Answer to Question #140068 in Calculus for Haley

Question #140068
f(x,y)=−2x^2+4y^2
find the value of the directional derivative at the point (3,4) in the direction given by the angle θ=2π/5. More specifically, find the directional derivative of f at the point (3,4) in the direction of the unit vector determined by the angle θ in polar coordinates.
1
Expert's answer
2020-10-25T19:13:54-0400

Given,

"f(x,y)=-2x^2+4y^2"


unit vector in the direction of "\\theta=\\dfrac{2\\pi}{5}" ,

"\\hat{u}=<cos(\\dfrac{2\\pi}{5}),sin(\\dfrac{2\\pi}{5})>"


"\\hat{u}=< 0.30 ,0.95>"


Directional Derivative of F(x,y) is

"D_uf(x,y)=f_x(x,y)""\\times 0.30+f_y(x,y)\\times 0.95"


"\\to D_uf(x,y)=-4x\\times0.30+" "8y\\times0.95"


"\\to D_uf(3,4)=-4\\times3\\times0.30+8\\times 4\\times0.95"


"\\to D_uf(3,4)=-3.6+30.4"


"\\to D_uf(3,4)=26.8"


Hence the required directional derivative is 26.8


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