Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

*(c) Copyright 1994,1995,1996 by Evans M. Harrell II and James V. Herod. All rights reserved.

At Georgia Tech, students were allowed to take one problem home to work out overnight. The rest of the test took 50 minutes (calculators allowed but no notes). Most students did reasonably well, but few finished early.

1.

a) Construct an orthonormal set of three functions on the interval -1 < x < 1, the span of which is the same as the span of {f₁(x) := 1, f₂(x) := sin( \pi x), f₃(x) := x⁴}. Call the resulting set {g₁(x), g₂(x), g₃(x)}.

g₁(x) = ______________________________________

g₂(x) = ______________________________________

g₃(x) = ______________________________________

b) Find the function f(x) in the span of these three functions, which is closest in the r.m.s. sense to the function |x|.

ANSWER:

The best approximation to |x| is ______ + ________sin([[pi]]x) + _________x⁴

(fill in with specific numbers).

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________

________________________________________________________________

2. Find the (full) Fourier series for the function f(x) = |sin([[pi]]x)| on the interval -1 < x < 1. Sketch the sum of this series for all x, -[[infinity]] < x < [[infinity]]. Find the formula for all coefficients.

ANSWER_______________________________________________________________

3. Let D f = f'(x), and consider the differential operator

A = D² + 3 D + 2 Id

a) The dimension of the null space of A is _______________________

b) A basis for the null space of A is ____________________________

FORMULA OR KEY FACT (FOR POSSIBLE PARTIAL CREDIT):_________

________________________________________________________________

Some help from Waterloo, Ontario:

* int (sin(a*x)*sin(b*x), x);

sin(a x - b x) sin(a x + b x)

(1/2) -------------- - (1/2) --------------

a - b a + b

* int (cos(a*x)*cos(b*x), x);

sin(a x - b x) sin(a x + b x)

(1/2) -------------- + (1/2) --------------

a - b a + b

* int (sin(a*x)*cos(b*x), x);

cos(a x + b x) cos(a x - b x)

- (1/2) -------------- - (1/2) --------------

a + b a - b

Note: Maple assumes here that a does not equal b. In case a=b, remember that the average of sin² or cos² over full cycles is 1/2. (Extra credit: Give an elementary argument why this is true, without using angle-sum formulae.)

See the solutions.

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