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mathcity.org
Merging man and maths
Exercise 2.1 (Solutions)
Calculus and Analytic Geometry, MATHEMATICS 12
Available online @ http://www.mathcity.org, Version: 1.0.1
Question # 1 (i)
Let 2
2 1y x= +
( )
2
2 1y y x xd dÞ + = + + ( )
2
2 1y x x yd dÞ = + + -
( )2 2 2
2 2 1 2 1y x x x x xd d dÞ = + + + - - 2
2 1y x= +Q
2 2 2
2 4 2 2y x x x x xd d dÞ = + + - 2 2 2
2 4 2 2y x x x x xd d dÞ = + + -
2
4 2y x x xd d dÞ = +
( )4 2x x xd d= +
Dividing by xd
4 2
y
x x
x
d
d
d
= +
Taking limit when 0xd ®
( )0 0
lim lim 4 2
x x
y
x x
xd d
d
d
d® ®
= +
4 2(0)
dy
x
dx
Þ = +
4
dy
x
dx
Þ = i.e. ( )2
2 1 4
d
x x
dx
+ =
Question # 1 (ii)
Let 2y x= -
2y y x xd dÞ + = - + 2y x x yd dÞ = - + -
2 2y x x xd dÞ = - + - + ( )
1 1
2 2y x x xd dÞ = - +
1
1 1
2
2 2 1
x
y x x
x
d
d
æ ö
Þ = - +ç ÷
è ø
( ) 21 1
2 2
1 1
2 2 11
1 ...
2 2!
x x
y x x
x x
d d
d
æ ö- æ ö
Þ = - + × + +ç ÷ç ÷ç ÷è øè ø
( ) 21 1 1
2 2 2
2
1 1
2 2
....
2 2
x x
x x x
x x
d dæ ö-
= - - + +ç ÷ç ÷
è ø
1
2
2
1 1
....
2 6
x
x x
x x
d
d
æ ö
= - - +ç ÷
è ø
Dividing by xd , we have
1
2
2
1 1
....
2 6
y x
x
x x x
d d
d
æ ö
= - - +ç ÷
è ø
Taking limit as
1
2
20 0
1 1
lim lim ....
2 6x x
y x
x
x x xd d
d d
d® ®
æ ö
= - - +ç ÷
è ø
1
2
1
0 0 ....
2
dy
x
dx x
æ ö
Þ = - + -ç ÷
è ø
1
2
1
2
x
x
= ×
1
1
2
1
2
x
-
=
1
2
1
2
dy
x
dx
-
Þ =
FSc-II / Ex- 2.1 - 2
Question # 1 (iii)
Let
1
y
x
= ( )
1
2y x
-
Þ =
Now do yourself
Question # 1 (iv)
Let 3
1
y
x
= 3
y x-
Þ =
( )
3
y y x xd d
-
Þ + = +
( )
3 3
y x x xd d
- -
Þ = + -
3
3
1 1
x
y x
x
d
d
-
-
é ùæ ö
Þ = + -ê úç ÷
è øê úë û
2
3 3 3( 3 1)
1 .... 1
2!
x x
x
x x
d d-
é ùæ ö- - - æ ö
= - + + -ê úç ÷ç ÷ç ÷è øê úè øë û
2
3 3 3( 4)
1 ..... 1
2
x x
x
x x
d d-
é ù- - æ ö
= - + + -ê úç ÷
è øê úë û
2
3 3 3( 4)
.....
