How Do You Calculate a Degree 3 Taylor Polynomial for e^x?

[caelestis]

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??

 

[Pere Callahan]

What is the definition of the Taylor polynomial of degree 3?

 

[caelestis]

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??

 

[Pere Callahan]

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.

 

[caelestis]

Oh ok, I didn't realize that...
So from there do you know how i can find the error??

 

[Pere Callahan]

Oh ok, I didn't realize 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

 

[caelestis]

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?

 

[Pere Callahan]

Yes, it gives you an expression which you have to estimate.

 

[caelestis]

ummm, I don't really understand... I've tried looking in my textbook but it doesn't help much...