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CHAB
Chapter Homework Assignment
S.Hogan
Name: ____________________________________
CHAB AP1 HW
Due on Wednesday 4/22/15 by 5:45PM.
Directions: Use this sheet as a cover sheet for your homework assignment. If
you do not staple this cover sheet to the front of your assignment, you will receive
a zero. You may submit your HW early but NO LATE HW will be accepted.
Assignment:
EXAM
2012
2010
2009 (Form B)
2006
2005 (Form B)
1998 MC
NUMBER OF
PROBLEMS
6 problems
6 problems
6 problems
6 problems
6 problems
45 problems
Grading:
Item
Possible Points
All Problems Completed
2
Problems Labeled and in Numerical Order,
Page Numbers are Labeled
First Random Problem (correct with sufficient work)
2
Second Random Problem (correct with sufficient work)
4
Third Random Problem (correct with sufficient work)
4
Fourth Random Problem (correct with sufficient work)
4
TOTAL
20
4
Score
2012 AP® CALCULUS AB FREE-RESPONSE QUESTIONS
CALCULUS AB
SECTION II, Part A
Time— 30 minutes
Number of problems— 2
A graphing calculator is required for these problems.
t (minutes)
0
4
9
15
20
W t (degrees Fahrenheit)
55.0
57.1
61.8
67.9
71.0
1. The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W,
where W t is measured in degrees Fahrenheit and t is measured in minutes. At time t 0, the temperature of
the water is 55F. The water is heated for 30 minutes, beginning at time t 0. Values of W t at selected
times t for the first 20 minutes are given in the table above.
(a) Use the data in the table to estimate W 12 . Show the computations that lead to your answer. Using correct
units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate
20
Ô0
W t dt. Using correct units, interpret the meaning of
20
Ô0
W t dt
in the context of this problem.
1 20
W t dt. Use a left Riemann sum
20 Ô0
1 20
W t dt. Does this
with the four subintervals indicated by the data in the table to approximate
20 Ô0
approximation overestimate or underestimate the average temperature of the water over these 20 minutes?
Explain your reasoning.
(c) For 0 t 20, the average temperature of the water in the tub is
(d) For 20 t 25, the function W that models the water temperature has first derivative given by
W t 0.4 t cos 0.06t . Based on the model, what is the temperature of the water at time t 25 ?
© 2012 The College Board.
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GO ON TO THE NEXT PAGE.
-2-
2. Let R be the region in the first quadrant bounded by the x-axis and the graphs of y
shown in the figure above.
ln x and y
5 x, as
(a) Find the area of R.
(b) Region R is the base of a solid. For the solid, each cross section perpendicular to the x-axis is a square.
Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid.
(c) The horizontal line y k divides R into two regions of equal area. Write, but do not solve, an equation
involving one or more integrals whose solution gives the value of k.
END OF PART A OF SECTION II
-3-
CALCULUS AB
SECTION II, Part B
Time— 60 minutes
Number of problems—4
No calculator is allowed for these problems.
3. Let f be the continuous function defined on > 4, 3@ whose graph, consisting of three line segments and a
semicircle centered at the origin, is given above. Let g be the function given by g x x
Ô1
f t dt.
(a) Find the values of g2 and g 2 .
(b) For each of g 3 and g 3 , find the value or state that it does not exist.
(c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these
points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a
maximum at the point. Justify your answers.
(d) For 4 x 3, find all values of x for which the graph of g has a point of inflection. Explain your
reasoning.
-4-
4. The function f is defined by f x 25 x 2 for 5 x 5.
(a) Find f x .
(b) Write an equation for the line tangent to the graph of f at x
3.
Î f x for 5 x 3
Ï
Ð x 7 for 3 x 5.
3 ? Use the definition of continuity to explain your answer.
(c) Let g be the function defined by g x Is g continuous at x
(d) Find the value of
5
Ô0 x
25 x 2 dx.
-5-
5. The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its
current weight. At time t 0, when the bird is first weighed, its weight is 20 grams. If Bt is the weight of the
bird, in grams, at time t days after it is first weighed, then
dB
dt
Let y
1
100 B .
5
Bt be the solution to the differential equation above with initial condition B0 20.
