MAT 2125 Mid-Term Examination 2015 Solutions
1. a) If cfw_an n
1
is a real sequence, and a 2 R, give the denition of
lim an = a.
n!1
Solution:
8" > 0 9N 2 N such that k
or
8" > 0 9N 2 N such that 8k
[2 point(s)]
Now dene a sequence cfw_an n
1
N ) |a
an |
University of Ottawa
MAT 2324 Midterm Exam
Feb 26, 2015. Duration: 80 Minutes. Instructor: Robert Smith?
Family Name:
First Name:
Do not write your student ID number on this front page. Please write your student ID
number in the space provided on the seco
ADM 2350A
October 31, 2014
Quiz #6 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
ADM 2350A
September 24, 2014
Quiz #2 Examination
Version #1 Solutions
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Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluati
20.
Find the Inverse Laplace transform of each
of the following functions.
1.
s2
(s2 +4)2
23. ln s+2
s 3
2s+3
s2 +9
24.
2s+1
s2 7s+6
3.
s
e 3s
s2 +6s 7
4.
3
2s
s (e
5.
2s+1
s2 +2s+1
6.
2s+3
e 3s
s2
7.
s2 +4
8.
1
e 2s
(s 1)2
9.
1
s2 +8s+25
10.
100(s+25)
s(
MAT 2324
2384-Practice Problems on Solving ODEs using Laplace TransformsUse Laplace transforms to solve each of the following IVPs.
1. y 00 + 10y 0 + 24y = 144t2 , y(0) = 19 , y 0 (0) = 5
12
(
0 if 0 < t < 1
2. y 00 + 3y 0 + 2y = r(t), r(t) =
, y(0) = 0,
MAT 2324
2384-Practice Problems on Systems of Dierential Equations
~
~
~
In each case, solve the Systems of Dierential Equations Y 0 = AY + F (x) for the given matrix
~
A and the vector F (x). If an initial condition is given, solve the corresponding IVP.
MAT 2324
2384-Practice Problems on Nonhomogeneous second order ODEs-Methods of
Undermined Coe cients and the Variation of Parameters
For each of the following ODEs, Find the General Solution. If an initial condition is given, nd also
the corresponding par
MAT 2324
2384-Practice Problems on Nonhomogeneous higher order ODEs-Methods of
Undermined Coe cients and the Variation of Parameters
For each of the following ODEs, Find the General Solution. If an initial condition is given, nd also
the corresponding par
20. te
MAT 2324
2384-Practice Problems on
3t cos t
21. f (t) =
(
0
t>2
t2
0<t<2
22. f (t) =
(
0
t>2
23. f (t) =
(
0
24. f (t) =
Laplace Transforms-
(
1
Find the Laplace transform of each of the
following functions.
1. e2
2.
3t
cos2 (2t)
3. et cosh(2t)
4.
MAT 2324
2384-Practice Problems on higher order homogeneous ODEs
Question 1 Find an ODE for which the given functions form a basis of solutions.
1. ex , e2x , e3x
2. ex , e
x
, cos x, sin x
3. 1, x, cos 2x, sin 2x
Question 2 Solve the given ODE.
1. y 000
ADM 2350A
October 1, 2014
Quiz #3 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
MAT 2324
2384-Practice Problems on independence of solutions of ODEs and the
Wronskian
Question 1 For each of the following higher orer ODEs, use the Wronskian to show that the given
functions form a basis of solutions.
1. y (4) = 0,
2. x2 y
3. y
000
000
MAT 2324
2384-Practice Problems on Exact ODEs and Integrating Factors
For each of the following ODEs, test for exactness. If exact solve. If not, use an
integrating factor to solve. If the ODE is equipped with an initial condition, use it
to nd the partic
MAT 2324
2384-Practice Problems on Linear Second Order ODEs with
constant coe cients
For each of the following ODEs, Find the General Solution. If an initial condition is
given, nd also the corresponding particular solution.
1. 4y 00
20y 0 + 25y = 0, y(0)
MAT 2324
2384-Practice Problems on First-order Separable- Homogeneous
ODEs
1. Find the general solution of each of the following ODEs.
(a) y 0 = 2 sec(2y)
(b) yy 0 + 25x = 0
(c) y 0 sin(x) = y cos(x)
(d) y 0 e
2x
= y2 + 1
(e) (x3 + y 3 )dx
3xy 2 dy = 0
(x
MAT 2324
2384-Practice Problems on Homogeneous Euler-Cauchy Equations
For each of the following ODEs, Find the General Solution. If an initial condition is
given, nd also the corresponding particular solution.
1. x2 y 00
6y = 0
2. x2 y 00
7xy 0 + 16y = 0
ADM 2350A
November 21, 2014
Quiz #8 Examination
Version #1
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation of that
ADM 2350A
November 7, 2014
Quiz #7 Examination
Version #1 Solutions
Name: _
Student ID #: _
Statement of Academic Integrity
The Telfer School of Management does not condone academic fraud, an act by a student that may result in a
false academic evaluation
2324
MAT 2384-Practice Problems on Linear First Order ODEs and Bernoulli
Equation
For each of the following ODEs, Find the General Solution. If an initial condition is
given, nd also the corresponding particular solution.
1
1. x2 y 0 + 3xy = x , y(1) =
1
MAT 2125 Assignment 5
Due 4:30pm, 13-Apr-2015.
Instructor: Barry Jessup
Family Name:
1
First Name:
2
Student number:
3
4
5
(For the markers use only !)
6
Total
PLEASE READ THESE INSTRUCTIONS CAREFULLY.
1. Only certain parts of the questions will be graded
2 HMszs Set I {Bags-4
1. For n E N \ {0}, dene open intervals Vn on the real line by Vn = (- .
a) Explain briey Why (0, 1) is not compact.
b) Prove that C 7- {Vn l n 2 1} is an open cover of (O, 1).
c) Without using the Heine-Borel theorem, prove that no
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 1
1. Prove by induction that the sum of the first n odd integers is n2 .
2. Prove by induction that 13n 8n is divisible by 5 for any natural n.
3. Let a1 = a2 = 5, and
an+1 = an + 6an1
for
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 5
P
x
1. Prove that
nxn = (1x)
2 for |x| < 1, and use this formula in order to find
P n P n Pn=1
n n
n=1 2n ,
n=1 3n ,
n=1 (1) 3n .
2. Prove that
X
(1)n 2n
x2
x
e
=
n!
n=0
Rx 2
for x R. B
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 2
Solutions
1. Prove without using the limit theorems for sequences that
lim
n
n+2
1
= .
2n 3
2
Solution. Denote our sequence by fn =
n+2
2n3
and its conjectural limit
by f = 21 . X Then w
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 3
1. Determine whether the series
X
(1)n sin n
n=1
converges and whether it absolutely converges.
2. Do the same for the series
2
X
n cos n
(1)
n3
n=1
3. Do the same for the series
X
n=1
4
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 3
Solutions
1. Determine whether the series
it absolutely converges.
n
n=1 (1)
P
sin n converges and whether
P
Solution. A necessary condition for convergence of a series
an is
that an 0.
MAT2125 Elementary real analysis (Winter 2016)
Homework assignment 4
Solutions
1. Prove (by using the definition) uniform continuity of the function
f (x) = 2x 3 on the real line R.
Solution. Since |f (x) f (x )| = 2|x x | by the definition of the
functio
#
#Question 1
#
def count_pos(anything):
'(list)->int
function takes a list l and returns the number of positive elements in
that list
'
count = 0
for i in anything:
if i> 0:
count+=1
return count
#main
lst = input("Please enter a list of numbers seperate
def abs_v1(x):
if x<0:
return -x
else:
return x
def abs_v2(x):
if x<0:
return -x
return x
def abs_v3(x):
if x<0:
x=-x
return x
def format_age(age):
'
(int)->string
Returns a string representing somebody's age'
if age < 20:
retval= str(age)
elif age<30:
re
def longest_run(l):
' (list)-> int
Function takes a list that is filled with only integers and returns
the longest run inside the lst
preconditions:
real numbers only
'
count_1 = 0
count_2 = 1
i = 0
while (i+1)<len(l):
if l[i] = l[i+1]:
count_1 = count_1
#
#Question 2
#
def two_length_run(l):
'(list) -> bool
function takes list l and returns True if the list l has
atleast one run with two identicle consequetive values
preconditions: numbers only
'
for i in range(len(l):
if (i+1)<len(l) and l[i] = l[i+1]:
class Point:
'class that represents a point in the plane'
def _init_(self, xcoord=0, ycoord=0):
' (Point,number, number) -> None
initialize point coordinates to (xcoord, ycoord)'
self.x = xcoord
self.y = ycoord
def setx(self, xcoord):
' (Point,number)->No