Quiz VI, Math 403, 17 November, 2010
Name:
Solutions
1. (30 points:) Solve the problem
2u
t2
u(0, y, t)
u(x, 0, t)
u(x, y, 0)
u
(x, y, 0)
t
2u 2u
, 0<x< , 0<y< ,
+
x2 y 2
= u(, y, t) = 0 , 0 y , t >
Quiz V, Math 403, 3 November, 2010
Name:
Solutions
1. (30 points:) Let h > 0 be constant. Solve the problem
u
2u
= k 2 hu , 0 < x < L ,
t
x
u
u
(0, t) =
(L, t) = 0 , t > 0 ,
x
x
u(x, 0) = f (x) , 0 <
Quiz IV, Math 403, 20 October, 2010
Name:
Solutions
1. (15 points:) Show that the set cfw_sin(nx), n = 1, 2, 3, . . ., is orthogonal on the
interval [0, ].
Solution: Suppose that m and n are distinct
Quiz III, Math 403, 15 September, 2010
Name:
Solutions
1. (30 points:) Find the eigenvalues and eigenvectors of the matrix
1 2
2
1
A=
Solution: Consider
.
1
2
2 1
A I =
.
Then the eigenvalues of A ar
Quiz II, Math 403, 15 September, 2010
Name:
Solutions
1. (10 points:) Use the Gram-Schmidt Process to convert B = cfw_u1 , u2 , u3 into
B = cfw_v1 , v2 , v3 , where B is an orthogonal set. Use the ve
Quiz I, Math 403, 1 September, 2010
Name:
Solutions
1. (10 points:) Let V = P3 , where P3 is the vector space (VS) of all polynomials of degree less than or equal to 3 and let W = cfw_f P3 | f (x) = c
Practice Midterm, Math 403
27 September, 2010
Name:
Version 1
1. Problems 11, 13, and 21, Section 8.10, page 406.
2. Problems 9, 11, 23, and 25, Section 8.8, page 395.
3. Given that the eigenvalues of
Practice Final, Math 403
24 November, 2010
Name:
Version 4
1. Show that
nx
p
sin
, n = 1, 2, 3, . . .
is orthogonal on the interval [0, p]. Find the norm of each function.
2. Show that
nx
p
1, cos
, n
Midterm Exam, Math 403, 4 October, 2010
Name:
Solutions
1. (10 points:) Consider the set B = cfw_u1 , u2 , where u1 , u2 R3 are given below.
Find an orthonormal set of vectors from R3 , call it B , su
Final Exam, Math 403
2 December, 2010
Name:
Solutions
1. (10 points): Compute the Fourier series of f (x) = x on [1, 1]. Then, without
doing any extra work, write down the Fourier Sine series of f (x)