AP Calculus Review - 1
Multiple choice. No calculators
1. What is the x-coordinate of the point of inflection of the graph of y =
(A) 5
(C) −
(B) 0
10
3
1 3
x + 5 x 2 + 24 ?
3
(D) –5
2. The graph of a piecewise linear function f, for –1 ≤ x ≤ 4, is
shown at right. What is the value of
(A) 1
(C) 4
(E) 8
3.
y
∫ f ( x ) dx ?
4
2
−1
(B) 2.5
(D) 5.5
1
x
–1
1
dx =
2
1 x
1
(A) −
2
∫
(E) –10
O
-1
1
2
3
4
2
(B)
7
24
(C)
1
2
(D) 1
(E) 2 ln 2
4. If f is continuous for a ≤ x ≤ b and differentiable for a < x < b, which of the following could be false?
f (b ) − f ( a )
for some c such that a < c < b.
(A) f ' ( c ) =
b−a
(B) f ' ( c ) = 0 for some c such that a < c < b.
(C) f has a minimum value on a ≤ x ≤ b.
(D) f has a maximum value on a ≤ x ≤ b.
(E)
∫ f ( x ) dx exists.
b
a
5.
∫
x
sin tdt =
0
(A) sin x
(B) –cos x
6. If x 2 + xy = 10 , then when x = 2,
(A) −
7
2
(B) –2
(C) cos x
(D) cos x – 1
(E) 1 – cos x
dy
=
dx
(C)
2
7
(D)
3
2
(E)
7
2
7.
 x2 − 1 
∫ 1  x dx =
1
(A) e −
e
e
(B) e 2 − e
(C)
e2
1
−e+
2
2
(D) e 2 − 2
(E)
e2 3
−
2 2
8. Let f and g be differentiable functions with the following properties:
(i) g ( x ) > 0 for all x
(ii) f ( 0 ) = 1
If h ( x ) = f ( x ) g ( x ) and h ' ( x ) = f ( x ) g ' ( x ) , then f ( x ) =
(C) ex
(B) g(x)
9. The flow of oil, in barrels per hour, through a pipeline on July
9 is given by the graph shown above. Of the following, which
best approximates the total number of barrels of oil that passed
through the pipeline that day?
(A) 500
(B) 600
(C) 2400
(D) 3000
(E) 4800
(D) 0
Barrels per Hour
(A) f '(x)
(E) 1
200
100
O
6
10. What is the instantaneous rate of change at x = 2 of the function f given by f ( x ) =
(A) –2
(B)
1
6
(C)
1
2
(D) 2
12
Hours
18
24
x2 − 2
?
x −1
(E) 6
Free Response. With calculators.
11. At a certain height, a tree trunk has a circular cross section. The radius R(t) of that cross section grows at a
rate modeled by the function
(
)
dR 1
=
3 + sin ( t 2 ) centimeters per year
dt 16
for 0 ≤ t ≤ 3, where time t is measured in years. At time t = 0, the radius is 6 centimeters. The area of the
cross section at time t is denoted by A(t).
a. Write an expression involving an integral for the radius R(t) for 0 ≤ t ≤ 3. Use your expression to find
R(3).
b. Find the rate at which the cross-sectional area A(t) is increasing at time t = 3 years. Indicate units of
measure.
c. Evaluate
∫
3
0
A ' ( t )dt . Using appropriate units, interpret the meaning of that integral in terms of cross-
sectional area.
12. 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 the road when the storm began and the storm lasted 5 hours. The derivative of f(t) is
1
f '(t ) =
− sin 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) = 1.007 in terms of the distance between the road and the
edge of the water.
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.
AP Calculus Review - 2
Multiple choice. No calculators
1. If f is a linear function and 0 < a < b, then
(A) 0
(B) 1
∫
b
a
f " ( x ) dx =
(C)
ab
2
 ln x for 0 < x ≤ 2
2. If f ( x ) =  2
, then lim f ( x ) is
x →2
 x ln x for 2 < x ≤ 4
(A) ln 2
(B) ln 8
(C) ln 16
3. The graph of the function f shown in the figure at right has a
vertical tangent at the point (2, 0) and horizontal tangents that
the points (1, –1) and (3, 1). For what values of x, –2 < x < 4,
if f not differentiable?
(A) 0 only
(B) 0 and 2only
(C) 1 and 3 only
(D) 0, 1 and 3 only
(E) 0, 1, 2, and 3
b2 − a 2
2
(D) b – a
(E)
(D) 4
(E) nonexistent
2
y
1
–2
–1 O
–1
1
2
3
4 x
–2
4. A particle moves along the x-axis so that its position at time t is given by x ( t ) = t 2 − 6t + 5 . For what value
of t is the velocity of the particle 0?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
5. If F ( x ) = ∫
(A) –3
x
t 3 + 1 dt , then F '(2) =
0
(B) –2
(C) 2
(D) 3
(E) 18
6. If f ( x ) = sin ( e − x ) , then f '(x) =
(A) − cos ( e − x )
(B) cos ( e − x ) + e − x
(C) cos ( e − x ) − e − x
7. The graph of a twice differentiable function f is shown the figure
at right. Which of the following is true?
(A) f(1) < f '(1) < f "(1)
(B) f(1) < f "(1) < f '(1)
(C) f ' (1) < f (1) < f "(1)
(D) f "(1) < f (1) < f '(1)
(E) f " (1) < f '(1) < f(1)
(D) e − x cos ( e − x )
(E) −e − x cos ( e − x )
y
O
x
1
8. An equation of the line tangent to the graph of y = x + cos x at the point (0, 1) is
(A) y = 2x + 1
(B) y = x + 1
(C) y = x
(D) y = x – 1
(E) y = 0
9. If f "(x) = x(x + 1)(x – 2)2, then the graph of f has inflection points when x =
(A) –1 only
(B) 2 only
(C) –1 and 0 only
(D) –1 and 2 only (E) –1, 0 and 2 only
10. What are all values of k for which
(A) –3
11. If
(B) 0
∫
k
x 2 dx = 0 ?
−3
(C) 3
(D) –3 and 3
(E) –3, 0 and 3
dy
= ky and k is a nonzero constant, then y could be
dt
(A) 2e kty
(B) 2e kt
(C) e kt + 3
(D) kty + 5
(E)
1 2 1
ky +
2
2
12. The function f is given by f(x) = x4 + x2 – 2. On which of the following intervals is f increasing?
1 
 1

