Name: __________________
Class:
Date: _____________
(copy A)
1 Find two positive numbers whose product is 36 and whose sum is a minimum.
a.
2, 18
b.
6, 6
c.
4, 9
2 Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L = 4 cm if one side of the rectangle lies
on the base of the triangle. Round the result to the nearest tenth.
a.
2 cm, 1.8 cm
b.
1.5 cm, 1.71 cm
c.
2 cm, 1.7 cm
d.
5 cm, 0.7 cm
e.
3 cm, 2.2 cm
f.
7 cm, 1.7 cm
3 The upper left hand corner of a piece of paper 12 in. wide by 14 in. long is folded over to the right hand edge as in the figure. How would you fold
it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
a.
x = 3 in
b.
x = 8.5 in
c.
x = 9 in
d.
x = 9.25 in
e.
x = 6 in
f.
x = 12 in
PAGE 1
Name: __________________
4
Class:
Date: _____________
Explain why Newton's method doesn't work for finding the root of the equation 6x
x = 2.
3
72x + 1 = 0 if the initial approximation is chosen to be
1
(use a separate sheet to answer if necessary)
5 Find the most general antiderivative of the function:
f(x) =
4
x
a.
1
F(x) = x
b.
F(x) =
1
x
c.
F(x) = 4
4
5
, x 0
+ C
+ C
1
x
6
+ C
6
The graph of a function f(x) is shown. Which graph is a possible graph for antiderivative of f(x)?
a.
2
b.
1
c.
3
PAGE 2
(copy A)
Name: __________________
Class:
Date: _____________
(copy A)
7 A stone was dropped off a cliff and hit the ground with a speed of 320 ft/s. What is the height of the cliff?
for g.
a.
51200 ft
b.
3200 ft
c.
1600 ft
d.
102400 ft
8 By reading values from the given graph of f, use five rectangles to find a lower estimate for the area from x = 0 to x = 10 under the given graph of
f. Round your answer to the nearest tenth.
a.
25.9
b.
21.9
c.
26.6
d.
23.5
e.
21.2
9 If f (x ) = sin sin x , 0 a.
1.53
b.
0.63
c.
0.35
d.
0.75
e.
0.83
f.
1.95
PAGE 3
x /2, approximate the area under the curve using ten approximating rectangles of equal widths and left endpoints.
Name: __________________
Class:
Date: _____________
10 Use the Midpoint Rule with n = 10 to approximate the integral.
a.
8.284406
b.
2.071102
c.
1.41838
2
2 + z
2
dz
1
11 Express the limit as a definite integral on the given interval.
n
lim
x
18
a.
2
i
11 x
i
x , [8,18]
)dx
( 3 x2
11 x
)
18 x
)dx
dx
8
11
c.
+ 11 x
3x
8
18
b.
( 3 x2
i = 1
( 8 x2
3
12 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
g(x) =
a.
dg(x) =
dx
2 + 3x
b.
dg(x) =
dx
3 + 3x
c.
dg(x)
3
=
dx
2 3 + 3x
7
13
Evaluate the integral.
5
a.
1
b.
1
c.
0
d.
2
PAGE 4
cos d x
3 + 3t d t
1
(copy A)
Name: __________________
Class:
Date: _____________
14 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
6x
g(x) =
2
t
10x 2
t
a.
b.
c.
dg(x) =
dx
100x
100x
2
+ 5
2
5
dg(x) =
dx
10 100x
dg(x) =
dx
10 x
100x
x
2
2
36x
+
36x
2
2
+ 5
2
5
+ 6 36x
36x
+ 6 x
5
x
2
2
5
+ 5
5
+ 5
2
+ 5 dt
2
2
+ 5
5
+ 5
5
15 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
6x
g(x) =
a.
dg(x) = cos((6x )8) + sin(x )cos(x )
dx
b.
dg(x) = 6cos((6x )8) + sin(x )cos(cos8(x ))
dx
c.
8
dg(x) =
6cos((6x ) ) dx
16 Find a function f (x ) such that
x
12 +
a
f (t )
t
2
dt = 6 x
for all x > 0 and some number a.
