Name __________________________________
From Euclid to Spigotry
Assignment 11 (outline 10)
Last modified: September 14, 2004
1. The binomial theorem, stated by Newton and proved by Euler, says that the functions

fr(x) = (1+x)r 
and gr(x)= 
are equal to one another.
a. By differentiating both fr(x) and gr(x), show that the theorem is true for exponent r if it
is true for exponent r-1. (If the derivatives of the two sides are equal, that only means that the two
sides differ by a constant, but you can easily show that the constant is zero.)

b. From this, prove the theorem by induction for the case where r is a positive integer, and explain why the proof does not apply when r is not an integer. 

2. The function f(x) = tan x satisfies the equation f(x) cos x = sin x.
By taking the derivative of both sides of this equation, show that f'(x) = 1 + (f(x))2.
3. Leonhard Euler, in the late 1700s, was the first to look at the infinite series

S(x) = 
C(x) = 
a. Without worrying about what it means to sum an infinite number of terms, show that
S'(x) = C(x) and that C'(x) = -S(x) 

b. Let f(x) = S(x)2 + C(x)2.  Use the chain rule to show that f'(x) = 0. What functions do you know that have the same properties as S(x) and C(x)?