mat01a1: functions and mathematical models · an algebraic function is one that can be formed by...
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MAT01A1: Functions and Mathematical Models
Dr Craig
4/5 March 2020
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Reminder: HW2 on WeBWorK
I On complex numbers
I Closes Friday at 23h59
Saturday class
I D-LES 101
I 09h00 – 12h00
I Revision to help prepare for Monday’s test
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Semester Test 1
I Full details in the pdf posted under
“Assessments”.
I The venue allocation is very important:
Surnames Arends – Moses:D1 LAB K08
Surnames Motaung – Zwane:D-LES 101
I You must be at your venue by 08h15.
I Test is 08h30–10h00.
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Consultation Hours
Thursday:
08h50 – 10h25 Dr Robinson (C-Ring 514)
11h20 – 12h15 Dr Craig (C-Ring 508)
15h00 – 16h15 Dr Craig (C-Ring 508)
Maths Learning Centre (C-Ring 512)
Wednesday 08h00 – 15h25
Friday 08h00 – 15h25
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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).
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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
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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.
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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.
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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]
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Examples
Sketch the following functions and find their
domain and range:
(a) f (x) = −3x + 4
(b) g(x) = x2 − 2
(c) h(x) = cscx
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Representations of functions
There are four ways to represent a function:
I Verbally (describe in words)
I Tables
I Graphically
I Algebraically
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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.
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The curve x2 + y2 = 4 is not a function:
The curve y =√4− x2 is a function:
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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
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Example: step functions
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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) =
8 if 0 6 x 6 5
9.60 if 5 < x 6 10
11.80 if 10 < x 6 15
13.90 if 15 < x 6 25
15 if 25 < x 6 35
16 if 35 < x
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Properties of functions
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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.
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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.
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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]
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Ch 1.2: a catalogue of essentialfunctions
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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.
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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?
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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
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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.
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More types of functions (mostlynon-linear)
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Polynomials
These are functions of the form:
f (x) = anxn+an−1x
n−1+. . .+a2x2+a1x+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
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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
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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
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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
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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
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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
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f (x) = 2x
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f (x) = 0.5x
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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.
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f(x) = log2 x
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Make sure that you are comfortablewith the sketches of exponential andlogarithmic functions. Know theirshape and where the intercepts occur.