162 5. Approximation and Location of Zeros
(^1) -3 7 -15 1
2 -2 10 -10
(^1) -1 5 -5 -9
2 2 14 I
1 1 7 9
2 6 I
1 3 13
2 1 w
1 5 13 9 -9
0.5 2.75 7.875 8.4375
1 5.5 15.75 16.875 -0.5625
0.5 3.00 9.375 I
(^1) 6.0 18.75 26.25
0.5 3.25
1 6.5 0.5 (^22 1) i
1 7 22 26.25 -0.5625
0.02 0.1404 0.4428 0.5339
1 7.02 22.1404 26.6928 -0.0286
...
(a) Argue that p(t) h as a zero between 2 and 3.
(b) Use Horner’s Table to verify that, if u = t-2, then p(t) = q(u) =
u4 + 5u3 + 13uz + 9u - 9.
(c) Show that q(u) h as a zero between 0.5 and 0.6, and therefore
that p(t) has a zero between 2.5 and 2.6.
(d) Justify the foregoing Horner’s tables as an attempt to estimate
2.52 = 2 + 0.5 + 0.02 as a zero of p(t). Carry the table further
to obtain an accuracy of three decimal places.- Newton’s Method. A method which can be regarded as an infinitesimal
version of that of Exercise 3 and a formalization of that of Exercise
4 can be formulated with the help of Taylor’s Theorem. We suppose
that we are given a number u which is believed to be close to an