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your
reasoning.
(b) Find
d2B
d2B
in
terms
of
B.
Use
to explain why the graph of B cannot resemble the following graph.
dt 2
dt 2
(c) Use separation of variables to find y
condition B0 20.
Bt , the particular solution to the differential equation with initial
-6-
6. For 0 t 12, a particle moves along the x-axis. The velocity of the particle at time t is given by
p
vt cos t . The particle is at position x 2 at time t 0.
6
(a) For 0 t 12, when is the particle moving to the left?
(b) Write, but do not evaluate, an integral expression that gives the total distance traveled by the particle from
time t 0 to time t 6.
(c) Find the acceleration of the particle at time t. Is the speed of the particle increasing, decreasing, or neither at
time t 4 ? Explain your reasoning.
(d) Find the position of the particle at time t
4.
STOP
END OF EXAM
-7-
CALCULUS AB
SECTION II, Part A
Time— 45 minutes
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 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 gt , in cubic feet per hour,
at which Janet removes snow from the driveway at time t hours after midnight is modeled by
g t for 0 t 6
Î0
Ñ
Ï125 for 6 t 7
ÑÐ108 for 7 t 9 .
(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 ht 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.?
WRITE ALL WORK IN THE PINK EXAM BOOKLET.
Visit the College Board on the Web: www.collegeboard.com.
-2-
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 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
Using correct units, explain the meaning of
1 8
E t dt.
8 Ô0
1 8
E t dt in terms of the number of entries.
8 Ô0
(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 t 3 30t 2 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.
-3-
3. 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
answer.
0 and t
3 ? Show the computations that lead to your
(b) Is the number of people waiting in line to get on the ride increasing or decreasing between t
t 3 ? Justify your answer.
2 and
(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
time t at which there is no longer a line for the ride.
-4-
CALCULUS AB
SECTION II, Part B
Time— 45 minutes
4. Let R be the region in the first quadrant bounded by the graph of y
the y-axis, as shown in the figure above.
2 x , the horizontal line y
6, and
(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.
-5-
5. The function g is defined and differentiable on the closed interval > 7, 5@ and satisfies g0 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 g3 and g 2 .
g x on the interval 7 x 5.
(b) Find the x-coordinate of each point of inflection of the graph of y
Explain your reasoning.
1 2
x . Find the x-coordinate of each critical point of h, where
2
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.
(c) The function h is defined by h x g x dy
d2y
xy3 also satisfy
dx
dx 2
dy
xy3 with f 1 2.
solution to the differential equation
dx
6. Solutions to the differential equation
(a) Write an equation for the line tangent to the graph of y
y3 1 3 x 2 y 2 . Let y
f x at x
f x be a particular
1.
(b) Use the tangent line equation from part (a) to approximate f 1.1 . Given that f x ! 0 for 1 x 1.1, is
the approximation for f 1.1 greater than or less than f 1.1 ? Explain your reasoning.
(c) Find the particular solution y
f x with initial condition f 1
2.
END OF EXAM
-6-
2009 AP® CALCULUS AB FREE-RESPONSE QUESTIONS (Form B)
CALCULUS AB
SECTION II, Part A
Time— 45 minutes
1. At a certain height, a tree trunk has a circular cross section. The radius Rt of that cross section grows at a rate
modeled by the function
dR
dt
centimeters per year
1
3 sin t 2
16
for 0 t 3, where time t is measured in years. At time t
cross section at time t is denoted by At .
0, the radius is 6 centimeters. The area of the
(a) Write an expression, involving an integral, for the radius Rt for 0 t 3. Use your expression to
find R3 .
(b) Find the rate at which the cross-sectional area At is increasing at time t
measure.
(c) Evaluate
3 years. Indicate units of
3
Ô0 At dt. Using appropriate units, interpret the meaning of that integral in terms of cross-
sectional area.
WRITE ALL WORK IN THE EXAM BOOKLET.
© 2009 The College Board. All rights reserved.
