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Methods bound reference 111

Mathematical Methods- Unit 3 (Best notes for high school - AU)

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2022 Method Mathemat ics :

Bound Reference
Lena Pham

Core: Functions and graph + algebra

Core: Calculus
Core: Probability and Statistics

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FUNCTIONS AND GRAPHS

FUNCTIONS AND RELATIONS.............................................................................................. 3

POLYNOMIALS .................................................................................................................... 5

CIRCULAR FUNCTIONS........................................................................................................ 8

TRANSFORMATIONS ......................................................................................................... 11

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Functions and Graphs

Functions and relations
Set Notation
A + B is the intersection of A and B , where the elements that are common in both
sets

A , B is the union of A and B, where there are all the elements of both A and B

A \ B is the set difference, where it states all the elements in A but not in B

B ¦ A every element of B is in an element of A, where B is the subset of A

Note:
- The ÔclosedÕ circle(¥)indicates that the number is included.
- The ÕopenÕ circle(ç) indicates that the number is not included.

The following are subsets of the real numbers for which we have special notation
- Positive real numbers: R+ ={x:x>0}
- Negative real numbers: R2 ={x:x<0}
- Real numbers excluding zero: R\{0}

Vertical line test Horizontal line test

If it only interests the graph a maximum of If it only interests the graph a maximum of
once, then the relation is a function once, then the function is one-to-one
(

Odd and Even functions

- A function f is odd if f (2x) = 2 f (x) for all x in the domain of f
- A function f is even if f (2x) = f (x) for all x in the domain of f

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Sum of functions

( f + g)(x) = f (x) + g(x), where dom( f + g) = dom f + domg

Difference of functions

( f 2 g)(x) = f (x) 2 g(x), where dom( f 2 g) = dom f + domg

Product of Functions

( f · g)(x) = f (x) · g(x) , where dom( f · g) = dom f + domg

Composite Functions
- If range of f ¦ domain of g, the composition g ç f is defined and

g ç f = g( f (x)) with dom( g ç f ) = dom f

- If range of f ¦ domain of f , the composition f ç g is defined and

f ç g = f (g(x)) with dom( f ç g ) = domg

Coordinate Geometry
- Distance between two points A B = (x 2 2 x1)2 + (y2 2 y1)2
- Midpoint between two points
- Gradient between two points
- Tangent of the angle of slope- For a straight line with gradient m, the angle of slope is
found using m = tan»
where » is the angle that the line makes with the positive direction of the x-axis.

Simultaneous equations
3 cases for a system of two linear equations in two variables
- Unique solution: lines intersect at a point, eg. y = 2x + 3 and y = 3x + 3
- Infinite solutions: lines coincide, eg. y = 2x + 3 and 2y = 4x + 6
- No solution: lines are parallel, eg. y = 2x + 3 and y = 2x + 4

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Polynomials
Polynomials- It is an equation or an expression that has terms that has been raised by powers
eg. y = x 3 + x 2 + x + 2

Graphing quadratics in turning point form

- General term y = a(x 2 h)2 + k, where vertex is ( h , k )
- y = a(x 2 h)2 + k also is y = a x 2 + bx + c through completing the square
- The axis of symmetry is x = h

The axis of symmetry

- Equation for axis of symmetry is x =
Completing the square
eg. 1. 2.

Quadratic formula

The discriminant
The discriminant — of a quadratic polynomial a x 2 + bx + c is
— = b2 2 4ac
For the equation a x 2 + bx + c = 0
- If — > 0 , there are two solutions
- If — = 0, there is one solution
- If — < 0, there are no real solution
Determining the rule for a parabola

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Division and factorisation of polynomials(

When we divide the polynomial P(x) by the polynomial D(x) we obtain two polynomials,
Q(x)the quotient and R(x) the remainder, such that
P(x) = D(x)Q(x) + R(x)

Dividend Divisor Quotient Remainder

and either R(x) = 0 or R(x) has degree less than D(x).

Eg. Dividing by a non-linear polynomial

The remainder theorem The factor theorem

When P(x) is divided by ³x + ³, If ³x + ³ is a factor of P(x) then
the remainder is

(
Sums and differences of cubes(
x 3 2 a 3 = (x 2 a)(x 2 + a x + a 2) x 3 + a 3 = (x + a)(x 2 2 a x + a 2)
(
(
(
(

Sign diagrams
For a cubic function with rulef (x) = (x 2 ³)(x 2 ³ )(x 2 ³) , where ³ < ³ < ³,
the sign diagram is as shown.

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Exponential and Logarithmic Functions

Index Laws

Logarithms
a x = y is equivalent to loga y =x
Logarithm Laws

Change of base

and

Law of exponential change

There are many situations in which a varying quantity can be modelled by an exponential
function. Let A be the quantity at time t. Then
kt
A = A0e
where A0 is the initial quantity and k is a constant. Growth corresponds to k > 0, and decay
corresponds to k < 0.

