Differentiability part 1

7/29/2019 Differentiability part 1

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Calculus 3rdsecondary Differentiability Part I -46-

Derivative of a functionIntroduction:

Last year , we have taken Differentiation, And we knew that there were various shapes of it :

st d f xdy1 derivative rate of change f ' y' Slope of the tangent dx dx

Also , we have taken the rules of differentiation which are :

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n n 1

n n 1

1 If f x x f ' x n x

2 If y a x , Where a is constant y' a n x

dy3 If y Constant Zerooooodx

4 If y

dySin x Cos x

dx

dy5 If y Cos x -Sin x

dx

6 If y Sin a x b y' aCos a x b

7 If y Cos a x b y' -a Sin a x b

dy dy dz 8 For Composite function:dx dz dx

9 The first derivative of the product of t

2

he two functions :

So If f x y Then f ' x y' y x'

9 The first derivative of the quotient "Divisions" of the two functions :

x y x' x y'So If f Then f '

y y

Also , we have taken the rule Containing the defi ni tion of Dif ferentiation of a single function

h 0

f x h f xlim

h

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For example

Use the definition of Differentiation to discuss the Differentiabilityof the following functions:

21 f x 2x

Answer

2 2 2 22 2 2 2

h 0 h 0 h 0

2

h 0 h 0 h 0

2 x 2xh h 2x2 x h 2x 2x 4xh 2h 2xlim lim lim

h h h

h 4x 2h4xh 2hlim lim lim 4x 2h 4x Then The derivative f ' x 4x

h h

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1

2 f x x at x 1x

Answer

h 0

22

h 0 h 0 h 0

2 2

h 0 h 0 h 0

1 1x h x

x h xlim Put x 1

h

1 h 11 1 1 h 1 2 1 h1 h 1 2

1 h 1 1 h 1 hlim lim limh h h

1 2h h 1 2 2h h h

lim lim lim 0h 1 h h 1 h 1 h

Then " f " is differentiable at

x 1 and its derivative f ' x 0

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3 f x 3x 2x at x 0 Answer

h 0

2

h 0 h 0 h 0

h 0

3 x h 2 x h 3x 2xlim Put x 0

h

h 9h 23h 2h 3h 2h 9h 2hlim lim lim

h 3h 2h h 3h 2h h 3h 2h

9h 2 -2lim Then " f " is not differentiable at x 0

03h 2h


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h 0

1 The left derivative the right derivative f ' a f ' a " using the

f x h f xDefinition of differentiation lim

h

2 If f ' a f ' a , Then f x is differentiable at x a, And f ' a f ' a f ' a3 To discus

Steps

s the differentiability of a function at a point , the point must belong to the domain

of this function.

4 When discussing the differentiability of a function at a point, it is not necessary to discuss

The Continuity of that function first, you can discuss the differentiability Directly.

x -1

f x

x

f -1 - x 1 f -1 x 1

h 0

h 0 h 0 h 0 h 0

h 0 h 0 h 0 h 0

x 1 x -1f x

- x 1 x -1

f x h f xAnd limh

-1 h 1 -1 1f -1 h f -1 hFor f ' -1 : lim lim lim lim 1 1

h h h

- -1 h 1 - -1 1f -1 h f -1 -hFor f ' -1 : lim lim lim lim -1 -1

h h h

So f ' -1 f ' -1

Then f x is Not differentiable at x -1

But what if we want to discuss the differentiability of more than

One function at a point Ex. Double fn

(5) Very very important note: (a) if f(x) is Continuousat x=a , then it is Not Necessaryto be

Differentiableat x= a BUT, if f(x) is differentiableat x=a, then it must beContinuousat x = a .(b) if f(x) is Discontinuousat x=a, then it is Not dif ferentiabletoo.

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Example (1)

Discuss the Differentiability of f x x 1 at x -1 Answer


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2 2

h 0 h 0

2 2

h 0 h 0 h 0

2 2

h 0 h 0

2

h 0 h 0

2 h 4 2 h 11 2 4 2 11f 2 h f 2For f ' 2 : lim lim

h h

4 4h h 8 4h 11 4 8 11 hlim lim lim h 0

h h

3 4 2 h 2 h 3 4 2 2f 2 h f 2For f ' 2 : lim lim

h h

3 8 4h 4 4h h 9 -hlim lim

h

2

h 0lim -h 0

h

So f ' 2 f ' 2 Then f x is differentiable at x -1 And its derivative is f ' 2 0

x 2

f x

x

2f 2 3 4x x 2f 2 x 4x 11

22

h 0

h 0 h 0 h 0 h 0

h 0 h 0 h 0 h 0

x 3 x 3f x x 6 x 9 x 3 x 3

- x 3 x 3

f x h f xAnd limh

3 h 3 3 3f 3 h f 3 hFor f ' 3 : lim lim lim lim 1 1

h h h

- 3 h 3 - 3 3f 3 h f 3 -hFor f ' 3 : lim lim lim lim -1 -1

h h h

So f ' 3 f ' 3

Then f x is Not differentiable at x 3

Example (2)

