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4. DIFFERENTITION

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4.1 Concept of Differentiation

cx

cfxfmPQ

=

)()(

4.1.1 Differentiation at one point

Introduction ( two topic in the same theme)

a. Tangent LinesThe secant line connecting P and Q has slope

c

f(c) P

x

f(x)

Q

x-c

f(x)-f(c)

Ifx c, then the secant line through P

and Q will approach the tangent line at P.Thus the slope of the tangent line is

cx

f(c)f(x)m

cx

=

lim

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b. Instantaneous VelocityLet a particle travel around an axis and positition of particle at time

t is s= f(t). If the particle has a coordinate f(c) at time c and f(c+h) at

time c+ h .

Thus, the average velocity during the time interval [c,c+h] is

c

c+h

Elapsed time distancetraveled

s

f(c)

f(c+h)

( ) ( )average

f c h f cv

h

+ =

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If h 0, we get instantaneous velocity at x= c:

Let x = c + h, instantaneous velocity can be written as

From two cases : The slope of the tangent line and instantaneous

velocity has the same formula

Definition : The first derivative off at x= c, denoted by is defined by

if limit exist

h

cfhcfvv

hrataratah

)()(limlim

00

+==

cx

f(c)f(x)v

cx

=

lim

)(' cf

cx

f(c)f(x)cf

cx

=

lim)('

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Other notation :

Example Let evaluate

)(',)(

cy

dx

cdf

x)x(f

1=

=

= 3

33

3 x

)f(f(x)lim)f'(

x 33

11

lim 3

xxx

=

=

)x(x

x

x 33

3lim

3

9

1

3

1lim

3==

xx

)3('f

)x(x

x

x 33

)3(lim

3

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4.1.2 Derivatives from the right and the left

Derivatives from the left at c, denoted by , is defined by

Derivatives from the right at c, denoted by , is defined by

A function f is said differentiable at c( exist) if

)c(f)c(f '' + =

cx

cfxf

cf cx

= )()(

lim)('

cx

f(c)f(x)

(c)f cx'

=

++ lim

)(' cf

' '

_'( ) ( ) ( ) f c f c f c

+= =

'( )f c

'( )f c

+

and

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Example : Let

+

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Theorem If fis differentiable at point cf is continuous at c.

Proof : We will proof

Since

Thus

The converse, however, is false , a function may be continuous ata point but not differentiable

)()(lim cfxfcx

=

cxcx

cx

cfxfcfxf

+= ,).(

)()()()(

+=

)(

)()()(lim)(lim cx

cx

cfxfcfxf

cxcx

)(lim.)()(lim)(lim cxcx

cfxfcfcxcxcx

+=

0).(')( cfcf += = f(c).

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Example Show that, f( x) = | x| is continuous at x= 0but f(x) is not differentiable at x= 0

Solution

We will show f(x)=|x| is continuous at x=0

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0

00

0

=

x

)(f)x(flim)(f

x

' 1lim0

lim00

=

=

=

x

x

x

x

xx

0

00

0

=

++

x

)(f)x(flim)(f

x

' .1lim0

lim00

==

=++ x

x

x

x

xx

Determine whether f is differentiable at x=0

1)0()0(1'' == + ffBecause

then f is not differentiable at x = 0.

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Example: Find the values of a and b so that f(x) will bedifferentiable at x= 1.

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1

)1()(lim)1(

1

'

=

x

fxff

x

2)1()1(''

== + aff

So that : a= 2 and b= 1.

1

2

1

+=

x

abxlim

x

1

12

1

+=

x

a)a(xlim

x 1

12

1

=

x

xlim

x

1

11

1

+=

x

)x)(x(lim

x21

1=+=

xlim

x

1

)1()(lim)1(

1

'

=

++

x

fxff

x 11

=

x

aaxlim

xa

x

xlima

x=

=

1

1

1

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Problems

f x a x x

x bx x( ) ;

;= +