methods expo 12

8/12/2019 Methods Expo 12

1/60

Maths Methods (CAS)


2/60


3/60


4/60


5/60


6/60


7/60

Copyright Neap VEMM12.FM 1

Contents

Section 1: A brief look at school-assessed coursework . . . . . 21.1 Unit 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Unit 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 How you will be assessed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Section 2: Neap programs . . . . . . . . . . . . . . . . . . . . . . . . . 4Section 2:The value of Neap programs . . . . . . . . . . . . . . . . . . . . . . . . 4

Section 3: Functions and graphs . . . . . . . . . . . . . . . . . . . . . 53.1 What you should know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3.2 What you will need to learn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Basic graph work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.4 Introduction to quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Basic trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Trigonometric graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.7 Logarithmic and exponential functions. . . . . . . . . . . . . . . . . . . . 22

3.8 Applications of exponential and logarithmic functions . . . . . . . . . 25

3.9 CAS calculators - an essential tool . . . . . . . . . . . . . . . . . . . . . . 27

3.10So what will happen this year?. . . . . . . . . . . . . . . . . . . . . . . . . 29

3.11School-assessed coursework . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.12Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Section 4: Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 What you should know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 What you will need to learn . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Section 5: Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1 What you should know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2 What you will need to learn . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Section 6: Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.1 What you should know . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 What you will need to learn . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


8/60


9/60

Section 1: A brief look at school-assessed coursework


The assessment program will be developed according to the following weightings.

Unit 3

Unit 4

1.4 How you will be assessed

1. Satisfactory completion i.e. S or N.

You need to display a satisfactory ability to apply the outcomes to the content specified in each unit.

If you complete the set exercises and all assessment tasks you are most likely to gain an S.

2. Level of achievement

Your ultimate study score for Maths Methods will be determined by your grades in each of:

School assessed course work

> Unit 3: 20%

> Unit 4: 14%

Exam 1 (technology free): 22%

Exam 2 (multiple choice and extended answer): 44%

Unit Outcome 1 Outcome 2 Outcome 3 Total

3 30 20 10 60

4 15 15 10 40

Total 45 35 20 100

Applications Task (40 marks) Tests (20 marks)

Outcome 1 Outcome 2 Outcome 3 Outcome 1 Outcome 3

15 20 5 15 5

Analysis Task 1 (20 marks) Analysis Task 2 (20 marks)

Outcome 1 Outcome 2 Outcome 3 Outcome 1 Outcome 2 Outcome 3

7 8 5 8 7 5


10/60


11/60

Section 3: Functions and graphs



3.1 What you should know

Unit 1

use and interpretation of graphs to express relationships;

sketch graphs of straight lines, quadratics and cubics (including the use of simple transformations);

domain and range of functions of a real variable;

using a sliding vertical line as a test for a function; and

recognition that circles of the form are not functions.

Unit 2

Circular functions

application of trigonometric ratios to right-angled triangles;

exact values of sin and cos of 30, 45 and 60;

definition of a radian and conversion between radians and degrees;

unit circle;

defining sine, cosine and tangent;

showing that and that , ;

special values; for example, ;

symmetry properties: ;

exact values of sin and cos of integer multiples of and ;

graphs of trigonometric functions of the form , , for cases of aand b, and

the graph of ;

applications of trigonometric functions such as tidal heights and temperature changes;

recognition and interpretation of period and amplitude;

solution of simple equations of the form , wherefis sin, cos, or tan, on a domain, by graphical

methods or by using a CAS calculator;

sketching graphs of and , and solving indicial equations related to graphs by CAS

calculator or by graphical methods;

sketching the graph of using CAS calculator generated values and relating graph to that

of ; informal discussion of their inverse relationship; and

mathematical modelling using exponential functions.

x h( )2

y k( )2

+ r2

=

x( )sin2 x( )cos2+ 1= 1 x( ) 1cos 1 x( ) 1sin

0( )sin 0,= ( )cos 1=

x( )sin x( )cos 2 x( )sin 2 x( )cos, , ,

6---

4---

y a bx( )sin= y a bx( )cos=

y a x( )tan=

f x( ) B=

y 10x

= y 2x

=

y x( )10log=

y 10x

=


12/60

VCE Units 3 & 4 Mathematical Methods (CAS): Coursework & Exam Preparation

6 VEMM12.FM Copyright Neap

3.2 What you will need to learn

Units 3 & 4

graphs of polynomials to degree 4 in factorised form;

graphs of the form where a, bis an element ofz. This includes graphs of the form ,

and , for example, including key features such as asymptotes and axes intercepts;

graphs of the form and (where ) including key features such as

asymptotes and axes intercepts;

transformations of functions (reflections, dilations and translations), addition of ordinates, products of

functions and modulus functions (graphs with oblique asymptotes are excluded);

graphs of inverse functions and relations; and

mathematical modelling of all these functions.

