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Chapter 12: Problem 50

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). $$Q(x, y, z)=\frac{10}{1+x^{2}+y^{2}+4 z^{2}}$$

Short Answer

Expert verified
Answer: The domain of Q(x, y, z) is all points (x, y, z) in the 3D space, represented as {(x, y, z) 鈭� 鈩澛硙.

Step by step solution

01

Check the denominator function

Observe that the denominator is a quadratic function of the form: $$1+x^{2}+y^{2}+4z^{2}$$ We need to find all points (x, y, z) that make this function non-zero to ensure Q(x, y, z) is defined.
02

Analyze the quadratic function

We can see that the function is always positive since the coefficients of \(x^{2}\), \(y^{2}\), and \(4z^{2}\) are all positive. Therefore, the only point of interest is when the quadratic function equals zero: $$1+x^{2}+y^{2}+4z^{2} = 0$$
03

Solve the equation for the critical points

Rearrange the equation: $$x^{2}+y^{2}+4z^{2} = -1$$ This equation has no real solutions since the sum of squares cannot be negative.
04

Determine the domain of the function

Since the denominator function is always positive and has no real solutions when it is equal to zero, the domain of Q(x, y, z) includes all points (x, y, z) in the 3D space. The function Q(x, y, z) is defined for every value of x, y, and z. Therefore, the domain of the function is: $$\{(x, y, z) \in \mathbb{R}^{3}\}$$

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Most popular questions from this chapter

Identify and briefly describe the surfaces defined by the following equations. $$y=4 z^{2}-x^{2}$$

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 \(h(x, y, z)=0\).

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