### Home > APCALC > Chapter 4 > Lesson 4.3.1 > Problem4-116

4-116.

A function has a derivative of $f^\prime\left(x\right) = 6x^{2} + 12x – 7$.

1. If $f\left(0\right) = 0$, what was the original function?

$f(x)=\int f'(x)dx=2x^{3}+6x^{2}-7x+C$

Solve for $C$ by evaluating $f\left(x\right)$ at $f\left(0\right) = 0$.

Note: you could also use the expression

$f(x)=f(0)+\int_{0}^{x}f(t)dt$

2. If $f\left(–2\right) = 25$, what was the original function?

$f\left(x\right) = 2x³ + 6x² − 7x + C$
Evaluate the given point to solve for $C$.

Note: you could also use the expression

3. Describe how you found the constant of integration in parts (a) and (b).

Refer to hints in part (a) and (b).