A certain college graduate borrows $8000 to buy a car. The lender charges interest at

In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = 0.9yn

In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = n + 1 n + 2 yn

In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = n + 3 n + 1 yn

In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = (1)n+1yn

In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = 0.5yn + 6

In each of Problems 1 through 6, solve the given difference equation in terms of the initial value y0. Describe the behavior of the solution as n yn+1 = 0.5yn + 6

Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance

An investor deposits $1000 in an account paying interest at a rate of 8%, compounded monthly, and also makes additional deposits of $25 per month. Find the balance in the account after 3 years

A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%.What monthly payment rate is required to pay off the loan in 3 years? Compare your result with that of Problem 9 in Section 2.3.

A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period.What monthly payment is required if the interest rate is (a) 9%, (b) 10%, (c) 12%?

A homebuyer takes out a mortgage of $100,000 with an interest rate of 9%.What monthly payment is required to pay off the loan in 30 years? In 20 years? What is the total amount paid during the term of the loan in each of these cases?

If the interest rate on a 20-year mortgage is fixed at 10% and if a monthly payment of $1000 is the maximum that the buyer can afford, what is the maximum mortgage loan that can be made under these conditions?

A homebuyer wishes to finance the purchase with a $95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed $900?

The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). Carry out the details in the linear stability analysis of the equilibrium solution un = ( 1)/. That is, derive the difference equation (26) in the text for the perturbation vn.

The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). (a) For = 3.2, plot or calculate the solution of the logistic equation (21) for several initial conditions, say, u0 = 0.2, 0.4, 0.6, and 0.8. Observe that in each case the solution approaches a steady oscillation between the same two values. This illustrates that the long-term behavior of the solution is independent of the initial conditions. (b) Make similar calculations and verify that the nature of the solution for large n is independent of the initial condition for other values of , such as 2.6, 2.8, and 3.4.

The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). Assume that > 1 in Eq. (21). (a) Draw a qualitatively correct stairstep diagram and thereby show that if u0 < 0, then un as n . (b) In a similar way, determine what happens as n if u0 > 1.

The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). The solutions of Eq. (21) change from convergent sequences to periodic oscillations of period 2 as the parameter passes through the value 3. To see more clearly how this happens, carry out the following calculations. (a) Plot or calculate the solution for = 2.9, 2.95, and 2.99, respectively, using an initial value u0 of your choice in the interval (0, 1). In each case estimate how many iterations are required for the solution to get very close to the limiting value. Use any convenient interpretation of what very close means in the preceding sentence. (b) Plot or calculate the solution for = 3.01, 3.05, and 3.1, respectively, using the same initial condition as in part (a). In each case estimate how many iterations are needed to reach a steady-state oscillation. Also find or estimate the two values in the steady-state oscillation.

The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). By calculating or plotting the solution of Eq. (21) for different values of , estimate the value of at which the solution changes from an oscillation of period 2 to one of period 4. In the same way, estimate the value of at which the solution changes from period 4 to period 8.

The Logistic Difference Equation. Problems 14 through 19 deal with the difference equation (21), un+1 = un(1 un). Let k be the value of at which the solution of Eq. (21) changes from period 2k1 to period 2k. Thus, as noted in the text, 1 = 3, 2 = 3.449, and 3 = 3.544. (a) Using these values of 1, 2, and 3, or those you found in Problem 18, calculate (2 1)/(3 2). (b) Let n = (n n1)/(n+1 n).It can be shown that n approaches a limit as n , where = 4.6692 is known as the Feigenbaum22 number. Determine the percentage difference between the limiting value and 2, as calculated in part (a). (c) Assume that 3 = and use this relation to estimate 4, the value of at which solutions of period 16 appear. (d) By plotting or calculating solutions near the value of 4 found in part (c), try to detect the appearance of a period 16 solution. (e) Observe that n = 1 + (2 1) + (3 2) ++ (n n1). Assuming that (4 3) = (3 2)1,(5 4) = (3 2)2, and so forth, express n as a geometric sum. Then find the limit of n as n . This is an estimate of the value of at which the onset of chaos occurs in the solution of the logistic equation (21).