Application of Differentiation
1. Find the equation of the normal to the curve y =

where x = 4.
at the point on the curve
[6]

y = 1/2 x + 10.

Step-by-step explanation:

To find the slope of  a tangent to the curve we find the derivative:

f(x) = (x^2 + 8) / (x - 2)

Using the quotient rule:

f'(x) =   (x - 2)(2x) - (x^2 + 8) / (x - 2)^2

= (2x^2 - 4x - x^2 - 8) / (x - 2)^2

= (x^2 - 4x - 8) / (x - 2)^2

At x = 4 , the slope of the tangent

= (16 - 16 - 8) / 4

= -2.

So the slope of the normal = -1/-2 = 1/2.

The equation of the normal is

y = 1/2 x + c

When x = 4, y =  (4^2 + 8)  / 4 -2 =  24/2 = 12, so

12 = 1/2(4) + c

c = 12 - 2 = 10.

The required equation is y = 1/2 x + 10.

y -5 = (-1/4)(x + 8)

step-by-step explanation: