Test 2

Bring

  • calculator
  • one 8 1/2 X 11 page of notes that you prepare ahead of time.

There will be two parts of the test.

    Part 1 will work like Test 1: You can bring a page of notes, but you may not consult any other resources / people, and you can't use CoCalc.

    Part 2. When you turn in the first part, I'll give you the last, Lagrange multipliers problem, and for that part, you may still refer to your notes. But you can also use CoCalc (or Mathematica if you're more comfortable with that, and have your own copy).

Topics--Functions of more than one variable

Parametric functions

Be able to parameterize a surface in terms of 2 parameters. Be able to parameterize a path (set of points along a curve) in terms of a single parameter. For example, if you want to look for a limit as you approach a point along a particular path. Using the chain rule.

Limits

Be able to test for the limits of a function of two variables as you approach a point by various paths.

Tabular Data

Understanding how to interpret tabular data for multivariable data. Know how to estimate partial derivatives from tabular data. Know how to use the methods of linear approximation (and tangent planes) to estimate values of functions of two variables from tabular data.

Contour Maps

Know the relationship between the 3D graph of a function and the contour plot of a function. Be able to sketch a contour plot. Know what a level curve is, what information can be obtained from looking at a plot of level curves (a contour map). Given a contour map, be able to determine the sign of first- and second-order partial derivatives. Given a contour map, be able to estimate values for first-order partial derivatives.

Derivatives

Given a function of two variables, be able to compute its partial derivatives, the gradient, the directional derivative in a specified direction. Understand the geometrical relationships between the contour plot of a function and the gradient. Understand the relationship between directional derivative and the gradient. Clairaut's theorem.

Maximization/minimization

Be able to find maximum, minimum, and saddle points. Be able to find critical points of a function. Be able to interpret a contour plot in terms of maximum and minimum values of a function. Be able to set up and do an optimization problem using the technique of Lagrange multipliers.

Double Integrals

Be able to do an integration estimate from a table of values or a contour plot. Be able to compute double integrals. Be able to determine limits of integration and change the order of integration. This includes changing a function of $x$, e.g. $y=f(x)=\frac 12 x^2$ into an equivalent function of $y$, e.g. $x=h(y)=\sqrt{2y}$.