AP Calculus BC - College Board

1 AP Calculus BCScoring Guidelines 2017 The College Board . College Board , Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board . Visit the College Board on the Web: Central is the official online home for the AP Program: Calculus AB/ Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 1 1 : units in parts (a), (c), and (d) (a) ( )() ( ) () ( ) ()()10020052210V552 cubic foeetlumeAhdhAAA + + = + + == { 1 : left Riemann sum2 : 1 : approximation (b) The approximation in part (a) is an overestimate because a left Riemann sum is used and A is decreasing. 1 : overestimate with reason (c) ( ) The volume is cubic feet. { 1 : integral2 : 1 : answer (d) Using the model, ( )( ) ( )( )5555 = = = = When 5,h= the volume of water is changing at a rate of cubic feet per minute.}}

2 2 : 3 : 1 : answerdVdt AP Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 2 (a) ()() = The area of R is { 1 : integral2 : 1 : answer (b) ( )()( )()()( )()( )()()222220012kgfdgfd = OR ( )()( )()()( )()( )()()222220kkgfdgfd = 1 : integral expression2 : for one region 1 : equation (c) ( )( ) ( )wgf = ( ) == The average value of ( )w on the interval 0,2 is ( ) 1 : 3 : 1 : integral 1 : average valuew (d) ( )Aww = for 02 = ( )Aww = at (or ). = () < ( )w is decreasing at = ( ) 1 : solves 2 : 1 : answer with reasonAww = AP Calculus AB/ Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 3 (a) ( )( )( )( )622662774 3fff x dxf x dx = += = = ( )( )( )5252723102fff x dx = += += ( )( ) 1 : uses initial condition3 : 1 : 6 1 : 5ff (b) ( )0fx > on the intervals [)6, 2 and ( )2, 5.]}

3 Therefore, f is increasing on the intervals []6, 2 and [ ]2, 5 . 2 : answer with justification (c) The absolute minimum will occur at a critical point where ( )0fx = or at an endpoint. ( )202,xfxx= = = x ( )fx 6 3 2 7 2 72 5 102 The absolute minimum value is ( )27 = { 1 : considers 2 : 1 : answer with justifica on2tix= (d) ( )( )2015262f == ( )( )33lim23xfx fx = and ( )( )33lim13xfx fx+ = ( )3f does not exist because ( )( )( )( ) ffx fxx + ( )( ) 1 : 52 : 1 : 3 does not exist,with explanationff AP Calculus AB/ Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 4 (a) ( )()109127164H = = ( )091H= An equation for the tangent line is 91 16 .yt= The internal temperature of the potato at time 3t= minutes is approximately 91 16 343 = degrees Celsius.}

4 1 : slope3 : 1 : tangent line 1 : approximation (b) ()()()()221111272744416dHdHHHdtdt= = = 27H> for 0t> ()22127016dHHdt= > for 0t> Therefore, the graph of H is concave up for > Thus, the answer in part (a) is an underestimate. 1 : underestimate with reason (c) ()()( )()()()( )()23231313133271273273 9127012327121227 for 1030dGdtGdGdtGGtCCCGttGtt= = = + =+ = = = + < The internal temperature of the potato at time 3t= minutes is ()312327543 += degrees Celsius. ( )( ) 1 : separation of variables 1 : antiderivatives 1 : constant of integration and5 : uses initial condition 1 : equation involving and 1 : and 3 GtGtG Note: max 25 [1-1-0-0-0] if no constant of integration Note: 05 if no separation of variables AP Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 5 (a) ( )()()223472 75xfxxx = + ( )( )( )()235182151534f =+ = ( )2 : 3f (b) ()()()223477042 75xxxxxf == =+ The only critical point in the interval << has x-coordinate f changes sign from positive to negative at Therefore, f has a relative maximum at 1 : -coordinate2 : 1 : relative maximum with justificationx (c) ( )()()()()()( )()( )255555321limlim2512 7525lim ln 25ln1255lnlnlim ln158limln 2lnln5144bbbbbbbbbfxdxdxdxxxxxxxxxbb == + = = == = 1 : antiderivative3 : 1 : limit expression 1 : answer (d) f is continuous, positive, and decreasing on [)5.]

5 The series converges by the integral test since 2532 75dxxx + converges. OR 2302 75nn> + and 210n> for Since 22332 75lim12nnnn += and the series 251nn= converges, the series 2532 75nnn= + converges by the limit comparison test. 2 : answer with conditions AP Calculus BC 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 6 (a) ( )00f= ( )01f = ( )( )0111f = = ( )( )021 2f = = ()()( )40326f= = The first four nonzero terms are !2!3!234xxxxxxxx ++= + ++ The general term is 1( 1).nnxn+ ( )( )( )( )4 1 : 0 ,0 , and 03 : 1 : verify terms 1 : general termfff (b) For 1,x= the Maclaurin series becomes () += The series does not converge absolutely because the harmonic series diverges. The series alternates with terms that decrease in magnitude to 0, and therefore the series converges conditionally. 2 : converges conditionally with reason (c) ( )( )( )()( )()100123423 4 5234510114151232 32 432411260121xnnxtxnntnnttftdtdtntttttttx xxtnnxxnn+=++=++ + = +++ = +++ + = + +++++ 1 : two terms3 : 1 : remaining terms 1 : general term (d) The terms alternate in sign and decrease in magnitude to 0.

6 By the alternating series error bound, the error ()()41122Pg is bounded by the magnitude of the first unused term, ( ) Thus, ()()( )5412111 20500Pg =< 1 : error bound