A Very Bumpy Wave
Move k up or down with the slider, and watch the horizontal plane corresponding to the height k move on the 3D plot.
For each value of k, you can visualize the intersection of k with the surface as showing you the level curve for that height.
How do the level curves differ as you approach a local max/min vs. a saddle point?
What can you say about the behavior of the partial derivatives at various heights?
You can rotate the 3D surface by right clicking and dragging to get a better view.