AP Calculus AB 2007 Scoring Guidelines

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1 AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 9, he associaion is composed of more han 5, schools, colleges, universiies, and oher educaional organizaions. Each year, he College Board serves seven million sudens and heir parens,, high schools, and,5 colleges hrough major programs and services in college admissions, guidance, assessmen, financial aid, enrollmen, and eaching and learning. Among is bes-known programs are he SAT, he PSAT/NMSQT, and he Advanced Placemen Program (AP ). The College Board is commied o he principles of ecellence and equiy, and ha commimen is embodied in all of is programs, services, aciviies, and concerns. 7 The College Board. All righs reserved. College Board, Advanced Placemen Program, AP, AP Cenral, SAT, and he acorn logo are regisered rademarks of he College Board. PSAT/NMSQT is a regisered rademark of he College Board and Naional Meri Scholarship Corporaion. 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 7 SCORING GUIDELINES Quesion Le R be he region in he firs and second quadrans bounded above by he graph of below by he horizonal line y. (a) Find he area of R. (b) Find he volume of he solid generaed when R is roaed abou he -ais. (c) The region R is he base of a solid. For his solid, he cross secions perpendicular o he -ais are semicircles. Find he volume of his solid. y and + + when ± : correc limis in an inegral in (a), (b), or (c) (a) Area d 7.96 or : { : inegrand : answer (b) Volume d : { : inegrand : answer (c) Volume + d d : { : inegrand : answer 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

3 7 SCORING GUIDELINES Quesion The amoun of waer in a sorage ank, in gallons, is modeled by a coninuous funcion on he ime inerval 7, where is measured in hours. In his model, raes are given as follows: (i) The rae a which waer eners he ank is f () sin ( ) gallons per hour for 7. (ii) The rae a which waer leaves he ank is 5 for < g () gallons per hour. for < 7 The graphs of f and g, which inersec a.67 and 5.76, are shown in he figure above. A ime, he amoun of waer in he ank is 5 gallons. (a) How many gallons of waer ener he ank during he ime inerval 7? Round your answer o he neares gallon. (b) For 7, find he ime inervals during which he amoun of waer in he ank is decreasing. Give a reason for each answer. (c) For 7, a wha ime is he amoun of waer in he ank greaes? To he neares gallon, compue he amoun of waer a his ime. Jusify your answer. 7 gallons : { : inegral (a) f() d 86 : answer (b) The amoun of waer in he ank is decreasing on he inervals.67 and 5.76 because f () < g() for <.67 and < < (c) Since f () g() changes sign from posiive o negaive only a, he candidaes for he absolue maimum are a,, and 7. (hours) gallons of waer f() d 5( ) f() d ( ) 5.87 : { : inervals : reason : idenifies as a candidae : inegrand 5 : : amoun of waer a : amoun of waer a 7 : conclusion The amoun of waer in he ank is greaes a hours. A ha ime, he amoun of waer in he ank, rounded o he neares gallon, is 57 gallons. 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

4 7 SCORING GUIDELINES Quesion f ( ) f ( ) g( ) g ( ) The funcions f and g are differeniable for all real numbers, and g is sricly increasing. The able above gives values of he funcions and heir firs derivaives a seleced values of. The funcion h is given by h ( ) f( g ( )) 6. (a) Eplain why here mus be a value r for < r < such ha hr ( ) 5. (b) Eplain why here mus be a value c for < c < such ha h ( c) 5. g( ) (c) Le w be he funcion given by w ( ) f( ) d. Find he value of w (. ) (d) If g is he inverse funcion of g, wrie an equaion for he line angen o he graph of y g ( ) a. (a) h() f( g() ) 6 f( ) h( ) f( g( ) ) 6 f( ) Since h( ) < 5 < h( ) and h is coninuous, by he Inermediae Value Theorem, here eiss a value r, < r <, such ha h( r ) 5. (b) h( ) h( ) 7 5 Since h is coninuous and differeniable, by he Mean Value Theorem, here eiss a value c, < c <, such ha h ( c) 5. (c) w ( ) f ( g( ) ) g ( ) f ( ) (d) g (), so g ( ). ( g ) ( ) g g ( ) g 5 ( ) () An equaion of he angen line is y ( ). 5 : h() and h( ) : : conclusion, using IVT h( ) h( ) : : : conclusion, using MVT : apply chain rule : { : answer : g ( ) : : ( g ) ( ) : angen line equaion 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

