3_-_Graphs_and_Functions v3

1
Graphs and Functions
Session #3
Carlos da Maia, PhD
Business School
UNIVERSITY OF ST. THOMAS OF MOZAMBIQUE
3L5ECONS
Advanced Mathematics for Economics, September 2014

2
Outline
1 Functions of One Variable
2 An Example of a Function
3 Domain of a Function
4 Graphs of Functions
5 Linear Functions
6 Linear Models
7 Non-linear Functions

3
Introduction
Important in every area of pure and applied mathematics
(including mathematics applied to economics);
The language of economic analysis is full of terms like demand
and supply functions, cost functions, production functions,
consumption functions, etc;
One variable is a function of another if this rst variable
depends upon the second;
Example
The area of a circle is a function of its radius; if radius r is given,
then the area A is determined;

4
Introduction
Example

5
Introduction
Example

6
Introduction, cont.
We don't need a mathematical formula to show that one
variable is a function of another;
The table denes consumption expenditure as a function of
the calendar year:
Example
Year 1998 1999 2000 2001 2002 2003
Consumption 5879.5 6282.5 6739.4 7055.0 7376.1 7760.9

7
Introduction, cont.
We don't need a mathematical formula to show that one
variable is a function of another;
The table denes consumption expenditure as a function of
the calendar year:
Example
Year 1998 1999 2000 2001 2002 2003
Consumption 5879.5 6282.5 6739.4 7055.0 7376.1 7760.9

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Introduction, cont.
The dependence between two variables can also be illustrated
by means of a graph.
Figure: The Laer curve

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Introduction, cont.
Functions are given letter names such as f, g, or F;
If f is a function and x is a number in its domain D, then f(x) denotes the
number that the function f assigns to x;
The symbol f(x) is pronounced f of x, or often just f x;
Note de dierence:
f is a symbol forthe function; and
f(x) denotes the value of f at x.
If f is a function, we sometimes ley y denote the value of f at x, so
Example
y = f(x)
x is the independent variable or the argument of f;
y is the dependent variable, because the y (in general) depends on the
value of x;

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Introduction, cont.
Note de dierence:
Example
y = f(x)
value of x;

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Introduction, cont.
Note de dierence:
Example
y = f(x)
value of x;

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Introduction, cont.
Note de dierence:
Example
y = f(x)
value of x;

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Introduction, cont.
Note de dierence:
Example
y = f(x)
value of x;

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Introduction, cont.
Note de dierence:
Example
y = f(x)
value of x;

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Introduction, cont.
Note de dierence:
Example
y = f(x)
value of x;

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Introduction, cont.
The domain of the function f is the set of all possible values of the
independent variable;
The range is the set of corresponding values of the dependent varaible;
In economics, x is often called exogenous variable, whereas y is the
endogenous variable;
A function is often dened by a formula such as: y = 2x2- 3x +8; the
function is then the rule that assigns the number 2x2- 3x +8 to each value of x.

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Introduction, cont.

18
Introduction, cont.

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Introduction, cont.

20
An Example of a Function
Example
The total dollar cost of producing x units of a product is given by
C(x) = 100x
√
x+500 for each nonnegative integer x. Find the cost
of producing:
1 16 units;
2 a units
3 Suppose the rm produces a units; Find the increase in the
cost from producing one additional unit.

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An Example of a Function, cont.
Example
Solution
1 C(16) = 100*16
√
16+500 = 100*16*4+500=6900
2 C(a) = 100a
√
a+500
3 The cost of producing a+1 units is C(a+1), so that the
increase in cost is C(a+1) - C(a) = 100(a+1)
√
a+1+500 -
(100a
√
a+500) = 100[(a+1)
√
a+1-a
√
a]

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Domain of a Function
The denition of a function is incomplete unless its domain is
specied;
The natural domain of the function f dened by f(x) = x3
is
the set of all real numbers;
For te case of our example where C(x) = 100x
√
x+500
denotes the cost of producing x units of a product, the domain
was not specied, but the natural domain is the set of
numbers 0,1,3,..., n, where n is the maximum number of items
the rm can produce.;
If output x is a continuous variable, the natural domain is the
closed interval [0, n].

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specied;
is
√
x+500

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specied;
is
√
x+500

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Domain of a Function, cont.
Examples
Find the domain of:
1 f(x)= 1
x+3
2 g(x) =
√
2x +2

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Domain of a Function, cont.
Examples
Solution:
1 f(x)= 1
x+3
; x= −3
2 g(x) =
√
2x +2; [-1, ∞]

27
Graphs of Functions
Figure: Some important graphs

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Graphs of Functions
Example
Consider the function f(x) = x2
- 4x + 3. The values of f(x) for
some special choices of x are given in the following table.
Table: Values of f(x) = x
2- 4x + 3
x 0 1 2 3 4
f(x) = x2
- 4x + 3
Fill the table, then plot the points obtained from the table in a
xy-plane and, and then draw a smooth curve through this point.
What is the name of the resulting graph?

