- Carbon dating -- Science News strikes again! Always there when I need it. From the 12/1/07 issue, the story Rolling Back the Years: Radiocarbon dating gets a remake:
-
"Even in the early days, Libby suspected that the carbon-12 to carbon-14 ratio
had not remained constant through time. Work on solar cycles and the Earth's
magnetic field proved him right. Both phenomena are known to influence
radiocarbon amounts by altering the level of cosmic radiation entering the
atmosphere. And radiocarbon traveling through the Earth's carbon cycle can do
so inconsistently. As cold, dense water sinks to the depths of the ocean, it
drags down radiocarbon that might get trapped for hundreds of years before
resurfacing. Scientists call this the marine offset, and it has important
implications for inferring carbon dioxide levels in past climates....
"Improvements in the particle-counting technique called accelerated mass spectroscopy (AMS) have greatly contributed to filling in the gaps of uncertainty in the time line, says John Southon of the University of California, Irvine. Scientists began using AMS in the 1970s, but recent advances mean that it can be used to measure radiocarbon in much smaller samples."
- Exponentials satisfy a very important differential equation:
- Doubling time: the time it takes for a growing exponential function to double in size.
- Half life: the time it takes for a dying exponential function to halve in size.
- Forget The rule of 70 (72, 71, 69.3,...):
When compounding is continuous (i.e. ), this reduces to the very lovely rule
- Examples:
- Finish
- Now, how do we know that
? 
So the question remains: how do we know that
? Let's rewrite it a little: we want to show that
- Here's the general situation:
- Examples:
- #1, p. 390
- #19, p. 390
- A better proof of the fact that
(take logs! Better than induction....)
- #53, p. 390