The formula sheet

Equation Sheet
εσ
0.5 10 n −
= ×
( n)
f′′ ( a) 2 f ( a)
n
f ( x) = f( a) + f′ ( a)( x− a) + ( x− a) + … + ( x− a) + R
2! n!
( n+
1)
f ( ξ )
Rn = ( x−a) ( n + 1)!
xl + xu
xr
=
2
l u
n+
1
f ( xu)( xl − xu)
xr = xu
−
f ( x ) − f( x )
n
2
r = ∑ i
i=
1
S e
∑ ∑ ∑
∑ ∑
n x y − x y
a =
r
i i i i
1
n
2
xi − (
2
xi)
S − S
=
S
2 t r
t
f( x ) − f( x )
f ( x) = f( x ) + ( x−x )
1 0
1 0
x1 − x0
0
f x x x x
ε
x − x
new old
a = r r
new
xr u l =
xu + xl
1
x − x
f ( xi
)
xi+ 1 xi
f ( x )
= − ′
i
n
| Δx |
≤ eps with eps = b
| x |
E
1−t
Δx
=
2
0
n
a n
− f′′ ( x )
E ≅ E
r 2
ti , + 1
2 f′ ( xr)
ti ,
n
t = ∑ (
i=
1
i −
2
)
T T
( Z Z) A= ( Z Y)
S y y
a0 = y − a1x f [ x , x ,..., x ] − f[ x , x ,..., x ]
n n−1 1 n−1 n−2
0
[ n, n−1,...,
1, 0]
=
xn−x0 r =
1 n
x = ∑ xi
n =
∑ ∑ ∑
∑ ∑ ∑ ∑
i 1
n xiyi − ( xi)( yi)
n x −( x ) n y −(
y )
2 2 2 2
i i i i
f ( x) = f( x ) + f[ x , x ]( x− x ) + f[ x , x , x ]( x−x )( x− x ) + f[ x , x ,..., x ]( x−x )( x−x )...( x− x )
n 0 1 0 0 2 1 0 0 1 n n−1 0 0 1 n−1
b
∫ ( )
b
∫ n ( )
a a
I= f xdx≅ f xdx
f ( a) + f( b)
I ≅( b− a)
2
n−1
∑
f ( x0) + 2 f( xi) + f( xn)
I ≅( b−a) i=
1
2n
f ( x0) + 4 f( x1) + f( x2)
I ≅( b− a)
6
3
( b−a) Et=− f′′ ( ξ )
12
3
( b−a) Et =− f ′′ 2
12n
( b−a) 2880
5
Et=− (4)
f ( ξ )

n−1 n−2
∑ ∑
f ( x0) + 4 f( xi) + 2 f( xj) + f( xn)
I ≅( b−a) i= 1,3,5
3n
j=
2,4,6
f ( x0) + 3 f( x1) + 3 f( x2) + f( x3)
I ≅( b− a)
8
5
Et=− (4)
f ( ξ )
2
( b−a) 6480
5
( b−a) Et =− f 4
180n
5
( b−a) Et =− f 4
80n
dy
f ( yt , )
dt = yi+ 1 yi f( yi, ti) h
= + h= ti+ 1 − ti
0 ⎧ yi+ 1 = yi + f( yi, ti) h
⎪
0
⎨ f( yi, ti) + f( yi+ 1, ti+
1)
⎪yi+
1 = yi +
h
⎩
2
yi+ 1 yi φh
1
φ ( k1 k2)
2
k1f( yi, ti)
k2 f( yi k1h, ti h)
= + ⎧
⎪
⎪ = +
⎨
⎪ =
⎪
⎪⎩ = + +
5 −1
R = xi+ 1 xi h f( xi)
2
= − ∇
(4)
(4)
yi+ 1 yi φh
1
φ ( k1 2k2 2 k3 k4)
6
k1f( yi, ti)
h h
k2 f( yi k1 , ti
)
2 2
h h
k3 f( yi k2 , ti
)
2 2
k4 f( yi k3h, ti h)
= + ⎧
⎪
⎪ = + + +
⎪
⎪ =
⎪
⎨
⎪ = + +
⎪
⎪
⎪
= + +
⎪
⎩ = + +
fs
= fN >
B
2