How Do You Calculate Error Bounds for Maclaurin Polynomial Approximations?

In summary, the conversation is about finding the upper bound on the error and an approximation for the integral I, with the given integrand and nodes. The 5th degree polynomial for the integrand is incorrect and should be calculated using the Maclaurin series. The conversation also discusses the method of finding the upper bound for the error and integrating the polynomial to approximate I.
  • #1
chief10
78
0
Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam...

Let's say:

I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5

The 5th degree polynomial is 1 + x^2 + (1/2)(x^4)

So my queries are:

How would I go about finding the upper bound on the error from 0 to 0.5? - My working gives 0.012
How do I get an approximation of I by integrating? - my working gives 0.545
How would I get an upper bound on the integration in the previous question?

Thanks a lot guys and girls.
 
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  • #2
chief10 said:
Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam...

Let's say:

I = integral[e^(x^2)]dx with nodes being x=0 to x=0.5

I calculated that the 5th degree poly of integrand e^(x^2) is
1+ (x^2) + (1/2)(x^4) + (24/25)(x^5) ----- correct me if I'm wrong here but i think it's okay
This is incorrect. The first few terms of the Maclaurin series for et are 1 + t + t2/2! + t3/3! + t4/4! + ...

If you replace t by x2, what do you get?


chief10 said:
So my queries are:

How would I go about finding the upper bound on the error?
How do I get an approximation of I? I'm guessing I'm going to have to integrate right?

Thanks a lot guys and girls.
 
  • #3
i was working under this assumption:

P(x) = f(a) + f'(a)(x-a) + [f''(a)(x-a)^2]/2! + ...etc..

where a=0 for Maclaurin

hmm
 
  • #4
Mark44 said:
This is incorrect. The first few terms of the Maclaurin series for et are 1 + t + t2/2! + t3/3! + t4/4! + ...

If you replace t by x2, what do you get?

alright going by that I've computed which i double checked with the taylor series for a=01 + x^2 + (1/2)(x^4)any ideas on the rest?
 
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  • #5
chief10 said:
alright going by that I've computed which i double checked with the taylor series for a=0


1 + x^2 + (1/2)(x^4)


any ideas on the rest?
So integrate! You know how to integrate that, don't you?
 
  • #6
HallsofIvy said:
So integrate! You know how to integrate that, don't you?

no need to snap at me.. lol.. of course i do - look at my markings next to the bolded questions

i just thought i could get some insight and to see if i was doing it correctly
 

FAQ: How Do You Calculate Error Bounds for Maclaurin Polynomial Approximations?

What is a Maclaurin Polynomial of an integral?

A Maclaurin Polynomial of an integral is a method used in calculus to approximate the value of an integral by using a polynomial function. This allows for easier calculation and can provide a close estimate to the exact value of the integral.

How do you find the Maclaurin Polynomial of an integral?

To find the Maclaurin Polynomial of an integral, you can use the Maclaurin series expansion method, which involves finding the derivatives of the function at the point where the integral is being evaluated. These derivatives are then used to construct the polynomial function.

What is the importance of Maclaurin Polynomials in calculus?

Maclaurin Polynomials are important in calculus because they allow for the approximation of complex integrals, which may not have a simple closed form solution. They also provide a way to evaluate integrals using only basic algebraic operations and are used in many real-world applications.

Can Maclaurin Polynomials be used to find the exact value of an integral?

No, Maclaurin Polynomials can only provide an approximation of the value of an integral. The accuracy of the approximation depends on the degree of the polynomial used. As the degree increases, the approximation becomes closer to the exact value, but it will never be an exact match.

Are there any limitations to using Maclaurin Polynomials for integrals?

Yes, there are limitations to using Maclaurin Polynomials for integrals. They can only be used for integrals that have a finite interval and can only approximate the value within that interval. They also may not provide an accurate approximation for highly oscillatory functions or functions with singularities.

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