Now find the number a such that
12 +
x
a
f (t )
t
2
dt = 6 x
for all x > 0 .
a =
PAGE 5
________
8
sin(x )cos(cos (x ))
8
cos(t ) d t
cos(x )
(copy A)
Name: __________________
17
Evaluate the integral.
18
a.
246
b.
113.6
c.
492
d.
104.8
9
x
2
1
Class:
Date: _____________
+ 2 dx
x
The area of the region that lies to the right of the y axis and to the left of the parabola x = 3 y by the integral
3
( 3y
y
2
) d y.
0
Find the area of the region.
a.
9
6
b.
27
c.
27
6
d.
2.25
19
If w ' (t ) is the rate of growth of a child in pounds per year, what does
(use a separate sheet to answer if necessary)
PAGE 6
(copy A)
8
w ' (t )dt represent?
6
y
2
(the shaded region in the figure) is given
Name: __________________
20
Class:
Date: _____________
(copy A)
2
The acceleration function ( in m/s ) and the initial velocity are given for a particle moving along a line.
a(t ) = t + 4 , v (0) = 2 , 0 t 10
Find the velocity at time t.
Find the distance traveled during the given time interval.
2
21 The acceleration function ( in m / s ) and the initial velocity are given for a particle moving along a line. Find the velocity at time t and the distance
traveled during the given time interval.
a (t ) = t + 4 , v ( 0 ) = 3 , 0 a.
v (t ) = t
2
2
b.
v (t ) = t
c.
v (t ) =
t
v (t ) = t
+ 4t + 3 m / s , 401 2 m
3
2
2
+ 3 m / s , 411 m
3
2
2
+ 4t + 3 m / s , 396 2 m
3
2
d.
t 2
2
2 m
+ 4t m / s , 406
3
22 Evaluate the integral by making the given substitution:
20
( 1 + 2x )
1
5
a.
( 1 + 2x )
b.
1
5
2
( 1 + 2x )
c.
10
+ C
1
( 1 + 2x )
5
d.
5
+C
4
1
( 1 + 2x )
PAGE 7
2
1
( 1 + 2x )
e.
+ C
4
2
+ C
3
dx , u = 1 + 2x
10
Name: __________________
Class:
Date: _____________
(copy A)
23 Evaluate the indefinite integral:
3 + 16x
8 + 3x + 8x
24
25
a.
2 8 + 3x + 8x
b.
8 + 3x + 8x
2
2
dx
2
+ C
+C
c.
2 8 + 3x + 8x
d.
e.
3 8 + 3x + 8x
2
2 8 + 3x + 8x
2
2
+ C
+ C
2
2
Sketch the region enclosed by x = 3 y and x = y 3 . Decide whether to integrate with respect to x or y. Draw a typical approximating
rectangle and label its height and width. Then find the area of the region.
a.
15.856406
b.
13.856406
c.
2.309401
d.
55.425626
e.
41.569219
f.
3.464102
2
Find the number b such that the line y = b divides the region bounded by the curves y = 3x and y = 7 into two regions with equal area.
7
a.
2/3
2
21
b.
3/2
2
7
c.
3/2
2
7
d.
2
3/2
21
e.
2/3
2
21
f.
2
PAGE 8
3/2
Name: __________________
Class:
Date: _____________
(copy A)
26 True or False:
The volume of the frustum of a right circular cone with height h = 6, lower base radius R = 3 and top radius r = 5 is 98 .
True
False
27 The volume of the frustum of a pyramid with square base of side b = 6, square top of side a = 3, and height h = 9 is 567 .
True or false?
True
28
False
The base of S is an elliptical region with boundary curve 25x
triangles with hypotenuse in the base.
2
+ 36y
2
= 16 . Cross sections perpendicular to the x axis are isosceles right
True or false:
The volume of S is
True
29
256 .
540
False
The base of S is the parabolic region
True or False:
The volume of S is 16 3 .
True
PAGE 9
False
{ (x
, y) | x
2
y 4
}
. Cross sections perpendicular to the y axis are equilateral triangles.
