3_-_graphs_and_functions v3
TRANSCRIPT
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 demandand supply functions, cost functions, production functions,consumption functions, etc;
One variable is a function of another if this �rst variabledepends 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
Important in every area of pure and applied mathematics(including mathematics applied to economics);
The language of economic analysis is full of terms like demandand supply functions, cost functions, production functions,consumption functions, etc;
One variable is a function of another if this �rst variabledepends 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;
5
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 demandand supply functions, cost functions, production functions,consumption functions, etc;
One variable is a function of another if this �rst variabledepends 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;
6
Introduction, cont.
We don't need a mathematical formula to show that onevariable is a function of another;
The table de�nes consumption expenditure as a function ofthe 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 onevariable is a function of another;
The table de�nes consumption expenditure as a function ofthe calendar year:
Example
Year 1998 1999 2000 2001 2002 2003
Consumption 5879.5 6282.5 6739.4 7055.0 7376.1 7760.9
8
Introduction, cont.
The dependence between two variables can also be illustratedby means of a graph.
Figure: The La�er curve
9
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
10
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
11
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
12
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
13
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
14
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
15
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 thenumber that the function f assigns to x;
The symbol f(x) is pronounced "f of x", or often just "f x";
Note de di�erence:
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 thevalue of x;
16
Introduction, cont.
The domain of the function f is the set of all possible values of theindependent variable;
The range is the set of corresponding values of the dependent varaible;
In economics, x is often called exogenous variable, whereas y is theendogenous variable;
A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.
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Introduction, cont.
The domain of the function f is the set of all possible values of theindependent variable;
The range is the set of corresponding values of the dependent varaible;
In economics, x is often called exogenous variable, whereas y is theendogenous variable;
A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.
18
Introduction, cont.
The domain of the function f is the set of all possible values of theindependent variable;
The range is the set of corresponding values of the dependent varaible;
In economics, x is often called exogenous variable, whereas y is theendogenous variable;
A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.
19
Introduction, cont.
The domain of the function f is the set of all possible values of theindependent variable;
The range is the set of corresponding values of the dependent varaible;
In economics, x is often called exogenous variable, whereas y is theendogenous variable;
A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.
20
An Example of a Function
Example
The total dollar cost of producing x units of a product is given byC(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 thecost 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 theincrease in cost is C(a+1) - C(a) = 100(a+1)
√a+1+500 -
(100a√a+500) = 100[(a+1)
√a+1-a
√a]
22
Domain of a Function
The de�nition of a function is incomplete unless its domain isspeci�ed;
The natural domain of the function f de�ned by f(x) = x3isthe 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 domainwas not speci�ed, but the natural domain is the set ofnumbers 0,1,3,..., n, where n is the maximum number of itemsthe �rm can produce.;
If output x is a continuous variable, the natural domain is theclosed interval [0, n].
23
Domain of a Function
The de�nition of a function is incomplete unless its domain isspeci�ed;
The natural domain of the function f de�ned by f(x) = x3isthe 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 domainwas not speci�ed, but the natural domain is the set ofnumbers 0,1,3,..., n, where n is the maximum number of itemsthe �rm can produce.;
If output x is a continuous variable, the natural domain is theclosed interval [0, n].
24
Domain of a Function
The de�nition of a function is incomplete unless its domain isspeci�ed;
The natural domain of the function f de�ned by f(x) = x3isthe 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 domainwas not speci�ed, but the natural domain is the set ofnumbers 0,1,3,..., n, where n is the maximum number of itemsthe �rm can produce.;
If output x is a continuous variable, the natural domain is theclosed interval [0, n].
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Graphs of Functions
Example
Consider the function f(x) = x2- 4x + 3. The values of f(x) forsome special choices of x are given in the following table.
Table: Values of f(x) = x2- 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 axy-plane and, and then draw a smooth curve through this point.What is the name of the resulting graph?
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Linear functions
They occur often in economics and are de�ned 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 = aa measures the change in the value of the function when x increasesby 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 thelarger the value of a, the steeper is the line;If a is negative, then the line slants downward to the right, andthe absolute value of a measures the steepness of the line.
Figure: Steepness of the line
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Linear functions, cont.The slope of a straight line
The slope of a straight line is:
a = y2−y1x2−x1 , x1 6=x2
where (x1, y1) and (x2, y2) are any two distinct points on thestraight line.
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Linear functions, cont.Exercises
Example
Determine the slopes of the 3 straight lines l, m, and n.
Figure: Determine the slopes
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Linear functions, cont.Exercises (Solutions)
Examples
al=3−24−1=
12
am=−2−21−2 = 4
an=−1−25−2 = -1
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Linear ModelsMost of the linear models in economics are approximations to more complicated models
Statistical methods have been devised to construct linearfunctions that approximate the actual data as closely aspossible
E
A UN report estimated that the European population was 641million in 1960, and 705 million in 1970. Use these estimates toconstruct a linear function of t that approximates the population inEurope (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 toestimate the population in 1975, 2000, and 1930.
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Linear ModelsSolution
S
*If P denotes the population (in millions), we construct an equationof 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−y1x2−x1 , x1 6=x2;
*So a = P2−P1
t2−t1 ,= 705−64110−0 =64/10;
*Therefore P = 64/10t + 641*What is P(15), P(40), and P(-30)?
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Linear ModelsOur 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 UNestimates. To correct for this we need to go beyond linearfunctions.
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Quadratic Functions
To obtain acceptable descriptions of economic phenomenaeconomists often have to use more complicated functions;
Many economic models involve functions that either decreasedown to some minimum value and the increase, or elseincrease 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, a6=0)
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Quadratic Functions
In order to investigate the function f(x) = ax2+ bx + c in moredetail, 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±√b2−4ac2a →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) = ax2+ bx + c2 f'(x) = 2ax + b3 Equate it to zero → 2ax + b = 0;
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Quadratic Functions, cont.
if a>0, then f(x) = ax2+ bx + c has a minimum at x = -b/2a
if a>0 f(x) has a maximum at x = -b/2a
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Quadratic Functions, cont.Example
Example
The price P per unit obtained by a �rm in producing and selling Qunits is P = 102 - 2Q, and the cost of producing and selling Qunits is C = 2Q + 1/2Q2.(1) What is the expression of the pro�t?(2) What is the value of Q which maximizes pro�ts, and thecorresponding maximum pro�t? (Sol: 20 and 1000)