Trigonometric functions

reflections, dilations and translations of graphs of trigonometric functions;

solution of trigonometric equations excluding those involving horizontal translations; and

recognition that equations of the form can be solved as .

Note: For more detail on assessable dot points refer to the VCAAs Mathematics Victorian Certificate of

Education Study Design 2010 page 153 (Area of Study 1. Functions and graphs). www.vcaa.vic.edu.au

y x

a

b---

= y x1

=

y x2

= y x

1

2---

= y x

3

2---

=

y ax

= y loga x( )= a 2 10 e, ,=

kx( )sin a kx( )cos= kx( )tan a=


13/60



3.3 Basic graph workWhat you should know:

use and interpretation of graphs to express relationships;

sketch graphs of straight lines, quadratics and cubics (including the use of simple transformations);

domain and range of functions of a real variable;

using a sliding vertical line as a test for a function; and

recognition that circles of the form are not functions.

We use graphs to represent many physical relationships:

This graph represents the cost of different amounts of petrol bought from a service station. It is a straight line

and the relationship is said to be linear. Such relationships have the general equation . You should

be able to both draw the graph of a given equation and, given the graph, find the equation.

This graph represents the way in which the surface area of a sphere increases as the radius increases. This is

known as a quadratic relationship and the graph is a parabola. In the present case, the formula is:

The general form of quadratic relationships is:

x h( )2

y k( )2

+ r2

=

Amount of petrol bought

Cost

y mx c+=

Radius

Surfacearea

A 4r2

=

y ax2

bx c+ +=


14/60


15/60



Domain and range

The horizontal extent of a graph is known as its domain and the vertical extent as the range:

You should recognise that the graph and the domain may be given to you in algebraic form:

has domain [0, 1] and range [1, 1] (look at the graph!).

It is mostimportant that you be able to find the implied domainwhen the rule for a function is given.

Example 1

Find the implied domain for .

Solution to Example 1

We require the function underthe square root to be positive or zero.

Consider the graph of .

Inspecting the graph shows that whenever or . This is the required domain fory.

It can be written as ( , .

Domain

Range

y 2x 1 0 x 1 =

y x2

4=

f x( ) x2

4=

2 2 x

y

f x( ) 0 x 2 x 2

2] [2,)


16/60



Exercises

Question 1

Find the implied domain in each of the following. See if you can state the range of each.

a.

________________________________________________________________________________

________________________________________________________________________________

b.

________________________________________________________________________________

________________________________________________________________________________

c.

________________________________________________________________________________

________________________________________________________________________________

d.

________________________________________________________________________________

________________________________________________________________________________

y x2

2x 3+=

y3

2x 5+( )2

---------------------- 6=

y2

x 4+----------------=

y1

4x2

1+-----------------=


17/60



Finally in this section, we will discuss the effects of transformations on the shapes of graphs. We will work

with the basic parabola, though the same effects apply to much more complex shapes.

The first example is a set of horizontal transformations:

While the distance of the horizontal transformation may be obvious, the direction is not and needs to be

handled with care.

Vertical transformations are rather more obvious:

Another common transformation is the dilation. The next set of examples illustrate the effect of changing the

parameterAin

AsAgets larger, the graph gets taller. These transformations can be confirmed with your CAS calculator.

-2-4-4

5

10

1515

2 44

5

10

1515

2 44

5

10

1515y x

2= y x 1( )

2= y x 2+( )

2=y

x

y

x

y

x

22-2-2

5

10

1515

22-2-2

5

1010

-5-5

y x2

5=

y x2

5+=

y

x

x22-2-2

5

10

1515

y x2

=

y

x

y Ax 2=

22-2-2

5

10

1515

22-2-2

5

10

1515

y 4x2

=y

x2

2-----=

22-2-2

5

10

1515

y x2

=

y

x

y

x

y

x


18/60



Example 2

Let , whereDis the largest subset ofRfor whichfis defined.

a. Express in the form .

b. StateD.

c. The graph offis translated 2 units to the left and 1 unit up. Write down the new equation for .

d. The graph offis reflected in they-axis and then dilated by a factor of 2 away from thex-axis.

Write down the new equation for .


f: D R f x( )2x

x 1+------------=

f x( ) f x( )A

x B+------------ C+=

f x( )

f x( )


19/60



Example 3

The diagram below shows a sketch of the part of the graph of for .

The linex = 2ais a line of symmetry of the graph.

On each of the axes below, sketch the graphs of

a. ,

b. ,

c. ,

d. ,


We will complete this during the lecture.

a. b.

c. d.

y f x( )= 0 x 2a

y

a 2ax

b

y f x( )= 0 x 4a

y f x a( )= a x 5a

y 2f x( ) b= 0 x 4a

y f2x( )= 0 x 2a


20/60



3.4 Introduction to quarticsA polynomial of degree 4 is a quartic, i.e.f(x) = ax4+ bx3+ cx2+ dx+ e.

The simplest type arises when b, c, d, eare all zero. Consider .

It is concave in shape, like a parabola.

The previous transformations shown for a parabola apply similarly here. Thus, functions of the type

f(x) = a(xh)4+ kwill have the shape for a> 0 and for a< 0 with a turning point

at (h, k).

Example 4

Sketch the graph off(x) = 16 (x + 1)4showing all relevant points.


Turning point at (1, 16). a< 0 so the turning point is a maximum.

Sox= 3 or 1.

f x( ) x4

=

(1, 0)

(0, 1)

(1, 0)x

y

yintercept: f 0( ) 16 14

15= =

xintercept: x 1+( )4

16=

x 1+ 164=

x 1+ 2=

x 1 2=


21/60


22/60



Your CAS calculator could be used to determine the coordinates of each local minimum. Clearly, the local

maximum atx= 1 hasy= 0 as we have a repeated factor of (x 1) in our expression.

Minima occur at (0.378, 7.533) and (4.6, 1E+2). Then confirm with CAS.

3.5 Basic trigonometryWhat you should know:

application of trigonometric ratios to right-angled triangles;

exact values of sin and cos of 30, 45 and 60;

definition of a radian and conversion between radians and degrees;

unit circle;

defining sine, cosine and tangent;

showing that and that , ;

special values; for example, , ; and

exact values of sin and cos of integer multiples of and .

Mostly, this is a revision of middle school work (using trigonometry in right-angled triangles). New work that

is fundamental to Year 12 mainly centres around the extension of the trigonometric ratios to angles larger than

a right angle and the use of radians:

This circular diagram contains the basic definition of sine and cosine for any angle (positive or negative).

The function tan() is defined as

x( )sin2 x( )cos2+ 1= 1 x( ) 1cos 1 x( ) 1sin

0( )sin 0= ( )cos 1=

6---

4---

cos

sin

( )sin

( )cos----------------


23/60



You should note that on almost all occasions when the trigonometric functions are used in Year 12 the angles

will be measured in radians. The definition of radian measure is:

The radian measure of the angle is:

This makes it easy to convert between the two systems of measuring angles because one full turn is 360 and

2radians.

Example 6

a. Convert to radians: 30, 45, 60, 120, 180.

b. Convert to degrees: , 3, .


a. This is a straight ratio problem:

, so .

The other answers are: .

b. In conversion to degrees, the process is reversed. The answers are: 180, 540 and 105.

There are some special angles with exact trigonometric ratios:

So for 45 or the sine and cosine are both and the tangent is 1.

For 30 the sine is the cosine is and the tangent is .

For 60 the sine is the cosine is and the tangent is .

Students who learn these ratios off by heart have a distinct advantage in exams.

r

L(arc length)

L

r---=

7

12------

1

180---------radians= 30

30

180---------

6---radians==

4---

3---

2

3------ , , ,

45

1

1

60

1

223

30

4---

1

2-------

1

2---

3

2-------

1

3-------

32

------- 12--- 3


24/60


25/60



The Year 12 course concentrates on the sine and cosine graphs and transformations of these. The following sets of

graphs illustrate the basic trigonometric graphs and some simple transformations of these. They are all .

