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Chapter 13: Problem 9

At what points of \(\mathbb{R}^{2}\) is a rational function of two variables continuous?

Short Answer

Expert verified
Answer: A rational function of two variables is continuous at every point $(x, y) \in \mathbb{R}^2$ such that the denominator, $q(x, y) \neq 0$.

Step by step solution

01

Define a Rational Function

A rational function of two variables can be written in the form: $$f(x,y) = \frac{p(x,y)}{q(x,y)}$$ where \(p(x,y)\) and \(q(x,y)\) are both polynomial functions in the variables \(x\) and \(y\).
02

Determine When the Denominator is Not Zero

A rational function is undefined when the denominator, \(q(x,y)\), is equal to zero. So, we need to find where \(q(x,y) \neq 0\). In general, this will require finding the roots of the polynomial equation \(q(x,y)=0\). However, we cannot provide an explicit solution for the roots of the given polynomial, as we do not have details on the specific polynomial. In general, finding the roots of a two-variable polynomial can be a complex task, and finding a general form for roots may not be possible.
03

Rational Function Continuity

A rational function is continuous at any point where it is defined and has a limit. Since it is undefined when \(q(x, y) = 0\), we can say that the function is continuous everywhere in \(\mathbb{R}^{2}\) except where the denominator is equal to zero. So, the rational function \(f(x,y)\) is continuous at every point \((x, y) \in \mathbb{R}^2\) such that \(q(x, y) \neq 0\).

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Most popular questions from this chapter

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(-1,0)} \frac{x y e^{-y}}{x^{2}+y^{2}}$$

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

Absolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=x^{2}-4 y^{2}+x y ; R=\left\\{(x, y): 4 x^{2}+9 y^{2} \leq 36\right\\}$$

Problems with two constraints Given a differentiable function \(w=f(x, y, z),\) the goal is to find its maximum and minimum values subject to the constraints \(g(x, y, z)=0\) and \(h(x, y, z)=0\) where \(g\) and \(h\) are also differentiable. a. Imagine a level surface of the function \(f\) and the constraint surfaces \(g(x, y, z)=0\) and \(h(x, y, z)=0 .\) Note that \(g\) and \(h\) intersect (in general) in a curve \(C\) on which maximum and minimum values of \(f\) must be found. Explain why \(\nabla g\) and \(\nabla h\) are orthogonal to their respective surfaces. b. Explain why \(\nabla f\) lies in the plane formed by \(\nabla g\) and \(\nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value. c. Explain why part (b) implies that \(\nabla f=\lambda \nabla g+\mu \nabla h\) at a point of \(C\) where \(f\) has a maximum or minimum value, where \(\lambda\) and \(\mu\) (the Lagrange multipliers) are real numbers. d. Conclude from part (c) that the equations that must be solved for maximum or minimum values of \(f\) subject to two constraints are \(\nabla f=\lambda \nabla g+\mu \nabla h, g(x, y, z)=0,\) and \(h(x, y, z)=0\)

Suppose \(n\) houses are located at the distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right) .\) A power substation must be located at a point such that the sum of the squares of the distances between the houses and the substation is minimized. a. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at \((0,0),(2,0),\) and (1,1) b. Find the optimal location of the substation in the case that \(n=3\) and the houses are located at distinct points \(\left(x_{1}, y_{1}\right)\) \(\left(x_{2}, y_{2}\right),\) and \(\left(x_{3}, y_{3}\right)\) c. Find the optimal location of the substation in the general case of \(n\) houses located at distinct points \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots\) \(\left(x_{n}, y_{n}\right)\) d. You might argue that the locations found in parts (a), (b), and (c) are not optimal because they result from minimizing the sum of the squares of the distances, not the sum of the distances themselves. Use the locations in part (a) and write the function that gives the sum of the distances. Note that minimizing this function is much more difficult than in part (a). Then use a graphing utility to determine whether the optimal location is the same in the two cases. (Also see Exercise 75 about Steiner's problem.)
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