Find the directional derivative of f ( x , y , z ) = x y + z 3 f(x,y,z)=xy+z3 at the point P = ( 4 , − 7 , − 2 ) P=(4,−7,−2) in the directio

Question

Find the directional derivative of f ( x , y , z ) = x y + z 3 f(x,y,z)=xy+z3 at the point P = ( 4 , − 7 , − 2 ) P=(4,−7,−2) in the direction pointing to the origin.

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Brielle 1 month 2021-10-19T05:48:54+00:00 1 Answer 0 views 0

Answers ( )

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    2021-10-19T05:50:26+00:00

    f(x,y,z)=xy+z^3

    has gradient

    \nabla f(x,y,z)=\langle y,x,3z^2\rangle

    The derivative of f(x,y,z) at P in the direction of a vector \vec v is

    D_{\vec u}f(P)=\nabla f(P)\cdot\dfrac{\vec u}{\|\vec u\|}

    In this case, \vec u is the vector pointing from P to the origin, which is

    \vec u=\langle0,0,0\rangle-\langle4,-7,-2\rangle=\langle-4,7,2\rangle

    and has norm

    \|\vec u\|=\sqrt{(-4)^2+7^2+2^2}=\sqrt{69}

    Then the derivative of f at P in the direction of \vec u is

    D_{\vec u}f(P)=\langle-7,4,12\rangle\cdot\dfrac{\langle-4,7,2\rangle}{\sqrt{69}}=\dfrac{80}{\sqrt{69}}

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45:7+7-4:2-5:5*4+35:2 =? ( )