## Math 285 Final Exam, Spring 2010

Here are the grade cut-offs for the course as a whole (after your two lowest quiz scores and two lowest homework scores have been dropped):

• A: 82-100
• B: 72-81
• C: 60-71
• D: 50-59
• F: 0-49
The grades might not show up on the course grades system until Saturday 15 May. Also, note that some grades will be higher than shown currently on the system (by a plus/minus type amount) due to upwards rounding. In particular, when you get official grades, the A- range will be 82-85, the A range will be 86-89, and the A+ range will be 90-100.
Tuesday 11 May, 7-10pm, in 228 Natural History Building. No books, notes, or electronic devices.

Conflict exam for Sec. F1 students: Thursday 13 May, 8-11am, in 245 Altgeld Hall.
(You must email Prof. Laugesen to ask permission to take the conflict exam, before May 6).

Office hours (in 376 Altgeld Hall): Thursday 6 May 3:30-4:30pm; Friday 7 May 12:30-1:30pm; Tuesday 11 May 11:30am-1:30pm and 4:30-5:30pm; Wednesday 12 May 4:30-5:30pm.

Review sessions (151 Everitt Hall): Monday 10 May, from 11:30am-12:30pm and 5:00-6:00pm. You may attend one or both of them.

Material
The entire course: 1.1-1.6, 2.2, 2.4, 3.1-3.6, 3.8, 9.1-9.7, as well as additional material introduced in class (e.g. Orthogonality Supplement) and in the homework assignments and projects. Section 3.8 and Chapter 9 will together count for about half of the final exam.

Formulas will be provided like on the last page of the Practice Exam. (Here are the Solutions for the Practice Exam.)

Re-read the study advice for Chapters 1-3, in the Test 1 Information and Test 2 Information. Print all the homework assignments and quiz problems, and make sure you have all the handouts.
To study the Iode projects, write a paragraph about each one, summarizing the main conclusions.

Basic topics to be very familiar with include: solution of y'=ky and y'=-ky, or x'=kx and x'=-kx, solution of y''-k2y=0 in terms of exponentials and in terms of hyperbolic sines/cosines, solution of y''+k2y=0 in terms of complex exponentials and in terms of sines/cosines.

Chapter 9. Here are a few study questions:

• Consider three types of periodic function: (i) with a jump, (ii) with a corner and no jump, (iii) with no corners and no jumps. For each type of function, explain (a) does its Fourier series show naked eye convergence? (b) if so, then roughly how many terms of the Fourier series would you expect to need for naked eye convergence? (c) roughly what kind of formula would you get if you computed the Fourier coefficients (1/n, or 1/n2, or 1/n3 or higher powers)?
• What do we use the Fourier Convergence Theorem for?
• How do we use Fourier series to solve the mass-spring-friction oscillator DE with periodic forcing function?
• Summarize the method of Separation of Variables for solving a partial differential equation.
• Describe some similarities and differences between the solutions of the heat equation for Dirichlet and Neumann boundary conditions, on the interval 0 < x < L. e.g. what happens as t tends to infinity?
• Suppose a small wave is released from rest (initial velocity = 0) near the middle of a long interval. Draw snapshots to illustrate what happens.
• Suppose a right-moving wave G(x-ct) of the wave equation hits a Dirichlet boundary condition. Draw snapshots to illustrate what happens. Do the same for hitting a Neumann boundary condition.