2

2

3 4x x x 2Discuss the differentiability of f x at x 2

x 4x 11 x 2

Answer

h 0

f x h f xlim

h

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Example (3)

2Discuss the differentiability of f x x 6x 9 at x 3. Answer

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x 0

f x

x

f 0 x x

332 32 1

2

h 0 h 0 h 0

The domain of x x is 0 ,

0 , Let a 0 ,

f a h f a a h a h a a a h a 3 3lim lim lim a a

h h a h a 2 2

3Then f x is differentiable at any point a 0 , And its derivative is f ' x a

2

f ' 0

For interval

At x = 0

h 0 h 0 h 0 h 0

f 0 h f 0 0 h 0 h 0 0 h h: lim lim lim lim h 0

h h h

Then f x is differentiable at x 0 And its derivative is f ' 0 0

So f x is Continous at x 0

13 3

1 1 1 1

3 3 3 3

h 0 h 0 h 0

The domain of x 2 is R So Let a R where f x x 2

f a h f a a h 2 a 2 a h 2 a 2lim lim lim

h h a h 2 a 2

1 21

3 3

23

1 1a 2 a 2

3 3

1Then f x is differentiable at any point a R And its derivative is f ' x3 a 2

So f x is Continous at a R -2

Example (4)

Discuss the Differentiability of f x x x at any point of its domain Answer

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Example (5)

3Discuss the Differentiability of f x x 2 at any point of its domain Answer

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h 0 h 0

1

a To discuss the Continuity at x 0

f 0 1 And f 0 lim 0 0 And f 0 lim 2 2

So f 0 f 0 f x is discontinous at x 0

Then the functions is Not differentiable at x 0

0 x 0

b g x x f x x x 0

2x x 0

1 1 1h 0 h 0

1 1

h 0

1h 0 h 0

To discuss the Continuity at x 0

g 0 0 And g 0 lim 0 0 And g 0 lim 2x 0

So g 0 g 0 g 0 g x is Continous at x 0f x h f x

To discuss the Differentiabiltiy at x 0 limh

f 0 h f 0For g ' 0 : lim lim

h

h 0 h 0

1h 0 h 0 h 0

1 1 1

2 2

2

2

2 0 h 2 0 2hlim lim 2 2

h h

f 0 h f 0 0 0For g ' 0 : lim lim lim 0 0

h h

So g ' 0 g ' 0 Then g x is Not differentiable at x 0

0 x 0

c g x x f x x x 0

2x x 0

To discuss the Differentiabiltiy at

2 2 2

2h 0 h 0 h 0 h 0

2 2

2 h 0 h 0 h 0

2 2 2 2

x 0

f 0 h f 0 2 0 h 2 0 2hFor g ' 0 : lim lim lim lim 2h 0

h h h

f 0 h f 0 0 0

For g ' 0 : lim lim lim 0 0h h

So g ' 0 g ' 0 Then g x is differentiable at x 0 and its derivative is g ' x =0

Then it is Continous at x 0

1

2

2

0 x 0

Let f x 1 x 0 , Then Prove that :

2 x 0a f x is discontinous and Not differentiable at x 0

b g x x f x is Continous and Not differentiable at x 0

c g x x f x is differentiable and Continous at x 0

Example (6)

Answer


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2 2

2 2

2 2

2 2

x 5 4 x 5 x 5 x 5 4 x 5 x 5f x f x

- x 5 4 - x 5 x 5 x 5 4 x 5 x 5

x 10x 25 4x 20 x 5 x 14x 45 x 5f x f x

x 10x 25 4x 20 x 5 x 6 x 5 x 5

And lim

h 0

2 2

h 0 h 0

2 2

h 0 h 0 h 0

2 2

h 0 h 0

h 0

f x h f x

h

5 h 14 5 h 45 5 14 5 45f 5 h f 5For f ' 5 : lim limh h

25 10h h 70 14h 45 25 70 45 h 4hlim lim lim h 4 -4

h h

5 h 6 5 h 5 5 6 5 5f 5 h f 5For f ' 5 : lim lim

h h

25lim

2 2

h 0 h 0

10h h 30 6h 5 25 30 5 h 4hlim lim h 4 4

h h

So f ' 5 f ' 5 Then f x is Not differentiable at x 5

2

h 0 h 0 h 0

2

h 0 h 0

2

h 0 h 0

3x -x 8 , x 0-3x 8 , x 0

f x f x3 x5 , x 0 8 , x 0

x

f 0 h f 0 8 8For f ' 0 : lim lim lim 0 0

h h

-3 0 h 8 -3 0 8f 0 h f 0For f ' 0 : lim lim

h h

-3h 8 8

lim lim -3h 0h

So f ' 0 f ' 0 Then f x

is differentiable at x 0 and its derivative is f ' 0 0

So f x is Continous at x 0

Example (7)