-2-
2. A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate
at which the distance between the road and the edge of the water was changing during the storm is modeled by
f t t cos t 3 meters per hour, t hours after the storm began. The edge of the water was 35 meters from
1
sin t.
the road when the storm began, and the storm lasted 5 hours. The derivative of f t is f t 2 t
(a) What was the distance between the road and the edge of the water at the end of the storm?
(b) Using correct units, interpret the value f 4 of the water.
1.007 in terms of the distance between the road and the edge
(c) At what time during the 5 hours of the storm was the distance between the road and the edge of the water
decreasing most rapidly? Justify your answer.
(d) After the storm, a machine pumped sand back onto the beach so that the distance between the road and the
edge of the water was growing at a rate of g p meters per day, where p is the number of days since
pumping began. Write an equation involving an integral expression whose solution would give the number
of days that sand must be pumped to restore the original distance between the road and the edge of the water.
-3-
3. A continuous function f is defined on the closed interval 4 x 6. The graph of f consists of a line segment
and a curve that is tangent to the x-axis at x 3, as shown in the figure above. On the interval 0 x 6, the
function f is twice differentiable, with f x ! 0.
(a) Is f differentiable at x
answer.
0 ? Use the definition of the derivative with one-sided limits to justify your
(b) For how many values of a, 4 a 6, is the average rate of change of f on the interval >a, 6@ equal to 0 ?
Give a reason for your answer.
(c) Is there a value of a, 4 a 6, for which the Mean Value Theorem, applied to the interval >a, 6@,
1
guarantees a value c, a c 6, at which f c ? Justify your answer.
3
(d) The function g is defined by g x x
Ô0 f t dt for 4 x 6. On what intervals contained in > 4, 6@
is the graph of g concave up? Explain your reasoning.
-4-
CALCULUS AB
SECTION II, Part B
Time— 45 minutes
4. Let R be the region bounded by the graphs of y
x and y
x
, as shown in the figure above.
2
(b) The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares.
Find the volume of this solid.
(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is rotated
about the horizontal line y 2.
-5-
5. Let f be a twice-differentiable function defined on the interval 1.2 x 3.2 with f 1 2. The graph of f ,
the derivative of f, is shown above. The graph of f crosses the x-axis at x 1 and x 3 and has
a horizontal tangent at x
e f x.
2. Let g be the function given by g x (a) Write an equation for the line tangent to the graph of g at x
1.
(b) For 1.2 x 3.2, find all values of x at which g has a local maximum. Justify your answer.
(c) The second derivative of g is g x 2
e f x ËÍ f x f x ÛÝ . Is g 1 positive, negative, or zero?
Justify your answer.
(d) Find the average rate of change of g, the derivative of g, over the interval >1, 3@.
-6-
t
(seconds)
0
8
20
25
32
40
v t (meters per second)
3
5
–10
–8
–4
7
6. The velocity of a particle moving along the x-axis is modeled by a differentiable function v, where the position x
is measured in meters, and time t is measured in seconds. Selected values of vt are given in the table above.
The particle is at position x 7 meters when t 0 seconds.
(a) Estimate the acceleration of the particle at t
Indicate units of measure.
(b) Using correct units, explain the meaning of
36 seconds. Show the computations that lead to your answer.
40
Ô20 vt dt
in the context of this problem. Use a trapezoidal sum
with the three subintervals indicated by the data in the table to approximate
40
Ô20 vt dt.
(c) For 0 t 40, must the particle change direction in any of the subintervals indicated by the data in the
table? If so, identify the subintervals and explain your reasoning. If not, explain why not.
(d) Suppose that the acceleration of the particle is positive for 0 t 8 seconds. Explain why the position of
the particle at t 8 seconds must be greater than x 30 meters.
END OF EXAM
-7-
CALCULUS AB
SECTION II, Part A
Time— 45 minutes
1. Let R be the shaded region bounded by the graph of y
ln x and the line y
x 2, as shown above.
(b) Find the volume of the solid generated when R is rotated about the horizontal line y
3.
(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated
when R is rotated about the y-axis.
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2
2. At an intersection in Thomasville, Oregon, cars turn left at the rate Lt time interval 0 t 18 hours. The graph of y
60 t sin 2
Lt is shown above.
