Date: Oct 23, 2012 5:36 PM
Author: Doug Kuhlmann
Subject: RE: [ap-calculus] e

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Erin.

That it collapses to 1 falls under the category "lies my calculator tells me." I will use the TI-84 as an example but they all do something like this. The TI-84 stores 14 digits plus exponent (i.e. power of ten) up to +/-99. (It only displays 10 digits, calling the last four "guard digits".) Try this with your kids: Have them type in 1?1+E-20 and hit ENTER. They will get E-20 (which is 10^-20) for an answer. Now have them type 1+E-20?1. Now the answer is 0. Why? In the first case it takes 1-1 getting 0 and then adds E-20 to it. In the second case it first adds 1+E-20 which should be 1.00000000000000000001 but it can only keep the first 14 digits so it stores 1.0000000000000. Now when it subtracts 1 from this it gets 0. The TI-84 cannot tell the difference between 1 and 1+E-20.

So when your students let x=10^18, say, in (1+1/x)^x, the calculator first does 1+1/10^18 but this gets stored as 1. Since 1 to any power is 1 your kids see the "answer" as 1. They are pushing the calculator beyond its capacity. As they do us often.

HTH

Doug
----------------------------
Doug K├╝hlmann
Math Dept
Phillips Academy
Andover, MA 01810

________________________________________
From: erinslong@yahoo.com [erinslong@yahoo.com]
Sent: Tuesday, October 23, 2012 7:37 AM
To: AP Calculus
Subject: [ap-calculus] e

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So I had a pre calculus student ask me why the lim x->inf (1+1/x)^x is e. I can prove it, but I've never noticed until he showed me this that in the calculator, the graph of (1+1/x)^x actually goes to one when you plug EXTREMELY large numbers into the table values. Does anyone know how to explain this?

Greatly appreciate your thoughts!

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