M161, Test 3, Spring 2003
1. Calulate the following limits.
(a) lim e x ln x
x!1
1
(b) lim ln(1 + 3x)
x! x
0
1
1
x7
() lim
x! x
4
1
2. Calulate
Z 1 1the following integrals. You must show your work.
(a)
dx
(3x + 1)
3
0
(b)
()
Z
1
1x
2
Z
5
3
1
dx
+1
1
p
x
2
dx
3. (a) Find the 4th order Taylor polynomial approximation of the funtion f (x) = os(x)
about the point a = =4.
(b) Use Taylor's Inequality (jRn (x)j nM jx ajn ) to estimate the auray of the
approximation f (x) T (x) when x satises 0 x =2.
+1
(
+1)!
4
() Find the seond order Taylor polynomial approximation of the funtion f (x) =
about the point a = 0.
1
1
p
x
4. Suppose you have derived the 10th order Taylor polynomial plus error term expansion of
the funtion f (x) about a = 0 and obtained
f (x)
=T
X1 k k
f (0)x .
(x) =
10
1
(x) +
10!
Zx
(t
x) f
10
(11)
(t) dt
0
10
where
T10
k
=0
k!
( )
Use integration by parts to obtain the 11th order Taylor
polynomial plus error term expansion of the funtion f (x) about a = 0.
5. Determine whether the following sequenes onverge or diverge. If the sequene onverges,
nd the limit.
3 + 5n
(a) an =
n+n
2
2
1
(b) an =
() an =
n3
e[ln(
n2
)+ln(
5n
5
)℄
.
( 1)n
.
3n + 1
(d) an = sin
1
n2
.
6. Solve the following dierential equations
dy
=2
(a) x (y + 1)
2
2
dx
(b)
dz
dt
+ et z = 0.
+
7. A bateria ulture grows at a rate proportional to its size. The ount was 400 after 2 hours
and 25,600 after 6 hours.
(a) Write a dierential equation that desribes the growth of the bateria ulture.
(b) Find an expression for the ulture population after t hours.
() What was the initial population?
(d) From some partiular time, t = t , how long does it take for the population to double?
0
8. The denition of lim an = L is as follows: A sequene fan g has the limit L and we write
n!1
lim an = L if for every > 0 there is a orresponding integer N suh that jan Lj < n!1
whenever n > N .
For jrj < 1 prove that lim = 0.
n!1
2