AP ® Calculus BC
2010 Free-Respon Questions
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CALCULUS BC
SECTION II, Part A
mutexTime—45 minutes
Number of problems—3
A graphing calculator is required for some problems or parts of problems.
1. There is no snow on Janet’s driveway when snow begins to fall at midnight. From midnight to 9 A .M ., snow accumulates on the driveway at a rate modeled by ()cos 7t f t te = cubic feet per hour, where t is measured in hours since midnight. Janet starts removing snow at 6 A .M . ()6.t = The rate (),g t in cubic feet per hour, at which Janet removes snow from the driveway at time t hours after midnight is modeled by
()0for 06125for 67108for 79.
t g t t t £<ÏÔ=£<ÌÔ££Ón是什么意思
(a) How many cubic feet of snow have accumulated on the driveway by 6 A .M .?
(b) Find the rate of change of the volume of snow on the driveway at 8 A .M .
(c) Let ()h t reprent the total amount of snow, in cubic feet, that Janet has removed from the driveway at time t hours after midnight. Express h as a piecewi-defined function with domain 09.t ££
(d) How many cubic feet of snow are on the driveway at 9 A .M .?
分析英语WRITE ALL WORK IN THE PINK EXAM BOOKLET.
t
(hours) 0 2 5 7 8
()E t (hundreds of
entries)
0 4 13 21 23
2. A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box between noon ()0t = and 8 P .M . ()8.t = The number of entries in the box t hours after noon is modeled by a differentiable function E for 08.t ££ Values of (),E t in hundreds of entries, at various times t are shown in the table above.
(a) U the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being deposited at time 6.t = Show the computations that lead to your answer.
(b) U a trapezoidal sum with the four subintervals given by the table to approximate the value of ()801.8E t dt Ú Using correct units, explain the meaning of ()80
18E t dt Ú in terms of the number of entries. (c) At 8
P .M ., volunteers began to process the entries. They procesd the entries at a rate modeled by the function P , where ()3230298976P t t t t =-+- hundreds of entries per hour for 812.t ££ According to the model, how many entries had not yet been procesd by midnight ()12?t =
(d) According to the model from part (c), at what time were the entries being procesd most quickly? Justify影评英文
your answer.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
3. A particle is moving along a curve so that its position at time t is ()()(),,x t y t where ()248x t t t =-+ and
()y t is not explicitly given. Both x and y are measured in meters, and t is measured in conds. It is known that 3 1.t dy te dt
-=- (a) Find the speed of the particle at time 3t = conds.
(b) Find the total distance traveled by the particle for 04t ££ conds.
(c) Find the time t , 04,t ££ when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer. (d) There is a point with x -coordinate 5 through which the particle pass twice. Find each of the following.
(i) The two values of t when that occurs
(ii) The slopes of the lines tangent to the particle’s path at that point
(iii) The y -coordinate of that point, given ()123y =+
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
END OF PART A OF SECTION II
CALCULUS BC
尚雯婕我是歌手唱的歌SECTION II, Part B
Time—45 minutes
Number of problems—3
No calculator is allowed for the problems.dead a
4. Let R be the region in the first quadrant bounded by the graph of y = the horizontal line 6,y = and the y -axis, as shown in the figure above.
(a) Find the area of R .
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is
rotated about the horizontal line 7.y =
(c) Region R is the ba of a solid. For each y , where 06,y ££ the cross ction of the solid taken
sp是什么意思perpendicular to the y -axis is a rectangle who height is 3 times the length of its ba in region R . Write, but do not evaluate, an integral expression that gives the volume of the solid.
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
5. Consider the differential equation 1.dy y dx
=- Let ()y f x = be the particular solution to this differential equation with the initial condition ()10.f = For this particular solution, ()1f x < for all values of x . (a) U Euler’s method, starting at 1x = with two steps of equal size, to approximate ()0.f Show the work
that leads to your answer.
treatment是什么意思(b) Find ()
31lim .1x f x x Æ- Show the work that leads to your answer.
(c) Find the particular solution ()y f x = to the differential equation 1dy y dx
=- with the initial condition ()10.f =
()2cos 1for 01for 0x x x f x x -ÏπÔ=ÌÔ-=Ó 6. The function f , defined above, has derivatives of all orders. Let g be the function defined by
()()01.x g x f t dt =+Ú
(a) Write the first three nonzero terms and the general term of the Taylor ries for cos x about 0.x = U this
ries to write the first three nonzero terms and the general term of the Taylor ries for f about 0.x = (b) U the Taylor ries for f about 0x = found in part (a) to determine whether f has a relative maximum, relative minimum, or neither at 0.x = Give a reason for your answer.
(c) Write the fifth-degree Taylor polynomial for g about 0.x =
(d) The Taylor ries for g about 0,x = evaluated at 1,x = is an alternating ries with individual terms that decrea in absolute value to 0. U the third-degree Taylor polynomial for g about 0x = to estimate the
value of ()1.g Explain why this estimate differs from the actual value of ()1g by less than 1.6!walkman怎么读
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END OF EXAM
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