2
x x
x
x x
d d-
é ù- - æ ö
= - + +ê úç ÷
è øê úë û
3
3 6 .....
x x
x
x x
d d- é ùæ ö
= × - + -ç ÷ê ú
è øë û
Dividing both sides by xd , we get
3 1
3 6 .....
y x
x
x x
d d
d
- - é ùæ ö
= - + -ç ÷ê ú
è øë û
Taking limit on both sided, we get
4
0 0
lim lim 3 6 .....
x x
y x
x
x xd d
d d
d
-
® ®
é ùæ ö
= - + -ç ÷ê ú
è øë û
[ ]4
3 0 0 .....
dy
x
dx
-
Þ = - + - +
4
3
dy
x
dx
-
Þ = - or 4
3dy
dx x
= -
Question # 1 (vi)
Let
1
y
x a
=
-
*Correction
( )
1
y x a
-
Þ = -
( )
1
y y x x ad d
-
Þ + = + -
( )
1
y x a x yd d
-
Þ = - + -
( ) ( )
1 1
y x a x x ad d
- -
Þ = - + - -
( )
1
1
1 1
x
x a
x a
d
-
- é ùæ ö
= - + -ê úç ÷
-è øê úë û
( )
( ) 2
1 1 1 1
1 .... 1
2!
x x
x a
x a x a
d d-
é ùæ ö- - - æ ö
= - - + + -ê úç ÷ç ÷ç ÷- -è øê úè øë û
FSc-II / Ex- 2.1 - 3
( )
( ) 2
1 1 1 1
1 .... 1
2!
x x
y x a
x a x a
d d
d
- é ù- - - æ ö
Þ = - - + + -ê úç ÷
- -è øê úë û
( )
( ) 2
1 1 2
....
2
x x
x a
x a x a
d d- é ù- - æ ö
= - - + +ê úç ÷
- -è øê úë û
( )
1
1 ....
x x
x a
x a x a
d d- é ùæ ö
= - × - + -ç ÷ê ú- -è øë û
Dividing by xd
( )
1 1
1 ....
y x
x a
x x a
d d
d
- - é ùæ ö
= - - + -ç ÷ê ú-è øë û
Taking limit when 0xd ® , we have
( )
1 1
0 0
lim lim 1 ....
x x
y x
x a
x x ad d
d d
d
- -
® ®
é ùæ ö
= - - + -ç ÷ê ú-è øë û
( ) [ ]2
1 0 0 ....
dy
x a
dx
-
Þ = - - + - +
( )2
1dy
dx x a
-
Þ =
-
Question # 1 (vi)
Let ( 3)y x x= -
2
3x x= -
Do yourself
Question # 1 (vii)
Let 4
2
y
x
= 4
2x-
=
4
2( )y y x xd d -
Þ + = +
Do yourself
Question # 1(viii)
Let 2
2
1
y x
x
= + 2 2
x x-
= +
( ) ( )
2 2
y y x x x xd d d
-
Þ + = + + +
( ) ( )
2 2 2 2
y x x x x x xd d d
- -
Þ = + + + - -
( ) ( )
2 22 2
x x x x x xd d
- -
= + - + + -
2
2 2 2 2
2 1 1
x
x x x x x x
x
d
d d
-
-
é ùæ ö
= + + - + + -ê úç ÷
è øê úë û
2
2 2 2 2( 2 1)
2 1 .... 1
2!
x x
x x x x
x x
d d
d d -
é ùæ ö- - - æ ö
= + + - + + -ê úç ÷ç ÷ç ÷è øê úè øë û
2
2 2 2 2( 3)
2 ....
2
x x
x x x x
x x
d d
d d -
é ù- - æ ö
= + + - + +ê úç ÷
è øê úë û
( ) 2
2 2 3 ....
x x
x x x x
x x
d d
d d - é ùæ ö
= + + × - + +ç ÷ê ú
è øë û
Dividing by xd
( ) 3
2 2 3 ....
y x
x x x
x x
d d
d
d
- é ùæ ö
= + + - + +ç ÷ê ú
è øë û
FSc-II / Ex- 2.1 - 4
Taking limit when 0xd ® , we have
( ) 3
0 0 0
lim lim 2 lim 2 3 ....
x x x
y x
x x x
x xd d d
d d
d
d
-
® ® ®
é ùæ ö
= + + - + +ç ÷ê ú
è øë û
( ) [ ]3
2 0 2 0 0....
