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Chapter 12: Q 52. (page 944)

For the partial derivatives $\frac{鈭� f}{鈭� y} = 0$ , find the most general form for a function of two variables $f (x, y)$ , with the given partial derivative.

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Most popular questions from this chapter

In Exercises $37 - 42$ , use the partial derivatives of role="math" localid="1650186853142" $g (x, y) = \tan^{- 1} (x y^{2})$ and the point role="math" localid="1650186870407" $(1, 0)$ 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 $(b)$ .

$Extrema : Find the local maxima, local minima, and saddle points of the given functions f (x, y) = x^{4} + y^{4} + 4 xy$

Given a function of three variables, $w = f (x, y, z),$ and a constraint equation $g (x, y, z) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

Describe the meanings of each of the following mathematical expressions:

$f_{z} (x_{0}, y_{0}, z_{0})$

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Find \frac{鈭� x}{鈭� s} and \frac{鈭� x}{鈭� t} .$

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For the partial derivatives 鈭�f鈭�y=0, find the most general form for a function of two variablesfx,y, with the given partial derivative.

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For the partial derivatives $\frac{鈭� f}{鈭� y} = 0$ , find the most general form for a function of two variables $f (x, y)$ , with the given partial derivative.