Mathematics

### Material on the Final Exam

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The final exam is Friday, Dec. 14, 8:00 - 11:00 a.m., in the usual room.

Calculators may be used for arithmetic only.

Please bring your text to the final exam. You can use the integral table inside the front cover, and the Laplace transform table and other information inside the back cover.

The exam will cover first-order equations (Secs. 1.1 - 1.3, 2.2 - 2.3, pp. 34 - 36, and 3.2 only), second-order linear equations (Secs. 4.2 - 4.6 only), Laplace transforms (Secs. 7.2 - 7.6, 7.8), linear systems (Secs. 9.2 - 9.7), and the phase plane (Secs. 5.4 and 12.1 - 12.4).

For first-order equations, second-order linear equations, Laplace transforms, linear systems, Secs. 5.4 and 12.2, you can use the review sheets for Tests 1, 2 and 3. Omit material from sections that are not listed above.

Things you should be able to do from Secs. 12.3 and 12.4:

Sec. 12.3. For x'=f(x,y), y'=g(x,y), find nullclines, find equilibria, find the matrix of the linearization at each equilibrium, find its eigenvalues and eigenvectors, use this information to help sketch the trajectories of the nonlinear system.

Sec. 12.4. For x''+g(x)=0, write as a system and sketch the phase portrait. I suggest reviewing Tests 1, 2 and 3.