AP Calculus BC 2011 Scoring Guidelines Form B

Size: px

Start display at page:

Download "AP Calculus BC 2011 Scoring Guidelines Form B"

Caren Ellis
6 years ago
Views:

1 AP Calculus BC Scorig Guidelies Form B The College Board The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad opportuity. Fouded i 9, the College Board is composed of more tha 5,7 schools, colleges, uiversities ad other educatioal orgaizatios. Each year, the College Board serves seve millio studets ad their parets,, high schools, ad,8 colleges through major programs ad services i college readiess, college admissio, guidace, assessmet, fiacial aid ad erollmet. Amog its widely recogized programs are the SAT, the PSAT/NMSQT, the Advaced Placemet Program (AP ), SprigBoard ad ACCUPLACER. The College Board is committed to the priciples of ecellece ad equity, ad that commitmet is embodied i all of its programs, services, activities ad cocers. The College Board. College Board, ACCUPLACER, Advaced Placemet Program, AP, AP Cetral, SAT, SprigBoard ad the acor logo are registered trademarks of the College Board. Admitted Class Evaluatio Service is a trademark owed by the College Board. PSAT/NMSQT is a registered trademark of the College Board ad Natioal Merit Scholarship Corporatio. All other products ad services may be trademarks of their respective owers. Permissio to use copyrighted College Board materials may be requested olie at: AP Cetral is the official olie home for the AP Program: apcetral.collegeboard.com.

2 SCORING GUIDELINES (Form B) Questio A cylidrical ca of radius millimeters is used to measure raifall i Stormville. The ca is iitially empty, ad rai eters the ca durig a 6-day period. The height of water i the ca is modeled by the fuctio S, where St () is measured i millimeters ad t is measured i days for t 6. The rate at which the height of the water is risig i the ca is give by S ( t) = si(.t) +.5. (a) Accordig to the model, what is the height of the water i the ca at the ed of the 6-day period? (b) Accordig to the model, what is the average rate of chage i the height of water i the ca over the 6-day period? Show the computatios that lead to your aswer. Idicate uits of measure. (c) Assumig o evaporatio occurs, at what rate is the volume of water i the ca chagig at time t = 7? Idicate uits of measure. (d) Durig the same 6-day period, rai o Mosoo Moutai accumulates i a ca idetical to the oe i Stormville. The height of the water i the ca o Mosoo Moutai is modeled by the fuctio M, where M () t = ( t t + t). The height M ( t ) is measured i millimeters, ad t is measured i days 4 for t 6. Let Dt ( ) = M ( t) S ( t). Apply the Itermediate Value Theorem to the fuctio D o the iterval t 6 to justify that there eists a time t, < t < 6, at which the heights of water i the two cas are chagig at the same rate. 6 (a) S( 6) = S ( t) dt = 7.8 mm : limits : itegrad (b) S( 6) S( ) 6 =.86 or.864 mm day (c) V() t = π S() t V ( 7) = π S ( 7) = 6.8 The volume of water i the ca is icreasig at a rate of 6.8 mm day. : relatioship betwee V ad S : { (d) D ( ) =.675 < ad D ( 6) = > Because D is cotiuous, the Itermediate Value Theorem implies that there is a time t, < t < 6, at which Dt ( ) =. At this time, the heights of water i the two cas are chagig at the same rate. : cosiders D( ) ad D( 6) : : justificatio : uits i (b) or (c) The College Board.

3 SCORING GUIDELINES (Form B) Questio The polar curve r is give by r( θ ) = θ + si θ, where θ π. (a) Fid the area i the secod quadrat eclosed by the coordiate aes ad the graph of r. π (b) For θ π, there is oe poit P o the polar curve r with -coordiate. Fid the agle θ that correspods to poit P. Fid the y-coordiate of poit P. Show the work that leads to your aswers. (c) A particle is travelig alog the polar curve r so that its positio at time t is ( ( t), y( t )) ad such that dθ dy π =. Fid at the istat that θ =, ad iterpret the meaig of your aswer i the cotet of dt dt the problem. π (a) Area = ( r( )) d 47.5 θ θ = π : itegrad : limits ad costat (b) = r( θ ) cosθ = ( θ + siθ) cosθ θ =.69 y = r( θ ) si ( θ ) = 6.7 : equatio : value of θ : y-coordiate (c) y = r( θ ) si θ = ( θ + si θ) si θ dy dy dθ = =.89 dt d θ dt θ= π θ= π : uses chai rule : iterpretatio The y-coordiate of the particle is decreasig at a rate of.89. The College Board.

4 SCORING GUIDELINES (Form B) Questio The fuctios f ad g are give by f ( ) = ad g( ) = 6. Let R be the regio bouded by the -ais ad the graphs of f ad g, as show i the figure above. (a) Fid the area of R. (b) The regio R is the base of a solid. For each y, where y, the cross sectio of the solid take perpedicular to the y-ais is a rectagle whose base lies i R ad whose height is y. Write, but do ot evaluate, a itegral epressio that gives the volume of the solid. (c) There is a poit P o the graph of f at which the lie taget to the graph of f is perpedicular to the graph of g. Fid the coordiates of poit P. (a) Area 4 = 4 d+ = = = + = : itegral : atiderivative (b) y = = y y = 6 = 6 y Width = ( 6 y) y Volume = ( ) y 6 y y dy { : itegrad (c) g ( ) = Thus a lie perpedicular to the graph of g has slope. f ( ) = = = 4 The poit P has coordiates ( ),. 4 : f ( ) : equatio The College Board.

