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66 2. Evaluation, Division, and Expansion

(a) Verify that if u(t) = t 5-4t3+2t2-7, then u’(t) = 5t4-12t2+4t,

u” t) = 20t3 - 24t + 4, d3)(t) = 60t2 - 24, ~(~1 = 120t and

UC5 \ (t) = 120.

(b) Show that, for any polynomial p(t) and any positive integer r

not exceeding degp, degp(‘) = (degp) - r.

(c) Show that, for any polynomial p(t) of positive degree n, p(“) is

a constant polynomial and ~(‘1 = 0 for r > n.

(d) Derive a formula for (pq)“(t) and generalize it to a formula for

(Pd”‘(t).

8. Show that for any positive integers m and k, with m 5 k, the mth
   derivative of (t - c)~ is equal to

k(k - l)(k-2)...(k-m+ l)(t -c)~-~.

9. We are now in a position to show that every polynomial has an ex-
   pansion in terms of powers of (t - c) and to identify the coefficients.
   The result is:
   Taylor’s ‘Theorem. Let p(t) b e any polynomial of degree n and c be
   a constant. Then

p(t) = P(C) + p’(c)(t - c) + p!!l& _ c)2 + p!f!$+, - c)3

+...+ p&?$+t _ C)k +...

where the sum on the right has at most n + 1 nonzero terms.

The right-hand side is called the Taylor expansion of p(t) about c.

Verify the theorem for the polynomial t4 + t2 - 3t + 7 expanded in

terms of powers of (t - 3)) checking your answer against the result of

Exercise 1.8.

Establish Taylor’s Theorem in the following steps:

(a) Use the Division Theorem to establish that p(t) can be written

in the form

p(t) = cn(t - c)” +pl(t) where degpr 5 n - 1.

(b) By repeated use of the Division Theorem on the remainders

resulting from the procedure in (a), show that

p(t) = co + q(t - c) + CZ(t - c)2 + cg(t - c)3 +....