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Systems of Equations & Matrices
1. (10 pts) Solve this system of equations by using ADVANCED ELIMINATION in a matrix.
Use extra paper to show the rest of your work.
cfw_ x + y + z = -3
4x + y 3z = 11
2x 3y + 2z = 9
2. (10 pts) Solve this system of equati
Functions
For #1-3, given that f(x) = x2 + 2, and g(x) = 4x, find
1. (f + g)(2)
2. (fg)(-3)
4. Find the inverse of f(x) = .
3. (g f)(5)
5. Find g -1(13) = x2 + 7
Given the starting function of f(x) = x2, describe the transformations:
7. g(x) = 2(x 3)2 + 6
1. Write a MATLAB function linind(varargin) that takes an arbitrary number of vectors
(matrices) of the same dimension and determines whether or not the inputted vectors
(matrices) are linearly independent. (Hint: Use
function span(v, varargin)
% Test whe
MATH 331-001 Prof. Victor Matveev Midterm Examination #1 October 4, 2007 Calculators not allowed. You must remain seated until you hand in the solution. Please read each problem carefully, and show all work. 1. (10) Consider the heat equation for a 1D rod
Math 331-001 Final Examination December 19, 2008 1. (35pts) Use separation of variables to solve the following PDE: 2u 2u 0 x 1 2 2 4 u, x t u u (1, t ) 0 (0, t ) x x u u ( x, 0) 0, t ( x, 0) 5 3cos 2 x a) Separate the variables to find the two ODEs. It i
Math 331 Midterm Exam
October 15, 2008
This is a closed-book test. Neither notes nor calculators are to be used. Check your answers
You can choose between problem 1a and problem 1b (no extra credit for doing both)
(1a, 15pts) The following equation descri
Math 331 Midterm Exam Solution
October 15, 2008
(1a, 15pts) The following equation describes the conservation of energy in a thin rod:
e
=
t
x
Here e(x,t) is the energy density. Derive this equation, and give the correct value of the sign of the
right-han
Math 331, Midterm Examination, Thursday October 12, 2006
1) Consider the following Initial Boundary-Value Problem for the heat equation with non-homogeneous boundary conditions:
ut = 2uxx ; 0 < x < L, t > 0
u(0, t) = 10 ; t > 0
ux (L, t) = 1 ; t > 0
u(x,
Math 331-001 * Final Exam Solution * December 19, 2008 u 2u 0 x 1 t 2 x 2 4 u , u ( x, t ) f ( x)h(t ) u u (1) (0, t ) f ( x) h '(t ) h(t ) f '( x) 4 f ( x)h(t ) f ( x) h(t ) (1, t ) 0 x x u h ' f ' 4 h ' 4 f ' u ( x, 0) 0, ( x, 0) 5 3cos 2 x h f h f t
2
MATH 331-001
Final Examination
December 14, 2007
1. (24) Solve the Laplaces equation in a half-disk, 0<r<R, 0<:
1 u 1 2 u
u=
=0
r +
r r r r 2 2
2
u (0, ) < ; u ( R , ) = T0 sin
u (r , 0) = u (r , ) = 0
a) Separate the variables to find the two ODEs
b) Wh
1. Prove cot(x) + tan(x) = sec(x)csc(x)
2. Prove
Solve for .
3. 6cos + 12 = 0
4. 4sin2 3 = 0
5. 2cos2 + cos 1 = 0
Trig Leftovers
Using the Law of Sines, solve for all missing sides and angles given that
1. a = 13, A = 37o, B = 84o
2. b = 5, c = 12, B = 51