5 SCORING GUIDELINES (Form B) Questio 4 The graph of the differetiable fuctio y = f( ) with domai is show i the figure above. The area of the regio eclosed betwee the graph of f ad the -ais for 5 is, ad the area of the regio eclosed betwee the graph of f ad the -ais for 5 is 7. The arc legth for the portio of the graph of f betwee = ad = 5 is, ad the arc legth for the portio of the graph of f betwee = 5 ad = is 8. The fuctio f has eactly two critical poits that are located at = ad = 8. (a) Fid the average value of f o the iterval 5. (b) Evaluate ( f ( ) + ) d. Show the computatios that lead to your aswer. 5 (c) Let g( ) = f( t) dt. O what itervals, if ay, is the graph of g both cocave up ad decreasig? Eplai your reasoig. (d) The fuctio h is defied by h ( ) = f( ). The derivative of h is h ( ) f ( ). the graph of y = h( ) from = to =. = Fid the arc legth of 5 (a) Average value = f( ) d 5 = = 5 ( ) 5 (b) ( f ( ) + ) d = f ( ) d + f ( ) d + 5 = ( + 7) + = 7 : aswer (c) g ( ) = f( ) g ( ) < o < < 5 g ( ) is icreasig o < < 8. The graph of g is cocave up ad decreasig o < < 5. : g ( ) = f( ) : aalysis ad reaso ( ) ( ( )) (d) Arc legth = + h ( ) d = + f d Let u =. The du = d ad ( ( )) ( ) + f d = + ( f u ) du = ( + 8) = 58 : itegral : substitutio The College Board.

6 SCORING GUIDELINES (Form B) Questio 5 t (secods) B( t ) (meters) vt ( ) (meters per secod) Be rides a uicycle back ad forth alog a straight east-west track. The twice-differetiable fuctio B models Be s positio o the track, measured i meters from the wester ed of the track, at time t, measured i secods from the start of the ride. The table above gives values for B( t ) ad Be s velocity, vt ( ), measured i meters per secod, at selected times t. (a) Use the data i the table to approimate Be s acceleratio at time t = 5 secods. Idicate uits of measure. 6 (b) Usig correct uits, iterpret the meaig of vt () dt i the cotet of this problem. Approimate 6 vt () dt usig a left Riema sum with the subitervals idicated by the data i the table. (c) For 4 t 6, must there be a time t whe Be s velocity is meters per secod? Justify your aswer. (d) A light is directly above the wester ed of the track. Be rides so that at time t, the distace Lt ( ) betwee Be ad the light satisfies ( Lt ()) = + ( Bt ()). At what rate is the distace betwee Be ad the light chagig at time t = 4? v( ) v( ). (a) a( 5) = =. meters sec 6 (b) vt () dt is the total distace, i meters, Be rides over the 6-secod iterval t = to t = 6. : meaig of itegral : { : approimatio 6 vt () dt. +.( 4 ) +.5( 6 4) = 9 meters B( 6) B( 4) 49 9 (c) Because = =, the Mea Value Theorem 6 4 implies there is a time t, 4 < t < 6, such that vt ( ) =. (d) Lt () L () t = Bt () B () t B( 4) v( 4) 9.5 L ( 4 ) = = = meters sec L( 4) : differece quotiet : { : coclusio with justificatio : derivatives : uses B () t = v() t : uits i (a) or (b) The College Board.

7 SCORING GUIDELINES (Form B) Questio 6 Let f ( ) = ( + ) l. 4 + (a) The Maclauri series for l ( + ) is ( ) +. Use the series to write 4 the first four ozero terms ad the geeral term of the Maclauri series for f. (b) The radius of covergece of the Maclauri series for f is. Determie the iterval of covergece. Show the work that leads to your aswer. (c) Write the first four ozero terms of the Maclauri series for f ( t ). If ( ) ( ) g = f t dt, use the first two ozero terms of the Maclauri series for g to approimate g (. ) (d) The Maclauri series for g, evaluated at =, is a coverget alteratig series with idividual terms that decrease i absolute value to. Show that your approimatio i part (c) must differ from g ( ) by less tha : first four terms (a) ( ) + : 4 { : geeral term (b) The iterval of covergece is cetered at =. At =, the series is, which 4 diverges because the harmoic series diverges. + At =, the series is ( ) +, the 4 alteratig harmoic series, which coverges. Therefore the iterval of covergece is <. (c) The Maclauri series for f ( ), ( ) f t, ad g( ) are f ( ) : ( ) = + + = ( ) ( ) f t : t = t t + t t + = g( ) : ( ) = = 8 Thus g() = (d) The Maclauri series for g evaluated at = is alteratig, ad the terms decrease i absolute value to. 7 8 Thus g() < = < : aswer with aalysis 4 : : two terms for f : other terms for : first two terms for : approimatio : aalysis ( t ) f ( t ) g( ) The College Board.

AP Calculus BC 2005 Scoring Guidelines

AP Calculus BC 2005 Scoring Guidelines AP Calculus BC 5 Scorig Guidelies The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success ad