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ÓO¯K§XJU~�$gêØ�!Om§JpOݧ
U~
�Ø��È\©
Ox255�§XJŦ254g¦{§�e�¤
x255 = x · x2 · x4 · x8 · x16 · x32 · x64 · x128
14g¦{$=©qXOõª
P (x) = anx
n + an−1xn−1 + · · ·+ a1x+ a0
�§e�Oakx
k2Åg\§�
n+ (n− 1) + · · ·+ 1 = 1
2
n(n+ 1)
g¦{Úng\{§eæ^Ê�{
Sn = an
Sk = xSk+1 + ak, k = n− 1, n− 2, · · · , 1, 0,
Pn(x) = S0
ng¦{Úng\{=ÑPn(x)�©
(2);üCê~
3êO¥üCê~k�êi¬î©
~Xx = 618.45Úy = 618.32Ñ´5 k�êi§�x− y = 0.13kü k�êi§¤±
ÐUCO{§;ùa$�u)©
X�x1Úx2��C§K
lnx1 − lnx2 = ln x1
x2
màªk�êiØ©�xé§U
√
x+ a−√x = a√
x+ a+
√
x
. 12 . 1Ù SØ
O(J�Щ�Of(x)− f(x0)�Cq§^TaylorÐmª
f(x)− f(x0) = f ′(x0)(x− x0) + 1
2
f ′′(x0)(x− x0)2 + · · ·
�màkCqà©XJÃ{UCª§Kæ^O\k� ê?1$©
(3)ê¯K�ê
3ê$¥kêþ?�é§
OÅik§XØ5¿$gSÒkU
Ñyê¯K�ê�y§KO(J�5©~X38 10?OÅþOx =
54 272 401 + 0.6§du3OÅSO§�¤2:/ª§
ké�§é�x =
54 272 401 = 0.542 724 01× 108§0.6 = 0.000 000 006× 10838 ÅþL«0§Ïd
x = 54 272 401 + 0.6 = 0.542 724 01× 108 + 0.000 000 00× 108
= 0.542 724 01× 108 = 54 272 401
(4)ýé���êبØê
ýéé��êØê¬KêO(J�°Ý§d
d
(
x
y
)
=
ydx− xdy
y2
,
�û�Ø�'Xª
e
(
x
y
)
=
ye(x)− xe(y)
y2
,
Ù¥e
(
x
y
)
, e(x), e(y)©OL«
x
y
, x, y�ýéØ�©
w,�|y|¿©�§e
(
x
y
)
¬é©;ù«¹u)�{´òÙzÙ¦�d�/
ª5?n©
~X§�x�Cu0§
1− cosx
sinx
�©f!©1Ñ�Cu0§
;ýé���êØ
ê§
;©füCê~§ò�ªCµ
1− cosx
sinx
=
sinx
1 + cosx
.
(5)Ø��DÂÈ\§À�ê½�Oúª
|^4íúª?1O,$L§'�5Æz§�õê4íúª7L5¿Ø��È
\©XJ4íL§¥Ø�È\O§õg4í¬�)Ø(J¶XJ4íL§¥Ø�~�§K
���(J'�©
1.3 OL§¥�Ø�9Ù . 13 .
1.3.5 ê{�½5
¤¢êêê{{{´|^OŧUX,êÆOúª5½�$gS§é®êâ?1k
goKâ$ÚÜ6$§¦Ñ¤'%êƯKCq)�{©ê{§3OL
§¥§XJØ��DÂéO(J�Ké�§½ö`§3OL§¥§Ø�D´�§
K¡ù{êêê½½½§ÄKê{§3OL§¥§XJØ��DÂéO(J�K
駽ö`§3OL§¥§Ø�D´Ø�§K`ùê{êêêØØؽ½½©
~X~1.2¥�{A§3OL§¥Ø�ÅìO§´Ø½�§
{B3OL§¥
Ø�Åì~�§´½�©
1.3.6 ¾�¯K^ê
3¢SêOL§¥§k
¯Kéê6Ä~¯a§k
¯Kéê6Äدa§
«OÚïÄù
¯K§½Â¯K�^êÚ¾�¯K�Vg©
½Â 1.6 ^ê
¯KÑÑCþ�éØ�Ñ\Cþ�éØ��û¡T¯K�^êcond(condition
number)©
~ 1.6 é½�x§O¼êy = f(x)§ek6Ä∆x = x− x∗§ÙéØ�∆x
x
§
¼êf(x∗)�éØ�
f(x)− f(x∗)
f(x)
©K¯K�^ê
cond =
∣∣∣∣f(x)− f(x∗)f(x)
∣∣∣∣ / ∣∣∣∣∆xx
∣∣∣∣ ≈ ∣∣∣∣xf ′(x)f(x)
∣∣∣∣ (1.9)
½Â 1.7 ê¯K�5�
éuê¯K§XJÑ\êâk�6Ä(Ø�)§KÚåÑÑêâ�éØ�(¯K
�^ê)駡ùê¯K´¾��©
ª(1.9)¡O¼ê¯K�^ê©gCþéØ�ج�§XJ^êcondé
§òÚå¼êéØ�é§Ñyù«¹�¯KÒ´¾�¯K©@Cond�¾�
�î(3©z[3]¥§@Cond � 1§¯K´¾�©3©z[9]¥§@Cond > 10§¯K
´¾�)©Ù¦¯K©Û´Ä¾�©~X5§|�ê)?دK�^ê9´
ľ�§ùò3AÙ!?10�©
. 14 . SK
SK
1.�êÆ[yÀQ±
355
113
�±Ç�Cq§¯dCqäkõ� k�êi?
