Problem
Points
1
10
2
5
3
10
4
10
5
5
6
5
7
10
8
7
9
10
10
5
11
8
12
5
13
10
Total
100
M161, Test 3, Fall 2003
NAME:
SECTION:
INSTRUCTOR:
You may use alulators.
No formulas an be
stored in your alulators.
1
Sore
1. Calulate the following limits.
ln x
(a) lim
x! x 1
1
(b) lim
x!
1
1
os(3x)
+ 3x
x3
2. Use the integral test to show that the harmoni series diverges.
2
Z
3. Calulate the following integrals. You must show your work.
p1 dx
(a)
3
x
0
(b)
Z
1
1
e
t dt
2
3
4. (a) Find the 3rd degree (n=3) Taylor polynomial approximation of the funtion f (x) = x
about the point a = 1.
2
(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 :9 x 1:1.
+1
(
+1)!
3
4
X
5. Determine whether the series is onvergent or divergent. In either ase justify your answer.
1
n
=1
n
n2
+n+1
Z
6. The 3rd order Taylor expansion for the funtion f (x) at a = 0 is given by f (x) =
1 x
f 00 (0)
f 000 (0)
0
x
(t x) f (t) dt. Find the 4th order Taylor
f (0) + f (0)x +
x +
2!
3!
3!
expansion of the funtion f about a = 0 by integration by parts.
2
3
3
0
5
(4)
7. Determine whether the following sequenes onverge or diverge. If the sequene onverges,
nd the limit.
(a) an = os(n=2)
(b) an =
3 + ( 1)n
5n
2
8. (a) Determine if the sequene an = 3=(n + 5) is inreasing or dereasing. Show why or
why not.
(b) Is it a monotoni sequene?
6
9. The rate of hange of enrollment at a university is proportional to the enrollment at any
time t. Supppose there were 20; 000 students in 1980 and 30; 000 students in 1990.
(a) Write a dierential equations that models the enrollment at the university.
(b) Find an expression for the enrollment at any time t (solve the dierential equation and
determine the onstants).
10. Solve the following dierential equation.
1+x
dy
=
, x > 0, y (1) = 4 (Solve for y ).
dx
xy
7
11. 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 exists an N so that if n > N then jan Lj < .
n!1
(a) Prove that lim rn = 0 when 1 < r < 1 and r =
6 0 by showing that for every > 0 there
n!1
exists an N so that if n > N then jrn 0j < .
(b) Prove that lim rn = 0 when r = 0 by showing that for every
n!1
that if n > N then jrn 0j < .
8
>
0 there exists an N so
12. Find the sum of the series
Xn
1
n
=3
1
.
2
13. Determine whether the series is absolutely onvergent, onditionally onvergent or divergent. In any ase explain why. You must justify your answers.
1
1
( 1)n p
(a)
n+2
n
X
=0
(b)
X
1
n
=1
( 1)n n
e
n
9