th ge ral term ofth Maclauri s ri-;.. f -,. l'\--t- l"'\-tl. (-) /). X X ) ( -+>\) :l(\. o... '.2. 0,, ) _ ({f L ' (f)- ilf) " -- : -: :-,- -.i:,i 'f\ \-, 'ZJf_J ( :_ ) (;) Uaulhorlzed copyl,i.or reuse of 8'\Y part of this page le Illegal. 20- Cotiue questio 6 o page 2..

6 . r 6"..., '. 6 6 ' () NO CALCULATOR ALLOWED ' '.a.,:.l,, g.. (b) Determie the iterval of covei:gece of the Maclauri series.for f. Show the work that leads to yol{... aswer. - )( = - Y- - -.)Y'\ -+ r / ) Y-') \. (-, \_ \.. V\* \ ,.,. / l ) : -\ \ ( - l )_; :.:+ l:>y'i(9;, q\\lv --t I ( c) Let P 4 () be the fourth-degree Taylor poty mial.:for f abo t == 0. Use the alteratig series erro.r boud tofi a upper 09utid f r E.2). - /(2) I-. 5 \ :r-, { J) - f (c?-).l: _L \-=-C )_... _... t., \j -f rkt -0oud, ls L l\.; 2 ;'Y 2 s q, 4 Uautt\orlzed copyig or reuse-of 8\Y part of this pag ts Illegal. GO ON TO THE NEXT PAG'E,

7 NO CALCULATOR ALLOWED 6 6. The Maclauri series for I ( J + ) is give by 2 4 X (-lf+l_ O its iterval of covergece, this series coverges to ( + ). Let/ be the fuctio defied by (a) Write the first four ozero terms ad the geeral term of the Maclauri series for f _f (Jc) Uauthorized copyig or reuse of ay part of this page Is Illegal Cotiue questio 6 o page 2.

8 b 0 NO :CALCULATOR ALLOWED (b) Determie the iterval f covergece of the Maclauri series for f Show the work that leads to your aswer. - [,Y'-i l (-r V\-r.. / vi? Ii ( r4o. \ <l-" l,._ ('MG¼ 'It. (c) Let {) be the fourth-degree Taylor poly_omial for f about = Q. Use the alteratig series error boud to fid a upper boud for IP 4, (2 - /(2) -... l 4...' \ 5"' _ : } i.,. \-''f ) L \-\b.. -, Uauthorized copyig or reuse ot ay-pert otthls,page ls Illegal. -2- GO ON TO THE'NEXT PAGE.

9 AP CALCULUS BC 208 SCORING COMMENTARY Questio 6 Overview I this problem the first four ozero terms ad the geeral term of the Maclauri series for l are give, ad the fuctio f is defied by f l. I part (a) studets were asked for the first four ozero terms ad the geeral term of the Maclauri series for f. A correct respose should substitute for i the supplied terms of the series for l, multiply the resultig terms by, ad epad so that each term is a costat multiple of a power of. The geeral term should also be icluded. I part (b) studets were asked to determie the iterval of covergece of the Maclauri series for f with supportig work for their aswer. A correct respose should demostrate the use of the ratio test to determie the radius of covergece of the series ad, the, a test of the edpoits of the iterval of covergece to determie which edpoits, if ay, are to be icluded i the iterval of covergece. I part (c) studets were asked to use the alteratig series error boud to fid a upper boud for P4 2 f 2, where P4 is the fourth-degree Taylor polyomial for f about 0. A correct respose should idicate that the alteratig series error boud bouds P4 2 f 2 by the magitude of the et term i the alteratig series formed by evaluatig the Taylor series for f about 0 at 2. For part (a) see LO 4.2B/EK 4.2B5. For part (b) see LO 4.A/EK 4.A, LO 4.A/EK 4.A6, LO 4.2C/EK 4.2C2. For part (c) see LO 4.2A/EK 4.2A5. 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. Sample: 6A Score: 9 The respose eared all 9 poits: 2 poits i part (a), 5 poits i part (b), ad 2 poits i part (c). I part (a) the respose would have eared the first poit i lie 2 for the first four ozero terms of the Maclauri series for f with o simplificatio. I this case, the simplificatio is correct, ad the poit was eared i lie. The respose would have eared the secod poit i lie 2 with the geeral term of the Maclauri series for f with o simplificatio. I this case, the simplificatio is correct, ad the poit was eared i lie. I part (b) the respose eared the first poit with the ratio i lie o the left. The respose eared the secod poit i lie 4 o the left with. The respose eared the third poit i lie 7 o the left with. The respose eared the fourth poit with the work o the right by substitutig the edpoits ad ito the geeral term of the Maclauri series for f. The respose eared the fifth poit for the work o the right with the aalysis for statig alteratig harmoic coverges coditioally ad for statig harmoic diverges ad the iterval of covergece i the last lie o the right. I part (c) the respose eared the first poit i lie o the right with use of the fifth-degree term. The respose eared the secod poit by substitutig 2 for i the fifth-degree term ad takig a absolute value to guaratee that a positive error boud is retured. Sample: 6B Score: 6 The respose eared 6 poits: o poits i part (a), 4 poits i part (b), ad 2 poits i part (c). I part (a) the respose did ot ear the first poit as the deomiators are icorrect i all of the first four ozero terms of the Maclauri series for f. The respose did ot ear the secod poit because of a icorrect geeral term of the Maclauri series for f. I part (b) the respose eared the first poit for a ratio i lie 2 that is cosistet with the

10 Questio 6 (cotiued) geeral term preseted i part (a). The respose eared the secod poit i lie 2 with. The respose eared the third poit with i lie ad a radius of covergece of the Maclauri series for f that is cosistet with the geeral term preseted i part (a). The respose eared the fourth poit by substitutig the edpoits ad ito the imported geeral term from part (a). The respose did ot ear the fifth poit as the respose states coverges ad diverges without justificatio. I part (c) the respose eared both poits with a epressio for the absolute value of the fifth-degree term evaluated at 2 that is cosistet with the fifthdegree term of the series i part (a). Sample: 6C Score: AP CALCULUS BC 208 SCORING COMMENTARY The respose eared poits: o poits i part (a), poits i part (b), ad o poits i part (c). I part (a) the respose did ot ear the first poit as the respose has icorrect deomiators i the secod, third, ad fourth terms of the Maclauri series for f. The respose did ot ear the secod poit because of a icorrect geeral term of the Maclauri series for f. I part (b) the respose eared the first poit for a ratio give i the top left corer that is cosistet with the geeral term preseted i part (a). The icorrect geeral term imported from part (a) results i a oversimplificatio of the questio, so the respose is ot eligible for the secod ad third poits. The respose eared the fourth poit by substitutig ad ito the geeral term imported from part (a). The respose eared the fifth poit with the aalysis for with harmoic diverges ad for with coverges alt. harmoic ad the iterval of covergece boed o the left that is cosistet with the geeral term preseted i part (a). I part (c) the respose did ot ear the first poit because the sithdegree term is used to fid the requested upper boud istead of the fifth-degree term. The respose did ot ear the secod poit because the error boud is ot cosistet with the preseted fifth-degree term i part (a).