Math 172
Prerequisite Worksheet
Due: 19 Jan. 2016
Show Appropriate Work, Justify Answers
1. (2 pts) Solve for x:
Name:
8
2=x
x
2. (6 pts) Sketch a graph for each. Clearly label the scale on the axes and provide appropriate points.
(a) y = sin x
(b) y = co
F=
N
5. (1 pt)
The plate in the gure is submerged in water with its top level
with the surface of the water. The water level is 3 6 m. The right
1
1
and left edges of the plate are the curves y = x 3 and y = x 3 .
Find the uid force on a side of the plate
1. (1 pt)
Calculate the arc length of y = 3x + 1 over the interval [0, 9].
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4. (1 pt)
Find the arc length of the curve y = ln(cos x) from x = 0 to
x = /4.
s=
s=
5. (1 pt)
Compute the surfa
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1. (1 pt)
Determine the convergence or divergence of the series
(1)n1
n0.8
n=1
3. (1 pt)
Determine the convergence or divergence of the series
This series is absolutely convergent.
sin n
n2 6
n=1
This series is conditionally convergent.
This s
1. (1 pt)
Calculate the rst four terms of the following sequence, starting with n = 1.
5 2n
cn =
n!
c1
c2
c3
c4
5. (1 pt)
Determine the limit of the sequence or type DIV if the sequence diverges.
=
=
=
=
an = 7
2. (1 pt)
Calculate the rst four terms of t
P(3 X 4) =
1. (1 pt)
Find a constant C such that p(x) =
3. (1 pt)
C
is a probability
(x + 1)3
density function on the interval [0, ).
C
Find a constant C such that p(x) =
is a probability
4 x2
density function on the interval [0, 2).
Then, compute the pr
1. (1 pt)
What trigonometric substitution should be used to evaluate
the following integral?
5. (1 pt)
Evaluate the integral.
1
81x2 1 dx ?
x (x 8)2
(Express the substitution in terms of t instead of theta)
dx =
6. (1 pt)
Evaluate the integral.
x=
dx
=
2
dx
=
(x + 5)3/2
3
1. (1 pt)
Determine whether the improper integral converges, and if
so, evaluate it. Possible answers are:
- a number if the integral is convergent
- or INF or -INF for or
dx
x18/19
1
5. (1 pt)
Determine whether the improper integral co
1. (1 pt)
Which of the following is the correct form of the partial fracx1
?
tion decomposition of 3
x + x2
Ax + B Cx + D
E
A.
+
+
2
x
x
x+1
A B
C
B. + 2 +
x x
x+1
Ax + B Cx + D Ex + F
+
+
C.
x
x2
x+1
Ax + B Cx + D Ex + F
D.
+
+
x
x2
x+1
5. (1 pt)
Use
1. (1 pt) For each of the indenite integrals below, select
which of the following trig substitutions would be most helpful
in evaluating the integral. Do not evaluate the integrals.
Then according to the triangle, sin =
5. (1 pt)
Evaluate using trigonomet
1. (1 pt)
A solid rectangular box with height 6 m and square base with
side lengths 2 m is built using a lightweight material whose denkg
sity is 650 m3 . Constructing this box requires work against gravity.
4. (1 pt)
The required work is
swer).
(include
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instead.
(For example, try nding the volume of a cone when the height
(h) is 4 and the radius is 2.)
1. (1 pt)
Let V be the volume of a pyramid of height 15 whose base is a
square of side 5.
3. (1 pt)
Find volume of the solid whose base is the sem
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the volume of a rotated solid using the washer method.
1. (1 pt)
Find the volume of the solid obtained by rotating the region
5
under the graph of the function f (x) = x 4 about the x-axis over
the interval [1, 4].
5. (1 pt)
Find the volume of the
6. (1 pt)
Evaluate the integral using Integration by Parts, substitution,
or both if necessary.
1. (1 pt)
Evaluate the integral using Integration by Parts. $
10x cos(9x) dx =
7xdx
=
7x + 1
7. (1 pt)
Evaluate the integral using Integration by Parts, substi
1. (1 pt)
the method of shells to find the volume of the generated solid.
Use the shell method to compute the volume of the solids
obtained by rotating the region enclosed by the graphs of the
functions
Volume =
y = x2 and y = 8 x2 for x 0.5
about the y-a