Math Review Manual
With a Brief
For Students Entering McMaster University
Department of Mathematics and Statistics
McMaster University
2008
Published by:
Miroslav Lovric
Department of Mathematics and Statistics
McMaster University
1280 Main St. West
Hamilton, Ontario, Canada L8S 4K1
© 2008 Miroslav Lovric
Mathematics Review Manual
With a Brief
First-year Survival Guide
For Students Entering McMaster University
Also available online at
www.math.mcmaster.ca/lovric/rm.html
written by
Miroslav Lovric
Department of Mathematics and Statistics
McMaster University
Hamilton 2008
This
booklet
will
help
you
prepare
for
your
academic
life
as
a
first‐year
university
student,
and,
in
particular,
for
the
first‐year
courses
in
mathematics
that
you
will
take.
Table of Contents
General Information
What’s This Manual and Survival Guide About ................................. iii
Why Background Knowledge Matters .................................................. v
Transition from High School to University ...................................... vii
How is Math in University Different from High School Math? .. ix
Learning Mathematics ......................................................................... xiii
Important Little Bits ............................................................................. xv
Mathematics
You have to know and be proficient in the material from the
following four chapters. Very little of it will be reviewed in class.
Chapter 1. Basic Algebra ......................................................................... 1
Chapter 2. Basic Formulas from Geometry ....................................... 9
Chapter 3. Equations and Inequalities ............................................... 13
Chapter 4. Elements of Analytic Geometry ..................................... 21
The material from the three chapters below will be taught
and discussed in your first-year calculus course. However,
you will have to spend extra time working on these areas,
to gain technical proficiency and confidence with the material.
Chapter 5. Functions ............................................................................. 30
Chapter 6. Trigonometry ...................................................................... 41
Chapter 7. Exponential and Logarithmic Functions ....................... 57
i
The material from the chapter below will be covered in depth in
your first-year calculus course. If you decide to skip something in
this manual, then skip this chapter.
Chapter 8. Intro to Calculus: Limits and Derivatives ................... 65
***
Answers to all Exercises ..................................................................... 74
***
This manual is also available online at (free download)
www.math.mcmaster.ca/lovric/rm.html
ii
What’s this manual and survival guide about?
A leap from secondary education to university environment will be, without doubt,
one of the most challenging and stressful events in your life. It is a true rite of
passage, with all of its anxieties, pains, hopes, frustrations, joys and rewards. You
have probably created a mental image of the new environment you will be
encountering soon - but it is blurry, lots of fine detail is missing. The better
prepared you are, the easier it will be to adjust to new situations, demands and
expectations that university life will place on you.
No matter which high school you came from, you have certain
strengths and certain weaknesses. There are things that you learnt well in
high school, things you know and are comfortable with. But, there are things that
you forgot, or you don’t know about or have very little experience with. In high
school you acquired certain skills, but need to brush up on some others.
This manual will tell you where you are; it will help you identify those areas
of mathematics that you are good at, and those areas that you need to learn, review
and work on. All you need is a little dedication, a pencil and paper, and about an hour
of your (uninterrupted) time per day (say, during the last three weeks of August).
Unplug the TV, turn off your cell phone, kick your sibling(s) and/or your parent(s)
out of your room, and tell them that you need to work on something really
important.
This manual has two parts. The first part is about the things you have thought
a lot lately. How is life in university different from high school? What should I
expect from my fist-year classes? How is university math different from math in
high school? Read, and reflect on the issues raised … discuss it with your parents,
friends, teachers, or older colleagues. Nobody can give you detailed and precise
answers to all questions that you have, but at least, you will get a good feeling about
the academic side of your first-year university experience.
What will my first-year professors assume that I know about
mathematics? The big part of this booklet is dedicated to answering this
question. Look at the table of contents to see what areas of mathematics are
covered.
As I said, have a pencil and paper handy. I suggest that you start with the first
section, and work from there, without skipping sections. Read the material
iii
presented in a section slowly, with understanding; make notes and try to solve
exercises as you encounter them (answers to all exercises are in the back of this
manual). Even if the material in some section looks easy, do not skip the whole
section – select several problems and test your knowledge.
If you realize that you have problems with certain material, read the section
carefully, twice or three times if needed. Work on problems slowly, making sure
that you understand what is going on. If needed, consult your high school
textbooks, or go to a local library and find a reference. Ask somebody who knows
the stuff to discuss it with you; if you prefer, hire a private math tutor for a few
sessions.
