AP Calculus AB 2015 Scoring Guidelines

1 AP Calculus AB 2015 Scoring Guidelines 2015 The College Board. College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered t rademarks of the College Board. Visit the College Board on the Web: AP Central is the official online home for the AP Program: AP Calculus AB/ Calculus BC 2015 Scoring Guidelines Question 1 2015 The College Board. Visit the College Board on the Web: The rate at which rainwater flows into a drainpipe is modeled by the function R, where ( )220 sin35tRt = cubic feet per hour, t is measured in hours, and The pipe is partially blocked, allowing water to drain out the other end of the pipe at a rate modeled by () ++ cubic feet per hour, for There are 30 cubic feet of water in the pipe at time (a) How many cubic feet of rainwater flow into the pipe during the 8-hour time interval 08?

2 T (b) Is the amount of water in the pipe increasing or decreasing at time 3t= hours? Give a reason for your answer. (c) At what time t, 08,t is the amount of water in the pipe at a minimum? Justify your answer. (d) The pipe can hold 50 cubic feet of water before overflowing. For 8,t> water continues to flow into and out of the pipe at the given rates until the pipe begins to overflow. Write, but do not solve, an equation involving one or more integrals that gives the time w when the pipe will begin to overflow. (a) ( ) { 1 : integrand2: 1 : answer (b) ( )( ) = < Since ( )( )33,RD< the amount of water in the pipe is decreasing at time 3t= hours. ( )( )3 and 3 1 : considers 2: 1 : answer and reasonRD (c) The amount of water in the pipe at time t, 08,t is ( )( )[] + ( )( )00, = = t Amount of water in the pipe 0 30 8 The amount of water in the pipe is a minimum at time (or ) hours.}

3 ( )( ) 1 : considers 3 : 1 : answer 1 : justification0 RtDt = (d) ( )( )[]05030wRtDtdt+ = { 1 : integral2: 1 : equation AP Calculus AB 2015 Scoring Guidelines Question 2 2015 The College Board. Visit the College Board on the Web: Let f and g be the functions defined by ( )221xxfxxe =++ and ( ) ++ Let R and S be the two regions enclosed by the graphs of f and g shown in the figure above. (a) Find the sum of the areas of regions R and S. (b) Region S is the base of a solid whose cross sections perpendicular to the x-axis are squares. Find the volume of the solid. (c) Let h be the vertical distance between the graphs of f and g in region S. Find the rate at which h changes with respect to x when (a) The graphs of ( )yfx= and ( )ygx= intersect in the first quadrant at the points ()0, 2 , ()2, 4 , and () (), 08.}

4 ,1AB= () ()[]( )( )[] aAAfxgxgxdxfxdx+ =+== 1 : limits4 : 2 : integrands 1 : answer (b) ( ) ()[] { 2 : integrand3: 1 : answer (c) ( )( ) ( )hxfxgx= ( )( )( )hxfxgx = ( )( )( ) (or )hfg = = { 1 : considers 2: 1 : answerh AP Calculus AB/ Calculus BC 2015 Scoring Guidelines Question 3 2015 The College Board. Visit the College Board on the Web: t (minutes) 0 12 20 24 40 ( )vt (meters per minute) 0 200 240 220 150 Johanna jogs along a straight path. For 040,t Johanna s velocity is given by a differentiable function v. Selected values of ( ),vt where t is measured in minutes and ( )vt is measured in meters per minute, are given in the table above. (a) Use the data in the table to estimate the value of ( )16 .v (b) Using correct units, explain the meaning of the definite integral ( )400v t dt in the context of the problem.}}

5 Approximate the value of ( )400v t dt using a right Riemann sum with the four subintervals indicated in the table. (c) Bob is riding his bicycle along the same path. For 010,t Bob s velocity is modeled by ( )326300,Bttt= + where t is measured in minutes and ( )Bt is measured in meters per minute. Find Bob s acceleration at time (d) Based on the model B from part (c), find Bob s average velocity during the interval (a) ( )22402005 meters/min201621v = 1 : approximation (b) ( )400v t dt is the total distance Johanna jogs, in meters, over the time interval 040t minutes. ( )()( )( )()4001212820424164012 2008 2404 22016 1502400192088024007600 metersv t dtvvvv +++= +++= + ++= 1 : explanation3 : 1 : right Riemann sum 1 : approximation (c) Bob s acceleration is ( )2312.

