Table Of Contente
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NEW FUNDAMENTAL MATHEMATICS
ADVANCED + EXTENSION 1 HSC COURSES
First edition.
ISBN: 978-09577428-9-5
Published: September 2019
© Terry Lee Enterprise
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Table of Contents
Graphing Techniques Applications of Calculus
1. 1 Transformation of graphs ................................. 6 7 .1 Review ......................................................... 300
1.2 Rational functions ............................................ 9 7 .2 Rates of change ........................................... 302
1.3 Using graphs .................................................. 13 7 .3 Displacement and Velocity as Integrals ....... 308
1.4 Review Exercise 1.. ........................................ 17 7.4 Differential Equations .................................. 310
Solutions .............................................................. 19 7 .5 Acceleration as a function of v or x .............. 316
Differential Calculus 7 .6 The Laws of Growth and Decay .................. 321
7 .7 Review Exercise 7 ........................................ 330
2.1 Review ........................................................... 50
Solutions ............................................................ 336
2.2 Geometrical significance off'(x) ................... 51
Introduction to Vectors
2.3 Geometrical significance off"(x) .................. 57
2.4 Maximum and minimum problems ................ 64 8.1 Two-dimensional vectors ............................. 384
2.5 Review Exercise 2 .......................................... 67 8.2 Geometrical applications .............................. 388
Solutions .............................................................. 71 8.3 Projectile motion .......................................... 390
Integral Calculus 8.4 Review Exercise 8 ........................................ 397
Solutions ............................................................ 400
3.1 Review ......................................................... 104
Financial Mathematics
3.2 More Integration Techniques ....................... 106
3. 3 Definite Integration ...................................... 110 9.1 Introduction .................................................. 420
3 .4 Volume of solids ofrevolution .................... 117 9.2 The partial sum of a series .......................... .424
3.5 Approximations to Definite Integrals ........... 121 9.3 The terms of an Arthmetic Sequence .......... .426
3.6 Review Exercise 3 ........................................ 122 9.4 The sum of an Arithmetic Series .................. 428
Solutions ............................................................ 125 9.5 The terms of a Geometric Sequence ............ 430
Exponential and Logarithmic 9.6 The sum of a Geometric Series ................... .433
9. 7 Infinite series ................................................ 4 3 5
Functions
9. 8 Financial Mathematics ................................. 4 3 7
4.1 Differentiation of exponential functions ...... 152 9.9 Review Exercise 9 ........................................ 443
4.2 Integration of exponential functions ............ 156 Solutions ............................................................ 447
4.3 Differentiation oflogarithmic functions ...... 158 Proof by Induction
4.4 Integration involving logarithmic functions. 161
10.1 Induction proofs ......................................... 488
4.5 Review Exercise 4 ........................................ 164
10.2 Review Exercise 10 .................................... 490
Solutions ............................................................ 167
Solutions ............................................................ 491
Trigonometric Functions
Data Analysis
5.1 Review 1 ...................................................... 194
11.1 Displaying data .......................................... 502
5.2 Review 2 ...................................................... 195
11.2 Central tendencies and disperson ............... 513
5.3 Graphs of trigonometric functions ............... 199
11.3 The normal distribution .............................. 519
5 .4 Trigonometric equations .............................. 202
11.4 Correlation and Regression ........................ 523
5.5 Small angles ................................................. 205
11.5 Review Exercise 11 .................................... 528
5 .6 Differentiation of Trigonometric functions .. 208
Solutions ............................................................ 531
5.7 Integration of Trigonometric functions ........ 210
Binomial Distribution
5.8 Review Exercise 5 ........................................ 213
Solutions ............................................................ 216 12.1 Binomial distribution ................................. 550
Inverse Trigonometric Functions 12.2 Normal approximation ............................... 555
12.3 Review Exercise 12 .................................... 560
6.1 Review ......................................................... 258
Solutions ............................................................ 563
6.2 Derivatives oflnverse Trig functions ........... 261
6.3 Integrals involving Inverse trig functions .... 266
Random number table ........................................ 573
6.4 Review Exercise 6 ........................................ 269
The z-score table ................................................ 574
Solutions ............................................................ 273
Preface
This combined book is written for the new Mathematics Advanced + Extension 1 courses, which are
being introduced into the NSW syllabus in 2020.
