AMath/PMath 331
1. (a) Note that S (x) =
1
1+x2
|SxSy |
|x y |
1
2 (1
+
x
)
x2 +1
Solutions to Assignment 6
lies in the interval
1
2 (1
1), 1 (1 + 1) = (0, 1).
2
Similarly, T (x) =
lies in (0, 1], and T (x) < 1 except for x = 0. So by the Mean
Value Theo

Reading, Discovering and Writing Proofs
Version 0.2.6
c Steven Furino
August 29, 2012
Contents
I
Introduction
11
1 In the beginning
1.1 What Makes a Mathematician a Mathematician?
1.2 How The Course Works . . . . . . . . . . . . . .
1.3 Why do we reason f

Math 137 Practice Final Exam
Waterloo Learning Centre
December 8, 2012
Question
1
2
3
4
5
6
7
8
Total
Mark
/12
/12
/11
/13
/7
/7
/10
/16
/88
1. (a) If f (x) =
x1/5
x+x2 ,
nd f (1) [3].
(b) If h(x) = ln( x), specify the domain of h. Then nd h (x) and speci

BU 111 Time Value of Money Formulae Sheet on Final Exam
FVsingle amount = PVSA(1 + r ) n
PVsingle amount =
FVsA
(1 + r ) n
FVordinary annuity
FVannuity due
(1 + r ) n 1
= PMT
r
(1 + r ) n 1
= PMT
(1 + r )
r
1
1
PVordinary annuity = PMT
n
r r (1 + r

Jared Adelstein
Practice Questions BU111 Midterm
1. Identify and describe the 4 steps in the managerial process.
2. Identify the six critical success factors and explain how they are connected.
3. Draw the Diamond-E Framework and explain how each are conn

MATH 249 NOTES
Ian Goulden April 4, 2008
1
Lecture of January 9
The rst six weeks of the course will be concerned with Enumerative Combinatorics, also referred to as Enumeration, Combinatorial Analysis or, simply, Counting. This subject concerns the basic

Copyright c 2005 by Karl Sigman
1
Fund theorems
2 In the Markowitz problem, we assumed that all n assets are risky; i > 0, i cfw_1, 2, . . . , n. This lead to the efficient frontier as a curve starting from the minimum variance point. We learned that in t

Note: In all questions the norm on Rn is assumed to be Euclidean norms.
[8]
1. Denitions and theorem statements:
(a) State the Cantor intersection theorem.
(b) State the Bolzano-Weierstrass Theorem.
(c) Dene : S is a complete subset of Rn .
(e) State the

Copyright c 2005 by Karl Sigman
1
Portfolio mean and variance
Here we study the performance of a one-period investment X0 > 0 (dollars) shared among
several dierent assets. Our criterion for measuring performance will be the mean and variance
of its rate

Copyright c 2007 by Karl Sigman
1
1.1
IEOR 4700: Introduction to stochastic integration
Riemann-Stieltjes integration
b Recall from calculus how the Riemann integral a h(t)dt is dened for a continuous function h over the bounded interval [a, b]. We partit

mlbaker.org presents
PMATH 352
Complex Analysis
Dr. Wentang Kuo Spring 2011 (1115) University of Waterloo
Disclaimer: These notes are provided as-is, and may be incomplete or contain errors.
Contents
1 Review
2
2 Holomorphic functions
2.1 Holomorphic func

1
Interest rates, and risk-free investments
Copyright c 2005 by Karl Sigman
1.1
Interest and compounded interest
Suppose that you place x0 ($) in an account that oers a xed (never to change over time)
annual interest rate of r > 0 ($ per year). x0 is call

Copyright c 2007 by Karl Sigman
Binomial lattice model for stock prices
Here we model the price of a stock in discrete time by a Markov chain of the recursive form
Sn+1 = Sn Yn+1 , n 0, where the cfw_Yi are iid with distribution P (Y = u) = p, P (Y = d)

