Math 237. Calculus II
Exam 4, Version 2
Fall, 2011
Name: _______________________________________
1. Find the indefinite integral
Substitute
u
= ln
x
.
This gives
du
=
dx
/
x
, and so
2.
Find the convergence set for the series
Starting off using the Absolute Ratio Test:
So the series will converge whenever
|3
x
–
2| < 1 or, equivalently,
whenever
–
1 < 3
x
–
2 < 1, which
simplifies to
.
To finish up, we need to know whether the series converges at either endpoint.
For
x
=
1
/
3
, the series is equal to
.
This series, the harmonic series, is known to
diverge.
For
x
= 1, the series is equal to
, which is the alternating harmonic
series and is known to converge.
Answer:
.
3.
Sketch the graph of the hyperbola whose equation is
9
x
2
–
16
y
2
+ 54
x
+ 64
y
–
127 = 0.
Include foci,
vertices, and asymptotes.
Here is the pre-sketch algebraic manipulation.
9(
x
2
+ 6
x
+ 9)
–
16(
y
2
–
4
y
+ 4) = 127 + 81
–
64 = 144
So this is a horizontal hyperbola with center (
–
3, 2),
a
= 4, and
b
= 3.
That means it has vertices at (
–
7, 2) and (1, 2) and asymptotes
.

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