This book has been written with two main objectives: it can be used as a textbook for classroom use,
as well as a step-by-step resource to be used independently by students for their own self-study
purposes. This book provides sufficiently clear explanations about each topic in the syllabus, with
worked out examples and alternative methods, where applicable. Questions are categorised by topic
and graded from easy to hard, to help guide students in their learning. Each chapter also contains a set
of review exercises and challenge problems, as well as fully worked solutions for each question. The
review exercises will help consolidate students' skills and knowledge, while improving their
competence and confidence. The book also features challenge problems. While they may extend
beyond the syllabus, they are designed to provide extra stimulus for highly motivated students and
increase confidence for the harder questions in the Higher School Certificate examination.
While this book is written specifically for the new HSC Mathematics Advanced and Extension 1
courses, it also covers various topics in the year 11 courses in the Review sections at the beginning of
the chapters. This book provides clear explanations of the basic concepts underpinning these topics to
help provide students with a strong foundation upon which they can develop a more thorough
understanding of the complex and challenging concepts in the Mathematics Advanced and Extension 1
courses.
This book also features colour-coding throughout to highlight various theorems and study tips - this
makes the book a study reference and more enjoyable to read. Students are advised to complete as
many questions in this book as possible to master the course.
This book builds upon what the Terry Lee series has been famous for: it includes many fully explained
tips and tricks to help students understand and solve problems efficiently, while ultimately developing
a greater enjoyment of the course.
Terry Lee
p h i n g
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-H--S~C- O--u-t~c-o~m-es - --- ~ ~~ ~ - --- - -
A student
uses detailed algebraic and graphical techniques to critically construct, model and evaluate arguments in a range of
familiar and unfamiliar contexts.
chooses and uses appropriate technology effectively in a range of contexts, models and applies critical thinking to
recognise appropriate times for such use.
constructs arguments to prove and justify results and provides reasoning to support conclusions which are appropriate to
the context.
In this chapter,
1.1 Transformation of graphs ................................ 6
1.2 Rational functions ............................................ 9
1.3 Using graphs .................................................. 13
1.4 Review Exercise I ......................................... 17
Solutions .............................................................. 19
6
New Fundamental Mathematics
1.1 Transformation of graphs
Example 1.1
Given f(x) = (x-1)2(2x+ 7). It has a maximum at (-2,27) and a minimum at (1,0). It meets they
axis at (0,7). The graph of y = f (x) is shown below.
X
Sketch the following graphs, showing stationary points and axes intercepts where possible.
(a) y = 2/(x). (b) y = f(2x). (c) y = f(-x). (d) y =- f(x).
(e) y=f(x)+l. (f) y= f(x+l). (g) y = f ( -x + 1)
The graph of y = k.f( x) stretches the graph of y = f (x) vertically.
The graph of y = f (ax) stretches the graph of y = f (x) horizontally.
* If x = a is a root of / ( x) , then x = a is a root of y = f ( ax) .
a
The graph of y = f(-x) is a horizontal flip of y = f(x) about the y-axis.
The graph of y =-f(x) is a vertical flip ofy = f(x) about the x-axis.
The graph of y = f(x) +c moves the graph of y = f (x) up c units.
The graph of y = f(x-b) shifts the graph of y = f(x) to the right b units.
The graph of y = f(-x+b) shifts the graph of y = f(-x) to the right b units.1
Properties of reciprocal functions:
1
* f ( x) and - - have the same sign for the same domain.
f(x)
➔ 1 ➔
* Asf(x) 0,-- oo, and vice versa.
f(x)
1
* When f ( x) is decreasing, - - is increasing, and vice versa.
f(x)
1
* f (x) = -- when/( x) = ±1.
/(x)
1 f(-x+b) = f(-(x-b)). It is the graph of y = f(x) flipped about the y-axis (i.e. f(-x) first), then shifted to
the right b units. It is NOT the graph of y = f (x-b) being flipped about the y-axis, which is f (-x-b) . A
simple curve such as /(x) = x and b = I (i.e. f (-x + 1) = -x + 1) will help you verify and remember this.
