###### Exploration 5.6.1

Here is an example shown both ways.

Let \(f(x,y,z) = z^2 –x^2 +y^2\text{.}\)

Determine equations for the level surfaces for this function with \(C = -2\) and \(C = 2\) and plot them separately in CalcPlot3D.

\(\large\textbf{Solution: Method 1:}\) Solving for \(z\) and graphing both parts of the level surface

Setting \(f (x, y, z) = z^2 – x^2 + y^2 = C\text{,}\) and solving for \(z\text{,}\) we obtain the following equations:

In a form for us to enter in CalcPlot3D these are:

`z = sqrt(C + x^2 - y^2)`

`z = -sqrt(C + x^2 - y^2)`

For \(C = -2\text{,}\) we enter:

`z = sqrt(-2 + x^2 - y^2)`

in the 1st function and `z = -sqrt(-2 + x^2 - y^2)`

in a 2nd function.

For \(C = 2\text{,}\) we enter:

`z = sqrt(2 + x^2 - y^2)`

in the 1st function and `z = -sqrt(2 + x^2 - y^2)`

in a 2nd function.

\(\large\textbf{Solution: Method 2}\) Using an Implicit Surface

Setting \(f (x, y, z) = z^2 – x^2 + y^2 = C\text{,}\) we can write a single implicit equation for each level surface.

For \(C = -2\text{,}\) we will plot the implicit equation: \(z^2 - x^2 + y^2 = -2\)

For \(C = 2\text{,}\) we will plot the implicit equation: \(z^2 - x^2 + y^2 = 2\)

To do this,

- We clear the screen by clicking on the clear screen button.
- Then we select to add an
*Implicit Surface*from the*Add to graph*menu. - Enter
`z^2 - x^2 + y^2 = -2`

in the corresponding textbox and select the checkbox (or press enter) to plot it. This is the level surface for \(C = -2\text{.}\) - Print it out, if desired, using the
*Print Plot*option on the app main menu. - To view the level surface for \(C = 2\text{,}\) just change the \(-2\) on the right side of the equation to a \(2\text{.}\)