Problem 35 - Calculus 3

Question

Find the domain of the following functions. $$ f(x, y, z)=\frac{1}{\sqrt{36-4 x^{2}-9 y^{2}-z^{2}}} $$

Step-by-Step Solution

Answer

The domain is \( 4x^2 + 9y^2 + z^2 < 36 \).

Step 1: Identify Restrictions

To find the domain of the function, recognize that the denominator cannot be zero and that the expression under the square root must be non-negative. This is because we cannot divide by zero and the square root of a negative number is not defined in the real number system.

Step 2: Set Up Inequality

The function is defined for values where the square root expression is greater than zero: \[ 36 - 4x^2 - 9y^2 - z^2 > 0 \] because the square root needs to be positive for the reciprocal function \( f(x, y, z) \) to be defined.

Step 3: Solve the Inequality

Rearrange the inequality to:\[ 4x^2 + 9y^2 + z^2 < 36 \]This condition represents an ellipsoid in the 3D space with semi-principal axes 3 along the x-direction, 2 along the y-direction, and 6 along the z-direction.

Key Concepts

Domain of a function

The domain of a function refers to all the possible input values (like x, y, z in our case) for which the function is defined and gives a valid output. It’s crucial because it sets the boundaries for using the function correctly. For the function \(f(x, y, z) = \frac{1}{\sqrt{36-4x^{2}-9y^{2}-z^{2}}}\), we need to ensure two things for it to work:

• The denominator (\(\sqrt{36-4x^{2}-9y^{2}-z^{2}}\)) cannot be zero - as division by zero is undefined.
• The expression inside the square root must be positive because the square root of negative numbers isn't real.

This results in needing the inequality \(36-4x^{2}-9y^{2}-z^{2} > 0\) to hold true. Thus, our domain consists of all \((x, y, z)\) that satisfy this inequality.

Inequality constraints

Inequality constraints describe the limitations or conditions placed on variables within a function or equation. In our context, they define where the function is actually allowed to "work" based on mathematical rules. Here, the constraint comes from the need for a non-zero and positive square root in the denominator:

• We set up the inequality \(36 - 4x^2 - 9y^2 - z^2 > 0\).
• This simplifies to \(4x^2 + 9y^2 + z^2 < 36\).

By solving this inequality, we identify which combinations of \((x, y, z)\) values make the function valid and operational. These values will form a 3D space wherein each set of coordinates does not violate the constraints, meaning we avoid zero or negative square roots.

Ellipsoid

An ellipsoid is a 3D shape, appearing much like a stretched or squashed sphere. Each axis of the ellipsoid stretches in a specific direction, creating distinct dimensions for the shape. It is defined by an equation similar to that which describes an ellipse, but in three dimensions. For our function's domain, the inequality \(4x^2 + 9y^2 + z^2 < 36\) forms an ellipsoid.

• This ellipsoid has semi-principal axes: 3 along the x-axis, 2 along the y-axis, and 6 along the z-axis.
• The length of these axes is derived from rearranging the inequality to its standard form, allowing us to interpret the limits in 3D space.

Being inside this ellipsoid essentially means staying within the bounds defined by these constraints, which ensures that our function remains valid and executable without hitting undefined values.