1
MATH 141B
Practice Problems for Quiz 2
1. Find the dominant (largest) eigenvalue of the matrix A =
A) 3,
B) 2,
C) 1,
1 6
.
2 6
D) 2,
E) none listed
2
6
=
, then use the denition of eigenvalue and eigenvector to determine which
1
3
of the following 3 sta
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MATH 141B
Practice Quiz 4
Date: Wed March 26, 2014
Print Name:
Student No.
[ 2 ] 1. Find the general solution to the dierential equation y 6y + 9y = 0 and put your answer
on the blank provided.
y=
[ 2 ] 2. Solve the initial value problem
y y 6y = 0
y (0
1
MATH 141B
Assignment #4
Print Name:
Due: Wednesday March 19, 2014
Student No.
INSTRUCTIONS: For full credit, your solutions must be submitted on this paper, with all your calculations
shown and nal answers on the blank lines provided. Please submit a sc
1
MATH 141B
Midterm 2
Model Summary
1) Malthusian Population Model
The growth rate (i.e., the rate of change of population) is proportional to the current population.
dP
= kP
dt
P (0) = P
0
This dierential equation is both separable and linear. The inte
1
MATH 141B
Assignment#2
Print Name:
Due: Friday Feb 7, 2014
Student No.
Instructions: Showing all your work in the space provided, answer the following questions.
Circle the correct answer on the question sheet, and bubble your answers on the scantron
pr
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Math 141B
Midterm 1
Practice Exam C
1
4xe2x dx.
[ 6 ] 1. Compute the integral
0
A) e2 + 1,
B) e2 1,
0
[ 6 ] 2.
Compute the integral
C) 1 e,
B)
E) none listed
x x + 1 dx.
1
2
A) ,
3
D) 1 2e,
2
,
15
C)
4
,
15
4
D) ,
5
E) none listed
[ 6 ] 3. Determine w
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Math 141B
1.
Midterm 1
Practice Exam B Solutions
Compute the anti-derivative rst, evaluate later. Let u = x and dv = ex1 dx. Then du = dx and v = ex1 .
Hence,
xex1 dx = uv
v du = xex1
2
xex dx = ex1 (x 1)
Evaluating, gives
1
2
1
ex1 dx = xex1 ex1 = ex
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MATH 141B
Practice Midterm C
Solutions
1
A
B
=
+
, or equivalently, 1 = A(x+1)+B (x1). Put
1
x1 x+1
1
1
1
1
1
x = 1 and get A = 1 . Put x = 1 and get B = 2 . Therefore, 2
=
.
2
x 1
2 x1 x+1
Now integrating both sides of this,
1. We need A and B so that
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Math 141B
Midterm 1
Practice Exam B
2
xex1 dx.
[ 6 ] 1. Compute the value of the integral
1
B) e 1,
A) 1/e,
C) e,
D) 2e,
[ 6 ] 2. Determine whether the improper integral
4
1
A) conv. to 2 ,
B) conv. to 1 ,
3
E) none listed
dx
converges or diverges, and
1
MATH 141B
1.
Practice Midterm B
Solutions
i) False. This is the von Bertalany model L(t) = L (L L0 )ekt .
ii) True. This is Allometric growth y = Cxk .
iii) False. This is the Weber-Fechner law R(S ) = k ln(S/S0 ).
Therefore, B) is the answer.
2. We hav
1
MATH 141B
[ 6 ] 1.
Practice Midterm B
Each of the following three functions are solutions to a dierential equation arising in one of the ve
named biological models listed in the line below:
3
a) y = 12 11e x/10
b) y = p
x 2/3
c) y = 1 ln x
3
2
3
4
Malt
1
MATH 141B
1.
Practice Midterm A
Solutions
i) True. y 1 = sin t is a pure-time DE since the right hand side of y = 1 + sin t is purely
a function of t.
dy
ii) False. For the dierential equation dx = sin(xy ), is not possible to rearrange so that all
the
1
MATH 141B
Practice Midterm B
[ 6 ] 1. Each of the following three functions are solutions to a dierential equation arising in one of the ve
named biological models listed in the line below:
a)
y = 12 11ex/10
Malthusian,
von Bertalany,
b) y =
Verhulst,
3
1
MATH 141B
Practice Midterm A
[ 6 ] 1. Which of the following 3 statements are true regarding the given dierential equations?
i) y 1 = sin t is pure-time.
dy
ii) dx = sin(xy ) is separable.
iii) y = y 3 + y 1 is autonomous.
A) i) only, B) ii) only, C) ii