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Interactive video lesson plan for: 2010 AP Calculus AB FRQ #4 Volumes Area Revolution Solids Known Cross section

Activity overview:

AP® CALCULUS AB
2010 SCORING GUIDELINES
Question 1
© 2010 The College Board.
Visit the College Board on the Web: www.collegeboard.com.
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 f (t ) = 7tecos t cubic feet per hour, where t is measured in
hours since midnight. Janet starts removing snow at 6 A.M. (t = 6). 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
( )
0 for0 6
125 for 6 7

(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 ) represent 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 piecewise-defined function with domain 0 t 9.
(d) How many cubic feet of snow are on the driveway at 9 A.M.?
(a) ( ) 6
0
∫ f t dt = 142.274 or 142.275 cubic feet 2 : { 1 : integral
1
Question 2
© 2010 The College Board.
Visit the College Board on the Web: www.collegeboard.com.
A zoo sponsored a one-day contest to name a new baby elephant. Zoo visitors deposited entries in a special box
between noon (t = 0) and 8 P.M. (t = 8). The number of entries in the box t hours after noon is modeled by a
differentiable function E for 0 t 8. Values of E(t ), in hundreds of entries, at various times t are shown in
the table above.
(a) Use the data in the table to approximate the rate, in hundreds of entries per hour, at which entries were being
deposited at time t = 6. Show the computations that lead to your answer.
(b) Use a trapezoidal sum with the four subintervals given by the table to approximate the value of ( ) 8
0
1 . 8 ∫ E t dt
Using correct units, explain the meaning of ( ) 8
0
18
∫ E t dt in terms of the number of entries.
(c) At 8 P.M., volunteers began to process the entries. They processed the entries at a rate modeled by the function
P, where P(t ) = t3 − 30t2 + 298t − 976 hundreds of entries per hour for 8 t 12. According to the model,
how many entries had not yet been processed by midnight (t = 12) ?
(d) According to the model from part (c), at what time were the entries being processed most quickly? Justify
your answer.



Entries are being processed most quickly at time t = 12.
3 :
1 : considers ( ) 0
1 : identifies candidates
1 : answer with justification
There are 700 people in line for a popular amusement-park ride
when the ride begins operation in the morning. Once it begins
operation, the ride accepts passengers until the park closes 8 hours
later. While there is a line, people move onto the ride at a rate of
800 people per hour. The graph above shows the rate, r(t ), at
which people arrive at the ride throughout the day. Time t is
measured in hours from the time the ride begins operation.
(a) How many people arrive at the ride between t = 0 and t = 3 ?
Show the computations that lead to your answer.
(b) Is the number of people waiting in line to get on the ride
increasing or decreasing between t = 2 and t = 3 ? Justify
your answer.
(c) At what time t is the line for the ride the longest? How many people are in line at that time? Justify your
answers.
(d) Write, but do not solve, an equation involving an integral expression of r whose solution gives the earliest
Let R be the region in the first quadrant bounded by the graph of y = 2 x, the horizontal line y = 6, 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 y = 7.
(c) Region R is the base of a solid. For each y, where 0 ≤ y ≤ 6, the cross section of the solid taken
perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base in region R. Write,
but do not evaluate, an integral expression that gives the volume of the solid.
The function g is defined and differentiable on the closed interval [−7, 5] and satisfies g(0) = 5. The graph of
y = g′( x), the derivative of g, consists of a semicircle and three line segments, as shown in the figure above.
(a) Find g(3) and g(−2).
(b) Find the x-coordinate of each point of inflection of the graph of y = g( x) on the interval −7 x 5.
Explain your reasoning.
(c) The function h is defined by h( x) = g( x) − 12 x2. Find the x-coordinate of each critical point of h, where
−7 x 5, and classify each critical point as the location of a relative minimum, relative maximum, or
neither a minimum nor a maximum. Explain your reasoning
Solutions to the differential equation ddyx xy3 = also satisfy ( ) 2
3 2 2
2 d y y 1 3x y .
dx
= + Let y = f ( x) be a
particular solution to the differential equation ddyx xy3 = with f (1) = 2.

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