Q. 25 In Exercises 21鈥�28, find the d... [FREE SOLUTION]

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

Short Answer

Expert verified

The directional derivative of the given

function is $鈭� \frac{7}{816} \sqrt{17}$

Step by step solution

Given data

To find the directional derivative function $f (x, y) = \sqrt{\frac{y}{x}}$

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

Solution

the directional derivative of

point P and in the direction of the given unit vector u given by

$鈭� f (P) 鈰� u = 鈭� f (4, 9) 鈰� u$

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

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

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

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

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

$= (鈭� \frac{9}{48} i + \frac{1}{12} j) 鈰� (鈭� \frac{\sqrt{17}}{17} i 鈭� \frac{4 \sqrt{17}}{17} j)$

Step 3

$= (\frac{9 鈰� \sqrt{17}}{48 鈰� 17} 鈭� \frac{4 \sqrt{17}}{12 鈰� 17})$

$= (\frac{9 鈰� \sqrt{17}}{16 鈰� 3 鈰� 17} 鈭� \frac{\sqrt{17}}{3 鈰� 17})$

$= \frac{1}{3 鈰� 17} (\frac{9 鈰� \sqrt{17}}{16} 鈭� \frac{\sqrt{17}}{1})$

$= \frac{1}{3 鈰� 17} (\frac{9 鈰� \sqrt{17} 鈭� 16 \sqrt{17}}{16})$

$鈭� f (P) 鈰� u = 鈭� \frac{7}{816} \sqrt{17}$

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

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=鈭�1717,鈭�41717

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

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Applied Mathematics

Calculus

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