numerical methods laboratory exercise
DESCRIPTION
Numerical Methods Laboratory Exercise.5 Samples of Taylor Series. Computations and in Excel.TRANSCRIPT
ADAMSON UNIVERSITY
College of Engineering
Electronics Engineering Department
2nd Sem. S. Y. 2015-2016
TAYLOR SERIES EXPANSION
Exercise #1
HERRERA, AILEEN MAY S.
Numerical Methods Lab.
59032
Friday 14:00-17:00
____________________________________
Engr. Alain Bernard C. Rañola
Given a: L6 =17
L1 =PI()/180
L2 =L6*L1
L3 =SIN(L2)
L4 =COS(L2)
L5 =(EXP(L2))
1.
f ( x )=sin x
I2 = SIN(B2*L1)
f ( x )=f (a )+ f ' (a ) (x−a )+ f ' ' (a ) ( x−a )2
2!+f ' ' ' (a ) ( x−a )3
3 !+f IV (a ) ( x−a )4
4 !
H2 = C2 + D2 + E2 + F2 + G2
f ( x )=sin x
f (a )=sin a
C2 = L3
f ' ( x )=cos x
f ' (a ) ( x−a )=cos a ( x−a )
D2 = L4 * (B2*L1 - L2)
f ' ' (x )=−sin x
f ' ' (a ) (x−a )2
2!=−sina
( x−a )2
2
E2 = -L3 * (B2*L1 - L2)^2 * ½
f ' ' ' (x)=−cos x
f ' ' ' (a ) ( x−a )3
3 !=−cosa
( x−a )3
6
F2 = -L4 * (B2*L1 - L2)^3 * 1/6
f IV ( x )=sin x
f IV (a ) ( x−a )4
4 !=sina
( x−a )4
24
G2 = L3 * (B2*L1 - L2)^4 * 1/24
2.
f ( x )=x sin x−1
I3 = ( B3*L1 * SIN(B3*L1) ) – 1
f ( x )=f (a )+ f ' (a ) (x−a )+ f ' ' (a ) ( x−a )2
2!+f ' ' ' (a ) ( x−a )3
3 !+f IV (a ) ( x−a )4
4 !
H3 = C3 + D3 + E3 + F3 + G3
f ( x )=x sin x−1
f (a )=a sina−1
C3 = L2*L3 - 1
f ' ( x )=x cos x+sin x
f ' (a ) ( x−a )=(acosa+sin a ) (x−a )
D3 = (L2*L4 + L3) * (B3*L1 - L2)
f ' ' ( x )=x (−sin x )+cos x (1 )+cos x=−x sin x+2cos x
f ' ' (a ) (x−a )2
2!=(−a sina+2cosa ) (x−a )2
2
E3 = (-L2*L3 + 2*L4) * (B3*L1 - L2)^2 * ½
f ' ' ' ( x )=−x cos x+sin x (−1 )+2 (−sin x )=−x cos x−3 sin x
f ' ' ' (a ) ( x−a )3
3 !=(−acos a−3 sina ) ( x−a )3
6
F3 = (-L2*L4 - 3*L3) * (B3*L1 - L2)^3 *1/6
f IV ( x )=−x (−sin x )+cos x (−1 )−3cos x=x sin x−4 cos x
f IV (a ) ( x−a )4
4 !=(a sin a−4 cosa ) ( x−a )4
24
H3 = (-L2*L3 - 4*L4) * (B3*L1 - L2)^4 * 1/24
3.
f ( x )=esin x
I4 = EXP(SIN(B4*L1))
f ( x )=f (a )+ f ' (a ) (x−a )+ f ' ' (a ) ( x−a )2
2!+f ' ' ' (a ) ( x−a )3
3 !+f IV (a ) ( x−a )4
4 !
