AP Calculus AB - College Board

1 AP Calculus ABScoring 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 AB 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 2 (a) ( ) pounds of bananas are removed from the display table during the first 2 hours the store is open. { 1 : integral2 : 1 : answer (b) () = (or ) After the store has been open 7 hours, the rate at which bananas are being removed from the display table is decreasing by (or ) pounds per hour per hour. { 1 : value2 : 1 : meaning (c) ( ) ( ) = < Because ( ) ( )550,gf < t he number of pounds of bananas on the display table is decreasing at time ( )( ) 1 : considers 5 and 52 : 1 : answer with reasonfg (d) ( )( ) + = pounds of bananas are on the display table at time { 2 : integrals3 : 1 : answer 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 AB 2017 SCORING GUIDELINES 2017 The College Board . Visit the College Board on the Web: Question 5 (a) ( )()222221201210 Ptttxtttt = ++= 22100tt +> for all t.

5 ( )01 Pxtt == ( )0 Pxt < for < Therefore, the particle is moving to the left for < ( ) 1 : 2 : 1 : intervalPxt (b) ( ) ()()53 Qvttt= ( )03,5 Qvttt= == Both particles move in the same direction for 13t<< and 58t< since ( )()PPvtxt = and ( )Qvt have the same sign on these intervals. ( )( ) 1 : intervals 2 : 1 : analysis using and PQvtvt Note: 12 if only one interval with analysis Note: 02 if no analysis (c) ()()28 QQatvtt == ( )222 84Qa= = ( )20Qa< and ()320Qv= > At time 2,t= the speed of the particle is decreasing because velocity and acceleration have opposite signs. ( ) 1 : 2 : 1 : speed decreasing with reason2Qa (d) Particle Q first changes direction at time ( )( )( )()()33200332035815154155936452330 QQQttxxvtdtttdtttt=== +=+ + =+ + =+ + = 1 : antiderivative3 : 1 : uses initial condition 1 : answer AP Calculus AB 2017 SCORING GUIDELINES 2017 The College Board .

6 Visit the College Board on the Web: Question 6 (a) ( )()sin2 sin 2cosxfxxxe = + ( )( )sin2 sin 2cos1fe = + = ( )2 : f (b) ( )( )()( )kxhfxfx = ( )( )()( )( ) ( )()( )2111133khffh == = = ( )( ) 1 : 2 : 1 : kxk (c) ( )( ) ( ) ( ) ( )222mxgxhxgxhx + = ()( )( ) ( ) ( )( )()()2 2424 221215333mghgh = + = + = ( )( ) 2 : 23 : 1 : mxm (d) g is differentiable. g is continuous on the interval [ 5, 3]. ( ) ( )( )352104235gg == Therefore, by the Mean Value Theorem, there is at least one value c, 53,c < < such that ( ) = ( ) ( )( ) 1 : 2 : 1 : justification, using Mean Value The35or5em3gg