Math 126
Carter
Sample Test 2 (version a) Spring 2011
General Instructions: In this document you will find 4 separate sample tests. The
actual test will be similar in format to any one of these tests. I anticipate making the problem
difficulty level about the same as it is here. But I cannot always guaranty the difficulty level.
I strongly suggest that you make yourself a 5th test of the same format; include problems
at this level of difficulty and a few problems that are more difficult.
1. (8 points each) Compute the following:
(a)
Z
x ln (x) dx
(b)
1
dx
1 + x2
Z
(c)
Z
ex sin (x) dx
(d)
Z
sin3 (x) dx
(e)
1
dx
x2 + 3x + 2
Z
(f)
Z ∞
1
dx
x
2. (7 points) Define
lim an = L.
n→∞
3. (10 points) Compute the arc length (
Rbq
a
1 + (f 0 (x))2 dx) of f (x) = x3/2 for x ∈ [1, 2].
4. (10 points) Compute the volume of the solid obtained by rotating the region bounded
by y = x2 and y = x about the y-axis.
1
5. (10 points) Compute the centroid of the region that is bounded by y = x2 and y =
Note that if f1 (x) ≤ f2 (x), then
√
x.
ρ ab x(f2 (x) − f1 (x)) dx
= Rb
ρ a (f2 (x) − f1 (x)) dx
R
xCM
and
1 ρ ab ((f2 (x))2 − (f1 (x))2 ) dx
=
R
2 ρ ab (f2 (x) − f1 (x)) dx
R
yCM
where ρ indicates the density measured in mass per unit area. An alternate formula
for yCM is
R
ρ ab y(g2 (y) − g1 (y)) dy
xCM = R b
ρ a (g2 (y) − g1 (y)) dy
Where the region is bounded by x = g1 (y) and x = g2 (y) for g1 (x) ≤ g2 (x).
6. (10 points) Compute the (Taylor) MacLaurin polynomial T5 (x) =
of the indicated function at the point a.
f (x) = ex ; a = 0.
2
P5
j=0
f (j) (a)
(x
j!
− a)j
Math 126
Carter
Sample Test 2 (version b) Spring 2011
General Instructions: In this document you will find 4 separate sample tests. The
actual test will be similar in format to any one of these tests. I anticipate making the problem
difficulty level about the same as it is here. But I cannot always guaranty the difficulty level.
I strongly suggest that you make yourself a 5th test of the same format; include problems
at this level of difficulty and a few problems that are more difficult.
1. (8 points each) Compute the following:
(a)
Z
(b)
xe4x dx
√
Z
4 − x2 dx
(c)
Z
e2x cos (x) dx
(d)
Z
cos3 (x) dx
(e)
Z
x2
1
dx
− 4x + 3
(f)
Z 1
0
dx
√
x
2. (7 points) Define
lim an = L.
n→∞
3. (10 points) Compute the arc length (
x ∈ [0, π/4].
Rbq
a
1 + (f 0 (x))2 dx) of f (x) = ln cos (x) for
4. (10 points) Compute
√ the volume of the solid obtained by rotating the region bounded
by y = x2 and y = x about the y-axis.
5. (10 points) Assuming a spring constant of k = 400kg/sec2 , compute the work in Joules
(kilogram meter-squared per second-squared) that is required to stretch the spring 10
centimeters beyond equilibrium.
3
6. (10 points) Compute the (Taylor) MacLaurin polynomial T5 (x) =
of the indicated function at the point a.
f (x) = sin (x); a = 0.
4
P5
j=0
f (j) (a)
(x
j!
− a)j
Math 126
Carter
Sample Test 2 (version c) Spring 2011
General Instructions: In this document you will find 4 separate sample tests. The
actual test will be similar in format to any one of these tests. I anticipate making the problem
difficulty level about the same as it is here. But I cannot always guaranty the difficulty level.
I strongly suggest that you make yourself a 5th test of the same format; include problems
at this level of difficulty and a few problems that are more difficult.
1. (8 points each) Compute the following:
(a)
Z
x cos (2x) dx
(b)
Z
√
t 1 − t2 dt
Z
sec3 (x) dx
Z
sec2 (x) dx
(c)
(d)
(e)
Z
x2
1
dx
+x+1
(f)
Z ∞
1
dx
x3/2
2. (7 points) Define
lim an = L.
n→∞
3. (10 points) Compute the arc length (
Rbq
1 + (f 0 (x))2 dx) of f (x) = 9−3x for x ∈ [1, 3].
a
4. (10 points) Compute
√ the volume of the solid obtained by rotating the region bounded
2
by y = x and y = x about the x-axis.
5. (10 points) Compute the work needed in building a brick (density 80 pounds per cubicfoot) structure that is a tower of height 20 feet and square base of side 10 feet.
6. (10 points) Compute the (Taylor) MacLaurin polynomial T5 (x) =
of the indicated function at the point a.
f (x) = cos (x); a = 0.
5
P5
j=0
f (j) (a)
(x
j!
− a)j
Math 126
Carter
Sample Test 2 (version d) Spring 2011
General Instructions: In this document you will find 4 separate sample tests. The
actual test will be similar in format to any one of these tests. I anticipate making the problem
difficulty level about the same as it is here. But I cannot always guaranty the difficulty level.
I strongly suggest that you make yourself a 5th test of the same format; include problems
at this level of difficulty and a few problems that are more difficult.
1. (8 points each) Compute the following:
(a)
Z
x sin (3x) dx
(b)
Z
√
1
dx
1 − x2
(c)
Z
ex sin (3x) dx
(d)
Z
tan2 (x) dx
(e)
1
dx
x(x − 1)2
Z
(f)
Z ∞
x−21/20 dx
1
2. (7 points) Define
lim an = L.
n→∞
Rbq
3. (10 points) Compute the arc length (
a
1 + (f 0 (x))2 dx) of f (x) = 3x+1 for x ∈ [0, 3].
4. (10 points) Compute the volume of the solid obtained by rotating the region bounded
by y = x2 and y = x about the y-axis. y = x2 and y = x about the x-axis.
5. (10 points) Water has a density of 104 kilograms per cubic meter. Compute the force
against a metal plate that is submerged in water and that is the shape of an isosceles
triangle with base 1 meters and height 2 meter. The vertex is at the water level.
6. (10 points) Compute the (Taylor) MacLaurin polynomial T5 (x) =
of the indicated function at the point a.
f (x) = ln (x + 1); a = 0.
6
P5
j=0
f (j) (a)
(x
j!
− a)j