Introduction to Calculus: Functions. Functions A function f is a rule that assigns an element f(x)...
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Transcript of Introduction to Calculus: Functions. Functions A function f is a rule that assigns an element f(x)...
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Introduction to Calculus: Functions
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Functions
A function f is a rule that assigns an element f(x) of a set, called the target set of the function f, to any element x of the domain of definition of the function. The symbol x is called the variable of the
function f.
DefinitionDefinition
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Domain of Definition of a Function
The set of values of the variable(s) of a function f for which the function is
defined is called the domain of definition of the function
f.
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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A Thermometer is a Function
This picture shows a reconstruction of a thermometer of Galileo. The
sealed glass cylinder contains clear liquid and objects which sink
differently as the cylinder is heated.
The variable of the thermometer function is time, and the value of
the function is the current temperature.
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Computer Programs are Functionsfunction putDataXML(xmlDoc,sDocPath)
{if(gnLoadFts==1){
var node=xmlDoc.lastChild;if(node){
var oChild=node.firstChild;
var aFCD=new Array();var aFTCD=new Array();while(oChild){
if(oChild.nodeName=="chunkinfo"){
.....
A computer program produces
an output of a given input, and is,
therefore, a function. Even more, programs use subroutines
that are functions themselves.
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Functions defined by Tables
A function whose domain of definition is a
finite set can also be defined by a table from which we can read the values of the function
for all possible values of the variable.
x f(x)
a α
b β
c γ
d δ
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Functions in Calculus
Functions in calculus are usually defined by algebraic expressions.
The function f, ,
is defined for all x ≠ 1.
f x( ) =
x2
x −1
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Graphs of FunctionsA function f, whose domain of definition is a subset of the set of real numbers and
whose values are real numbers, can
be pictured by drawing the points (x, f(x)) for some
values of x. x
f(x)
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Notations of FunctionsThe notation
f: A →Bindicates that the domain of the
definition of the function f is the set A, and that to each element of the set A the function f assigns an element of
the set B. The set B is the target set of the function f.
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Functions
It defines, therefore, a function, whose variable takes values in the interval [-1,1].
The values of this function are real numbers.
The expression takes real values if -1 ≤ x ≤ 1.
1−x2
f x( ) = 1−x2
We indicate this by writing
f: [-1,1]→R, .
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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FunctionsThe function f: A → B assigns, to any
element a of the set A an element f(a) of the set B.
The set A is the domain of definition of the function f, the set B is the target set of the function
f.
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
The variable a of the function f varies in the set A. For any a ∈ A, f(a) ∈ B.
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History of Functions
The concept of a function was defined by Leonhard Euler in the
mid 18th century.
Using functions one can develop general theories.
Preliminaries/Introduction to Calculus/Functions by M. Seppälä
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Leonhard Euler (1707 - 1783)