 1 1 

(A)  −
,∞
(B)  −
,
(C) (0, ∞)
(D) (–∞, 0)
(E)  −∞, −


2 
2 2
2



13. The maximum acceleration attained in the interval 0 ≤ t ≤ 3 by the particle whose velocity is given by
v(t) = t3 – 3t2 + 12t + 4 is
(A) 9
(B) 12
(C) 14
(D) 21
(E) 40
14. What is the area of the region between the graphs of y = x2 and y = –x from x = 0 to x = 2?
2
8
14
16
(A)
(B)
(C) 4
(D)
(E)
3
3
3
3
y
a
O
b
x
15. The graph of f is shown in the figure above. Which of the following could be the graph of the derivative of
f?
y
(A)
a
O
b
x
a
y
(C)
a
O
y
(B)
x
b
x
y
(D)
b
O
a
O
b
x
y
(E)
a
O
b
x
x
f(x)
0
1
1
k
2
2
16. The function f is continuous on the closed interval [0, 2] and has values that are given in the table above.
1
The equation f ( x ) = must have at least two solutions in the interval [0, 2] if k =
2
1
(A) 0
(B)
(C) 1
(D) 2
(E) 3
2
17. What is the average value of y = x 2 x 3 + 1 on the interval [0, 2]?
26
52
26
52
(A)
(B)
(C)
(D)
9
9
3
3
π 
18. If f(x) = tan (2x), then f '   =
6
(A) 3
(B) 2 3
(C) 4
(D) 4 3
(E) 24
(E) 8
Free Response. With calculators.
19. 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 at right. On the interval
0 ≤ x ≤ 6, the function is twice differentiable
with f "(x) > 0.
y
.
.
(6, 1)
.
–4
x
–3 –2 1 O
1
2
3
4
5
6
Graph of f
a. Is f differentiable at x = 0? Use the
definition of the derivative with one-sided
limits to justify your answer.
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 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 ) = ∫ f ( t ) dt for –4 ≤ x ≤ 6. On what intervals contained in [–4, 6] is
x
0
the graph of g concave up? Explain your reasoning.
AP Calculus Review - 3
Multiple choice. With calculators.
1. The graph of a function f is shown at right. Which of the following
statements about f is false?
(A) f is continuous at x = a
(B) f has a relative maximum at x = a
(C) x = a is in the domain of f
(D) lim+ f ( x ) is equal to lim− f ( x )
x →a
(E) lim f ( x ) exists
y
O
a
x
x →a
x→a
2. Let f be the function given by f ( x ) = 3e2 x and let g be the function given by g(x) = 6x3. At what value of x
do the graphs of f and g have parallel tangent lines?
(A) –0.701
(B) –0.567
(C) –0.391
(D) –0.302
(E) –0.258
3. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the
circumference C, what is the rate of change of the area of the circle, in square centimeters per second?
( 0.1) C
(A) –(0.2)πC
(B) –(0.1)C
(C) −
(D) (0.1)2C
(E) (0.1)2πC
2π
4. The first derivative of the function f is given by f ' ( x ) =
cos 2 x 1
− . How many critical values does f have
x
5
on the open interval (0, 10)?
(A) One
(B) Three
(C) Four
(D) Five
(E) Seven
y
a
O
y
y = f '(x)
y
y = g '(x)
x
b
a
O
y = h '(x)
x
x
O
a
b
b
5. The graphs of the derivatives of the functions f, g and h are shown above. Which of the functions f, g or h
have a relative maximum on the open interval a < x < b?
(A) f only
(B) g only
(C) h only
(D) f and g only
(E) f, g and h
6. Let f be the function given by f(x) = |x|. Which of the following statements about f are true?
I. f is continuous at x = 0
II. f is differentiable at x = 0
III. f has an absolute minimum at x = 0
(A) I only
(B) II only
(C) III only
(D) I and III only (E) II and III only
7. If f is a continuous function and if F '(x) = f(x) for all real numbers x, then
∫ f ( 2 x )dx =
3
1
(A) 2F(3) – 2F(1)
(B)
(D) F(6) – F(2)
1
1
F ( 3) − F (1)
2
2
(C) 2F(6) – 2F(2)
(E)
1
1
F (6) − F ( 2)
2
2
Free Response. No calculators.
y
8. Let R be the region in bounded by the graphs of y = x and
2
x
y = as shown in the figure at right.
2
a. Find the area of R.
1
b. The region R is the base of a solid. For this solid, at each
R
x the cross section perpendicular to the x-axis is a square.
Find the volume of this solid.
O
1
2
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.
.(4, 2)
x
3
4
9. Let f be a twice-differentiable function defined on the interval –
y
1.2 < x < 3.2 with f(1) = 2. The graph of f ', the derivative of f,
is shown at right. The graph of f ' crosses the x-axis at x = –1
and x = 3 and has a horizontal tangent at x = 2. Let g be the
O
f ( x)
x
–1
1
2
3
function given by g ( x ) = e .
a. Write an equation for the line tangent to the graph of g at x
(1, -4)
= 1.
Graph of f '
b. For –1.2 < x < 3.2, find all values of x at which g has a local
maximum. Justify your answer.
2
c. The second derivative of g is g " ( x ) = e f ( x ) ( f ' ( x ) ) + f " ( x )  . 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].
AP Calculus Review - 4
Multiple choice. With calculators.
x2 − a2
1. If a ≠ 0, then lim 4
is
x→a x − a 4
1
1
(B)
(A) 2
a
2a 2
(C)
1
6a 2
(D) 0
(E) nonexistent
dy
= ky , where k is a constant and t is measured in years. If
dt
the population doubles every 10 years, then the value of k is
(A) 0.069
(B) 0.200
(C) 0.301
(D) 3.322
(E) 5.000
2. Population y grows according to the equation
x
f(x)
2
10
5
30
7
40
8
20
3. The function f is continuous on the closed interval [2, 8] and has values that are given in the table above.
Using the subintervals [2, 5], [5, 7] and [7, 8], what is the trapezoidal approximation of
∫ f ( x )dx ?
8
2
(A) 110
(B) 130
(C) 160
(D) 190
(E) 210
4. Which of the following is an equation of the line tangent to the graph of f(x) = x4 + 2x2 at the point
where f '(x) = 1?
(A) y = 8x – 5
(B) y = x + 7
(C) y = x + 0.763
(D) y = x – 0.122 (E) y = x – 2.146
5. Let F(x) be an antiderivative of
(A) 0.048
( ln x )
x
3
. If F(1) = 0, then F(9) =
(B) 0.144
(C) 5.827
(D) 23.308
(E) 1640.250
y
4
O
8
x
6. The base of a solid is the region in the first quadrant bounded by the x-axis, the y-axis, the line x + 2y = 8,
as shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what
is the volume of the solid?
(A) 12.566
(B) 14.661
(C) 16.755
(D) 67.021
(E) 134.041
7. If g is a differentiable function such that g(x) < 0 for all real numbers x and if f '(x) = (x2 – 4)g(x), which of
the following is true?
(A) f has a relative maximum at x = –2 and a relative minimum at x = 2
(B) f has a relative minimum at x = –2 and a relative maximum at x = 2
(C) f has relative minima at x = –2 and x = 2
(D) f has relative maxima at x = –2 and x = 2
(E) It cannot be determined if f has any relative extrema.
8. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a
rate of 3 inches per minute, which of the following must be true about the area A of the triangle?
(A) A is always increasing.
(B) A is always decreasing.
(C) A is decreasing only when b < h.
(D) A is decreasing only when b > h.
(E) A remains constant.
9. Let f be a function that is differentiable on the open interval (1, 10). If f(2) = –5, f(5) = 5 and f(9) = –5,
which of the following must be true?
I. f has at least two zeros.
II. The graph of f has at least one horizontal tangent.
III. For some c, 2 < c < 5, f(c) = 3
(A) None
10. If 0 ≤ k <
(B) I only
π
(A) 1.471
2
(C) I and II only
(D) I and III only
and the area under the curve y = cos x from x = k to x =
(B) 1.414
(E) I, II and III
π
is 0.1, then k =
2
(D) 1.120
(E) 0.436
(C) 1.277
Free Response. No calculators.
t
(seconds)
v(t)
(meters per second)
0
8
20
25
32
40
3
5
−10
−8
−4
7
11. 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 v(t) 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 = 36 seconds. Show the computations that lead to your
answer.
b. Using correct units, explain the meaning of
∫ v ( t ) dt
40
in the context of the problem. Use a trapezoid
20
sum with the subintervals indicated by the data to approximate
∫ v ( t ) dt .
40
20
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 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.
AP Calculus Review - 5
Multiple choice. No calculators.
1. If y = ( x3 + 1) , then
2
( 3x )
2 2
(A)
2.
1
∫e
−4 x
dy
=
dx
(B) 2 ( x 3 + 1)
(C) 2 ( 3 x 2 + 1)
(D) 3 x 2 ( x 3 + 1)
(B) −4e −4
(C) e −4 − 1
(D)
(E) 6 x 2 ( x 3 + 1)
dx =
0
(A)
−e −4
4
1 e −4
−
4 4
(E) 4 − 4e −4
3. For x ≥ 0, the horizontal line y = 2 is an asymptote for the graph of the function f. Which of the following
statements must be true?
(A) f(0) = 2
(B) f(x) ≠ 2 for all x ≥ 0
(C) f(2) is undefined
(E) lim f ( x ) = 2
(D) lim f ( x ) = ∞
x→2
2x + 3
dy
, then
=
3x + 2
dx
12 x + 13
12 x − 13
(A)
(B)
2
2
( 3x + 2 )
( 3x + 2 )
x →∞
4. If y =
5.
∫
π /4
(C)
5
( 3x + 2 )
2
(D)
−5
( 3x + 2 )
(E)
2
2
3
sin xdx =
0
(A) −
2
2
2
2
(B)
(C) −
2
−1
2
2
+1
2
(D) −
2
−1
2
(E)
x3 − 2 x 2 + 3 x − 4
=
x →∞ 4 x 3 − 3 x 2 + 2 x − 1
6. lim
(A) 4
(B) 1
(C)
1
4
7. The graph of f ', the derivative of the function f, is shown
above. Which of the following statements is true about the
function f?
(A) f is decreasing for –1 ≤ x ≤ 1
(B) f is increasing for –2 ≤ x ≤ 0
(C) f is increasing for 1 ≤ x ≤ 2
(D) f has a local minimum at x = 0
(E) f is not differentiable at x = –1 and x = 1
(D) 0
(E) –1
y
x
–2
–1
O
1
2
Graph of f '
8. If f ( x ) = ln ( x + 4 + e −3 x ) , then f '(0) =
(A) −
2
5
(B)
1
5
(C)
1
4
(D)
2
5
(E) nonexistent
9.
∫x
2
cos ( x 3 ) dx =
1
(A) − sin ( x 3 ) + C
3
B)
1
sin ( x 3 ) + C
3
C) −
x3
sin ( x 3 ) + C
3
 x4 
x3
sin   + C
E)
3
 4
x3
(D)
sin ( x 3 ) + C
3
10. The function f has the property that f(x), f '(x) and f "(x) are negative for all real values of x. Which of the
following could be the graph of f?
(A)
(B)
(C)
y
y
y
O
x
O
y
(D)
O
x
(E)
x
O
x
y
O
x
Free response. With calculators.
11. Let R be the region in the first quadrant bounded by the graphs of y = x and y =
x
.
3
a. Find the area of R.
b. Find the volume of the solid generated when R is rotated about the vertical line x = –1.
c. The region R is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are
squares. Find the volume of this solid.
12. For 0 ≤ t ≤ 6, a particle is moving along the x-axis. The particle’s position, x(t), is not explicitly given.
The velocity of the particle is given by v ( t ) = 2sin ( et /4 ) + 1 . The acceleration of the particle is given by
1 t/4
e sin ( et / 4 ) and x(0) = 2.
2
Is the speed of the particle increasing or decreasing at time t = 5.5 ? Give a reason for your answer.
Find the average velocity of the particle for the time period 0 ≤ t ≤ 6.
Find the total distance traveled by the particle from time t = 0 to t = 6.
For 0 ≤ t ≤ 6, the particle changes direction exactly once. Find the position of the particle at that time.
a (t ) =
a.
b.
c.
d.
AP Calculus Review - 6
Multiple choice. No calculators.
1. Using the substitution u = 2x + 1,
(A)
1 1/ 2
udu
2 ∫ −1/ 2
(B)
∫
2
2 x + 1dx is equivalent to
0
1 2
udu
2 ∫0
(C)
1 5
udu
2 ∫1
(D)
∫
2
(E)
udu
0
∫
5
udu
1
2. The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to the
square root of the volume. Which of the following is a differential equation that describes this
relationship?
dV
k
dV
dV
(B) V ( t ) = k V
(C)
(D)
=
(E)
(A) V ( t ) = k t
=k t
=k V
dt
dt
dt
V
y
3. The graph of a function f is shown at right.
At which value of x is f continuous, but not
differentiable?
(A) a
(B) b
(C) c
(D) d
(E) e
O
4. If y = x2sin 2x, then
a
b
c
d
e
x
dy
=
dx
(A) 2x cos 2x
(B) 4x cos 2x
(C) 2x (sin 2x + cos 2x)
(D) 2x (sin 2x – x cos 2x)
(E) 2x (sin 2x + x cos 2x)
2
5. Let f be the function with derivative given by f ' ( x ) = x 2 − . On which of the following intervals is f
x
decreasing?
(E)  3 2, ∞
(A) (–∞, –1] only (B) (–∞, 0)
(C) [–1, 0) only
(D) 0, 3 2 
(
)
6. If the line tangent to the graph of the function f at the point (1, 7) passes through the point (–2, –2), then
f '(1) is
(A) –5
(B) 1
(C) 3
(D) 7
(E) undefined
7. Let f be the function given by f ( x ) = 2 xe x . The graph of f is concave down when
(A) x < –2
(B) x > –2
(C) x < –1
(D) x > –1
x
g'(x)
–4
2
–3
3
–2
0
–1
–3
0
–2
1
–1
2
0
3
3
(E) x < 0
4
2
8. The derivative g' of a function g is continuous and has exactly two zeros. Selected values of g' are given in
the table above. If the domain of g is the set of all real numbers, then g is decreasing on which of the
following intervals?
(A) –2 ≤ x ≤ 2 only (B) –1 ≤ x ≤ 1
(C) x ≥ –2
(D) x ≥ 2 only
(E) x ≤ –2 or x ≥ 2
9. A curve has a slope 2x + 3 at each point (x, y) on the curve. Which of the following is an equation for this
curve if it passes through the point (1, 2)?
(A) y = 5x – 3
(B) y = x2 + 1
(C) y = x2 + 3x
(D) y = x2 + 3x – 2 (E) y = 2x2 + 3x – 3
 x + 2 if x ≤ 3
f ( x) = 
4 x − 7 if x > 3
10. Let f be the function given above. Which of the following statements are true about f?
I. lim f ( x ) exists
II. f is continuous at x = 3
III. f is differentiable at x = 3
x →3
(A) None
(B) I only
11. The second derivative of the function f is
given by f "(x) = x(x – a)(x – b)2. The
graph of f " is shown at right. For what
values of x does the graph of f have a point
of inflection?
(A) 0 and a only
(B) 0 and m only
(C) b and j only
(D) 0, a and b
(E) b, j and k
(C) II only
(D) I and II only
y
a
j
O
(E) I, II, and III
y = f "(x)
k
m
x
b
12. A particle moves along the x-axis so that at time t ≥ 0 its position is given by x(t) = 2t3 – 21t2 + 72t – 53.
At what time t is the particle at rest?
7
7
(A) t = 1 only
(B) t = 3 only
(C) t =
only
(D) t = 3 and t =
(E) t = 3 and t = 4
2
2
13. The graph of f ', the derivative of f, is the line shown in the figure at right. If
f(0) = 5, then f(1) =
(A) 0
(B) 3
(C) 6
(D) 8

