AP Calculus BC 2010 Scoring Guidelines

Transcription

1 AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board is composed of more han 5,7 schools, colleges, universiies and oher educaional organizaions. Each year, he College Board serves seven million sudens and heir parens,, high schools, and,8 colleges hrough major programs and services in college readiness, college admission, guidance, assessmen, financial aid and enrollmen. Among is widely recognized programs are he SAT, he PSAT/NMSQT, he Advanced Placemen Program (AP ), SpringBoard and ACCUPLACER. The College Board is commied o he principles of excellence and equiy, and ha commimen is embodied in all of is programs, services, aciviies and concerns. The College Board. College Board, ACCUPLACER, Advanced Placemen Program, AP, AP Cenral, SAT, SpringBoard and he acorn logo are regisered rademarks of he College Board. Admied Class Evaluaion Service is a rademark owned by he College Board. PSAT/NMSQT is a regisered rademark of he College Board and Naional Meri Scholarship Corporaion. All oher producs and services may be rademarks of heir respecive owners. Permission o use copyrighed College Board maerials may be requesed online a: Visi he College Board on he Web: AP Cenral is he official online home for he AP Program: apcenral.collegeboard.com.

2 SCORING GUIDELINES Quesion There is no snow on Jane s driveway when snow begins o fall a midnigh. From midnigh o A.M., snow cos accumulaes on he driveway a a rae modeled by f() = 7e cubic fee per hour, where is measured in hours since midnigh. Jane sars removing snow a 6 A.M. ( = 6. ) The rae g (), in cubic fee per hour, a which Jane removes snow from he driveway a ime hours afer midnigh is modeled by for < 6 g () = 5 for 6 < 7 8 for 7. (a) How many cubic fee of snow have accumulaed on he driveway by 6 A.M.? (b) Find he rae of change of he volume of snow on he driveway a 8 A.M. (c) Le h () represen he oal amoun of snow, in cubic fee, ha Jane has removed from he driveway a ime hours afer midnigh. Express h as a piecewise-defined funcion wih domain. (d) How many cubic fee of snow are on he driveway a A.M.? 6 or.75 cubic fee : { : inegral (a) f() d =.7 : answer (b) Rae of change is f( 8) g( 8) = 5.58 or 5.58 cubic fee per hour. : answer (c) h ( ) = For < 6, h () = h( ) + gs ( ) ds= + ds=. For 6 7, < h () = h( 6) + gs ( ) ds= + 5 ds= 5( 6 ). For 7, 6 6 < h () = h( 7) + g( s) ds = ds = 5 + 8( 7 ). 7 7 for 6 Thus, h () = 5( 6) for 6 < ( 7) for 7 < : h () for 6 : h () for 6 < 7 : h () for 7 < : inegral (d) Amoun of snow is f() d h( ) = 6. or 6.5 cubic fee. : h( ) : answer The College Board. Visi he College Board on he Web:

3 SCORING GUIDELINES (hours) E() (hundreds of enries) Quesion A zoo sponsored a one-day cones o name a new baby elephan. Zoo visiors deposied enries in a special box beween noon ( = ) and 8 P.M. ( = 8. ) The number of enries in he box hours afer noon is modeled by a differeniable funcion E for 8. Values of E(), in hundreds of enries, a various imes are shown in he able above. (a) Use he daa in he able o approximae he rae, in hundreds of enries per hour, a which enries were being deposied a ime = 6. Show he compuaions ha lead o your answer. 8 (b) Use a rapezoidal sum wih he four subinervals given by he able o approximae he value of (). 8 E d 8 Using correc unis, explain he meaning of () 8 E d in erms of he number of enries. (c) A 8 P.M., voluneers began o process he enries. They processed he enries a a rae modeled by he funcion P, where P () = hundreds of enries per hour for 8. According o he model, how many enries had no ye been processed by midnigh ( = )? (d) According o he model from par (c), a wha ime were he enries being processed mos quickly? Jusify your answer. E( 7) E( 5) (a) E ( 6) = hundred enries per hour (b) () 8 E d E( ) + E( ) E( ) + E( 5) E( 5) + E( 7) E( 7) + E( 8) =.687 or () 8 E d is he average number of hundreds of enries in he box beween noon and 8 P.M. (c) P () d= 6 = 7 : answer : rapezoidal sum : approximaion : meaning hundred enries : 8 { : inegral (d) P () = when =.85 and =.867. P() Enries are being processed mos quickly a ime =. : answer : considers P () = : idenifies candidaes : answer wih jusificaion The College Board. Visi he College Board on he Web:

