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Q1: 

Given: $$f(x, y)=x y^{3}-x^{2},(1,2), \theta=\pi / 3$$

Req: Direction Derivative:

Sol: 

[1] $$D_{v}=\nabla f(1,2) \cdot u$$

[2] $$\nabla f(x, y)=\left\langle f_{x}(x, y), f_{y}(x, y)\right\rangle$$

$$f(x, y)=y^{3}-2 x$$
$$f(x, y)=3 x y^{2}$$

$$\nabla f(x, y)=\langle y^{3}-2 x, 3 x y^{2}\rangle$$

[3] $$\nabla\left(1, 2\right)=\langle 6,12\rangle$$

[4] $$u=\langle\cos \theta, \sin \theta\rangle$$

$$u=\left\langle\frac{1}{2}, \frac{\sqrt{3}}{2}\right\rangle$$

[5] $$D_{v}=\langle 6,12\rangle \cdot\left\langle\frac{1}{2}, \frac{\sqrt{3}}{2}\right\rangle$$

$$D_{v}=3+6 \sqrt{3} $$

Q2:

Given: (1) $$f\left(x, y\right)=\sqrt{2 x+3 y} $$

(2) Point $$(3,1)$$

(3) $$\theta=-\pi / 6$$

Sol:

(1) $$D_{v}=\nabla f(3,1) \cdot u$$

(2) $$\nabla f(x, y)=\left\langle\frac{1}{\sqrt{2 x+2 y}}, \frac{3}{2 \sqrt{2 x+3 y}}\right\rangle$$

$$\nabla f(3, 1) =\left\langle\frac{1}{3}, \frac{1}{2}\right\rangle$$

(3) $$u=\langle\cos \theta, \sin \theta\rangle$$

$$=\left\langle\frac{\sqrt{3}}{2},-\frac{1}{2}\right\rangle$$

(4) $$D_{v}=\left\langle\frac{1}{3}, \frac{1}{2}\right\rangle \cdot\left\langle\frac{\sqrt{3}}{2},-\frac{1}{2}\right\rangle$$

$$D_{V}=\left \langle \frac{\sqrt{3}}{6},-\frac{1}{4} \right\rangle$$

Q3: $$f(x, y)=\frac{x}{x^{2}+y^{2}},\quad (1,2) , \quad V=\langle 3,5\rangle$$

Req: $$D_{v}$$?

Sol: 

(1) $$D_{v}=\nabla f(x, y) \cdot \vec{u} $$

(2) $$f_{x}=\frac{y-x}{\left(x^{2}+y^{2}\right)^{2}} $$

$$f_{y}=\frac{-2 x y}{\left(x^{2}+y^{2}\right)^{2}} $$

$$f_{x}(1,2)=\frac{3}{25} $$

$$f_{y}(1,2)=\frac{-4}{25} $$

$$\nabla f(1,2)=\left\langle\frac{3}{25}, \frac{-4}{25}\right\rangle$$

(3) $$|\vec{v}|=\sqrt{9+25}=\sqrt{34} $$

$$\vec u=\frac{\vec{v}}{|\vec v|}=\left\langle\frac{3} {\sqrt {34}}, \frac{5}{\sqrt{34}}\right\rangle$$

(4) $$D_{v}=\left \langle \frac{3}{25}, \frac{-4}{25}\right \rangle \cdot\left \langle \frac{3}{\sqrt{34}}, \frac{5}{\sqrt{34}}\right \rangle$$

$$D_{v}=\frac{-11}{25 \sqrt{34}} $$

Q4: 

Given: $$f(x, y)=4 y \sqrt{x} $$

Req: Max Rate of change at $$P :(4,1)$$

Sol:

(1) Max rate of change $$=|\nabla f(4,1)|$$

(2) $$\nabla f(x, y)=\left\langle\frac{2 y}{\sqrt{x}}, 4 \sqrt{x}\right\rangle$$

$$\nabla f(4, 1)=\langle 1,8\rangle$$

(3) $$|\nabla f(4,1)|=\sqrt{1+64}=\sqrt{65} $$

(4) Max Rate of change $$\sqrt{65} $$ in the direction $$\langle 1,8 \rangle$$

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