d  x
3
 ∫ sin ( t ) dt  =
dx  0

6
y
(E) 11
y = f '(x)
2
14.
(A) − cos ( x 6 )
(D) 2 x sin ( x 3 )
(B) sin ( x 3 )
(E) 2 x sin ( x 6 )
(C) sin ( x 6 )
O
1
15. Let f be the function defined by f(x) = 4x3 – 5x + 3. Which of the following is an equation of the line
tangent to the graph of f at the point where x = –1?
(A) y = 7x – 3
(B) y = 7x + 7
(C) y = 7x + 11
(D) y = –5x – 1
(E) y = –5x – 5
16. What is the slope of the line tangent to the curve 3y2 – 2x2 = 6 – 2xy at the point (3, 2)?
4
7
6
5
(A) 0
(B)
(C)
(D)
(E)
9
9
7
3
x
17. Let f be the function defined by f(x) = x3 + x. If g(x) = f –1(x) and g(2) = 1, what is the value of g'(2)?
1
1
7
(B)
(C)
(D) 4
(E) 13
(A)
13
4
4
18. Let g be a twice-differentiable function with g'(x) > 0 and g"(x) > 0 for all real numbers x, such that
g(4) = 12 and g(5) = 18. Of the following, which is a possible value for g(6)?
(A) 15
(B) 18
(C) 21
(D) 24
(E) 27
Free response. With calculators.
Distance from the
river’s edge (feet)
Depth of water
(feet)
0
8
14
22
24
0
7
8
2
0
19. A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location.
The measurements are taken in a straight line perpendicular to the edge of the river. The data are shown in
the table above. The velocity of the water at Picnic Point, in feet per minute, is modeled by
v ( t ) = 16 + 2sin
(
)
t + 10 for 0 ≤ t ≤ 120 minutes.
a. Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the
area of the cross section of the river at Picnic Point, in square feet. Show the computations that lead to
your answer.
b. The volumetric flow at a location along the river is the product of the cross-sectional area and the
velocity of the water at that location. Use your approximation from part (a) to estimate the average
value of the volumetric flow at Picnic Point, in cubic feet per minute, from t = 0 to t = 120 minutes.
(Problem continued on next page.)
π x 
c. The scientist proposed the function f, given by f ( x ) = 8sin 
 as a model of the depth of the water,
 24 
in feet, at Picnic Point x feet from the river’s edge. Find the area of the cross section of the river at
Picnic Point based on this model.
d. Recall that the volumetric flow is the product of the cross sectional area and the velocity of the water at
a location. To prevent flooding, water must be diverted if the average value of the volumetric flow at
Picnic Point exceeds 2100 cubic feet per minute for a 20-minute period. Using your answer from part
(c), find the average value of the volumetric flow during the time interval 40 ≤ t ≤ 60 minutes. Does
this value indicate that water must be diverted?
AP Calculus Review - 7
Multiple choice. With calculators.
1. A particle moves along the x-axis so that at time t ≥ 0, its velocity is given by v(t) = 3 + 4.1cos(0.9t). What
is the acceleration of the particle at time t = 4?
(A) –2.016
(B) –0.677
(C) 1.633
(D) 1.814
(E) 2.978
2. The radius of a circle is increasing at a constant rate of 0.2 meters per second. What is the rate of increase
in the area of the circle at the instant when the circumference of the circle is 20π meters?
(A) 0.04π m2/sec (B) 0.4π m2/sec
(C) 4π m2/sec
(D) 20π m2/sec
(E) 100π m2/sec
3. The regions A, B and C in the figure above are bounded by the graph
of the function f and the x-axis. If the area of each region is 2, what
is the value of
y
y = f(x)
∫ ( f ( x ) + 1) dx ?
3
−3
(A) –2
(D) 7
B
(B) –1
(E) 12
(C) 4
C
O
–3 A
x
3
4. For which of the following does lim f ( x ) exist?
x →4
y
I.
1
x
O
4
1
(A) I only
(B) II only
II. y
III. y
1
1
O
x
(C) III only
x
O
4
1
4
1
(D) I and II only
(E) I and III only
6. Let f be the function with derivative given by f '(x) = sin(x2 + 1). How many relative extrema does f have
on the interval 2 < x < 4?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
5. The function f is continuous for –2 ≤ x ≤ 1 and differentiable for –2 < x < 1. If f(–2) = –5 and f(1) = 4,
which of the following statements could be false?
(A) There exists c, where –2 < c < 1, such that f(c) = 0.
(B) There exists c, where –2 < c < 1, such that f '(c) = 0.
(C) There exists c, where –2 < c < 1, such that f(c) = 3.
(D) There exists c, where –2 < c < 1, such that f '(c) = 3.
(E) There exists c, where –2 ≤ c ≤ 1, such that f(c) ≥ f(x) for all f on the closed interval –2 ≤ x ≤ 1.
7. The rate of change of the altitude of a hot-air balloon is given by r(t) = t3 – 4t2 + 6 for 0 ≤ t ≤ 8. Which of
the following expressions gives the change in altitude of the balloon during the time the altitude is
decreasing?
(A)
∫
3.514
1.572
r ( t )dt
(B)
∫ r ( t )dt
8
(C)
0
∫
2.667
0
r ( t )dt
(D)
∫
3.514
1.572
r ' ( t )dt
(E)
∫
2.667
0
r ' ( t )dt
Free response. No calculators.
8. The functions f and g are given by f ( x ) = ∫
3x
0
4 + t 2 dt and g ( x ) = f ( sin x ) .
a. Find f '(x) and g'(x).
b. Write an equation for the line tangent to the graph of y = g(x) at x = π.
c. Write, but do not evaluate, an integral expression that represents the maximum value of g on the
interval 0 ≤ x ≤ π. Justify your answer.
9. Let g be a continuous function with g(2) = 5. The
graph of the piecewise-linear function g', the
derivative of g, is shown in the graph at right for
–3 ≤ x ≤ 7.
y
(1, 1)
(7, 1)
(–1, 0)
x
a. Find the x-coordinate of all points of inflection
of the graph of y = g(x) for –3 < x < 7. Justify
your answer.
b. Find the absolute maximum value of g on the
interval –3 ≤ x ≤ 7. Justify your answer.
c. Find the average rate of change of g(x) on the
interval –3 ≤ x ≤ 7.
(4, –2)
(–3, –4)
Graph of g'
d. Find the average rate of change of g'(x) on the interval –3 ≤ x ≤ 7. Does the Mean Value Theorem
applied on the interval –3 ≤ x ≤ 7 guarantee a value of c, for –3 < x 7, such that g"(c) is equal to this
average rate of change? Why or why not?
AP Calculus Review - 8
Multiple choice. With calculators.
1. The velocity, in ft/sec, of a particle moving along the x-axis is given by the function v(t) = et + tet. What is
the average velocity of the particle from time t = 0 to time t = 3?
(A) 20.086 ft/sec
(B) 26.447 ft/sec
(C) 32.809 ft/sec
(D) 40.671 ft/sec (E) 79.342 ft/sec
2. A pizza, heated to a temperature of 350° Fahrenheit (°F), is taken out of an oven and placed in a 75°F room
at time t = 0 minutes. The temperature of the pizza is changing at a rate of −110e −0.4t degrees Fahrenheit
per minute. To the nearest degree, what is the temperature of the pizza at time t = 5 minutes?
(A) 112°F
(B) 119°F
(C) 147°F
(D) 238°F
(E) 335°F
3. If a trapezoidal sum overapproximates
∫
4
0
f ( x ) dx and a right Riemann sum underapproximates
, which of the following could be the graph of y = f(x)?
(A)
(B)
4
3
2
1
O 1 2 3 4
y
4
3
2
1
4
3
2
1
x
(D) y
O 1 2 3 4
x
(E)
y
O 1 2 3 4
4
3
2
1
4
3
2
1
O 1 2 3 4
0
(C)
y
y
∫ f ( x ) dx
4
x
O 1 2 3 4
x
x
4. The base of a solid is the region in the first quadrant bounded by the y-axis, the graph of y = tan–1 x, the
horizontal line y = 3 and the vertical line x = 1. For this solid, each cross section perpendicular to the x-axis
is a square. What is the volume of the solid?
(A) 2.561
(B) 6.612
(C) 8.046
(D) 8.755
(E) 20.773
5. The function f has its first derivative given by f ' ( x ) =
point of the graph of f?
(A) 1.008
(D) –0.278
x
. What is the x-coordinate of the inflection
1 + x + x3
(B) 0.473
(C) 0
(E) The graph of f has no inflection point.
6. A particle moves along the x-axis so that at any time t > 0, its acceleration is given by a(t) = ln(1 + 2t). If
the velocity of the particle is 2 at time t = 1, then the velocity of the particle at time t = 2 is
(A) 0.462
(B) 1.609
(C) 2.555
(D) 2.886
(E) 3.346
7. On the closed interval [2, 4], which of the following could be the graph of a function f with the property
4
1
that
f ( t )dt = 1 ?
∫
4−2 2
(A) y
(B) y
(C) y
4
3
2
1
O 1 2 3 4
4
3
2
1
4
3
2
1
x
O 1 2 3 4
(D) y
O 1 2 3 4
x
(E) y
4
3
2
1
O 1 2 3 4
x
4
3
2
1
x
O 1 2 3 4
x
8. Let f be a differentiable function with f(2) = 3 and f '(2) = –5 and let g be the function defined by
g(x) = xf(x). Which of the following is an equation of the line tangent to the graph of g at the point where
x = 2?
(A) y = 3x
(B) y – 3 = –5(x – 2)
(C) y – 6 = –5(x – 2)
(D) y – 6 = –7(x – 2)
(E) y – 6 = –10(x – 2)
9. For all x in the closed interval [2, 5], the function f has a positive first derivative and a negative second
derivative. Which of the following could be a table of values for f?
(A) x f(x)
(B) x f(x)
(C) x f(x)
(D) x f(x)
(E) x f(x)
2
7
2
7
2 16
2 16
2 16
3
9
3 11
3 12
3 14
3 13
4 12
4 14
4
9
4 11
4 10
5 16
5 16
5
7
5
7
5
7
10. Let g be the function given by g ( x ) = ∫ sin ( t 2 ) dt for –1 ≤ x ≤ 3. On which of the following intervals is g
x
0
decreasing?
(A) –1 ≤ x ≤ 0
(B) 0 ≤ x ≤ 1.772
(D) 1.772 ≤ x ≤ 2.507
(C) 1.253 ≤ x ≤ 2.171
(E) 2.802 ≤ x ≤ 3
Free response. No calculators.
11. Consider the closed curve in the xy-plane given by x2 + 2x + y4 + 4y = 5.
dy − ( x + 1)
a. Show that
=
.
dx 2 ( y 3 + 1)
b. Write an equation for the line tangent to the curve at the point (–2, 1).
c. Find the coordinates of two points on the curve where the line tangent to the curve is vertical.
d. Is it possible for this curve to have a horizontal tangent at points where it intersects the x-axis? Explain
your reasoning.
AP Calculus Review - 9
Multiple choice. No calculators.
( 2 x − 1)( 3 − x )
x →∞ ( x − 1)( x + 3 )
1. lim
(A) –3
2.
1
∫x
2
is
(B) –2
(C) 2
(D) 3
(E) Nonexistent
(B) − ln x 2 + C
(C) x −1 + C
(D) − x −1 + C
(E) −2x −3 + C
dx =
(A) ln x 2 + C
3. If f ( x ) = ( x − 1) ( x 2 + 2 ) , then f ' ( x ) =
3
(A) 6 x ( x 2 + 2 )
2
(D)
4.
(x
2
(B) 6 x ( x − 1) ( x 2 + 2 )
+ 2) ( 7 x2 − 6 x + 2)
2
2
(E)
( x + 2) ( x
−3 ( x − 1) ( x + 2 )
(C)
2
2
2
2
+ 3 x − 1)
2
∫ ( sin ( 2 x ) + cos ( 2 x ) ) dx =
1
1
cos ( 2 x ) + sin ( 2 x ) + C
2
2
(C) 2 cos ( 2 x ) + 2sin ( 2 x ) + C
(A)
(E) −2 cos ( 2 x ) + 2sin ( 2 x ) + C
5x4 + 8x 2
is
x → 0 3 x 4 − 16 x 2
1
(A) −
2
1
1
(B) − cos ( 2 x ) + sin ( 2 x ) + C
2
2
(D) 2 cos ( 2 x ) − 2sin ( 2 x ) + C
5. lim
(B) 0
(C) 1
(D)
5
3
(E) Nonexistent
 x2 − 4
if x ≠ 2