~
Further, f (- ax + b) = f ( -a ( x -~)) -It is the curve f (- ax) moved to the right units.
7
Chapter 1: Graphing technigues
= =
(a) y 2f(x). (b) y f(2x) .
(c) y =f(-x). (d) y=-f(x).
y
X
-27
(e) y =f(x)+l . (f) y = f (x + 1) .
y
27
X
-3
(g) y =f(-x+l).
\ 7,
X 1:I X
~ I
I
:IIII IIIII
I
8
New Fundamental Mathematics
Exercise 1 . 1
1 Given the graph of y = f(x) as shown below. Sketch the following graphs.
(i) y = 2/( x). (ii) y = f (2x) . (iii) y = f (-x). (iv) y = -f( x).
... ) 1
(v) y= f(x)+l. (vi) y = f(x+ 1) . (vii) y=f(-x+l). ( Vlll y=-- .
f(x)
2 Repeat question 1 for the following curves y = f(x).
(a) (b) (c)
y
X - 1 2 X
-1
(d) (e) y (f)
y y
_________ 2 ----------
X 4 X X
3 The graph of y = log x is shown below. Sketch the following graphs.
3
1
(d) y=--.
log x
3
X
4 Sketch the following graphs, given that y = 3x and y = log x are inverses of each other.
3
(a) y=1+3x. (b) y =Ji". (c) y=2x3-x. (d) y=-32x.
'..
9
Chapter 1: Graphing technigues
5 Given the graph of y = f(x) as shown. Sketch the following graphs.
(a) y=2f(x)+l. (b) y=f(-2x+4) . (c) y= ✓ f(x). 2 (d) y = f2(x) .3
(e) y=l/(-x)I, (1) y=/(lxl) .4 (g) y=jJ(lxl)j . (h) IYI = f(x) .5
y
X
----------- f-2
6 Sketch the graphs of the following reciprocal functions, showing any turning points, axes intercepts
and asymptotes.
1 -1 1 1
(a) y=-2-· (b) y=-2-. (c) y= (d) y=-2-·
2 •
X +1 X +2 X +4x+5 X -l
1 1 1 1
(e) y = x2 -x-12 · (t) y = 5-4x-x2 • (g) y = x2 -2x-3 (h) y = 2x-x2 •
(i) y=secx,-2n-~x~2.1r. U) y=cosecx,-2n-<x<2n-. (k) y = cotx,-n-< x < n-.
7 Given f (x) = 2x(2 - x) . Sketch the following curves, showing all important features.
(a) y = f(x). (b) y = f(2x). (c) y = 2/(x). (d) y = f(x-l).
(e) y = f(2x-1). (t) y = f(l-x). (g) y2 = f(x). (h) y = J(lxl).
(i) Y = IJ(x)I. (i) IYI = f(x) · (k) IYI = f (lxl) · (1) Y = f(x) + IJ(x)I ·
1.2 Rational functions
Example 1.2
Sketch the following curves, showing the asymptotes and any stationary points. Do not use Calculus.
(a) y =--1 -. (b) y = 9(x - 3)
(x- 3)(x+l) (x-2)(x+l)
2 Review: To draw y = ✓ f(x).
Step 1: Remove the part of the curve where f(x) < 0.
Step 2: If f(x) < I, ✓ f(x) is higher than/(x).
Step 3: The square root of a single root has a vertical tangent. The square root of a double root has a sharp point.
3 Review: To draw y = f 2(x).
Step 1: / 2 (x) ~ 0, :. all x-intercepts become minimum turning points.
Step 2: If f(x) < l,f2(x) is lower than f(x).
4 Review: To draw y = J(lxl).
Step 1: Remove the part of the curve where x < 0.
Step 2: Flip the curve about the y-axis so that the curve is symmetrical about the y-axis.
5 Review: To draw IYI = f(x).
Step 1: Remove the part of the curve where y < 0.
Step 2: Flip the curve about the x-axis so that the curve is symmetrical about the x-axis.
