Math 237. Calculus II
Exam 4, Version 1
Fall, 2011
Name: SOLUTIONS
.
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
2
x
–
3 < 1 or, equivalently,
whenever
–
1 < 2
x
–
3 < 1, which
simplifies to 1 <
x
< 2.
To finish up, we need to know whether the series converges at either endpoint.
For
x
= 1, the series is equal to
.
This series, the harmonic series, is known to
diverge.
For
x
= 2, the series is equal to
, which is the alternating harmonic
series and is known to converge.
Answer:
(1, 2].
3.
Sketch the graph of the hyperbola whose equation is
x
2
–
4
y
2
–
14
x
–
32
y
–
11 = 0.
Include foci, vertices,
and asymptotes.
The first thing to do is to complete the squares.
And then take it from there . . . .
, or in a more standard form
.
So it is a vertical hyperbola with center (7,
–
4),
a
= 1, and
b
= 2.
Adding and subtracting
a
= 1
to/from
the
y
coördinate of the center shows that the vertices are (7,
–
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 Fall '1
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 Calculus, Derivative

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