**Unformatted text preview: **H1
Mathematics
Textbook
CHOO YAN MIN & Answers.
Covers 8865 (revised) syllabus.
Includes TYS This version: 19th August 2016.
The latest version will always be at this link. Recent changes: First complete draft done.
Upcoming changes: None planned. Page 2, Table of Contents , Errors? Feedback? Email me! ,
With your help, I plan to keep improving this textbook. Page 3, Table of Contents This book is licensed under the Creative Commons license CC-BY-NC-SA 4.0. You are free to:
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For example, other rights such as publicity, privacy, or moral rights may limit how you use
the material. Author: Choo, Yan Min.
Title: H1 Mathematics Textbook.
ISBN: 978-981-11-0755-9 (e-book).
Page 4, Table of Contents The first thing to understand is that mathematics is an art.
Paul Lockhart (2009, A Mathematician’s Lament, p. 22). A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are
more permanent than theirs, it is because they are made with ideas. ... Beauty is the
first test: there is no permanent place in the world for ugly mathematics.
- G.H. Hardy (1940 [1967], A Mathematician’s Apology, pp. 84-85). The scientist does not study nature because it is useful to do so. He studies it because
he takes pleasure in it, and he takes pleasure in it because it is beautiful.
- Henri Poincaré (1908 [1914], Science and Method, English trans., p. 22). Page 5, Table of Contents About This Book
This textbook is for Singaporean H1 Maths students. It is based exactly on the revised
(8865) syllabus (also reproduced from p. 317 of this book), which will be examined for
the first time only in 2017.1
I assume that if you’re an H1 Maths student, you
•
•
•
•
• have passed O-Level Mathematics;
may or may not have taken O-Level Additional Mathematics;
are somewhat weaker or less interested in maths than the average H2 Maths student;
want to learn or do the minimum amount of maths necessary to get an A;
won’t be studying such subjects as mathematics, physics, engineering, or economics at
university. This textbook is thus written simply and non-rigorously. For example, there are few formal
definitions or proofs. 2
Simple and non-rigorous as this textbook may be, I fully intend that a careful study of this
textbook (complemented by a capable teacher) will easily earn you your A in H1 Maths.
You might also be interested in reading this brief 3-page document: H1 Maths vs H2 Maths:
What’s the Difference? Which Should I Take?
• FREE! This book is free. But if you paid any money for it, I certainly hope your money
is going to me! This book is free because:
1. It is a shameless advertising vehicle for my awesome tutoring services.
2. The marginal cost of reproducing this book is zero.
• DONATE! This book may be free, but donations are more than welcome! Donation
methods in footnote.3
It’s irrational for Homo economicus to donate. But please consider donating because:
1. You’re a nice human being , [*emotional_manipulation*].
2. Your donations will encourage me and others to continue producing awesome free content
for the world.
The old syllabus is 8864, to be examined for the last time in 2017. It is not very different from 8865.
This is in contrast to my H2 Mathematics Textbook, which is an authoritative reference that the interested H1 Maths
student should look at. That H2 Mathematics Textbook covers the same topics (and more) as this textbook, but much
more rigorously and thoroughly. Indeed a good deal of this H1 Mathematics Textbook is simply a diluted version of that
H2 Mathematics Textbook.
3
Singapore. POSB Savings Account 174052271 or OCBC Savings Account 5523016383 (Name: Choo Yan Min). International. Bitcoin wallet: 1GDGNAdGZhEq9pz2SaoAdLb1uu34LFwViz. Paypal [email protected] (Name: Yan Min Choo,
USD preferred because this account was set up in the US). USA. Venmo link (Name: Yanmin Choo). 1 2 Page 6, Table of Contents • HELP ME IMPROVE THIS BOOK! Feel free to email me if:
1. There are any errors in this book. Please let me know even if it’s something as trivial
as a spelling mistake or a grammatical error.
2. You have absolutely any suggestions for improvement.
3. Any part of this book is less than crystal clear.
Here’s an anecdote about Richard Feynman, the great teacher and physicist:
Feynman was once asked by a Caltech faculty member to explain why spin
1/2 particles obey Fermi-Dirac statistics. He gauged his audience perfectly
and said, “I’ll prepare a freshman lecture on it.” But a few days later he
returned and said, “You know, I couldn’t do it. I couldn’t reduce it to
the freshman level. That means we really don’t understand it.”
