Q. 25 In Exercises 21-28, find the dir... [FREE SOLUTION]

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Chapter 12: Q. 25 (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 $u$ .
$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Short Answer

Expert verified

Directional derivative for the given function is $鈭� f (P) 脳 u = - \frac{7}{816} \sqrt{17}$ .

Step by step solution

Given information

Directional derivative of function is,

$f (x, y) = \sqrt{\frac{y}{x}}$

Given is,

$P = (x_{0}, y_{0}) = (4, 9)$ and $u = (伪, 尾) = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Calculation for directional derivative

Direction derivative of function,

$鈭� f (P) 脳 u = 鈭� f (4, 9) 脳 u$

$= ({\frac{df}{dx}|}_{(4, 9)} i + {\frac{df}{dy}|}_{(4, 9)} j) 脳 (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [{(\frac{1}{2 \sqrt{\frac{y}{x}}} \frac{d}{dx} (\frac{y}{x}))}_{(4, 9)} i + {(\frac{1}{2 \sqrt{\frac{y}{x}}} \frac{d}{dx} (\frac{y}{x}))}_{(4, 9)} j] 脳 (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [{(\frac{\frac{- y}{x^{2}}}{2 \sqrt{\frac{y}{x}}})}_{(4, 9)} i + {(\frac{\frac{1}{x}}{2 \sqrt{\frac{y}{x}}})}_{(4, 9)} j] 脳 (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [(\frac{\frac{- 9}{4^{2}}}{2 \sqrt{\frac{9}{4}}}) i + (\frac{\frac{1}{4}}{2 \sqrt{\frac{9}{4}}}) j] 脳 (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [(\frac{- \frac{9}{16}}{2 脳 \frac{3}{2}}) i + (\frac{\frac{1}{4}}{2 脳 \frac{3}{2}}) j] 脳 (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= (- \frac{9}{48} i + \frac{1}{12} j) 脳 (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= (\frac{9 脳 \sqrt{17}}{48 脳 17} - \frac{4 \sqrt{17}}{12 脳 17})$

$= (\frac{9 脳 \sqrt{17}}{16 脳 3 脳 17} - \frac{\sqrt{17}}{3 脳 17})$

$= \frac{1}{3 脳 17} (\frac{9 脳 \sqrt{17}}{16} - \frac{\sqrt{17}}{1})$

$= \frac{1}{3 脳 17} (\frac{9 脳 \sqrt{17} - 16 \sqrt{17}}{16})$

role="math" localid="1650565813026" $鈭� f (P) 脳 u = - \frac{7}{816} \sqrt{17}$

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91影视

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 $u$ .
$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Short Answer

Step by step solution

Given information

Calculation for directional derivative

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91影视

In Exercises 21-28, find the directional derivative of the given function at the specified point Pand in the direction of the given unit vectoru .f(x,y)=yxat P=(4,9),u=-1717,-41717

Short Answer

Step by step solution

Given information

Calculation for directional derivative

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Pure Maths

Mechanics Maths

Applied Mathematics

Geometry

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 $u$ .
$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$