MAT01A1: Functions and Mathematical Models

Dr Craig

20 February 2018

Introduction

Who:

Dr Craig

What:

Lecturer & course coordinator for MAT01A1

Where:

C-Ring 508 acraig@uj.ac.za

Web:

http://andrewcraigmaths.wordpress.com

Important information

Lecture times: Tuesday 08h50 – 10h25

Wednesdays 17h10 – 18h45

Lecture venues: C-LES 102, C-LES 103

Tutorials: Tuesday afternoons

13h50 – 15h25: D-LES 104 or D-LES 106

OR

15h30 – 17h05: C-LES 203 ONLY

Lecturers’ Consultation Hours

Monday:

12h00 – 13h30 Ms Richardson (C-503)

Wednesday:

15h00 – 16h00 Ms Richardson (C-503)

Thursday:

11h20 – 12h55 Dr Craig (C-508)

Friday:

11h20 – 12h55 Dr Craig (C-508)

e-Quizzes

I e-Quiz 1 (questions about the content of

the Learning Guide) closes at 23h59 on

Wednesday night.

I e-Quiz 2 (questions about Appendices)

closes at 23h59 on Thursday night.

Complex numbers

We don’t cover Complex Exponentials at the

end of Appendix H.

Note: these lecture slides should be inused in conjunction with the textbook.These slides do not tell the full story.You must read the sections from thetextbook and complete the prescribedtutorial exercises in order to fullyunderstand each section.

Introduction to functions

Four examples of functions:

I The area of a circle depends on the

radius: A = πr2.

I Population of the world depends on time:

P (1950) = 2, 560, 000, 000.

I The cost of posting a package depends on

the weight: C(w).

I The vertical ground acceleration during

an earthquake: S(t).

Example:

A rectangular box with an open top has a

volume of 10m3. The length of the base is

twice its width. Material for the base costs

R10/m2 and material for the sides costs

R6/m2. Express the cost of the materials as

a function of the width of the base.

Solution:

C(w) = 20w2 +180

w, w > 0

Domain and range of functions

A function f is a rule which assigns to each

element x in a set D exactly one element,

f (x), in a set E.

The set D is the domain of f . The range of

f is the set of all possible values of f (x) as

x varies through the domain.

A symbol representing an arbitrary element

of the domain is called an independent

variable and a symbol representing an

arbitrary element of the range is a dependent

variable. In the example of the circle: r is

the independent variable while A is the

dependent variable.

Graphs of functions: a common way of

representing a function is by a graph.

Formally, the graph of the function f is the

set of ordered pairs {(x, f (x)

)| x ∈ D }.

7

1

4

1 4 7

Dom(f)= [1,6] and Ran(f)=[1,7]

Examples

Sketch the following functions and find their

domain and range:

(a) f (x) = −3x + 4

(b) g(x) = x2 − 2

(c) h(x) = secx

Representations of functions

There are four ways to represent a function:

I Verbally (describe in words)

I Tables

I Graphically

I Algebraically

The vertical line test

How do we know if a curve is a function?

A curve in the xy-plane is a function of x if

and only if no vertical line intersects the

curve more than once.

The curve x2 + y2 = 4 is not a function:

The curve y =√4− x2 is a function:

Piecewise defined functions

We are already familiar with one example of

a piecewise defined function, the absolute

value function:

f (x) = |x| =

{x if x > 0

−x if x < 0

Now sketch the function

f (x) =

{−x + 2 if x 6 1

x2 if x > 1

Example: step functions

Consider the cost of travelling on the Rea

Vaya bus. Here x is the number of km

travelled and C(x) is in rands:

C(x) =

7.0 if 0 6 x 6 5

8.5 if 5 < x 6 10

10.5 if 10 < x 6 15

12.5 if 15 < x 6 25

13.5 if 25 < x 6 35

14.5 if 35 < x

Properties of functions

Symmetry in functions

If a function f satisfies

f (−x) = f (x) for all x ∈ D

then f is an even function.