Name: __________________
30
The base of S is the parabolic region
Class:
{ (x
, y) | x
2
Date: _____________
y 2
}
(copy A)
. Cross sections perpendicular to the y axis are squares.
True or False:
The volume of S is 8 .
True
False
31 True or False:
The volume of a solid torus (the donut shaped solid shown in the figure) with r = 3 and R = 17 is 306
True
2
.
False
32 Cavalieri's Principle states that if a family of parallel planes gives equal cross sectional areas for two solids S and S then the volumes of S
1
2
1
and S are equal.
2
True or False:
If r = 7 and h = 5, then the volume of the oblique cylinder shown in the figure is 245 .
True
False
33 True or False:
If the center of each of two spheres, each having radius r = 2, lies on the surface of the other sphere, then the volume common to both spheres is
40 .
12
True
PAGE 10
False
Name: __________________
Class:
Date: _____________
(copy A)
34 Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x axis
x = 5 + y
a.
V = 85 b.
V = 80 c.
V = 160 d.
V = 78 2
, x = 0,y = 1,y = 3
35 Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified
axis.
x =
a.
V =
b.
c.
V =
y ) sin y dy
(5 y ) sin y dy
y , x = 0 ; about y = 5.
0
V =
2 (5 sin y , 0 0
2 (5 y ) sin y dy
0
V = (5 d.
y ) sin y dy
0
36 Find the work done in pushing a car a distance of 9 m while exerting a constant force of 300 N.
a.
W = 4200 J
b.
W = 2400 J
c.
W = 2200 J
d.
W = 2700 J
37 True or False:
If a force of 12 lbs is required to hold a spring stretched 5 inches beyond its natural length, then 76.8 lb in. of work is done in stretching it from its
natural length to 8 in. beyond its natural length.
True
PAGE 11
False
Name: __________________
Class:
Date: _____________
(copy A)
38 A spring has a natural length of 16 cm. If a force of 31 N force is required to keep it stretched to a length of 26 cm, how much work is required to
stretch it from 16 cm to 29 cm?
a.
3.6195 J
b.
2.1195 J
c.
2.6195 J
d.
3.1195 J
e.
4.1195 J
39 A cable that weighs 14 lb/ft is used to lift 650 lb of coal up a mineshaft 300 ft deep.
True or False:
The work required is 825000 ft lb.
True
False
40 When gas expands in a cylinder with radius R , the pressure at any given time is a function of the volume: P = P (V ). The force exerted by the
2
gas on the piston ( see the figure ) is the product of the pressure and the area: F = R P . Show that the work done by the gas when the volume
expands from volume V to volume to volume V is
1
V
W =
V
2
2
P dV
1
(use a separate sheet to answer if necessary)
41 True or False:
The average value of the function f (t ) =
8
2
(4 + t )
True
PAGE 12
False
on the interval [7, 8] is less than 1.068182.
Name: __________________
Class:
Date: _____________
(copy A)
42 True or False:
The average value of the function f (x ) = 7 True
x
2
on the interval [0, 8] is 8.21.
False
43 True or False:
If u(t) is continuous and
True
6
u(t )dx = 21 , then u(t) takes on the value 7 at least once on the interval [3, 6].
3
False
44 The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is
v (r ) =
P
4 l
(R 2
r
2
)
where P is the pressure difference between the ends of the vessel and is the viscosity of the blood.
2
Suppose that blood vessel has length 1.2 cm and outer radius 0.01 cm, and P = 4000 dynes/cm . Find the average velocities (with respect to r
) over the interval 0 r R for the vessel. Please round your answer to the nearest thousandth.
v
ave
=
________ cm/s
Compare the average velocity with the maximum velocity.
(use a separate sheet to answer if necessary)
45 True or False:
The Mean Value Theorem for Integrals says that if f(z) is continuous on [c, g], then there exists a number m in [c, g] such that
f (m) = f
True
PAGE 13
False
ave
=
g
1
c g
c
f(z) dz
Name: __________________
Class:
Date: _____________
46 True or False:
If f
([c , b]) denotes the average value of the function f on the interval [c, b] and c < m < b, then
ave
f
True
PAGE 14
False
ave
([c , b]) = m b c f ([c , m]) + b c ave
b m f ([m, b])
c ave
(copy A)
ANSWER KEY
Name: __________________
1.