dy
x x
dx
-
Þ = + + - + -
3
2
2
dy
x
dx x
Þ = -
Question # 1(ix)
Let ( )
1
34y x= +
( )
1
34y y x xd dÞ + = + +
( )
1
34y x x yd dÞ = + + -
( ) ( )
1 1
3 34 4x x xd= + + - +
( )
1
1 3
34 1 1
4
x
x
x
d
é ù
æ öê ú= + + -ç ÷ê ú+è ø
ë û
( )
( ) 21
3
1 1
3 3 11
4 1 .... 1
3 4 2! 4
x x
x
x x
d dé ùæ ö- æ öê úç ÷= + + + + -ç ÷ç ÷+ +è øê úè øë û
( )
( ) 21
3
1 2
3 3
4 ....
3( 4) 2 4
x x
x
x x
d dé ù- æ ö
= + + +ê úç ÷
+ +è øê úë û
( )
1
3
1 1
4 ....
4 3 9 4
x x
x
x x
d dé ùæ ö
= + × - +ç ÷ê ú+ +è øë û
Dividing by xd
( )
1
1
3
1 1
4 ....
3 9 4
y x
x
x x
d d
d
- é ùæ ö
= + - +ç ÷ê ú+è øë û
Taking limit when 0xd ®
( )
2
3
0 0
1 1
lim lim 4 ....
3 9 4x x
y x
x
x xd d
d d
d
-
® ®
é ùæ ö
= + - +ç ÷ê ú+è øë û
( )
2
3
1
4 0 0 ....
3
dy
x
dx
- é ù
Þ = + - + -ê úë û
( )
2
3
1
4
3
dy
x
dx
-
Þ = +
Question # 1 (x)
Let
3
2y x=
( )
3
2y y x xd dÞ + = +
( )
33
22y x x xd dÞ = + -
3
3
2
2
1 1
x
x
x
d
é ù
æ öê ú= + -ç ÷ê úè ø
ë û
( ) 23
2
3 3
2 2 13
1 .... 1
2 2!
x x
x
x x
d dé ùæ ö- æ öê úç ÷= + + + -ç ÷ç ÷è øê úè øë û
FSc-II / Ex- 2.1 - 5
( ) 23
2
3 1
2 23
....
2 2
x x
x
x x
d dé ùæ ö
= + +ê úç ÷
è øê úë û
3
2
3 3
....
2 8
x x
x
x x
d dé ùæ ö
= × + +ç ÷ê ú
è øë û
Dividing by xd
3
1
2
3 3
....
2 8
y x
x
x x
d d
d
- é ùæ ö
= + +ç ÷ê ú
è øë û
Taking limit when 0xd ®
1
2
0 0
3 3
lim lim ....
2 8x x
y x
x
x xd d
d d
d® ®
é ùæ ö
= + +ç ÷ê ú
è øë û
1
2
3
0 0 ....
2
dy
x
dx
é ù
Þ = - + -ê úë û
1
2
3
2
dy
x
dx
Þ =
Question # 1 (xi)
Let 5 2
y x=
Do yourself as above.
Question # 1 (xii)
Let m
y x=
( )
m
y y x xd dÞ + = +
( )
m m
y x x xd dÞ = + -
1 1
m
m x
x
x
dé ùæ ö
= + -ê úç ÷
è øê úë û
( ) 2
1
1 .... 1
2!
m m mx x
x m
x x
d dé ùæ ö- æ ö
= + × + + -ê úç ÷ç ÷ç ÷è øê úè øë û
( ) 2
1
....
2
m m mm x x
x
x x
d dé ù- æ ö
= + +ê úç ÷
è øê úë û
( )1
....
2
m m mx x
x m
x x
d d-é ùæ ö
= × + +ç ÷ê ú
è øë û
Dividing by xd
( )1 1
....
2
m m my x
x m
x x
d d
d
- -é ùæ ö
= + +ç ÷ê ú
è øë û
Taking limit when 0xd ®
( )1
0 0
1
lim lim ....
