The investment of money at compound interest is an example of a physical
problem which requires analysis of discrete values. Say, $1,000.00 is to be invested
at Rpercent interest compounded quarterly. How does one determine the discrete
values representing the amount of money available at the end of each compound
period? To solve this problem, let P 0 denote the amount of money initially invested,
Rthe percent interest yearly with^14100 R =ithe quarterly interest and let Pn denote
the principal due at the end of the nth compound period. The equations for the
determination of Pn can be found by examining the discrete values produced. For
P 0 the initial amount invested, one finds
P 1 =P 0 +P 0 i=P 0 (1 + i)
P 2 =P 1 +P 1 i=P 1 (1 + i) = P 0 (1 + i)^2
P 3 =P 2 +P 2 i=P 2 (1 + i) = P 0 (1 + i)^3
..
.
Pn=Pn− 1 +Pn− 1 i=Pn− 1 (1 + i) = P 0 (1 + i)n
For i=^14100 R and P 0 = 1 , 000. 00 ,figure 10-7 illustrates a graph of Pn vs time, for
a 30 year period, where one year represents four payment periods. In this figure
values of Rfor 4% , 5 .5% ,7% , 8 .5% and 10% were used in the above calculations.
Let us investigate some techniques that can be used in the analysis of discrete
phenomena like the compound interest problem just considered.
Figure 10-7.
Return from $1,000 investment compounded quarterly over 30 year period.
The study of calculus has demonstrated that derivatives are the mathematical
quantities that represent continuous change. If derivatives (continuous change) are
