Final Exam Review

Scheduling Information:

Exam Format:

The exam will be given in two 80 minute parts. The first part will begin at 1:00 and must be completed by 2:20. Following that, there will be a 20 minute break before the second part begins at 2:40. The second part will conclude at 4:00.

Once each of the two parts begins, you may not leave the room until you have finished working on that part. Once you leave the room you will not be permitted to return to work on that part of the exam.

Reading:

Review Questions:

1. What does it mean to say that T is a "period" of a function f?
2. What does it mean to say that T is the "fundamental period" of f?
3. What are the Fourier coefficients of a piecewise continuous function with period 2L?
4. What is the Fourier series of a piecewise continuous function with period 2L?
5. What does the Fourier Convergence Theorem (Theorem 11.2.1) tell us about convergence of a Fourier series?
6. What is an even function? An odd function?
7. What useful facts do you know about integrals of even functions? Odd functions?
8. If f:[0,L]->R, how can you extend this to an even function on [-L,L]? On odd function on [-L,L]?
9. What is the Fourier sine series of a function f:[0,L]-R?
10. What is the Fourier cosine series of a function f:[0,L]->R?
11. What is a partial differential equation?
12. What is the wave equation?
13. What do the boundary conditions u(0,t)=u(L,t)=0 say about our vibrating string?
14. What does the initial condition u(x,0)=f(x) say about our vibrating string?
15. What does the initial condition du/dt(x,0)=g(x) say about our vibrating string?
16. What is separation of variables? What do we assume about our solutions u(x,t)?
17. How does the the separation of variables lead to the boundary value problem X"+\lambda X=0, X(0)=X(L)=0?
18. What are the solutions to the boundary value problem X"+\lambda X=0, X(0)=X(L)=0?
19. How do we solve our "wave equation problem" when g(x)=0? When f(x)=0? When both f(x) and g(x) are non-zero?

Exercises:

Old Exam Problems:

Here are some old exam problems I have wrotten for Differential Equations students in previous semesters. Also here is the table of Laplace transforms you will be given on the exam.

Before the questions come in - no, I don't have solutions available for these problems. You have a large number of problems from the text with answers, and it's important for you to get used to working on problems where the answers are not available. You'll have a chance to ask questions at the Review Session, and during your TA's office hours, too.