Question

What is the equation of the tangent line of `f(x) =-2x^3+6x^2+2x-3` at `x=3`?

Answer 1

y = - 16x + 51

Explanation:

The equation of the line is y - b = m (x - a ) where m is gradient
and (a , b ) a point on the line.

To find m (gradient ) we differentiate f(x) as f'(x) gives the gradient of the tangent to the curve.

`f'(x) = - 6x^2 + 12x + 2`

and m = f'(3) =`-6(3)^2`+ 12(3) + 2 = - 54 + 36 + 2 = - 16

a = 3 : b = f(3) `= -2(3)^3 + 6(3)^2 + 2(3) - 3`

= - 54 + 54 + 6 - 3 = 3

equation of tangent : y - b = m(x - a ) , m= - 16 , (a , b ) = (3 , 3 )

so y - 3 = - 16(x - 3 ) hence y - 3 = - 16x + 48

`rArr y = - 16x + 51`