f ( x) =  x − 2
 1
if x = 2

6. Let f be the function defined above. Which of the following statements about f are true?
I. f has a limit at x = 2
II. f is continuous at x = 2
III. f is differentiable at x = 2
(A) I only
(B) II only
(C) III only
(D) I and II only (E) I, II and III
7. A particle moves along the x-axis with velocity given by v(t) = 3t2 + 6t for time t ≥ 0. If the particle is at
position x = 2 at time t = 0, what is the position of the particle at time t = 1 ?
(A) 4
(B) 6
(C) 9
(D) 11
(E) 12
π 
8. If f ( x ) = cos ( 3x ) , then f '   =
9
3 3
3
(A)
B)
2
2
3
2
C) −
(D) −
3
2
E) −
3 3
2
9. The graph of the piecewise linear function f is shown in the figure at
right. If g ( x ) = ∫
x
−2
f ( t ) dt , which of the following values is
1
greatest?
(A) g(–3)
(B) g(–2)
(D) g(1)
(C) g(0)
(E) g(2)
O
1
Graph of f
10. The graph of the function f is shown at right for 0 ≤ x ≤ 3.
Of the following, which has the least value?
(A)
y
∫ f ( x )dx
3
1
(B) Left Riemann sum approximation of
∫ f ( x )dx
3
with
1
1
four subintervals of equal length.
(C) Right Riemann sum approximation of
∫ f ( x )dx
3
with
1
O
x
1
Graph of f
four subintervals of equal length.
(D) Midpoint Riemann sum approximation of
∫ f ( x )dx
3
with four subintervals of equal length.
1
(E) Trapezoidal sum approximation of
∫ f ( x )dx
3
1
with four subintervals of equal length.
Free response. With calculators.
y
2
(4, 2)
y = f(x)
y = g(x)
1
x
O
1
2
3
4
5
6
11. The functions f and g are given by f ( x ) = x and g ( x ) = 6 − x . Let R be the region bounded by the xaxis and the graphs of f and g, as shown in the figure above,
a. Find the area of R.
b. The region R is the base of a solid. For each y, where 0 ≤ y ≤ 2, the cross section of the solid taken
perpendicular to the y-axis is a rectangle whose base lies in R and whose height is 2y. Write, but do not
evaluate, an integral expression that gives the volume of the solid.
c. There is a point P on the graph of f at which the line tangent to the graph of f is perpendicular to the
graph of g. Find the coordinates of P.
dy
= x 2 ( y − 1) .
dx
a. On the axes at right, sketch a slope field for the given
differential equation at the twelve points indicated.
b. While the slope field in part (a) is only drawn at twelve
points, it is defined at every point in the xy-plane. Describe
all points in the xy-plane for which the slopes are positive.
c. Find the particular solution y = f(x) to the given differential
equation with the initial condition f(0) = 3.
12. Consider the differential equation
AP Calculus Review - 10
Multiple choice. No calculators.
1. If f ( x ) = e(
(A) 2e(
2/ x )
2/ x )
, then f '(x) =
ln x
2. If f(x) = x2 + 2x, then
(A)
2 ln x + 2
x
(B) e(
2/ x )
( −2/ x )
2
(C) e
(D) −
2 ( 2/ x )
e
x2
(E) −2 x 2 e(
d
( f ( ln x ) ) =
dx
(B) 2 x ln x + 2 x
(C) 2ln x + 2
(D) 2 ln x +
2
x
(E)
2x + 2
x
2/ x )
y
x
3. The graph of a function f is shown above. Which of the following could be the graph of f ', the derivative
of f ?
y
y
y
(A)
(B)
(C)
x
(D)
x
y
(E)
x
y
x
x
f "( x )
0
5
x
1
0
2
–7
3
4
4. The polynomial function f has selected values of its second derivative f " given in the table above. Which
of the following statements must be true?
(A) f is increasing on the interval (0, 2)
(B) f is decreasing on the interval (0, 2)
(C) f has a local maximum at x = 1
(D) The graph of f has a point of inflection at x = 1
(E) The graph of f changes concavity in the interval (0, 2)
5.
∫x
2
(A)
x
dx =
−4
−1
4 ( x2 − 4)
2
+C
(B)
(D) 2 ln | x 2 − 4 | +C
1
2 ( x − 4)
2
+C
(C)
(E)
1
x
arctan   + C
2
2
1
ln | x 2 − 4 | +C
2
6. If sin(xy) = x, then
(A)
1
cos ( xy )
dy
=
dx
(B)
1
x cos ( xy )
(C)
1 − cos ( xy )
cos ( xy )
(D)
1 − y cos ( xy )
x cos ( xy )
(E)
y (1 − cos ( xy ) )
x
4
3
2
1
O
–1
–2
1
2
3 4
5 6
7. The graph of the function f shown above has horizontal asymptotes at x = 2 and x = 5. Let g be the
function defined by g ( x ) = ∫ f ( t ) dt . For what values of x does the graph of g have a point of inflection?
x
0
(A) 2 only
(B) 4 only
(C) 2 and 5 only
(D) 2, 4 and 5
(E) 0, 4 and 6
8. In the xy-plane, the line x + y = k, where k is a constant, is tangent to the graph of y = x2 + 3x + 1. What is
the value of k?
(A) –3
(B) –2
(C) –1
(D) 0
(E) 1
5 + 2x
in the xy-plane?
1 − 2x
(B) y = 0 only
(C) y = 5 only
(D) y = –1 and y = 0
(E) y = –1 and y = 5
9. What are all horizontal asymptotes of the graph of y =
(A) y = –1 only
10. Let f be a function with a second derivative given by f "(x) = x2(x – 3)(x – 6). What are the x-coordinates of
the points of inflection of the graph of f?
(A) 0 only
(B) 3 only
(C) 0 and 6 only
(D) 3 and 6 only (E) 0, 3, and 6
11. A rumor spreads among a population of N people at a rate proportional to the product of the number of
people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the
number of people who have heard the rumor, which of the following differential equations could be used to
model this situation with respect to time t, where k is a positive constant?
dp
dp
dp
(A)
(B)
(C)
= kp
= kp ( N − p )
= kp ( p − N )
dt
dt
dt
dp
dp
(D)
(E)
= kt ( N − t )
= kt ( t − N )
dt
dt
12. The function f is twice differentiable with f ( 2 ) = 1 , f ' ( 2 ) = 4 and f " ( 2 ) = 3 . What is the value of the
approximation of f (1.9 ) using the line tangent to the graph of f at x = 2?
(A) 0.4
(B) 0.6
(C) 0.7
(D) 1.3
(E) 1.4
x(t)
2
1
O
–1
t
1
2
3
4
5
6
–2
13. A particle moves along a straight line. The graph of the particle’s position x ( t ) at time t is shown above
for 0 < t < 6. The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2. For
what values of t is the velocity of the particle increasing?
(A) 0 < t < 2
(B) 1 < t < 5
(C) 2 < t < 6
(D) 3 < t < 5 only
(E) 1 < t < 2 and 5 < t < 6
14. Which of the following is the solution to the differential equation
dy x 2
=
with the initial condition
dx y
y ( 3) = −2 ?
(A) y = 2e−9+ x
3
(B) y = −2e−9+ x
3
/3
(D) y =
2 x3
− 14
3
(C) y =
/3
(E) y = −
2 x3
3
2 x3
− 14
3
cx + d if x ≤ 2
f ( x) =  2
 x − cx if x > 2
15. Let f be the function defined above, where c and d are constants. If f is differentiable at x = 2, what is the
value of c + d ?
(A) –4
(B) –2
(C) 0
(D) 2
(E) 4
16. Let f be a differentiable function such that f ( 3) = 15 , f ( 6 ) = 3 , f ' ( 3) = 8 and f ' ( 6 ) = −2 . The function
g is differentiable and g ( x ) = f −1 ( x ) for all x. What is the value of g ' ( 3) ?
1
1
1
1
(B) −
(C)
(D)
2
8
6
3
(E) The value of g ' ( 3) cannot be determined from the information given.
(A) −
17. What is the slope of the line tangent to the curve y = arctan(4x) at the point at which x =
(A) 2
(B)
1
2
(C) 0
(D) −
1
2
1
?
4
(E) –2
y
5
–5
5
x
–5
18. Shown above is a slope field for which of the following differential equations?
dy
dy
dy
dy
(A)
(B)
(C)
(D)
= xy
= xy − y
= xy + y
= xy + x
dx
dx
dx
dx
(E)
dy
2
= ( x + 1)
dx
Free response. With calculators.
19. A cylindrical can of radius 10 millimeters is used to measure rainfall in Stormville. The can is initially
empty, and rain enters the can during a 60-day period. The height of water in the can is modeled by the
function S, where S(t) is measured in millimeters and t is measured in days for 0 ≤ t ≤ 60. The rate at
which the height of the water is rising in the can is given by S ' ( t ) = 2sin ( 0.03t ) + 1.5
a. According to the model, what is the height of the water in the can at the end of the 60-day period?
b. According to the model, what is the average rate of change in the height of the water in the can over the
60-day period? Show the computations that lead to your answer. Indicate units of measure.
c. Assuming no evaporation occurs, at what rate is the volume of water in the can changing at time t = 7?
Indicate units of measure.
d. During the same 60-day period, rain on Monsoon Mountain accumulates is a can identical to the one in
Stormville. The height of the water in the can on Monsoon Mountain is modeled by the function M,
1
where M ( t ) =
( 3t 3 − 30t 2 + 330t ) . The height M(t) is measured in millimeters and t is measured in
400
days for 0 ≤ t ≤ 60. Let D ( t ) = M ' ( t ) − S ' ( t ) . Apply the Intermediate Value Theorem to the function
