Taylor Series/Error Term

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trouty323

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Derive the following formulas using Taylor series and then establish the error terms for each.

f ' (x) = (1/2*h) [4*f(x+h) - 3*f(x) - f(x+2h)]

f ' (x) = (1/4*h) [f(x+2h) - f(x-2h)]

I honestly have no idea how to go about solving this. I worked with Taylor series a long time ago, but I am stuck on this. Any help would be greatly appreciated! :D
 
Last edited:
trouty323 said:
Derive the following formulas using Taylor series and then establish the error terms for each.

f ' (x) = (1/2*h) [4*f(x+h) - 3*f(x) - f(x+2h)

f ' (x) = (1/4*h) [f(x+2h) - f(x-2h)

I honestly have no idea how to go about solving this. I worked with Taylor series a long time ago, but I am stuck on this. Any help would be greatly appreciated! :D
Click to expand...

What does your text-book say?

Did you try to use Google - to find out about error-term in Taylor's series?

Please tell us what you found.
 
Well, the text book does not go into much detail about this. It is very concise and does not show the steps. Also, my teacher's accent is very hard to understand. That is why I came here :). I did try to Google this, but had a hard time finding examples that are like mine. Even if I did find an example, it just looks like a bunch of letters and numbers to me. I do not remember Taylor series being this hard, which is why I'm very frustrated and confused.
 
Please tell us the Taylor's series expansion of:

f(x+h) and f(x+2h)

in terms of f(x), f'(x) and h.

We will take care of the error terms as we progress.
 
Is this what you mean?

f(h) + (f '(x+h)) + ((f ''(x+h)*x2)/(2)) + ((f '''(x+h)*x3)/(6)) + ... + ((f k(x+h)*xk)/(k!))

f(2h) + (f '(x+2h)) + (( f ''(x+2h)*x2)/(2)) + ((f '''(x+2h)*x3)/(6)) + ... + ((f k(x+2h)*xk)/(k!))
 
trouty323 said:
Is this what you mean?

f(h) + (f '(x+h)) + ((f ''(x+h)*x2)/(2)) + ((f '''(x+h)*x3)/(6)) + ... + ((f k(x+h)*xk)/(k!)) = ?

f(2h) + (f '(x+2h)) + (( f ''(x+2h)*x2)/(2)) + ((f '''(x+2h)*x3)/(6)) + ... + ((f k(x+2h)*xk)/(k!))= ?
Click to expand...

Yes and NO...

Read my question carefully....

also in your original post you have several open "[" - please correct those
 
I'm not sure what you mean by = ?

Should it be:

f(h) + (f '(x+h)) + ((f ''(x+h)*x2)/(2)) + ((f '''(x+h)*x3)/(6)) + ... + ((f k(x+h)*xk)/(k!)) = f(x+h)

f(2h) + (f '(x+2h)) + (( f ''(x+2h)*x2)/(2)) + ((f '''(x+2h)*x3)/(6)) + ... + ((f k(x+2h)*xk)/(k!)) = f(x+2h)
 
The form you want is:

f(x+h) = f(x) + h * f'(x).....
 
I'm sorry, I'm not following. I don't see how this coincides with my original formulas.
 
Then probably you need a face-to-face tutor - beyond me.....
 
All I really need is to see it worked out. I can only learn through examples. I'm not too concerned with fully understanding every aspect down to the last detail. This homework is due tomorrow, and after that we are moving on to different things.
 
f(x+h) = f(x) +h*f'(x) + (1/2!)(h2)f''(x) + (1/3!)(h3)f'''(x) + ...

Is this correct?
 
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