Let D represent duration, a term in finance that measures the length of time an investor must wait to receive half of the value of a cash flow stream totaling S dollars. Let r be the rate of interest and V the value of the investment. The value of S can be calculated by two formulas that are approximately equal: S ≈ V(1 + rD) and S ≈ V(1 + r)^D.

(a) Show that the first approximation follows from the second by using the Taylor polynomial of degree 1 for the function ƒ(r) = V(1 + r)^D.

(b) For V = +1000, r = 0.1, and D = 3.2, calculate and compare the two expressions for S.