2.
3.
Class:
Date: _____________
(copy A)
b
c
c
f ( x ) = 6x
3
72x + 1 = 0 4.
is horizontal. Attempting to find x
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
a
b
c
e
e
b
b
b
c
b
b
16. f ( x ) =3x
4
2
f ' ( x ) = 18x
2
72 . If x
1
= 2 , then f '
( x 1)
= 0 and the tangent line used for approximating x
2
results in trying to divide by zero.
3
2
17. d
18. c
If w ' (t ) is the rate of change of weight in pounds per year, then w (t ) represents the weight in pounds of the child at age t. We know from the
Net Change Theorem that
19.
8
w ' (t )dt = w (8) w (6), so the integral represents the increase in the child's weight (in pounds) between the ages
6
of 6 and 8.
2
20. v ( t ) = t +4t+2, 386 2
2
3
21. c
22. e
23. a
24. b
25. c
26. T
27. F
28. T
29. F
30. T
31. T
32. F
33. T
34. b
35. c
36. d
37. T
38. c
39. T
2
V = R x , so V is a function of x and P can also be regarded as a function of x. If V
x
40. W =
x
x
2
F ( x ) dx =
1
the Substitution Rule.
PAGE 1
x
2
x
2
R P (V ( x ) ) dx =
1
x
2
2
1
= R x
1
and V
2
2
= R x , then
V
2
P (V ( x ) ) dV ( x ) [Let V ( x ) = R x , so dV ( x ) = R
1
2
2
dx .] =
V
2
P (V ) dV by
1
ANSWER KEY
Name: __________________
Class:
Date: _____________
41. T
42. F
43. T
2.058
44. Since v ( r ) decreasing on
45. F
46. T
PAGE 2
( 0, R , v max = v ( 0 ) = P R
2
4 l
. Thus v
ave
= 2v
.
3 max
(copy A)
Name: __________________
Class:
Date: _____________
(copy B)
1 Find two positive numbers whose product is 144 and whose sum is a minimum.
a.
2, 72
b.
12, 12
c.
4, 36
2 Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L = 6 cm if one side of the rectangle lies
on the base of the triangle. Round the result to the nearest tenth.
a.
3 cm, 2.7 cm
b.
8 cm, 2.6 cm
c.
4 cm, 3.1 cm
d.
6 cm, 1.6 cm
e.
3 cm, 2.6 cm
f.
2.5 cm, 2.61 cm
3 The upper left hand corner of a piece of paper 12 in. wide by 17 in. long is folded over to the right hand edge as in the figure. How would you fold
it so as to minimize the length of the fold? In other words, how would you choose x to minimize y?
a.
x = 6 in
b.
x = 9 in
c.
x = 3 in
d.
x = 8.5 in
e.
x = 12 in
f.
x = 9.25 in
PAGE 1
Name: __________________
4
Class:
Date: _____________
Explain why Newton's method doesn't work for finding the root of the equation 4x
x = 2.
3
48x + 7 = 0 if the initial approximation is chosen to be
1
(use a separate sheet to answer if necessary)
5 Find the most general antiderivative of the function:
f(x) =
2
x
a.
F(x) =
1
x
b.
F(x) = 2
F(x) = , x 0
+ C
1
x
c.
3
2
1
x
4
+ C
+ C
6
The graph of a function f(x) is shown. Which graph is a possible graph for antiderivative of f(x)?
a.
3
b.
2
c.
1
PAGE 2
(copy B)
Name: __________________
Class:
Date: _____________
(copy B)
7 A stone was dropped off a cliff and hit the ground with a speed of 128 ft/s. What is the height of the cliff?
for g.
a.
16384 ft
b.
512 ft
c.
256 ft
d.
8192 ft
8 By reading values from the given graph of f, use five rectangles to find a lower estimate for the area from x = 0 to x = 10 under the given graph of
f. Round your answer to the nearest tenth.
a.
27.7
b.
25.1
c.
