Principles and Techniques of Applied Mathematics

MATH 373, Exam #1

Thursday, March 1, 6:00 - 7:30 pm, Swords 359

The first exam covers all of Chapter 1 and Sections 9 - 14 in Chapter 2, of the course text. This is all of the material we have covered on Fourier Series except for Gibbs phenomenon discussed on pp. 49 - 50. It is highly recommended that you go over homework problems and your class notes, including the proofs of some of our key results concerning Fourier series.

Note: You will be allowed one "cheat sheet," a single 8.5 x 11 piece of paper, front and back, full of whatever formulas, graphs, etc. you wish.

Exam Review: We will review for the exam during class on Wednesday, Feb. 28th. Please come prepared with specific questions.

The following concepts, definitions, equations, formulae, theorems and corollaries are important material for the exam:

  1. Analysis: piecewise continuous functions, piecewise smooth functions, left and right-hand limits, left and right-sided derivatives, definition of continuity, definition of the derivative, infinite series (definition of convergence, nth-term test, harmonic series, p-series, series of functions)

  2. Fourier Series: definitions of, finding coefficients of (for cosine, sine and full Fourier series on the standard intervals (0,Pi) or (-Pi,Pi) or on the non-standard intervals (0,c), (-c,c)), linearity of Fourier coefficients, how to go from a Fourier series on a standard interval to one on a non-standard interval

  3. Convergence Theorems: even periodic extensions, odd periodic extensions, general periodic extensions, Fourier coefficients go to zero, Bessel's inequalities, the Dirichlet kernel, the Fourier convergence theorem and its corollaries, convergence on non-standard intervals, Gibbs phenomenon

Some Practice Problems:

Chapter 1
pp. 11 - 13 :   2, 6
pp. 17 - 18 :   1, 5, 7
pp. 21 - 22 :   2, 4, 5, 9

Chapter 2
pp. 39 - 40 :   1, 3, 4
pp. 42 - 45 :   2, 4, 6

Additional Problems:

  1. Show that if the usual derivative f'(x) exists at a point x, then f'_R(x) = f'_L(x) = f'(x).

  2. Consider the function f defined as f(x) = x^2 sin(1/x) for x not equal to 0 and f(0) = 0.
    (a) Is f continuous at x = 0? If so, prove it rigorously.
    (b) Does f'(0) exist? If so, find its value and prove it rigorously.
    (c) What are f'_R(0) and f'_L(0)?
    (d) What are f'(0+) and f'(0-)?
    (e) Is f piecewise smooth on (-Pi,Pi)? Explain.

  3. Derive Bessel's inequality for the Fourier sine series of a piecewise continuous function on 0 < x < c by amending your proof to HW problem #3, page 31, accordingly.