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

In summary: I'm not sure what you are asking. Can you explain it a bit more? ]Yes, Lagrange's form is a form of the error that gives you an estimate for the error. But it is not the real error. The real error is what you get after you use Lagrange's form to estimate the error.
  • #1
caelestis
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??
 
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  • #2
What is the definition of the Taylor polynomial of degree 3?
 
  • #3
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??
 
  • #4
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:
  • #5
Oh ok, I didn't realize that...
So from there do you know how i can find the error??
 
  • #6
caelestis said:
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
 
Last edited by a moderator:
  • #7
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?
 
  • #8
Yes, it gives you an expression which you have to estimate.
 
  • #9
ummm, I don't really understand... I've tried looking in my textbook but it doesn't help much...
 

1. What are Taylor Polynomials?

Taylor Polynomials are mathematical expressions used to approximate a function around a specific point. They are a series of terms involving the derivatives of the function evaluated at the point of interest.

2. How do you use Taylor Polynomials to approximate a function?

To use Taylor Polynomials, you first choose a point around which you want to approximate the function. Then, you calculate the derivatives of the function at that point. Finally, you plug these values into the Taylor Polynomial formula to get the approximation.

3. What is the purpose of using Taylor Polynomials?

The purpose of using Taylor Polynomials is to approximate complicated functions with simpler ones. This can make it easier to perform calculations or analyze the behavior of the function.

4. Can Taylor Polynomials be used for any function?

Yes, Taylor Polynomials can be used for any function. However, the accuracy of the approximation depends on the smoothness of the function and the number of terms in the polynomial.

5. How do you know if a Taylor Polynomial is a good approximation?

The accuracy of a Taylor Polynomial can be determined by comparing it to the original function. Generally, the more terms included in the polynomial, the better the approximation will be. Additionally, the closer the point of approximation is to the function's point of interest, the more accurate the approximation will be.

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