29.6
d.
28.8
e.
25.9
9 If f (x ) = sin sin x , 0 a.
2.15
b.
0.71
c.
0.87
d.
0.83
e.
0.93
f.
2.47
PAGE 3
x /2, approximate the area under the curve using ten approximating rectangles of equal widths and left endpoints.
Name: __________________
Class:
Date: _____________
10 Use the Midpoint Rule with n = 10 to approximate the integral.
a.
14.038474
b.
2.438679
c.
3.509618
2
10 + y
2
dy
1
11 Express the limit as a definite integral on the given interval.
n
lim
x
11
a.
13 z
)
2
i
13 z
i
z , [6,11]
dz
( 5 z2
+ 13 z
)dz
6
13
c.
5z
6
11
b.
( 5 z2
i = 1
( 6 z2
11 z
)dz
5
12 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
g(x) =
a.
dg(x) =
dx
6 + 4x
b.
dg(x) =
dx
7 + 4x
c.
dg(x)
4
=
dx
2 7 + 4x
8
13
Evaluate the integral.
6
a.
2
b.
1
c.
1
d.
0
PAGE 4
cos d x
7 + 4t d t
1
(copy B)
Name: __________________
Class:
Date: _____________
14 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
10x
g(x) =
7x
a.
b.
c.
dg(x) =
dx
7 49x
dg(x) =
dx
7 x
dg(x) =
dx
2
49x
x
49x
49x
2
2
2
2
2
+ 10
10
+ 10
100x
+ 10 x
10
+ 10
+ 10 100x
x
100x
+
10
100x
2
2
t
2
2
+ 10 d t
10
+ 10
10
+ 10
2
2
2
2
t
10
+ 10
10
15 Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
3x
g(x) =
a.
dg(x) = cos((3x )3) + sin(x )cos(x )
dx
b.
dg(x) = 3cos((3x )3) dx
c.
3
3
dg(x) =
3cos((3x ) ) + sin(x )cos(cos (x ))
dx
16 Find a function f (x ) such that
x
18 +
a
f (t )
t
2
dt = 6 x
for all x > 0 and some number a.
Now find the number a such that
18 +
x
a
f (t )
t
2
dt = 6 x
for all x > 0 .
a =
PAGE 5
________
3
sin(x )cos(cos (x ))
3
cos(t ) d t
cos(x )
(copy B)
Name: __________________
17
Evaluate the integral.
18
a.
70
b.
37.2
c.
35
d.
20.4
4
x
2
1
Class:
Date: _____________
+ 4 dx
x
The area of the region that lies to the right of the y axis and to the left of the parabola x = 2 y by the integral
2
( 2y
y
2
) d y.
0
Find the area of the region.
a.
8
b.
1
c.
8
6
d.
4
6
10
19
If w ' (t ) is the rate of growth of a child in pounds per year, what does
(use a separate sheet to answer if necessary)
PAGE 6
(copy B)
w ' (t )dt represent?
6
y
2
(the shaded region in the figure) is given
Name: __________________
20
Class:
Date: _____________
(copy B)
2
The acceleration function ( in m/s ) and the initial velocity are given for a particle moving along a line.
a(t ) = t + 4 , v (0) = 5 , 0 t 10
Find the velocity at time t.
Find the distance traveled during the given time interval.
2
21 The acceleration function ( in m / s ) and the initial velocity are given for a particle moving along a line. Find the velocity at time t and the distance
traveled during the given time interval.
a (t ) = t + 5 , v ( 0 ) = 4 , 0 a.
v (t ) = t
2
2
b.
v (t ) = t
c.
v (t ) =
t
v (t ) = t
+ 4 m / s , 471 2 m
3
2
2
+ 5t + 4 m / s , 461 m
3
2
2
+ 5t m / s , 446 2 m
3
2
d.
t 2
2
2 m
+ 5t + 4 m / s , 456
3
22 Evaluate the integral by making the given substitution:
64
( 1 + 8x )
a.
1
4
( 1 + 8x )
4
b.
2
+ C
1
( 1 + 8x )
c.
4
1
( 1 + 8x )
8
d.