2
m
x x
m my x
x m
x xd d
d d
d
-
® ®
-é ùæ ö
= + +ç ÷ê ú
è øë û
[ ]1
0 0....mdy
x m
dx
-
Þ = + + 1mdy
m x
dx
-
Þ =
Question # 1 (xii)
Let
1 m
m
y x
x
-
= =
Do yourself as above, just change the m by m- in above question.
FSc-II / Ex- 2.1 - 6
Question # 1 (xvi)
Let 40
y x=
( )
40
y y x xd dÞ + = +
( )
40 40
y x x xd dÞ = + -
40 39 38 2 40 4040 40 40 40
....
0 1 2 40
x x x x x x xd d d
é ùæ ö æ ö æ ö æ ö
= + + + + -ç ÷ ç ÷ ç ÷ ç ÷ê úè ø è ø è ø è øë û
( ) 40 39 38 2 40 4040 40 40
1 ....
1 2 40
x x x x x x xd d d
æ ö æ ö æ ö
= + + + + -ç ÷ ç ÷ ç ÷
è ø è ø è ø
39 38 2 4040 40 40
....
1 2 40
x x x x xd d d
æ ö æ ö æ ö
= + + +ç ÷ ç ÷ ç ÷
è ø è ø è ø
Dividing by xd
39 38 3940 40 40
....
1 2 40
y
x x x x
x
d
d d
d
æ ö æ ö æ ö
= + + +ç ÷ ç ÷ ç ÷
è ø è ø è ø
Taking limit as 0xd ®
39 38 39
0 0
40 40 40
lim lim ....
1 2 40x x
y
x x x x
xd d
d
d d
d® ®
é ùæ ö æ ö æ ö
= + + +ç ÷ ç ÷ ç ÷ê úè ø è ø è øë û
3940
0 0 .... 0
1
dy
x
dx
é ùæ ö
= + + + +ç ÷ê úè øë û
3940
1
dy
x
dx
æ ö
Þ = ç ÷
è ø
or 39
40
dy
x
dx
=
Question # 1 (xiii)
Let 100
y x-
=
Do yourself Question # 1(xii), Replace m by 100- .
Question # 2 (i)
Let ( )
1
22 2y x x= + = +
Now do yourself as Question # 1(ix)
Question # 2 (ii)
Let ( )
1
2
1
y x a
x a
-
= = +
+
Now do yourself as Question # 1 (ix)
× The End Ø
Made by: Atiq ur Rehman (mathcity@gmail.com), http://www.mathcity.org

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Ex 2 1_fsc_part2

  • 1. mathcity.org Merging man and maths Exercise 2.1 (Solutions) Calculus and Analytic Geometry, MATHEMATICS 12 Available online @ http://www.mathcity.org, Version: 1.0.1 Question # 1 (i) Let 2 2 1y x= + ( ) 2 2 1y y x xd dÞ + = + + ( ) 2 2 1y x x yd dÞ = + + - ( )2 2 2 2 2 1 2 1y x x x x xd d dÞ = + + + - - 2 2 1y x= +Q 2 2 2 2 4 2 2y x x x x xd d dÞ = + + - 2 2 2 2 4 2 2y x x x x xd d dÞ = + + - 2 4 2y x x xd d dÞ = + ( )4 2x x xd d= + Dividing by xd 4 2 y x x x d d d = + Taking limit when 0xd ® ( )0 0 lim lim 4 2 x x y x x xd d d d d® ® = + 4 2(0) dy x dx Þ = + 4 dy x dx Þ = i.e. ( )2 2 1 4 d x x dx + = Question # 1 (ii) Let 2y x= - 2y y x xd dÞ + = - + 2y x x yd dÞ = - + - 2 2y x x xd dÞ = - + - + ( ) 1 1 2 2y x x xd dÞ = - + 1 1 1 2 2 2 1 x y x x x d d æ ö Þ = - +ç ÷ è ø ( ) 21 1 2 2 1 1 2 2 11 1 ... 