Re: TI-M: e^x, sin x, cos x
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Re: TI-M: e^x, sin x, cos x
Sure... The examples you state are just specific applications of Taylor's
Formula. The premise behind Taylor's formula is that you can approximate
any function by a polynomial. The more terms you add to this polynomial,
the closer you can approach the actual function. As the number of terms
goes to infinity the error between the function approaches 0. The formula
for this is
n
____
\ (k)
\ f (a) k
T f(x) = / --------- (x-a)
n / k!
----
k=0
Where f^(k) is the kth derivative of f and a is the x value about which
the approximation is centered at. One can derive this in the special case
where a=0 quite easily (this is the case your examples use). Tom M.
Apostol's Calculus Volume I derives this using the following strategy:
We have a function f that has n derivatives at point x=0. We want a
polynomial we will call P(x) that will agree with f and its n derivatives
at 0. This implies we need to satisfy several equations (in fact n+1)
P(0) = f(0), P'(0) = f'(0), P''(0) = f''(0), ..., P^(n)(0)=f^(n)(0).
The polynomial we will write will be of the general form
P(x) = C0 + C1 X + C2 X^2 + ... + CN X^n
where C0, C1, ..., CN are coefficients that we will determine.
1. take P(0) then it is clear C0 = f(0).
2. take P'(0) then C1 = f'(0).
3. take P''(0) then 2 C2 = f''(0) => C2 = f''(0)/2
4. take P'''(0) then 6 C3 = f'''(0) => C3 = f'''(0)/6 = f'''(0)/3!
5. ...
n+1. take P^(n)(0) then n! Cn = f^(n)(a)
Thus we can write the polynomial, because we know the constants...
P(x) = f(0) + f'(0) x + f''(0) x^2 / 2 + ... + f^(n) x^n/n!
= sum( d(f(0),x,k) x^k / k!, k, 0, n)
or in TeX form :)
$$ \sum_0^n \frac{f^{(n)}(a)}{k!} x^k $$
This proof can be extended to other center points easily. There is also
another form of taylor's approximation that has a remainder term, and that
can be proved using integration by parts...
:)
--
Andy Selle <aselle@ticalc.org>
Programming and System Administration, Survey Editor, Accounts Manager
the ticalc.org project - http://www.ticalc.org/
On Thu, 18 May 2000 JayEll64@aol.com wrote:
>
> I know one definition of e^x is:
> e^x = x^0/0! + x^1/1! + x^2/2! + x^3/3! + ... + x^n/n! + ...
> = 1 + x + x^2/2 + x^3/3! + x^4/4! + ... x^n/n! + ...
>
> Similarly, sin x and cos x can be defined by an infinite sum:
> sin x = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! - ...
> cos x = 1 - x^2/2 + x^4/4! - x^6/6! + x^8/8! - ...
>
> Does anyone know any proofs of these? I sort of remember the e^x infinite
> sum coming from the definition of e, but I can't seem to recall it and don't
> want to bother finding it ;)
>
> Thanks for the insight,
>
> JayEll
>
> P.S.: I'm not 100% sure these infinite sums are correct; they're coming from
> memory, so correct them if they're wrong...
>
>
References: