Differentiability part 1
-
Upload
sherif-yehia-al-maraghy -
Category
Documents
-
view
227 -
download
0
Transcript of Differentiability part 1
-
7/29/2019 Differentiability part 1
1/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
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 :
----------------------------------------------------------------------------------------- --------------
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
----------------------------------------------------------------------------------------- --------------
-
7/29/2019 Differentiability part 1
2/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -4-
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
---------------------------------------------------------------------------------------------------------------------
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
----------------------------------------------------------------------------------------------------------------------
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
-
7/29/2019 Differentiability part 1
3/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -44-
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.
----------------------------------------------------------------------------------------------------------------------
Example (1)
Discuss the Differentiability of f x x 1 at x -1 Answer
-
7/29/2019 Differentiability part 1
4/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -4-
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
----------------------------------------------------------------------------------------------------------------------
Example (3)
2Discuss the differentiability of f x x 6x 9 at x 3. Answer
----------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
5/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -4-
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
----------------------------------------------------------------------------------------------------------------------
Example (5)
3Discuss the Differentiability of f x x 2 at any point of its domain Answer
----------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
6/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -4-
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
-
7/29/2019 Differentiability part 1
7/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -7-
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
----------------------------------------------------------------------------------------------------------------------
Example (8)
3x x 8 , x 0
Discuss the differentiability of f x , at x 03 x5 , x 0
x
Answer
-
7/29/2019 Differentiability part 1
8/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
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
-
7/29/2019 Differentiability part 1
9/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I --
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
----------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
10/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I --
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
----------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
11/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -6-
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
---------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
12/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I --
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
---------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
13/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I -4-
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
---------------------------------------------------------------------------------------------------------------------
-
7/29/2019 Differentiability part 1
14/14
Facebook page : https://www.facebook.com/MrSherifYehiaAlMaraghy
Mr . Sher if Yehia Al Maraghy https://twitter.com/Mr_Sheri f_yehia
0100998883601009988826 Email : Rooshery@hotmai l.com
Calculus 3rdsecondary Differentiability Part I --
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
----------------------------------------------------------------------------------------------------------------------