I agree: If you can’t explain something simply, you don’t understand it well enough.4 And
as a corollary, the best way to gauge whether you understand something is to see if you
can explain it simply to someone else.
If at any point in this textbook, you have read the same passage a few times, tried to reason
it through, and still find things confusing, then it is a failure on MY part. Please let me
know and I will try to rewrite it so that it’s clearer. (There is also the possibility that I
simply messed up! So please let me know if there’s anything confusing!)
I deeply value any feedback, because I’d like to keep improving this textbook
for the benefit of everyone! I am very grateful to all the kind folks who’ve already
written in, allowing me to rid this book of more than a few embarrassing errors.
• LyX rocks!
This book was written using LYX.5
• Is the font size big enough?
You’re probably reading this on some device. So I’ve tried to set the font sizes and stuff so
that one can comfortably read this on a device as small as a seven-inch tablet. It should
also be possible to read this on a phone, though somewhat less comfortably. (Please let me
know if you have any feedback about this!)
(I’ll probably be contacting some publishers to see if they want to do a print version of
this, for anyone who prefers it in print.)
This quote or some similar variant is often (mis)attributed to Einstein. But as Einstein himself once said, “73% of Einstein
quotes are misattributed.”
5 A
L TEX is the typesetting programme used by most economists and scientists. But LATEX can be difficult to use. LYX is a
user-friendly GUI version of LATEX. LYX has boosted my productivity by countless hours over the years and you should use
LYX too! 4 Page 7, Table of Contents Tips for the Student
• Read maths slowly.
Reading maths is not like reading Harry Potter. Most of Harry Potter is fluff. There is
little fluff in maths.
So go slowly. Dwell upon and carefully consider every sentence in this textbook. Make sure
you completely understand what each statement says and why it is true. Reading maths
is very different from reading any other subject matter.
If you don’t quite understand some material, you might be tempted to move forward anyway.
Don’t. In maths, later material usually builds on earlier material. So if you simply move
forward, this will usually cost you more time and frustration in the long run.
Better then to stop right there. Keep working on it until you “get” it. Ask a friend or
a teacher for help. Feel free to even email me! (I’m always interested to know what the
common points of confusion are and how I can better clear them up.)
• Examples and exercises are your best friends. So work through them. A good stock of examples, as large as possible, is indispensable for a
thorough understanding of any concept, and when I want to learn something
new, I make it my first job to build one.
- Paul Halmos (1983, Google Books).
Work through all the examples and exercises. Merely moving your eyeballs is not the same
as working. Working means having pencil and paper by your side and going through each
example/exercise word-by-word, line-by-line.
For example, I might say something like “x2 − y 2 = 0. Thus, (x − y)(x + y) = 0.” If it’s not
obvious to you why the first sentence implies the second, stop right there and work on it
until you understand why. Don’t just let your eyeballs fly over these sentences and pretend
that your brain is “getting” it.
I will often not bother to explain some steps, especially if they simply involve some simple
algebra.
• You get a List of Formulae during the A-level exam.
It’s called List of Formulae MF26. It’s reproduced from p. 331 of this book and is also
available at this link (MF26). (I cannot guarantee though that your JC will give you the
List during your JC common tests and exams.) Page 8, Table of Contents • Online Calculators
Google is probably the quickest for simple calculations. Type in anything into your
browser’s Google search bar and the answer will instantly show up: Wolfram Alpha is somewhat more advanced (but also slower). Enter “sin x” for example
and you’ll get graphs, the derivative, the indefinite integral, the Maclaurin series, and a
bunch of other stuff you neither know nor care about.
The Derivative Calculator and the Integral Calculator are probably unbeatable for the
specific purposes of differentiation and integration. Both give step-by-step solutions for
anything you want to differentiate or integrate.
Here is a collection of spreadsheets I made. These spreadsheets are for doing tedious and
repetitive calculations that H2 Maths students (and hence also H1 Maths students) will
often encounter. As with anything I do, I welcome any feedback you may have about
these spreadsheets. Perhaps in the future I will make a more attractive version of it.
