Write down the recursion equation for the frequency of A at time t+1 as a function of its frequency at time t (use p[t] for the frequency of A).

What is the change in frequency of allele A from one generation to the next (ie find p)?

Perform a Taylor Series on p with respect to s and, assuming that selection is weak, ignore s² and higher terms. By this method, show that p = s p (1-p) + O(s²)

[Hint: This is a somewhat different use of the Taylor Series than what we've done before. Remember that the Taylor Series says that a function f(x) can be written as f(a) + x f'(a) + x² f''(a)/2 + ... Now, we say that p is a function of selection, f(s), and we write f(s) = f(0) + s f'(0) + s² f''(0)/2 + ... Dropping s² and higher order terms, we get that f(s) is approximately f(0) + s f'(0) when s is small, where f(0) is p when s equals 0 and f'(0) is the derivative of p evaluated at the point s=0.]

Using the fact that p is approximately s p (1-p), when will the gene frequency change at the fastest rate?

[Hint: To find a maximum or a minimum of a function, take the derivative of the function with respect to the variable of interest and set this derivative to zero. Then solve the equation for the variable of interest.]