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RE: [apcalculus] e
Posted:
Oct 23, 2012 5:36 PM


NOTE: This apcalculus EDG will be closing in the next few weeks. Please sign up for the new AP Calculus Teacher Community Forum at https://apcommunity.collegeboard.org/gettingstarted and post messages there.  Erin.
That it collapses to 1 falls under the category "lies my calculator tells me." I will use the TI84 as an example but they all do something like this. The TI84 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+E20 and hit ENTER. They will get E20 (which is 10^20) for an answer. Now have them type 1+E20?1. Now the answer is 0. Why? In the first case it takes 11 getting 0 and then adds E20 to it. In the second case it first adds 1+E20 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 TI84 cannot tell the difference between 1 and 1+E20.
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 Kühlmann 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: [apcalculus] e
NOTE: This apcalculus EDG will be closing in the next few weeks. Please sign up for the new AP Calculus Teacher Community Forum at https://apcommunity.collegeboard.org/gettingstarted and post messages there.  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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