COURSE OUTLINE
MATH1141
HIGHER MATHEMATICS 1A
Semester 1, 2015
Cricos Provider Code: 00098G Copyright 2015 School of Mathematics and Statistics, UNSW
1
CONTENTS OF THE
MATH1131/1141 COURSE PACK 2015
Your course pack should contain the following four item
Chapter 2: Vector Geometry
Daniel Chan
UNSW
Semester 1 2017
Daniel Chan (UNSW)
Chapter 2: Vector Geometry
Semester 1 2017
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Goals of this chapter
In this chapter, we will answer the following geometric
Questions
1
2
1 1
How do you define and then
Chapter 1: Introduction to Vectors
(based on Ian Dousts notes)
Daniel Chan
UNSW
Semester 1 2017
Daniel Chan (UNSW)
Chapter 1: Introduction to Vectors
Semester 1 2017
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A typical problem
Chewie points his crossbow southeast. If bolts fly at 5ms1 , it
Chapter 3: Complex Numbers
Daniel Chan
UNSW
Semester 1 2017
Daniel Chan (UNSW)
Chapter 3: Complex Numbers
Semester 1 2017
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Philosophical discussion about numbers
Q In what sense is 1 a number? DISCUSS
Q Is 1 a number?
A from your Kindergarten teache
Chapter 1: Introduction to Vectors
Denition
A vector is a quantity that has both a size, or magnitude, and a
direction.
In physics, a vector is contrasted with a scalar quantity, which
posseses a magnitude but no direction.
A vector is represented geometr
Chapter 2: Vector Geometry
Denition
The dot product (or scalar product) of two vectors
b1
a1
b2
a2
a = . and b = .
.
.
.
.
bn
an
in Rn is
a b = a1 b1 + a2 b2 + .an bn .
Properties of the dot product:
a =
aa
ab=ba
(a) b = (a b)
a (b + c) = a b +
Sets, inequalities and functions
Sets of numbers
A set is a collection of distinct objects. The objects in a set are
called the elements or members of the set.
The set N of natural numbers is given by
N = cfw_0, 1, 2, 3, 4, . . ..
The set Z of integers
Lecturer: Professor Fedor Sukochev
To contact me:
Room 5109 East Wing Red Centre
email: f.sukochev @ unsw.edu.au
Oce hour: .
Chapter 1
Sets, inequalities and functions.
Sets of numbers
A set is a collection of distinct objects. The objects in a set are
ca
Limits
Limit is the fundamental concept in calculus. There two main types of limits.
Limits at . What is the long term behaviour of the function f ?
y
f (x)
x
Limits at a point. What is the local behaviour of f for x near some point
a R?
y
f (x)
a
x
1
Lim
Limits
Limit is the fundamental concept in calculus. There two main types of limits.
Limits at . What is the long term behaviour of the function f ?
y
f (x)
x
Limits at a point. What is the local behaviour of f for x near some point
a R?
y
f (x)
a
x
1
Lim
Dierentiable functions
If you are travelling at a constant speed, then it is easy to nd this
speed by measuring a distance x and how long it takes for you to
x
travel that distance t: speed =
.
t
If you graph the distance x covered against time t, you jus
Continuous functions
Recall that: f is continuous at x0 R if
lim f (x) = f (x0).
xx0
The value f (x0) needs to be dened and the limit needs to exist!
Formal denition is: f is continuous at x0 R if for every > 0, there
exists > 0 such that f (x) f (x0) <
Inverse functions
We often think of a function as a rule which takes in an input and
assigns to it an output.
Usually we have a nice formula or recipe which tells us how to calculate the output for a given input.
Many hard and interesting problems go the
The mean value theorem
This section is about an odd looking result that stands at the foundations
of much of calculus, the Mean Value Theorem (MVT).
f (x)
C
B

f (b)

f (a) A
a

c


b
x
The mean value theorem. Suppose that a function f is continuous
The mean value theorem
This section is about an odd looking result that stands at the foundations
of much of calculus, the Mean Value Theorem (MVT).
f (x)
C
B

f (b)

f (a) A
a

c


b
x
The mean value theorem. Suppose that a function f is continuous
Limits
Limit is the fundamental concept in calculus. There two main types of limits.
Limits at . What is the long term behaviour of the function f ?
y
f (x)
x
Limits at a point. What is the local behaviour of f for x near some point
a R?
y
f (x)
a
x
1
Lim
Sets, inequalities and functions
Sets of numbers
A set is a collection of distinct objects. The objects in a set are
called the elements or members of the set.
The set N of natural numbers is given by
N = cfw_0, 1, 2, 3, 4, . . ..
The set Z of integers
Curve sketching.
1 .
You all know how to sketch the graph of y = x2 4x or y = 1x
2
In this section we will look at
Additional information you can put into a sketch.
Sketching curves that dont come in Cartesian form
y2
2
implicitly defined curves, such
Integration
R
Ill assume that you can find sin(x) dx,
R3 3
or 0 (x + ex) dx.
In this chapter well look at:
R
What area actually means.
Z b
The formal definition of
f (x) dx.
a
Z x
The area function A(x) =
f (t) dt.
0
The Fundamental Theorems of Calcul
Differentiable functions
If you are travelling at a constant speed, then it is easy to find this
speed by measuring a distance x and how long it takes for you to
x
travel that distance t: speed =
.
t
If you graph the distance x covered against time t, you
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH 1141
HIGHER MATHEMATICS 1A ALGEBRA.
Section 5:  Vector Geometry.
In Chapter 2, see looked at the rudiments of vector geometry. We now have enough machinery to study this subject in g
THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS AND STATISTICS
MATH1141 Algebra
Section 2:  Introduction to Vectors.
You may have already met the notion of a vector in physics. There you will have thought
of it as an arrow, that was used to repre
UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
MATH 1141
HIGHER MATHEMATICS 1A ALGEBRA.
Section 4:  Matrices.
In our last topic we used matrices to solve systems of linear equations. A matrix was simply
a block of numbers, which, in that context, re