(Instructions: Click “Make a copy” to open up your own independent copy of
this spreadsheet. Enter your input in the yellow cells. Output is produced in
the blue cells. If you mess up anything, simply click the same link and “Make
a copy” again.) Page 9, Table of Contents Use of Graphing Calculators
You are required to know how to use a graphing calculator.6
This textbook will give only a very few examples involving graphing calculators.
There is no better way of learning to use it than to play around with it yourself. By the
time you sit down for your A-level exams, you should have had plenty of practice with it.
You can also use any of the seven calculators in the list below (last updated by SEAB
on March 1st, 2016, PDF). But this textbook will stick with the TI-84 PLUS Silver Edition (which I’ll simply call the TI84). (My understanding is that most students use a TI
calculator and that the five approved TI calculators are pretty similar.)
I’ll always start each example with the calculator freshly reset. 6 Pretty bizarre that in this age of the smartphone, they want you to learn how to use these clunky and now-useless devices
from the ’80s and ’90s. It is the equivalent of learning to program a VCR.
IMHO it’d be much better to teach you to some simple programming or Excel (or whatever spreadsheet program). “B-b-but
... how would such learning be tested in an exam format?” Ay, there’s the rub. In the Singapore education system, anything
that cannot be “examified” is not worth learning. Page 10, Table of Contents Contents
About This Book 6 Tips for the Student 8 Use of Graphing Calculators 10 I 18 Functions and Graphs 1 Dividing By Zero 19 2 Functions 20 3 Graphs: Introduction 22 4 Graphs: Intercepts 24 5 Graphs: Turning Points 26 6 Quadratic Equations 28 7 Graphs: Asymptotes 33 8 Exponents: Laws 36 9 Exponents: Graphs 37 10 Exponential Growth and Decay 39 11 Logarithms: Introduction 43 12 Logarithms: Laws 44 13 Logarithms: Graphs 46 14 Logarithmic Growth 48 Page 11, Table of Contents 15 Graphs: Symmetry 50 16 Graphing with the TI84 52 17 Simultaneous Equations: One Linear and One Quadratic 54 18 Solving Equations Using Your TI84 57 19 Quadratic Inequalities 58 20 Solving Inequalities Using Your TI84 59 21 Formulating an Equation or a System of Linear Equations from a Problem
Situation
62 II Calculus 63 22 Equations of Lines 64 23 The Derivative as Slope of the Tangent 66 24 Chain Rule 73 25 Increasing, Decreasing, and f ′ 75 26 Finding Turning Points (the First Derivative Test) 77 27 Inflexion Points 81 28 Finding Max/Min Points on the TI84 86 29 Finding the Derivative at a Point on the TI84 88 30 Connected Rates of Change Problems 89 31 Integration as the Reverse of Differentiation 91 Page 12, Table of Contents 32 The Constant of Integration 94 33 Basic Rules of Integration 95 34 The Definite Integral as the Area Under a Graph 98 35 Area between a Curve and Lines Parallel to Axes 102 36 Area between a Curve and a Line 103 37 Area between Two Curves 104 38 Finding Definite Integrals on your TI84 105 III 106 Probability and Statistics 39 How to Count: Four Principles 107 39.1 How to Count: The Addition Principle . . . . . . . . . . . . . . . . . . . . . . . 108
39.2 How to Count: The Multiplication Principle . . . . . . . . . . . . . . . . . . . 111
39.3 How to Count: The Inclusion-Exclusion Principle . . . . . . . . . . . . . . . . 115
39.4 How to Count: The Complements Principle . . . . . . . . . . . . . . . . . . . . 117
40 How to Count: Permutations 118 40.1 Permutations with Repeated Elements . . . . . . . . . . . . . . . . . . . . . . . 122
40.2 Partial Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
40.3 Permutations with Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
41 How to Count: Combinations 129 41.1 Pascal’s Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
41.2 The Combination as Binomial Coefficient . . . . . . . . . . . . . . . . . . . . . 133
41.3 The Number of Subsets of a Set is 2n . . . . . . . . . . . . . . . . . . . . . . . . 136 Page 13, Table of Contents 42 Probability: Introduction 138 42.1 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
42.2 The Experiment as a Model of Scenarios Involving Chance . . . . . . . . . . . 140
42.3 Mutually Exclusive Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
42.4 Complementary Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
42.5 The Union of Two Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
