Chapter 12: Q. 63 (page 954)

$f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$

Short Answer

Expert verified

The equation's solution $f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$ is $9 x - y + 6 z = 18$

Step by step solution

Step 1:Given data

$f (x, y) = \sqrt{\frac{y}{x}} at point P = (x_{0}, y_{0}) = (1, 9)$

The line of tangent equation is

$f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0}) = z - f (x_{0}, y_{0})$

$f_{x} (1, 9) (x - 1) + f_{y} (1, 9) (y - 9) = z - f (1, 9)$ $(1)$

Step 2:Find fx

$f_{x} (x_{0}, y_{0}) = \frac{d}{d x} (\sqrt{\frac{y}{x}}) \frac{d}{d x} {(\frac{y}{x})}_{(x_{0}, y_{0})}$

$f_{x} (x_{0}, y_{0}) = {\frac{1}{2 \sqrt{\frac{y}{x}}} (- \frac{y}{x^{2}})|}_{(x_{0}, y_{0})}$

$f_{x} (1, 9) = {\frac{1}{2 \sqrt{\frac{y}{x}}} (- \frac{y}{x^{2}})|}_{(1, 9)}$

$f_{x} (1, 9) = \frac{1}{2 \sqrt{\frac{9}{1}}} (- \frac{9}{1^{2}})$

$f_{x} (1, 9) = - \frac{9}{6}$ $(2)$

Step 3:Find fy

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} f (x, y)|}_{(x_{0}, y_{0})} = {\frac{d}{d y} \sqrt{\frac{y}{x}}|}_{(x_{0}, y_{0})}$

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} (\sqrt{\frac{y}{x}}) \frac{d}{d y} (\frac{y}{x})|}_{(x_{0}, y_{0})}$ $f_{y} (x_{0}, y_{0}) = {\frac{1}{2 \sqrt{\frac{y}{x}}} \cdot \frac{1}{x}|}_{(x_{0}, y_{0})}$

localid="1650515198116" $f_{y} (1, 9) = \frac{1}{{2 x \sqrt{\frac{y}{x}}|}_{(1, 9)}}$

localid="1650515201090" $f_{y} (1, 9) = \frac{1}{6}$ localid="1650515204300" $(3)$

Findf(1,9)

$f (x_{0}, y_{0}) = f (1, 9) = \sqrt{\frac{9}{1}}$

$f (1, 9) = 3$ $(4)$

Step 5:Solution

Substituting equation $f_{x} (1, 9) = - \frac{9}{6}$ , $f_{y} (1, 9) = \frac{1}{6}$ and $f (1, 9) = 3$ in $f_{x} (1, 9) (x - 1) + f_{y} (1, 9) (y - 9) = z - f (1, 9)$ get

$- \frac{9}{6} (x - 1) + \frac{1}{6} (y - 9) = z - 3$

$- \frac{9}{6} x + \frac{9}{6} + \frac{1}{6} y - \frac{9}{6} = z - 3$

$- \frac{9}{6} x + \frac{1}{6} y - z = - 3$

Multiple $6$ on both side

$\begin{aligned} - 9 x + y - 6 z = - 18 \\ 9 x - y + 6 z = 18 \end{aligned}$

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$f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$

Short Answer

Step by step solution

Step 1:Given data

Step 2:Find fx

Step 3:Find fy

Findf(1,9)

Step 5:Solution

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f(x,y)=yx,P=(1,9)

Short Answer

Step by step solution

Step 1:Given data

Step 2:Find fx

Step 3:Find fy

Findf(1,9)

Step 5:Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Applied Mathematics

Pure Maths

Discrete Mathematics

Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

$f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$