Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe geometrically the level surfaces for the functions defined in Problems 33-38. $$ f(x, y, z)=100 x^{2}+16 y^{2}+25 z^{2} ; k>0 $$

Step-by-Step Solution

Answer

The level surfaces of the function are ellipsoids centered at the origin with axes lengths depending on \( k \).

Step 1: Identify the Form of the Function

The given function is \( f(x, y, z) = 100x^2 + 16y^2 + 25z^2 \). This resembles the standard form of an ellipsoid equation, \( Ax^2 + By^2 + Cz^2 = D \), where \( A = 100 \), \( B = 16 \), \( C = 25 \), and \( D = k > 0 \).

Step 2: Analyze the Level Surface Equation

The level surfaces are given by setting \( f(x, y, z) = k \). This results in the equation \( 100x^2 + 16y^2 + 25z^2 = k \), which is the equation of an ellipsoid centered at the origin (0,0,0).

Step 3: Simplify to Match Standard Ellipsoid Form

Divide the entire equation by \( k \) to get the standard form: \( \frac{x^2}{k/100} + \frac{y^2}{k/16} + \frac{z^2}{k/25} = 1 \). This confirms it's an ellipsoid with semi-principal axes lengths dependent on the value of \( k \).

Step 4: Determine the Semi-Principal Axes

Identify the semi-principal axes from the equation: they are \( \sqrt{\frac{k}{100}} \), \( \sqrt{\frac{k}{16}} \), and \( \sqrt{\frac{k}{25}} \) for the x, y, and z axes, respectively. These represent the radii of the ellipsoidal level surfaces.

Key Concepts

Level Surfaces

A level surface is a set of points \((x, y, z)\) in three-dimensional space where a function \(f(x, y, z)\) of three variables remains constant. For example, consider the function \(f(x, y, z) = 100x^2 + 16y^2 + 25z^2\).
Setting this function equal to a constant \(k\) results in the equation of a level surface: \(100x^2 + 16y^2 + 25z^2 = k\). Level surfaces are crucial in visualizing how a function behaves in three dimensions.

• When the function is set equal to different constants \(k\), different level surfaces are formed.
• In our example, each level surface represents an ellipsoid centered at the origin.

Thus, level surfaces offer a way to break down a complex 3D function into understandable slices.

Function of Three Variables

In calculus, a function of three variables \(f(x, y, z)\) assigns a real number to every point \((x, y, z)\) in three-dimensional space. Much like functions of a single variable trace a line or curve, functions of three variables define surfaces or solids in space.
The function \(f(x, y, z) = 100x^2 + 16y^2 + 25z^2\) is an example of such a function. Here:

• Each variable \(x\), \(y\), and \(z\) contributes differently to the function due to their respective coefficients \(100\), \(16\), and \(25\).
• The result is a complex 3D shape, in this case, ellipsoids for different values of \(k\).

These functions are particularly useful in physics and engineering, where multiple factors influence a given outcome.

Geometric Interpretation

The geometric interpretation of level surfaces often simplifies understanding a complex function of three variables. By visualizing the equation of level surfaces, you can understand the shape and orientation of surfaces or solids in space.
For our function \(100x^2 + 16y^2 + 25z^2 = k\), the level surfaces are ellipsoids.

• These ellipsoids are centered at the origin \((0, 0, 0)\) since there are no linear terms in \(x\), \(y\), or \(z\).
• They stretch differently along each axis, controlled by the coefficients \(100\), \(16\), and \(25\).

Delving deeper, you can see how changes in \(k\) affect the size of each ellipsoid, making the concept of level surfaces more tangible.

Semi-Principal Axes

Semi-principal axes are significant characteristics of ellipsoids, determining the shape and size along each dimension. When you deal with the equation of an ellipsoid in standard form, the coefficients divided by the constant \(k\) relate to the semi-principal axes.
Specifically, for the ellipsoid defined by the function \(100x^2 + 16y^2 + 25z^2 = k\), you find the semi-principal axes as:

• \(\sqrt{\frac{k}{100}}\) along the \(x\)-axis.
• \(\sqrt{\frac{k}{16}}\) along the \(y\)-axis.
• \(\sqrt{\frac{k}{25}}\) along the \(z\)-axis.

These values indicate the radii or lengths the ellipsoid extends from the center along each axis, illustrating how each axis stretches differently.
Understanding semi-principal axes helps students to visualize how three-dimensional shapes vary with changes in parameters.