Question

What is the equation of the tangent line of `f(x) =x^3-3x^2+4x-9` at `x=20`?

Answer 1

y - 1084x + 14809 = 0

Explanation:

To find the equation of the tangent , y - b = m(x - a ) , require to know m and (a , b ) , a point on the line.
Differentiating f(x) and evaluating f'(20) will give value for m , and evaluating f(20) , will give (a , b ).

hence : `f'(x) = 3x^2 - 6x + 4`

and `f'(20) = 3(20)^2 - 6(20) + 4 = 1084 color(black)(" = m of tangent ")`

now f(20) `= (20)^3 - 3(20)^2 + 4(20) - 9 = 6871`

hence (a , b ) = (20 , 6871 ) and m = 1084

equation of tangent : y - 6871 = 1084(x - 20 )

y - 6871 = 1084x - 21680

hence y - 1084x + 14809 = 0