Q. 26 In Exercises 21鈥�28, find the d... [FREE SOLUTION]

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Chapter 12: Q. 26 (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.
$f (x, y) = \sqrt{\frac{y}{x}} at P = (4, 9), u = \frac{15}{17} i + 8 / 17 j$

Short Answer

Expert verified

The directional derivative is $= 鈭� \frac{103}{816}$

Step by step solution

Given data

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

The given points is $P = (x_{0}, y_{0}) = (4, 9) and u = (伪, 尾) = (\frac{15}{17} i + \frac{8}{17} j)$

Solution

Directional derivative of function

$鈭� f (P) 鈰� u = 鈭� f (4, 9) 鈰� u = ({\frac{d f}{d x}|}_{(4, 9)} i + {\frac{d f}{d y}|}_{(4, 9)} j) 鈰� (\frac{15}{17} i + \frac{8}{17} j)$

$= [{(\frac{1}{2 \sqrt{\frac{y}{x}}} \frac{d}{d x} (\frac{y}{x}))}_{(4, 9)} i + {(\frac{1}{2 \sqrt{\frac{y}{x}}} \frac{d}{d x} (\frac{y}{x}))}_{(4, 9)} j] 鈰� (\frac{15}{17} i + \frac{8}{17} j)$

$= {[\frac{\frac{鈭� y}{x^{2}}}{2 \sqrt{\frac{y}{x}}})}_{(4, 9)} i + {(\frac{\frac{1}{x}}{2 \sqrt{\frac{y}{x}}})}_{(4, 9)} j 鈰� (\frac{15}{17} i + \frac{8}{17} j)$

$= [(\frac{\frac{鈭� 9}{4^{2}}}{2 \sqrt{\frac{9}{4}}}) i + (\frac{\frac{1}{4}}{2 \sqrt{\frac{9}{4}}}) j] 鈰� (\frac{15}{17} i + \frac{8}{17} j)$

role="math" $= [(\frac{鈭� \frac{9}{16}}{2 鈰� \frac{3}{2}}) i + (\frac{\frac{1}{4}}{2 鈰� \frac{3}{2}}) j] 鈰� (\frac{15}{17} i + \frac{8}{17} j)$

Step 3

$= (鈭� \frac{9}{48} i + \frac{1}{12} j) 鈰� (\frac{15}{17} i + \frac{8}{17} j)$

$= (\frac{鈭� 9 鈰� 15}{48 鈰� 17} + \frac{8}{12 鈰� 17})$

$= (\frac{鈭� 135}{16 脳 3 脳 17} + \frac{2}{3 脳 17})$

$= \frac{1}{3 脳 17} (\frac{鈭� 135}{16} + 2)$

$= \frac{1}{3 脳 17} (\frac{鈭� 103}{16})$

$= 鈭� \frac{103}{816}$

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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.
$f (x, y) = \sqrt{\frac{y}{x}} at P = (4, 9), u = \frac{15}{17} i + 8 / 17 j$

Short Answer

Step by step solution

Given data

Solution

Step 3

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91影视

In Exercises 21鈥�28, find the directional derivative of the givenfunction at the specified point P and in the direction of thegiven unit vector u.f(x,y)=yxatP=(4,9),u=1517i+8/17j

Short Answer

Step by step solution

Given data

Solution

Step 3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Decision Maths

Geometry

Discrete Mathematics

Calculus

Logic and Functions

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.
$f (x, y) = \sqrt{\frac{y}{x}} at P = (4, 9), u = \frac{15}{17} i + 8 / 17 j$