H4 = C4 + D4 + E4 + F4 + G4
f ( x )=esin x
f (a )=esin a
C4 = EXP(L3)
f ' ( x )=esin x cos x
f ' (a ) ( x−a )=esin a cosa ( x−a )
D4 = EXP(L3) * L4 * (B4*L1 - L2)
f ' ' (x)=esin x (−sin x )+cos x (esin x) (cos x )=−esin xsin x+esin xcos2 x
f ' ' ( x )=−esin x (sin x−cos2 x )
f ' ' (a ) (x−a )2
2!=−esin a (sin a−cos2a ) ( x−a )2
2
E4 = -EXP(L3) * (L3 - L4^2) * (B4*L1 - L2)^2 * ½
f ' ' ' ( x )=−esin x [cos x−2 (−sin x ) (cos x ) ]+ (sin x−cos2 x ) (−esin x ) (cos x )
f ' ' ' ( x )=−esin x (cos x+2 sin xcos x )−esin x (sin xcos x−cos3 x )
f ' ' ' ( x )=−esin x (cos x+3 sin xcos x−cos3 x )=−esin x cos x ( 1+3 sin x−cos2 x )
f ' ' ' (a ) ( x−a )3
3 !=−esina cosa (1+3 sin a−cos2a ) ( x−a )3
6
F4 = -EXP(L3) * L4 * (1 + 3*L3 - L4^2) * (B4*L1 - L2)^3 * 1/6
fIV ( x )= d
dx[−esin x (cos x+3 sin xcos x−cos3 x ) ]
f IV ( x )=−esin x [−sin x+3 {sin x (−sin x )+cos x cos x }−3(−sin x) (cos2 x ) ]+( cos x+3 sin x cos x−cos3 x ) (−esin x ) cos x
f IV ( x )=−esin x (−sin x+4 cos2 x−3sin2 x+6cos2 x sin x−3cos4 x )
f IV (a ) ( x−a )4
4 !=−esin a (−sina+4cos2a−3 sin2a+6cos2a sina−3 cos4a ) ( x−a )4
24
G4 = -EXP(L3) * (-L3 + 4*L4^2 - 3*L3^2 + 6*L3*L4^2 - 3*L4^4) * (B4*L1 - L2)^4 * 1/24
L8 = EXP(L2)
L9 = L8^2
L10 = L8^3
L11 = L8^4
L12 = SIN(L8*L1)
L13 = COS(L8*L1)
L14 = (B5*L1 - L2)
4.
f ( x )=cosex
I5 =COS(EXP(B5*L1)*L1)
f ( x )=f (a )+ f ' (a ) (x−a )+ f ' ' (a ) ( x−a )2
2!+f ' ' ' (a ) ( x−a )3
3 !+f IV (a ) ( x−a )4
4 !
H5 = C5 + D5 + E5 + F5 + G5
f ( x )=cosex
f (a )=cos ea
C5 = L13
f ' (x)=−ex sin ex
f ' (a ) ( x−a )=−ea sin ea (x−a )
D5 = -L8*L12 * L14
f ' ' ( x )=−ex cosex ex+sin ex (−ex )=−e2 xcos ex−ex sin ex
f ' ' (a ) (x−a )2
2!=(−e2a cosea−ea sin ea ) ( x−a )2
2
E5 = (-L9*L13 - L8*L12) * (L14^2) * 1/2
f ' ' ' ( x )=−e2 x (−sin ex) (ex)+cosex (−e2x )(2)−ex cose xex+sin ex (−ex )
f ' ' ' (x)=e3xsin ex−3e2 xcos ex−ex sin ex
f ' ' ' (a ) ( x−a )3
3 !=(e3asin ea−3 e2a cosea−ea sin ea ) ( x−a )3
6
F5 = (L10*L12 - 3*L9*L13 - L8*L12) * (L14^3) * 1/6
f IV ( x )=e3 xcos exex+sin ex e3x (3 )−3 {e2x (−ex sin ex )+cose x (e2x ) (2 ) }−excos ex ex+sin ex (−ex)
f IV ( x )=e4x cosex+6e3x sin ex−7e2x cosex−e xsin ex
f IV (a ) ( x−a )4
4 !=(e4 acos ea+6e3a sin ea−7e2a cosea−ea sin ea ) ( x−a )4
24
G5 = (L11*L13 + 6*L10*L12 - 7*L9*L13 - L8*L12) * (L14^4) * 1/24