Copyright c 2006 by Karl Sigman
1
IEOR 4700: Notes on Brownian Motion
We present an introduction to Brownian motion, an important continuous-time stochastic process that serves as a continuous-time analog to the simple symmetric random walk on the one
han

Copyright c 2005 by Karl Sigman
1
1.1
Internal rate of return, bonds, yields
Internal rate of return
Given a deterministic cash ow steam, (x0 , x1 , . . . , xn ), where xi (allowed to be positive, 0 or
negative) denotes the ow at time period i (years say)

Copyright c 2005 by Karl Sigman
1
Capital Asset Pricing Model (CAPM)
We now assume an idealized framework for an open market place, where all the risky assets refer to (say) all the tradeable stocks available to all. In addition we have a risk-free asset

Copyright c 2006 by Karl Sigman
1
Geometric Brownian motion
Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is not appropriate for modeling stock prices. Inste

Midterm exam rooms for AM/PM331 Fall 1014
November 6th 4:00 to 6:00
DWE 1502: All students registered in PM331 only. (20 seats)
DWE 2527: Students registered in AM331 Alphabetical block: A to R (48 seats)
DWE 3517: Students registered in AM331 Alphabetica

Topics to review for AmPm331 midterm, Fall 2014.
1) The real numbers .
Countable and uncountable subsets of .
Convergent sequences in .
Basic limit properties.
Bounded and unbounded subsets of .
Supremum and infimum of a subset of .
The Least upper

PMath/AMath 331, Fall 2014 - Assignment 2
Handed out on Friday, September 19; due on Friday, September 26.
Topics: Upper and lower bounds, Least upper bound principle, Monotone convergence,
Bolzano Weierstrass theorem, Cauchy sequences.
Please submit for

PMath/AMath 331, Fall 2014 - Assignment 7
Handed out on Friday October 24. Due Friday October 31.
Topics: Lipschitz functions, linear transformations, contractions.
Problem 1.
a) Use the Mean Value Theorem to show that any dierentiable real valued functio

PMath/AMath 331, Fall 2014 - Assignment 9
Handed out on Friday November 7; due on Friday November 14.
Topics: Intermediate Value theorem, continuity in normed vector spaces, uniformly
continuous functions.
Problem 1. (a) Show that, if f : Rn Rm and g : Rn

PMath/AMath 331, Fall 2014 - Assignment 10
Handed out on Friday November 14; due on Friday November 21.
Topics: Finite dimensional vector spaces, Pointwise convergence of functions, uniform
convergence of functions.
Problem 1. Review denitions 10.3 and 10

PMath/AMath 331, Fall 2014 - Assignment 11
Handed out on Friday November 21; due on Friday November 28.
Topics: Pointwise convergence of functions, uniform convergence of functions.
Problem 1. Review denitions 28.1, 28.2 and theorem 28.3. Consider the seq

Friday, October 24 Lecture 20 : Contractions and fixed points.
Expectations:
1. Verify in the case of a simple dynamical system (X, T) whether T is a contraction or
not.
20.1 Proposition Suppose S V. Suppose T : S V is a contraction on S with a fixed
poin

Wednesday, September 10 Lecture 2 : Limits
1. Define the limit L of a sequence cfw_xn in .
2. Use the definition of the limit to determine whether a sequence converges or not
in .
3. Use the Squeeze theorem to show a sequence converges.
Objectives:
2.0 In

Wednesday, October 22 Lecture 19 : Applications: Dynamical systems.
Expectations:
1.
2.
3.
4.
Define orbit O(x) in a discrete dynamical system (X, T).
For a system (X, T ), define a fixed point of T.
For a system (X, T ) define T is a contraction on X .
D

Topics to review for AmPm331 midterm, Fall 2014.
1) The real numbers .
Countable and uncountable subsets of .
Convergent sequences in .
Basic limit properties.
Bounded and unbounded subsets of .
Supremum and infimum of a subset of .
The Least upper