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Math Help - Can someone check my work?

  1. #1
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    Can someone check my work?

    Im studying for a final and I just need some reassurance.

    1. Find the directional derivative of f(x,y)=x^2+y^2+1  at  (1,2)  in  the  direction  from  (1,2)  to  (3,6)

    I found partial derivatives, found the direction vector, found the unit vector, My answer is 16/sqrt (5)

    2.Find the area of the surface z=16-x^2-y^2 over the region {(x,y)|x^2+y^2=16}
    I setup double integral using polar coordinates, r from 0 to 4, theta from 0 to 2pi.
    My final answer is 256pi/3.
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  2. #2
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    The unit vector I obtain is \frac{1}{\sqrt{5}}<1,2>, the gradient is <2x,2y> so the dot product is the function D_uf(x,y)=2x\frac{1}{\sqrt{5}}+4y\frac{1}{\sqrt{5}  }, at the point you're given I'm obtaining \frac{10}{\sqrt{5}}
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  3. #3
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    pshh..dur dur, I had an arithmetic brain fart. Yep thats what I have. lol
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  4. #4
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    I'm obtaining a different answer for the surface area:

    S=\int_R\int\sqrt{1+[f_x(x,y)]^2+[f_y(x,y)]^2}dA

    f_x(x,y)=-2x
    f_y(x,y)=-2y

    S=\int_R\int\sqrt{1+4x^2+4y^2}dA

    =\int_{0}^{2\pi}\int_0^4\sqrt{1+4r^2}rdrd\theta

    u=1+4r^2\rightarrow dr=8rdu

    =\frac{1}{8}\int_0^{2\pi}\int_1^{65}\sqrt{u}dud\th  eta

    =\frac{1}{8}\frac{2}{3}\int_0^{2\pi}[u^{\frac{3}{2}}]_1^{65}d\theta

    =\frac{1}{12}\int_0^{2\pi}(65\sqrt{65}-1)d\theta

    =\frac{1}{12}(65\sqrt{65}\theta-\theta)]_0^\frac{2\pi}

    \frac{65\sqrt{65}\pi-\pi}{6}
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  5. #5
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    Thanks, got it now. My brain is just on overload.
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