Review guide for the final exam

Some Math 421
exam axioms
Exam information
Math 421:01 Friday December 16 8 AM-11 AM SEC 216
The final exam will be cumulative, but will have somewhat more emphasis on material which has been covered since the last exam.
Calculators, notes,
formula sheets, etc.

Review session(s)
Certainly all students are welcome at all review sessions. The sessions may work better if they are nominally devoted to specific topics. So I will offer to review certain topics and answer questions about these topics. When that's done, I'll try to answer questions about other 421 material.

Wednesday, December 14 at 4-6 PM in Hill 425 Laplace transforms & linear algebra.

Thursday, December 15 at 4-6 PM in Hill 425 Fourier series, the wave/heat equations, & associated boundary value problems.

What's next?
The Goals below are copied from the syllabus. The remarks after Reality attempt to describe what sections in the text were actually covered and give links to suggested review material. Since I've written so much here, students might like a short cut. The final exam will likely resemble in length and difficulty the exam I gave last spring: here.

Laplace transforms
Goals Students should know the definition of the Laplace transform. They should be able to compute simple Laplace transforms "by hand" and should be able to use a table of Laplace transforms to solve initial value problems for constant coefficient ODE's. They should be able to write "rough" data in terms of Dirac and Heaviside functions. They should be able to recognize and use simple results regarding Laplace transforms.

Reality We covered sections 4.1-4.6. Sources for review material:

Linear algebra
Goals Students should know the principal definitions of linear algebra and recognize and work with these ideas. Students should know how to compute the rank of a matrix, the determinant of a matrix, and be able to invert and diagonalize simple matrices "by hand".

Reality We covered sections 8.1-8.8, 8.10, and 8.12. Students should be able to state the principal definitions. Sources for review material:

  • This semester
    The second exam and its answers are available. Most of the questions on this exam were about linear algebra, as were a few of questions on the first exam (answers here). The principal definitions are listed here, along with appropriate links to the course diary and references to the text.
  • Fall 2004
    The second exam and its answers which I gave are available. Please note especially problem 3, which indicates my expectation that students should be able to apply the ideas of linear algebra to varied situations. The first 4 problems are relevant here. Problems 6 and 7 on the first exam (with answers here) deal with linear algebra. Problems 3 and 4 on this final exam are directly about linear algebra.
  • Spring 2004
    This course used a different text and covered slightly different topics. You can look at review problems and answers. See especially problem 14 and its answers. That problem asks for very brief definitions of common terms in linear algebra. I think students in this semester's class should also be able to answer that sort of question. A version of the second exam I gave and answers to that exam are available. You may look at these problems, too.
Fourier series
Goals Students should be able to compute the Fourier coefficients of simple functions "by hand". They should know what orthogonality means for functions. Given a function's graph on [0,A] they should be able to graph the sum of the Fourier sine and cosine series on [-A,A]. They should have some qualitative understanding of what partial sums of Fourier series look like compared to the original function, including an idea of the Gibbs phenomenon.

Reality Certainly we discussed the material in sections 12.1-12.3. The ideas of 3.9 and 12.5 are essential in the last portion of the course. The specific content needed from section 12.5 is brief. Sources for review material:

  • This semester
    The last three questions on the second exam are relevant (its answers).
  • Fall 2004
    The second exam and its answers are available. Problems 5, 6, 7 and 10 are about Fourier series. Problem 5 on this final exam test Laplace transforms.
  • Spring 2004
    The first seven review problems here cover material to be tested on this exam, and here are answers. Problems 1, 2, and 3 on this exam could be asked here.
Boundary value problems for ODE's and PDE's
Goals Students should be able to solve simple boundary and initial value problems for the classical PDE's using separation of variables combined with the appropriate trigonometric series. They should also understand some of the qualitative aspects of solutions (smoothness for heat and possible shocks for wave), and simple asymptotics for some BVP's.

Reality Here the difference between what is declared in the syllabus and what we've done is greater than in the other topics of the course. I think the exam should cover 13.1-13.5 and 13.8. I am writing this before the last two meetings of the course. I hope to cover in lecture some discussion of steady-state heat flows on a rectangular plate (Laplace's equation) and some discussion of vibrations on a rectangular plate. Sources for review material:

  • This semester
    Nothing yet. I will probably give out an updated version of the material linked directly below. Here is a draft version of a Fourier series/PDE formula sheet.
  • Fall 2004
    Here are some questions about two-dimensional material: heat, wave, and Laplace's equations, and double sine series. This material was not covered in the spring 2004 version of the course. Answers to these questions and to some questions from the textbook are given here.
    Problems 5, 6, 7, 8, and 9 on this final exam are about the PDE material we covered.
  • Spring 2004
    Problems 8 through 17 of this review set cover material to be tested on this exam, and here are answers. Problems 4 through 7 on this exam could be asked here.

I try to write and grade exams carefully. As you prepare for and work on the final exam, I hope that you will:

  1. Do my problems, not problems you invent.
  2. Realize that I cannot read your mind or imagine your answers: what you write is what I will read (and grade).
  3. Realize that if you do extensive computation, you are probably not doing "my" problem (see 1) or you are doing "my" problem inefficiently.

Formula sheets, etc.
At the exam, students will get a formula sheet with Laplace transform information and with information about Fourier series and boundary value problems. If needed, I will likely also give you information about some matrices and their RREF's. You may not use any calculators or other notes during the exam. Although I can add more formulas, I would rather not make the formula sheet(s) unwieldy. Also, please keep in mind that if you have more formulas, legitimate test questions can be more difficult.

Vocabulary of the day
the last vocabulary word!

cumbersome, clumsy, or hard to manage, owing to size, shape, or weight

Maintained by and last modified 12/6/2005.