In this case, larger as compress the graph horizontally.

Exactly the same principles of horizontal and vertical transformations that we used with simple polynomialsalso apply to trigonometric graphs. This set of graphs is of for various values of a.

Vertical translations and dilations are shown on this set of examples that are based on the equation .

The Year 12 course includes graphs of . Again, larger values of acompress the graph

horizontally as seen below.

y ax( )sin=

5 1010-5-10-10

0.5

1.01.0

-0.5

-1.0-1.0

5 1010-5-10-10

0.5

1.01.0

-0.5

-1.0-1.0

5 1010-5-10-10

0.5

1.01.0

-0.5

-1.0-1.0

a 0.5= a 1= a 2=

x xx

y yy

y x a+( )cos=

5 1010-5-5

0.5

1.01.0

-0.5

-1.0-1.0

5 1010-5-5

0.5

1.01.0

-0.5

-1.0-1.0

5 1010-5-5

0.5

1.01.0

-0.5

-1.0-1.0

a 0=

x

ya 2=

x

ya 1=

x

y

y A x( )sin b+=

10 2020-10-10

-2

-4

-6-65 1010-5-5

1

2

33

5 10 1515-5-5

11

-1

-2-2

A 2 b, 1= = A 0.5 b, 3= =

x

y

x

y

A 3 b, 4= =

x

y

y ax( )tan=


26/60



a= 0.5 a= 1 a= 2

The sine and cosine functions are particularly useful in modelling periodic phenomena such as tides, water

waves, alternating current electricity etc. It is quite common to encounter functions of the type ,

where n is a multiple of . The period of such a function is . So, for example, the function

has the graph (with period 1):

and the function has the graph (with period 4):

Note that the period can be made to have integer values in this way. This is particularly useful in modelling

since most periodic phenomena do not have periods that are multiples of .

You should also be able to solve some simple trigonometric equations.

y nx( )cos=

2n

------ y 2x( )sin=

2 2

0.5

0.5

y

x

y x

2------

cos=

2 2

0.5

0.5

y

x


27/60



Example 8

a. Solve for exactly.

b. Solve , correct to two decimal places.

c. Solve , .


a. Get the basic solution (firstly in degrees, if necessary and then radians) using the inverse sine function

on your CAS calculator (if necessary): . The other solution can either be found from

the graph of the sine function or the unit circle:

b. Using your CAS calculator, (there are numerous approaches) let and .

These two graphs can be plotted together. Thexvalues of the points of intersection of the curve and the

line will give the solutions.

Choose a window of XMin: 0 and XMax: with appropriate

Yvalues.

The solutions are .

c. This is an example of solving equations of the type:

We approach such equations by dividing each side by .

Thus

i.e.

Now, given , we get .

So

i.e.

Hence

2x( )sin1

2---= 0 x

3 2x( )sin 1= 0 x

3 2x( )cos 3 2x( )sin= 0 x 2

2x

6---= x

12------=

2x5

6------= x

5

12------=

f1 x( ) 3 2x( )sin= f2 x( ) 1=

x 1.7 and 3=

asin ( ) b ( )cos=

( )cos

asin ( )

( )cos------------------

b ( )cos

( )cos--------------------=

( )tanb

a---=

atan ( ) b=

3 2x( )cos 3 2x( )sin=3

3------- 2x( )tan=

2x 6---

6---+ 2

6---+ 3

6---+, , ,=

2x

6---

7

6------

13

6---------

19

6---------, , ,=

x

12------

7

12------

13

12---------

19

12---------, , ,=


28/60



Exercises

Question 2

Write down the minimum value of and find the smallest positive value of for which

it occurs.

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

Question 3Given that , find the exact values of and for

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

_____________________________________________________________________________________

3.7 Logarithmic and exponential functionsWhat you should know:

sketching graphs of and , and solving indicial equations related to graphs by calculator

or by graphical methods;

sketching the graph of using calculator generated values and relating graph to that of

informal discussion of their inverse relationship; and

mathematical modelling using exponential functions.

Relations of the form are known as exponential functions. Their general graphs are of the form:

Altering the value of aalters the slope of the curve: making the value of alarger makes the graph steeper. The

yintercept is 1 for all values of a.

