Deret Taylor

KAKULUS 1 MODUL 11 X. DERET MACLAURIN, MACLAURIN, BINOMIAL, BINOMIAL, TAYLOR TAYLOR 10.1. Deret Maclaurin

Misalkan f(x) = a + bx + cx2 + dx3 + ex4 + fx5 + ........ Masukkan x = 0, maka f(0) = a + 0 + 0 + 0 + ........... ...... .....

a

= f(0)

Diferensialkan: f '(x) = b + c.2x + d.3x2 + e.4x3 + f.5x4 + .......... Masukkan x = 0, maka f' (0) = b + 0 + 0 + ...........+.... ....... ....+....b = f' (0) Diferensialkan: f ''(x) = c.2.1 + d.3.2x + e.4.3x2 + f.5.4x3 + ......... Masukkan x = 0, maka f(0) = c.2! + 0 + 0 + ............... ........... ....c =

f  ′′(0) 2!

Diferensialkan: f '''(x) = d.3.2.1 + e.4.3.2.x + f.5.4.3.x2 + ......... Masukkan x = 0, maka f'''(0) = d.3! + 0 + 0 + ...........d =

f  ′′′(0) 3!

Dan seterusnya .....sehingga diperoleh Deret Maclaurin : f(x) = f(0) + f'(0) x +

f  ′′(0) 2!

2

x +

f  ′′′(0) 3!

3

x + .....+

f  ( n ) (0) n!

xn + ......

Contoh:

1. Perderetkan f(x) = ex dalam deret Maclaurin. Jawab: f(x) = f'(x) = f''(x) = f'''(x) = ....... = f (n)(x) = ex f(0) = f'(0) = f''(0) = f'''(0) = ...... = f (n)(0) = 1 ........................................................................ x

Maka e = 1 + x +

x2 2!

+

x3 3!

+ .....

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2. Perderetkan f(x) = sin x dalam deret Maclaurin. Jawab: f(0) = 0 f'(x) = cos x



f'(0) = 1

f''(x) = -sin x



f''(0) = 0

f'''(x) = -cos x



f'''(0) = -1

f'(iv)(x) = sin x



f'(iv)(0) = 0

f (v)(x) = cos x



f (v)(0) = 1

.................................................. Maka sin x = x -

x3 3!

+

x5

.....

5!

3. Perderetkan f(x) = cos x dalam deret Maclaurin. Jawab: f(0) = 1 f'(x) = - sin x



f'(0) = 0

f''(x) = -cos x



f''(0) = -1

f'''(x) = sin x



f'''(0) = 0

f'(iv)(x) = cos x



f'(iv)(0) = 1

f (v)(x) = - sin x



f (v)(0) = 0

.................................................. Maka cos x = 1 -

x2 2!

+

x4 4!

- .....

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4. Perderetkan f(x) = ln (1 + x) dalam deret Maclaurin. Jawab: f(0) = ln 1 = 0 f'(x) = (1+x)-1



f'(0) = 1

f''(x) = -(1+x)-2



f''(0) = -1 = 1 !

f'''(x) = 2(1+x)-3



f'''(0) = 2 = 2 !

f'(iv)(x) =-6(1+x)-4



f'(iv)(0) = - 3 = - 3!

f (v)(x) = 24(1+x)-5



f (v)(0) = 24 = 4!

.................................................. x

Maka ln (1 + x) = x -

2

+

2

x3 3

-

x

4

4

+

x

5

5

- ...............

Rangkuman

1. ex

x2

=1+x+

2. sin x = x -

x

3

3. cos x = 1 -

x2 2!

x5 5!

2

2

7!

+

+ x

-

3

x8

4

-∞