2 2! x x y x x x x d d d æ ö- æ ö Þ = - + × + +ç ÷ç ÷ç ÷è øè ø ( ) 21 1 1 2 2 2 2 1 1 2 2 .... 2 2 x x x x x x x d dæ ö- = - - + +ç ÷ç ÷ è ø 1 2 2 1 1 .... 2 6 x x x x x d d æ ö = - - +ç ÷ è ø Dividing by xd , we have 1 2 2 1 1 .... 2 6 y x x x x x d d d æ ö = - - +ç ÷ è ø Taking limit as 1 2 20 0 1 1 lim lim .... 2 6x x y x x x x xd d d d d® ® æ ö = - - +ç ÷ è ø 1 2 1 0 0 .... 2 dy x dx x æ ö Þ = - + -ç ÷ è ø 1 2 1 2 x x = × 1 1 2 1 2 x - = 1 2 1 2 dy x dx - Þ =
  • 2. FSc-II / Ex- 2.1 - 2 Question # 1 (iii) Let 1 y x = ( ) 1 2y x - Þ = Now do yourself Question # 1 (iv) Let 3 1 y x = 3 y x- Þ = ( ) 3 y y x xd d - Þ + = + ( ) 3 3 y x x xd d - - Þ = + - 3 3 1 1 x y x x d d - - é ùæ ö Þ = + -ê úç ÷ è øê úë û 2 3 3 3( 3 1) 1 .... 1 2! x x x x x d d- é ùæ ö- - - æ ö = - + + -ê úç ÷ç ÷ç ÷è øê úè øë û 2 3 3 3( 4) 1 ..... 1 2 x x x x x d d- é ù- - æ ö = - + + -ê úç ÷ è øê úë û 2 3 3 3( 4) ..... 2 x x x x x d d- é ù- - æ ö = - + +ê úç ÷ è øê úë û 3 3 6 ..... x x x x x d d- é ùæ ö = × - + -ç ÷ê ú è øë û Dividing both sides by xd , we get 3 1 3 6 ..... y x x x x d d d - - é ùæ ö = - + -ç ÷ê ú è øë û Taking limit on both sided, we get 4 0 0 lim lim 3 6 ..... x x y x x x xd d d d d - ® ® é ùæ ö = - + -ç ÷ê ú è øë û [ ]4 3 0 0 ..... dy x dx - Þ = - + - + 4 3 dy x dx - Þ = - or 4 3dy dx x = - Question # 1 (vi) Let 1 y x a = - *Correction ( ) 1 y x a - Þ = - ( ) 1 y y x x ad d - Þ + = + - ( ) 1 y x a x yd d - Þ = - + - ( ) ( ) 1 1 y x a x x ad d - - Þ = - + - - ( ) 1 1 1 1 x x a x a d - - é ùæ ö = - + -ê úç ÷ -è øê úë û ( ) ( ) 2 1 1 1 1 1 .... 1 2! x x x a x a x a d d- é ùæ ö- - - æ ö = - - + + -ê úç ÷ç ÷ç ÷- -è øê úè øë û
  • 3. FSc-II / Ex- 2.1 - 3 ( ) ( ) 2 1 1 1 1 1 .... 1 2! x x y x a x a x a d d d - é ù- - - æ ö Þ = - - + + -ê úç ÷ - -è øê úë û ( ) ( ) 2 1 1 2 .... 