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.^^ 참고!


1.2 Answers for Numerical Methods 633
ANSWERS FOR NUMERICAL METHODS
Exercise Set 1.2 (Page 000)
1. For each part, f ∈ C[a, b] on the given interval. Since f(a) and f(b) are of opposite
sign, the Intermediate Value Theorem implies a number c exists with f(c) 〓 0.
3. For each part, f ∈ C[a, b], f exists on (a, b), and f(a) 〓 f(b) 〓 0. Rolles Theorem
implies that a number c exists in (a, b) with f(c) 〓 0. For part (d), we can use
[a, b] 〓 [−1, 0] or [a, b] 〓 [0, 2].
5. a. P2(x) 〓 0
b. R2(0.5) 〓 0.125; actual error 〓 0.125
c. P2(x) 〓 1+3(x − 1) + 3(x − 1)2
d. R2(0.5) 〓 −0.125; actual error 〓 −0.125
7. Since
P2(x) 〓 1+x and R2(x) 〓
−2eξ(sin ξ + cos ξ)
6 x3
for some number ξ between x and 0, we have the following:
a. P2(0.5) 〓 1.5 and f(0.5) 〓 1.446889. An error bound is 0.093222 and |f(0.5)−
P2(0.5)| ≤ 0.0532
b. |f(x) − P2(x)| ≤ 1.252
c.
1
0 f(x) dx ≈ 1.5
d. |
1
0 f(x) dx−
1
0 P2(x) dx| ≤
1
0
|R2(x)| dx ≤ 0.313, and the actual error is 0.122.
9. The erro
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