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Chapter 12: TF. 4 (page 946)

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Substitute the expressions for x and y into f (x, y) .$

Short Answer

Expert verified

$f (x, y) = s^{5} \cos^{2} t \sin^{3} t$

Step by step solution

01

Step 1. Given information

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t .$

02

Step 2. Substitute the expressions for x and y into f(x, y)

$f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t 鈬� f (x, y) = {(s \cos t)}^{2} 脳 {(s \sin t)}^{3} 鈬� f (x, y) = (s^{2} \cos^{2} t) 脳 (s^{3} \sin^{3} t) 鈬� f (x, y) = s^{5} \cos^{2} t \sin^{3} t$

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Most popular questions from this chapter

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y, z) 鈫� (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}}$

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

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 $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Sketch the level curves f(x, y) = c of the following functions for c = 鈭�3, 鈭�2, 鈭�1, 0, 1, 2, and 3:

$f (x, y) = y^{2} 鈭� x^{2}$

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

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