2.Uo�Ê\�K§òe�ê�¤5 k�êi.
816.856 7, 6.000 015, 17.322 50, 1.235 651, 93.182 13, 0.015 236 23
3.e�ê´Uo�Ê\�K���Cq§§kA k�êi?
81.897, 0.008 13, 6.320 05, 0.180 0
4.e
1
4
^0.25L«§¯§kõ� k�êiº
5.O
√
10− pi�§°(�5 k�êi.
6.ea∗ = 1.106 2, b∗ = 0.947´²Lo�Ê\���Cq§¯a∗+ b∗, a∗b∗kA k�
êiº
7.�x∗1 = 0.986 3, x
∗
2 = 0.006 2´²Lo�Ê\���Cq§¯
1
x∗1
,
1
x∗2
�OÚý
�éØ�9x∗1, x
∗
2Úý�éØ�.
8.UCe�ª§¦O(J'�O(µ
(1) lnx1 − lnx2, x1 ≈ x2¶ (2) 1
1− x −
1− x
1 + x
, |x| � 1¶
(3)
√
x+
1
x
−
√
x− 1
x
, 1� x¶ (4) 1− cosx
x
, x 6= 0, |x| � 1¶
(5)
1
x
− cotx, x 6= 0, |x| � 1¶ (6)
w n+1
n
1
1 + x2
dx, n¿©©
9.Of = (
√
2 − 1)6§�√2 = 1.4§|^e�ªO§=���O(J
к
(1)
1
(
√
2 + 1)6
¶ (2) (3− 2√2)3¶
(3)
1
(3 + 2
√
2)3
¶ (4) 99− 70√2¶
Ü©SKY
SK
1.k7 k�êi
2. 816.96, 6.0000, 17.323, 1.2357, 93.182, 0.015236.
3. 5 §3 §6 §4 .
4. 2 k�êi.
5. 0.020685.
6. 3 §3 .
7. 0.5× 10−4, 0.8× 10−2.
8.
(1) ln
x1
x2
, x1 ≈ x2¶ (2) 3x− x
2
1− x2 , |x| � 1¶
(3)
2
x
√
x+ 1x +
√
x− 1x
, 1� x¶ (4) x
1 + cosx
, x 6= 0, |x| � 1¶
(5)
x
3
, x 6= 0, |x| � 1¶ (6) arctan 1
1 + n(n+ 1)
, n¿©©
SK�
1. (1)3.6320, (3)1.3247, (4)1.2599, (5)0.6412, (6)0.2575.
10. x2 = 1.365230013. 14. 0.257530286.
SKn
. 16 . Ü©SKY
2.x1 = 2, x2 = 1, x3 = 0.5.
4.x1 = 0.8333333, x2 = 0.6666666, x3 = 0.4999999, x4 = 0.3333333, x5 = 0.1666666
5. 30.
6.
1 0 02 1 0
3 −1 1
−2 4 80 10 −32
0 0 −76
x1x2
x3
=
58
7
.
SKo
1.(1)5,
(
−13
50
, 1,− 6
25
)T
; (2)
17
5
,
(
10
17
,
3
17
, 1
)T
(3)9.6058,(1, 0.6056,−0.3945)T;(4)8.86951,(−0.50422, 1, 0.15094)T.
2.
252
101
.
3.
2
√
2 0√
2 1 0
0 0 3
, 0, 3, 3.
4.
2 1 01 −1 2
0 2 3
.
6.
1 −3 0 0
−3 7
3
−
√
2
3
0
0 −
√
2
3
7
6
−3
2
0 0 −3
2
1
2
.
7.7.288,(1, 0.5229, 0.2422)T.
9.A1 =
1 0 0
0 −3
5
−4
5
0 −4
5
3
5
, A2 =
1 −5 0
−5 77
25
14
25
0
14
25
−23
25
.
10.(1)
1
2
+
√
33
2
2,
1
2
−
√
33
2
; (2)2 +
√
3, 2, 2−√3.
SKÊ
SK8 . 17 .
10.
5x2
6
+
3x
2
− 7
2
.
11.-0.620219,-0.616839.
SK8
1.B1(f, x) = x,B3(f, x) = 1.5x− 0.402x2 − 0.098x3.
3.(1)
3
4
,
3
4
+
√
2
2
− 1
2
x;(2)P2(x) = −1.1430 + 1.3828x− 0.2335x2.
5.P1(x) = (e− 1)x+ 1
2
(e− (e− 1) ln(e− 1)).
