Question

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

Answer 1

y = 14x + 12

Explanation:

to find the gradient (slope) of the tangent we differentiate f(x) and evaluate for x = - 2.

`f(x) = 2x - 3x^2 rArr f'(x) = 2 - 6x`

f'(-2) = - 2 - 6(- 2 ) = - 2 + 12 = 14 =gradient of tangent.

the equation of the tangent is : y - b = m( x - a ) where m represents the slope and (a , b ) a point on the line.

we have x= - 2 and require the y-coordinate. Evaluate f(- 2 ) for this.

f(- 2 ) = #2(-2) - 3(- 2 )^2 = - 4 - 12 = - 16

Now y - b = m(x - a ) where m = 14 and (a , b ) =(- 2 , - 16 )

`rArr y + 16 = 14(x + 2 ) rArr y + 16 = 14x + 28`

`rArr y = 14x + 12`