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AP®Calculus BC
2003 Free-Respon Questions
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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.
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1. Let R be the shaded region bounded by the graphs of y x =
and y e x =−3 and the vertical line x =1,
as shown in the figure above.
(a) Find the area of R .
(b) Find the volume of the solid generated when R is revolved about the horizontal line y =1.
(c) The region R is the ba of a solid. For this solid, each cross ction perpendicular to the x -axis is a rectangle who height is 5 times the length of its ba in region R . Find the volume of this solid.
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2. A particle starts at point A on the positive x -axis at time t =0 and travels along the curve from A to B to C to D , as shown above. The coordinates of the particle’s position x t y t ()(),05
are differentiable functions of t, where ()==- + x t dx dt t t 9612cos sin p p and ()=y t dy dt
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is not explicitly given. At time t =9, the particle reaches its final position at point D on the positive x -axis. (a) At point C , is dy dt positive? At point C , is dx dt
positive? Give a reason for each answer. (b) The slope of the curve is undefined at point B . At what time t is the particle at point B ?
(c) The line tangent to the curve at the point x y 88()(),05 has equation y x =
-59 2. Find the velocity vector and the speed of the particle at this point.
(d) How far apart are points A and D , the initial and final positions, respectively, of the particle?
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3. The figure above shows the graphs of the line x y =53 and the curve C given by x y =+12. Let S be the shaded region bounded by the two graphs and the x -axis. The line and the curve interct at point P . (a) Find the coordinates of point P and the value of dx dy
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 221-=. Show that x y 221-= can be written as the polar equation
r 2221=-cos sin .q q
(d) U the polar equation given in part (c) to t up an integral expression with respect to the polar angle q that reprents the area of S .
END OF PART A OF SECTION II
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CALCULUS BC
地骨皮功效SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for the problems.
4. Let f be a function defined on the clod interval - 34x with f 0305=. The graph of ′f , the derivative of f , consists of one line gment and a micircle, as shown above.
(a) On what intervals, if any, is f increasing? Justify your answer.
(b) Find the x -coordinate of each point of inflection of the graph of f on the open interval -<<34x . Justify your answer.
(c) Find an equation for the line tangent to the graph of f at the point 03,.05
(d) Find f -()3 and f 405. Show the work that leads to your answers.