2 3

4--- +

cos

( )sin1

3---= ( )cos ( )tan

2--- .< 4 ,( )

R

0 1,( )

2 2

3----------

2

4-------,

x2

y x2

2x+=

x x 2( )=

x 0= x 2=

yx

2 2x

4---------------------=

yx

4--- x 2+( )=

x 0= x 2=


49/60



Question 6(page 29)

Consider the graph shown

Significant points for restricted domain are the endpoints (2, 5) and (1, 2) as well as the local

minimum (0, 1).

Question 7(page 30)

For the graph shown the asymptote is hence there has been no translation applied to .

Eliminate Band C. Thexintercept of has changed from (1, 0) to (2, 0). This is a dilation in thexdirection by a factor of 2. Eis correct.

Question 8(page 30)

Ais correct.

y x2

1+=

(2, 5)

(0, 1)

(1, 2)

y

x

2 5[ , ]

Range 1 5[ , ]=

B

y 0= y log10 x( )=

y log10 x( )=

4

3---

cos 1 3=

4

3---

cos 4=

3---

cos 1=

3--- 2=

6=


50/60



Question 9(page 30)

Sine is negative in 3rd and 4th quadrants.

Ais correct.

Question 10(page 31)

Enter data into a Lists and Spreadsheet page to graph a scatterplot. On the CAS calculator:

From the scatterplot the function is obviously logarithmic.

Select a new Lists and Spreadsheet page.

Press enter. Rename lists A and B appropriately.

Enter the data as per the screen shot:

To create a scatterplot, create a new Graphs and

Geometry page, select Menu 3: Graph Type 4:

Scatter Plot

Press the var button on the calculator and select the

appropriate lists.

Press enter and adjust the window to suit (Menu

4:Window/Zoom).

2 2x( ) 1+sin 0=

2 2x( )sin 1=

2x( )sin1

2------=

Basic angle sin1 1

2---

=

6---=

2x , 2 +=

2x

6---, 2

6---+=

2x 7

6------, 11

6---------=

x7

12------,

11

12---------=


51/60




a.

, yintercept is (0, 4)

b.

c.

f x( ) 4 2 e0.01x

( )=

f x( ) 8 4e0.01x

=

asymptote isy 8=

x

y

(0, 4)

y= 8

f 10( ) 4 2 e0.01 10( )

( )=

4 2 e0.1

( )=

4.38=

5 4 2 e0.01x

( )=

1.25 2 e0.01x

=

e0.01x

2 1.25=

e0.01x

0.75=

0.01x loge 0.75( )=

xloge 0.75( )

0.01-------------------------=

x 28.77=


52/60




For the modulus function , we plot the hybrid function:

The linear function is also plotted. Use addition of ordinates to plot on the same set of

axes.


x( ) 2x x2

= x 1 3[ , ]

f x( )

x2

2x 1 x 0

2x x2 0 x 2

x2

2x 2 x 3

=

g x( ) 4 2x= h x( )

1 2 3

2

1

1

2

3

4

5

6

7

8

9

x

y

1

f(x)

g(x)h(x)

(4, 2)

(0, 14)

y

x

4 2+ 0( , )4 2 0( , )


53/60


54/60



f.

t

x y,

1000

50 1 2 3 4

x t( )

y t( )900


55/60

Section 4: Algebra


Section 4: Algebra


Units 1 & 2

substitution in and rearrangement of formulas;

identifying key features of polynomials: variables, coefficients, degree, and so on;

use of notation ; substitution and evaluation of , where ais real;

expansion of quadratics and cubics from factors;

factorisation

> connections between factors, solutions and corresponding graphs

> quadratic trinomials

> factor theorem

> factorisation of a cubic with at least one factor of the form where ais an integer;

solving quadratic equations: by systematic trial and error, by graphing, by completion of the square

(for cases where the coefficient of is 1 only), and by the quadratic formula to include cases with

irrational solutions;

use and interpretation of the discriminant to identify the number of solutions;

completing the square with quadratics and using this to specify the transformations applied to ;

solving cubic equations by any of the following methods

> graphing (including cases which do not have three solutions)

> systematic trial and error

> algebraic methods; for example, factorisation of cubics that have at least one integer solution;

solving simultaneous equations (including two linear and one linear with one quadratic) using

algebraic and graphical methods; and

finding the correct polynomial model for a data set.

