COMPUTER EXPLORATIONS(Continuation of Exercise 109.) In

Chapter 3, Problem 110E

(choose chapter or problem)

Problem 110E

COMPUTER EXPLORATIONS

(Continuation of Exercise 109.) In Exercise 109, the trigonometric polynomial that approximated the sawtooth function g(t) on had a derivative that approximated the derivative of the sawtooth function. It is possible, however, for a trigonometric polynomial to approximate a function in a reasonable way without its derivative approximating the function’s derivative at all well. As a case in point, the “polynomial”

graphed in the accompanying figure approximates the step function s = k(t) shown there. Yet the derivative of h is nothing like the derivative of k.

a. Graph dk/dt. (where defined) over[–π, π].

b. Find dh/dt.

c. Graph dh/dt. to see how badly the graph fits the graph of dk/dt. Comment on what you see.

Trigonometric Polynomials

As the accompanying figure shows, the trigonometric “polynomial”  gives a good approximation of the sawtooth function s = g(t) on the interval [–π, π]. How well does the derivative of ƒ approximate the derivative of g at the points where dg/dt is defined? To find out, carry out the following steps.

a. Graph dg/dt (where defined) over[–π, π].

b. Find df/dt.

c. Graph df/dt . Where does the approximation of dg/dt by df/dt seem to be best? Least good? Approximations by trigonometric polynomials are important in the theories of heat and oscillation, but we must not expect too much of

them, as we see in the next exercise.