MAT 135 (Section L5102) - Lecture 1
Announcements (Administrative)
1. Pick a copy of course outline, read it carefully (not during the
class!).
2. The OFFICIAL website for MAT135:
www.math.utoronto.ca/lam/
3. Sidney Smith Math Aid Centre (SS1071) starts operating the
coming Monday (September 19). The hours of operation are
posted on the MAT135Y Website above.
4. Starting from tomorrow night, the lecture notes (no pictures!)
for this section will be posted online at
www.math.toronto.edu/∼pinaki/teaching/mat135/
Note: This website is NOT the official one. For any important
information about MAT135 (e.g., Tutorial List, Test marks,
etc.), you MUST visit the official site.
Administrative Announcements - Contd.
5. Every MAT135F student MUST enrol in a Lecture section
AND a Tutorial section. If you have not yet picked a Tutorial
Time, pick one as soon as possible - using SWS (Student Web
Service). After you pick a Tutorial Time, a Tutorial section
will be assigned to you. For details, read the Course Outline.
Right now, there are approximately 2,050 students enrolled in
the course. But about 160 still have not picked a Tutorial
Time. Beware, if you are one of them!!!
Real Announcements
1. ASK questions - by all means! It is a crime to sit still and not
ask questions if you do not understand. If you do not
understand something, chances are that most of the others do
not understand it as well, but are not coming forward. Shame
on those criminals! Be a hero(ine) instead, and ask!
He who asks a question is a fool for five minutes; he who does
not remains a fool forever.
2. Do not sleep. Please!
Principal topic of the course:
Real-valued Functions of Real Numbers
• Integers: 1, 2, 3, ..., 0, -1, -2, ... The set of all integers is
usually denoted by Z (for zahlen: German for numbers).
Now visualize the number line.
Why is “right” the direction of increase?
Convention! Or a right handed conspiracy?
• Rational Numbers: Ratios or quotients of integers. Introduces
new numbers in the number line which fall in the gaps
between integers. The set of all rationals is usually denoted by
Q (for quotient).
• Real Numbers: “limits” of rationals, denoted by R (for real).
They fill out all the “gaps” in the number line.
Remember the picture of the number line!
Some Random Things
• At first people (i.e. who cared) thought all numbers were
rationals, i.e. given any two sticks you can find two integers,
say r1 and r2 such that if you divide the first stick in r1
segments of equal length and the second one by r2 segments
of equal length, then each part of the first one has the same
length as the each part of the second one.
• By
√ 5th century B.C. some people (e.g. Pythagoreans) knew
2 is an irrational number.
• In late 1800s Georg Cantor (an absolutely brilliant man)
showed that there are “more” irrational numbers than
rationals.
Food for thought:
√
• What is 2?
√
• What does it mean that √
2 is not rational? Ans: there are
no integers p, q such that 2 = qp . How does one prove it?
• Are there more irrational people than rationals?
Effects of Arithmetic Operations on Number Line
• Addition = Translation
• Addition by a positive number ↔ Translation to the right
• Addition by a negative number ↔ Translation to the left
Subtraction
• Multiplication by a positive number = Stretching/Shrinking
• Multiplication by c, c > 1 ↔ Stretching by a factor of c
• Multiplication by c, c < 1 ↔ Stretching by a factor of c
• Multiplication by 1 ↔ ?
• Negation = Multiplication by −1= Reflection about 0
• If x < 0, is −x negative or positive?
• Multiplication by a negative number = Combination of the
preceding two operations!
• So, if x is a real number, then −(−x) = x !
Topic: Real-valued Functions of Real Numbers
• A function is a process which produces exactly one output for
every input.
• Almost all processes in the real world produces more than one
output for all inputs.
Input
Food
Electricity
Process
−→ Food Processor
Coffee Bean −→ Coffee Grinder
Electricity
Humans
Output
−→ Processed Food
Noise
−→ Ground Coffee
Noise
−→ Education System −→ Educated Humans, Uneducated
Humans, Robots, Sociopaths, ...
• So functions are simpler than all the real world processes.
• You should be happy - Calculus is about a simple thing!
Topic: Real-valued Functions of Real Numbers
Real-valued functions of real numbers = functions whose inputs
and outputs (i.e. values) are real numbers
Usually we will denote functions by f , g , h, ...
Terminology:
• Domain = Set of valid inputs
• Domain of a coffee grinder: coffee beans, not green beans, or
any other kind of beans or anything else
• Domain of a function may be all of R or only a subset of it.
• Range = Set of valid outputs
• Range of a coffee grinder: ground coffee
• Range of a function may be all of R or only a subset of it.
Examples of functions
• Safe Keeper: f (x) = x
• Does nothing, outputs exactly what the input is.
• Usually called the identity function.
• Domain = ? Range = ?
• Lazy Function: f (x) = 0.5
• Usually called a constant function.
• Domain = ? Range = ?
• Absolute Value: f (x) = |x|
• |x| (pronounced “Absolute Value” of x) is the distance
between x and 0 in the number line.
• distance is always non-negative!
• Domain = ? Range = ?
(
x
if x ≥ 0
|x| =
− x if x < 0