AP CALCULUS BC 2014 SCORING GUIDELINES

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2 SCORING GUIDELINES Question The graphs of the polar curves r = and r = sin ( θ ) are shown in the figure above for θ. (a) Let R be the shaded region that is inside the graph of r = and inside the graph of r = sin ( θ ). Find the area of R. (b) For the curve r = sin ( θ ), find the value of at θ =. 6 (c) The distance between the two curves changes for < θ <. Find the rate at which the distance between the two curves is changing with respect to θ when θ = (d) A particle is moving along the curve r = sin ( θ ) so that = for all times t. Find the value of at θ = (a) Area = + ( sin ( )) θ = 9.78 (or 9.77) : integrand : : limits : answer (b) x = ( sin ( θ) ) cos θ =.66 θ = d θ 6 : expression for x : { : answer (c) The distance between the two curves is D = ( sin( θ) ) = sin( θ). : expression for distance : { : answer dd = θ = (d) = = = ( )( ) = 6 θ = 6 : chain rule with respect to t : { : answer The College Board. Visit the College Board on the Web:

3 SCORING GUIDELINES Question The graphs of the polar curves r = and r = sin θ are shown in the figure above. The curves intersect when θ = and 6 (a) Let S be the shaded region that is inside the graph of r = and also inside the graph of r = sin θ. Find the area of S. θ = (b) A particle moves along the polar curve r = sin θ so that at time t 5. 6 seconds, θ = t. Find the time t in the interval t for which the x-coordinate of the particle s position is. (c) For the particle described in part (b), find the position vector in terms of t. Find the velocity vector at time t = (a) Area = 6 + ( sin θ) =.79 (or.78) : 6 : integrand : limits and constant : answer (b) x = r cos θ x( θ) = ( sin θ) cos θ xt ( ) = ( sin ( t )) cos( t ) : x( t) = when t =.8 (or.7) : x( θ ) or xt ( ) : x( θ ) = or xt ( ) = : answer (c) y = r sin θ y( θ ) = ( sin θ) sin θ ( ) = ( sin ( t )) sin ( t ) yt : position vector : { : velocity vector Position vector = xt ( ), yt ( ) ( sin ( t )) cos( t ),( sin ( t )) sin ( t ) = v(.5) = x (.5 ), y (.5) = 8.7,.67 ( or 8.7,.67 ) The College Board. Visit the College Board on the Web:

4 SCORING GUIDELINES (Form B) Question The polar curve r is given by r( θ ) = θ + sin θ, where θ. (a) Find the area in the second quaant enclosed by the coordinate axes and the graph of r. (b) For θ, there is one point P on the polar curve r with x-coordinate. Find the angle θ that corresponds to point P. Find the y-coordinate of point P. Show the work that leads to your answers. (c) A particle is traveling along the polar curve r so that its position at time t is ( x( t), y( t )) and such that =. Find at the instant that θ =, and interpret the meaning of your answer in the context of the problem. (a) Area = ( r( )) d 7.5 θ θ = : : integrand : limits and constant : answer (b) = r( θ ) cosθ = ( θ + sinθ) cosθ θ =.69 y = r( θ ) sin ( θ ) = 6.7 : : equation : value of θ : y-coordinate (c) y = r( θ ) sin θ = ( θ + sin θ) sin θ = =.89 d θ θ= θ= : : uses chain rule : answer : interpretation The y-coordinate of the particle is decreasing at a rate of.89. The College Board. Visit the College Board on the Web:

5 9 SCORING GUIDELINES (Form B) Question The graph of the polar curve r = cos θ for θ is shown above. Let S be the shaded region in the third quaant bounded by the curve and the x-axis. (a) Write an integral expression for the area of S. (b) Write expressions for and in terms of. θ (c) Write an equation in terms of x and y for the line tangent to the graph of the polar curve at the point where θ =. Show the computations that lead to your answer. (a) r ( ) = ; r( θ ) = when θ =. Area of S = ( cosθ) : limits and constant : { : integrand (b) x = rcos θ and y = rsin θ sinθ = = cos θ r sin θ = sin θcos θ sin θ = sin θ + r cos θ = sin θ + ( cos θ) cos θ : : uses x = rcos θ and y = rsin θ : : answer (c) When θ =, we have x =, y =. d = θ = θ = θ = The tangent line is given by y = x. : : values for x and y : expression for : tangent line equation 9 The College Board. All rights reserved. Visit the College Board on the Web:

