Question

In Exercises \(19-26,\) use a computer algebra system to graph the surface. (Hint: It may be necessary to solve for \(z\) and acquire two equations to graph the surface.) $$ 4 x^{2}-y^{2}+4 z^{2}=-16 $$

Short answer

The equation represents a 3D surface shaped like a cylinder. The equations for graphing are \(y=2*\sqrt{x^{2}+z^{2}+4}\) and \(y=-2*\sqrt{x^{2}+z^{2}+4}\).

Step-by-step solution

Rewrite Equation in Standard Form

Rewrite the equation \(4 x^{2}-y^{2}+4 z^{2}=-16\) in a more standard form: \(4 x^{2}+4 z^{2}=y^{2}-16\). Divide the equation by 4 in order to simplify it to \(x^{2}+z^{2}=(1/4)*y^{2}-4\). This is now in the form of \(x^{2}+z^{2}=k\), resembling an equation of a cylinder.

Preparing for Graphing

To graph this 3D surface, solve it for y to get two separate equations: \(y=2*\sqrt{x^{2}+z^{2}+4}\) and \(y=-2*\sqrt{x^{2}+z^{2}+4}\). The two equations are needed because square root function is not defined for negative values.

Graphing the Surface

Use a computer algebra system to graph the two equations obtained from step 2. These two will combine to form the 3D surface represented by the given equation.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is an invaluable tool for modern students tackling complex mathematical problems, especially those involving calculus and 3D graphing. A CAS allows you to perform algebraic operations symbolically, which is particularly helpful when dealing with equations that are daunting to solve by hand.

Imagine trying to manually graph a complicated surface equation; it is not only time-consuming but also prone to errors. CASs make this easy by providing commands for 3D plotting, simplifying expressions, and even solving equations symbolically. For example, entering the equation from our exercise, you can promptly visualize the 3D surface without performing tedious algebraic manipulations. Furthermore, CASs are capable of handling both numerical and symbolic computations, which means it can graph an abstract function or evaluate an expression at specific points.

For students, a CAS can be a learning aid that helps to check solutions to complex problems or to visualize concepts that are difficult to grasp in a purely abstract form. For instance, understanding the shape and the orientation of the 3D surface is much easier when you can see a visual representation.

3D Surface Graphing

Graphing in three dimensions is a powerful way to visualize mathematical functions and surfaces. While 2D graphing deals with functions of a single variable, 3D surface graphing involves functions of two variables, representing surfaces within three-dimensional space. To graph a 3D surface, you typically need to understand the relationship between the three axes, X, Y, and Z.

The surface given by the equation in our exercise can be graphed on a computer using a CAS. When graphing, it's essential to recognize the domain and the range of the function and to interpret the roles that the X, Y, and Z variables play. In the given equation after rewriting, you can read the relationship between X and Z that forms an infinite cylinder. By using a CAS to graph these surfaces, you can rotate the graph and see the structure from different angles, allowing for a deeper comprehension of its geometric properties.

Cylindrical Equations

In calculus, cylindrical equations refer to a class of equations that describe shapes with circular cross sections and symmetry about an axis. These shapes are three-dimensional extensions of two-dimensional circles. The general form of a cylinder's equation is given by the variables x, y, and z. For example, the equation of a right circular cylinder, whose axis coincides with the y-axis, can be written as:

\( x^2 + z^2 = r^2 \).

It defines a surface where any point is at a constant distance r from the y-axis. When provided with an equation like the one in our exercise, you can decipher the shape of a surface by identifying the symmetrical properties. After you express the original equation in a standard form that resembles an equation of a cylinder, you unearth the symmetry along the Y-axis, characterized by the cylinder's radius depending on the Y coordinate. Understanding the dynamics of cylindrical equations equips you with the knowledge to tackle a plethora of real-world applications, ranging from physics to engineering, where cylindrical symmetry plays a critical role.