Find the directional derivative of at the point (-2, -3) in the direction .

The gradient of f is :∇f=⟨?,? ⟩

∇f(−2,−3)=⟨?,? ⟩

The directional derivative is:?

Explanation:

Given: [tex]f(x, y) = \sin(x + 2y)[/tex] P(-2, -3)

The unit vector in the direction of [tex]\theta = 3\pi/4[/tex] is given by

[tex]\vec{\textbf{u}} = \left(-\dfrac{\sqrt{2}}{2}\right)\hat{\textbf{i}} + \left(\dfrac{\sqrt{2}}{2}\right)\hat{\textbf{j}}[/tex]

The gradient of f(x, y) [tex]\nabla{f(x, y)}[/tex] is defined as

[tex]\nabla{f(x, y)} = \dfrac{\partial{f}}{\partial{x}}\hat{\textbf{i}} + \dfrac{\partial{f}}{\partial{y}}\hat{\textbf{j}}[/tex]

[tex]= \cos(x + 2y)\hat{\textbf{i}} + 2\cos(x+ 2y)\hat{\textbf{j}}[/tex]

so the gradient of f(x, y) at (-2,-3) is

[tex]\nabla{f(-2, -3)} = \cos(-8)\hat{\textbf{i}} + 2\cos(-8)\hat{\textbf{j}}[/tex]

Therefore, the directional derivative [tex]D_uf(x, y)[/tex] is

[tex]D_uf(x, y) = \nabla{f(x, y)}\cdot \vec{\textbf{u}}[/tex]

[tex]\:\:\:\:\:\:\:=\cos(-8)\left(-\frac{\sqrt{2}}{2}\right) + 2\cos(-8)\left(\frac{\sqrt{2}}{2}\right)[/tex]

[tex]\:\:\:\:\:\:\:\:=\cos(-8)\left(\frac{\sqrt{2}}{2}\right) = -0.103[/tex]

Note: the arguments of the cosines are in radian measure.

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Prompt

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...


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