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Chapter 13: Problem 46

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$p(x, y)=e^{x-y}$$

Short Answer

Expert verified
Answer: The function \(p(x, y)=e^{x-y}\) is continuous at every point \((x, y)\) in its domain, which is \(\mathbb{R}^{2}\).

Step by step solution

01

Break down the function

Given the function \(p(x, y) = e^{x - y}\), we can break it down into its constituent operations: addition, subtraction and exponentiation. The function can be rewritten as $$p(x, y) = e^{f(x, y)}$$ where \(f(x, y) = x - y\).
02

Determine the continuity of the constituent functions

We have the following elementary functions and their corresponding continuity properties: 1. The addition and subtraction functions, \(f(x, y) = x \pm y\), are continuous in \(\mathbb{R}^{2}\). 2. The exponential function, \(g(x) = e^x\), is continuous in \(\mathbb{R}\).
03

Use rules of continuity for composite functions

Now that we know our constituent functions are continuous, we must determine whether the composite function, \(p(x, y)\), is continuous. If a function \(g \circ f(x, y)\) is the composition of two functions \(f(x, y)\) and \(g(u)\), the following must hold for it to be continuous: - \(f\) is continuous at each point \((x, y)\) in its domain. - \(g\) is continuous at each point \(u = f(x, y)\) in its domain. If these conditions are satisfied, then the composite function \(g \circ f(x, y)\) will also be continuous at each point \((x, y)\) in its domain.
04

Analyze continuity based on rules

We know that both constituent functions are continuous individually, so let's check the conditions mentioned in Step 3: 1. The function \(f(x, y) = x - y\) is continuous in \(\mathbb{R}^{2}\). 2. The function \(g(u) = e^u\) is continuous in \(\mathbb{R}\), and since the range of \(f(x, y)\), which is \(\mathbb{R}\), is contained in the domain of \(g(u)\), we can conclude that \(g(f(x, y))\) is continuous. Therefore, the composite function \(p(x, y) = e^{x - y}\) is continuous at every point \((x, y)\) in its domain, which is \(\mathbb{R}^{2}\).

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Most popular questions from this chapter

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$

Consider the ellipse \(x^{2}+4 y^{2}=1\) in the \(x y\) -plane. a. If this ellipse is revolved about the \(x\) -axis, what is the equation of the resulting ellipsoid? b. If this ellipse is revolved about the \(y\) -axis, what is the equation of the resulting ellipsoid?

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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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