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212 7. Approximations and Inequalities

The polynomial obtained should be the same as the Lagrange polynomial
of degree n determined by the data. Investigate whether this is indeed so.

E.56. Propagation of Error. In making a certain physical observation,
we measure the values of a function as follows:

f(l) = f(2) = f(3) = o, f(4) = 1, f(5) = f(6) = f(7) = 0.

We have reason to believe that the function should vanish identically and
that f(4) is in error. Keeping f(4) as indicated, construct a difference
table. Investigate various ways of interpolating the value of f(3.5) using
polynomials which fit some of the data.

E.57. Summing by Differences. Let g(t) = Af(t). Show that

&g(i) = f(b+ 1) -f(l).

i=a

In Exploration E.18, this formula was invoked in the case g(i) = ick). Apply
it to find the following sums:

(a) a + ar + ur2 +... + urn-’

(b) a + (u + d) + (u + 2d) +... + (u + (n - l)d)

(c) CLzl 1”’ where m = 1,2,3,4,5.

For (c), express the power k” a.9 a sum of terms involving factorial powers.
See Explorations E.18 and E.54.
The operators A and C are analogous to differentiation and integration
in calculus. What correspond to the following results?

(i) -$ It f (u)du = f(t)

a

(ii) J” fl(u)du = f(b) -f(u)

a

(iii) J” f ‘(u)g(u)du = f (b)db) - f (ah(a) - /” f (u)d(u)du.

a a

E.58. The Absolute Value F+unction. Let f(t) = ItI for -1 5 t 5.1,

and let f,,(t) be the unique polynomial of degree at most n for which

fn(t) = f(t) when t is one of the n + 1 equally spaced points (-1) + F