Which is Greater?

Date: 2/12/96 at 21:37:0
From: Raul Daniel Digestani
Subject: Help

Please help me to solve the following problem:

Let be A = 10001^9999  and  B = 10000^10000 - Which is greater? Why?

Thanks.
Raul Daniel Digestani
Buenos Aires - Republica Argentina

Date: 9/1/96 at 18:36:36
From: Doctor James
Subject: Re: Help

Let B = 10000^10000=10000 * 10000^9999

We can use the approximation f(x+h) ~= f(x) + d(f(x))/dx * h, where
x = 10000 and h=1, and f(x) = x^9999

d(f(x))/dx = 9999 * x^9998, so
f(1001) ~= 10000^9999 + 9999 * 10000^9998 * 1,  where ~= means 
approximately
        ~= 10000^9999 + 10000 * 10000^9998      equal to.
         = 10000^9999 + 10000^9999
         = 2 * 10000^9999,

which is much less than

10000^10000 = 10000 * 10000^9999.
So B > A.

-Doctor James,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/