numerical methods laboratory exercise

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