# إ ج½¼¼ ث ءث ؤ ئج إؤ ت

``` !
&quot; # \$# % &amp; # '
# % ( &quot; ) #
# &quot; * # +
) \$\$) &quot; &amp;) * )
\$# % # &quot;' \$# # ,# \$! -./)
/0 # /%
-
Qi,j (x) =
(x − xi−j )Qi,j−1 (x) − (x − xi )Qi−1,j−1(x)
xi − xi−j
/0% 1 -
Pn (x) = f [x0 ] +
n
f [x0 , x1 , &middot; &middot; &middot; , xk ](x − x0 )(x − x1 ) &middot; &middot; &middot; (x − xk−1 )
k=1
1 -
Pn (x) = f (x0 ) +
n s
k=1
k
Δk f (x0 )
- 1 -
Pn (x) =P2m+1 (x) = f (x0 ) +
sh
(f [x−1 , x0 ] + f [x0 , x1 ]) + s2 h2 f [x−1 , x0 , x1 ]
2
s(s2 − 1)h3
(f [x−2 , x−1 , x0 , x1 ] + f [x−1 , x0 , x1 , x2 ])
+
2
+ &middot; &middot; &middot; + s2 (s2 − 1)(s2 − 4) &middot; &middot; &middot; (s2 − (m − 1)2 )h2m f [x−m , &middot; &middot; &middot; , xm ]
s(s2 − 1) &middot; &middot; &middot; (s2 − m2 )h2m+1
(f [x−m−1 , . . . , xm ] + f [x−m , &middot; &middot; &middot; , xm+1 ])
+
2
2 1 -
n
k −s
Pn (x) = f (xn ) +
(−1)
∇k f (xn )
k
k=1
1 1
f (x0 ) =
f (x0 + h) − f (x0 ) h − f (ξ)
h
2
3'. 4' 1
1
h2
[−3f (x0 ) + 4f (x0 + h) − f (x0 + 2h)] + f (ξ)
2h
3
3'. 5 1
f (x0 ) =
1
h2
[f (x0 + h) − f (x0 − h)] − f (ξ)
2h
6
/%. 5 1
f (x0 ) =
1
h4
[f (x0 − 2h) − 8f (x0 − h) + 8f (x0 + h) − f (x0 + 2h)] + f (v) (ξ)
12h
30
/%. 4' 1
f (x0 ) =
1
h4
[−25f (x0 ) + 48f (x0 + h) − 36f (x0 + 2h) + 16f (x0 + 3h) − 3f (x0 + 4h)] + f (v) (ξ)
12h
5
&amp; ' 5 1
f (x0 ) =
1
h2 (iv)
[f
(x
−
h)
−
2f
(x
)
+
f
(x
+
h)]
−
f (ξ)
0
0
0
h2
12
6\$ 7# 1
h
h
1
Nj (h) = Nj−1
+ j−1
Nj−1
− Nj−1 (h)
2
2 −1
2
f (x0 ) =
15 f (3) 8.9 &quot;
% &quot; % &quot; %
i xi
Qi,0
0
Qi,1
1
3
a
1.828
b
2
√
1.707
1.747
c
1.764
1.726
1.737
8
8
:
5
Qi,2
Qi,3
/ )0 a! b c ) # Qi,4
1.691
20
% &quot; xi
f (xi )
0.0
0.2
0.4
0.6
0.8
1 1.2214 1.4918 1.8221 2.2255
f (0.1) f (0.7) ; 1! ; - 1 ; 2
1 \$) % #&quot;! &quot;# % 8
[0, 2] f (x) * S(x) &quot; # #
0≤x≤1
1 + 2x − x3 ,
S(x) =
2
3
2 + b(x − 1) + c(x − 1) + d(x − 1) , 1 ≤ x ≤ 2
% )0 b! c d &quot; :
15
15
% &quot; xi
−0.92
f (xi ) 84.64
−0.86 −0.84
73.96 70.56
−0.78
60.84
−0.74 −0.72
54.76 51.84
−0.66
43.56
f (−0.84) f (−0.78) % #&quot; ' ) &quot;#! % &lt;
15 f (x) = xe2x * f (0.5) h = 0.01 &quot; %
# % \$ ' &quot; &quot; =
20
h &gt; 0 ' * *
f (x0 + h) − f (x0 ) h h2 f (x0 ) =
− f (x0 ) − f (x0 ) + O(h3 )
h
2
6
* &quot; / )0 f (x) = ln x &middot; tan x * x0 = 3 .
h = 0.4 * * O(h3 ) %
&quot; &gt;
```