How Do You Calculate Error Bounds for Maclaurin Polynomial Approximations?

[chief10]

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.

 

[Mark44]

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 e^t are 1 + t + t²/2! + t³/3! + t⁴/4! + ...

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

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.

 

[chief10]

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

 

[chief10]

This is incorrect. The first few terms of the Maclaurin series for e^t 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?

 

[HallsofIvy]

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?

 

[chief10]

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