3t cars per hour over the
(a) To the nearest whole number, find the total number of cars turning left at the intersection over the time
interval 0 t 18 hours.
(b) Traffic engineers will consider turn restrictions when Lt 150 cars per hour. Find all values of t for
which Lt 150 and compute the average value of L over this time interval. Indicate units of measure.
(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the
total number of cars turning left and the total number of oncoming cars traveling straight through the
intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight
through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to
your conclusion.
3
3. The graph of the function f shown above consists of six line segments. Let g be the function given
by g x x
Ô0
f t dt.
(a) Find g 4 , g 4 , and g 4 .
(b) Does g have a relative minimum, a relative maximum, or neither at x
1 ? Justify your answer.
(c) Suppose that f is defined for all real numbers x and is periodic with a period of length 5. The graph above
shows two periods of f. Given that g 5 2, find g 10 and write an equation for the line tangent to the
graph of g at x 108.
4
CALCULUS AB
SECTION II, Part B
Time— 45 minutes
t
(seconds)
0
10
20
30
40
50
60
70
80
vt (feet per second)
5
14
22
29
35
40
44
47
49
4. Rocket A has positive velocity vt after being launched upward from an initial height of 0 feet at time t 0
seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 t 80 seconds, as
shown in the table above.
(a) Find the average acceleration of rocket A over the time interval 0 t 80 seconds. Indicate units of
measure.
(b) Using correct units, explain the meaning of
70
Ô10 vt dt in terms of the rocket’s flight. Use a midpoint
Riemann sum with 3 subintervals of equal length to approximate
70
Ô10 vt dt.
3
feet per second per second. At time
t 1
t 0 seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of
the two rockets is traveling faster at time t 80 seconds? Explain your answer.
(c) Rocket B is launched upward with an acceleration of at WRITE ALL WORK IN THE PINK EXAM BOOKLET.
5
5. Consider the differential equation
1 y
, where x 0.
x
dy
dx
(a) On the axes provided, sketch a slope field for the given differential equation at the eight points indicated.
(Note: Use the axes provided in the pink exam booklet.)
(b) Find the particular solution y
state its domain.
f x to the differential equation with the initial condition f 1
1 and
6. The twice-differentiable function f is defined for all real numbers and satisfies the following conditions:
f 0 2, f 0 4, and f 0 3.
(a) The function g is given by g x e ax f x for all real numbers, where a is a constant. Find g 0 and
g 0 in terms of a. Show the work that leads to your answers.
(b) The function h is given by h x cos kx f x for all real numbers, where k is a constant. Find h x and
write an equation for the line tangent to the graph of h at x 0.
END OF EXAM
6
AP® CALCULUS AB
2005 SCORING GUIDELINES (Form B)
Question 1
Let f and g be the functions given by f ( x ) = 1 + sin ( 2 x ) and
g ( x ) = e x 2 . Let R be the shaded region in the first quadrant enclosed by
the graphs of f and g as shown in the figure above.
(b) Find the volume of the solid generated when R is revolved about the
x-axis.
(c) The region R is the base of a solid. For this solid, the cross sections
perpendicular to the x-axis are semicircles with diameters extending
from y = f ( x ) to y = g ( x ) . Find the volume of this solid.
The graphs of f and g intersect in the first quadrant at
( S , T ) = (1.13569, 1.76446 ) .
S
³0 ( f ( x ) − g ( x ) ) dx
S
= ³ (1 + sin ( 2 x ) − e x 2 ) dx
0
(a) Area =
1 : correct limits in an integral in (a), (b),
or (c)
1 : integrand
2: ®
¯ 1 : answer
= 0.429
(b) Volume = π
³0 ( ( f ( x ) )
S
µ
= π´
S
¶0
2
)
− ( g ( x ) )2 dx
((1 + sin ( 2x ))
2
(
− ex
)
2 2
2 : integrand
°
−1 each error
°
°
Note: 0 2 if integral not of form
3: ®
b
°
c ³ R 2 ( x ) − r 2 ( x ) dx
a
°
° 1 : answer
¯
) dx
(
= 4.266 or 4.267
)
S
2
´ π § f ( x) − g( x) ·
(c) Volume = µ
dx
¨
¸
2
¹
¶0 2 ©
S
2 : integrand
3: ®
¯ 1 : answer
2
´ π § 1 + sin ( 2 x ) − e x 2 ·
=µ
¸ dx
2¨
2
¹
¶0 ©
= 0.077 or 0.078
Copyright © 2005 by College Board. All rights reserved.
