Problem 2 - The Calculus with Analytic Geometry

Question

Let the function \(g\) of three variables, \(x, y\), and \(z\), be the set of all ordered pairs of the form \((P, w)\) such that \(w=\) \(\sqrt{4-x^{2}-y^{2}-z^{2}}\). Find: (a) \(g(1,-1,-1) ;\) (b) \(g\left(-a, 2 b, \frac{1}{2} c\right) ;\) (c) \(g(y,-x,-y) ;\) (d) the domain of \(g ;\) (e) the range of \(g\) (f) \([g(x, y, z)]^{2}-[g(x+2, y+2, z)]^{2}\). Draw a sketch showing as a shaded solid in \(R^{3}\) the set of points in the domain of \(g\).

Step-by-Step Solution

Answer

(a) 1. (b) \sqrt{4 - a^2 - 4b^2 - \frac{1}{4}c^2}\. (c) \sqrt{4 - x^2 - 2y^2}\. (d) Sphere of radius 2 centered at the origin. (e) [0, 2]. (f) -4x -4y - 8.

Step 1 - Understand the function

The function given is a function of three variables: \(g(x, y, z) = \sqrt{4 - x^2 - y^2 - z^2}\). This implies that the value of the function is determined by the expression under the square root.

Step 2 - Calculate g(1, -1, -1)

Substitute \(x = 1\), \(y = -1\), and \(z = -1\): \[g(1, -1, -1) = \sqrt{4 - (1)^2 - (-1)^2 - (-1)^2} = \sqrt{4 - 1 - 1 - 1} = \sqrt{1} = 1\].

Step 3 - Calculate g\left(-a, 2b, \frac{1}{2} c\right)

Substitute \(x = -a\), \(y = 2b\), and \(z = \frac{1}{2} c\): \[g\left(-a, 2b, \frac{1}{2}c\right) = \sqrt{4 - (-a)^2 - (2b)^2 - \left(\frac{1}{2}c\right)^2} = \sqrt{4 - a^2 - 4b^2 - \frac{1}{4}c^2}\].

Step 4 - Calculate g(y, -x, -y)

Substitute \(x = -y\), \(y = -x\), and \(z = -y\): \[g(y, -x, -y) = \sqrt{4 - y^2 - (-x)^2 - (-y)^2} = \sqrt{4 - y^2 - x^2 - y^2} = \sqrt{4 - x^2 - 2y^2}\].

Step 5 - Determine the domain of g

The function \(g(x, y, z)\) is defined as long as the expression under the square root is non-negative. Therefore, the domain is the set of points where: \[4 - x^2 - y^2 - z^2 \geq 0 \Rightarrow x^2 + y^2 + z^2 \leq 4\]. This represents a solid sphere of radius 2 centered at the origin.

Step 6 - Determine the range of g

The function \(g(x, y, z)\) outputs real values. Since \( \sqrt{4 - x^2 - y^2 - z^2}\) must be non-negative and the maximum value inside the square root is 4, the range of \(g\) is \[ [0, 2]\].

Step 7 - Calculate \([g(x, y, z)]^2 - [g(x+2, y+2, z)]^2\)

First, express \([g(x, y, z)]^2\) as: \[ g^2(x, y, z) = 4 - x^2 - y^2 - z^2\]. For \(g(x+2, y+2, z)\): \[ g^2(x+2, y+2, z) = 4 - (x+2)^2 - (y+2)^2 - z^2\]. Therefore, the difference is: \[ g^2(x, y, z) - g^2(x + 2, y + 2, z) \ = 4 - x^2 - y^2 - z^2 - (4 - (x + 2)^2 - (y + 2)^2 - z^2) \ = x^2 + y^2 + z^2 - (x^2 + 4x + 4 + y^2 + 4y + 4 + z^2) \ = -4x -4y - 8\].

Key Concepts

Functions of Several Variables

A function of several variables depends on more than one input to determine its output. In this exercise, the function is given as
\[g(x,y,z) = \sqrt{4 - x^2 - y^2 - z^2}\].
Here, the function depends on three variables: \(x\), \(y\), and \(z\). This means that the value of the function will vary depending on the values of these three variables.
The main idea is to understand how changing each input affects the output.
For instance, when you substitute specific values for \(x\), \(y\), and \(z\), you can calculate the exact value of \(g\).
This is shown in the solution steps where different values are substituted.

Domain and Range

To work with functions of several variables, it is crucial to understand their domain and range.

Domain:
The domain of a function is the set of all possible input values. For the function \(g(x,y,z)\), the domain is determined by the condition under the square root must be non-negative. This means:
\[4 - x^2 - y^2 - z^2 \geq 0\]
Simplifying this, we get:
\[x^2 + y^2 + z^2 \leq 4\]
This represents a solid sphere with a radius of 2 centered at the origin in 3D space.

Range:
The range of a function is the set of all possible output values. For \(g(x,y,z)\), the minimum value is 0 when \(x^2 + y^2 + z^2 = 4\), and the maximum is 2 when \(x^2 + y^2 + z^2 = 0\). Therefore, the range of the function is:
[0, 2]
This means \(g(x, y, z)\) will output values within this range for any valid input.

3D Geometry

Understanding 3D geometry helps visualize functions of several variables. Here, the function \(g(x, y, z)\) is connected to the geometric shape of a sphere.
The domain of the function is represented by points inside a sphere of radius 2 centered at the origin. This sphere is given by the inequality:
\[x^2 + y^2 + z^2 \leq 4 \].
The boundary of this domain forms a sphere in 3D space. This visualization makes it easier to understand where the function is defined and how the inputs are spatially distributed.
The solutions often involve visualizing these points and regions in 3D to get a clear understanding of where the function is applicable.
Understanding these geometric interpretations helps tackle complex multivariable problems more effectively.