Q. 23 In Exercises 21鈥�28, find the d... [FREE SOLUTION]

Chapter 12: Q. 23 (page 953)

In Exercises $21 鈥� 28$ , find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u .$
$f (x, y) = \frac{x}{y^{2}} at P = (鈭� 2, 1), u = \{\frac{\sqrt{10}}{10}, 鈭� \frac{3 \sqrt{10}}{10}>$

Short Answer

Expert verified

The directional derivative of the function $f (x, y) = \frac{x}{y^{2}}$ is $D_{n} f (鈭� 2, 1) = \frac{鈭� 11 \sqrt{10}}{10}$ . $\frac{鈭� 11 \sqrt{10}}{10}$

Step by step solution

Given data

The function is $f (x, y) = \frac{x}{y^{2}}$

The given points is $P = (x_{0}, y_{0}) = (鈭� 2, 1) and u = (伪, 尾) = (\frac{\sqrt{10}}{10}, 鈭� \frac{3 \sqrt{10}}{10})$

Solution

Consider directional derivative

$D_{u} f (x_{0}, y_{0}) = {Lim}_{h 鈫� 0} 鈦� \frac{f (x_{0} + 伪 h, y_{0} + 尾 h) 鈭� f (x_{0}, y_{0})}{h}$

$D_{w} f (鈭� 2, 1) = {Lim}_{h 鈫� 0} 鈦� \frac{f (鈭� 2 + \frac{\sqrt{10}}{10} h, 1 鈭� \frac{3 \sqrt{10}}{10} h) 鈭� f (鈭� 2, 1)}{h}$ Equation $(1)$

Therefore,

$f (鈭� 2 + \frac{\sqrt{10}}{10} h, 1 鈭� \frac{3 \sqrt{10}}{10} h) = \frac{鈭� 2 + \frac{\sqrt{10}}{10} h}{{(1 鈭� \frac{3 \sqrt{10}}{10} h)}^{2}}$

$= \frac{鈭� 2 + \frac{h}{\sqrt{10}}}{{(1 鈭� \frac{3}{\sqrt{10}} h)}^{2}}$

$= \frac{鈭� 2 \sqrt{10} + h}{(1 鈭� \frac{6}{\sqrt{10}} h + \frac{9}{10}} h^{2})$

Solve

$= \frac{\sqrt{10}}{(\frac{10 鈭� 6 \sqrt{10} h + 9 h^{2}}{10})}$

$= \frac{\sqrt{10} (鈭� 2 \sqrt{10} + h)}{10 鈭� 6 \sqrt{10} h + 9 h^{2}}$

$= \frac{鈭� 2 脳 10 + h \sqrt{10}}{10 鈭� 6 \sqrt{10} h + 9 h^{2}}$ Equation $(2)$

And

$f (x_{0}, y_{0}) = f (鈭� 2, 1) = \frac{鈭� 2}{1^{2}} = - 2$ Equation $(3)$

Substitute

Substituting Equation $(2), (3)$ in $(1)$

$D_{n} f (鈭� 2, 1) = {Lim}_{h 鈫� 0} 鈦� \frac{\frac{鈭� 2 鈭� 10 + h \sqrt{10}}{10 鈭� 6 \sqrt{10} h + 9 h^{2}} 鈭� (鈭� 2)}{h}$

$= {Lim}_{h 鈫� 0} 鈦� \frac{鈭� 20 + h \sqrt{10} + 20 鈭� 12 \sqrt{10} h + 18 h^{2}}{h (10 鈭� 6 \sqrt{10} h + 9 h^{2})}$

$= {Lim}_{h 鈫� 0} 鈦� \frac{鈭� 11 h \sqrt{10} + 18 h^{2}}{h (10 鈭� 6 \sqrt{10} h + 9 h^{2})}$

$= {Lim}_{h 鈫� 0} 鈦� \frac{h (鈭� 11 \sqrt{10} + 18 h)}{h (10 鈭� 6 \sqrt{10} h + 9 h^{2})}$

$= {Lim}_{h 鈫� 0} 鈦� \frac{(鈭� 11 \sqrt{10} + 18 h)}{(10 鈭� 6 \sqrt{10} h + 9 h^{2})} = \frac{鈭� 11 \sqrt{10}}{10}$

$D_{n} f (鈭� 2, 1) = \frac{鈭� 11 \sqrt{10}}{10}$

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91影视

In Exercises $21 鈥� 28$ , find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u .$
$f (x, y) = \frac{x}{y^{2}} at P = (鈭� 2, 1), u = \{\frac{\sqrt{10}}{10}, 鈭� \frac{3 \sqrt{10}}{10}>$

Short Answer

Step by step solution

Given data

Solution

Solve

Substitute

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91影视

In Exercises 21鈥�28, find the directional derivative of the givenfunction at the specified pointP and in the direction of thegiven unit vectoru.f(x,y)=xy2atP=(鈭�2,1),u=1010,鈭�31010

Short Answer

Step by step solution

Given data

Solution

Solve

Substitute

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Calculus

Geometry

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises $21 鈥� 28$ , find the directional derivative of the given
function at the specified point $P$ and in the direction of the
given unit vector $u .$
$f (x, y) = \frac{x}{y^{2}} at P = (鈭� 2, 1), u = \{\frac{\sqrt{10}}{10}, 鈭� \frac{3 \sqrt{10}}{10}>$