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Chapter 13: Problem 42

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

Short Answer

Expert verified
Answer: The function \(f(x,y) = e^{x^2+y^2}\) is continuous at every point (x, y) in the 2-dimensional real plane \(\mathbb{R}^2\).

Step by step solution

01

Recall the definition of continuity

A function is continuous at a point (a, b) if the following conditions are met: 1. The function is defined at (a, b). 2. The limit of the function as (x, y) approaches (a, b) exists. 3. The limit of the function as (x, y) approaches (a, b) equals the value of the function at (a, b). Mathematically, this can be written as: $$\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$ We need to check whether the given function \(f(x, y) = e^{x^2+y^2}\) satisfies these conditions for every point (x, y) in \(\mathbb{R}^2\).
02

Check if the function is defined at all points in \(\mathbb{R}^2\)

The function we are given, \(f(x, y) = e^{x^2+y^2}\), is an exponential function. Exponential functions are always defined for any real input, and in this case, the input consists of x and y. Therefore, the function is indeed defined at all points in \(\mathbb{R}^2\).
03

Check if the limit of the function exists

In order to check if the limit exists, we can look at the function terms \(x^2+y^2\). Since both of the terms are squared, their values will always be non-negative. Therefore, the input to the exponential function (x^2+y^2) will range from \(0\) to positive infinity. The exponential function is continuous and defined for all real input values, so as long as the input \((x^2 + y^2)\) is continuous, the output \(e^{x^2+y^2}\) will also be continuous. The limit of the function as (x, y) approaches (a, b) is guaranteed to exist, since the exponential function behaves nicely for all real input values.
04

Show that the limit equals the function value

From Step 3, we know that the limit of the function as (x, y) approaches (a, b) exists. We have: $$\lim_{(x, y) \to (a, b)} e^{x^2+y^2} = e^{a^2+b^2}$$ The right-hand side of the equation above is exactly the value of the function at the point (a, b), which is \(f(a, b)\). Therefore, the limit of the function as (x, y) approaches (a, b) equals the value of the function at (a, b).
05

Conclude the function continuity

Since the function \(f(x, y) = e^{x^2+y^2}\) is: 1. Defined at all points in \(\mathbb{R}^2\) 2. The limit of the function as (x, y) approaches (a, b) exists 3. The limit of the function as (x, y) approaches (a, b) equals the value of the function at (a, b) we can conclude that the function is continuous at every point (x, y) in the 2-dimensional real plane \(\mathbb{R}^2\).

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Most popular questions from this chapter

Describe the set of all points at which all three planes \(x+2 y+2 z=3, y+4 z=6,\) and \(x+2 y+8 z=9\) intersect.

Maximizing a sum Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).

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.)

a. Show that the point in the plane \(a x+b y+c z=d\) nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right),\) where \(D^{2}=a^{2}+b^{2}+c^{2} .\) Conclude that the least distance from the plane to the origin is \(|d| / D\). (Hint: The least distance is along a normal to the plane.) b. Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=d\) is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\).

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$g(x, y, z)=\frac{10}{x^{2}-(y+z) x+y z}.$$
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