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Chapter 12: Q. 52 (page 954)

Find all points where the first-order partial derivatives of the functions in Exercises 43鈥�54 are continuous. Then use Theorems 12.28 and 12.31 to determine the sets in which the functions are differentiable $f (x, y) = \ln (x y^{2})$ .

Short Answer

Expert verified

The answer for the equation $f (x, y) = \ln (x y^{2})$ is $3 x - 2 y - 3 z = 2.4084$ .

Step by step solution

01

Given.

Given $f (x, y) = \ln (x y^{2})$ at point $P = (x_{0}, y_{0}) = (1, - 3)$

The equation of line of tangent is

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

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

02

Considering.

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

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

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

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

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

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

03

Equation 3.

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

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

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

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

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

04

Equation 4.

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

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

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

05

Substituting equation(2),(3) and (4)in (1)to get.

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

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

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

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

Multiply by 3 on both sides, get

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

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Most popular questions from this chapter

Given a function of n variables, $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$ and a constraint equation, $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

Explain the steps you would take to find the extrema of a function of two variables, $f (x, y), if (x, y)$ is a point in the rectangle $R$ defined by role="math" localid="1649881836115" $a 鈮� x 鈮� b and c 鈮� y 鈮� d .$

In Exercises $37 - 42$ , use the partial derivatives of role="math" localid="1650186824938" $f (x, y) = x^{y}$ and the point $(e, 3)$ specified to

$(a)$ find the equation of the line tangent to the surface defined by the function in the $x$ direction,

$(b)$ find the equation of the line tangent to the surface defined by the function in the $y$ direction, and

$(c)$ find the equation of the plane containing the lines you found in parts $(a)$ and $(b)$ .

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.

$\frac{鈭� z}{鈭� s} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Solve the exact differential equations in Exercises 63鈥�66. $e^{y} + (x e^{y} 鈭� 7) \frac{d y}{d x} = 0$

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