302
Instructor’s Manual
This implies
u
e
(
x
2
)
u
5
u
r
(
x
2
)
x
2
u
5
u
r
(
x
1
)
x
2
u
%
(0.005/39.95)
•
0.0506
,
0.00001.
18.
61.2
2
7.5
•
15.5
1
11.2
•
3.94
1
2.80
5
61.2
2
116
1
44.1
1
2.80
52
7.90
((
x
2
7.5)
x
1
11.2)
x
1
2.80
5
(
2
3.56
•
3.94
1
11.2)3.94
1
2.80
5
(
2
14.0
1
11.2)3.94
1
2.80
11.0
1
2.80
8.20
Exact:
2
8.336016
SECTION 19.2. Solution of Equations by Iteration, page 787
Purpose.
Discussion of the most important methods for solving equations ƒ(
x
)
5
0,
a very important task in practice.
Main Content, Important Concepts
Solution of ƒ(
x
)
5
0 by iteration (3)
x
n
1
1
5
g
(
x
n
)
Condition sufficient for convergence (Theorem 1)
Newton (–Raphson) method (5)
Speed of convergence, order
Secant, bisection, false position methods
Comments on Content
Fixed-point iteration gives the opportunity to discuss the idea of a
fixed point,
which is
also of basic significance in modern theoretical work (existence and uniqueness of
solutions of differential, integral, and other functional equations).
The less important
method of bisection
and
method of false position
are included in the
problem set.
SOLUTIONS TO PROBLEM SET 19.2, page 796
2.
x
0
5
1,
x
1
5
0,
x
2
5
1,
x
3
5
0,
•••
x
0
5
0.5,
x
1
5
0.875,
x
2
5
0.330,
x
0
5
2,
x
1
7,
x
2
5
344,
x
3
40 707 583,
4.
g
5
Ï
4
x
2
0.
w
2
w
, 1, 0.9457, 0.9293, 0.9241, 0.9225, 0.9220, 0.9218, 0.9217, 0.9217
6.
x
5
x
/(
e
0.5
x
sin
x
), 1, 0.7208, 0.7617, 0.7541, 0.7555, 0.7553, 0.7553, 0.7553
8. CAS Project.
(a) This follows from the intermediate value theorem of calculus.