3D Monkey Saddle Graph
A three-way saddle with three valleys and three ridges.
z = x^3 - 3*x*y^2Teacher prompt
Why can this surface confuse the second derivative test?
The usual quadratic test degenerates at the origin, so higher-order behavior matters.
What this graph represents
It shows how higher-order terms can create shapes that are not visible from second derivatives alone.
Where it appears in calculus
Use this after students learn the second derivative test.
Embed this graph
Use the Embed button in the calculator to copy a ready iframe for blogs, LMS pages, and lesson notes.
Open embed pageRelated graphs
Open another surface page and compare shape, slices, and contour behavior.
Saddle Surface
z = x^2 - y^2A saddle surface curves up in one direction and down in the perpendicular direction.
Gaussian Surface
z = exp(-(x^2 + y^2))A smooth bell-shaped surface centered at the origin.
Elliptic Paraboloid
z = x^2 + y^2A bowl-shaped surface that opens upward.
Inverted Paraboloid
z = 12 - x^2 - y^2A dome-shaped surface with a highest point at the center.