6 Bttt = ( ) ( )( )253 2512 515 meters/minB = = ( ) 1 : uses 2: 1 : answerBt (d) ()103201043016300101230010 41 1000020003000350 meters/Avg v0emin14lttdtttt + = + = + = = 1 : integral3 : 1 : antiderivative 1 : answer AP CALUCLUS AB/ Calculus BC 2015 Scoring Guidelines Question 4 2015 The College Board. Visit the College Board on the Web: Consider the differential equation (a) On the axes provided, sketch a slope field for the given differential equation at the six points indicated. (b) Find 22dydx in terms of x and y. Determine the concavity of all solution curves for the given differential equation in Quadrant II. Give a reason for your answer. (c) Let ( )yfx= be the particular solution to the differential equation with the initial condition ( ) Does f have a relative minimum, a relative maximum, or neither at 2?

7 X= Justify your answer. (d) Find the values of the constants m and b for which ymxb= + is a solution to the differential equation. (a) {0 1 : slopes where 2: 1 : slopes whe 1rexx== (b) 2222(2)22dydyxyxydxdx= = = + In Quadrant II, 0x< and 0,y> so +> Therefore, all solution curves are concave up in Quadrant II. 22 1 : 2: 1 : concave up with reasondydx (c) ( ) ( )( ),2, 322 301xydydx== ==/ Therefore, f has neither a relative minimum nor a relative maximum at ( ) ( ),2, 3 1 : considers 2: 1 : conclusion with justificationxydydx= (d) ()()() ()222020 22dmxbmdxxymxmdyymxbxbmmxmbmmbbdxm+= = += + == += = == = Therefore, 2m= and () 1 : 3: 1 : 2 1 : answerdmxbmdxmxy+ = = AP Calculus AB 2015 Scoring Guidelines Question 5 2015 The College Board. Visit the College Board on the Web: The figure above shows the graph of ,f the derivative of a twice-differentiable function f, on the interval []3, 4.}

8 The graph of f has horizontal tangents at 1,1,xx= = and The areas of the regions bounded by the x-axis and the graph of f on the intervals []2, 1 and [ ]1, 4 are 9 and 12, respectively. (a) Find all x-coordinates at which f has a relative maximum. Give a reason for your answer. (b) On what open intervals contained in 34x < < is the graph of f both concave down and decreasing? Give a reason for your answer. (c) Find the x-coordinates of all points of inflection for the graph of f. Give a reason for your answer. (d) Given that ( )13,f= write an expression for ( )fx that involves an integral. Find ( )4f and ( ) (a) ()0fx = at 2,x= 1,x= and ( )fx changes from positive to negative at Therefore, f has a relative maximum at { 1 : identifies 22: 1 : answer with reasonx= (b) The graph of f is concave down and decreasing on the intervals 12x << and 13x<< because f is decreasing and negative on these intervals.}

9 { 1 : intervals2: 1 : reason (c) The graph of f has a point of inflection at 1x= and 3x= because f changes from decreasing to increasing at these points. The graph of f has a point of inflection at 1x= because f changes from increasing to decreasing at this point. { 1 : identifies 1, 1, and 32: 1 : reasonx= (d) ( )( )13xfxftdt = + ( )( )( )41933124fftdt= += += ( )( )( )( )21122339123fftdtftdt = = =+= ( )( )( ) 1 : integrand3 : 1 : expression for 1 : 4 2 andfxff AP Calculus AB 2015 Scoring Guidelines Question 6 2015 The College Board. Visit the College Board on the Web: Consider the curve given by the equation = It can be shown that (a) Write an equation for the line tangent to the curve at the point ()1, 1 . (b) Find the coordinates of all points on the curve at which the line tangent to the curve at that point is vertical.}}

10 (c) Evaluate 22dydx at the point on the curve where 1x= and (a) ( ) ()( ) ( ),1, 12114311xydydx= == An equation for the tangent line is () ++= { 1 : slope2: 1 : equation for tangent line (b) 22033xxyy = = So, ( )( )3322 32 1yxyyyyy = = = ( )( )31123xx = = The tangent line to the curve is vertical at the point ()3, 1 . 2 1 : sets 3 : 1 : equation in one variable 1 : coordinates30yx = (c) ()()()222223613dydyyxyydydxdxdxyx = ( ) ()( )()()( )()22222,1, 1113111 6114431111121632xydydx= = == 2 : implicit differentiation4 : 1 : substitution for 1 : answerdydx}