Math 1220
Practice Final Exam
April 27, 2012
Name:
Instructions:
• Answer the questions in the space provided.
• You must show your work in order to get credit! Writing just an answer
is worth 0 points, even if the answer is correct.
• Partial credit will be awarded.
• The instructor has extra scratch paper if you need it.
• Graphing and scientific calculators are allowed, but smartphones and
computers are not allowed.
• This exam is closed book and closed notes, except you may use two
double sided 8.5 by 11 inch page of notes.
1. [10 points] Suppose a, b > 0, compute the following integral:
Z
1
dx.
ax − b
2. [10 points] Compute
Z
6e1/x
dx.
x2
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April 27, 2012
3. [10 points] Completely solve the initial value problem (differential equation), showing
all steps.
dy
= ky,
y(0) = y0
dx
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4. [10 points] Use integration by parts to compute
Z
arctan(x) dx.
5. [10 points] Use integration by parts to compute
Z
ln(x) dx.
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6. [10 points] Use the tabular method (i.e. integration by parts) to find:
Z
x3 e2x dx.
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7. [10 points] Use a trigonometric substitution to integrate
Z
dx
√
.
x 2 9 − x2
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8. [10 points] Use l’Hôpital’s rule to determine the following limit with indeterminate form:
0
.
0
tan x
lim
x→0
x
9. [10 points] Use l’Hôpital’s rule to determine the following limit with indeterminate form:
(∞ − ∞).
1
lim csc x −
x→0
x
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10. [10 points] Evaluate the improper definite integral:
Z∞
2
xe−x dx.
−∞
11. [10 points] Determine whether the following series converges or diverges. Justify your
answer.
∞
X
n3n
(n + 1)!
n=0
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April 27, 2012
12. [10 points] Find the radius of convergence for the following power series. Do not bother
to determine whether the endpoints of the interval are contained in the set
of convergence.
∞
X
(n!)2 n
x
(2n)!
n=0
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April 27, 2012
13. [10 points] Use the following two facts:
d
[sinh x] = cosh x
dx
d
[cosh x] = sinh x
dx
to compute the first four nonzero terms of the Maclaurin series for sinh x. Leave any
factorial quantities in their factorial notation: n!, i.e. don’t bother computing a number
for n!.
14. [10 points] Sketch the graph of the following polar function.
r = 2 + 5 cos θ
0 ≤ θ ≤ 2π
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April 27, 2012
15. [10 points] Find the area of the first petal of the four–leaved rose determined by
r = 4 sin 2θ
0 ≤ θ ≤ π/2.
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Question Points Score
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
11
10
12
10
13
10
14
10
15
10
Total:
150
April 27, 2012