Question

What is the equation of the tangent line of `f(x)=4x^3+12x^2+9x+7` at `x=-3`?

Answer 1

y = 45x + 115

Explanation:

To find the equation of the tangent in the form y = mx +c.

We require to find m (gradient) and the value of c.

Evaluating f'(-3) will give us m and evaluating f(-3) gives c.

Differentiate using the `color(blue)" power rule "`

`f'(x) =12x^2+24x+9`

and `f'(-3)=12(-3)^2+24(-3)+9=108-72+9=45`

Now f(-3) `=4(-3)^3+12(-3)^2+9(-3)+7=-20`

partial equation is y = 45x + c , and to find c substitute (-3 ,-20) into partial equation.

hence : -20 = -135 + c → c = 115

Equation of tangent : y = 45x + 115