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

Short Answer

Expert verified

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

Step by step solution

Given data

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

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

Solution

Directional derivative of function

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

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

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

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

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

Step 3

$= (- \frac{9}{48} i + \frac{1}{12} j) \cdot (\frac{15}{17} i + \frac{8}{17} j)$

$= (\frac{- 9 \cdot 15}{48 \cdot 17} + \frac{8}{12 \cdot 17})$

$= (\frac{- 135}{16 \times 3 \times 17} + \frac{2}{3 \times 17})$

$= \frac{1}{3 \times 17} (\frac{- 135}{16} + 2)$

$= \frac{1}{3 \times 17} (\frac{- 103}{16})$

$= - \frac{103}{816}$

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

Short Answer

Step by step solution

Given data

Solution

Step 3

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In Exercises 21–28, 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

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