ECONOMICS 207
SPRING 2006
EXAM 2
Problem 1 (18 Points).
Find the derivatives of each of the following functions with respect to x.
(i) y = 10x3
5x2 + ln (x) 4x + 5
y 0 = 30x2
10x +
y 0 = (2x
(v) y = 15x5=8
y0 =
11) (2x5 + 4x3
(4x2 )2
+ 7x)5 + 2e3x
h
1
5 2
ECONOMICS 207
FALL 2006
PROBLEM SET 9
Problem 1. Consider a consumer with the following utility function:
1=2 1 =3
U (x1 ; x2 ) = x1 x2
and facing the following prices for x1 and x2 ; respectively. w is the consumers income.
p1 = 2
p2 = 3
w = 1000
Form th
ECONOMICS 207
FALL 2006
PROBLEM SET 10
Problem 1. Consider the following matrix and vectors.
2
3
1
1
2
3
65
A = 43 0
5
5
11
3
23
2
c= 495
11
(i) Find the inverse of the matrix A.
2
1
43
5
So, A(
1)
1
3
0
5
3
3
2100
R2 ! ^ + R2
3R 1
6 0 1 05
R 3 ! 5R 1 + R
ECONOMICS 207
FALL 2006
PROBLEM SET 6
Problem 1. Consider the following matrices.
5
1
A=
3
;
2
2
1
E =4 3
8
23
4
a = 455 b
6
Compute the following
(i) [a.]
(ii) A + 2C
1
7
2
=
30
;
56
3
2
9
4
65 F = 4 3
0
0
23
0
455 c = 5
5
9
B=
A + 2C =
1
2
53
+2
12
C=
4
ECONOMICS 207
FALL 2006
PROBLEM SET 7
01
x1
or @x2 A
x3
by appending the right-hand side vector to the coefcient matrix and performing row reduction.
(i)
11
x1
2
=
2
3
x2
2
x1
Problem 1. For each of the following systems of equations, nd the solution vect
ECONOMICS 207
FALL 2006
PROBLEM SET 5
Problem 1. Find the second derivative of each of the following functions with respect to x
(i) y = 160x1=4 z 3=5
45x
35z
y0
y
2
(ii) f (x) = 9x3 e3x
f 0 (x) =
f 00 (x) =
h2
9x3 e3x
h
54x4
x
00
=
40x
=
3
4
30x
3
z5
7
4
ECONOMICS 207
FALL 2006
EXAM 4
Problem 1 (40 pts). Find the inverse of the matrix A =
3
4
5
:
7
Problem 2. Use the inverse matrix you found above to solve the following system of equations:
35
47
x1
=
x2
Date: November 14, 2006.
1
1
1
2
ECONOMICS 207 FALL
ECONOMICS 207
FALL 2006
EXAM 4
Problem 1. (17 Points) Find the inverse of the matrix A =
A
1
=
1
1
7
4
5
=
3
7
4
3
4
5
:
7
5
3
Problem 2. (17 Points) Use the inverse matrix you found above to solve the following system of
equations:
35
47
x1
=A
x2
1
1
=
1
ECONOMICS 207
SPRING 2006
EXAM 2
Problem 1 (18 Points).
Find the derivatives of each of the following functions with respect to x.
(i) y = 10x3
5x2 + ln (x) 4x + 5
(ii) y = (2x
(iii) y =
5x2 )(6x2
2x5 + 4x3
4x2
2).
11x
Date: 1 October 2006.
1
2
ECONOMICS