Problem 34 - Calculus with Concepts in Calculus

Question

Sketch the level surface \(f(x, y, z)=c\). \(f(x, y, z)=x^{2}+y^{2}+z^{2} ; c=2\)

Step-by-Step Solution

Answer

The level surface is a sphere centered at the origin with radius \(\sqrt{2}\).

Step 1: Understanding the Level Surface Equation

The equation given is a level surface defined by the function \(f(x, y, z) = x^2 + y^2 + z^2\). A level surface is a set of points where the function has the same constant value, which in this case is \(c = 2\). This means \(x^2 + y^2 + z^2 = 2\).

Step 2: Identifying the Shape of the Level Surface

The equation \(x^2 + y^2 + z^2 = 2\) is a representation of a sphere centered at the origin with a radius \(r\). The general form of a sphere is \(x^2 + y^2 + z^2 = r^2\). By comparing, we identify \(r^2 = 2\), thus \(r = \sqrt{2}\).

Step 3: Sketching the Level Surface

To sketch the level surface, draw a sphere centered at the origin (0, 0, 0). The radius of this sphere should be \(\sqrt{2}\). Visualize it in three dimensions where every point \( (x, y, z) \) on the surface satisfies \( x^2 + y^2 + z^2 = 2 \).

Step 4: Analyzing the Dimensions

Since \(r = \sqrt{2} \approx 1.41\), the diameter of the sphere is approximately 2.82. Therefore, the sphere will intersect each axis at points approximately \( \pm 1.41 \), helping to guide the accuracy of the sketch.

Key Concepts

Understanding the Sphere Equation

The equation of a sphere \[ x^2 + y^2 + z^2 = r^2 \]is fundamental in three-dimensional geometry. It describes a symmetrical object in space comprised of all points equidistant from a central point, known as the center of the sphere. In our exercise, we recognize this form as \[ x^2 + y^2 + z^2 = 2 \].

• The equation is centered at the origin, which is \((0, 0, 0)\).
• The right-hand side of the equation, here 2, gives us the square of the radius, \( r^2 = 2 \).

It is important to note that variations in values on the right side of the equation alter the size of the sphere but not its center.

Three-Dimensional Graphing of the Sphere

Graphing a sphere like \( x^2 + y^2 + z^2 = 2 \) in three dimensions involves visualizing the symmetry about the origin. Each point \((x, y, z)\) that satisfies this equation lies exactly on the surface of the sphere.
To sketch or graph it:

• Imagine the sphere centered at the origin.
• The distance from the center to any point on the surface is the radius, \( \sqrt{2} \).

This sphere will symmetrically intersect the \(x\), \(y\), and\(z\) axes around the origin. Helping to visualize involves understanding that the sphere is like a uniform bubble in space, centered at the origin, touching each coordinate axis at plus/minus \( \sqrt{2} \approx 1.41 \).

Calculating the Radius of the Sphere

Calculating the radius of a sphere from its equation involves a straightforward application of the formula for a sphere's equation. Given \[ x^2 + y^2 + z^2 = r^2 \],we find the radius \( r \) of a sphere by taking the square root of the constant term, \( r^2 \).In the example \( x^2 + y^2 + z^2 = 2 \):

• Identify \( r^2 = 2 \), then solve for \( r \).
• The radius \( r = \sqrt{2} \), approximately valued at 1.41.

Understanding this ensures accurate measurement for modeling or graphical representation of spheres in mathematics and real-world contexts.