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Graphs of Functions
Example
Find some of the points of the graph g(x) = 2x - 1, and sketch it.

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Linear functions
They occur often in economics and are dened as follows:
y = ax + b (a and b are constants)
The graph of the equation is a straight line.
a is called the slope of the function:
Proof.
f(x+1) - f(x) = a(x+1) + b - ax - b = a
a measures the change in the value of the function when x increases
by 1 unit. For this reason, the number a is the slope of the line,
and so is called the slope of the function.

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Linear functions, cont.
If a is positive, the line slants upward to the right, and the
larger the value of a, the steeper is the line;
If a is negative, then the line slants downward to the right, and
the absolute value of a measures the steepness of the line.
Figure: Steepness of the line

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The slope of a straight line
The slope of a straight line is:
a = y2−y1
x2−x1
, x1=x2
where (x1, y1) and (x2, y2) are any two distinct points on the
straight line.

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Exercises
Example
Determine the slopes of the 3 straight lines l, m, and n.
Figure: Determine the slopes

34
Exercises (Solutions)
Examples
al = 3−2
4−1
=1
2
am= −2−2
1−2
= 4
an= −1−2
5−2
= -1

35
Linear Models
Most of the linear models in economics are approximations to more complicated models
Statistical methods have been devised to construct linear
functions that approximate the actual data as closely as
possible
E
A UN report estimated that the European population was 641
million in 1960, and 705 million in 1970. Use these estimates to
construct a linear function of t that approximates the population in
Europe (in millions), where t is the number of years from 1960
(t=0 is 1960, t=1 is 1961, and so on). Then use the function to
estimate the population in 1975, 2000, and 1930.

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Linear Models
Solution
S
*If P denotes the population (in millions), we construct an equation
of the form P = at + b;
*We just need 2 points: (t1,P1) = (0,641) and (t2,P2) = (10, 705);
*So P is our y and t our x;
*From the point-slope formula we know that: in y = ax + b, a =
y2−y1
x2−x1
, x1=x2;
*So a = P2−P1
t2−t1
,= 705−641
10−0
=64/10;
*Therefore P = 64/10t + 641
*What is P(15), P(40), and P(-30)?

37
Linear Models
Our estimates versus UN forecasts
Table: UN estimates versus our forecasts
Year 1930 1975 2000
t -30 15 40
UN estimates 573 728 854
Our forecasts 449 737 897
Our formula does not give very good results compared to UN
estimates. To correct for this we need to go beyond linear
functions.

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Quadratic Functions
To obtain acceptable descriptions of economic phenomena
economists often have to use more complicated functions;
Many economic models involve functions that either decrease
down to some minimum value and the increase, or else
increase up to some maximum value and then decrease;
These are the quadractic functions:
Example
f(x) = ax2
+ bx + c (a, b, and c are constants, a=0)

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Quadratic Functions
In order to investigate the function f(x) = ax2
+ bx + c in more
detail, we should nd the answers to the following questions:
1 For which values of x is ax2
+ bx + c = 0? → if and only if x
= −b±
√
b2−4ac
2a →MEMORIZE THIS FORMULA
2 What are the coordinates of the maximum/minimum point P,
also called the vertex of the parabola? → Use derivates
3 What are the coordinates of the maximum/minimum point P,
also called the vertex of the parabola? → Use derivates
1 f(x) = ax
2+ bx + c
2 f'(x) = 2ax + b
3 Equate it to zero → 2ax + b = 0;

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Quadratic Functions, cont.
if a0, then f(x) = ax2
+ bx + c has a minimum at x = -b/2a
if a0 f(x) has a maximum at x = -b/2a

41
Example
Example
The price P per unit obtained by a rm in producing and selling Q
units is P = 102 - 2Q, and the cost of producing and selling Q
units is C = 2Q + 1/2Q2
.
(1) What is the expression of the prot?
(2) What is the value of Q which maximizes prots, and the
corresponding maximum prot? (Sol: 20 and 1000)

42
Exercises
Determine the zeros and maximum/minimum points:
1 x2
+ 4x
2 x2
+ 6x + 18
3 -3x2
+ 30x - 3
4 -x2
- 200x - 30000

3_-_Graphs_and_Functions v3

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