2014 AP CALCULUS AB EXAM FREE RESPONSE QUESTIONS AND SOLUTIONS2014 Free Response Questions Solutions2014 AP Calculus Free Response #1
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2014 AP CALCULUS AB FREE RESPONSE QUESTIONS
2014 AP CALCULUS AB SCORING GUIDELINES
Grass clippings are placed in a bin, where they decompose. For 030,t≤≤ the amount of grass clippings remaining in the bin is modeled by ()()6.6870.931,tAt= where ()At is measured in pounds and t is measured in days.
(a) Find the average rate of change of ()At over the interval 030.t≤≤ Indicate units of measure.
(b) Find the value of ()15.A′ Using correct units, interpret the meaning of the value in the context of the problem.
(c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval 030.t≤≤
(d) For 30,t> (),Lt the linear approximation to A at 30,t= is a better model for the amount of grass clippings remaining in the bin. Use ()Lt to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.
(a)
Let R be the region enclosed by the graph of ()432.34fxxx=−+ and the horizontal line 4,y= as shown in the figure above.
(a) Find the volume of the solid generated when R is rotated about the horizontal line 2.y=−
(b) Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid.
(c) The vertical line xk= divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k.The function f is defined on the closed interval []5,4.− The graph of f consists of three line segments and is shown in the figure above. Let g be the function defined by ()()3.xgxftdt−=∫
(a) Find ()3.g
(b) On what open intervals contained in 54x−<< is the graph of g both increasing and concave down? Give a reason for your answer.
(c) The function h is defined by ()().5gxhxx= Find ()3.h′
(d) The function p is defined by ()()2.pxfxx=− Find the slope of the line tangent to the graph of p at the point where 1.
Train A runs back and forth on an east-west section of railroad track. Train A’s velocity, measured in meters per minute, is given by a differentiable function (),Avt where time t is measured in minutes. Selected values for ()Avt are given in the table above.
(a) Find the average acceleration of train A over the interval 28.t≤≤
(b) Do the data in the table support the conclusion that train A’s velocity is 100− meters per minute at some time t with 58?t<< Give a reason for your answer.
(c) At time 2,t= train A’s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time 12.t= Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time 12.t=
(d) A second train, train B, travels north from the Origin Station. At time t the velocity of train B is given by ()256025,Bvttt=−++ and at time 2t= the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train B is changing at time 2.
The twice-differentiable functions f and g are defined for all real numbers x. Values of f, ,f′ g, and g′ for various values of x are given in the table above.
(a) Find the x-coordinate of each relative minimum of f on the interval []2,3.− Justify your answers.
(b) Explain why there must be a value c, for 11,c−<< such that ()0.fc′′=
(c) The function h is defined by ()()()ln.hxfx= Find ()3.h′ Show the computations that lead to your answer.
Consider the differential equation ()3cos.dyyxdx=− Let ()yfx= be the particular solution to the differential equation with the initial condition ()01.f= The function f is defined for all real numbers.
(a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point ()0,1.
(b) Write an equation for the line tangent to the solution curve in part (a) at the point ()0,1. Use the equation to approximate ()0.2.f
(c) Find (),yfx= the particular solution to the differential equation with the initial condition ()01.
(d) Evaluate ()()()32.
(a) Find the average rate of change of ()At over the interval 030.t≤≤ Indicate units of measure.
(b) Find the value of ()15.A′ Using correct units, interpret the meaning of the value in the context of the problem.
(c) Find the time t for which the amount of grass clippings in the bin is equal to the average amount of grass clippings in the bin over the interval 030.t≤≤
(d) For 30,t> (),Lt the linear approximation to A at 30,t= is a better model for the amount of grass clippings remaining in the bin. Use ()Lt to predict the time at which there will be 0.5 pound of grass clippings remaining in the bin. Show the work that leads to your answer.
(a)
Let R be the region enclosed by the graph of ()432.34fxxx=−+ and the horizontal line 4,y= as shown in the figure above.
(a) Find the volume of the solid generated when R is rotated about the horizontal line 2.y=−
(b) Region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in R. Find the volume of the solid.
(c) The vertical line xk= divides R into two regions with equal areas. Write, but do not solve, an equation involving integral expressions whose solution gives the value k.The function f is defined on the closed interval []5,4.− The graph of f consists of three line segments and is shown in the figure above. Let g be the function defined by ()()3.xgxftdt−=∫
(a) Find ()3.g
(b) On what open intervals contained in 54x−<< is the graph of g both increasing and concave down? Give a reason for your answer.
(c) The function h is defined by ()().5gxhxx= Find ()3.h′
(d) The function p is defined by ()()2.pxfxx=− Find the slope of the line tangent to the graph of p at the point where 1.
Train A runs back and forth on an east-west section of railroad track. Train A’s velocity, measured in meters per minute, is given by a differentiable function (),Avt where time t is measured in minutes. Selected values for ()Avt are given in the table above.
(a) Find the average acceleration of train A over the interval 28.t≤≤
(b) Do the data in the table support the conclusion that train A’s velocity is 100− meters per minute at some time t with 58?t<< Give a reason for your answer.
(c) At time 2,t= train A’s position is 300 meters east of the Origin Station, and the train is moving to the east. Write an expression involving an integral that gives the position of train A, in meters from the Origin Station, at time 12.t= Use a trapezoidal sum with three subintervals indicated by the table to approximate the position of the train at time 12.t=
(d) A second train, train B, travels north from the Origin Station. At time t the velocity of train B is given by ()256025,Bvttt=−++ and at time 2t= the train is 400 meters north of the station. Find the rate, in meters per minute, at which the distance between train A and train B is changing at time 2.
The twice-differentiable functions f and g are defined for all real numbers x. Values of f, ,f′ g, and g′ for various values of x are given in the table above.
(a) Find the x-coordinate of each relative minimum of f on the interval []2,3.− Justify your answers.
(b) Explain why there must be a value c, for 11,c−<< such that ()0.fc′′=
(c) The function h is defined by ()()()ln.hxfx= Find ()3.h′ Show the computations that lead to your answer.
Consider the differential equation ()3cos.dyyxdx=− Let ()yfx= be the particular solution to the differential equation with the initial condition ()01.f= The function f is defined for all real numbers.
(a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point ()0,1.
(b) Write an equation for the line tangent to the solution curve in part (a) at the point ()0,1. Use the equation to approximate ()0.2.f
(c) Find (),yfx= the particular solution to the differential equation with the initial condition ()01.
(d) Evaluate ()()()32.