I know that doing math is not the coolest thing to do in summer
- BUT think a bit about the future. Change from high school to university is a big
change; the better prepared you are, the easier it will be for you to adjust
successfully to your new life as a university student. Student life is a busy life. It
will be quite difficult for you (I did not say impossible!) to find time to do two
things: learn new material presented in a lecture and, at the same time, review
background material that you are assumed to know and be comfortable with. Not to
mention that, without adequate preparation, you will have difficulties following
lectures. Review your math now, while you have lots of free time
on your hands!
One thing is certain: the more math you do, the easier it gets - experience
helps! Do as many problems as you can, don’t give up because the stuff
looks difficult or you feel bored with it. Little investment of your time now,
in summer, will make studying mathematics in the fall a whole lot easier.
Good luck!
See you in September,
Miroslav Lovric, Associate Professor
First-year instructor
Department of Mathematics and Statistics, McMaster University
iv
Why Background Knowledge Matters
Mathematics is cumulative, new material builds upon the previously
covered (i.e., understood, learnt) material. It is not possible to truly
understand and apply an advanced concept (say, derivatives) without understanding
all basic concepts that are used to define it (fractions, limits, graphs, etc.).
Many times, the reason why students lose marks on tests in first-year
Calculus (and other math courses!) is due to a problem with something
elementary, such as fractions, simplifying, solving equations, or recalling basic
properties of exponents, trigonometric functions. etc. Let us look at a few samples
of actual test solutions.
In the case below, the student chose the appropriate integration method (which is
taught in the first-year Calculus course), but then did not simplify correctly the
fraction in the integral (see the last two lines). This error cost the student 50% of
the credit for the question.
Look how mach effort was put into simplifying the expression for f’(x) below – not
to mention how much valuable time was lost! Moreover, the student made a
v
mistake in simplifying and got two (of three) correct values for x. Penalty for this
mistake: 25% of the credit for the question. As in the previous case, note that
the credit lost was not due to a new concept learnt in the university
Calculus course, but due to errors related to high school material.
Most-often-heard comment about a test is that ‘there was not enough time.’
Certainly, if it takes you more than 5 minutes to do this question, you
will not have enough time to complete the test.
In the case below, the student tried to analyze the expression for f’(x) by looking
at the graphs of sin x and cos x (excellent idea!). However, the graphs of the two
functions are incorrect, and the answer does not make sense. The student lost all
credit for the question.
vi
TRANSITION FROM HIGH SCHOOL TO UNIVERSITY
What is new and different in university? Well, almost everything: new people
(your peers/colleagues, teaching and lab assistants, teachers, administrators, etc.),
new environment, new social contexts, new norms, and – very important new demands and expectations. Think about the issues raised below.
How do you plan to deal with it? Read tips and suggestions, and try
to devise your own strategies.
First-year lectures are large – you will find yourself in a huge auditorium,
surrounded by 300, 400, or perhaps even more students. Large classes create
intimidating situations. You listen to a professor lecturing, and hear something
that you do not understand. Do you have enough courage to rise your hand and ask
the lecturer to clarify the point? Keep in mind that you are not alone – other
students feel the same way you do. It’s hard to break the ice, but you have to try.
Other students will be grateful that you asked the question – you can be sure that
lots of them had exactly the same question in mind.
Remember, learning is your responsibility. Come to classes regularly, be
active, take notes, ask questions. Find a quiet place to study. Use all resources
available to you. Discuss material with your colleagues, teaching and lab assistants,
and/or professors.
Courses have different requirements and restrictions with regards to calculators
and computer software. You will find the information about it in the course
syllabus that will be given to you (usually) in the first lecture of a course.
The amount of personal attention you get from your teachers, compared to
high school, is drastically lower. If you have a question, or a problem, you will have
to make an effort to talk to your lecturer, or to contact the most appropriate
person.
Consider taking courses that help you develop research skills (such as: critical use
of electronic resources, logical and critical thinking, library search skills,
communication and presentation skills etc.). Have you heard of inquiry courses?
vii
Good time management is essential. Do not leave everything for the last
moment. Can you complete three assignments in one evening? Or write a major
essay and prepare for a test in one weekend? Plan your study time carefully. Eat
well, exercise regularly, plan social activities - have a life! Amount of material
covered in a unit of time increases at least three-fold in university courses,
compared to high school. This means that things happen very quickly. If you miss
classes and do not study regularly you will get behind in your courses. Trying to
catch up is not easy. For each hour of lecture plan to spend (at least) three
hours studying, reviewing, doing assignments, etc.
