300672 Mathematics 1A
Lecture 2 Complex
Numbers
COMPLEX NUMBERS
Complex numbers came into existence when it was found that
the solution to the simple quadratic equation
x2
1
0
had no solution in the real number system.
The general quadratic equation ax bx

300672 Mathematics 1A
Lecture 11 Still more on
Applications of
Differentiation
Applied Maximum and Minimum Problems
The maximum or minimum values of a function can
occur at the end points (if the domain is restricted) or at
the turning points. Use of this

300672 Mathematics 1A
Lecture 9 Applications
of Differentiation
Maximum and Minimum Values
A function f has an absolute maximum at c if f(c)
all x in the domain, D, of the function f.
The number f(c) is the maximum value of f in D.
f(x) for
A function f h

300672 Mathematics 1A
Lecture 10 More of
Applications of
Differentiation
Consider two functions f and g such that
f a 0, g a 0
f ' , g' are continuous
g' a 0
f x f a
f x f a
f ' x f ' a x a
xa
xa
lim
lim
x a g' x
g' a lim g x g a x a g x g a
xa
xa
xa

300672 Mathematics 1A
Lecture 8 Last one on
Differentiation
Related Rates
If we are concerned with rates of change with respect to
time we differentiate implicitly with respect to t.
Sometimes different quantities are related to each other
through time. C

300672 Mathematics 1A
Lecture 5 - Differentiation
CALCULUS
Calculus is divided into two main areas, namely:
Differential calculus
Differentiation is used to compute the rate of change at
which one variable changes in relation to another, at any
particular

300672 Mathematics 1A
Lecture 7 Still more on
Differentiation
Using techniques from lecture 7,
d yx
a
dx
d ln a y x
e
dx
d ln a y( x )
e
dx
dy
ln a. eln a y( x )
dx
So:
d yx
dy
a
a y( x ) ln a
dx
dx
Derivatives of Logarithms
Let
So
300672 Week

300672 Mathematics 1A
Lecture 3 - Limits
Limits
We write
lim f ( x ) L
x c
if the function, f, has a limit, L, if x approaches the value
c from the left hand side.
We write
lim f ( x ) L
x c
if the function, f, has a limit, L, if x approaches the value
c

300672 Mathematics 1A
Lecture 6 More on
Differentiation
Product Rule
Consider two differentiable functions u = f(x) and
v = g(x). The derivative of their product is given by:
d
dv
du
( uv ) u v
dx
dx
dx
or
d
d
d
( f ( x )g( x ) f ( x ) [ g( x )] g( x ) [

300672 Mathematics 1A
Lecture 4 More on
Limits
Review of Limits
involving x approaching infinity
Last week we noted the following:
lim
Note:
x
Also:
lim
x
1
0
x
4
0
x
lim
1
0
x2
lim
5
0
x2
x
x
Examples
(1)
5x
5x
5
5
5
x lim
lim
lim
x 3 7x
x 3
x 3
7x

300672 Mathematics 1A
Lecture 1 Functions and
Inverse Functions
This lecture revises/introduces the functions and their
inverses which will be used throughout this unit.
FUNCTIONS AND INVERSE FUNCTIONS
Consider:
y=x2
x
y
2 1
4 1
y = 2x 1
0
0
1
1
2
4
x
y
3

300672 Mathematics 1A
Lecture 12 Integration
(This section is non-examinable but is included to give a taster for 300673 Maths 1B)
Antiderivatives
A function F is called an antiderivative of f on an interval
I if
F' x f x
for all x in I.
Examples
(1)
F x