D on 0 ≤ t ≤ 60 to justify that there exists a time t, 0 < t < 60 at which the heights of the water in the
two cans are changing at the same rate.
AP Calculus Review - 11
Multiple choice. With calculators.
y
2
1
–2
x
–1 O
–1
1
2
3
4
5
1. The graph of f ' , the derivative of f, is shown above for –2 ≤ x ≤ 5. On what intervals is f increasing?
(A) [–2, 1] only
(B) [–2, 3]
(C) [3, 5] only
(D) [0, 1.5] and [3, 5]
(E) [–2, –1], [1, 2] and [4, 5]
y
O
x
1
2
3
4
2. The figure above shows the graph of a function f with domain 0 ≤ x ≤ 4. Which of the following statements
are true?
II. lim+ f ( x ) exists
III. lim f ( x ) exists
I. lim− f ( x ) exists
x →2
(A) I only
x →2
x →2
(B) II only
(C) I and II only
(D) I and III only
(E) I, II and III
3. The first derivative of the function f is defined by f ' ( x ) = sin ( x 3 − x ) for 0 ≤ x ≤ 2. On what intervals is f
increasing?
(A) 1 ≤ x ≤ 1.445 only
(C) 1.445 ≤ x ≤ 1.875
(E) 0 ≤ x ≤ 1 and 1.691 ≤ x ≤ 2
4. If
∫
2
−5
f ( x ) dx = −17 and
(A) –21
(B) 1 ≤ x ≤ 1.691
(D) 0.577 ≤ x ≤ 1.445 and 1.875 ≤ x ≤ 2
∫ f ( x ) dx = −4 , what is the value of ∫
2
5
5
−5
(B) –13
(C) 0
f ( x ) dx ?
(D) 13
(E) 21
5. The derivative of the function f is given by f ' ( x ) = x 2 cos ( x 2 ) . How many points of inflection does the
graph of f have on the open interval (–2, 2) ?
(A) One
(B) Two
(C) Three
(D) Four
6. A particle moves along a straight line with velocity given by v ( t ) = 7 − (1.01)
acceleration of the particle at time t = 3 ?
(A) -0.914
(B) 0.055
(C) 5.486
(D) 6.086
(E) Five
−t 2
at time t ≥ 0. What is the
(E) 18.087
7. If G ( x ) is an antiderivative f ( x ) and G ( 2 ) = −7 , then G ( 4 ) =
(A) f '(4)
(B) –7 + f '(4)
∫ f ( t ) dt
4
(C)
2
(D)
∫ ( −7 + f ( t ) ) dt
(E) −7 + ∫ f ( t ) dt
4
4
2
2
Free response. No calculators.
8. For 0 ≤t ≤ 12, a particle moves along the x-axis. The velocity of the particle at time t is given by
π 
v ( t ) = 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.
(−2, 3)
(−4, 1)
(1, 0)
O
(3, −1)
Graph of f
9. 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 ) = ∫ f ( t ) dt .
x
1
a. Find the values of g(2) 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
those 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 at which the graph of g has a point of inflection. Explain your
reasoning.
AP Calculus Review - 12
Multiple choice. With calculators.
1. What is the area enclosed by the curves y = x3 – 8x2 + 18x – 5 and y = x + 5?
(A) 10.667
(B) 11.833
(C) 14.583
(D) 21.333
3
2
1
(E) 32
y
–3 –2 –1 O
−1
–2
1
2
3 4 5
x
2. The graph of the derivative of a function f is shown in the figure above. The graph has horizontal tangent
lines at x = –1, x = 1 and x = 3. At which of the following values of x does f have a relative maximum?
(A) –2 only
(B) 1 only
(C) 4 only
(D) –1 and 3 only (E) –2, 1 and 4
x
f ( x)
–4
0.75
–3
−1.5
–2
–2.25
–1
–1.5
–3
–1.5
0
1.5
f '( x)
3. The table above gives values of a function f and it derivative at selected values of x. If f ' is continuous on
the interval [–4, –1], what is the value of
(A) –4.5
(B) –2.25
∫
−1
−4
f ' ( x ) dx ?
(C) 0
(D) 2.25
(E) 4.5
4. An object traveling in a straight line has position x ( t ) at time t. If the initial position is x ( 0 ) = 2 and the
velocity of the object is v ( t ) = 3 1 + t 2 , what is the position of the object at time t = 3?
(A) 0.431
(B) 2.154
(C) 4.512
(D) 6.512
(E) 17.408
5. The radius of a sphere is decreasing at a rate of 2 centimeters per second. At the instant when the radius of
the sphere is 3 centimeters, what is the rate of change, in square centimeters per second, of the surface area
of the sphere? (The surface area S of a sphere with radius r is S = 4πr2)
(A) –108π
(B) –72π
(C) –48π
(D) –24π
(E) –16π
6. The function f is continuous for –2 ≤ x ≤ 2 and f ( −2 ) = f ( 2 ) = 0 . If there is no c, where –2 < c < 2, for
which f ' ( c ) = 0 , which of the following statements must be true?
(A) For –2 < k < 2, f ' ( k ) > 0 .
(B) For –2 < k < 2, f ' ( k ) < 0 .
(C) For –2 < k < 2, f ' ( k ) exists.
(D) For –2 < k < 2, f ' ( k ) exists but f ' is not continuous.
(E) For some k, where –2 < k < 2, f ' ( k ) does not exist.
t
v (t )
0
–1
1
2
2
3
3
0
4
–4
7. The table gives selected values of the velocity, v ( t ) , of a particle moving along the x-axis. At time t = 0,
the particle is at the origin. Which of the following could be the graph of the position, x ( t ) , of the
particle for 0 ≤ t ≤ 4 ?
x(t)
(A)
x(t)
(B)
1
O
1
O
t
1
t
1
x(t)
(D)
1
O
1
O
t
1
O
t
1
x(t)
(E)
1
x(t)
(C)
1
t
cos x
on the closed interval [–1, 3]?
x + x+2
(B) 0.090
(C) 0.183
(D) 0.244
8. What is the average value of y =
(A) –0.085
2
(E) 0.732
9. The function of is continuous on the closed interval [2, 4] and twice differentiable on the open interval
(2, 4). If f ' ( 3) = 2 and f " ( x ) < 0 on the open interval (2, 4), which of the following could be a table of
values for f ?
(A)
(B)
(C)
(D)
(E)
x f ( x)
x f ( x)
x f ( x)
x f ( x)
x f ( x)
2.5
5
6.5
2
3
4
2.5
5
7
2
3
4
3
5
6.5
2
3
4
3
5
7
10. A city located beside a river has a rectangular boundary as shown in the figure at
right. The population density of the city at any point along a strip x miles from
the river’s edge is f ( x ) persons per square mile. Which of the following
expressions gives the population of the city?
(A)
∫ f ( x ) dx
(B) 7 ∫ f ( x ) dx
4
4
0
0
(D)
∫ f ( x ) dx
7
0
(C) 28∫ f ( x ) dx
4
0
(E) 4 ∫ f ( x ) dx
7
0
2
3
4
3.5
5
7.5
4 miles
River
2
3
4
City
7 miles
Free response. No calculators.
11. 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.
1
c. The function h is defined by h ( x ) = g ( x ) − x 2 . Find the x-coordinate of each critical point of h,
2
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.
AP Calculus Review - 13
Multiple choice. No calculators
1.
∫ ( 4x
2
1
3
− 6 x ) dx =
(A) 2
(B) 4
(C) 6
(D) 36
(E) 42
2. If f ( x ) = x 2 x − 3 then f '(x) =
3x − 3
(A)
3. If
2x − 3
(B)
x
2x − 3
1
(C)
2x − 3
(D)
−x + 3
2x − 3
(E)
5x − 6
2 2x − 3
∫ f ( x ) dx = a + 2b , the ∫ ( f ( x ) + 5 ) dx =
b
b
a
a
(A) a + 2b + 5
(B) 5b – 5a
(C) 7b – 4a
(D) 7b – 5a
(E) 7b – 6a
1 2t
4.
e dt =
2∫
(A) e − t + C
5.
(B) e
d
cos 2 ( x 3 ) =
dx
(A) 6 x 2 sin ( x 3 ) cos ( x 3 )
−
t
2
+C
t
(C) e 2 + C
(B) 6 x 2 cos ( x 3 )
(D) −6 x 2 sin ( x 3 ) cos ( x 3 )
t
(D) 2e 2 + C
(E) et + C
(C) sin 2 ( x 3 )
(E) −2sin ( x 3 ) cos ( x 3 )
Problems 6 and 7 refer to the following situation.
A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 ≤ t ≤ 8, is given by
the function whose graph is shown at right.
6. At what value of t does the bug change direction?
(A) 2
(B) 4
(C) 6
(D) 7
(E) 8
7. What is the total distance the bug traveled from t = 0 to t = 8?
(A) 14
(B) 13
(C) 11
(D) 8
(E) 6
1
8. If f ( x ) = − x 3 + x + , then f '(–1)=
x
(A) 3
(B) 1
(D) –3
(E) –5
(C) –1
9. The graph of y = 3 x 4 − 16 x 3 + 24 x 2 + 48 is concave down for
(A) x < 0
(B) x > 0
(C) x < –2 ∨ x > −
2
2
2
(D) x <
∨ x > 2 (E)
<x<2
3
3
3
Free Response. With calculators.
t (minutes)
H(t) (degrees Celsius)
0
66
2
60
5
52
9
44
10
43
10. As a pot of tea cools, the temperature of the tea is modeled by a differentiable function H for 0 ≤ t ≤ 10
where time t is measured in minutes and temperature H(t) is measured in degrees Celsius. Values of H(t)
at selected values of time t are shown in the table above.
a. Use the data in the table to approximate the rate at which the temperature of the tea is changing at time
t = 3.5. Show the computations that lead to your answer.
1 10
b. Using correct units, explain the meaning of
H ( t ) dt in the context of this problem. Use a
10 ∫ 0
1 10
trapezoidal sum with the four subintervals indicated by the table to estimate
H ( t ) dt .
10 ∫ 0
c. Evaluate
∫
10
0
H ' ( t ) dt . Using correct units, explain the meaning of the expression in the context of this
problem.
d. At time t = 0, biscuits with temperature 100°C were removed from an oven. The temperature of the
biscuits at time t is modeled by a differentiable function B for which it is known that
B ' ( t ) = −13.84e−0.173t . Using the given models, at time t = 10, how much cooler are the biscuits than
the tea?
AP Calculus Review - 14
Multiple choice. No calculators
1. An equation of the line tangent to the graph of y = cos ( 2 x ) at x =
π

