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       From Euclid to Spigotry
Assignment 12 (outlines 10 and 12)
Last modifed: September 14, 2004
1. In "calculus review" topic 2 it was shown that the rule "if f(x) = xr, then f'(x) = r xr-1
is true if r is any positive integer k.  In topic 5 the rule was established for the case where r = 
for any positive integer n. Now let r =  where k and n are positive integers and use the product rule to prove the rule for any positive rational exponent. This is an alternative to the proof using the chain rule that was done in topic 5. 
2. Using the binomial expansion formula, find all six terms in the expansion of (1+x)5 .

3. Using the binomial expansion formula, find the first four terms in the expansion of f(x) = (1+x)-1
and of g(x) = (1+x)-2. Show that these are consistent with the fact that f'(x) = - g(x).
4. Before Newton, John Wallis got a good approximation to p by what he called "interpolation."
The mysterious tables on p. 75 of the Pi source book show his results.  The key line is the third line of table 3. In modern notation, what Wallis calculated was the integral
In =  for the case where n = 0, 1, and 2.  The entries in the table show 1 /In, so you should get I0=1, I1=2/3, and I2=8/15. Wallis's clever conjecture was that In =  In-1. Confirm that this is also correct for n = 3. You may want to use Mathematica to add fractions.