Chapter 12: Q. 64 (page 954)

$f (x, y) = \ln (x y^{2}), P = (1, - 3)$

Short Answer

Expert verified

The solution's response $f (x, y) = \ln (x y^{2}), P = (1, - 3)$ is $3 x - 2 y - 3 z = 2.4084$

Step by step solution

Given data

The equation of line of segment is

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

$f_{x} (1, - 3) (x - 1) + f_{y} (1, - 3) (y + 3) = z - f (1, - 3)$ $(1)$

Find fx

$f_{x} (x_{0}, y_{0}) = {\frac{d}{d x} f (x, y)|}_{(x_{0}, y_{0})} = {\frac{d}{d x} \ln (x y^{2})|}_{(x_{0}, y_{0})}$

$f_{x} (x_{0}, y_{0}) = {\frac{d}{d x} (\ln (x y^{2})) \frac{d}{d x} (x y^{2})|}_{(x_{0}, y_{0})}$

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

$f_{x} (x_{0}, y_{0}) = {\frac{1}{x}|}_{(x_{0}, y_{0})}$

$f_{x} (1, - 3) = {\frac{1}{1}|}_{(1, - 3)}$

$f_{x} (1, - 3) = 1$ $(2)$

Find fy

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} f (x, y)|}_{(x_{0}, y_{0})} = {\frac{d}{d y} \ln (x y^{2})|}_{(x_{0}, y_{0})}$

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} (\ln (x y^{2})) \frac{d}{d y} (x y^{2})|}_{(x_{0}, y_{0})}$

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

$f_{y} (1, - 3) = {\frac{2}{y}|}_{(1, - 3)}$

$f_{y} (1, - 3) = - \frac{2}{3}$ $(3)$

Find f(1,-3)

$f (x_{0}, y_{0}) = f (1, - 3) = \ln (1 \times (- 3)^{2})$

$f (1, - 3) = \ln (9)$

$f (1, - 3) = 2.1972$ $(4)$

Step 5:Solution

Subsututing Equation $f_{x} (1, - 3) = 1$ , $f_{y} (1, - 3) = - \frac{2}{3}$ and $f (1, - 3) = 2.1972$ in $f_{x} (1, - 3) (x - 1) + f_{y} (1, - 3) (y + 3) = z - f (1, - 3)$ get

$1 (x - 1) - \frac{2}{3} (y + 3) = z - 2.1972$

$x - 1 - \frac{2}{3} y - 2 - z = - 2.1972$

$x - \frac{2}{3} y - z = - 2.1972 + 1 + 2$

$x - \frac{2}{3} y - z = 0.8028$

multiple $3$ on both side ,get

$3 x - 2 y - 3 z = 2.4084$

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$f (x, y) = \ln (x y^{2}), P = (1, - 3)$

Short Answer

Step by step solution

Given data

Find fx

Find fy

Find f(1,-3)

Step 5:Solution

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f(x,y)=ln⁡xy2,P=(1,−3)

Short Answer

Step by step solution

Given data

Find fx

Find fy

Find f(1,-3)

Step 5:Solution

One App. One Place for Learning.

Most popular questions from this chapter

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$f (x, y) = \ln (x y^{2}), P = (1, - 3)$