2

Discuss the differentiability of f x x 5 4 x 5 at x 5

Answer

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Example (8)

3x x 8 , x 0

Discuss the differentiability of f x , at x 03 x5 , x 0

x

Answer


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Calculus 3rdsecondary Differentiability Part I --

22 2

2 2 2

2

2

2

2

2

h 0 h

x x 6 x 3 x 2 0x x 6 x x 6 0f x f x

-x x 6 x x 6 0 -x x 6 x 3 x 2 0

x x 6 x 2x x 6 x -3 , x 2

f x f x x x 6 x -3-x x 6 x -3 , x 2

-x x 6 -3 x 2

-3 ,2 Let a -3 , 2

f a h f alim lim

h

For in terval

2 2

0

2 2 2

h 0 h 0 h 0

h 0

- a h a h 6 - a a 6

h

-h 2a h 1-a 2ah h a h 6 a a 6 lim lim lim -2a h 1 -2a 1

h h

Then f x is differentiable at any point a -3 , 2 And its derivative is f ' x -2a 1

f -3 h f -3-3 for f ' -3 : lim

At x =

2 2

h 0

2 2

h 0 h 0 h 0 h 0

2 2

h 0 h 0

2

h 0 h 0

-3 h -3 h 6 -3 -3 6 limh h

h -5 h9 6h h 3 h 6 9 3 6 -5h hlim lim lim lim - 5 h -5

h h h

- -3 h -3 h 6 - -3 -3 6 f -3 h f -3for f ' -3 : lim lim

h h

-9 6 h h 3 h 6 9 3 6 5lim lim

h

2

h 0 h 0

2 2

h 0 h 0

2 2

h 0 h 0

h 5 hh hlim lim 5 h 5

h h

So f ' -3 f ' -3 Then f x is Not differentiable at x -3

2 h 2 h 6 2 2 6 f 2 h f 22 for f ' 2 : lim lim

h h

4 4h h 2 h 6 4 2 6 5h hlim lim li

h h

At x =

h 0 h 0

2 2

h 0 h 0

2 2

h 0 h 0 h 0 h 0

h 5 hm lim 5 h 5

h

- 2 h 2 h 6 - 2 2 6 f 2 h f 2for f ' 2 : lim lim

h h

-h 5 h-4 4h h 2 h 6 4 2 6 -5h hlim lim lim lim - 5 h -5

h h h

So f ' 2 f ' 2 Then f x is Not differentia

ble at x 2

x 2

f x

x

2x x 6

x -3

2x x 6 2-x x 6

Example (9)

2Discuss the differentiability of f x x x 6 over its domain

Answer


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x 1

f x

x

2x 1 2x a

2

x 1 x 1

2

2 2

h 0 h 0

f x is Continous at x 1

Then f 1 f 1 f 1

f 1 2 1 1 3

f 1 lim 2x 1 3 And f 1 lim x a 1 a

2x 1 x 1

So 1 a 3 a 2 f x x 2 x 1

To discuss the differentiabilty at x 1 :

1 h 2 1f 1 h f 1For f ' 1 : lim lim

h

2

h 0 h 0

h 0 h 0 h 0 h 0

2

h

1 2h h 2 1 2lim lim 2 h 2

h

f 1 h f 1 2 1 h 1 2 1 1 2hFor f ' 1 : lim lim lim lim 2 2

h h h

So f ' 1 f ' 1 Then f x is differentiable at x 1 and its derivative is f ' 1 2

Example (10)

22x 1 x 1

If f x is Continous at x 1 , Then find a and discuss the differentiabiltiyx a x 1

at x 1

Answer

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22

x 2

x 2

2

f x is Continous at x 2 Then f 2 f 2 f 2

f 2 4 " Given

f 2 lim x a x 2 2 2a 2 2a 2

f 2 lim a b x a 2b

3So 2a 2 4 a 1 And a 2b 4 1 2b 4 b2

x x 2 x 2

Then f x1

2 2

h 0 h 0

2 2

h 0 h 0 h 0 h 0

h 0

3x x 2

2

To discuss the differentiabilty at x 2 :

2 h 2 h 2 2 2 2f 2 h f 2For f ' 2 : lim lim

h hh 5 h4 4h h 2 h 2 4 2 2 5h h

lim lim lim lim 5 h 5h h h

f 2 h f 2For f ' 2 : lim lim

h

h 0

h 0 h 0 h 0

3 31 2 h 1 2

2 2

h

3 31 3 h 1 3 h

3 32 2lim lim limh h 2 2

So f ' 2 f ' 2 Then f x is Not differentiable at x 2

Example (11)

