Problem 2
Question
What is the domain of \(f(x, y)=x^{2} y-x y^{2} ?\)
Step-by-Step Solution
Verified Answer
Answer: The domain of the function is all real numbers for both x and y, which can be represented in set notation as \(\{(x, y) \in \mathbb{R}^2\}\).
1Step 1: Identify Undefined Operations
In the given function \(f(x, y) = x^2y - xy^2\), there are no square roots, logarithms, or fractions with variables in the denominator. That means the function is defined for all real values of x and y.
2Step 2: Determine the Domain
Since there are no undefined operations in the function, the domain of the function \(f(x, y)\) is all real numbers for both x and y.
3Step 3: Write the Domain in Set Notation
In set notation, the domain of the function \(f(x, y)\) can be written as:
$$
\text{Domain} = \{(x, y) \in \mathbb{R}^2\}
$$
Hence, the domain of \(f(x, y) = x^2y - xy^2\) is all real numbers for both x and y.
Other exercises in this chapter
Problem 2
Find a vector normal to the plane \(-2 x-3 y+4 z=12\)
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Let \(z\) be a function of \(x\) and \(y,\) while \(x\) and \(y\) are functions of \(t\). Explain how to find \(\frac{d z}{d t}\).
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Find \(f_{x}\) and \(f_{y}\) when \(f(x, y)=3 x^{2} y+x y^{3}.\)
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If \(f(x, y, z)=x^{2}+y^{2}+z^{2}\) and \(g(x, y, z)=\) \(2 x+3 y-5 z+4=0,\) write the Lagrange multiplier conditions that must be satisfied by a point that max
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