Consider f ' (x) = 6x^2 + 2 and (-1,2) lies on the graph of f. Find the equation for f(x).

Answer 1

To solve this you must integrate the function using the power rule. When you do so, a constant is kicked out, and then you can use the point to solve for the constant.

The indefinite integral of the given function is 2x^3+2x+c (because if you differentiate this you get f'(x) given in the problem. Note we have to include c because any constant would differentiate to 0 making it indeterminant.

Now, use the point given (-1.2) to solve for c: 2 = 2*(-1)^3 + 2*(-1) + c

2 = -2 -2 +c

2 = -4 + c

c = 6

Therefore your function is f(x) = 2x^3+2x+6