lecture 0-1: introduction to calculus i, functions
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Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Lectures 0–1:Introduction to Calculus I, Functions
Thomas Calculus Section 1.1
Oscar F Bandtlow
School of Mathematical SciencesQueen Mary University of London
Calculus I — Week 1
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
What is Calculus?
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
How will you be taught?
Main point of accessQMplus (don’t forget to log in!)
Live teaching events4 lectures per week (2 on campus, 2 online, all of themrecorded)1 tutorial per week, starting in Week 2 (details to follow)
ResourcesThomas Calculus: the module textbookMyLab Math: virtual learning environment, for practiceand assessment
AssessmentFinal exam in January, worth 80%5 quiz-courseworks on MyLab Math, each worth 4%
Queries, help and supportUse the student forumAsk during the live lecturesTalk to your tutorEmail o.bandtlow@qmul.ac.uk.
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Please wear a mask when talking to me afterclass...
...I will wear one as well!
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Notation
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Equations and implications
Equations
The equality sign ‘=’ is used to denote equality of expressions.
Example
Implications
The implication sign ‘ =⇒ ’ connects to assertions and means‘therefore’ or ’implies’.
Example
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Sets
Sets may be written using curly brackets ‘{. . .}’. Some setshave established symbols.
{1, 2, 3, . . .} = N the set of natural numbers;
{−3,−2,−1, 0, 1, 2, 3, . . .} = Z the set of integers;
R the set of real numbers;
[a, b] closed interval, the set of all real numbers x withx ≥ a and x ≤ b
[a, b] = {x ∈ R | x ≥ a and x ≤ b};
(a, b) open interval, the set of all real numbers x withx > a and x < b
(a, b) = {x ∈ R | x > a and x < b}.
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Exercise
Write the following set using interval notation:
A = {x ∈ R | x2 − 1 < 0} .
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Exercise (ctd)
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Another exercise
Write the following set using interval notation:
B = {t ∈ R | t2 − t − 2 ≥ 0} .
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Another exercise (ctd)
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Functions
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Functions
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Functions
Definition
A function f from a set D to a set Y is a rule that assigns aunique value y ∈ Y to each element x ∈ D. We write this
f : D → Y
x 7→ f (x).
D is called the domain of f ;
Y is called the codomain of f .
The range R of f is the set of all values taken f (x), thatis,
R = {y ∈ Y | y = f (x), x ∈ D}.
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Examples
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Examples
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Natural domain
The natural domain of a function f is a the largest set D forwhich the rule y = f (x) makes sense (that is, gives realoutput).
Example
Find the natural domain of the following functions:
1 f (x) = x2;
2 g(x) =√
4− x2;
3 h(x) =√x2 − 4.
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Example 1
f (x) = x2
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Example 2
g(x) =√
4− x2
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Example 3
h(x) =√x2 − 4
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Representation of functions
Table:
Graph:
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Graph of a function
The graph of a function f : D → Y are the points (x , y) in theCartesian plane whose coordinates are the input-output pairs(x , f (x)) of f with x ∈ D.
Lectures 0–1
OF Bandtlow
Introductionto Calculus I
What is Calculus?
How will you betaught?
Notation
Functions
Notation
Functions, domain,range
Vertical Line Test
Vertical Line Test
The graph of a function cuts a vertical line at most once.
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