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Circular functions
Degrees and radians

Circular functions

Graphs of y = a sin(nt) and y = a cos(nt)(

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The tangent function

Asymptotes:

k=0, =

k=1,

k=2,

k=3,

Hence the asymptotes are

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General solution of trigonometric equations

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Transformations
Translations

Dilations

Reflections

Combination of transformations

Determining transformations

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Matrices

Addition between two matrices

Note: Subtraction is defined in a similar way

Multiplication between matrices example:

Using matrices for transformations( Example:

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(
Note: If a transformation is defined by a matrix of the form where then

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FUNCTIONS AND GRAPHS

APPLICATIONS OF DIFFERENTITATION................................................................................ 3

POLYNOMIALS .................................................................................................................... 5

CIRCULAR FUNCTIONS........................................................................................................ 8

TRANSFORMATIONS ......................................................................................................... 11

CACULUS .......................................................................................................................... 15

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Applications of Differentiation

Average rate of change

Derivative of a function(first principle)

Note: This definition of the derivative assumes that the limit exists. For polynomial functions, such limits always exist. But it
is not true that for every function you can find the derivative at every point of its domain.

Leibniz notation
An alternative notation for the derivative is the following:

An angle associated with the gradient of a curve at a point

Derivatives of circular functions(

Note

Example-

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The chain rule(using Leibniz notation)

Example-

x
Differentiation of e (

Example-

1. 2.

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The product rule (Leibniz notation)

Example-

The quotient rule (Leibniz notation)

Example-

Differentiation of the natural logarithm function

Example-

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Algebra of limits

Example-)

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Stationary Points

Example-

Types of Stationary Points

)

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Tangents and normals

Equation of a tangent
Suppose (x1, y1) is a point on the curve y = f (x). Then, if f is differentiable at x = x1, the equation of the
tangent to the curve at (x1, y1) is given by y2y1 = f2(x1)(x2x1).
Example-

Example-

Gradient of normal
If a tangent has gradient m, the normal has gradient
Example-

Absolute maximum and minimum values

For a function defined on an interval:
- the actual maximum value of the function is called the absolute maximum
- the actual minimum value of the function is called the absolute minimum
Example-

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Rates of Change
For the function with rule f (x):(

Example-

Motion in a straight line(

Example-

Note: further derives velocity

or
Derives original equation twice

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Example-

Velocity 1. 2. Sub into

Acceleration

Example-

Velocity

When

Sub into to find where

Where

Maximum and minimum problems

Example-

Also, y= 50 - x

Sub x=25 into y= 50 - x, y=25

Example-

x + y =4 Quadratic formula
y= 4 - x

Sub x= 4/3 into

y= 4 - x 23

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Example-

SA= 150
length= x , width = x , height= h

SA= 2lw + 2lh + 2hw

= 2x 2 + 2xh + 2hx
= 2x 2 + 4xh

V= lwh
= x2 h

Example-

Quadratic formula

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Example-

Area= (a+b) x h

Pythagorean identity

Also, -

Factorised

Look at the

sub into

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Example-)

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Example-

Example-

Example-

Example-

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Integration
The area under a graph
Divide the interval [a, b] on the x-axis into n equal subintervals [a, x1],
[x1, x2], [x2, x3], . . . , [xn21, b] as illustrated. Estimates for the area
under the graph of y= f(x) between x=a and x=b:

Left-endpoint estimate

Right-endpoint estimate

Example-

Example-

a Apply the right-end point rule with strips of 1 width 0.25 to estimate the area under the curve.
b Hence find an approximate value for Ã. How could you improve the
approximation? (

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Antidifferentiation(
Indefinite integrals
r
The antiderivative of x where r b 1

Properties of Antidifferentiation

Example- Example-

r
The antiderivative of (ax + b)
r
The antiderivative of (ax + b) where r b -1
(

r
The antiderivative of (ax + b) where r = -1

Example-
Example-

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kx
The antiderivative of e

Example-

Example-

The fundamental theorem of calculus and the definite integral

b
For any continuous function f on an interval [a, b], the definite integral f (x)d x gives the signed
a
area enclosed by the graph of y = f (x) between x = a and x = b .

Fundamental theorem of calculus

Example-

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Finding the area under a curve

Finding the area of a region

Example-

CanÕt use
a=0
b=6 As f(x) is not bigger than 0 for (c,b) so must use
c=4 the rule

Example-

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Example-

Integration of circular functions

Example-

sin à = 0

Miscellaneous exercises
Example-

Product rule

c o s2 x + s i n2 x = 1

= sec 2 x + c

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Example-

The area of a region between two curves

Example-

Example-

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Applications of integration

Average value of a function(

Example-

Example-

Use rule

Use rule a

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Example-

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