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Find-the-directional-derivative-of-f-x-y-4x-3-3x-2-y-2-in-the-direction-given-by-the-angle-pi-3-and-also-Evaluate-directional-derivatives-at-the-point-1-2-




Question Number 210078 by Spillover last updated on 29/Jul/24
Find the directional derivative of  f(x,y)=4x^3 −3x^2 y^2    in the direction given  by the angle θ=(π/3)   and also Evaluate directional derivatives  at the point (1,2)
$${Find}\:{the}\:{directional}\:{derivative}\:{of} \\ $$$${f}\left({x},{y}\right)=\mathrm{4}{x}^{\mathrm{3}} −\mathrm{3}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \:\:\:{in}\:{the}\:{direction}\:{given} \\ $$$${by}\:{the}\:{angle}\:\theta=\frac{\pi}{\mathrm{3}}\: \\ $$$${and}\:{also}\:{Evaluate}\:{directional}\:{derivatives} \\ $$$${at}\:{the}\:{point}\:\left(\mathrm{1},\mathrm{2}\right) \\ $$
Answered by Spillover last updated on 30/Jul/24
  D_v f(x,y)=f_x (x,y)a+f_y (x,y)b  f_x =12x^2 −3y^2   f_y =−6xy  v=(a,b)=(cos θ,sin θ)  v=(a,b)=(cos (π/3),sin(π/3)  )=((1/2),((√3)/2))  D_v f(x,y)=f_x (x,y)a+f_y (x,y)b  =(12x^2 −3y^2 )((1/2))+(−6xy)(((√3)/2))  D_v f(x,y)=6x^2 −(3/2)y^2 −3(√3) xy  D_v f(1,2)=6(1^2 )−(3/2)(2^2 )−3(√3)  2×1=−6(√(3 ))
$$ \\ $$$${D}_{{v}} {f}\left({x},{y}\right)={f}_{{x}} \left({x},{y}\right){a}+{f}_{{y}} \left({x},{y}\right){b} \\ $$$${f}_{{x}} =\mathrm{12}{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \\ $$$${f}_{{y}} =−\mathrm{6}{xy} \\ $$$${v}=\left({a},{b}\right)=\left(\mathrm{cos}\:\theta,\mathrm{sin}\:\theta\right) \\ $$$${v}=\left({a},{b}\right)=\left(\mathrm{cos}\:\frac{\pi}{\mathrm{3}},\mathrm{sin}\frac{\pi}{\mathrm{3}}\:\:\right)=\left(\frac{\mathrm{1}}{\mathrm{2}},\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right) \\ $$$${D}_{{v}} {f}\left({x},{y}\right)={f}_{{x}} \left({x},{y}\right){a}+{f}_{{y}} \left({x},{y}\right){b} \\ $$$$=\left(\mathrm{12}{x}^{\mathrm{2}} −\mathrm{3}{y}^{\mathrm{2}} \right)\left(\frac{\mathrm{1}}{\mathrm{2}}\right)+\left(−\mathrm{6}{xy}\right)\left(\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\right) \\ $$$${D}_{{v}} {f}\left({x},{y}\right)=\mathrm{6}{x}^{\mathrm{2}} −\frac{\mathrm{3}}{\mathrm{2}}{y}^{\mathrm{2}} −\mathrm{3}\sqrt{\mathrm{3}}\:{xy} \\ $$$${D}_{{v}} {f}\left(\mathrm{1},\mathrm{2}\right)=\mathrm{6}\left(\mathrm{1}^{\mathrm{2}} \right)−\frac{\mathrm{3}}{\mathrm{2}}\left(\mathrm{2}^{\mathrm{2}} \right)−\mathrm{3}\sqrt{\mathrm{3}}\:\:\mathrm{2}×\mathrm{1}=−\mathrm{6}\sqrt{\mathrm{3}\:}\: \\ $$$$ \\ $$

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