chapter 1 -part 1 real functions
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Chapter 1Calculus of One VariableEEM1016 Engineering Mathematics I
Trimester 2 Session 2013/14
Prepared by:Nasrin Sadeghianpour
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1. Real functions of a real variable2. Limits and continuity3. Differentiation & its applications4. Integration & its applications
Outline of Chapter 1
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Part 1 Real functions of a real variable
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If a variable y depends on variable x in such a way that each value of x determines exactly one value of y, then we say that y is a function of x.
Two commonly used methods of representing functions are:
1. By formulas 2. By graph
Definition of a function
)(xfy
Dependant variable Independent
variable (argument)
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Let X and Y be any nonempty sets. A function or mapping f from X to Y denoted by is a rule that assigns to each element of Y exactly one element of X. We say that X is the domain of f.
We write or to indicate that the element is the value assigned by the function to element In this case, we say that y is the image of x. The set of all images is called the range or image set of f, denoted by
Another definition of a function
YXf :
y f x ( )
( ) f x x X
y Y
R f .
YXf :
x X .f
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Example 1X:0123
Y:0149
)(xfy YXf :
Domain
Range2xy
Note that if the domain and the range of a function are both real, then is called a real-valued function of a real variable, or simply a real function.
fYXf :
In order to specify a function completely, the domain must be stated explicitly. Otherwise the domain is taken as the largest possible subset of R for which the real-valued function can be defined.
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Example 2
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Example 2 cont….
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Let be a function. The graph of is the set consisting of all points in the Cartesian coordinate plane, for all , i.e., the graph is the set
Graphs of Functionsf f
( , ( ))x f x,fx D
. )( and ),( xfyDxyx f
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Not every curve is the graph of a function! A curve in the xy-plane is the graph of a function
when it satisfies the vertical line property: any vertical line (a line parallel to the y-axis) intersects the curve at most once.
Graph of functions- Vertical line test
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Example 3
Solution (a) y is a function of x
Solution (b) y is a function of x
Solution (c) y is not a function of x
Solution (d) y is not a function of x
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Odd & Even functions
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Exercise
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Exercise
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Arithmetic Operations on functions
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Example 4
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Composition of functions
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Example 5
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Monotone functions Let A be a subset of R and let f be a
function. We say that is increasing on A if for all
such that decreasing on A if for all
such that
Some properties of functions
( ) ( )f x f y ,x y A;x y
( ) ( )f x f y ,x y A;x y
)(xf
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One-to-One or Injective Functions
Some properties of functions
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Some properties of functions
Inverse function
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Example 6
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Condition for existence of inverse
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How to find an inverse of a function
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Example 7
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
The tangent line to this
graph at (0.1) has slope 1.
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions
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Elementary functions