(A) y − 1 = −  x − 
4

π

(B) y − 1 = −2  x − 
4

π

(D) y = −  x − 
4

π
4
is
π

(C) y = 2  x − 
4

π

(E) y = −2  x − 
4

1 2
x is the tangent line parallel to the line 2x – 4y = 3 ?
2
1
1 1

 1
(B)  , 
(C)  1, − 
(D)  1, 
(E) (2, 2)
4
 2 8

 2
3. At what point on the graph of y =
1 1
(A)  , − 
2 2
4. Let f be a function defined for all real numbers x. If f ' ( x ) =
(B) (–∞, ∞)
(A) (–∞, 2)
(C) (–2, 4)
4 − x2
, then f is decreasing on the interval
x−2
(D) (–2, ∞)
(E) (2, ∞)
y
y = f '(x)
2
O
2
x
2. The graph of the derivative of f is shown in the figure above. Which of the following could be the
graph of f?
y
y
(A)
(B)
–2
O
2
x
–2
O
y
2
x
y
(C)
(D)
–2
O
2
x
y
(E)
–2
O
2
x
–2
O
2
x
5. Let f be a differentiable function such that f(3) = 2 and f '(3) = 5. If the tangent line to the graph of f at x =
3 is used to find an approximation to a zero of f, that approximation is
(A) 0.4
(B) 0.5
(C) 2.6
(D) 3.4
(E) 5.5
y
3
2
1
O
a
x
b
6. The graph of the function f is shown in the figure above. Which of the following statements about f is true?
(A) lim f ( x ) = lim f ( x )
(B) lim f ( x ) = 2
(C) lim f ( x ) = 2
x →a
x →b
x→a
(D) lim f ( x ) = 1
x →b
x →b
(E) lim f ( x ) does not exist
x→a
7. The area of the region enclosed by the graph of y = x2 + 1 and the line y = 5 is
14
16
28
32
(A)
(B)
(C)
(D)
3
3
3
3
d2y
8. If x + y = 25, what is the value of
at the point (4, 3)?
dx 2
25
7
7
(B) −
(C)
(A) −
27
27
27
2
9.
∫
π
4
0
(E) 8π
2
(D)
3
4
(E)
25
27
e tan x
dx is
cos 2 x
(A) 0
(B) 1
(C) e – 1
(D) e
(E) e + 1
10. If f ( x ) = ln x 2 − 1 , then f '(x) =
(A)
2x
x −1
2
(B)
2x
x −1
2
(C)
2x
x −1
2
11. The average value of cos x on the interval [–3, 5] is
sin 5 − sin 3
sin 5 − sin 3
sin 3 − sin 5
(A)
(B)
(C)
8
2
2
(D)
2x
x −1
(E)
1
x −1
(D)
sin 3 + sin 5
2
(E)
sin 3 + sin 5
8
2
2
x
is
x →1 ln x
12. lim
(A) 0
(B)
1
e
(C) 1
(D) e
(E) nonexistent
13. What are all values of x for which the function f defined by f ( x ) = ( x 2 − 3) e − x is increasing?
(A) There are no such values of x.
(D) –1 < x < 3
(B) x < –1 and x > 3
(C) –3 < x < 1
(E) All values of x.
14. If the region enclosed by the y-axis, the line y = 2 and the curve y = x is revolved about the y-axis, the
volume of the solid generated is
32π
16π
16π
8π
(A)
(B)
(C)
(D)
(E) π
5
3
5
3
15. The expression
(A)
∫
1
0
x
dx
50
1  1
2
3
50 
+
+
+ ... +

 is a Riemann sum approximation for
50  50
50
50
50 
(B)
∫
1
(C)
xdx
0
1 1 x
dx
50 ∫ 0 50
(D)
1 1
xdx
50 ∫ 0
(E)
1 50
xdx
50 ∫ 0
Free Response. With calculators.
16. Ben rides a bicycle back and forth along a straight east-west track. The twice differentiable function B
models Ben’s position on the track, measured in meters from the western end of the track, at time t,
measured in seconds from the start of the ride. The table above gives values of B(t) and Ben’s velocity,
v(t), measured in meters per second, at selected time t.
a. Use the data in the table to approximate Ben’s acceleration at time t = 5 seconds. Indicate units of
measure.
b. Using correct units, interpret the meaning of
∫
60
0
v ( t ) dt in the context of this problem. Approximate
c. For 40 ≤ t ≤ 60, must there be a time t when Ben’s velocity is 2 meters per second? Justify your
answer.
d. A light is directly above the western end of the track. Ben rides so that at time t, the distance L(t)
between Ben and the light satisfies ( L ( t ) ) = 122 + ( B ( t ) ) . At what rate is the distance between Ben
and the light changing at time t = 40?
2
2
AP Calculus Review - 15
Multiple choice. With calculators.
e2 x
1. If f ( x ) =
, then f '(x) =
2x
e 2 x (1 − 2 x )
(A) 1
(B)
2x2
(C) e 2 x
(D)
e 2 x ( 2 x + 1)
e 2 x ( 2 x − 1)
(E)
x2
2x2
2. The graph of the function y = x3 + 6x2 + 7x – 2cos x changes concavity at x =
(A) –1.58
(B) –1.63
(C) –1.67
(D) –1.89
(E) –2.33
5. Let f be the function given by f ( x ) = 2e 4 x . For what value of x is the slope of the tangent line to the
2
graph of f at (x, f(x)) equal to 3?
(A) 0.168
(B) 0.276
(C) 0.318
(D) 0.342
(E) 0.551
3. The graph of f is shown in the figure at right. If
∫ f ( x )dx = 2.3 and
y
3
1
F ' ( x ) = f ( x ) , then F(3) – F(0) =
(A) 0.3
(B) 1.3
(D) 4.3
(E) 5.3
4. Let f be the function such that lim
f ( 2 + h ) − f ( 2)
h →0
3
(C) 3.3
h
2
1
O
= 5 . Which of the
following must be true?
I. f is continuous at x = 2.
II. f is differentiable at x = 2/
III. The derivative of f is continuous at x = 2
(A) I only
(B) II only
(C) I and II only
(D) I and III only
1
x
2 3 4
(E) II and III only
6. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the
crossing and watches a train moving 60 meters per second. At how many meters per second is the train
moving away from the observer 4 seconds after it passes through the crossing?
(A) 57.60
(B) 57.88
(C) 59.20
(D) 60.00
(E) 67.40
Free Response. No calculators.
7. Let R be the region in the first quadrant enclosed by the graphs of y = 2x and y = x2, as
shown in the figure at right.
a. Find the area of R.
b. The region R is the base of a solid. For this solid, at each x the cross section
π 
perpendicular to the x-axis has area A ( x ) = sin  x  . Find the volume of the solid.
2 
c. Another solid has the same base R. For this solid, the cross sections perpendicular to
the y-axis are squares. Write, but do not evaluate, an integral expression for the
volume of the solid.
x
2
3
5
8
13
f ( x)
1
4
−2
3
6
y
(2, 4)
4
.
3
2
1
R
O
1
2 x
8. Let f be a function that is twice differentiable for all real numbers. The table above gives values of f for
selected points in the closed interval 2 ≤ x ≤ 13.
a. Estimate f '(4). Show the work that leads to your answer.
b. Evaluate
∫ ( 3 − 5 f ' ( x ) ) dx .
13
Show the work that leads to your answer.
2
c. Use a left Riemann sum with subintervals indicated by the data in the table to approximate
∫
13
2
f ( x ) dx .
Show the work that leads to your answer.
d. Suppose f ' ( 5) = 3 and f " ( x ) < 0 for all x in the closed interval 5 ≤ x ≤ 8. Use the line tangent to the
graph of f at x = 5 to show that f ( 7 ) ≤ 4. Use the secant line for the graph of f on 5 ≤ x ≤ 8 to show
that f ( 7 ) ≥
4
.
3
AP Calculus Review - 16
Multiple choice. With calculators.
1. If y = 2x – 8, what is the minimum value of the product xy?
(A) –16
(B) –8
(C) –4
(D) 0
(E) 2
2. What is the area of the region in the first quadrant enclosed by the graphs of y = cos x, y = x and the y-axis.
(A) 0.127
(B) 0.385
(C) 0.400
(D) 0.600
(E) 0.947
3. The base of a solid S is the region enclosed by the graph of y = ln x , the line x = e and the x-axis. If cross
sections of S perpendicular to the x-axis are squares, then the volume of S is
1
2
1
(A)
(B)
(C) 1
(D) 2
(E) ( e3 − 1)
2
3
3
4. If the derivative of f is given by f ' ( x ) = e x − 3x 2 , at which of the following values of x does f have a
relative maximum value?
(A) –0.46
(B) 0.20
(C) 0.91
(D) 0.95
(E) 3.73
5. Let f ( x ) = x . If the rate of change of f at x = c is twice its rate of change at x = 1, then c =
(A)
1
4
(B) 1
(C) 4
(D)
1
2
(E)
1
2 2
6. At time t ≥ 0, the acceleration of a particle moving on the x-axis is a(t) = t + sin t. At t = 0, the velocity of
the particle is –2. For what value of t will the velocity of the particle be 0?
(A) 1.02
(B) 1.48
(C) 1.85
(D) 2.81
(E) 3.14
x
f(x)
0
3
0.5
3
1.0
5
1.5
8
2.0
13
7. A table of values for a continuous function f is shown above. If four equal subintervals of [0, 2] are used,
which of the following is the trapezoidal approximation of
∫ f ( x )dx ?
2
0
(A) 8
(B) 12
(C) 16
(D) 24
8. Which of the following are antiderivatives of f(x) = sin x cos x?
sin 2 x
cos 2 x
cos 2 x
I. F ( x ) =
II. F ( x ) =
III. F ( x ) = −
2
2
4
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) 32
(E) II and III only
9. Let f ( x ) = ∫ h ( t ) dt , where h has the graph shown at right. Which of the
x
y
a
following could be the graph of f?
a O
y
(A)
a O
y
(B)
c x
b
a O
y
y
(D)
a O
cx
cx
(C)
b
c x
a O
b
cx
y
(E)
b
b
a O
b
cx
y
Free Response. No calculators.
10. The derivative of a function f is defined by
g ( x)
for − 4 ≤ x ≤ 0
f '( x) =  −x/3
.
− 3 for 0 < x ≤ 4
 5e
The graph of the continuous function f ', shown in the figure at right,
5
has x-intercepts at x = –2 and x = 3ln   . The graph of g on
3
–4 ≤ x ≤ 0 is a semicircle, and f(0) = 5.
O
Graph of f '
a. For –4 ≤ x ≤ 4, find all the values of x at which the graph of f has a point of inflection. Justify your
answer.
b. Find f(–4) and f(4).
c. For –4 ≤ x ≤ 4, find the value of x at which f has an absolute maximum. Justify your answer.
x
Answers
HW - 1
1. (D)
8. (E)
2. (B)
3. (C)
9. (D)
10. (D)
1 t
11. a. R(t) = 6 +
3 + sin ( x 2 ) dx
∫
16 0
R(3) = 6.61084766
(
2
3
0
6. (A)
7. (E)
dA
= 2π R ( t ) R '(t )
dt
R '(3) = 0.213257405 so
∫
5. (E)
)
b. A ( t ) = π ( R ( t ) ) →
c.
4. (B)
dA
dt
= 8.858 cm2/year
t =3
A ' ( t )dt = A(3) – A(0) = (6.61084766)2 π– 36π = 24.201 cm2. This is the increase in cross sectional
area from t = 0 to t = 3 years.
12. a. d =35 +
∫(
5
)
t + cos t − 3 dt = 26.495 m.
0
b. At t = 4 hours, the rate of chance of the distance between the road and the water’s edge was increasing
at a rate of 1.007 meters per hour per hour.
c. We want f(t) to be a minimum. f '(t) = 0 at t = 0.66186548 =A and at t = 2.8403832 = B. f ' changes
from negative to positive at t = B so this and t = 0 are candidates for the minimum.
f(0) = –2 and f(B) = –2.2696 so the distance between the road and the water’s edge is decreasing most
rapidly at t = 2.840 hours.
d. We need to add 35 – 26.495 = 8.505 meters of beach.
∫ g ( p ) dp = 8.505
t
0
HW - 2
1. (A)
8. (B)
15. (A)
2. (E)
3. (B)
4. (C)
5. (D)
6. (E)
7. (D)
9. (C)
10. (A)
11. (B)
12. (C)
13. (D)
14. (D)
16. (A)
17. (A)
18. (E)
 f ( x ) − f (0) 
19. a. lim− 
 > 0 since f is increasing for x < 0
x →0
x−0


