Given $ f(x, y) = \frac{x^2 y}{x^2 + y^4} $:a. Find and sketch the domain.b. Find $ \lim_{(x, y) \to (0, 0)} f(x, y) $ or show that it does not exist.c. If $ g(t) = \sqrt{t} $, then find the set of points where $ h(x, y) = g(f(x, y)) $ at which $ h $ is continuous.d. Find the first partial derivatives of $ f $.

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Given $ f(x, y) = \frac{x^2 y}{x^2 + y^4} $:

a. Find and sketch the domain.

b. Find $ \lim_{(x, y) \to (0, 0)} f(x, y) $ or show that it does not exist.

c. If $ g(t) = \sqrt{t} $, then find the set of points where $ h(x, y) = g(f(x, y)) $ at which $ h $ is continuous.

d. Find the first partial derivatives of $ f $.

Expert Solution

Step 1

The function $f (x, y) = \frac{x^{2} y}{x^{2} + y^{4}}$ can be defined only when the denominator is not equal to zero.

For any real values of x and y, the denominator should not be negative. So, the function exists only when the expression in the denominator is greater than zero, that is, $x^{2} + y^{4} > 0$ .

Therefore, the domain is $\{(x, y) \in R^{2} : x^{2} + y^{4} > 0\}$ .

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