6.P (x) = 3x3 + x2 + 34 .
7.P ∗3 (x) = 5x
3 − 5
4
x2 +
1
4
x− 129
128
.
8.P (x) =
4
15
+
4
5
x.
9.(1)s1 = −0.2958x + 1.1410;(2)s1 = 0.1878x + 1.6244;(3)s1 = −0.24317x +
1.2159;(4)s1 = 0.6822x− 0.6371.
10.(2)P2(x) = −1.1430 + 1.3828x− 0.2335x2.
11.S∗3(x) = 1.5531913x− 0.5622285x3.
12.
1
4
+
1
2
√
2− 1
2
x.
13.P3(x) = 0.20183(x− 0.38268)(x+ 0.38268)(x+ 0.92388)
+ 0.23877(x− 0.92388)(x+ 0.38268)(x+ 0.92388)
+ 0.23877(x− 0.92388)(x− 0.38268)(x+ 0.92388)
+ 0.20183(x− 0.92388)(x− 0.38268)(x+ 0.38268).
14.P2(x) = −0.2320x2 + 1.3823x− 1.1459.
15.R22(x) = 2− 4
x+ 0.5+
1.25
x+ 1.5
.
16.
2
pi
+
∞∑
j=1
4(−1)j−1
(2j − 1)(2j + 1)Tj(x).
17.1− pi
2
+
∞∑
j=1
2
j2pi
((−)j−1 + 1)Tj(x).
18.y = 2.014 + 2.25x, y = 1.9983 + 2.25x+ 0.0314x2.
19.y = 11.436e0.2912x.
20.y = 0.050035 + 0.972555x2, ||r||2 = 0.1226.
. 18 . Ü©SKY
21.y =
x
2.0158x+ 1.0061
.
22.y = 2.973 + 0.531 lnx.
SKÔ
1.(1)T8 = 0.11140, S4 = 0.11157;(2)T10 = 1.39148, S5 = 1.45471;
(3)T4 = 17.222774, S2 = 17.32222;(4)S1 = 0.63233§Ø�0.00035.
2.(1)T8 = 0.8347, S4 = 0.8357§(2):T6 = 1.6355, S3 = 1.6360.
3.(1)0.9461;(2)0.7468245.
4.(1)0.843;(2):0§(3)10.1517434.
8.n = 2, I = 10.9484;n = 3, I = 10.95014;°(I = 10.9517032.
11.
a0 =
5
9
, a1 =
8
9
, a2 =
5
9
,
w 1
−1
f(x)dx ≈ 5
9
f
(
−
√
3
5
)
+
8
9
f(0) +
5
9
f
(√
3
5
)
,
x = 2
(
t− 1
2
)
,
w 1
0
√
t
(1 + t)2
dt =
√
2
w 1
−1
√
1 + x
(x+ 3)2
dx
≈
√
2
5
9
√
−
√
3
5
+ 1(
−
√
3
5
+ 3
)2 + 89 132 + 59
√√
3
5
+ 1(√
3
5
+ 3
)2
= 0.2885.
12.
w 1
−1
f(x)dx ≈ f
(
− 1√
3
)
+ f
(
1√
3
)
, I ≈ 1.3987.
SKl
1.n:úªµ-0.247,-0.217,-0.187, 2.Ê:úªµ2.644225.
SKÊ
SKÊ . 19 .
1.
L 1.3 11KO(J
x 0.1 0.2 0.3 , 0.8 0.9 1.0
(1) y 1.0000 0.2005 0.3022 , 0.8458 0.9625 1.0815
(2) y 1.0000 1.0000 1.0000 , 1.0000 1.0000 1.0000
(3) y 0.0010 0.0050 0.0143 , 0.4224 2.2703 53.8920
(4) y 1.8000 1.6200 1.4580 , 0.8609 0.7748 0.6974
2.(2)
L 1.4 12K(2)O(J
x 0.1 0.2 0.3 , 0.8 0.9 1.0
Eulerwª y 1.0000 1.0100 1.0304 , 1.3601 1.5081 1.7129
U?Euler y 1.0050 1.0204 1.0470 , 1.4684 1.6758 1.9881
F/{ y 1.0051 1.0205 1.0474 , 1.4766 1.6926 2.0264
3.(1)-(4)
L 1.5 13K(1)(3)O(J
x 1.1 1.2 1.3 , 1.8 1.9 2.0
(1) y 1.0000 1.0363 1.0714 , 1.2540 1.2793 1.3030
(3) y 1.0000 1.2401 1.5873 , 18.0306 34.4383 72.8124
L 1.6 13K(2)(4)O(J
x 0.1 0.2 0.3 , 0.8 0.9 1.0
(2) y 1.0000 1.1052 1.2214 , 1.8221 2.0137 2.2255
(4) y 1.0000 1.1103 1.2428 , 2.6511 3.0192 3.4366
4.
L 1.7 14KO(J
x -1.0000 -0.9000 -0.8000 , -0.2000 -0.1000 0
y 0 0.0900