2
European Style Derivatives
2.1 Asset Price Models and Its Lemma
o
2.1.1 Models for Asset Prices
MSFT 3/30/996/8/00
115
110
105
100
95
90
85
80
75
70
65
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1/2/00
Fig. 2.1. Stock price of Microsoft Inc.
As examples, in Figs. 1.11.7 we showed how the
Summative Assessment
tech n ical B u lleti n #1
Version 3, August 2014
actaspire.org
ACT endorses the Code of Fair Testing Practices in Education and the Code of Professional Responsibilities in
Educational Measurement, guides to the conduct of those invo
Integration
by substitution
There are occasions when it is possible to perform an apparently difficult piece of integration
by first making a substitution. This has the effect of changing the variable and the integrand.
When dealing with definite integral
AP CalculusIntegration Practice
I. Integration by substitition.
Basic Idea: If u = f (x), then du = f 0 (x)dx.
Example. We have
Z
x dx
x4 + 1
u = x2
=
dx = 2x dx
=
=
1
2
Z
du
+1
u2
1
tan1 u + C
2
1
tan1 x2 + C
2
Practice Problems:
Z
1.
x3 4 + x4 dx
Z
dx
Integration By Partial Fractions
Objectives
Understand the Concept of a Partial Fractions Decomposition
Use Partial Fractions Decomposition with Linear Factors to Integrate
a Rational Function
Use Partial Fractions Decomposition with Quadratic Factors
Partial fractions
3.6
Introduction
It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions.
1
3
For example it can be shown that x24x+7
has the same value as x+2
+ x+1
for any value of x.
+3x+2
We say that
1
3
4x
Foundations of Accounting (ACC 310F)
Assignment 4
Part A
A1. The Williams Company had the following accounting framework available:
Required:
a. If $240 of insurance expired, what dollar amount of ending prepaid expenses would the company
report?
b. If th
Question1 2 pts
If a bank was trying to determine whether our company should be issued a longterm loan, it would most likely use which of the following ratios?
Currentratio
Daysinaccountsreceivable
Netincomepercentage
Liabilitiestostockholdersequity
Q
SOLUTIONS TO EXAM 1, MATH 10560
Problem 1. Evaluate the integral
Z
2
2
dx
.
4 + x2
Solution.
Z
2
2
Now put u =
x
2
1
dx =
4 + x2
Z
2
2
1
4 1+
1
dx =
x2
4
4
Z
2
2
1
2 dx.
1 + x4
then dx = 2du. Also x = 2 u = 1 and x = 2 u = 1. So
Z
2
2
Z
1
1 1 2
dx =
du
4
Table of Useful Integrals, etc.
e
ax 2
0
1
dx =
2 a
2 ax 2
x e
0
0
n ax
xe
dx =
0
ax 2
dx =
0
1
2
3 ax 2
x e
dx =
0
(
)
1 3 5 2n 1
dx =
a
2 n+1 a n
x e
2
1
dx =
4a a
2n ax 2
x e
1
1
2
2
1
2a
1
2a 2
2n+1 ax
x e dx =
0
n!
2a n+1
n!
a n+1
Integra
Integration Worksheet - Substitution Method Solutions
The following are solutions to the Math 229 Integration Worksheet - Substitution Method. Heres
the link to that worksheet http:/www.math.niu.edu/courses/math229/misc/int_prac.pdf
Z
1.
(5x + 4)5 dx
(a)
Partial Fractions
A rational function is a fraction in which both the numerator and denominator are polynomials.
x + 26
4
3
For example, f ( x) =
, g( x) =
, and h( x) =
are rational functions. You
2
x 2
x +5
x + 3x 10
should already be quite familiar wit
LH
opitals rule practice problems
21-121: Integration and Differential Equations
Find the following limits. You may use LHopitals rule where appropriate. Be aware that LHopitals
rule may not apply to every limit, and it may not be helpful even when it doe
1.3
Sum of discrete random variables
Let X and Y represent independent Bernoulli distributed random variables
B(p).
Find the distribution of their sum
Let Z = X + Y . The probability P (Z = z) for a given z can be
written as a sum of all the possible com
1
Probability
1.1
Normal distribution
The hemoglobin values of an undoped athlete1 are known to be normally
distributed with = 148 and variance 2 = 85. Let X denote a measured
value of hemoglobin, i.e. X N (148, 85).
Calculate the probability that the he
1.6
The distribution of the average
As another example from the doping case, we now consider the mean value
of each individual. As it turns out, each individual may have his own mean
value and these values can vary considerably between individuals. Say th
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Mathematical Techniques : Differentiation SAMPLE
The total number of marks is 30. The marks available for each question are indicated in
parentheses. The pass mark is 24 or above. 1 alculators must not be used.
Section A. Differentiate the following exp
Math 273
Practice problems for exam 2
1. Evaluate each double integral:
a)
Z 2Z
0
4x
(x2 + y) dy dx
b)
x
Z
2
0
Z
Z 2 Z 4y2
cos(x2 + y 2 ) dx dy
c)
3
r2 sin() dr d
0
0
0
2. For each part of #1, sketch the 2-dimensional region over which the integration is
DIFFERENTIATING UNDER THE INTEGRAL SIGN
KEITH CONRAD
I had learned to do integrals by various methods shown in a book that my high
school physics teacher Mr. Bader had given me. [It] showed how to differentiate
parameters under the integral sign its a cer
MATHEMATICS IA CALCULUS
TECHNIQUES OF INTEGRATION
WORKED EXAMPLES
Find the following integrals:
Z
1.
3x2 2x + 4 dx. See worked example Page 2.
Z
2.
Z
3.
Z
4.
Z
5.
Z
6.
Z
7.
Z
8.
Z
9.
Z
1
1
dx. See worked example Page 4.
+ 2
2
x
x +1
x(x + 1)2 dx. See work
Double integrals
Notice: this material must not be used as a substitute for attending
the lectures
1
0.1
What is a double integral?
Recall that a single integral is something of the form
Z b
f (x) dx
a
A double integral is something of the form
ZZ
f (x, y
5
Indefinite integral
The most of the mathematical operations have inverse operations: the
inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse operation of exponentation is rooting. The
inverse op
Lecture Notes
Integrating by Parts
page 1
Sample Problems
Compute each of the following integrals. Please note that arcsin x is the same as sin
as tan 1 x
1.
Z
xex
dx
2.
Z
x cos x dx
3.
Z
4.
Z
xe
4x
dx
ln x dx
5.
Z
arcsin x dx
6.
Z
arctan x dx
7.
Z
ex sin
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effectiveness in Ontario schools.
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TJs Study Guide for Math 212
The best way to feel confident for the exam is to know your stuff cold. What
do I mean by that? Most concepts and mathematical objects weve learned
about have several facets. For each topic you should know:
The definition/ th