THE UNIVERSITY OF NEvV SOUTH \iVALES
SCHOOL OF NIATHEMATICS AND STATISTICS
Semester 2 2014
MATH1231
MATHEMATICS lB
(1) TIME ALLOvVED - TWO (2) HOURS
(2) TOTAL NUMBER OF QUESTIONS - 4
(3) ANSvVER ALL QUESTIONS
(4) THE QUEST IONS ARE OF EQUAL VALUE
(5) ANSW
Maple Lab Test Solutions
Oliver Tan
MATH123/41 - 2013s2
This is based off the all inclusive maple lab test questions (the one with 40 marks).
Notes and Tips
*x* means to plug x in place of it
This resource is not responsible for you losing marks. Do not
MATH1231
2011 Semester Two
SUMMARY
Tommy Sailing
Table of Contents
Important Things to Memorise.3
Calculus Component.4
Functions of Several Variables.4
Trigonometric Integration Techniques.5
Integrating Rational Functions.7
1st Order Differential Equation
Solutions to MATH1241 Test 1
These solutions were written up and typed up by Evgeny Martynov. If you spot any errors, hit
me up via FB or whatever.
MATH1231/1241 Algebra S2 2007 Test 1 Version 1B
1.
(i) We have S = cfw_x R4 | x1 5x3 = 2x4 .
Clearly, 0 S:
MATH1231 Mathematics 1B
MATH1241 Higher Mathematics 1B
INFORMATION BOOKLET
Semester 2 2015
Copyright 2015 School of Mathematics and Statistics, UNSW
1
CONTENTS OF THE
MATH1231/1241 COURSE PACK 2015
Your course pack should contain the following four items:
Maple Lab Test Solutions
Oliver Tan
MATH123/41 2013s2
This is based off the all inclusive maple lab test questions (the one with 40 marks).
Notes and Tips
0 *x* means to plug 3: in place of it
o This resource is not responsible for you losing marks. Do no
Maple Hos
Math 1B Maple Lab Test
Question 1: Score 1/1
Before attempting this question you should work through the self-directed learning Module
"Lesson 10 - Further Linear Algebra" on UNSW Blackboard.
-Use Maple to find the eigenvalues of
.
Enter the eig
MATH1231 Mathematics 1B
MATH1241 Higher Mathematics 1B
CALCULUS PROBLEMS
Semester 2 2012
Copyright 2012 School of Mathematics and Statistics, UNSW
Preface
Please read carefully.
These Notes form the basis for the calculus strand of MATH1231 and MATH1241.
Linear Maps and Matrices
Theorem
For each m n matrix A, the function TA : Rn Rm , defined by
TA (x) = Ax
for x Rn ,
is a linear map.
Proof.
Recall that, for
x1
x2
x = . Rn ,
.
xn
the vector Ax is defined by the formula
(Ax)k =
n
X
akl xl ,
1 6 k 6
MATH1231 Mathematics 1B
Chapter 2
Techniques of Integration
Lecture 6
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Techniques of Int
Applications
Finding power of a diagonal matrix is easy.
Example
1 0
0
0 . Find D 2 and D n .
Let D = 0 3
0 0 2
Solution
We have
2
1
0
0
0,
D 2 = 0 32
0 0 (2)2
n
1
0
0
0.
D n = 0 3n
0 0 (2)n
8.3 Applications
September 14, 2017
1 / 27
Recall that a matri
MATH1231 Mathematics 1B
Chapter 2
Techniques of Integration
Lecture 4
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Techinques of Int
MATH1231 Mathematics 1B
Chapter 3
Ordinary Differential Equations
Lecture 8
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats) MATH1231 Ordinary Di
Notes
MATH1231 Mathematics 1B
Chapter 1
Functions of Several Variables
Lecture 3
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Functi
MATH1231 Mathematics 1B
Chapter 2
Techniques of Integration
Lecture 5
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Techniques of Int
MATH1231 Mathematics 1B
Chapter 1
Functions of Several Variables
Lecture 3
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Functions of
MATH1231 Mathematics 1B
Chapter 1
Functions of Several Variables
Lecture 2
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Functions of
MATH1231 Mathematics 1B
Chapter 2
Techniques of Integration
Lecture 5
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Techniques of Int
MATH1231 Algebra, 2017
Dr. Dmitriy Zanin
School of Mathematics and Statistics
University of New South Wales
[email protected]
Dmitriy Zanin (UNSW)
MATH1231 Algebra
1 / 22
Contact details
Contact Details
Office: Red Centre, East Wing, Room 4075
Phone: 93
Subspaces Associated with Linear Maps
Definition (Kernel and Image of a Linear Transformation)
Let T : V W be a linear map.
