Download Homework 6 Solutions Exercise 1. The air in a room with volume

Homework 6
Solutions
Exercise 1. The air in a room with volume 180 m3 contains 0.15% carbon dioxide
initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of
2 m3 /min and the mixed air flows out at the same rate. Find the percentage of carbon
dioxide in the room as a function of time. What happens in the long run?
1
Exercise 2. Solve the differential equation.
1. (x2 + 1)y 0 = xy
2.
dz
+ et+z = 0
dt
Exercise 3. Find the solution of the differential equation that satisfies the given initial
condition.
1.
dy
y cos x
=
,
dx
1 + y2
y(0) = 1
2. x cos x = (2y + e3y )y 0 ,
y(0) = 0
2
Exercise 4. Determine whether the sequence converges or diverges. If it converges, find
the limit.
1. an =
3 + 5n2
n + n2
√
2. an =
n
√
1+ n
3. an =
n
√
1+ n
4. an =
(−1)n−1 n
n2 + 1
3
Exercise 5. Determine whether the series is convergent or divergent. If it is convergent,
find its sum.
1.
∞
X
1
2n
n=1
2.
∞
X
k=1
3.
k2
k2 − 1
∞
X
1 + 2n
n=1
3n
4
Exercise 6. Determine whether the series is convergent or divergent, using either the
Comparison or Integral Test as appropriate.
1.
∞
X
n=2
2.
1
n ln n
∞
X
cos2 n
n2 + 1
n=1
3.
∞
X
4 + 3n
n=1
4.
∞
X
2n
ne−n
n=1
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