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005 AP CALCULUS BC FREE-RESPONSE QUESTIONS CALCULUS BC SECTION II, Part A Time 45 minutes Number of problems 3 A graphing calculator is required for some problems or parts of problems. 1 = Let R be the shaded region in 4 the first quadrant enclosed by the y-axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) Find the area of S. -x 1. Let f and g be the functions given by f ( x) = + sin( p x) and gx ( ) 4. (c) Find the volume of the solid generated when S is revolved about the horizontal line y =- 1. GO ON TO THE NEXT PAGE.
005 AP CALCULUS BC FREE-RESPONSE QUESTIONS. The curve above is drawn in the xy-plane and is described by the equation in polar coordinates r = q + sin( q) for 0 q p, where r is measured in meters and q is measured in radians. The derivative of r with respect dr to q is given by 1 cos( q). dq = + (a) Find the area bounded by the curve and the x-axis. (b) Find the angle q that corresponds to the point on the curve with x-coordinate -. p p (c) For < q < dr, is negative. What does this fact say about r? What does this fact say about the 3 3 dq curve? p (d) Find the value of q in the interval 0 q that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer. 3 GO ON TO THE NEXT PAGE.
005 AP CALCULUS BC FREE-RESPONSE QUESTIONS Distance x (cm) Temperature C T( x ) ( ) 0 1 5 6 8 100 93 70 6 55 3. A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature T( x ), in degrees Celsius ( C, ) of the wire x cm from the heated end. The function T is decreasing and twice differentiable. (a) Estimate T ( 7. ) Show the work that leads to your answer. Indicate units of measure. (b) Write an integral expression in terms of T( x ) for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure. 8 (c) Find Ú T ( x) dx, and indicate units of measure. Explain the meaning of ( ) 0 temperature of the wire. (d) Are the data in the table consistent with the assertion that T ( x ) > 0 Explain your answer. 8 Ú T x dx in terms of the 0 for every x in the interval 0 < x < 8? END OF PART A OF SECTION II 4
005 AP CALCULUS BC FREE-RESPONSE QUESTIONS CALCULUS BC SECTION II, Part B Time 45 minutes Number of problems 3 No calculator is allowed for these problems. dy 4. Consider the differential equation x y. dx = - (a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point ( 0, 1 ). (Note: Use the axes provided in the pink test booklet.) (b) The solution curve that passes through the point ( ) y-coordinate of this local minimum? (c) Let y f( x) ( 0) 1. 3 0, 1 has a local minimum at ( ) x = ln. What is the = be the particular solution to the given differential equation with the initial condition f = Use Euler s method, starting at 0 Show the work that leads to your answer. (d) Find d y dx greater than ( ) x = with two steps of equal size, to approximate f (- ) in terms of x and y. Determine whether the approximation found in part (c) is less than or f - 0.4. Explain your reasoning. 0.4. 5 GO ON TO THE NEXT PAGE.
005 AP CALCULUS BC FREE-RESPONSE QUESTIONS 5. A car is traveling on a straight road. For 0 t 4 seconds, the car s velocity vt (), in meters per second, is modeled by the piecewise-linear function defined by the graph above. 4 Ú Using correct units, explain the meaning of () 0 Ú 0 v and ( 0 ), (a) Find vt () dt. (b) For each of ( 4) 4 vt dt. v find the value or explain why it does not exist. Indicate units of measure. (c) Let at () be the car s acceleration at time t, in meters per second per second. For 0 < t < 4, write a piecewise-defined function for at (). (d) Find the average rate of change of v over the interval 8 t 0. Does the Mean Value Theorem guarantee a value of c, for 8 < c < 0, such that v () c is equal to this average rate of change? Why or why not? 6. Let f be a function with derivatives of all orders and for which f ( ) = 7. When n is odd, the nth derivative of f at x = is 0. When n is even and n, the nth derivative of f at x = is given by ( n ) ( n - ) f ( ) = (a) Write the sixth-degree Taylor polynomial for f about x =. (b) In the Taylor series for f about, x = what is the coefficient of ( ) x - for n 1? n 1!. n 3 (c) Find the interval of convergence of the Taylor series for f about x =. Show the work that leads to your answer. END OF EXAM 6