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2. (Newton's Method of Computing the Square Root of a Positive Number) The equation x2 = a can be written in the form x = 1/2 (x + a/x). This form leads to Newton's method . (a) Show that this difference equation has two equilibrium points, -va and a. (b) Sketch a stair step diagram for a ====== 3,x(0) = 1, and x(0) = -1. (c) What can you conclude from (b)? 3. (Pielou's Logistic Equation) E.C. Pielou [119] referred to the following equation as the discrete logistic equation: x(n+1)= = ax(n) B > 0. 1 + Bx(n) (a) Find the positive equilibrium point. (b) Demonstrate, using the stair step diagram, that the positive equi- librium point is asymptotically stable, taking a = 2 and B ====== 1. 11. Suppose that the supply and demand equations are given by D(n) = -2p(n) + 3 and S(n + 1) = p2(n)+1 (a) Assuming that the market price is the price at which supply equals demand, find a difference equation that relates p(n + 1) to p(n). (b) Find the positive equilibrium value of this equation. (c) Use the stair step diagrams to determine the stability of the positive equilibrium value.

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