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7/31/2019 1. Functions and Their Properties
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FunctionsandTheirBaProperties
FunctionsandtheirGraphsTangentsandAsymptotesI j i i d S j i i
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I j ti it d S j ti it
FunctionsA function is rule that assigns an element of a set, calt a r g e t d o m a in of the function, to any element of thethe function is defined. This set is the d o m a in o f d e fthe function.
Examples
The scale is a function. It assigns to every per
stepping on it, the weight of the person.1
The thermometer is a function, it can be used toutside temperature at any moment. It measurtemperature of its location continuously. The d
d fi iti i ti i t l d th t t d
2
fGraphicallyDomain ofdefinition
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LengthoftheDay24
18
12
6
o
urs
HelsinkiTo understand how thelength of the day varies,one can measure thelength of the first day of
each month. That datacan be plotted.
Joining the plotted pointsby straight or slightlycurved lines yields agraph that allows one tounderstand how thelength of the day varies.The l en g t h o f t h e d ay
i f t i d th
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DefinitionofFunctionsGiven sets A and B. A f u n c t i o n f : A
which assigns an element f(a) of the set B for every
If the sets A and Bare finite, then this rule can be ex
terms of a table or a diagram.Usually the sets A and Bare not finite. In such a casquestion is usually expressed in terms of an algebraic einvolving possibly special functions, for f(a).
Alternatively the rule to compute f(a), for a given a,program taking a as input and producing f(a) as its
Definition
Example ( ) ( )sinf1xx
x=
is a function which is defin
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GraphsofFunctions
Examples
In calculus we are usually concerned with functions f:real number to a real number. Such functions are usuaexplicit expression for f(x).
The product 2 = {(x,y)| x, y} is called the p lane
pictured by drawing the x axis horizontally, and the yThe graph of the function f: is the graph of the set
Below are graphs of the functions f(x) = s
g(x) = x4
2x3
x2
+ 2xand h(x) = 2sin(x)
Which is which?
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CurvesandGraphsProblem Which of the following curves in the plane
graphs of functions?
A Th fi t t t h f f
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SecantandTangentLinesA line intersecting the graph of afunction at two points is a secantl i ne.
If you modify a secant line by
rotating it around the firstintersection point so that thesecond intersection pointapproaches the first one, you get
a t a n g en t l i n e at the limit.
A tangent lin
It h th t
No tangentat this point
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AsymptotesAn asymptote of a curve is a st ra i g h t l i n ewhich the curve approaches arbitrarilyclose as one moves sufficiently far along thecurve.
Here we see the graph of the functionand its asymptotes, the line y= x,and the vertical asymptote x= 2.
1
2y xx= +
A graph may intersect its asinfinitely often. Asymptotes
i f ti b t th b h
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InjectiveFunctionsDefinition
A one-to-one function associates at most one point in t
to any given point in the set B; i.e. f(x1) = f(x2) x1 =
Problem Which of the following graphs are graphs oone functions?
A function f: A B is i n j e c t i v e or o n e - t odifferent elements ofA have different imagi.e. if x1 x2 f(x1) f(x2).
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SurjectiveandBijectiveFunctDefinition A function f: A B is su r jec t i ve or o n t o
covers all of the target domain; i.e. if y B: x A such that f(x) = y.
A function f: A B is b i j ec t i ve if it is both surjective i.e. if y B: ! x A such that f(x) = y.
The notation ! x A in the above means that there
element in the set A having the given property.
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Increasing andDecreasingFuDefinition
A function f is inc reas ing a> b f(a)
A function, whose values grow as the value of the variacalled increasing. If the values of the function decreasvariable grows, the function is decreasing.
A function f is decreas ing a> b f(a) < f(b).
A function f is m o n o t o n i c if it is everywhere increasi
everywhere decreasing.
Theorem A monotonic function is one-to-one (inje
Proof We have to show that, if x1 x2, then also f
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Summary
Functions are rules that assign an element of the targeto any element of the domain of definition of the funct
Functions are often defined by algebraic expressions li
In such a case the domain of definition is the set of nuwhich the given algebraic expression is defined.
Tangen t l i nes of the graph of a function give local infabout the function near the point of tangency. A s y m pglobal information about the graph of a function.
Functions are i n j e c t i v e, or o n e - t o -o n e , if they assignelements of the target set to different elements of the definition. Functions are b i j ec t i ve , if they are injectiv
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Continuity Gently
Definition of Continuity
I di V l Th
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Simplest way to define the continuity of functions is tofunction is continuous if one can draw its graph withouthe pen from the paper.
C ti F ti Di ti
Definition of Continuity
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Simplest way to define the continuity of functions is tofunction is continuous if one can draw its graph withouthe pen from the paper.
C i F i Di i
Definition of Continuity
Continuity
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Continuity
DefinitionA function f is continuous at a number a if
limxa
f(x) = f(a).
Definition of Continuity
Continuity
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Continuity
DefinitionA function f is continuous at a number a if
limxa
f(x) = f(a).
In other words...
1 f(a) is defined;
2 the limit, limxa f(x), exists; and,
3
these two values coincide.
Definition of Continuity
Continuity
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Continuity
What is continuity?
Speaking loosely, a function is continuous if we can trace its graph
without having to lift our pencil from the paper. In other words, the
graph is connected.
What is the mathematical definition of continuity? How do weexpress the connectedness of a graph?
It is easier to look at some discontinuous examples to see what can
go wrong.
