Application to Time Value of Money. The following is a model of money for a loan that is being paid off by making a payment every month. We wish to calculate the amount of the loan for any month k given a fixed interest rate, r, compounded monthly. The amount at the end of the next month will be the old amount plus the interest on the old amount minus the payment. Later we shall return to this application.
Many problems have the same formulation, and another model is a to heat transfer.
Application to Discrete Form of Newton's Law of Cooling. This problem concerns the cooling of a well-stirred liquid such as a cup of coffee. Here we want to predict the temperature of the liquid based on some initial observations. Newton's law of cooling is based on the observation that for small changes of time, h, the change in the temperature is nearly equal to the product of the c, the h and the difference in the room temperature and the present temperature of the coffee. Let y k equal the temperature of a well stirred cup of coffee at time t k .