Answer to Question #219990 in Calculus for Hari

Question #219990

Find the directional derivative of f (x, y,z) = x2y3 − 4xz in the direction of f (x, y,z) = x2y3 − 4xz


1
Expert's answer
2021-07-26T14:55:27-0400
f(x,y,z)=x2y34xzf(x, y, z)=x^2y^3-4xz

v=1,2,0,Point(0,0,1)\vec v=\langle-1,2,0\rangle, Point(0,0,1)

f=fxi+fyj+fzk\nabla f=\dfrac{\partial f}{\partial x}\vec i +\dfrac{\partial f}{\partial y}\vec j +\dfrac{\partial f}{\partial z}\vec k

=(2xy34z)i+(3x2y2)j4xk=(2xy^3-4z)\vec i +(3x^2y^2)\vec j -4x\vec k

f(0,0,1)=4i\nabla f|_{(0,0,1)}=-4\vec i

v=(1)2+(2)2+(0)2=5|\vec v|=\sqrt{(-1)^2+(2)^2+(0)^2}=\sqrt{5}


Dv=f(0,0,1)vv=4ii+2j+0k5=455D_{\vec v}=\nabla f|_{(0,0,1)}\cdot\dfrac{\vec v}{|\vec v|}=-4\vec i\cdot\dfrac{-\vec i +2\vec j +0\vec k }{\sqrt{5}}=\dfrac{4\sqrt{5} }{5}



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