Mathematical Studies WorkBook & Revision Guide SACE Stage 2

331 Pages·2013·13.735 MB·English

by  Warren Brown

mathematical_Studiescover.pdf 1 20/03/13 11:50 AM
SACE 2
THIRD EDITION
MATHEMATICAL
Warren Brown
STUDIES
WORKBOOK &
REVISION
GUIDE
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SACE STAGE 2
THIRD EDITION
Warren Brown BSc (Hons), MSc, Grad Dip Teaching
PUBLISHED BY GREG EATHER
in association with
the author
CONTENTS 
Warren Brown BSc (Hons), MSc, Grad Dip Teaching
Working with graphs and functions using calculus            1 
• Warren has taught Mathematics and Physics at all secondary levels, including Year 12 in Maths Studies,
Differentiation                     2 
Specialist Maths, Maths Methods, Maths Applications and Physics.

Using derivatives                    10 
• He has co-authored the Essentials Maths Studies text book, written trial exams, and taken a number of
revision classes for Adelaide Tuition Centre. He also has extensive experience with SACE exam marking
Exponential and logarithmic functions              32 

Integral calculus                    44 

Working with statistics                  65 
Acknowledgements
Graeme Payze - For his previous work and dedication to make this publication possible
Statistics                      66 
SACE Board of SA - For permission to use their past exam questions
Continuous random variables and the normal distribution          76 
publishing information
Sampling and statistical inference                88 
This publication is part of the Essentials series, designed to support the teaching of SACE Stage 1 and 2 subjects in South
Australia. It is designed to meet the requirements of the SACE Stage 2 Physics Course.
Hypothesis testing                    105 
The Essentials education series is published by Greg Eather in association with
Working with linear equations and matrices            113 
Adelaide Tuition Centre,
21 Fourth Street, Bowden 5007.
Solving systems of linear equations                114 
Telephone [phone]
Facsimile [phone]
Matrices                      125 
www.essentialseducation.com.au
Leslie Matrices                    132 
Library catalogue:
Brown, Warren. Solutions to exercises                   143 
1. Mathematical Studies SACE 2 2. Essentials Workbook & Revision Guide
2010 SACE Board of SA Mathematical Examination           219 
ISBN 978-1-921548-05-5
2010 SACE Board of SA Mathematical Examination Solutions        258 
First published 1999; second edition published 2007. This third edition printed 2013.
Copyright © W.Brown. 2011 SACE Board of SA Mathematical Examination           270 
2011 SACE Board of SA Mathematical Examination Solutions        308 
copyright information
All rights reserved except under the conditions described in the Copyright Act 1968 of Australia and subsequent
admendments. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form
or by any means, without prior permission of the publishers. While every care has been taken to trace and acknowledge
copyright, the authors tender their apologies for any accidental infringement where copyright has proved untraceable.
the author
CONTENTS 
Warren Brown BSc (Hons), MSc, Grad Dip Teaching
Working with graphs and functions using calculus            1 
• Warren has taught Mathematics and Physics at all secondary levels, including Year 12 in Maths Studies,
Differentiation                     2 
Specialist Maths, Maths Methods, Maths Applications and Physics.

Using derivatives                    10 
• He has co-authored the Essentials Maths Studies text book, written trial exams, and taken a number of
revision classes for Adelaide Tuition Centre. He also has extensive experience with SACE exam marking
Exponential and logarithmic functions              32 

Integral calculus                    44 

Working with statistics                  65 
Acknowledgements
Graeme Payze - For his previous work and dedication to make this publication possible
Statistics                      66 
SACE Board of SA - For permission to use their past exam questions
Continuous random variables and the normal distribution          76 
publishing information
Sampling and statistical inference                88 
This publication is part of the Essentials series, designed to support the teaching of SACE Stage 1 and 2 subjects in South
Australia. It is designed to meet the requirements of the SACE Stage 2 Physics Course.
Hypothesis testing                    105 
The Essentials education series is published by Greg Eather in association with
Working with linear equations and matrices            113 
Adelaide Tuition Centre,
21 Fourth Street, Bowden 5007.
Solving systems of linear equations                114 
Telephone [phone]
Facsimile [phone]
Matrices                      125 
www.essentialseducation.com.au
Leslie Matrices                    132 
Library catalogue:
Brown, Warren. Solutions to exercises                   143 
1. Mathematical Studies SACE 2 2. Essentials Workbook & Revision Guide
2010 SACE Board of SA Mathematical Examination           219 
ISBN 978-1-921548-05-5
2010 SACE Board of SA Mathematical Examination Solutions        258 
First published 1999; second edition published 2007. This third edition printed 2013.
Copyright © W.Brown. 2011 SACE Board of SA Mathematical Examination           270 
2011 SACE Board of SA Mathematical Examination Solutions        308 
copyright information
All rights reserved except under the conditions described in the Copyright Act 1968 of Australia and subsequent
admendments. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form
or by any means, without prior permission of the publishers. While every care has been taken to trace and acknowledge
copyright, the authors tender their apologies for any accidental infringement where copyright has proved untraceable.

Working with Functions and Graphs using Calculus 1
Working with
Functions and
Graphs using
Calculus
2 Mathematical Studies Essentials Stage 2
Differentiation
The process of finding the derivative of a function y=f(x) is called differentiation.
dy
- differentiate y produces
dx
dy
- differentiate f(x) produces f′(x). = f′(x)
dx

Derivatives from first principles
1) At a point:

The slope of the tangent at x=a, called the
derivative at x=a, is defined as

f(x)− f(a)
f′(a)=lim
x→a x−a

f′(a) measures the instantaneous rate of change
in y (with respect to x) at the point where x=a

OR

f(a+h)− f(a)
f′(a)=lim
h→0 h

f(a+h)− f(a)
Note: represents the slope of the
h
chord AB. As B→A, h→0 and the slope
of chord AB→ slope of tangent at A.

2) As a function:

The slope function of f(x), also called the derived
function or just derivative, is defined as

f(x+h)− f(x)
f′(x)=lim
h→0 h
Working with Functions and Graphs using Calculus 3

Example
3x
P and Q are two point on the graph of y= f(x)= , with X-coordinates of a and a+h respectively.
x−2
−6
a) Show that the slope of the chord PQ is equal to .
(a+h−2)(a−2)
b) Explain how this expression can be used to determine the slope of the tangent to the graph at
point P.
c) Evaluate f′(3).

f(a+h)− f(a)
a) Slope of chord PQ =
note: a quick way of
a+h−a
making a common
3(a+h) 3a
− denominator is a
a+h−2 a−2
= cross-multiply

h
p r
3(a+h)(a−2)−3a(a+h−2) +
= ÷h
q s
(a+h−2)(a−2) ↔
3a2 −6a+3ah−6h−3a2 −3ah+6a
= = ps + qr

(a+h−2)(a−2)h
rs

−6h
=
(a+h−2)(a−2)h
−6
= (h≠0)
(a+h−2)(a−2)

get rid of all terms not

b) As h→0, slope of chord PQ→ slope of tangent at P. involving h and then
factorize and cancel
−6
∴ Slope of tangent at P =lim out h
h→0 (a+h−2)(a−2)
−6
=
(a−2)2

−6
c) f′(a)=
(a−2)2
−6
∴ f′(3)= =−6
(3−2)2