1. Find each limit.
(a)
lim
(10%)
1 2
In(1- x) + x + - x
3
x
r-+O
n
2
(b)
lim I ~
n-->ao
j=!
2. Evaluate each of the following integrals.
(a)
.
2
S
!
x 2 + 3 dx
2x+x 3
(b)
·4
S: Jx
n
(10%)
2-6x+81
dx.
N
3. Find the equation of the tangent line to the curve. 2(x 2 + y2)2 = 25(x 2 .; y2)at the
point (3,1) .
(10%)
C
4. Find the constants A and B so that the curve y
of inflection at(;r / 6, 5).
=
A cos 2x + B sin 3x will have a point
(10%)
HU
5. Find the radius of convergence and interval of convergence of the series.
(10%)
--
1. Find
A-I for A =
.
l~ ~ ~J.
(10%)
2 2 3
Ilxll 00= m~x Ixil. One could define a norm on
=
Find (i)
C
N
2. The uniform norm is defined as
Ilxll,
.
(t,14
Ilxll",
f
Let
X
(ii) II x il 2 (iii)
l:Sl:Sn
be the vector (4,-5,3)' in
IIXII
I•
(15 %)
HU
3. Show hat if A and B are similar matrices then det (A)
4. Find the least square solution to the system (15 %)
-Xl +x 2 = 10
2x J +x2 = 5
xl - 2 x2 =20
R',
= det (B). (10 %)
R"
by