Download Formal Functions

MATHS 102 Mathematics 2
Module 0
Models & Functions
Lecture 3
Formal Functions
MATHS 102 – Introductory Module
In this third lecture we will discuss
one of the key concepts of mathematics,
that of a function.
The idea of a function is used
repeatedly in mathematics. You will
already have met many functions, or
mathematical relationships which can be
expressed as functions.
but first … your work:
Cost
of running a flat …...
Braking distance of a car …...
Final 102 mark …...
Post-Lecture
Exercise …...
Factory Production
TOTAL COST
Number Units Produced
Factory Production
TOTAL COST
Number Units Produced
MATHS 102
Lecture 3/0
Administration
An
Example
Formal Functions
Types of Functions
Administration
Reminders
•
•
Assumed knowledge
Sources of help / Office hours
David:
MTWTh 8-9
Bill:
Ye Yoon:
MTWTh 9-10
MTWTh 10-11
New
•
•
Items
Class Reps
Computer access
MATHS 102
Lecture 3/0
Administration
An
Example
Formal
Functions
Types of Functions
Preliminary Exercise
x
x
+ 16
2(x + 16)
2(x + 16) – 14
2{2(x + 16) – 14}
2{2(x + 16) – 14} – x
[2{2(x + 16) – 14} – x]/3 – x
[2{2(x + 16) – 14} – x]/3 – x + 4
√([2{2(x + 16) – 14} – x]/3 – x + 4)
Preliminary Exercise
√([2{2(x
+ 16) – 14} – x]/3 – x + 4)
√([2{2x + 32 – 14} – x]/3 – x + 4)
√([2{2x + 18} – x]/3 – x + 4)
√([4x + 36 – x]/3 – x + 4)
√([3x + 36]/3 – x + 4)
√(x + 12 – x + 4)
√(16)
4
Surface area of a cylinder of
height 10cm
S = 2πr2 + 20πr
Radius
2cm
4cm
6cm
8cm
10cm
Surface Area
150.8cm2
351.9cm2
cm2
MATHS.102
Lecture 3/0
Administration
An
Example
Formal
Types
Functions
of Functions
Domains & Games
The
domain is a set (of numbers) which is
the starting set.
The second part of a function is the rule
which associates a particular value with
every member of the domain.
What are sensible domains for:
The
cost of n tomatoes ?
The cost of n kilos of tomatoes ?
The length of a piece of metal at a given
temperature ?
The function g(x) = 100/x
Algebraic formulations ...
= 2πr2 + 20πr
S: r —> 2πr2 + 20πr
S(r) = 2πr2 + 20πr
S
MATHS.102
Lecture 3/0
Administration
An
Example
Formal Functions
Types
of Functions
Functions can be expressed ...
Algebraically
Graphically
As
a Table.
Functions with which you may
be familiar.
What shape are they ?
linear
quadratic
cubic
& other polynomial
exponential & logarithmic
periodic (sines, cosines)
hyperbolic
Lecture 3/0 – Summary
Functions
are a key idea in
mathematics.
The idea of a function can be used in
many different situations
Functions map objects (numbers)
from one set (the domain) to another
(the range).
MATHS 102
Before
Lecture 3/0
the next lecture........
Go over Lecture 3/0 in your notes
Do the Post-Lecture exercise p13
Do the Preliminary Exercise p14
See you tomorrow ........