mat01a1: functions and mathematical models · 2017. 2. 24. · important information lecture...
TRANSCRIPT
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MAT01A1: Functions and Mathematical Models
Dr Craig
21 February 2017
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Introduction
Who:
Dr Craig
What:
Lecturer & course coordinator for MAT01A1
Where:
C-Ring 508 [email protected]
Web:
http://andrewcraigmaths.wordpress.com
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Important information
Course code: MAT01A1
NOT: MAT1A1E, MAT1A3E, MATE0A1,
MAEB0A1, MAA00A1, MAT00A1,
MAFT0A1
Learning Guide: available on Blackboard.
Please check Blackboard twice a week.
Student email: check this email account
twice per week or set up forwarding to an
address that you check frequently.
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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 or D1 LAB 408
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Other announcements
I No tuts for MAT01A1 on Wednesdays. If
you see this on your timetable, it is an
error. (To move your Chem. prac., email
Mr Kgatshe [email protected])
I CSC02A2 students: tut on Thu 09h40 to
11h15.
I Maths Learning Centre in C-Ring 512:10h30 – 14h35 Mondays
08h00 – 15h30 Tuesday to Thursday
08h00 – 12h55 Fridays
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Lecturers’ Consultation Hours
Monday:
10h30 – 11h30 Ms Richardson (C-503)
Wednesday:
14h30 – 16h00 Ms Richardson (C-503)
Thursday:
11h00 – 13h00 Dr Craig (C-508)
13h30 – 14h00 Ms Richardson (C-503)
Friday:
11h30 – 13h00 Dr Craig (C-508)
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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.
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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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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 ofrepresenting 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) = secx
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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
Consider the function
f (x) =
{−x + 2 if x 6 1x2 if x > 1
Example: The absolute value function.
f (x) = |x| =
{x if x > 0
−x if x < 0
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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) =
6.2 if 0 6 x 6 5
8.1 if 5 < x 6 10
10.1 if 10 < x 6 15
12 if 15 < x 6 25
12.9 if 25 < x 6 35
14.1 if 35 < x
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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 ∈ DExamples:
I f (x) = xI f (x) = sin(x)I f (x) = x3
An odd function is a reflection about the
origin. Thus, 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) = cosxincreasing, 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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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: The length of a newly born snakeis 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− 2005Therefore m = 200, 000.
To be more accurate, we can use a statistical
technique known as linear regression.
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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 apolynomial 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.