Maclaurin Series
SOUND COMPLICATED???
ITS PROBABLY THE EASIEST THING IN FP3 IF YOU KNOW YOUR WAY AROUND DIFFERENTIATION!!!!
BASICALLY, MACLAURIN SAID THAT:

Where f(x) is a function we want to find an expansion for.
EASY EXAMPLE:
Q: Find the first 3 non-zero terms in the Maclaurin Expansion of f(x) = Sin x
Well f(x) = Sin x
So f(0) = Sin 0 = 0
f'(x) = Cos x
So f'(0) = Cos 0 = 1
f''(x) = -Sin x
So f''(0) = - Sin 0 = 0
f'''(x) = - Cos x
So f'''(0) = - Cos 0 = -1
f''''(x) = Sin x
So f''''(0) = Sin 0 = 0
f'''''(x) = Cos x
So f'''''(0) = Cos 0 = 1
WE HAVE NOW FOUND OUR 3 NON-ZERO TERMS!!!!!
AS A SUMMARY:
f(0) = 0
f'(0) = 1
f''(0) = 0
f'''(0) = -1
f''''(0) = 0
f'''''(0) = 1
MACLAURIN SERIES:

So:
%20%2B%20%5Cfrac%7B0%7D%7B2%7Dx%5E2%20%2B%20%5Cfrac%7B-1%7D%7B3!%7Dx%5E3%20%2B%20%5Cfrac%7B0%7D%7B4!%7Dx%5E4%20%2B%20%5Cfrac%7B1%7D%7B5!%7Dx%5E5)
%20%2B%200%20%2B%20%5Cfrac%7B-1%7D%7B3!%7Dx%5E3%20%2B%200%20%2B%20%5Cfrac%7B1%7D%7B5!%7Dx%5E5)
THEREFORE THE ANSWER IS :


NOW LETS CHECK OUR ANSWER.
USING A CALCULATOR, SIN 0.5 = 0.479425538
PLUGGING 0.5 INTO THE EXPANSION, WE GET 0.479427083
EASY, ISN'T IT?????????
ALL YOU NEED TO KNOW IS DIFFERENTIATION
PLEASE FEEL FREE TO LEAVE HARDER EXAMPLES AND/OR CORRECT ANY MISTAKES ON THIS PAGE
Comments (2)
Jason said
at 1:25 pm on May 11, 2009
PLEASE LEAVE COMMENTS IF I MADE A MISTAKE :)
Steph Richards said
at 7:44 pm on May 11, 2009
Thank you for your contribution Jason. It is REALLY appreciated!
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