M161, Test 2, Spring 2004
NAME:
SECTION:
INSTRUCTOR:
You may use alulators. No formulas an be
stored in your alulators.
Problem
1
Points
10
2
5
3ab
15
3def
15
4
15
5
15
6
15
7
10
Total
100
Sore
1. Find the length of the urve x = sin(2y ), 0 y =2.
2. Determine whether the series
X1
( 1)n 2n
is onvergent or divergent. If it onverges, nd the sum.
5n+1
n=0
X
3. Determine whether the following series onverge or diverge. Give reasons for your answer.
1 n+1
(a)
2p
n=1
(b)
()
n
n
X1
n(n + 1)2n
X1
( 1)n
3n
n=1
n=1
n2
(d)
(e)
(f)
X1
n=1
X1
n=1
X1
n
n+1
2n + 1
n2 + 2n + 1
2n+1
3n
n=1
4. Find the enter of mass of a thin plate of onstant density Æ overing the region enlosed by the parabolas
x = y2 .
(a) On the piture inluded below, draw an arbitary approximating retangle R (dierential element).
y
= x2 and
1.5
y
y
1
x=y2
y=x2
0.5
x
0
−0.5
−0.5
0
0.5
x
1
1.5
(b) Find the enter of mass of the dierential element R.
() Find the moment of the dierential element R about the y -axis.
(d) Summing the moments over all of the approximating retangles and letting
about the y -axis.
n
(e) Repeat steps () and (d) to nd the moment of the region R about the x-axis.
(f) Find the enter of mass of the region R.
! 1, nd the moment of the region R
5. The denition of onvergene of a sequene is as follows: The sequene fan g onverges to the number L if to every positive
number there orresponds an integer N suh that for all n,
n>N
)
jan
j
L < :
If no suh number L exists, we say that fan g diverges.
n
(a) an =
n+1
. Write the rst six terms of the sequene fan g, beginning with a1 .
(b) Plot, on the axis given below, the rst six terms of fan g.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
0
1
2
3
4
5
6
7
x
n
() Does the sequene dened by an =
onverge or diverge? Explain your answer. Inlude in your explanation a
n+1
graphial demonstration using the denition given above.
6. Determine whether eah sequene onverges or diverges. If it onverges, nd the limit. Support your answer with work or
an explanation.
5n3 + 1
(a) an =
10n2 + n 3
(b) an =
ln(n)
ln(2n)
() an = 1 + ( 1)n
X1
1
1
and the funtion f (x) = 3=2 , whih is graphed below. Illustrate the proof of the integral test
x
n=1
by drawing appropriate retangles on the graph to dedue that
7. Consider the series
n3=2
X
n
i=2
ai
1
1
1
= 3=2 + 3=2 + + 3=2
2
3
n
Explain in your own words how this shows that the series
y
X1
n=1
Z
n
1
dx:
1 x3=2
1
is onvergent.
n3=2
1
y=
1
5
1
x3=2
10
x