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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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