2x a x 2 x 2If f x where f 2 4 , and if it is Continous at x 2

a b x x 2

Then find a and b then discuss the differentiabiltiy of f x at x 2

Answer

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2a x +b x 7 x 2

find the value of a and b if f x is differentiable at x 2 .5x 2x 1

Example (12)

Answer

2

h 0 h 0

2 2

h 0 h 0

h 0 h 0

h

f x is differentiable at x 2 , Then f ' 2 f ' 2

f 2 h f 2 a 2 h b 2 h 7 4a 2b 7for f ' 2 : lim lim

h h

a 4 4h h 2b hb 7 4a 2b 7 4ah ah hblim lim

h h

h 4a ah blim lim 4a ah b 4a b 1h

for f ' 2 : lim

0 h 0 h 0 h 0

h 0

2

x 2

5 5 h 15 5 5 5-5h2 h 1 2 1 1 h 1 1 hlim lim lim

h h h h h 1

-5lim -5 2

h 1

Then from 1 and 2 : 4a b -5 3

So f x is differentiable at x 2 , Then it is Continous at x 2 f 2 f 2

for f 2 : lim a x +b x

x 2

57 4a 2b 7 And for f 2 : lim 5

x 1

Then 4a 2b 7 5 4a 2b -2 4

So from 3 and 4 and by subtraction : b 3 Then a -2

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Example (13)

2

2

a x x x 1 8let f x is Contiunous at x 1 and the average rate of change is

3x b x 2 x 1

when x changes from 0 to 3 ,Then find the values of a , b and discuss the differentiabilty at x 1

Answer

2

x 1

2

x 1

2

f x is Continuous at x 1 , Then F 1 F 1 F 1

for F 1 : lim a x x a 1

for F 1 : lim x b x 2 b 3

a 1 b 3 a b 4 1

8When x 0 and h 3 0 3 Then A h

3

f x h f x f 0 3 f 0 f 3 f 0 8And A h

h 3 3 3

f 1 x b x 2 f 3 9 3b 2

2

2

2

h 0

h 0

3b 11

f 1 a x x f 0 0

f 3 f 0 3b 11 8A h 3b 11 8 3b -3 b -1

3 3 3

3x x x 1Then by substitute in 1 : a 3 f x

x x 2 x 1

f 1 h f 1To discuss the Differentiability at x 1 : lim

h

for f ' 1 : lim

2 22

h 0

2

h 0 h 0 h 0

2 22

h 0 h 0

2

h 0 h 0 h 0

3 1 h 1 h 3 1 1 3 3h 1 2h h 3 1lim

h h

h 1 hh hlim lim lim 1 h 1h h

1 h 1 h 2 1 1 2 1 2h h 1 h 2 2for f ' 1 : lim lim

h h

h 1 hh hlim lim lim 1 h 1

h h

So f ' 1 f ' 1 Then f x

is differentiable at x 1

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Example (14)

Sin x , x

2Discuss the differentiability and the Continuity of f x at x

21 Cos x , x2

Answer

h 0

h 0 h 0 h 0

h 0

f x h f xDiscuss the differentiability : lim

h

f h f Sin h SinCos h 1 02 2 2 2

f ' : lim lim lim2 h h h 0

1lim

2 h2Sin 12

2

h 0 h 0 h 0

h 0 h 0

h h-2Sin -Sinh2 2lim lim limSin -1 0 0

hh h 2

2

1 Cos h 1 Cos-Sinh 02 2

f ' : lim lim -12 h h

So f ' f ' Then F x is not differentiable at x

2 2 2Discuss the Con

x x2 2

tinuity :

f Sin 1 And f lim Sin x 1 And f lim 1 Cos x 12 2 2 2

And f f Then F x is Continous at x2 2 2

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Example (15)

Cos2 x , x

2Discuss the differentiability of f x at x

2-Sin x , x2

Answer

h 0

h 0 h 0 h 0

h 0

f x h f xDiscuss the differentiability : lim

h

Cos2 h Cos2Cos 2h Cos -Cos2h 1 02 2

f ' : lim lim lim2 h h h 0

-1lim

22Sin h 1

2

h 0 h 0 h 0

h 0 h 0

h 0

2Sin h 2Sin hlim lim limSin h 2 1 Sin0 0h h h

-Sin h Sin-Cosh 12 2

f ' : lim lim2 h h

-1lim

2 h2Sin 12

2

h 0 h 0 h 0

h h2Sin Sin

h2 2lim lim limSin 1 Sin0 0hh h 2

2

So f ' f ' Then F x is differentiable at x2 2 2

Then F x is Continous at x2

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Differentiability part 1

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