2 x x x a x a x a d d- é ù- - æ ö = - - + +ê úç ÷ - -è øê úë û ( ) 1 1 .... x x x a x a x a d d- é ùæ ö = - × - + -ç ÷ê ú- -è øë û Dividing by xd ( ) 1 1 1 .... y x x a x x a d d d - - é ùæ ö = - - + -ç ÷ê ú-è øë û Taking limit when 0xd ® , we have ( ) 1 1 0 0 lim lim 1 .... x x y x x a x x ad d d d d - - ® ® é ùæ ö = - - + -ç ÷ê ú-è øë û ( ) [ ]2 1 0 0 .... dy x a dx - Þ = - - + - + ( )2 1dy dx x a - Þ = - Question # 1 (vi) Let ( 3)y x x= - 2 3x x= - Do yourself Question # 1 (vii) Let 4 2 y x = 4 2x- = 4 2( )y y x xd d - Þ + = + Do yourself Question # 1(viii) Let 2 2 1 y x x = + 2 2 x x- = + ( ) ( ) 2 2 y y x x x xd d d - Þ + = + + + ( ) ( ) 2 2 2 2 y x x x x x xd d d - - Þ = + + + - - ( ) ( ) 2 22 2 x x x x x xd d - - = + - + + - 2 2 2 2 2 2 1 1 x x x x x x x x d d d - - é ùæ ö = + + - + + -ê úç ÷ è øê úë û 2 2 2 2 2( 2 1) 2 1 .... 1 2! x x x x x x x x d d d d - é ùæ ö- - - æ ö = + + - + + -ê úç ÷ç ÷ç ÷è øê úè øë û 2 2 2 2 2( 3) 2 .... 2 x x x x x x x x d d d d - é ù- - æ ö = + + - + +ê úç ÷ è øê úë û ( ) 2 2 2 3 .... x x x x x x x x d d d d - é ùæ ö = + + × - + +ç ÷ê ú è øë û Dividing by xd ( ) 3 2 2 3 .... y x x x x x x d d d d - é ùæ ö = + + - + +ç ÷ê ú è øë û
  • 4. FSc-II / Ex- 2.1 - 4 Taking limit when 0xd ® , we have ( ) 3 0 0 0 lim lim 2 lim 2 3 .... x x x y x x x x x xd d d d d d d - ® ® ® é ùæ ö = + + - + +ç ÷ê ú è øë û ( ) [ ]3 2 0 2 0 0.... dy x x dx - Þ = + + - + - 3 2 2 dy x dx x Þ = - Question # 1(ix) Let ( ) 1 34y x= + ( ) 1 34y y x xd dÞ + = + + ( ) 1 34y x x yd dÞ = + + - ( ) ( ) 1 1 3 34 4x x xd= + + - + ( ) 1 1 3 34 1 1 4 x x x d é ù æ öê ú= + + -ç ÷ê ú+è ø ë û ( ) ( ) 21 3 1 1 3 3 11 4 1 .... 1 3 4 2! 4 x x x x x d dé ùæ ö- æ öê úç ÷= + + + + -ç ÷ç ÷+ +è øê úè øë û ( ) ( ) 21 3 1 2 3 3 4 .... 3( 4) 2 4 x x x x x d dé ù- æ ö = + + +ê úç ÷ + +è øê úë û ( ) 1 3 1 1 4 .... 4 3 9 4 x x x x x d dé ùæ ö = + × - +ç ÷ê ú+ +è øë û Dividing by xd ( ) 1 1 3 1 1 4 .... 3 9 4 y x x x x d d d - é ùæ ö = + - +ç ÷ê ú+è øë û Taking limit when 0xd ® ( ) 2 3 0 0 1 1 lim lim 4 .... 3 9 4x x y x x x xd d d d d - ® ® é ùæ ö = + - +ç ÷ê ú+è øë û ( ) 2 3 1 4 0 0 .... 3 dy x dx - é ù Þ = + - + -ê úë û ( ) 2 3 1 4 3 dy x dx - Þ = + Question # 1 (x) Let 3 2y x= ( ) 3 2y y x xd dÞ + = + ( ) 33 22y x x xd dÞ = + - 3 3 2 2 1 1 x x x d é ù æ öê ú= + -ç ÷ê úè ø ë û ( ) 23 2 3 3 2 2 13 1 .... 1 2 2! x x x x x d dé ùæ ö- æ öê úç ÷= + + + -ç ÷ç ÷è øê úè øë û
