AP Calculus BC 2003 Scoring Guidelines Form B


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1 AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet Program. Teachers may reproduce them, i whole or i part, i limited quatities for ocommercial, facetoface teachig purposes. This permissio does ot apply to ay thirdparty copyrights cotaied herei. This material may ot be mass distributed, electroically or otherwise. These materials ad ay copies made of them may ot be resold, ad the copyright otices must be retaied as they appear here. These materials were produced by Educatioal Testig Service (ETS ), which develops ad admiisters the examiatios of the Advaced Placemet Program for the College Board. The College Board ad Educatioal Testig Service (ETS) are dedicated to the priciple of equal opportuity, ad their programs, services, ad employmet policies are guided by that priciple. The College Board is a atioal oprofit membership associatio whose missio is to prepare, ispire, ad coect studets to college ad opportuity. Fouded i 9, the associatio is composed of more tha, schools, colleges, uiversities, ad other educatioal orgaizatios. Each year, the College Board serves over three millio studets ad their parets,, high schools, ad,5 colleges through major programs ad services i college admissios, guidace, assessmet, fiacial aid, erollmet, ad teachig ad learig. Amog its bestkow programs are the SAT, the PSAT/NMSQT, ad the Advaced Placemet Program (AP ). The College Board is committed to the priciples of equity ad excellece, ad that commitmet is embodied i all of its programs, services, activities, ad cocers. For further iformatio, visit Copyright College Etrace Examiatio Board. All rights reserved. College Board, Advaced Placemet Program, AP, AP Vertical Teams, APCD, Pacesetter, PreAP, SAT, Studet Search Service, ad the acor logo are registered trademarks of the College Etrace Examiatio Board. AP Cetral is a trademark owed by the College Etrace Examiatio Board. PSAT/NMSQT is a registered trademark joitly owed by the College Etrace Examiatio Board ad the Natioal Merit Scholarship Corporatio. Educatioal Testig Service ad ETS are registered trademarks of Educatioal Testig Service. Other products ad services may be trademarks of their respective owers. For the College Board s olie home for AP professioals, visit AP Cetral at apcetral.collegeboard.com.
2 SCORING GUIDELINES (Form B) Questio Let f be the fuctio give by fx ( ) = x x, ad let be the lie y = 8 x, where is taget to the graph of f. Let R be the regio bouded by the graph of f ad the xaxis, ad let S be the regio bouded by the graph of f, the lie, ad the xaxis, as show above. (a) Show that is taget to the graph of y = f() x at the poit x =. Fid the area of S. (c) Fid the volume of the solid geerated whe R is revolved about the xaxis. (a) f ( x) = 8x x ; f () = 7 = f () = 6 7 = 9 Taget lie at x = is y = ( x ) + 9 = x + 8, which is the equatio of lie. : fids f() ad f() fids equatio of taget lie or : shows (,9) is o both the graph of f ad lie fx ( ) = at x = The lie itersects the xaxis at x = 6. Area = ()(9) ( x x ) dx = 7.96 or 7.97 OR : itegral for otriagular regio : limits : itegrad : area of triagular regio : aswer Area = (( 8 ) ( )) x x x dx + ()(8 ) = 7.96 or 7.97 (c) Volume = ( ) x x dx = 56.8 or 9.8 : limits ad costat : : itegrad : aswer Copyright by College Etrace Examiatio Board. All rights reserved.
3 The figure above shows the graphs of the circles AP CALCULUS BC SCORING GUIDELINES (Form B) Questio x + y = ad ( x ) + y =. The graphs itersect at the poits (,) ad (, ). Let R be the shaded regio i the first quadrat bouded by the two circles ad the xaxis. (a) Set up a expressio ivolvig oe or more itegrals with respect to x that represets the area of R. Set up a expressio ivolvig oe or more itegrals with respect to y that represets the area of R. (c) The polar equatios of the circles are r = ad r = cos, respectively. Set up a expressio ivolvig oe or more itegrals with respect to the polar agle that represets the area of R. (a) Area = Area = ( ) ( x ) dx + x dx OR + x dx : itegrad for larger circle : itegrad or geometric area : for smaller circle : limits o itegral(s) Note: < > if o additio of terms Area = ( ( )) y y dy : : limits itegrad < > reversal < > algebra error i solvig for x < > add rather tha subtract < > other errors (c) Area = ( ) d + (cos ) d OR Area = ( ) (cos ) d 8 + : itegrad or geometric area for larger circle : : itegrad for smaller circle : limits o itegral(s) Note: < > if o additio of terms Copyright by College Etrace Examiatio Board. All rights reserved.
