Page 157 - Calculus Demystified

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EXAMPLE 5.25 CHAPTER 5 Indeterminate Forms
Because of inflation, the value of a dollar decreasesastime goeson.Indeed,
thisdecrease in the value of money isdirectly related to the continuous
compounding of interest. For if one dollar today is invested at 6% contin-
uously compounded interest for ten years then that dollar will have grown
to e 0.06·10 = $1.82 (see Section 6.5 for more detail on this matter). This
meansthat a dollar in the currency of ten yearsfrom now correspondsto
only e −0.06·10 = $0.55 in today’scurrency.
Now suppose that a trust is established in your name which pays 2t +50
dollarsper year for every year in perpetuity, where t istime measured in
years (here the present corresponds to time t = 0). Assume a constant
interest rate of 6%, and that all interest is re-invested. What is the total
value, in today’sdollars, of all the money that will ever be earned by your
trust account?

SOLUTION
Over a short time increment [t j−1 ,t j ], the value in today’s currency of the
money earned is about

(2t j + 50) · e −0.06·t j · 8t j .
The corresponding sum over time increments is

(2t j + 50) · e −0.06·t j 8t j .
j

This in turn is a Riemann sum for the integral

(2t + 50)e −0.06t dt.

If we want to calculate the value in today’s dollars of all the money earned from
now on, in perpetuity, this would be the value of the improper integral

+∞
(2t + 50)e −0.06t dt.
0
This value is easily calculated to be $1388.89, rounded to the nearest cent.

You Try It: A trust is established in your name which pays t +10 dollars per year
for every year in perpetuity, where t is time measured in years (here the present
corresponds to time t = 0). Assume a constant interest rate of 4%. What is the total
value, in today’s dollars, of all the money that will ever be earned by your trust
account?