Inquire about learning resources available to you. Do you know where the science
(engineering, humanities) library is? When are computer labs open? Do you know
how the Centre for Student Development can help you deal with academic issues?
Before coming to Mac, browse through its internet site. Bookmark the sites that
link to learning resources.
‘In university grades drop by 30%.’ Not necessarily. Study regularly (do you
know how to study math? Chemistry? Physics? Why not discuss it with your
lecturer?). Most probably, you will have to adjust/modify your present study habits.
One thing is certain: the amount of work that earned you good marks in high school
will not suffice to keep those marks in university.
It is possible to study hard and still fail a test. If you fail a test, react
immediately. Identify reasons for your poor performance. Visit your professor
during her/his office hours, bring your test and discuss it. Be ready to re-examine
and modify your study strategies. Do not get discouraged by initial bad marks that
you might get.
If you have problems, react and deal with it immediately. Ask your professor for
advice. Talk to an adviser in your faculty office.
What constitutes academic dishonesty? Copying stuff from internet and pasting
into your assignment could be considered academic dishonesty. If you are caught,
you might fail the course. What other practices are considered academic
dishonesty? Be informed about it, so that you don’t get in trouble. The course
syllabus (for any of your courses) will provide information and links to McMaster
policy regarding academic dishonesty.
viii
HOW IS MATH AT UNIVERSITY DIFFERENT FROM HIGH
SCHOOL MATH?
Lectures move at a faster pace. Usually, one lecture covers one section from
your textbook. Although lectures provide necessary theoretical material, they
rarely present sufficient number of worked examples and problems. You have to do
those on your own.
Certain topics (trigonometry, exponential and logarithm functions, vectors,
matrices, etc.) will be taught and/or reviewed in your first-year calculus and
linear algebra courses. However, the time spent reviewing in lectures will not
suffice to cover all details, or to provide sufficient number of routine exercises –
you are expected to do it on your own. Use this manual! Don’t leave it at home, bring
it with you to McMaster.
You have to know and be proficient with the material covered in the first
four chapters of this booklet:
• Basic Algebra
• Basic Formulas from Geometry
• Equations and Inequalities
• Elements of Analytic Geometry.
For instance, computing common denominators, solving equations involving fractions,
graphing the parabola y=x2, or solving a quadratic equation will not be reviewed in
lectures.
In university, there is more emphasis on understanding than on technical
aspects. For instance, your math tests and exams will include questions that will ask
you to quote a definition, or to explain a theorem, or answer a ‘theoretical question.’
Here is a sample of questions that appeared on past exams and tests in the firstyear calculus course:
• Is it true that f’(x)=g’(x) implies f(x)=g(x)? Answering ‘yes’ or ‘no’ only will
not suffice. You must explain your answer.
• State the definition of a horizontal asymptote.
• Given the graph of 1/x, explain how to construct the graph of 1+1/(x-2).
ix
•
•
•
•
Using the definition, compute the derivative of f(x)=(x-2)-1.
Can a polynomial of degree 3 have two inflection points? You must explain
your answer to get full credit.
Given below is the graph of the function f(x). Make a rough sketch of the
graph of its derivative f’(x).
Below are the graphs of two functions, f(x) and g(x). Compute the
composition g(f(3)).
You will be allowed - and encouraged - to use your (graphing) calculator and/or
computer software (such as Maple) to study mathematics, to do homework
assignments and computer labs. On tests and exams, either no calculator will be
allowed (would you really need a calculator to answer any of the above test/exam
questions?), or you will be asked to use the calculator that McMaster chose as a
standard (this way, everybody uses the same calculator). Calculators and software
are an aid, but not a replacement for your brain, and you should treat them as such.
If a calculator says something, it is not necessarily a correct answer.
Mathematics is not just formulas, rules and calculations. In university
courses, you will study definitions, theorems, and other pieces of ‘theory.’ Proofs
are integral parts of mathematics, and you will meet some in your first-year
x
courses. You will learn how to approach learning ‘theory,’ how to think about proofs,
how to use theorems, etc.
Layperson-like attitude towards mathematics (and other disciplines!) - accepting
facts, formulas, statements, etc. at face value - is no longer acceptable in
university. Thinking (critical thinking!) must be (and will be) integral part of
your student life. In that sense, you must accept the fact that proofs and
definitions are as much parts of mathematics as are computations of derivatives
and operations with matrices.
