Unknown

7.2. Approximation on an Interval 219

6. (a) Show that the mapping f -+ B(f, n; t) is linear: i.e.

B(f+g,n;t) = B(f,n;t)+B(g,n;t)

B(cf, n; t) = cB(f, n; t)

for any functions f and g and any constant c.

(b) For n = 1,2,3,4, find those values of k and those functions f

for which

W, n;t) = W(t).

[Such functions f(t) are called eigenjunctions of the operator B

and the corresponding values of k eigenvalues.]

Explorations

E.59. Taylor Approximation. (Knowledge of calculus required.) Let a
function f(t) be given which is defined for values of t near 0 and which
possesses derivatives of all order at t = 0. This means that the function has
a derivative, its derivative also has a derivative, and so on indefinitely. One
can try to approximate f(t) by a polynomial which, up to some order, has
exactly the same derivative values as f(t) does when t = 0.
Show that a polynomial p(t) of d e g ree not exceeding n is determined
uniquely by specifying the n + 1 values p(O), p’(O),... , p(^)(O).
Suppose the values f(O), f’(O),... , f(“)(O) are given. What is the poly-
nomial p(t) for which p(“)(O) = fck)(0) when 0 5 k 5 n? We call this the
Taylor approximant of order n at the value 0.
Sketch the graphs of log(1 + t) an d sin t along with their Taylor approx-
imants of orders 1, 2 and 3 at the value 0.
How should the Taylor approximant of order n at the general value c be
defined?

E.60. In this chapter, we have discussed a number of approximation tech-
niques. We can compare their effectiveness on a particular example. Con-
sider the problem of approximating t 1/Z for 100 < - 3: < 200. On the same
axis, with a large scale, sketch the graphs of the followmg functions:

(a) x1f2

(b) the linear function whose values agree with x1i2 when x = 100 and

x = 196

(c) the quadratic function whose values agree with x112 at x = 100,

x = 144 and x = 196

(d) the linear polynomial p(t) which minimizes

max{ lt’j2 -p(t)1 : 100 < t 6 200)