AP® Calculus AB
2007 Scoring Guidelines
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Question 1
Let R be the region in the first and cond quadrants bounded above by the graph of 2
20
迈克尔杰克逊怎么死的
1y x
=
+ and below by the horizontal line 2.y =
(a) Find the area of R .
(b) Find the volume of the solid generated when R is rotated about the x -axis.
(c) The region R is the ba of a solid. For this solid, the cross ctions perpendicular to the
x -axis are micircles. Find the volume of this solid.
Question 2
The amount of water in a storage tank, in gallons, is modeled by a continuous function on the time interval 07,t ≤≤ where t is measured in hours. In this model, rates are given as follows:
(i) The rate at which water enters the tank is
(
)2100sin f t t = gallons per hour for 07.t ≤≤ (ii) The rate at which water leaves the tank is
()250 for 03
2000 for 37
t g t t ≤<⎧=⎨
<≤⎩ gallons per hour. The graphs of f and g , which interct at 1.617t = and 5.076,t = are shown in the figure above. At time 0,t = the amount of water in the tank is 5000 gallons. (a) How many gallons of water enter the tank during the time interval 07?t ≤≤ Round your answer to擦玻璃舞蹈
the nearest gallon. (b) For 07,t ≤≤ find the time intervals during which the amount of water in the tank is decreasing.
Give a reason for each answer. (c) For 07,t ≤≤ at what time t is the amount of water in the tank greatest? To the nearest gallon,
compute the amount of water at this time. Justify your answer.
(a)
员工个人工作总结()7
8264f t dt ≈∫ gallons
2 :
{1 : integral
1 : answer
(b) The amount of water in the tank is decreasing on the
intervals 0 1.617t ≤≤ and 3 5.076t ≤≤ becau ()()f t g t < for 0 1.617t ≤< and 3 5.076.t << 2 : {
1 : intervals
1 : reason
(c) Since ()()f t g t − changes sign from positive to negative
only at 3,t = the candidates for the absolute maximum are at 0,3,t = and 7.
三年级作文难忘的生日
t (hours) gallons of water
0 5000
3
()()3
0500025035126.591f t dt +
−=∫
7 ()()7
3我的滑板车
5126.591200044513.807腊肉有营养吗
f t dt +−=∫
The amount of water in the tank is greatest at 3 hours. At that time, the amount of water in the tank, rounded to the nearest gallon, is 5127 gallons.
5 : 1 : identifies 3 as a candidate 1 : integrand 1 : amount of water at 3 1 : amount of water at 7 1 : conclusion t t t =⎧⎪⎪
=⎨⎪=⎪
⎩
Question 3
x ()f x ()f x ′()g x ()g x ′1
6 4
2 5
2 9 2 3
1
3 10 –
4 4 2 4 –1
3
6 7
The functions f and g are differentiable for all real numbers, and g is strictly increasing. The table
above gives values of the functions and their first derivatives at lected values of x . The function h is given by ()()() 6.h x f g x =−
(a) Explain why there must be a value r for 13r << such that () 5.h r =− (b) Explain why there must be a value c for 13c << such that () 5.h c ′=− (c) Let w be the function given by ()()()
1
.g x w x f t dt =
∫ Find the value of ()3.w ′
(d) If 1g − is the inver function of g , write an equation for the line tangent to the graph of ()1y g x −=
at 2.x =
一十二年Question 4
A particle moves along the x -axis with position at time t given by ()sin t x t e t −= for 02.t π≤≤ (a) Find the time t at which the particle is farthest to the left. Justify your answer.
搭积木的拼音(b) Find the value of the constant A for which ()x t satisfies the equation ()()()0Ax t x t x t ′′′++=
for 02.t π<<
