Exercise 26

In this exercise, we will examine the graph of $z = 3x^2 + 4y^2 - 2$. You do not need to know what this graph looks like, but we will look at the intersection of this graph with several types of fundamental planes.

1. Determine a vector-valued function $\vr$ that parameterizes the curve which is the $x = 2$ trace of $z = 3x^2 + 4y^2 - 2$. Plot the resulting curve. Do likewise for $x = -2, -1, 0,$ and $1$.

2. Determine a vector-valued function $\vr$ that parameterizes the curve which is the $y = 2$ trace of $z = 3x^2 + 4y^2 - 2$. Plot the resulting curve. Do likewise for $y = -2, -1, 0,$ and $1$.

3. Determine a vector-valued function $\vr$ that parameterizes the curve which is the $z = 2$ contour of $z = 3x^2 + 4y^2 - 2$. Plot the resulting curve. Do likewise for $z = -2, -1, 0,$ and $1$.

4. Use the traces and contours you've just investigated to create a wireframe plot of the surface generated by $z = 3x^2 + 4y^2 - 2$. In addition, write a couple sentences to describe the characteristics of the surface.