4
2
1
( 1 + 8x )
PAGE 7
2
1
( 1 + 8x )
e.
4
4
+ C
+ C
+ C
3
dx , u = 1 + 8x
10
Name: __________________
Class:
Date: _____________
(copy B)
23 Evaluate the indefinite integral:
3 + 8x
6 + 3x + 4x
24
25
a.
b.
2 6 + 3x + 4x
c.
2 6 + 3x + 4x
2 6 + 3x + 4x
2
2
d.
6 + 3x + 4x
e.
3 6 + 3x + 4x
2
dx
2
+ C
+ C
2
+C
2
+ C
2
2
Sketch the region enclosed by x = 10 y and x = y 2 . Decide whether to integrate with respect to x or y. Draw a typical approximating
rectangle and label its height and width. Then find the area of the region.
a.
156.767344
b.
41.191836
c.
13.063945
d.
39.191836
e.
7.838367
f.
78.383672
2
Find the number b such that the line y = b divides the region bounded by the curves y = 8x and y = 4 into two regions with equal area.
4
a.
3/2
2
32
b.
2/3
2
32
c.
3/2
2
32
d.
2
3/2
4
e.
2
3/2
4
f.
2/3
2
PAGE 8
Name: __________________
Class:
Date: _____________
(copy B)
26 True or False:
The volume of the frustum of a right circular cone with height h = 6, lower base radius R = 2 and top radius r = 2 is 24 .
True
False
27 The volume of the frustum of a pyramid with square base of side b = 5, square top of side a = 2, and height h = 5 is 195 .
True or false?
True
28
False
The base of S is an elliptical region with boundary curve 36x
triangles with hypotenuse in the base.
2
+ 9y
2
= 16 . Cross sections perpendicular to the x axis are isosceles right
True or false:
The volume of S is
True
29
256 .
162
False
The base of S is the parabolic region
True or False:
The volume of S is 25 3 .
True
PAGE 9
False
{ (x
, y) | x
2
y 5
}
. Cross sections perpendicular to the y axis are equilateral triangles.
Name: __________________
30
The base of S is the parabolic region
Class:
{ (x
, y) | x
2
Date: _____________
y 8
}
(copy B)
. Cross sections perpendicular to the y axis are squares.
True or False:
The volume of S is 128 .
True
False
31 True or False:
The volume of a solid torus (the donut shaped solid shown in the figure) with r = 2 and R = 17 is 136
True
2
.
False
32 Cavalieri's Principle states that if a family of parallel planes gives equal cross sectional areas for two solids S and S then the volumes of S
1
2
1
and S are equal.
2
True or False:
If r = 3 and h = 7, then the volume of the oblique cylinder shown in the figure is 63 .
True
False
33 True or False:
If the center of each of two spheres, each having radius r = 3, lies on the surface of the other sphere, then the volume common to both spheres is
135 .
12
True
PAGE 10
False
Name: __________________
Class:
Date: _____________
(copy B)
34 Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x axis
x = 5 + y
a.
V = 85 b.
V = 78 c.
V = 160 d.
V = 80 2
, x = 0,y = 1,y = 3
35 Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified
axis.
x =
a.
2 (2 V =
sin y , 0 y , x = 0 ; about y = 2.
y ) sin y dy
0
V = (2 b.
y ) sin y dy
0
c.
V =
d.
V =
2 (2 y ) sin y dy
0
(2 y ) sin y dy
0
36 Find the work done in pushing a car a distance of 8 m while exerting a constant force of 200 N.
a.
W = 1600 J
b.
W = 3100 J
c.
W = 1900 J
d.
W = 2100 J
37 True or False:
If a force of 21 lbs is required to hold a spring stretched 5 inches beyond its natural length, then 210 lb in. of work is done in stretching it from its
natural length to 10 in. beyond its natural length.
True
PAGE 11
False
Name: __________________
Class:
Date: _____________
(copy B)
38 A spring has a natural length of 14 cm. If a force of 25 N force is required to keep it stretched to a length of 24 cm, how much work is required to
stretch it from 14 cm to 36 cm?
a.
7.55 J
b.