***
Have you given any thought to mathematics as a career? Attend information
sessions organized by Mathematics and Statistics Department (will be advertised in
lectures), learn about programs and careers in mathematics and statistics. Get
informed, keep your options open!
***
If, for some reason, you developed negative attitude and
feelings towards mathematics in high school, then leave them
there! You will have a chance to start fresh at a university. First-year
math courses at McMaster start at a level that is appropriate for most high
school graduates. Use this manual, read it from cover to cover, get prepared!
xi
xii
LEARNING MATHEMATICS
Learning
mathematics
requires
seriousness,
dedication,
discipline, concentration, significant amount of time and hard
work. Your teachers will help you learn how to learn
mathematics.
To learn mathematics means to understand
AND to memorize.
To understand something means to be able to correctly and effectively
communicate it to somebody else, in writing and orally; to be able to answer
questions about it, and to be able to relate it to known mathematics material.
Understanding is a result of a thinking process. It is not a mere transfer from the
one who understands (your lecturer) to the one who is supposed to understand
(you).
How do you make yourself understand math? Ask questions about the
material and answer them (either by yourself, or with the help of your colleague,
teaching assistant or lecturer). Approach material from various perspectives, study
solved problems and work on your own on problems and exercises. Make connections
with previously taught material and apply what you just learnt to new situations.
It is necessary to memorize certain mathematics facts, formulas and
algorithms. Memorizing is accomplished by exposure: by doing drill exercises, by
using formulas and algorithms to solve exercises, by using mathematics facts in
solving problems.
The only way to master basic technical and computational skills is to solve a large
number of exercises. Drill.
It is impossible to understand new mathematics unless one has mastered (to a
certain extent) the required background material.
xiii
understand so that you won’t have to
memorize
wrong approach to learning math:
memorize so that you won’t have to understand
think about it!!!
xiv
IMPORTANT LITTLE BITS
Come to classes, tutorials and labs regularly, be active! Think, ask
questions in class, give feedback to your lecturer.
Lecture by itself will not suffice. You need to spend time on your own doing
math: studying, working on assignments, preparing for tests and exams, etc. Rule of
thumb: three hours on your own for each hour of lecture.
Plan your study time carefully. Don’t underestimate the amount of time you
need to prepare for a test, or to work on an assignment; try not to do everything in
the last minute (the fact that your hard drive crashed night before your
assignment is due is not an acceptable excuse for a late assignment).
Make sure that you are aware of (and use!) learning resources available to you.
Here are some of them:
• Lectures, tutorials, and review sessions
• Your lecturer’s office hours
• Your teaching assistant’s office hours
• E-mail and course internet page
• Mathematics help centre in Hamilton Hall
• Thode Library (Science Library)
• Centre for Student Development
Always learn by understanding. Memorizing will not get you too far. Think, do
not just read; highlighting every other sentence in your textbook is not studying!
If you are able to explain something to a colleague and answer their questions about
it, then you have learnt it!
Drill is essential for a success (not just in math!). It’s boring, but it works!
Solving hundreds of problems will help you gain routine and build confidence you
need (together with a few other things) to write good exams.
You will be allowed to use a (graphing) calculator and/or computer
software for assignments and labs. On tests and exams, you might not be allowed
xv
to use a calculator, or will have to use a model that is accepted as a standard at
McMaster. Your instructor will give you detailed information about this.
Note about your lecture notes
Your lecture notes will be your most valuable resource. You will refer to them when
you do homework, a computer lab, or prepare for a test or an exam. So:
• during a lecture, take notes
• later, read the notes; make sure that you have correct statements of all
definitions, theorems, and other important facts; make sure that all
formulas and algorithms are correct, and illustrated by examples
• fill in the gaps in your notes, fix mistakes; supplement with additional
examples, if needed
• add your comments, interpret definitions in your own words; restate
theorems in your own words and pick exercises that illustrate their use
• write down your questions, and attempts at answering them; discuss your
questions with your colleague, lecturer or teaching assistant, write down the
answers
• it is a waste of time to try again and yet again to understand a concept; so,
once you understood it, write it down correctly, in a way that you will be able
to understand later; this way, studying for an exam consists of re-calling and
not re-learning; re-calling takes less time, and is easier than re-learning
• keep your notes for future reference: you might need to recall a formula, an
algorithm or a definition in another mathematics course.
xvi
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