Problem 28 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 27-32, describe geometrically the domain of each of the indicated functions of three variables. $$ f(x, y, z)=\sqrt{x^{2}+y^{2}-z^{2}-9} $$

Step-by-Step Solution

Answer

The domain is the region outside and including a double cone centered at the origin with axis along the z-axis, opening in the direction of the x-y plane.

Step 1: Understand the Domain of a Function

To find the domain of the function, we need to identify the set of all possible input values (combinations of \(x\), \(y\), and \(z\)) for which the function is defined. For the square root to be defined in real numbers, the expression inside must be non-negative.

Step 2: Analyze the Expression Under the Square Root

The function is \( f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9} \). For the square root to be real, the expression inside must satisfy: \[ x^2 + y^2 - z^2 - 9 \geq 0 \] which simplifies to: \[ x^2 + y^2 - z^2 \geq 9 \].

Step 3: Rewriting the Condition

The inequality \(x^2 + y^2 - z^2 \geq 9\) represents a set of points in 3D space. This can be rewritten as: \[ x^2 + y^2 \geq z^2 + 9 \] This inequality implies that any combination of \(x\), \(y\), and \(z\) that satisfied this condition is part of the function's domain.

Step 4: Geometric Interpretation

The expression \(x^2 + y^2 - z^2 = 9\) describes a double cone centered at the origin (\((0,0,0)\)) with its axis along the z-axis. The inequality \(x^2 + y^2 - z^2 \geq 9\) indicates the region outside and on the surface of this cone.

Key Concepts

Geometric Interpretation

When we interpret the domain of the function geometrically, it provides us with a visual understanding of how the variables interact in space. For the function \(f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}\), the inequality \(x^2 + y^2 - z^2 \geq 9\) helps us identify the domain. This expression is a description of three-dimensional space in terms of cones.
In geometry, the equation \(x^2 + y^2 - z^2 = 9\) represents a double cone. These cones are centered at the origin \((0, 0, 0)\) and extend outwards symmetrically along the z-axis.
The inequality \(x^2 + y^2 - z^2 \geq 9\) tells us that the points we are interested in lie on or outside this double cone. Visually, you can imagine this as a space outside this conical surface, indicating where the function is valid. Understanding this geometric interpretation is key to grasping how multivariable functions with inequalities form regions in space.

Inequalities

Inequalities are crucial for determining the domain of functions, especially those involving square roots. Here, the function is \(f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}\). Inequalities help us specify which input values keep the expression under the square root non-negative.
For our function, the inequality \(x^2 + y^2 - z^2 \geq 9\) indicates that the term inside the square root, \(x^2 + y^2 - z^2\), must be at least 9 to keep the values real and non-negative.

• \(x^2 + y^2\) represents the squared distance from the z-axis in the xy-plane.
• \(-z^2\) accounts for the effect of the z variable subtracted from the total.
• The inequality thus geometrically represents a boundary, beyond or including which all points \((x, y, z)\) are in the domain of the function.

Understanding this helps identify which parts of the three-dimensional space satisfy such functions involving square roots.

Three-Dimensional Space

A deep understanding of three-dimensional space is important when dealing with multivariable calculus, especially when visualizing domains. In our example of the function \(f(x, y, z) = \sqrt{x^2 + y^2 - z^2 - 9}\), it demonstrates how a condition describes a spatial region.
The inequality \(x^2 + y^2 - z^2 \geq 9\) identifies a region in this 3D space that forms and extends from a double cone. Understanding this space means visualizing not just lines or flat surfaces, but entire regions that curve and extend through three dimensions.

• The domain is represented by points in space.
• These points represent possible input values for the function where it results in a real number.
• Visualizing such regions is important for anyone working with functions of multiple variables.

Three-dimensional space, when combined with algebraic expressions and inequalities, offers a comprehensive picture of where a multivariable function is defined, guiding real-world applications in fields such as physics and engineering.