Objectives
- Understand the geometric intuition behind second partials of multivariable functions.
- Examine the connection between second partials and concavity.
- Explore the geometric interpretation of mixed partial derivatives.
Section 6.9 Second Partial Derivatives
In this lesson, the second partial derivatives are defined and motivated in terms of how the geometry of the partial derivatives is changing. In particular, the second partials with respect to each variable are linked to the ideas of concavity, similar to the discussion in earlier calculus courses. Further, there are several animated figures that show how changes in the first partial derivatives shows the geometric meaning of mixed partials. These animations show how by changing the value of the variable held constant when computing a first partial derivative, the mixed partial derivatives are shown on the surface.
Several questions are posed asking the students to identify whether the a second partials is positive, negative, or zero.
