e u n a m e n a • I ,A DVANCED + EXTENSION 1 HSC COURSES C -,c:; u t::.rl. NEW FUNDAMENTAL MATHEMATICS ADVANCED + EXTENSION 1 HSC COURSES First edition. ISBN: 978-09577428-9-5 Published: September 2019 © Terry Lee Enterprise Visit our website to buy this book online or to download free of charge Terry's solutions to past HSC papers. advancedmathematics.com.au Printed m Australia Copyright© 2019 by Terry Lee. All rights reserved. Reproducing copyright material without the copyright owner's permission will usually be an infringement of copyright. Dealing with part of a work may also infringe copyright if that part is important to the work - it needs not be a proportionally large part. Teachers can legally copy up to 10% of this book for own use and/or students use but please show your respect to our intellectual properties and support us by buying our books. Contact us for your special discounts. We thank you for your support. .. ,, e w un amenta - a ema 1cs ADVANCED + EXTENSION 1 HSC COURSES . - . .... •. ... - . - .· 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 • n1ques -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.