maclaurins theorem

[watford73]

Hi,I need help! We’re doing second order approximation which is ok but to find more accurate results, we have to use MACLAURINS THEOREM, I’ve been looking at books for weeks now and can’t get a grip of it.

The question is “using Maclaurins Theorem or otherwise, obtain the power series for

e to the power of -kh

We have to show how we differentiate it for f `, f `` etc..
i.e f `=, f``=
We then have to put it into a polynominal to recheck our results,I hope this means something to someone out there!!
THANKS FOR YOUR TIME!!

[krtxmrtz]

...
We have to show how we differentiate it for f `, f `` etc..
i.e f `=, f``=
...
We then have to put it into a polynominal to recheck our results,I hope this means something to someone out there!!
THANKS FOR YOUR TIME!!

Welcome the the forums!

So that's 2 questions, how you calculate the derivatives of f for various orders and then use them to build the polynome. Do you need any help with the former? Once you've got the derivatives all you have to do is apply McLaurin's formula:

f(x) = f(0)*x⁰ / 0! + f'(0)*x¹ / 1! + f''(0)*x² / 2!

there are more terms but that's enough for second order.

In your case you have to use h instead of x, I assume k is a constant.

[watford73]

Hi ther krtxmrtz, thanks for the warm welcome!

Yes you've guessed it I need help for the former too,is it possible that e to the -kh is all constant? the question is about atmospheric pressure in the form of p(pressure)=A(intercept(109.95))e to the power of
-k(gradient(.00014)h(height)

we've already done a basic second order approximation to find a table of results for p at intervals of 1000m from 0m to 10000m but we now have to do it using Maclaurins,everytime I try to I end up with either similar answers to the basic second order approximation or some other crazy numbers,the basic second order approximation we used was

1-kh+(kh squared) / 2!

apologies for the way this is written but I can't find a superscript function!!
thanks again for your time,I hope this is a little clearer to you than it is to me.

[zaza]

So do you know how to differentiate at all? If not, then this is going to be tricky, although the derivative of e^ax is about as easy as it gets, namely ae^ax.

For Maclaurin's series expansions, you evaluate this when your variable is 0, which for e^x is again easy because it is going to give you 1.

Your variable is h; you are calculating how pressure changes with height. You should find that the Maclaurin series gives you the same result as your second order approximation.


zaza

[watford73]

zaza,thats all I wanted to hear!! everytime for the last month I tried something different and got the same answers,I was looking for a different set of answers,I can now finish, eternally grateful..