 f ( x ) − f ( 0) 
lim+ 
 < 0 since f '(3) = 0 and f "(x) > 0 on (0, 3) means f '(x) < 0 on (0, 3).
x →0
x
−
0


Thus f '(0) DNE and f is not differentiable there.
f (6) − f ( a )
b. We need
= 0 or f(a) = f(6) = 1. This happens for two values of a in the given interval.
6−a
f ( 6) − f ( a )
1
for a > 0 (MVT does not apply for a < 0). This happens at a = 3.
6−a
3
d. g"(x) = f '(x) > 0 on (–4 to 0) and (3, 6). g is concave up on [–4, 0] and [3, 6] because f ' is increasing
on those intervals
c. We need
=
HW - 3
1. (A)
2. (C)
3. (B)
4. (B)
5. (A)
6. (D)
7. (E)
4
2
x
x2 
2
4

8. a. A = ∫  x − dx =  x3/ 2 −  = ( 8 ) − 4 =
0
2
4
3
3
3
4
0
4
2
4
 x 2 2 5/ 2 x3 
x
x2 
64 64 8

3/ 2
=
b. V = ∫  x −  dx = ∫  x − x + dx =  − x +  = 8 − +
0
0
2
4
12 
5 12 15
 2 5
0
4
4
c. V = π ∫  2 − x
0

(
9. a. g(1) = e
f (1)
)
2
x

−2− 
2

e2. g'(1) = e
f (1)
2

 dx

f ' (1) = –4e2.
Tangent line equation: y – e2 = –4e2(x – e2)
b. g ' ( x ) = e ( ) f ' ( x ) . e ( ) > 0, g' can change from positive to negative only where f ' does so. This is at
x = –1, only.
c. Since f ' is decreasing at x = 1, we have f "(–1) < 0. Thus g"(–1) = (positive)[0 + (negative)] < 0.
f x
f x
d.
g ' ( 3 ) − g ' (1)
3 −1
=
e
f ( 3)
f ' ( 3) − e
f (1)
f ' (1)
2
HW - 4
1. (B)
8. (D)
2. (A)
3. (C)
9. (E)
10. (D)
v ( 40 ) − v ( 32 ) 11
=
m/s2
11. a. a ( 36 ) ≈
40 − 32
8
b.
∫ v ( t ) dt
40
=
0 − ( − 4e 2 )
4. (D)
2
= 2e 2
5. (C)
6. (C)
7. (B)
is the displacement of the particle in meters from time t = 20 minutes to time t = 40 minutes.
20
1
1
1
∫ v ( t ) dt ≈ 2 ( 5)( −10 − 8) + 2 ( 7 )( −8 − 4 ) + 2 (8)( 7 − 4 ) = −75 meters
40
20
c. Since v(8) > 0 and v(20) < 0, the particle changes direction sometime during the interval (8, 20). It
changes direction again during (32, 40) since v(32) < 0 and v(40) > 0.
d. x ( 8 ) = x ( 0 ) + ∫ v ( t )dt = 7 + ∫ v ( t )dt
Approximating
8
8
0
0
∫ v ( t )dt
8
0
using a left Riemann “sum” with one interval gives
∫ v ( t )dt
8
0
“greater than” is because v is increasing on that interval. Thus x(8) > 7 + 24 = 31 > 30
> 24, where the
HW - 5
1. (E)
8. (A)
2. (D)
3. (E)
9. (BA)
10. (B)
9
x
11. a. A = ∫  x − dx = 4.5
0
3
5. (D)
6. (C)
7. (B)
((3 y + 1) − ( y + 1) )dy = 41.4π = 130.062
V = ∫ ( ( 3 y − y ) )dy = 8.1
b. A = π ∫
c.
4. (D)
3
2
2
2
0
3
2 2
0
12a. v(5.5) = −0.453 and a(5.5) = −1.359. Since v(5.5) and a(5.5) have same sign, speed increasing at t = 5.5.
b.
1 6
v ( t ) dt = 1.949
6 ∫0
c.
∫ v ( t ) dt = 12.573
6
0
d. v(t) changes sign at t = 5.1955223 = b
x ( b ) = x ( 0 ) + ∫ v ( t ) dt = 14.135
b
0
HW - 6
1. (C)
2. (E)
3. (A)
4. (E)
5. (D)
6. (C)
8. (A)
9. (D)
10. (D)
11. (A)
12. (E)
13. (D)
15. (C)
16. (B)
17. (B)
18. (E)
19. a. Let D(x) represent the depth of the water x feet from river’s edge.
7. (A)
14. (E)
A = ∫ D ( x ) dx ≈ 0.5 ( 8 )( 0 + 7 ) + 0.5 ( 6 )( 7 + 8 ) + 0.5 ( 8 )( 8 + 2 ) + 0.5 ( 2 )( 2 + 0 ) = 115 ft2
24
0
b. F =
c.
(
)
1 120
(115) 16 + 2 sin t + 10 dt = 1807.170 ft3/min
∫
120 0
A = ∫ f ( x ) dx = 122.231 ft2
24
0
(
)
1 60
(122.231) 16 + 2 sin t + 10 dt = 2181.913 ft3/min
∫
20 40
Since this value is greater that 2100, it indicates water must be diverted.
d. F =
HW - 7
1. (C)
2. (C)
3. (C)
8. a. f ' ( x ) = 3 4 + x 2
g ' ( x ) = 3 4 + 9sin 2 ( x ) cos x
b. g (π ) = f ( 0 ) = 0
g ' (π ) = −6
Tangent line: y = –6(x – π)
4. (D)
5. (D)
6. (B)
7. (A)
c. In [0, π], g' = 0 at x = π/2 only and g' changes from positive to negative at that point. Thus g has its
only and absolute maximum at x = p/2.
3
π 
g   = f (1) = ∫ 4 + t 2 dt
0
2
9. a. The graph of g has inflection points at x = 1 and x = 4 because g' has relative extrema at those points.
b. g' changes from positive to negative at x = 2 only. That and the endpoints are the candidates for
absolute maximum.
−3
2
2
−3
g ( −3) = 5 + ∫ g ' ( x ) dx = 5 − ∫ g ' ( x ) dx = 5 − ( −4 + 1.5 ) = 7.5
g(2) = 5
g ( 7 ) = 5 + ∫ g ' ( x ) dx = 5 + ( −4 + 0.5 ) = 1.5
7
2
On –3 ≤ x ≤ 7, g has an absolute maximum value of 7.5
c.
∆g g ( 7 ) − g ( −3) 1.5 − 7.5
=
=
= −0.6
∆x
7 − ( −3 )
10
d.
∆ ( g ' ) g ' ( 7 ) − g ' ( −3) 1 − ( −4 )
=
=
= 0.5
∆x
7 − ( −3 )
10
No. The MVT does not apply since g' is not differentiable on this interval.
HW - 8
1. (A)
8. (D)
2. (A)
9. (B)
3. (A)
10. (D)
4. (B)
5. (B)
11. a. 2x + 2 + 4y3y' + 4y' = 0 → 4y'(y3 + y) = –2(x + 1) → y ' =
6. (E)
7. (C)
− ( x + 1)
2 ( y 3 + 1)
1
1
. So the tangent line is y – 1 = ( x + 2 )
4
4
3
c. Tangent line is vertical when y + 1 = 0 → y = –1.
When y = –1, x2 + 2x + 1 – 4 = 5 → x2 + 2x – 8 = 0 → (x – 4)(x + 2) = 0 → x = –2 and = 4.
The points are (–2, –1) and (4, –1).
d. No. The curve has a horizontal tangent only at x = –1. But when x = –1, y ≠ 0.
b. y'(–2, 1) =
HW - 9
1. (B)
8. (C)
2. (D)
9. (D)
11. a. Area = ∫
(=
4
0
3. (D)
10. (C)
4. (B)
4
5. (A)
6
6. (A)

