Calculus 111: The Next Generation (cont)

Calculus 111: The Next Generation (cont)

What if we continued this exercise? How many direct ancestors, according to the formula we've been using, would you have in the year 1400? __________ In the year 1250? __________ Now what do you think is going on?

Only after writing down your thoughts above, should to turn to look at page 3.
Now study page 3. How do you reconcile these apparently contradictory pieces of information? Discuss within your group which explanations you might offer for this. Try to be as specific as possible, and back up your assertions. Then write down an explanation (a) that a student taking regular first semester calculus could understand (b) that a lay-person could understand.

Recall that the human race is well over 100,000 years old, and that the boundary between the Old Stone Age and New Stone Age is roughly 8,000 years ago. If we use the formula we have for the number of ancestors you had 6,640 years ago, what would we get? Can you quantify this number? Without carrying out any computations directly, comment on what you think we would get for the number of ancestors 10,000 years ago? 20,000? 50,000? 100,000?

Discussion.

(1) What did you learn about:

The size and rate of growth of exponential functions?
The ways in which trying to apply exponential functions (or mathematical models, generally) to real life can mislead?

(2) For your finale, discuss the following issues: Does all of this suggest something about the extent to which we are all related to one another? Have we proved this conclusively? What other interpretations might there be? How would you further investigate these issues to be more sure? What additional data, and what types of mathematical analysis, would you want?