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2
AP® CALCULUS AB
Question 2
A water tank at Camp Newton holds 1200 gallons of water at time t = 0. During the time interval
0 ≤ t ≤ 18 hours, water is pumped into the tank at the rate
W ( t ) = 95 t sin 2
( 6t ) gallons per hour.
During the same time interval, water is removed from the tank at the rate
R( t ) = 275sin 2
( 3t ) gallons per hour.
(a) Is the amount of water in the tank increasing at time t = 15 ? Why or why not?
(b) To the nearest whole number, how many gallons of water are in the tank at time t = 18 ?
(c) At what time t, for 0 ≤ t ≤ 18, is the amount of water in the tank at an absolute minimum? Show the
work that leads to your conclusion.
(d) For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R( t )
until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not
solve, an equation involving an integral expression that can be used to find the value of k.
(a) No; the amount of water is not increasing at t = 15
since W (15 ) − R (15 ) = −121.09 < 0.
(b) 1200 +
18
³0
(W ( t ) − R( t ) ) dt = 1309.788
1 : limits
°
3 : ® 1 : integrand
°¯ 1 : answer
1310 gallons
(c)
W ( t ) − R( t ) = 0
t = 0, 6.4948, 12.9748
1 : interior critical points
°° 1 : amount of water is least at
3: ®
t = 6.494 or 6.495
°
¯° 1 : analysis for absolute minimum
t (hours) gallons of water
0
6.495
12.975
18
1 : answer with reason
1200
525
1697
1310
The values at the endpoints and the critical points
show that the absolute minimum occurs when
t = 6.494 or 6.495.
(d)
k
³18 R( t ) dt = 1310
1 : limits
2: ®
¯ 1 : equation
3
AP® CALCULUS AB
Question 3
A particle moves along the x-axis so that its velocity v at time t, for 0 ≤ t ≤ 5, is given by
(
)
v( t ) = ln t 2 − 3t + 3 . The particle is at position x = 8 at time t = 0.
(a) Find the acceleration of the particle at time t = 4.
(b) Find all times t in the open interval 0 < t < 5 at which the particle changes direction. During which
time intervals, for 0 ≤ t ≤ 5, does the particle travel to the left?
(c) Find the position of the particle at time t = 2.
(d) Find the average speed of the particle over the interval 0 ≤ t ≤ 2.
(a)
5
a( 4 ) = v′( 4 ) =
7
(b)
v( t ) = 0
1 : answer
1 : sets v( t ) = 0
°
3 : ® 1 : direction change at t = 1, 2
°¯ 1 : interval with reason
2
t − 3t + 3 = 1
t 2 − 3t + 2 = 0
( t − 2 ) ( t −1) = 0
t = 1, 2
v( t ) > 0 for 0 < t < 1
v( t ) < 0 for 1 < t < 2
v( t ) > 0 for 2 < t < 5
The particle changes direction when t = 1 and t = 2.
The particle travels to the left when 1 < t < 2.
(c)
(
³0 ln ( u − 3u + 3) du
2
s ( 2 ) = 8 + ³ ln ( u 2 − 3u + 3) du
0
s( t ) = s( 0 ) +
t
= 8.368 or 8.369
(d)
)
1 : 2 ln u 2 − 3u + 3 du
³0
°°
3: ®
1 : handles initial condition
°
¯° 1 : answer
2
1 2
v( t ) dt = 0.370 or 0.371
2 ³0
1 : integral
2: ®
¯ 1 : answer
4
AP® CALCULUS AB
Question 4
The graph of the function f above consists of three line
segments.