6 7 SCORING GUIDELINES Question The graphs of the polar curves r = and r = + cos θ are shown in the figure above. The curves intersect when θ = and θ =. (a) Let R be the region that is inside the graph of r = and also inside the graph of r = + cos θ, as shaded in the figure above. Find the area of R. (b) A particle moving with nonzero velocity along the polar curve given by r = + cos θ has position ( x() t, y() t ) at time t, with θ = when t =. This particle moves along the curve so that =. Find the value of at θ = and interpret your answer in terms of the motion of the particle. (c) For the particle described in part (b), =. Find the value of at θ = and interpret your answer in terms of the motion of the particle. (a) Area = ( ) + ( cos ) d + θ θ =.7 : : area of circular sector : integral for section of limaçon : integrand : limits and constant : answer (b) =.7 = θ= θ= The particle is moving closer to the origin, since < and r > when θ =. : : θ= : interpretation (c) y = rsin θ = ( + cos θ) sin θ.5 = = θ= θ= The particle is moving away from the x-axis, since > and y > when θ =. : expression for y in terms of θ : : θ = : interpretation 7 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for students and parents).

7 5 SCORING GUIDELINES Question The curve above is awn in the xy-plane and is described by the equation in polar coordinates r = θ + sin ( θ) for θ, where r is measured in meters and θ is measured in radians. The derivative of r with respect to θ is given by cos( θ ). = + (a) Find the area bounded by the curve and the x-axis. (b) Find the angle θ that corresponds to the point on the curve with x-coordinate. (c) For < θ <, is negative. What does this fact say about r? What does this fact say about the curve? (d) Find the value of θ in the interval θ that corresponds to the point on the curve in the first quaant with greatest distance from the origin. Justify your answer. (a) Area = r = ( θ + sin( θ) ) =.8 : : limits and constant : integrand : answer (b) = r cos( θ) = ( θ + sin( θ) ) cos( θ) θ =.786 : { : equation : answer (c) Since < for < θ <, r is decreasing on this interval. This means the curve is getting closer to the origin. : information about r : { : information about the curve (d) The only value in, where = is θ =. θ r.9.57 : : θ = or.7 : answer with justification The greatest distance occurs when θ =. Copyright 5 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and (for AP students and parents).

8 The figure above shows the graphs of the circles AP CALCULUS BC SCORING GUIDELINES (Form B) Question x + y = and ( x ) + y =. The graphs intersect at the points (,) and (, ). Let R be the shaded region in the first quaant bounded by the two circles and the x-axis. (a) Set up an expression involving one or more integrals with respect to x that represents the area of R. (b) Set up an expression involving one or more integrals with respect to y that represents the area of R. (c) The polar equations of the circles are r = and r = cos, respectively. Set up an expression involving one or more integrals with respect to the polar angle that represents the area of R. (a) Area = Area = ( ) ( x ) + x OR + x : integrand for larger circle : integrand or geometric area : for smaller circle : limits on integral(s) Note: < > if no addition of terms (b) Area = ( ( )) y y : : limits : integrand < > reversal < > algebra error in solving for x < > add rather than subtract < > other errors (c) Area = ( ) d + (cos ) d OR Area = ( ) (cos ) d 8 + : integrand or geometric area for larger circle : : integrand for smaller circle : limits on integral(s) Note: < > if no addition of terms Copyright by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com.

9 The figure above shows the graphs of the line AP CALCULUS BC SCORING GUIDELINES Question 5 x = y and the curve C given by x = + y. Let S be the shaded region bounded by the two graphs and the x-axis. The line and the curve intersect at point P. (a) Find the coordinates of point P and the value of for curve C at point P. (b) Set up and evaluate an integral expression with respect to y that gives the area of S. (c) Curve C is a part of the curve x y =. Show that x y = can be written as the polar equation r =. cos sin (d) Use the polar equation given in part (c) to set up an integral expression with respect to the polar angle that represents the area of S. 5 (a) At P,, y = + y so y =. 5 5 Since x = y, x =. : : coordinates of P : at P y y = =. + y x At P,. = 5 = 5 5 (b) Area = ( + y ) =.6 or.7 y : : limits : integrand : answer (c) x = r cos ; y = r sin x y = r cos r sin = r = cos sin : substitutes x = r cos and : y = r sin into x y = : isolates r (d) Let be the angle that segment OP makes with y the x-axis. Then tan = = =. x 5 5 tan ( 5 ) Area = r d tan ( 5) = d cos sin : limits : : integrand and constant Copyright by College Entrance Examination Board. All rights reserved. Available at apcentral.collegeboard.com.