AP Calculus AB 2010 Scoring Guidelines

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1 AP Calculus AB 1 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 1, 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, 3, high schools, and 3,8 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. 1 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 1 SCORING GUIDELINES Quesion 1 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 () = 15 for 6 < 7 18 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 1.75 cubic fee : { 1 : inegral (a) f() d = 1.7 (b) Rae of change is f( 8) g( 8) = 5.58 or cubic fee per hour. (c) h ( ) = For < 6, h () = h( ) + gs ( ) ds= + ds=. For 6 7, < h () = h( 6) + gs ( ) ds= + 15 ds= 15( 6 ). For 7, 6 6 < h () = h( 7) + g( s) ds = ds = ( 7 ). 7 7 for 6 Thus, h () = 15( 6) for 6 < ( 7) for 7 < 1 : h () for 6 1 : h () for 6 < 7 1 : h () for 7 < 1 : inegral (d) Amoun of snow is f() d h( ) = 6.33 or cubic fee. 1 : h( ) 1 The College Board. Visi he College Board on he Web:

3 1 SCORING GUIDELINES 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. (b) Use a rapezoidal sum wih he four subinervals given by he able o approximae he value of (). 8 E d 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 3 P, where P () = hundreds of enries per hour for 8 1. According o he model, how many enries had no ye been processed by midnigh ( = 1 )? (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 7 5 (b) () 8 E d 1 E( ) + E( ) E( ) + E( 5) E( 5) + E( 7) E( 7) + E( 8) = or () 8 E d is he average number of hundreds of enries in he box beween noon and 8 P.M. (c) 3 P () d= 3 16 = 7 (hours) E() (hundreds of enries) 1 : rapezoidal sum 1 : approximaion 1 : meaning 1 hundred enries : 8 { 1 : inegral (d) P () = when = and = P() Enries are being processed mos quickly a ime = : considers P () = 1 : idenifies candidaes wih jusificaion 1 The College Board. Visi he College Board on he Web:

4 1 SCORING GUIDELINES Quesion 3 There are 7 people in line for a popular amusemen-park ride when he ride begins operaion in he morning. Once i begins operaion, he ride acceps passengers unil he park closes 8 hours laer. While here is a line, people move ono he ride a a rae of 8 people per hour. The graph above shows he rae, r (), a which people arrive a he ride hroughou he day. Time is measured in hours from he ime he ride begins operaion. (a) How many people arrive a he ride beween = and = 3? Show he compuaions ha lead o your answer. (b) Is he number of people waiing in line o ge on he ride increasing or decreasing beween = and = 3? Jusify your answer. (c) A wha ime is he line for he ride he longes? How many people are in line a ha ime? Jusify your answers. (d) Wrie, bu do no solve, an equaion involving an inegral expression of r whose soluion gives he earlies ime a which here is no longer a line for he ride (a) r () d= + = 3 people : { 1 : inegral (b) The number of people waiing in line is increasing because people move ono he ride a a rae of 8 people per hour and for < < 3, r () > 8. wih reason (c) r () = 8 only a = 3 For < 3, r () > 8. For 3 < 8, r() < 8. Therefore, he line is longes a ime = 3. There are = 15 people waiing in line a ime = 3. 1 : idenifies = 3 1 : number of people in line 1 : jusificaion (d) = 7 + rs ( ) ds 8 1 : 8 1 : inegral 1 The College Board. Visi he College Board on he Web:

5 1 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 3 imes he lengh of is base in region R. Wrie, bu do no evaluae, an inegral expression ha gives he volume of he solid. 3 (a) Area ( x x ) x= = 6 = 6 = 18 3 x= 1 : inegrand 1 : aniderivaive ( ) { : inegrand (b) Volume = π ( 7 x ) ( 7 6) 1 : limis and consan y (c) Solving y = x for x yields x =. y y 3 Each recangular cross secion has area 3 y. = 16 Volume = y dy { : inegrand 1 The College Board. Visi he College Board on he Web:

6 1 SCORING GUIDELINES Quesion 5 g = The graph of y = g, he derivaive of g, consiss of a semicircle and hree line segmens, as shown in he figure above. The funcion g is defined and differeniable on he closed inerval [ 7, 5] and saisfies ( ) 5. (a) Find g ( 3) and g(. ) (b) Find he x-coordinae of each poin of inflecion of he graph of y = g on he inerval 7 < x < 5. Explain your reasoning. 1 (c) The funcion h is defined by hx ( ) = gx ( ) x. Find he x-coordinae of each criical poin of h, where 7 < x < 5, and classify each criical poin as he locaion of a relaive minimum, relaive maximum, or neiher a minimum nor a maximum. Explain your reasoning. 3 π 3 13 (a) g( 3) = 5 + g = = + π g( ) = 5 + g = 5 π (b) The graph of y = g has poins of inflecion a x =, x =, and x = 3 because g changes from increasing o decreasing a x = and x = 3, and g changes from decreasing o increasing a x =. (c) h = g x = g = x On he inerval x, g = x. On his inerval, g = x when x =. The only oher soluion o g = x is x = 3. h = g x > for x < h = g x for < x 5 Therefore h has a relaive maximum a x =, and h has neiher a minimum nor a maximum a x = 3. 1 : uses g( ) = 5 1 : g( 3) 1 : g( ) 1 : idenifies x =,, 3 : { 1 : explanaion : 1 : h 1 : idenifies x =, 3 for wih analysis for 3 wih analysis 1 The College Board. Visi he College Board on he Web:

7 Soluions o he differenial equaion paricular soluion o he differenial equaion AP CALCULUS AB 1 SCORING GUIDELINES Quesion 6 dy 3 xy = also saisfy d y 3 y ( x y ) dy 3 = xy wih () = Le y = f be a f 1 =. (a) Wrie an equaion for he line angen o he graph of y = f a x = 1. (b) Use he angen line equaion from par (a) o approximae f ( 1.1 ). Given ha f( x ) > for 1 < x < 1.1, is he approximaion for f ( 1.1) greaer han or less han f ( 1.1 )? Explain your reasoning. (c) Find he paricular soluion y = f wih iniial condiion f () 1 =. (a) () 1 dy f = = 8 ( 1, ) An equaion of he angen line is y = + 8( x 1 ). 1 : f () 1 : (b) f ( 1.1).8 Since y = f > on he inerval 1 x < 1.1, ( x y ) d y 3 = y 1+ 3 > on his inerval. : { 1 : approximaion 1 : conclusion wih explanaion Therefore on he inerval 1 < x < 1.1, he line angen o he graph of y = f a x = 1 lies below he curve and he approximaion.8 is less han f ( 1.1 ). (c) dy 3 xy = 1 dy = x 3 y 1 x = + C y 1 = 1 + C C = y = 5 x 5 5 f =, < x < 5 x 5 : 1 : separaion of variables 1 : aniderivaives 1 : consan of inegraion 1 : uses iniial condiion 1 : solves for y Noe: max 5 [1-1---] if no consan of inegraion Noe: 5 if no separaion of variables 1 The College Board. Visi he College Board on he Web:

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