mathcity.org Merging man and maths

Exercise 2.2 (Solutions) Calculus and Analytic Geometry, MATHEMATICS 12

Available online @ http://www.mathcity.org, Version: 1.0.0

Question # 1 (i) Let ( )3y ax b= +

( )3( )y y a x x bd d + = + + ( )3y ax b a x yd d = + + -

( )3 3( ) ( )ax b a x ax bd= + + - + 3 2 2 3 3( ) 3( ) ( ) 3( )( ) ( ) ( )ax b ax b a x ax b a x a x ax bd d d = + + + + + + - +

2 2 2 3 33 ( ) 3 ( )a ax b x a ax b x a xd d d= + + + + ( )2 2 3 23 ( ) 3 ( )x a ax b a ax b x a xd d d= + + + +

Dividing by xd

2 2 3 23 ( ) 3 ( )y a ax b a ax b x a xx

dd d

d= + + + +

Taking limit as 0xd 2 2 3 2

0 0lim lim 3 ( ) 3 ( )x x

y a ax b a ax b x a xxd d

dd d

d = + + + +

2 2 3 23 ( ) 3 ( )(0) (0)dy a ax b a ax b adx

= + + + +

23 ( ) 0 0dy a ax bdx

= + + + 23 ( )dy a ax bdx

= +

Question # 1 (ii) Let 5(2 3)y x= +

( )( )52 3y y x xd d + = + + ( )52 2 3y x x yd d = + + -

( )5 5(2 3) 2 (2 3)x x xd= + + - +

5 4 3 25 5 5(2 3) (2 3) (2 ) (2 3) (2 ) ....0 1 2x x x x xd d = + + + + + +

5 55... (2 ) (2 3)5 x xd + - +

( ) 5 4 3 25 51 (2 3) 2 (2 3) 4 (2 3) ....1 2x x x x xd d = + + + + + +

5 55... 32 (2 3)5 x xd + - +

4 3 2 55 5 52 (2 3) 4 (2 3) .... 321 2 5x x x x xd d d = + + + + +

Dividing by xd

4 3 45 5 52 (2 3) 4 (2 3) .... 321 2 5y x x x xx

dd d

d = + + + + +

Taking limit as 0xd 4 3 4

0 0

5 5 5lim lim 2 (2 3) 4 (2 3) .... 321 2 5x xy x x x xxd d

d d dd

= + + + + +

FSc-II / Ex- 2.2 - 2

452 (2 3) 0 0 .... 01dy xdx

= + + + + +

42(5)(2 3)dy xdx

= + or 410(2 3)dy xdx

= +

Question # 1 (iii) Let ( ) 13 2y t -= +

( ) 23( ) 2y y t td d - + = + + ( ) 23 3 2y t t yd d - = + + - ( ) ( )2 2(3 2) 3 3 2y t t td d - - = + + - +

( )2

2 33 2 1 13 2

tttd -- = + + - +

( ) ( )2

2 2 2 13 33 2 1 ( 2) .... 13 2 2! 3 2

t ttt td d- - - - = + + - + + - + +

( ) ( )2

2 2 363 2 1 .... 13 2 2 3 2

t ty tt td d

d - - - = + - + - + - + +

( )2

2 6 33 2 3 ....3 2 3 2

t ttt td d- = + - + - + + +

( ) 1 3 33 2 2 3 ....3 2 3 2

t ttt td d- = + - + - + + +

Dividing by td

( ) 2 1 33 3 2 2 ....3 2

y ttt t

d dd

- - = + - + - + +

Taking limit when 0td , we have

( ) 30 0

3lim lim 3 3 2 2 ....3 2t t

y ttt td d

d dd

-

= + - + - +

( ) [ ]33 3 2 2 0 0 ....dy tdx

- = + - + - + ( ) 36 3 2dy tdx

- = - +

Question # 1 (iv) Let 5( )y ax b -= +

Do yourself

Question # 1 (vii)

Let 71

( )y

az b=

- 7( )az b -= -

( ) 7( )y y a z z bd d - + = + - ( ) ( )7 7( )y az b a z az bd d - - = - + - -

( )7

7 1 1( )

a zy az baz b

dd-

- = - + - -

Now do yourself

Made by: Atiq ur Rehman (mathcity@gmail.com), http://www.mathcity.org