7
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f(x) , <x<b.
, (1) , f(x) (, , ..) , .
*, f(x) , .. ,
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(1) . (1), .
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2) .
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.
. (1)
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φ()=+λf().
[α,β] 0 ,
1=φ(0),
2=φ(1),
n n-1
n=φ(n-1). (3)
n (n=1,2,3,) (3) .
[α,β], =* (2), 1,2,n,, ,
|φ()≤q<1,
, .. n , *.
, n-1 n
|
|
|n - n-1|≤ ε; (4)
|* - n|≤ε,
ε *.
q≤0,5, ≥1 (4)
|n - n-1|≤ε, (5)
, *.
(1) (2) φ() , φ′() , , 1.
. f(x)=0 [α,β], f′(x) f′′(x) , [α,β].
. 0(0,f(x0)), :
y=f(x0)+f′(x0)(x-x0).
=0, 1 :
1=0- ,
:
2=1- ,
............ (6)
n=n-1- .
|n - n-1|≤ ,
m - |f′(x)| [α,β]; M |f′′(x)| [α,β].
|* - n|≤ε,
ε *.
0 , f(x0) f′′(x0)>0.
. f(x)=0, f(x) [α,β] f(α)f(β)<0.
(1), [α,β], , .. 0=(α+β)/2. f(x0)=0, x0 . f(x0)≠0, [α,x0] [x0,β], f(x) . ..
, , , ε.
, | ||||
3sin +0,35 3,8=0 | [2;3] | 2,2985 | ||
0,253+ 1,2502=0 | [0;2] | 1,0001 | ||
+ + 2,5 =0 | [0,4;1] | 0,7376 | ||
=0 | [0;0,85] | 0,2624 | ||
0,12 lnx=0 | [1;2] | 1,1183 | ||
tg x tg3 x + tg5 x =0 | [0;0,8] | 0,3333 | ||
arcos x =0 | [0;1] | 0,5629 | ||
3x 4 ln x 5 =0 | [2;4] | 3,2300 | ||
cos 2 sin + = 0 | [1;2] | 1,8756 | ||
arcsinx = 0 | [0;1] | 0,7672 | ||
ex e -x 2 = 0 | [0;1] | 0,8814 | ||
Sin(lnx) cos(lnx) + 2lnx = 0 | [1;3] | 1,3749 | ||
x 2 + sin = 0 | [1,2;2] | 1,3077 | ||
ex +lnx 10x =0 | [3;4] | 3,5265 | ||
osx e-x + x 1 = 0 | [1;2] | 1,0804 | ||
1 x + sinx ln(1+x)=0 | [0;1,5] | 1,1474 | ||
3x 14 + ex e-x =0 | [1;3] | 2,0692 | ||
tgx = 0 | [0;1] | 0,5768 | ||
x+cos(x0,52+2)=0 | [0,5;1] | 0,9892 | ||
3ln2x+6lnx 5 =0 | [1;3] | 1,8832 | ||
Sinx2 + cos x2 10x = 0 | [0;1] | 0,101 | ||
x2 ln(1+x) 3 =0 | [2;3] | 2,0267 | ||
2x sinx cosx =0 | [0,4;1] | 0,6533 | ||
ex + 2 = 0 | [-1;0] | 0,2877 | ||
lnx x +1,8 = 0 | [2;3] | 2,8459 | ||
tgx =0 | [0,2;1] | 0,5472 | ||
tg ctg +x =0 | [1;2] | 1,0769 | ||
0,4+arctg x = 0 | [1;2] | 1,2388 | ||
cos =0 | [0;1] | 0,4538 | ||
0,6 3x 2,3x 3 =0 | [2;3] | 2,42 |
|
|
8.
.
.
:
h(y0 + 2y1 + + 2yn-1 + yn)/2,
h= , x0=a, x1=a+h, ,xn=b.
yi=f(xi), xi=a+ih, i=0,1, , n.
.
:
h(y0 + 4y1 +2y2 ++ 4y2n-1 + y2n)/6,
h= , x0=a, x1=a+h, ,x2n=b.
yi=f(xi), xi=a+ih, i=0,1, , 2n.
[a;b] | ||||
[1;3,5] | (lnx +1)3/2 2(lnx+1)1/2 + | |||
tg2x + ctg2x | [π/6; π/3] | tgxctgx 2x tg ctg + | ||
[2;3] | 2,3026(lnlnx ln ln2) | |||
[1;4] | ln3x | |||
[0;ln2] | 2 2arctg | |||
xexsinx | [0;1] | (x ex (sin x cos x) + ex cosx 1))/2 | ||
x shx | [0;2] | - | ||
[0;2] | ln(x+ ) ln3 | |||
sin | [1;2,5] | cos cos1 | ||
x arctgx | [0; ] | arctgx - + arctgx | ||
arcsin | [0;3] | arcsin - + arctg | ||
(1+lnx) | [1;3] | - 1 | ||
[0;1] | ln | |||
[1;2] | - 0,4 arccos - - + 0,4 arccos 0,4 | |||
23x | [0;1] | (23x 1) | ||
[0;1] | arctg x ln(x+ ) | |||
[0;2] | - ex + x + 0,5 | |||
sin2x | [0; ] | - sin 2x | ||
x2 | [0;1,99] | 2arcsin - sin(4 arcsin ) | ||
ex cos2 x | [0;π] | (1+ ) 0,6 | ||
(x lnx)2 | [1;e] | (9 ln2x 6 lnx + 2) - | ||
Arcsin | [0;3] | x arcsin - + arctg | ||
[0;1] | - arcsin() | |||
sinx ln(tgx) | [1;1,5] | ln(tg ) (cosx)(ln(tgx)) lntg0,5+ +(cos1) lntg1 | ||
[0;1,5] | ex tg | |||
[0;3/4] | (ln ) | |||
[0;1] | ||||
28 | [1;2] | |||
[1;2] | ||||
[1;2] |
9
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