Chapter 12: Q. 23 (page 953)

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

Short Answer

Expert verified

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

Step by step solution

Given data

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

The given points is $P = (x_{0}, y_{0}) = (- 2, 1) and u = (α, β) = (\frac{\sqrt{10}}{10}, - \frac{3 \sqrt{10}}{10})$

Solution

Consider directional derivative

$D_{u} f (x_{0}, y_{0}) = {Lim}_{h \to 0} \frac{f (x_{0} + α h, y_{0} + β h) - f (x_{0}, y_{0})}{h}$

$D_{w} f (- 2, 1) = {Lim}_{h \to 0} \frac{f (- 2 + \frac{\sqrt{10}}{10} h, 1 - \frac{3 \sqrt{10}}{10} h) - f (- 2, 1)}{h}$ Equation $(1)$

Therefore,

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

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

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

Solve

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

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

$= \frac{- 2 \times 10 + h \sqrt{10}}{10 - 6 \sqrt{10} h + 9 h^{2}}$ Equation $(2)$

And

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

Substitute

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

$D_{n} f (- 2, 1) = {Lim}_{h \to 0} \frac{\frac{- 2 * 10 + h \sqrt{10}}{10 - 6 \sqrt{10} h + 9 h^{2}} - (- 2)}{h}$

$= {Lim}_{h \to 0} \frac{- 20 + h \sqrt{10} + 20 - 12 \sqrt{10} h + 18 h^{2}}{h (10 - 6 \sqrt{10} h + 9 h^{2})}$

$= {Lim}_{h \to 0} \frac{- 11 h \sqrt{10} + 18 h^{2}}{h (10 - 6 \sqrt{10} h + 9 h^{2})}$

$= {Lim}_{h \to 0} \frac{h (- 11 \sqrt{10} + 18 h)}{h (10 - 6 \sqrt{10} h + 9 h^{2})}$

$= {Lim}_{h \to 0} \frac{(- 11 \sqrt{10} + 18 h)}{(10 - 6 \sqrt{10} h + 9 h^{2})} = \frac{- 11 \sqrt{10}}{10}$

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

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

Short Answer

Step by step solution

Given data

Solution

Solve

Substitute

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

Geometry

Logic and Functions

Pure Maths

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

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