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Chapter 12: Problem 48

Find the domain of the following functions. If possible, give a description of the domains (for example, all points outside a sphere of radius 1 centered at the origin). $$p(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}-9}$$

Short Answer

Expert verified
Answer: The domain of the function is the set of all points outside a sphere of radius 3 centered at the origin, including those on the surface of the sphere.

Step by step solution

01

Identify the condition for the function to be defined

In this case, the function will be defined if the expression under the square root is non-negative, which means: $$x^{2} + y^{2} + z^{2} - 9 \geq 0$$
02

Rearrange the inequality

Rearrange the inequality to better understand the geometric interpretation of the domain: $$x^{2} + y^{2} + z^{2} \geq 9$$
03

Geometric interpretation

The equation \(x^{2} + y^{2} + z^{2} = 9\) represents the equation of a sphere centered at the origin with radius 3. Given the inequality \(x^{2} + y^{2} + z^{2} \geq 9\), the domain of the function consists of the points that are outside or on the surface of the sphere. So, the domain of the function is the set of all points outside the sphere of radius 3 centered at the origin (including those on the surface of the sphere).
04

Write the final answer

The domain of the function \(p(x, y, z) = \sqrt{x^{2} + y^{2} + z^{2} - 9}\) is the set of all points outside a sphere of radius 3 centered at the origin, including those on the surface of the sphere.

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Most popular questions from this chapter

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$

Find an equation of the line passing through \(P_{0}\) and normal to the plane \(P\). $$P_{0}(2,1,3) ; P: 2 x-4 y+z=10$$

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 \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\) (Hint: Find the point \(P\) on the plane closest to \(P_{0}\).)

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).

Consider the following functions \(f\). a. Is \(f\) continuous at (0,0)\(?\) b. Is \(f\) differentiable at (0,0)\(?\) c. If possible, evaluate \(f_{x}(0,0)\) and \(f_{y}(0,0)\). d. Determine whether \(f_{x}\) and \(f_{y}\) are continuous at (0,0). e. Explain why Theorems 12.5 and 12.6 are consistent with the results in parts \((a)-(d)\). $$f(x, y)=\sqrt{|x y|}$$
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