Paper 4: Fundamentals of Business Mathematics & Statistic

3.38 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

FOR IMPLICIT FUNCTION AND PARAMETRIC FORMS :

Example 82 : For

2 2

2 2

x y 1,

a b+ = find

2

2

d y

dx

Solution:

Diff. Both sides w.r.t. x we get a b2x 2 +2y.y 21 =^0 or 2x 2ya b 2 + 2 .y 0 1 = or

2

(^12)
y b x.
= −a y
2
2 2 1 2 2
2 2 2 2 2
y x .b x
d y b .y.1 x.y b a y
dx a y a y

+^

−

= − = − [putting the value of y 1 ]

(^2) (^22 2 2) 2 2 2
2 2 3 2 2 3
b .a y b x b a b.
a a y a a y

= −^ + = −^

[from the given expression]

4

2 3

b.

=a y

Example 83 : If y = log x

Solution:

2y

1 2 2 2

y dy 1, y d^1.

=dx x= =dx x= −

Example 84:

If y = t^2 + t^3 , x = t – t^4 , find

2

2

d y.

dx

Solution:

dy 2t 3t , 1 4t 2 dx 3

dt= + dt= −

2

3

dy dy dt. dy dx 2t 3t

dx dt dx dt dt 1 4t

= = = +

−

( )

(^223)
2 3
d y d dy d dy dt d dy dx d 2t 3t. 1 4t
dx dx dx dt dx dx dt dx dt dt 1 4t

=^ =^ =^ =^ +^ −

(^) − (^)

( )( ) ( )( )

( ) ( ) ( )

(^32243)
32 2 3 2
1 4t 2 6t 2t 3t 12t 12t 16t 6t 2
1 4t. 1 4t 1 4t

− + − + − + + +

= =

− − −