# Study Guide - Final 2

## Math 221

- Total derivatives and extrema: Find the total derivative for a scalar function of three variables.
Find the total second derivative (a definition is provided). Find and classify all critical points
(an extended second derivative test is provided).
- Change of variables: Study the area of an ellipse which we worked in class. This is an
extended problem, but is worked much the same way. Evaluate a triple integral to find the
volume where you need to change variables. This is like the problems where you took the
sides of a region, let your u and v be those - except this is a triple integral so it won't be sides
of regions, it will be planes which are edges of the solid.
- Minimization / Maximization: Use the method of Lagrange multipliers to find a minimum or
maximum value. Concentrate on the production problems from economics. Find the least
squares regression line for a set of data.
- Quadric Surfaces: Identify and sketch six quadric surfaces. Part seven is to summarize the
things that you're looking for when identifying the surfaces. A sentence is began and you have
to finish it with the type of surface.
- Center of Mass / Moments of Inertia: Find the center of mass of an solid with the given
density. Find the moment of inertia and the radius of gyration about an axis for a different
solid.
- Differential Equations: Solve two non-homogeneous second order linear differential equations.
One can be done by undetermined coefficients, the other must be done by variation of
parameters.
- Parametric Surfaces: Find a vector valued function whose graph is the surface described. Find
the area of a surface over the specified region. Find the equation of the tangent plane to the
surface.

There are seven problems which must be completed. You have two hours. You may work
together. You may use your notes. It is not important that each person complete every problem,
just that all problems are completed between the members of the class.