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Please help me with Taylor Polynomials

  • Thread starter caelestis
  • Start date
12
0
Hello,

I'm having trouble with this question and was wondering if someone could give me hints or suggestions on how to solve it. Any help would be greatly appreciated thankyou! :)


Find the Taylor polynomial of degree 3 of f (x) = e^x

about x = 0 and hence find an approximate value for e. Give an estimate for the error in the approximation.



I know from following the Big O Notation...

e^x = 1 + x + (x^2) / 2! + (x^3) / 3! + ... + (x^n) / n! + O (x^(n+1))

So I'm thinking for a polynomial of degree 3 we have...

e^x = 1 + x + (x^2) / 2! + (x^3) / 3! + O (x^4)


And so from there i'm really not sure what comes next?? Could someone help me please??
 

Answers and Replies

What is the definition of the Taylor polynomial of degree 3?
 
12
0
I think the Taylor polynomial of degree 3 is the last line I have above... after substituting
n = 3 in the Taylor formulae (next equation up).

But I'm not sure if I'm doing it the right way?? Any help please??
 
Your last line is not a polynomial (due to the big O term).

If you omit it, your expression is indeed the Taylor polynomial of degree three.
 
Last edited:
12
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Oh ok, I didn't realise that...
So from there do you know how i can find the error??
 
Oh ok, I didn't realise that...
So from there do you know how i can find the error??
Yes, I do.
Are you familiar with the various forms of the remainder term in a Taylor approximation?
If not you should look it up in your textbook or on wikipedia

"[URL [Broken]
 
Last edited by a moderator:
12
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Ah yes! We use Lagrange's form for the error. But doesn't that just give me an expression for the error? And not a value for the error which the question asked?
 
Yes, it gives you an expression which you have to estimate.
 
12
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ummm, I don't really understand... I've tried looking in my text book but it doesn't help much...
 

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