Given = (
1.
2 6
0
) and = (
3 8
1
1
8
APPLIED MATH FINAL STUDY GUIDE
SECTION 3
2
), find each of the following:
3
2. 1
3.
Determine whether the given set of functions satisfies the given system of differential equations.
3 2
4
= (
) , = 3 ( ) 2
2 2
2
4.
APPLIED MATH FINAL STUDY GUIDE
SECTION 1
= ( + 1), subject to the initial condition (0) = 3.
1.
Find the solution to
2.
Find the general solution to = 2 + 1.
3.
A 500-gallon tank contains 120 pounds of salt in solution. A salt solution with a concentrati
1.
APPLIED MATH FINAL STUDY GUIDE
SECTION 2
Find the general solution to (4) 6 + 9 = 4 2 .
2.
Set up but do not solve the following problem:
A mass of 60 grams stretches a spring 4cm. The mass is also attached to a viscous damper that exerts a force of 10
1. Find the solution to
dy
= x ( y+ 1 ) , subject to the initial condition y ( 0 )=3 .
dx
y
y ' =2 x +1 .
x
2. Find the general solution to
3. A 500-gallon tank contains 120 pounds of salt in solution. A salt solution with a
concentration of 0.35 pound/ga
Section 7.1
In the following problem, transform the given initial value problem into an initial value problem
for two first order equations.
u' ' +0.25 u' + 4 u=2 cos 3 t , u ( 0 ) =1,u' ( 0 )=2
Systems of first order equations can sometimes be transforme
Section 3.5
In each of the following problems, find the general solution of the given differential equation.
2t
y ' '2 y '3 y=3 e
'
'
t
y 2 y 3 y=3 t e
In each of the following problems, find the solution of the given initial value problem.
'
'
t
'
y ( 0
MATH 247
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Group.Quiz.12
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10.9.13
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1. (3 points) Determine whether
0
3 5
0
2
A= 1
4 9 7
is invertible. Provide 6 distinct reasons to justify your answer.
2. (3 points) Let U be a square matrix such that U T U = I. A matrix that satis
MATH 247
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Group.Quiz.14
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10.21.13
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3
2
4 2 1
3
2
2 5 7 3, u = , v = 1.
1. (5 points) Let A =
1
3
7 8 6
3
0
(a) Is u N(A)? Why or why not? Could u be in Col(A)? Why or why not?
(b) Is v Col(A)? Why or why not? Could v be in N(A)? W
MATH 247
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Group.Quiz.13
10.14.13
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1. (2 points each) Let V denote a vector space and H a subset of V . Determine if H is a subspace
of V in the following items. Be sure to justify your answer.
(a) V = R2 . H = the union of the rst
MATH 247
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Group.Quiz.15
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1
1
1. (5 points) Let A =
2
3
10.23.13
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2 3 4 8
2 0
2 8
.
4 3 10 9
6 0
6 9
(a) (3 points) Find a basis for N(A). Justify your answer. In complete sentences, give a recipe
for nding a basis for the null spac
MATH 247
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Group.Quiz.7
9.18.13
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1. (1 point) In your own words (whatever makes the most sense for you), state a working denition
of linear independence.
2. (2 points) Let
1
v1 = 2 ,
3
4
v2 = 5 ,
6
2
v3 = 1 .
0
Determine if cf
MATH 247
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Group.Quiz.11
10.7.13
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1. (3 points) Prove the following.
(a) If A is invertible, then A1 is also invertible and A1
1
= A.
(b) If A and B are n n invertible matrices, then so is AB, and (AB)1 = B 1 A1 .
1
T
(c) If A is in
MATH 247
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Group.Quiz.9
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9.30.13
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We now want to look at the solutions of the matrix equation Ax = b by examining the properties of
the transformation A. Particularly, I want you to look for the connection between the one-to-oneness
MATH 247
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Group.Quiz.10
10.2.13
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Rating:
1. (2 points) Use very little energy to multiply
1
1 0 0 0
0 2 3 4 5
0 0 3 0 1
3
0 0 0 4
2
6
0
1
3
7
2
6
4
8
4
0
and justify your reasoning.
2. (1 point) Suppose the third column of B is the sum of
MATH 247
Group.Quiz.8
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9.25.13
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1. (4 points) Let
1 3
3
3
2
4 , u =
A= 3
, b = 2 , c = 2 ,
1
1 7
5
5
and dene a transformation T : R2 R3 by T (x) = Ax so that
1 3
x1 3x2
x
4 1 = 3x1 + 4x2 .
T (x) = Ax = 3
x2
1 7
x1 + 7x2
(a) Fi
MATH 247
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Group.Quiz.6
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9.16.13
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1. (2 points) Let A be an m n matrix and b Rm . What can you say about the system Ax = b
if
(a) A has a pivot position in every row.
(b) the columns of A span Rm .
Summarize what you noticed from doi
MATH 247
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Group.Quiz.2
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8.28.13
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1. (4 points) Suppose CheapNFast have ights that originate from seven cities: Los Angeles, San
Francisco, Seattle, Chicago, Denver, Las Vegas, and New York with destination cities indicated
by the fo
MATH 247
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Group.Quiz.5
9.11.13
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1. (1 point) Give an example of an augmented matrix of size 4 7 that is in the row echelon form
but not in the reduced row echelon form. Justify your answer.
2. (4 points) Given a linear system of eq
MATH 247
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1. (4 points) Let u =
9.9.13
Rating:
1
2
and v =
. Answer the following questions.
1
1
5
1
(a) (1 point) display the vectors w = u v and z = 3u + 3v on the graph paper.
2
2
www.PrintablePaper.net
(b) (1 point) What
MATH 247
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Group.Quiz.3
9.4.13
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1. (3 points) Solve the following system of equations in matrix notations. What does your solution
set mean geometrically?
x1 + x2 + x3 =
2
2x1 + 3x2 + x3 =
3
x1 x2 2x3 = 6
2. (3 points) Given an augm
MATH 247
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Group.Quiz.1
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8.26.13
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1. (4 points) List four possible expected student learning outcomes.
2. (4 points) Brainstorm and describe a winning strategy for you to get an A in this class.
3. (2 points) List two biggest challen