Math Help - maclaurins theorem

1. maclaurins theorem

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-kh

We have to show how we differentiate it for f , f  etc..
We then have to put it into a polynominal to recheck our results,I hope this means something to someone out there!!

2. Originally Posted by 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-kh

We have to show how we differentiate it for f , f  etc..
We then have to put it into a polynominal to recheck our results,I hope this means something to someone out there!!
Mclaurin series for a function $f(x)$ is:

$f(x) = f(0) + f'(0)x + f''(0) x^2/2 + ... +f^{(n)}x^n/n!+ ..$

Now if:

$f(x)=e^{-kx}$

(I am assuming $h$ is your variable and $k$ a constant and using $x$ instead of $h$)

Then:

$
f'(x)=-ke^{-kx}
$

$
f''(x)=-k^2 e^{-kx}
$

and the n-th derivative:

$
f^{(n)}=(-k)^{n} e^{-kx}
$

so:

$e^{-kx} = 1 + (-k)x + (-k)^2 x^2/2 + ... +(-k)^nx^n/n!+ ..$

or:

$e^{-kx} = 1 + (-1)(kx) + (-1)^2 (kx)^2/2 + ... +(-1)^n(kx)^n/n!+ ..$

RonL

3. maclaurins theorem

hello there captain black,
I feel I'm inching closer to what my tutor wants.

The assignment is about atmospheric pressure using the equation

p = Ae to the power of -kh

We were doing second order approximations to find the values of p pressure
at different heights (h) from 0m to 10000m at intervals of 1000m using our intercept (A)(109.95) and our gradient k(-.00014),the second order approximation we were using was;

1-kh+kh2/2! but every time I try to use Maclaurins for it,I end up with similar answers or just a constant answer?

I'm tearing my hair out!!
p.s I think e to the power of -kh may be constant (if this is possible)

thanks again..

4. Originally Posted by watford73
hello there captain black,
I feel I'm inching closer to what my tutor wants.

The assignment is about atmospheric pressure using the equation

p = Ae to the power of -kh

We were doing second order approximations to find the values of p pressure
at different heights (h) from 0m to 10000m at intervals of 1000m using our intercept (A)(109.95) and our gradient k(-.00014),the second order approximation we were using was;

1-kh+kh2/2! but every time I try to use Maclaurins for it,I end up with similar answers or just a constant answer?

I'm tearing my hair out!!
p.s I think e to the power of -kh may be constant (if this is possible)

thanks again..
Do you know $A$ and $k$?

$p(h) \approx A[1-k h +(kh)^2/2]$