1. ## Find the function.

Find the function with the given derivative whose graph passes through the point P.

1.) $f'(x)=sin(x)+cos(x), P(\pi,3)$
2.) $f'(x)=x^\frac{1}{3}+x^2+x+1, P(1,0)$

Can someone please explain what I have to do here. I'm pretty sure it involves anti-derivatives, but I've never done them before. If you could, could you explain step by step how to do them. Any help is appreciated, thanks.

2. Yes, you basically need to integrate. If you've never done an antiderivative before, then I suggest you look at a tutorial like this:

Pauls Online Notes : Calculus I - Integrals

For the first one you have:

$\int[sin(x)+cos(x)]dx = -cos(x)+sin(x)+C$

If you differentiate the right side, you'll get the left side for any value of C, which is a constant. The point $(\pi,3)$ allows you to find C so you can determine the exact function:

$3=-cos(\pi)+sin(\pi)+C$

Solve for C...

3. Here are three hints for the first problem:

\begin{aligned}
\frac{d}{dx}(-\cos x)=-(-\sin x)&=\sin x\\
\frac{d}{dx}\sin x&=\cos x\\
\frac{d}{dx}(f(x)+C)=\frac{d}{dx}f(x).
\end{aligned}