Definition of Continuity
Continuity
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Continuity
Examples
Continuity
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Continuity
Examples
Continuity
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Continuity
Examples
Continuity
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u y
Examples
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Examples of Continuous Func
1 f(x)
= x3
x
is continuous everywhere.
2Continuous for x
0, and d
at x
= 0.
3 h(x)
= sin(x)/x
is defined and continuous for
Setting h(0) = 1 extends the function h to a cfunction defined for allx.
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Rules of Continuous FunctionsAssume that both functions f and g are continuous aLet c R.
Theorem The following functions are continuous atx
1 f(x) + g(x) 2 cf(x)
3 f(x) g(x) 4 f(x)/g(x) assuming tha
Proof The result follows immediately from thecorresponding properties of limits of functio
Statement of the properties of limits.Rigorous proof of the properties of limits.
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Basic Continuous Functions
Since clearly the function f(x) =x
is continuous, the
Continuous Functions implies that:
1 . A ll po l ynom ia ls are continuous functions.
2 . A ll r a t i ona l f unc t i ons, i.e., all quotients R = P/Qpolynomial P and Q are continuous at all pointsfor which Q(x0
)
0.
On e ca n sh o w f u r t h e r t h a t :
1 . A ll p ow e r f u n ct i o n s xr, rReal, are continuous wh
2 . A ll f un ct i ons f ( x) = ax, a
> 0, are continuous. I
the Ex pon en t ia l Fun ct ion e x is continuous.
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Examples
1 Where the function tanxis continuous?
Solution
By the previous remarks, tanx
is
continuous everywhere where it is defined.
The function tanx
= sin(x)/cos(x) is defined
for all valuesx for which cosx 0.
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Examples
2 Where the function f(x) =
x
+
x
is continu
Recall that
x
= largest integer
x.
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Examples
2 Where the function f(x) =
x
+
x
is continu
Recall that
x
= largest integer
x.
Solution
Observe that if n
1
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Examples
3 Where the function is continuou
SolutionObserve that the numerator isdefined and continuous for x
> 0.
The denominator lnx
1 is also
defined for allx, x > 0.
The denominator takes the value0 ifx= e. At this point thefunction is undefined, and hence
e
Grapg. Tline
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Examples
4 Study the continuity of the function
Solution
Since x2
and the Sine function are both continuo
composed function sin(x2
) is continuous.
1
Since 1 + sin(x2
) 0 for allx, is dealso continuous for allx.2
The numerator is defined and confor allx. The denominator x2 is continuous, and
3
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Examples
5 The function is defined
Is it possible to define f(0) so that the function continuous atx
= 0?
Solution
Since
=1 + sin x2( )( ) 1
=sin x2( )
x0
1
We need to find out whether f has a limit
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Examples
5 The function is defined
Is it possible to define f(0) so that the function continuous atx
= 0?
Solution(contd)
We have concluded that, if we set f(0) = function f is continuous at x
= 0.
Problems of this type can usually besolved by computing the limit (if oneexists) of the function in question at the
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Tangents, Velocity, andthe Derivative
Tangents as Limits of Secant Lines
Tangent Lines
Linear Approximations of Functions
Velocity
Rates of Change
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Differentiation/Introduction to Differentiation/Tangents, Velocity, and the Derivative
Tangents as Limits of Secant Lines
The basic problem that leads to differentiation is to compute theslope of a tangent line of the graph of a given function f at a given
point x0. The key observation, which allows one to compute slopes
of tangent lines is that the tangent is a certain limit of secant lines
as illustrated in the picture below.
x0 x0+h
f
A secant line intersects the graph of afunction f at two or more points.
The figure on the left shows secantlines intersecting the graph at the
points corresponding to x=x0 and
x=x0 + h.
As h approaches 0, the secant line
in question approaches the tangent
line at the point (x0,f(x0)).
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Differentiation/Introduction to Differentiation/Tangents, Velocity, and the Derivative
Slopes of Secant Lines
The slope of a secant line intersecting the graph of a function f atpoints corresponding tox=x0 and x=x0 + h can readily be
computed using the notations defined in the picture below.
x0 x0+h
f
h
f(x0+h)
f(x0) f(x0+h)- f(x0)
As h approaches 0 (through
positive numbers), the secant
in the pictures approaches the
tangent to the graph of f atthe point (x0,f(x0)).
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Differentiation/Introduction to Differentiation/Tangents, Velocity, and the Derivative
1
1
Tangent Lines (1)
Compute the slope of the
tangent line, at the point
(1,1), of the graph of the
function x2.
The tangent line of the graph of the function f atthe point (x0,f(x0)) is the line passing through this
point and having the slope
provided that the limit exists and is finite.
limh0
fx0+ h( ) fx
0( )
h
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Differentiation/Introduction to Differentiation/Tangents, Velocity, and the Derivative
Tangent Lines (2)
limh0
fx0 + h( ) fx0( )h
= limh0
1 + h( )2
1
2
h.
Compute the slope of thetangent line, at the point
(1,1), of the graph of the
function x2.
limh0
1+h
2
12
h= limh0
12 +2h+h2 12h
= limh0
2h+h2
h= limh0
2+h=2.
Equation of the
tangent line is
y-1=2(x-1), i.e.,
y=2x-1.
1
1
By the definition, the slope is thelimit
This can readily be computed by expandingthe brackets:
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Differentiation/Introduction to Differentiation/Tangents, Velocity, and the Derivative
Linear Approximations of Functions
The following pictures show, in different scales, the graph ofthe function x2 and that of its tangent line at the point (1,1).
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