• عربي

Need Help?

Subscribe to Calculus C

###### \${selected_topic_name}
• Notes

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 $\langle 1,8 \rangle$