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Chapter 13: Problem 1

A function is defined by \(z=x^{2} y-x y^{2} .\) Identify the independent and dependent variables.

Short Answer

Expert verified
Answer: In the given function, \(x\) and \(y\) are independent variables, while \(z\) is the dependent variable.

Step by step solution

01

Identify the Variables

The function given is \(z = x^2y - xy^2\). The variables involved are \(x\), \(y\), and \(z\). Now we need to determine their roles in the function.
02

Identify the Inputs and Output

In a function, the independent variables are typically considered the inputs, while the dependent variable is considered the output. In this case, we find that: 1. \(x\) and \(y\) can both be changed (or manipulated) without affecting the other. Thus, they are both independent variables. 2. The value of \(z\) is based on the values of \(x\) and \(y\). It depends on them, making it the dependent variable.
03

State the Independent and Dependent Variables

After analyzing the function \(z = x^2y - xy^2\), we have found that the independent variables are \(x\) and \(y\), while the dependent variable is \(z\).

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Most popular questions from this chapter

Consider the curve \(\mathbf{r}(t)=\langle\cos t, \sin t, c \sin t\rangle,\) for \(0 \leq t \leq 2 \pi,\) where \(c\) is a real number. a. What is the equation of the plane \(P\) in which the curve lies? b. What is the angle between \(P\) and the \(x y\) -plane? c. Prove that the curve is an ellipse in \(P\).

Match equations a-f with surfaces A-F. a. \(y-z^{2}=0\) b. \(2 x+3 y-z=5\) c. \(4 x^{2}+\frac{y^{2}}{9}+z^{2}=1\) d. \(x^{2}+\frac{y^{2}}{9}-z^{2}=1\) e. \(x^{2}+\frac{y^{2}}{9}=z^{2}\) f. \(y=|x|\)

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2}}{x^{2}+y^{2}}$$

Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid \(36 x^{2}+4 y^{2}+9 z^{2}=36\).

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(0,0)} \frac{|x-y|}{|x+y|}$$
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