2
x2 
2
36 
16 
xdx + ∫ ( 6 − x )dx = x 3/2 +  6 x −  = 43/ 2 + 36 − −  24 − 
4
3
2 4 3
2 
2
0

6
22
but it is NOT necessary to simplify.)
3
7. (B)
b. V = ∫ 2 y ( 6 − y − y 2 ) dy
2
0
df
1
1
=
= 1 when x = . Coordinates of P are
dx 2 x
4
c.
1 1
 , 
4 2
b. Slopes are positive at points (x, y) where x ≠ 0 and y > 0.
3a.
c.
1
dy = x 2 dx
y −1
1
ln | y − 1|= x 3 + C
3
1 3
x
C 3
| y − 1|= e e
1
x3
y − 1 = Ke 3 , K = ±eC
2 = Ke0 = K
1
y = 1 + 2e 3
HW - 10
1. (D)
8. (A)
15. (B)
2. (A)
9. (E)
16. (A)
3. (B)
10. (D)
17. (A)
4. (E)
11. (B)
18. (C)
x3
5. (C)
12. (E)
6. (D)
13. (A)
7. (C)
14. (E)
19. a. S ( 60 ) = ∫ S ' ( t ) dt = 171.813 mm
60
0
b.
S ( 60 ) − S ( 0 )
60 − 0
= 2.864 mm/day
c. V = π r 2 h = 100π h →
dV
dh dV
.
= 100π
dt
dt
dt
t =7
= 100π S ' ( 7 ) = 602.218 mm3/day
1
9t 2 − 60t + 330 ) − S ' ( t ) is continuous
(
400
D(0) = −675, D(60) = 69.377.
Since D is continuous and D(0) < 0 < D(60), the IVT guarantees a time t = c, 0 < c < 60, at which
D ( t ) = M ' ( t ) − S ' ( t ) = 0. At that time, M ' ( c ) = S ' ( c ) so the heights are changing at the same rate.
d. D ( t ) =
HW – 11
1. (B)
2. (C)
3. (B)
4. (B)
5. (E)
6. (B)
7. (E)
8. a. v ( t ) = 0 when t = 3 and t = 9. Particle moving to the left when v ( t ) < 0 ; 3 < t < 9
b.
∫ v ( t ) dt
6
0
π
3
1
π

π 
π

π < 0 ; v ( 4 ) = cos  ( 4 )  = − < 0
sin  t  ; a ( 4 ) = − sin  ( 4 )  = −
6
12
6
2
6 
6

6

The speed is increasing at t = 4 b/c velocity and acceleration have the same sign.
c. a ( t ) = −
π
4
6
6
π 
 π 
 2π
d. x ( 4 ) = −2 + ∫ cos  t  dt = −2 +  sin  t   = −2 + sin 
0
π
6 
 6 
 3
π
4
3 3

 = −2 +
π

0
−2
1
1 1
1
3 π
π 3
(1)   = − ; g ( −2 ) = ∫ f ( t ) dt = − ∫ f ( t ) dt = −  −  = −
1
1
−2
2 2
4
2 2 2 2
b. g ' ( x ) = f ( x ) ⇒ g ' ( −3) = 2 ; g " ( x ) = f ' ( x ) ⇒ g " ( −3) = 1
9. a.
g ( 2 ) = ∫ f ( t ) dt = −
2
c. g has a horizontal tangent where g ' ( x ) = f ( x ) = 0 : x = -1 and x = 1.
g has a relative maximum at x = −1 b/c g ' ( x ) changes from positive to negative there.
g ' does not change sign at x = 1 so g has neither a relative maximum nor minimum there.
d. The graph of g has inflection points at x = −2, x = 0 and x = 1 b/c g " ( x ) = f ' ( x ) changes sign at each
of these points.
HW – 12
1. (B)
8. (C)
2. (C)
9. (A)
3. (B)
4. (D)
5. (C)
6. (E)
7. (C)
10. (B)
3
1
1
13
2
11. a. g ( 3) = 5 + ∫ g ' ( x ) dx = 5 + π ( 2 ) + (1)( 3 ) = + π
0
4
2
2
−2
0
1
2
g ( −2 ) = 5 + ∫ g ' ( x ) dx = 5 − ∫ g ' ( x ) dx = 5 − π ( 2 ) = 5 − π
0
−2
4
b. g has points of inflection at x = 0, x = 2 and x = 3 because g' changes from increasing to decreasing or
vice versa at those points.
c. h'(x) = g'(x) – x = 0 at x = 2 and x = 3.
h has a relative maximum at x = 2 because h' changes from positive to negative there.
h has neither a max nor a min at x = 3 because h' does not change sign there.
HW - 13
1. (C)
8. (D)
2. (A)
3. (C)
4. (C)
5. (D)
6. (C)
7. (B)
9. (E)
H ( 5 ) − H ( 2 ) 52 − 60
8
10. a. H ' ( 3.5 ) =
=
=−
5−2
3
3
10
1
1 1
1
1
1

b.
E ( t ) dt ≈  ( 2 )( 66 + 60 ) + ( 3)( 60 + 52 ) + ( 4 )( 52 + 44 ) + (1)( 44 + 43) 
∫
10 0
10  2
2
2
2

(= 52.95. But if you’re smart, you won’t bother to work it out.)
c.
∫
10
0
H ' ( t ) dt = H (10 ) − H ( 0 ) = 43 − 66 = −23°C . This represents the net change in the temperature of
the tea in degrees Celsius from t = 0 to t = 10 minutes.
d. B (10 ) = B ( 0 ) + ∫ B ' ( t ) dt = 34.183
10
0
43 – 34.183 = 8.817. The biscuits are 8.817°C cooler than the
HW - 14
1. (E)
8. (A)
13. (D)
2. (B)
3. (A)
9. (C)
10. (D)
14. (A)
15. (B)
v (10 ) − v ( 0 )
16. a. a ( 5 ) ≈
= 0.03 m/s 2
10 − 0
b.
∫
60
∫
60
0
0
4. (E)
11. (E)
5. (C)
12. (E)
7. (D)
v ( t ) dt represents the distance Ben traveled, in meters, from time t = 0 to t = 60.
v ( t ) dt ≈ 10 ( 2 ) + 30 ( 2.3) + 20 ( 2.5 ) = 139 (evaluating the answer is NOT required)
c. Yes. B is differentiable on (40, 60) and
B ( 60 ) − B ( 40 )
60 − 40
time in 40 < t < 60 when Ben’s velocity was 2 m/s.
d. 2 L
6. (B)
= 2 so by the Mean Value Theorem there is a
( 49 )( 4.6 )
dL
dL
dB
so
=
= 2B
dt
dt
dt t = 40
12 2 + 492
HW - 15
1. (E)
2. (D)
3. (A)
4. (D)
5. (C)
6. (A)
2

x3 
8 4
7. a. A = ∫ ( 2 x − x ) dx =  x 2 −  = 4 − =
(Note: Last step optional.)
0
3
3 3

0
2
2
2
 2
2
4
π 
 π 
b. V = ∫ sin  x  dx =  − cos  x   = − ( cos π − cos 0 ) =
0
π
π
2 
 2 
 π
2
0
2
2
y
c. V = ∫  y −  dy
0
2
8. a. f '(4) ≈
b.
c.
f ( 5 ) − f ( 3)
5−3
=
∫ ( 3 − 5 f ' ( x ) ) dx = ∫
∫
−6
= −3
2
13
13
2
2
13
2
3dx − 5∫
13
2
f ' ( x ) dx = 3 (11) − 5 ( f (13) − f ( 2 ) ) = 33 − 5 ( 6 − 1) = 8
f ( x ) dx ≈ (1)(1) + ( 2 )( 4 ) + ( 3)( −2 ) + ( 5 )( 3 ) = 18
d. Tangent line: y + 2 = 3 ( x − 5 ) → y(7) = 4. Since f is concave down on [5, 8], the tangent line will
overestimate of f(7). Thus f(7) ≤ 4.
5
4
( x − 5) → y(7) = . Since f is concave down on [5, 8], the secant line will
3
3
4
underestimate f(7). Thus f(7) ≥ .
3
Secant line: y + 2 =
HW - 16
1. (B)
2. (C)
3. (C)
4. (C)
5. (A)
6. (B)
7. (B)
8. (D)
9. (E)
10. a. f will have inflection points where f ' changes from increasing to decreasing or vice versa. This happens
at x = –2 and x = 0.
0
1
2

b. f ( −4 ) = f ( 0 ) − ∫ f ' ( x ) dx = 5 −  ( 2 )( 4 ) − π ( 2 )  = 2π − 3
−4
2


f ( 4 ) = f ( 0 ) + ∫ f ' ( x ) dx = 5 + ( −15e − x / 3 − 3 x ) = 5 − 15e −4 / 3 − 12 + 15 = 8 − 15e −4 / 3
4
4
0
0

 5 
c. f ' > 0 on (–4, –2) and  −2, 3ln    and f ' < 0 on
 3 

5
x = 3ln   .
3

5 
 3ln  3  , 4  so f has its maximum value at
  
