Consider the function f(x) whose second derivative is f''(x)=8x+4sin(x). If f(0)=3 and f'(0)=2, what is f(x)?
f"(x) = 8x + 4sinx
f'(x) = 4x^2 - 4cosx + C
since f'(0) = 2,
-4+C = 2
C = 6 and
f'(x) = 4x^2 - 4cosx + 6
f(x) = 4/3 x^3 + 6x - 4sinx + C
since f(0) = 3,
C=3, and
f(x) = 4/3 x^3 - 14x + 3 - 4sinx