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Math Help - Find the function.

  1. #1
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    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.
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  2. #2
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    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...
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  3. #3
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    Here are three hints for the first problem:

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