(a) Let g be the function given by g ( x ) =
x
³− 4 f ( t ) dt.
For each of g ( −1) , g ′( −1) , and g ′′( −1) , find the
value or state that it does not exist.
(b) For the function g defined in part (a), find the
x-coordinate of each point of inflection of the graph
of g on the open interval − 4 < x < 3. Explain
your reasoning.
(c) Let h be the function given by h( x ) =
3
³ x f ( t ) dt. Find all values of x in the closed interval
− 4 ≤ x ≤ 3 for which h( x ) = 0.
(d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your
reasoning.
(a)
g ( −1) =
−1
1
15
³− 4 f ( t ) dt = − 2 ( 3)( 5) = − 2
g ′( −1) = f ( −1) = −2
g ′′( −1) does not exist because f is not differentiable
at x = −1.
1 : g ( −1)
°
3 : ® 1 : g ′( −1)
°¯ 1 : g ′′( −1)
(b)
x =1
g ′ = f changes from increasing to decreasing
at x = 1.
1 : x = 1 (only)
2: ®
¯ 1 : reason
(c)
x = −1, 1, 3
2 : correct values
−1 each missing or extra value
(d)
h is decreasing on [ 0, 2]
h′ = − f < 0 when f > 0
1 : interval
2: ®
¯ 1 : reason
5
AP® CALCULUS AB
Question 5
Consider the curve given by y 2 = 2 + xy.
(a) Show that
dy
y
=
.
dx 2 y − x
(b) Find all points ( x, y ) on the curve where the line tangent to the curve has slope
1
.
2
(c) Show that there are no points ( x, y ) on the curve where the line tangent to the curve is horizontal.
(d) Let x and y be functions of time t that are related by the equation y 2 = 2 + xy. At time t = 5, the
dy
dx
= 6. Find the value of
at time t = 5.
value of y is 3 and
dt
dt
(a)
(b)
2 y y′ = y + x y′
( 2 y − x ) y′ = y
y
y′ =
2y − x
1 : implicit differentiation
2: ®
¯ 1 : solves for y ′
y
1
=
2y − x 2
2y = 2y − x
x=0
y=± 2
y
1
1:
=
°
2y − x 2
2: ®
°¯ 1 : answer
y
=0
2y − x
y=0
The curve has no horizontal tangent since
1: y = 0
2: ®
¯ 1 : explanation
( 0,
(c)
2 ) , ( 0, − 2 )
02 ≠ 2 + x ⋅ 0 for any x.
(d) When y = 3, 32 = 2 + 3 x so x =
7
.
3
1 : solves for x
°
3 : ® 1 : chain rule
°¯ 1 : answer
dy dy dx
y
dx
=
⋅
=
⋅
2 y − x dt
dt
dx dt
3
9 dx
dx
At t = 5, 6 =
⋅
=
⋅
7 dt 11 dt
6−
3
dx
22
=
dt t = 5
3
6
AP® CALCULUS AB
Question 6
dy − xy 2
. Let
=
dx
2
y = f ( x ) be the particular solution to this differential
equation with the initial condition f ( −1) = 2.
Consider the differential equation
(a) On the axes provided, sketch a slope field for the
given differential equation at the twelve points
indicated.
(Note: Use the axes provided in the test booklet.)
(b) Write an equation for the line tangent to the graph of
f at x = −1.
(c) Find the solution y = f ( x ) to the given differential equation with the initial condition f ( −1) = 2.
(a)
1 : zero slopes
2: ®
¯ 1 : nonzero slopes
− ( −1) 4
=2
2
y − 2 = 2 ( x + 1)
(b) Slope =
(c)
1 : equation
1
x
dy = − dx
2
2
y
1 : separates variables
° 2 : antiderivatives
°
6 : ® 1 : constant of integration
° 1 : uses initial condition
°
¯° 1 : solves for y
1
x2
=−
+C
4
y
1
1
1
− = − + C; C = −
2
4
4
1
4
y= 2
= 2
1
x
x +1
+
4
4
−
Note: max 3 6 [1-2-0-0-0] if no
constant of integration
Note: 0 6 if no separation of variables
7
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