Examples:

I f (x) = x2

I f (x) = cos(x)

I f (x) = |x|Another way of defining an even function is

to say that it is a reflection about the y-axis.

Symmetry in functions

A function is odd if

f (−x) = −f (x) for all x ∈ D

Examples:

I f (x) = x

I f (x) = sin(x)

I f (x) = x3

An odd function is a reflection about the

origin. A necessary condition for a function f

to be odd is that it must have f (0) = 0.

Increasing and decreasing functions

A function f is increasing on an interval I if

whenever x1 < x2, we have f (x1) < f (x2).

A function f is decreasing on an interval I if

whenever x1 < x2 we have f (x1) > f (x2).

Example: Is the function f (x) = cosx

increasing, decreasing, or neither over the

following intervals:

(a) x ∈ [π, 3π/2]

(b) x ∈ [0, π/2]

(c) x ∈ [π/2, 3π/2]

Ch 1.2: a catalogue of essentialfunctions

Mathematical models

A mathematical model is a mathematical

description, using a function or an equation,

of a real-world problem.

If y is a linear function of x then the graph is

a straight line:

y = mx + c

where m is the slope of the graph, and c is

the y-intercept.

Example of a linear model: The length

of a newly born snake is 10cm and after 3

months the length is 25cm. It grows the

same amount each month.

(a) Express length as a function of time

(in months) where the D = [0, 12].

(b) Draw the graph of the length

function.

(c) What is the length of the snake

after 9 months?

Functions from data: If we don’t have a functionto work from, we can try to determine a functionusing empirical data.

Year Number of registered cars in SA

2005 4,500,000

2006 4,670,000

2007 4,890,000

2008 5,100,000

2009 5,310,000

2010 5,540,000

2011 5,770,000

2012 5,900,000

For the last example we can try to find a

function which fits the data by calculating

the slope of the line which goes through the

first and last points.

m =5, 900, 000− 4, 500, 000

2012− 2005

Therefore m = 200, 000.

To be more accurate, we can use a statistical

technique known as linear regression.

More types of functions (mostlynon-linear)

Polynomials

These are functions of the form:

f (x) = an.xn+an−1.x

n−1+. . .+a2.x2+a1.x+a0

where a0, . . . , an are constants. If the

leading coefficient an 6= 0 then f (x) is a

polynomial of degree n.

I polynomial of degree 1 = linear function

I polynomial of degree 2 = quadratic

function

I polynomial of degree 3 = cubic function

Power functions

These are functions of the form

f (x) = xa

Note that a can be

I a positive integer

I a = 1n where n is a positive integer

I a negative integer, so f (x) = 1xa

Rational functions

A rational function f is a ratio of two

polynomials

f (x) =P (x)

Q(x)

Example:

f (x) =2x4 − x2 + 1

x2 − 4

Algebraic functions

An algebraic function is one that can be

formed by using the algebraic operations of

addition, subtraction, multiplication, powers,

division and taking roots.

Note: any rational function is automatically

an algebraic function.

Examples:

f (x) =√x2 + 1 h(x) =

x4 − 16x2

x +√x

Trigonometric functions

Functions that express the ratio between x, y

and r when angles are plotted on the

xy-plane.

I sinx

I cosx

I tanx

I cscx

I secx

I cotx

Exponential functions

These are functions of the form

f (x) = ax

where a > 0 is constant.

Examples:

I f (x) = 2x

I f (x) = 0.5x

I f (x) = ex

f (x) = 2x

f (x) = 0.5x

Logarithmic functions

f (x) = loga x where a > 0 is a constant

(also known as the base). Logarithmic

functions are the inverse of exponential

functions. That is,

if y = ax then loga y = x.

f(x) = log2 x

Make sure that you are comfortablewith the sketches of exponential andlogarithmic functions. Know theirshape and where the intercepts occur.