The kernel of T (written ker(T ) is the set of all zeroes of T , that is,
ker(T ) = cfw_v V : T (v) = 0.
The image of T is the set
Chapter 8
Eigenvalues and Eigenvectors
Let T : Rn Rn be a linear map.
If we restrict the domain to a subspace S of Rn , the restricted function is
also a linear map. (why?)
We also know that the image of a linear map is a subspace of the
codomain. Hence,
Notes
MATH1231 Mathematics 1B
Chapter 2
Techniques of Integration
Lecture 4
Dr. Jonathan Kress, Dr. Joshua Capel
School of Mathematics and Statistics
University of New South Wales
Session 2, 2017
JM Kress, J Capel (UNSW Maths & Stats)
MATH1231 Techinques
The Binomial Distribution
Definition
Let n N and let p (0, 1). Binomial distribution B(n, p) is a function
from cfw_0, , n to R defined by the formula
n
B(n, p, k) =
p k (1 p)nk for all k = 0, 1, , n.
k
Binomial distribution is a probability distributio
Further Applications and Examples
Example
The function T : R3 P1 is defined by
a
T b (x) = (a + 2b) + (b 2c)x,
c
a
for all b R3 .
c
(a) Prove that T is linear.
(b) Find ker(T ) and Im(T ).
August 29, 2017
1 / 15
Solution
Let u = (a1 , b1 , c1 )> R3 a
Probability
We assume that you can reasonably master NSW HSC Mathematics
Extension 1 probability and counting.
We consider probability as a theory which has been developed to analyse
the outcomes of repeated experiments, where an experiment should
usually
Continuous Random Variables
In contrast to the discrete random variables, random variables such as the
height or weight of an individual cannot be specified by the probability
P(X = x) for each real value x because the probability is 0. Instead, we
will d
Chapter 9
Probability and Statistics
Statistics is the science of production, analysis and interpretation of
numerical data for an objective.
Data production
What should be measured?
Size of the sample?
Ensure randomness.
Data analysis
Organise the data i
Random Variables
In a random process, we want to measure some quantities of an outcome.
For example, in each game, both Jack and Jill toss two one-dollar coins
simultaneously (4 coins are tossed altogether). Then the coins showing
heads are belong to Jack
Diagonalisation
Why eigenvectors?
Let V be an n-dimensional vector space and T : V V be a linear
transformation. If T has n linearly independent eigenvectors, they form a
basis for V . This basis is tailor made for T .
If we compute T n : V V in an arbitr
Normal Distribution
An important continuous distribution which is widely used in statistics is
the normal distribution. It models a lot of real life statistical situations; in
particular, it is the limiting case of the binomial distribution.
Definition
A
Chapter 7
Linear Transformations
In this chapter, we are going to study a special type of functions between
two vector spaces over the same set of scalars. Suppose that V and W are
vector spaces over the same set of scalars F, but they may have different
MapleTA
week2.docx
MapleTA Week
5.docx
Maple TA
Week10.docx
MapleTA
week6.docx
mapleTA
Week4.docx
MapleTA
week3.docx
MapleTA
Week1
Vector space property
mapleTA
Week9.docx
MapleTA
week8.docx
MapleTA
week7.docx
Subset of a space vector
Examples and counter
MATH1231/1241 Algebra S2 2007 Test 1
v1B
Full Solutions
August 18, 2017
These solutions were written by Treves Li, typed up by Brendan Trinh and edited by Henderson
Koh and Aaron Hassan. Please be ethical with this resource. It is for the use of MathSoc
m
MATH1231/1241 Calculus S2 2009 Test 1
v7a
Full Solutions
August 18, 2017
These solutions were written and typed up by Gary Liang, and also edited by Matthew Yan,
Henderson Koh and Aaron Hassan. Please be ethical with this resource. It is for the use of
Ma