42.6 The Intersection of Two Events . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
42.7 Properties of Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
43 Probability: Conditional Probability 147 44 Probability: Independence 149 45 Probability: Not Everything is Independent 153 46 Random Variables: Introduction 155 47 Random Variables: Probability Distribution 156 48 Random Variables: Independence 160 49 Random Variables: Expectation 162 49.1 The Expectation Operator is Linear . . . . . . . . . . . . . . . . . . . . . . . . 165
50 Random Variables: Variance 167 50.1 Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
50.2 Properties of the Variance Operator . . . . . . . . . . . . . . . . . . . . . . . . 174
51 The Binomial Distribution 176 51.1 Probability Distribution of the Binomial R.V. . . . . . . . . . . . . . . . . . . 177
51.2 The Mean and Variance of the Binomial Random Variable Page 14, Table of Contents . . . . . . . . . . 178 52 The Continuous Uniform Distribution 180 52.1 The Continuous Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . . 180
52.2 Important Digression: P (X ≤ k) = P (X < k) . . . . . . . . . . . . . . . . . . . 182
52.3 The Cumulative Distribution Function (CDF) . . . . . . . . . . . . . . . . . . 183
52.4 The Probability Density Function (PDF) . . . . . . . . . . . . . . . . . . . . . 184
53 The Normal Distribution 185 53.1 The Normal Distribution, in General . . . . . . . . . . . . . . . . . . . . . . . . 191
53.2 Sum of Independent Normal Random Variables . . . . . . . . . . . . . . . . . 200
54 The Central Limit Theorem and The Normal Approximation 204 55 Sampling 207 55.1 Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
55.2 Population Mean and Population Variance . . . . . . . . . . . . . . . . . . . . 208
55.3 Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
55.4 Distribution of a Population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
55.5 A Random Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
55.6 Sample Mean and Sample Variance . . . . . . . . . . . . . . . . . . . . . . . . . 213
55.7 Sample Mean and Sample Variance are Unbiased Estimators . . . . . . . . . 219
55.8 The Sample Mean is a Random Variable . . . . . . . . . . . . . . . . . . . . . 222 55.9 The Distribution of the Sample Mean . . . . . . . . . . . . . . . . . . . . . . . 223
55.10Non-Random Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
56 Null Hypothesis Significance Testing (NHST) 225 56.1 One-Tailed vs Two-Tailed Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
56.2 The Abuse of NHST (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
56.3 Common Misinterpretations of the Margin of Error (Optional) . . . . . . . . 233
56.4 Critical Region and Critical Value . . . . . . . . . . . . . . . . . . . . . . . . . . 236
56.5 Testing
of
a
Population
Mean
2
(Small Sample, Normal Distribution, σ Known) . . . . . . . . . . . . . . . . . 238
Page 15, Table of Contents 56.6 Testing
of
a
Population
Mean
2
(Large Sample, Any Distribution, σ Known) . . . . . . . . . . . . . . . . . . . 240
56.7 Testing
of
a
Population
Mean
2
(Large Sample, Any Distribution, σ Unknown) . . . . . . . . . . . . . . . . . 242
56.8 Formulation of Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
57 Correlation and Linear Regression 245 57.1 Bivariate Data and Scatter Diagrams . . . . . . . . . . . . . . . . . . . . . . . . 245
57.2 Product Moment Correlation Coefficient (PMCC) . . . . . . . . . . . . . . . . 247
57.3 Correlation Does Not Imply Causation (Optional) . . . . . . . . . . . . . . . . 253
57.4 Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
57.5 Ordinary Least Squares (OLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
57.6 TI84 to Calculate the PMCC and the OLS Estimates . . . . . . . . . . . . . . 261
57.7 Interpolation and Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
57.8 The Higher the PMCC, the Better the Model? . . . . . . . . . . . . . . . . . . 271 IV Ten-Year Series 273 58 Past-Year Questions for Section A: Pure Mathematics 274 59 Past-Year Questions for Section B: Prob. & Stats 289 V 315 Syllabus and List of Formulae 60 Revised (8865) Syllabus 316 61 New List of Formulae (MF26) 330 VI 343 Answers to Exercises 62 Answers to Exercises in Part I: Functions and Graphs Page 16, Table of Contents 344 63 Answers to Exercises in Part II: Calculus 360 64 Ans...

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