y f x( )= f a( )

x a( )

x2

y x2

=


56/60


57/60

Section 5: Calculus


Section 5: Calculus


Unit 1

idea of a rate of change and practical applications of this;

the gradient of a linear function as a rate of change;

calculation of average rates of change using chords;

the concept of instantaneous rates of change and the use of the gradient of the tangent to find these;

the description of a graph in terms of its rate of change being positive, negative or zero;

relating gradient functions to their original function;

relating velocity-time graphs to displacement-time graphs as an application of rates of change; and

using the average rate of change between two points very close to each other on a graph as a means of

approaching the instantaneous rate of change.

Unit 2

the concept of the derivative function being a gradient function;

the notation and for derivatives;

using first principles to find gradient functions in simple cases;

finding derivatives of polynomial functions;

using derivatives to find rates of change;

using derivatives to assist in curve sketching by finding stationary points and their nature;

the concept of antidifferentiation being the reverse process to differentiation; and

finding simple antiderivatives using rules for antidifferentiation.

f x( )dy

dx------


58/60




Units 3 & 4

identifying graphs of derivative functions given the original function;

knowing and applying the rules for derivatives of (nis rational), , , and ;

the product, quotient and chain rules for differentiation and manipulation of expressions to apply

these rules;

using derivatives to assist in curve sketching by finding stationary points and their nature;

finding equations of tangents and normals;

solving maximum/minimum problems;

applications of rates of change including related rates of change;

small change approximations using and its geometric interpretation;

identifying graphs of antiderivative functions given the original function;

calculating the approximate areas under curves using the left rectangle and right rectangle methods;

using geometric methods to approximate areas under curves;

simple algebraic manipulations using the fundamental theorem of calculus;

finding the general rule for the antiderivative and calculating the value of the definite integral for the

following functions and where nis rational, , , and ;

deriving and using the knowledge that to find where

is not able to be transformed into one of the functions in the dot point above; and

using integration to find areas under curves and areas between curves.

how to use your CAS calculator to perform all of the above.


Education Study Design 2009, page 154 (Area of Study 3. Calculus). www.vcaa.vic.edu.au

xn

ex

loge x( ) x( )sin x( )cos

x h+( ) f x( ) hf' x( )+

xn

ax b+( )n

eax

ax( )sin ax( )cos ax b+( )1

g x( ) f' x( )= g x( ) xd f x( ) c+= g x( ) xdg x( )


59/60

Section 6: Probability


Section 6: Probability


Unit 1

definition of chance and probability;

long-term relative frequency as an estimate of probability;

complementary events probability;

multiple events probability;

appropriate notation;

multiple events and the use of tree diagrams, lattice diagrams to calculate probabilities;

Venn diagrams;

mutually exclusive events; and

independent events.

Unit 2

addition and multiplication principles;

permutations: concept of ordered samples, ;

combinations: concept of unordered samples, ;

evaluation of and and establishing that ;

relating combinations to Pascals triangle; and

applications of permutations and combinations to probability.

Pn r

Cn

r

Pn

r Cn

r Pn

r Cn

r r!=


60/60



Units 3 & 4

DISCRETERANDOMVARIABLES

definition of a discrete random variable;

producing a discrete probability distribution;

calculating and explaining the mean, variance and standard deviation of a discrete random variable;

and

calculation and interpretation of the property that approximately 95% of the distribution is within two

standard deviations of the mean.

BINOMIALDISTRIBUTION application of the binomial distribution to the number of successes in a fixed number, n, of Bernoulli

trials with probabilitypof success;

the effect of the parameters nandpon the graph of the probability function;

calculation of probabilities; and

use of formulas for the expectation and variance of a binomial random variable.

CONTINUOUSRANDOMVARIABLES

the concept of a probability distribution function; and

how to calculate the central measures (mean, median and mode) of a probability density function using

integration.

NORMALDISTRIBUTION

the normal curve as the limit of the histogram using examples such as weights and heights of people

(for large samples); and

the effect of the mean and variance on the shape of the normal distribution.

You will need to learn how to use your statistical probability functions of your CAS calculator to support

all of the above.


Education Study Design 2009 page 155 (Area of Study 4. Probability). www.vcaa.vic.edu.au

methods expo 12

Documents