anti-derivative

[legacyofpiracy]

Hello everyone, I just got back from a nice long winter break which gave me just enough time to forget everything I have ever learned in math, including anti-derivatives. :roll: Here is the problem:

Consider the function F(x) whose second derivative is F''(x)=9x + 9 sin(x). If F(0)=4 and F'(0)=4 what is F'(x)? What is F(4)

Now first off wouldn't the second derivative be the same as F(x), or perhaps I have just been mistaken. Anyway the real problem, sadly enough, was just finding the anti derivative of 9x+9 sin(x). I determined that the 9x would be 4.5x^2 but I am lost as to how to do the 9 sin(x) even though it seems to be realtively simple. Any suggestions would be greatly appreciated

-thanks

 

[galactus]

They give you the 2nd derivative. Integrate to arrive at the first derivative.

\(\displaystyle F'(x)=\int{9x+9sin(x)}dx=\frac{9x^{2}}{2}-9cos(x)+C\)

Set this function equal to 4, sub in x=0 and solve for C.

Integrate again to arrive at the original function, F(x), and don't forget the

constant.

Set it equal to 4, sub in x=0 and solve for the constant...you should have it.

Then to check, take the derivative of F(x) abd see if you arrive at what you came

up with. Do it again for the second check.