Thomas Calculus 13th [Solutions]

Section 8.7 Numerical Integration 613
2
4 2 4 4
15. (a) M 12 (see Exercise 5): Then x 2 E 2 2 (12) 8
T
10 n 8 10 n 8 10
n 12 n 2
n
n 282.8, so let n 283
(b) M 0 (see Exercise 5): Then n 2 (n must be even )
2 4 4
x 1 E s (1) (0) 0 10
180
2
4 2 4 4
16. (a) M 6 (see Exercise 6): Then x 2 E 2 2 (6) 4
T
10 n 4 10 n 4 10
n 12 n 2
n
200, so let n 201
(b) M 0 (Exercise 6): Then n 2 (n must be even)
2 4 4
x 1 E s (1) (0) 0 10
180
2
4 2 4 4
17. (a) M 6 (see Exercise 7): Then x 1 E 1 1 1 1 1
T (6) 10 n 10 n 10
n 12 n 2
2n
2 2
n 70.7, so let n 71
4
4 4 4
(b) M 120 (see Exercise 7): Then x 1 E 1 1 2 2
s (120) 10 n 10
n 180 n 4
3n
3
4 2 4
n 3
10 n 9.04, so let n 10 (n must be even)
18. (a) M 6 (see Exercise 8): Then x 2
2 2 2 4 4 2 4 4
(6) 10 4 10 4 10
n E T 12 n n n
2
n
n 200, so let n 201
4
4 4 4
(b) M 120 (see Exercise 8): Then x 2 E 2 2
64 64
s (120) 10 n 10
n 180 n 4
3n
3
4 64 4
n 3
10 n 21.5, so let n 22 (n must be even)
1/2 3/2
19. (a) f ( x ) x 1 f ( x ) 1 1 1 1 1
2 ( x 1) f ( x ) 4 ( x 1) M
3 3 4
.
4 x 1 4 1
2
4 2 4 4
Then x 3 E 3 3 1 9 9 9
T
10 n 10 n 10 n 75, so let
n 12 n 4 2
16n
16 16
n 76
(3) 5/2 (4) 7/2
(b) f ( x ) 3 15 15 15 15
8 ( x 1) f ( x ) 16 ( x 1) M
7 7 16
. Then
16 x 1 16 1
5 4 5 4
4 5
3 (15) 4 4
3 (15) 10 3 (15) 10
4
180 16 4
16(180) 16(180)
x 3 3 3 15
n Es
n 10 n n n 10.6, so let
16(180) n
n 12 (n must be even)
20. (a) f ( x) 1 f ( x) 1 3/2 3 5/2 3 3 3
1 2 ( x 1) f ( x) 4 ( x 1) M 4 1 5 4 1
5 4
. Then
x
x
4 4 4 4
2
4
3 10 3 10
4 2
x 3 E 3 3 3 3
T
10 n n n 129.9, so let n 130
n 12 n 4 2
48n
48 48
(3) 7/2 (4) 9/2
(b) f ( x ) 15 105 105 105 105
8 ( x 1) f ( x ) 16 ( x 1) M
9 9 16
. Then
16 x 1 16 1
5 4 5 4
4 5
3 (105) 4 4
3 (105) 10 3 (105) 10
4
180 16 4
16(180) 16(180)
x 3 3 3 105
n Es
n 10 n n n 17.25,
16(180) n
so let n 18 (n must be even)
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