4 SCORING GUIDELINES (Form B) Questio A blood vessel is 6 millimeters (mm) log Distace with circular cross sectios of varyig diameter. x (mm) Diameter The table above gives the measuremets of the B(x) (mm) diameter of the blood vessel at selected poits alog the legth of the blood vessel, where x represets the distace from oe ed of the blood vessel ad Bx () is a twicedifferetiable fuctio that represets the diameter at that poit. (a) Write a itegral expressio i terms of Bx () that represets the average radius, i mm, of the blood vessel betwee x = ad x = 6. Approximate the value of your aswer from part (a) usig the data from the table ad a midpoit Riema sum with three subitervals of equal legth. Show the computatios that lead to your aswer. 75 Bx () (c) Usig correct uits, explai the meaig of dx 5 i terms of the blood vessel. (d) Explai why there must be at least oe value x, for < x < 6, such that B ( x) =. (a) 6 Bx () dx 6 : limits ad costat : itegrad B(6) B(8) B() + + = 6 [ 6( + + )] = 6 : B(6) + B(8) + B() : aswer (c) Bx ( ) Bx ( ) is the radius, so is the area of the cross sectio at x. The expressio is the volume i mm of the blood vessel betwee 5 : volume i mm : betwee x = 5 ad x = 75 mm ad 75 mm from the ed of the vessel. (d) By the MVT, B ( c) = for some c i (6, 8) ad B ( c) = for some c i (, 6). The MVT applied to B ( x) shows that B () x = for some x i the iterval ( c c ),. : explais why there are two values of x where B( x) has the same value : explais why that meas B ( x) = for < x < 6 Copyright by College Etrace Examiatio Board. All rights reserved. Note: max / if oly explais why B ( x) = at some x i (, 6).
5 SCORING GUIDELINES (Form B) Questio A particle moves i the xyplae so that the positio of the particle at ay time t is give by ( ) 7 x t e t e t = + ad ( ) t t y t = e e. (a) Fid the velocity vector for the particle i terms of t, ad fid the speed of the particle at time t =. Fid dy dy i terms of t, ad fid lim. dx t dx (c) Fid each value t at which the lie taget to the path of the particle is horizotal, or explai why oe exists. (d) Fid each value t at which the lie taget to the path of the particle is vertical, or explai why oe exists. 7 (a) x () t = 6e t 7e y () t = 9e + e t t t t 7t t t Velocity vector is < 6e 7 e, 9e + e > : : x( t) : y( t) : speed Speed = x() + y() = ( ) + = dy dy dt 9e + e = = dx dx 6e 7e dt t t t 7t dy : i terms of t dx : limit t t dy 9e + e 9 lim = lim = = dx t t 7t 6e 7e 6 t (c) Need y t t () t =, but 9e + e > for all t, so oe exists. : cosiders y( t) = : explais why oe exists (d) Need x () t = ad y() t. t 7t e = e t e = 6 7 t = l 6 ( ) : cosiders x( t) = : solutio Copyright by College Etrace Examiatio Board. All rights reserved. 5
6 SCORING GUIDELINES (Form B) Questio 5 Let f be a fuctio defied o the closed iterval [,7]. The graph of f, cosistig of four lie segmets, is show above. Let g be the x fuctio give by gx ( ) = ftdt ( ). (a) Fid g (, ) g ( ), ad g ( ). Fid the average rate of chage of g o the iterval x. (c) For how may values c, where < c <, is g () c equal to the average rate foud i part? Explai your reasoig. (d) Fid the xcoordiate of each poit of iflectio of the graph of g o the iterval < x < 7. Justify your aswer. (a) g() = f( t) dt = ( + ) = g () = f() = g() = f() = = : : g() : g() : g() g() g() = () ftdt 7 ()() + ( + ) = = ( ) : g() g() = f( t) dt : aswer (c) There are two values of c. We eed 7 = g( c) = f( c) The graph of f itersects the lie places betwee ad. 7 y = at two : aswer of : reaso Note: / if aswer is by MVT (d) x = ad x = 5 because g = f chages from icreasig to decreasig at x =, ad from decreasig to icreasig at x = 5. : x = ad x = 5 oly : justificatio (igore discussio at x = ) Copyright by College Etrace Examiatio Board. All rights reserved. 6
7 SCORING GUIDELINES (Form B) Questio 6 The fuctio f has a Taylor series about x = that coverges to fx ( ) for all x i the iterval of ( +! ) ( ) covergece. The th derivative of f at x = is give by f ( ) = for, ad f ( ) =. (a) Write the first four terms ad the geeral term of the Taylor series for f about x =. Fid the radius of covergece for the Taylor series for f about x =. Show the work that leads to your aswer. (c) Let g be a fuctio satisfyig g ( ) = ad g ( x) = f( x) for all x. Write the first four terms ad the geeral term of the Taylor series for g about x =. (d) Does the Taylor series for g as defied i part (c) coverge at x =? Give a reaso for your aswer.!!! (a) f () = ; f () = ; f () = ; f () =!! fx ( ) = + ( x ) + ( x ) + ( x ) +!! ( + )! ( ) + + x +! = + ( x ) + ( x ) + ( x ) + + ( ) + + x + : ( f ) () : coefficiets i! first four terms : powers of ( x ) i first four terms : geeral term + ( ) + + x + lim = lim x + ( ) x + = x < whe x < The radius of covergece is. : : sets up ratio : limit : applies ratio test to coclude radius of covergece is (c) g () = ; g () = f() ; g() = f() ; g() = f() gx ( ) = + ( x ) + ( x ) + ( x ) + : first four terms : geeral term ( ) x (d) No, the Taylor series does ot coverge at x = because the geometric series oly coverges o the iterval x <. : aswer with reaso Copyright by College Etrace Examiatio Board. All rights reserved. 7
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