5 7 SCORING GUIDELINES Quesion A paricle moves along he -ais wih posiion a ime given by () e sin for. (a) Find he ime a which he paricle is farhes o he lef. Jusify your answer. (b) Find he value of he consan A for which () saisfies he equaion A () + () + () for < <. (a) () sin e + e cos e ( cos sin ) () when cos sin. Therefore, () on 5 for and. The candidaes for he absolue minimum are a 5,,, and. 5 : : () : ses () : answer : jusificaion () e sin ( ) 5 e e ( ) > ( ) sin 5 5 sin < e sin ( ) The paricle is farhes o he lef when 5. (b) () e ( cos sin ) + e ( sin cos ) e cos A () + () + () ( ) ( ) A e cos + e cos sin + e sin ( A + ) e cos : : () : subsiues (), (), and () ino A () + () + () : answer Therefore, A. 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

6 7 SCORING GUIDELINES Quesion 5 (minues) () (fee per minue) The volume of a spherical ho air balloon epands as he air inside he balloon is heaed. The radius of he balloon, in fee, is modeled by a wice-differeniable funcion r of ime, where is measured in minues. For < <, he graph of r is concave down. The able above gives seleced values of he rae of change, (), of he radius of he balloon over he ime inerval. The radius of he balloon is fee when 5. (Noe: The volume of a sphere of radius r is given by V r. ) (a) Esimae he radius of he balloon when 5. using he angen line approimaion a 5. Is your esimae greaer han or less han he rue value? Give a reason for your answer. (b) Find he rae of change of he volume of he balloon wih respec o ime when 5. Indicae unis of measure. (c) Use a righ Riemann sum wih he five subinervals indicaed by he daa in he able o approimae () d. Using correc unis, eplain he meaning of () d in erms of he radius of he balloon. (d) Is your approimaion in par (c) greaer han or less han () d? Give a reason for your answer. (a) r( 5.) r( 5) + ( 5) Δ + (.).8 f : Since he graph of r is concave down on he inerval { : esimae : conclusion wih reason 5 < < 5., his esimae is greaer han r ( 5. ). dv d dv d r (b) ( ) 5 dr d ( ) 7 f min (c) ( ) d (.) + (.) + (.) + (.6) + (.5) 9. f () d is he change in he radius, in fee, from o minues. (d) Since r is concave down, is decreasing on < <. Therefore, his approimaion, 9. f, is less han () d. : dv : d : answer : { : approimaion : eplanaion : conclusion wih reason Unis of f min in par (b) and f in par (c) : unis in (b) and (c) 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

7 7 SCORING GUIDELINES Quesion 6 Le f be he funcion defined by f ( ) k ln for >, where k is a posiive consan. (a) Find f ( ) and f ( ). (b) For wha value of he consan k does f have a criical poin a? For his value of k, deermine wheher f has a relaive minimum, relaive maimum, or neiher a. Jusify your answer. (c) For a cerain value of he consan k, he graph of f has a poin of inflecion on he -ais. Find his value of k. (a) k f ( ) : : : f ( ) f ( ) f ( ) k + (b) f () k k When k, f () and f () + >. f has a relaive minimum value a by he Second Derivaive Tes. : : ses f () or f ( ) : solves for k : answer : jusificaion (c) A his inflecion poin, f ( ) and f ( ). k f ( ) + k ln f( ) k ln k : : f ( ) or f( ) : equaion in one variable : answer Therefore, ln ln e k e 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens).

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