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Chapter 13: Problem 38

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\frac{x^{2}+y^{2}}{x\left(y^{2}-1\right)}$$

Short Answer

Expert verified
Answer: The function \(f(x, y)=\frac{x^{2}+y^{2}}{x\left(y^{2}-1\right)}\) is continuous for all points in \(\mathbb{R}^2\), except for the lines x = 0, y = 1, and y = -1.

Step by step solution

01

Determine the discontinuity points

To find the discontinuous points, let's set the denominator of the given function to zero. $$x\left(y^{2}-1\right) = 0$$ Now, we can find the possible values of x and y for which this equation holds true.
02

Find the possible values of x and y

To find the values of x and y, we can consider two cases: when x = 0, and when y虏 - 1 = 0. Case 1: When x = 0 The function is discontinuous along the line x = 0, regardless of the value of y. Case 2: When y虏 - 1 = 0 $$y^2 = 1$$ $$y = 卤\sqrt{1}$$ We have two values, y = 1, and y = -1. The function is discontinuous along the lines y = 1 and y = -1, regardless of the value of x.
03

Identify the continuous points

Now we have found the functions discontinuous points, which are along the lines x = 0, y = 1, and y = -1. Since these are only lines, we can see that the function will be continuous for all other points in \(\mathbb{R}^{2}\), excluding these lines. In conclusion, the function \(f(x, y)=\frac{x^{2}+y^{2}}{x\left(y^{2}-1\right)}\) is continuous for all points in \(\mathbb{R}^2\), except for the lines x = 0, y = 1, and y = -1.

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Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(B\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3}\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\). d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\). e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0)
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