4 SCORING GUIDELINES Quesion A paricle is moving along a curve so ha is posiion a ime is ( x(), y() ), where x () = + 8 and y () is no explicily given. Boh x and y are measured in meers, and is measured in seconds. I is known ha = e. d (a) Find he speed of he paricle a ime = seconds. (b) Find he oal disance raveled by he paricle for seconds. (c) Find he ime,, when he line angen o he pah of he paricle is horizonal. Is he direcion of moion of he paricle oward he lef or oward he righ a ha ime? Give a reason for your answer. (d) There is a poin wih x-coordinae 5 hrough which he paricle passes wice. Find each of he following. (i) The wo values of when ha occurs (ii) The slopes of he lines angen o he paricle s pah a ha poin (iii) The y-coordinae of ha poin, given y( ) = + e (a) Speed = ( x ( ) ) + ( y ( ) ) =.88 meers per second : answer (b) x () = = + e d =.587 or.588 meers Disance ( ) ( ) : { : inegral : answer d (c) = = when e = and d This occurs a =.7. Since x (.7) >, he paricle is moving oward he righ a ime =.7 or.8. : considers = : =.7 or.8 : direcion of moion wih reason (d) x () = 5 a = and = A ime =, he slope is A ime =, he slope is y() = y() = + d e + d = = d.. d = = = = d. d = = = : = and = : slopes : y-coordinae The College Board. Visi he College Board on he Web:

5 SCORING GUIDELINES Quesion Le R be he region in he firs quadran bounded by he graph of y = x, he horizonal line y = 6, and he y-axis, as shown in he figure above. (a) Find he area of R. (b) Wrie, bu do no evaluae, an inegral expression ha gives he volume of he solid generaed when R is roaed abou he horizonal line y = 7. (c) Region R is he base of a solid. For each y, where y 6, he cross secion of he solid aken perpendicular o he y-axis is a recangle whose heigh is imes he lengh of is base in region R. Wrie, bu do no evaluae, an inegral expression ha gives he volume of he solid. (a) Area ( x) ( x x ) x= = 6 = 6 = 8 x= : inegrand : aniderivaive : answer ( ) { : inegrand (b) Volume = π ( 7 x ) ( 7 6) : limis and consan y (c) Solving y = x for x yields x =. y y Each recangular cross secion has area y. = 6 Volume = 6 6 y { : inegrand : answer The College Board. Visi he College Board on he Web:

6 SCORING GUIDELINES Quesion 5 Consider he differenial equaion y. = Le y = f( x) be he paricular soluion o his differenial equaion wih he iniial condiion f () =. For his paricular soluion, f( x ) < for all values of x. (a) Use Euler s mehod, saring a x = wih wo seps of equal size, o approximae f (. ) Show he work ha leads o your answer. (b) Find lim f ( x). Show he work ha leads o your answer. x x (c) Find he paricular soluion y = f( x) o he differenial equaion = y wih he iniial condiion f () =. f f + x Δ ( ) ( ), = + = (a) ( ) () f ( ) f( ) + Δx (, ) 5 + ( ) = (b) Since f is differeniable a x =, f is coninuous a x =. So, ( ) ( ) lim f x = = lim x and we may apply L Hospial s x x Rule. f( x) f ( x) lim f ( x) x lim = lim = = x x x x lim x x : Euler s mehod wih wo seps : { : answer : use of L Hospial s Rule : { : answer (c) y = y = ln y = x + C ln = + C C = ln y = x y e x = 5 : : separaion of variables : aniderivaives : consan of inegraion : uses iniial condiion : solves for y Noe: max 5 [----] if no consan of inegraion Noe: 5 if no separaion of variables f ( x) = e x The College Board. Visi he College Board on he Web:

7 SCORING GUIDELINES Quesion 6 cos x for x f( x) = x for x = The funcion f, defined above, has derivaives of all orders. Le g be he funcion defined by g( x) f( ) d. = + x (a) Wrie he firs hree nonzero erms and he general erm of he Taylor series for cos x abou x =. Use his series o wrie he firs hree nonzero erms and he general erm of he Taylor series for f abou x =. (b) Use he Taylor series for f abou x = found in par (a) o deermine wheher f has a relaive maximum, relaive minimum, or neiher a x =. Give a reason for your answer. (c) Wrie he fifh-degree Taylor polynomial for g abou x =. (d) The Taylor series for g abou x =, evaluaed a x =, is an alernaing series wih individual erms ha decrease in absolue value o. Use he hird-degree Taylor polynomial for g abou x = o esimae he value of g (). Explain why his esimae differs from he acual value of g () by less han. 6! n x x n x (a) cos( x) = + + ( ) +! ( n)! n x x n+ x f( x) = ( ) +! 6! ( n + )! (b) f ( ) is he coefficien of x in he Taylor series for f abou x =, so f ( ) =. f ( ) = is he coefficien of x in he Taylor series for f abou!! x =, so f ( ) =. Therefore, by he Second Derivaive Tes, f has a relaive minimum a x =. : erms for cos x : erms for f : firs hree erms : general erm : deermines f ( ) : : answer wih reason 5 x x x : wo correc erms (c) P5 ( x ) = + :! 5 6! { : remaining erms 7 (d) g() + =! 7 Since he Taylor series for g abou x = evaluaed a x = is alernaing and he erms decrease in absolue value o, we know g 7 (). 7 < 5 6! < 6! : { : esimae : explanaion The College Board. Visi he College Board on he Web: