Question
Max
Your grade
1
6
2
4
3
6
4
4
5
6
2b &6 (Bonus)
4
Solution: HW #2 : solutions
1. (6 points) Find the supremum of each of the following sets in R, if it exists. Prove your assertions.
(a) S = [1,
2) [3, 7)
(b) T = cfw_ 2r|r Q (0, 1)
(c) U = cfw_
MAT1325 : Homework #2
Prof : Monica Nevins
Due Thursday, February 4, 2016 by 9pm
at the Department of Mathematics and Statistics (KED)
Staplers are not available at the Department.
Instructions :
(1) Read each question carefully, and prepare your solution
Question
Max score
Your score
1
4
2
4
3
6
4
4
5
6
6 (Bonus)
2
HW #1
1. (4 points) Prove that the axioms imply that for every x R, (1)x = x.
Recall that 1 is dened as the additive inverse of 1, and x is the additive inverse of x. You
may use any other resu
MAT1325 : DGD#1 LOGIC AND MATHEMATICAL PROOF
A useful notation :
n
X
ai = a1 + a2 + + an ; so for example
i=1
4
X
f (2i) = f (4) + f (6) + f (8).
i=2
1. (Proof by cases) An integer n is even if there exists an integer k such that n = 2k, and it is odd if
MAT1325: DGD#8 SERIES
1. Let
X
an be a convergent series. Prove that if you define a new series by grouping the terms in pairs,
n=0
like:
X
(a1 + a2 ) + (a3 + a4 ) + =
(a2n + a2n+1 )
n=0
then this new series is also convergent, and has the same sum=limit.
MAT1325-DG4
Mohammad Bardestani
March 13, 2014
These notes might have many typos and there are not really polished. The only intention of
writing these notes is just to provide a set of questions and solutions, and an example of how
to write a proof. So I
MAT1325-DG3
Mohammad Bardestani
February 11, 2014
These notes might have many typos and there are not really polished. The only intention of
writing these notes is just to provide a set of questions and solutions, and an example of how
to write a proof. S
MAT1325-DG2
Mohammad Bardestani
January 22, 2014
These notes might have many typos and there are not really polished. The only intention of
writing these notes is just to provide a set of questions and solutions, and an example of how
to write a proof. So
MAT1325-DG1
Mohammad Bardestani
January 16, 2014
These notes might have many typos and there are not really polished. The only intention of
writing these notes is just to provide a set of questions and solutions, and an example of how
to write a proof. So
MAT 1325, Winter 2014
Assignment 1-Solution
(10 points)
Instructor: Mohammad Bardestani
Question 1 [2 points] In this assignment we want to show that
2 + 3 is not a rational number. We
prove this by contradiction.
a) Show that if 2 + 3 is rational then 2
MAT1325: SOME EXERCISES TO PREPARE FOR QUIZ #1 ON FEBRUARY 13
This list is representative of the kinds of questions you may anticipate for the quiz on the 13th of February,
or the midterm. The remarks in italics are additional, and would not normally be p
MAT1325 : Calculus II and an Introduction to Analysis:
Winter 2015
Professor: Rebecca deBoer, [email protected]
Lectures: Mondays 10:0011:30 and Wednesdays 8:3010:00 in MRT 205.
Discussion group: Fridays 1:002:30. The rst DGD will take place on Friday,
MAT1325: PREPARATORY EXERCISES FOR THE TEST ON MARCH 27, 2015
This list of questions is representative of the times of questions that you may see on the test. The remarks in
italics are additional and would not typically appear on the test. The test will
MAT1325: QUIZ #2, APRIL 1, 2016
PROFESSOR: MONICA NEVINS
1. (a) (2 points) State the Comparison TestPfor series. (Be
careful to include all hypotheses.)
P
Solution: Comparison test: Suppose n=1 an and n=1 bn are two series such that for all n,
0 anP bn .
MAT1325 : QUIZ #1, FEBRUARY 12, 2016
PROFESSOR : MONICA NEVINS
Multiple solutions are possible ; here are some.
1. (a) (2 pts) Let S be a nonempty subset of R. Define the supremum of S.
Solution: Let s R. Then s = sup(S), the supremum of S, if : (a) for e
1. (4 points) For each of the following series of positive terms, decide if the series converges or diverges. State
explicitly which test you apply, verify its hypotheses explicilty and be sure to use complete sentences
P to explain
1
your answer. (Marks
1. (4 points) (Area between curves)
(a) Let r > 0. Find the area of the region enclosed by the curve x2 +y 2 = r2 . (Hint: trigonometric substitution)
(b) Let a, b > 0. Find the area of the region enclosed by the curve ax2 + by 2 = 1.
Solution: (a) We evi
Question
Max score
Your score
1
4
2
4
3
6
4
4
5
6
6 (Bonus)
2
HW #1
1. (4 points) Prove that the axioms imply that for every x R, (1)x = x.
Recall that 1 is defined as the additive inverse of 1, and x is the additive inverse of x. You
may use any other re
MAT1325 PRACTICE MIDTERM
The midterm will have the following instructions :
(1) This is a closed-book exam. No notes, calculators or other devices allowed. A language dictionary
is permitted.
(2) You have 80 minutes for this exam.
(3) This exam is out of
MAT1325 : Calculus II and an Introduction to Analysis
Winter 2016
Professor: Monica Nevins, [email protected], Office KED 102 (585 King Edward).
Office hours: The hour before and after each lecture. I will schedule additional office hours
each week, as a
MAT1325 : Homework #3 : Part B (Part A is on Blackboard
Learn)
Prof : Monica Nevins
Due Thursday, February 25, 2016 by 9pm
at the Department of Mathematics and Statistics (KED)
Staplers are not available at the Department.
This assignment is excellent pra
MAT1325 : PRACTICE QUESTIONS FOR THE FINAL EXAM
1. State the following definitions or theorems :
(a) A bounded subset S R
(b) The supremum of a set S R
(c) The limit of a sequence cfw_an nN
(d) limn an =
(e) A decreasing sequence
(f) The Bolzano-Weierstr
Solution:
HW #2
1. (3 points) Prove the following sequence is convergent, and find its limit, using theorems from
class. A rigourous and well-written answer is required.
an =
n43n + 5
6n54n
(n 1).
Solution:
We write
n(43 )n
5
1
an =
+
=
4
n
4n
6n(5 )
6n(5
MAT1325 : Homework #4
Prof : Monica Nevins
Due : Thursday March 10, 2016 before 9pm,
at the Department of Mathematics and Statistics (KED)
There are no staplers available at the department.
Instructions :
(1) Read each question carefully, and prepare your
MAT1325 : Homework #2
Prof : Monica Nevins
Due Thursday, February 4, 2016 by 9pm
at the Department of Mathematics and Statistics (KED)
Staplers are not available at the Department.
Instructions :
(1) Read each question carefully, and prepare your solution
UNIVERSITY OF OTTAWA
MAT1325 : MIDTERM EXAMINATION
MARCH 2, 2016
PROFESSOR : MONICA NEVINS
Solutions. Please note that most solutions can be found in the course notes (either from class, or
on Blackboard Learn).
1. (4 points) The following are statements
MAT1325 : Homework #4
Prof : Monica Nevins
Due : Thursday April 7, 2016 before 9pm,
at the Department of Mathematics and Statistics (KED)
There are no staplers available at the department.
Instructions :
(1) Read each question carefully, and prepare your
1
1
1. Find the arc length of the curve y = x2 ln(x) from x = 2 to x = 4. Sketch the curve, to check that your
2
4
answer seems reasonable.
Compute f (x) = x
1
4x .
Then the arc length is given by
4
4
1 + (f (x)2 dx =
L=
1+ x
2
2
4
x2 +
=
2
Since x +
1
4
MAT1325 : SOME EXERCISES TO PREPARE FOR THE MIDTERM ON MARCH 4
This list is representative of the kinds of questions you may anticipate for the midterm. The remarks in
italics are additional, and would not normally be part of the question on an exam. The
MAT1325 : Homework # 1
Instructor: Rebecca deBoer
Due : Friday January 23 before 1 pm
in the department of mathematics and statistics (KED)
A stapler is not provided in the department.
Directions :
(1) Read each question carefully. If you have questions,