  • 5. FSc-II / Ex- 2.1 - 5 ( ) 23 2 3 1 2 23 .... 2 2 x x x x x d dé ùæ ö = + +ê úç ÷ è øê úë û 3 2 3 3 .... 2 8 x x x x x d dé ùæ ö = × + +ç ÷ê ú è øë û Dividing by xd 3 1 2 3 3 .... 2 8 y x x x x d d d - é ùæ ö = + +ç ÷ê ú è øë û Taking limit when 0xd ® 1 2 0 0 3 3 lim lim .... 2 8x x y x x x xd d d d d® ® é ùæ ö = + +ç ÷ê ú è øë û 1 2 3 0 0 .... 2 dy x dx é ù Þ = - + -ê úë û 1 2 3 2 dy x dx Þ = Question # 1 (xi) Let 5 2 y x= Do yourself as above. Question # 1 (xii) Let m y x= ( ) m y y x xd dÞ + = + ( ) m m y x x xd dÞ = + - 1 1 m m x x x dé ùæ ö = + -ê úç ÷ è øê úë û ( ) 2 1 1 .... 1 2! m m mx x x m x x d dé ùæ ö- æ ö = + × + + -ê úç ÷ç ÷ç ÷è øê úè øë û ( ) 2 1 .... 2 m m mm x x x x x d dé ù- æ ö = + +ê úç ÷ è øê úë û ( )1 .... 2 m m mx x x m x x d d-é ùæ ö = × + +ç ÷ê ú è øë û Dividing by xd ( )1 1 .... 2 m m my x x m x x d d d - -é ùæ ö = + +ç ÷ê ú è øë û Taking limit when 0xd ® ( )1 0 0 1 lim lim .... 2 m x x m my x x m x xd d d d d - ® ® -é ùæ ö = + +ç ÷ê ú è øë û [ ]1 0 0....mdy x m dx - Þ = + + 1mdy m x dx - Þ = Question # 1 (xii) Let 1 m m y x x - = = Do yourself as above, just change the m by m- in above question.
  • 6. FSc-II / Ex- 2.1 - 6 Question # 1 (xvi) Let 40 y x= ( ) 40 y y x xd dÞ + = + ( ) 40 40 y x x xd dÞ = + - 40 39 38 2 40 4040 40 40 40 .... 0 1 2 40 x x x x x x xd d d é ùæ ö æ ö æ ö æ ö = + + + + -ç ÷ ç ÷ ç ÷ ç ÷ê úè ø è ø è ø è øë û ( ) 40 39 38 2 40 4040 40 40 1 .... 1 2 40 x x x x x x xd d d æ ö æ ö æ ö = + + + + -ç ÷ ç ÷ ç ÷ è ø è ø è ø 39 38 2 4040 40 40 .... 1 2 40 x x x x xd d d æ ö æ ö æ ö = + + +ç ÷ ç ÷ ç ÷ è ø è ø è ø Dividing by xd 39 38 3940 40 40 .... 1 2 40 y x x x x x d d d d æ ö æ ö æ ö = + + +ç ÷ ç ÷ ç ÷ è ø è ø è ø Taking limit as 0xd ® 39 38 39 0 0 40 40 40 lim lim .... 1 2 40x x y x x x x xd d d d d d® ® é ùæ ö æ ö æ ö = + + +ç ÷ ç ÷ ç ÷ê úè ø è ø è øë û 3940 0 0 .... 0 1 dy x dx é ùæ ö = + + + +ç ÷ê úè øë û 3940 1 dy x dx æ ö Þ = ç ÷ è ø or 39 40 dy x dx = Question # 1 (xiii) Let 100 y x- = Do yourself Question # 1(xii), Replace m by 100- . Question # 2 (i) Let ( ) 1 22 2y x x= + = + Now do yourself as Question # 1(ix) Question # 2 (ii) Let ( ) 1 2 1 y x a x a - = = + + Now do yourself as Question # 1 (ix) × The End Ø Made by: Atiq ur Rehman (mathcity@gmail.com), http://www.mathcity.org