Chapter Notes (2)

Therefore it is often referred to as the

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Unformatted text preview: the limit of this average rate of change as the increment ∆x approaches zero. Therefore, it is often referred to as the instantaneous rate of change of y . Differential equations arise when we know something about the rate at which a function changes and want to deduce the behavior of the function itself. In physics, because of Newton's law relating force to acceleration (which is the rate of change of velocity), differential equations are at the core of the subject. We will see in Chapters 5and 6 that many problems concerning biological populations can be given reasonable formulations as differential equations. Here we want to consider a different sort of aggregate, but variable quantity, - money. Unlike populations, for which our mathematical descriptions represent a simplification of reality, an exact mathematical process is usually at the bottom of many financial transactions. We consider the question of continuous compounding of interest and show how equation (2.6) leads us to a description of this process as a differential equation. Example 2.8: Suppose you have $1000 in a bank account paying 10% annual interest, compounded daily. After 30 days what would be your total accumulation? Solution: The daily compounding of interest means that on any day the interest rate would be 1/365th of the annual rate or 0.10/365. The amount of interest earned in a day, I , is the current principal Pold multiplied by this daily rate. The new principal is then given by Pnew = Pold + I from which we can derive the next day's interest. The table below shows the results obtained at the beginning and end of the 30-day period. The interest is rounded to three decimal places. The change in principal from one day to the next is just the interest earned during the day in question. Notice that the daily interest does change, though it only increases slightly over the entire 30-day period. 23 2 Differential Equations Day Principal ($) Interest ($) Day Principal ($) Interest ($) 1 1,000.00 0.274...
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This document was uploaded on 04/06/2014.

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