Finding equation of the normal

Question

let f(x)= (g(x)/h(x) where g(2)=18, h(2)=6, g'(2)=5 and h'(2)=2. Find the equation of the normal to the graph of f at x=2.

Answer 1

Let us first find the slope of the tangent line at x=2.

 

m_tangent = f'(x) = (g'(x)h(x)-h'(x)g(x))/h²(x) = f'(2) = ((5)*(6)-(2)*(18))/(6)² =-1/6

Note: we found f'(x) by the quotient rule.

The slope of the normal line has the negative-reciprocal of the slope of the tangent line.

 

m_normal = -(m_tangent)^-1 = 6

 

The normal line takes the form of 

y = m_normalx + b

 

we know x = 2, y = f(x) = f(2) = (18)/(6) = 3

 

hence,

(3) = (6)(2) + b

 

solve for b... b = -9

 

So the equation of the normal to the graph of f at x = 2 is...

 

y = 6x - 9