lecture 0-1: introduction to calculus i, functions

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


What is Calculus?


Notation

Functions

Notation


What is Calculus?

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


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 [email protected].

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Please wear a mask when talking to me afterclass...

...I will wear one as well!

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Notation

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


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


What is Calculus?


Notation

Functions

Notation


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


What is Calculus?


Notation

Functions

Notation


Exercise

Write the following set using interval notation:

A = {x ∈ R | x2 − 1 < 0} .

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Exercise (ctd)

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Another exercise

Write the following set using interval notation:

B = {t ∈ R | t2 − t − 2 ≥ 0} .

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Another exercise (ctd)

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Functions

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


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


What is Calculus?


Notation

Functions

Notation


Examples

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


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


What is Calculus?


Notation

Functions

Notation


Example 1

f (x) = x2

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Example 2

g(x) =√

4− x2

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Example 3

h(x) =√x2 − 4

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


Representation of functions

Table:

Graph:

Lectures 0–1

OF Bandtlow


What is Calculus?


Notation

Functions

Notation


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


What is Calculus?


Notation

Functions

Notation


Vertical Line Test

Vertical Line Test

The graph of a function cuts a vertical line at most once.

lecture 0-1: introduction to calculus i, functions

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