7.05 J
c.
5.55 J
d.
6.05 J
e.
6.55 J
39 A cable that weighs 8 lb/ft is used to lift 510 lb of coal up a mineshaft 540 ft deep.
True or False:
The work required is 1441800 ft lb.
True
False
40 When gas expands in a cylinder with radius R , the pressure at any given time is a function of the volume: P = P (V ). The force exerted by the
2
gas on the piston ( see the figure ) is the product of the pressure and the area: F = R P . Show that the work done by the gas when the volume
expands from volume V to volume to volume V is
1
V
W =
V
2
2
P dV
1
(use a separate sheet to answer if necessary)
41 True or False:
The average value of the function f (v ) =
2
2
(9 + v )
True
PAGE 12
False
on the interval [0, 9] is less than 1.018519.
Name: __________________
Class:
Date: _____________
(copy B)
42 True or False:
The average value of the function u (t ) = 5 True
t
2
on the interval [0, 10] is 7.73.
False
43 True or False:
If z(v) is continuous and
True
8
z(v )dx = 12 , then z(v) takes on the value 2 at least once on the interval [2, 8].
2
False
44 The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is
v (r ) =
P
4 l
(R 2
r
2
)
where P is the pressure difference between the ends of the vessel and is the viscosity of the blood.
2
Suppose that blood vessel has length 2.3 cm and outer radius 0.015 cm, and P = 4000 dynes/cm . Find the average velocities (with respect to
r) over the interval 0 r R for the vessel. Please round your answer to the nearest thousandth.
v
ave
=
________ cm/s
Compare the average velocity with the maximum velocity.
(use a separate sheet to answer if necessary)
45 True or False:
The Mean Value Theorem for Integrals says that if f(x) is continuous on [a, g], then there exists a number n in [a, g] such that
f (n) = f
True
PAGE 13
False
ave
=
g
1
a g
a
f(x) dx
Name: __________________
Class:
Date: _____________
46 True or False:
If u
([c , b]) denotes the average value of the function u on the interval [c, b] and c < n < b, then
ave
u
True
PAGE 14
False
ave
([c , b]) = n b c u ([c , n]) + b c ave
b n u ([n, b])
c ave
(copy B)
ANSWER KEY
Name: __________________
1.
2.
3.
Class:
Date: _____________
(copy B)
b
e
b
f ( x ) = 4x
3
48x + 7 = 0 4.
is horizontal. Attempting to find x
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
b
b
c
e
d
c
a
b
d
a
c
16. f ( x ) =3x
9
2
f ' ( x ) = 12x
2
48 . If x
1
= 2 , then f '
( x 1)
= 0 and the tangent line used for approximating x
results in trying to divide by zero.
3
2
17. d
18. c
If w ' (t ) is the rate of change of weight in pounds per year, then w (t ) represents the weight in pounds of the child at age t. We know from the
10
Net Change Theorem that
19.
w ' (t )dt = w (10) w (6), so the integral represents the increase in the child's weight (in pounds) between the
6
ages of 6 and 10.
2
20. v ( t ) = t +4t+5, 416 2
2
3
21. d
22. c
23. b
24. d
25. a
26. T
27. F
28. T
29. F
30. T
31. T
32. F
33. T
34. d
35. c
36. a
37. T
38. d
39. T
2
V = R x , so V is a function of x and P can also be regarded as a function of x. If V
x
40. W =
x
x
2
F ( x ) dx =
1
the Substitution Rule.
PAGE 1
x
2
x
2
R P (V ( x ) ) dx =
1
x
2
2
1
= R x
1
and V
2
2
= R x , then
V
2
P (V ( x ) ) dV ( x ) [Let V ( x ) = R x , so dV ( x ) = R
1
2
2
dx .] =
V
2
P (V ) dV by
1
2
ANSWER KEY
Name: __________________
Class:
Date: _____________
41. T
42. F
43. T
2.415
44. Since v ( r ) decreasing on
45. F
46. T
PAGE 2
( 0, R , v max = v ( 0 ) = P R
2
4 l
. Thus v
ave
= 2v
.
3 max
(copy B)