Page 191 - Calculus Demystified

CHAPTER 6
Transcendental Functions
178 Here C is a (negative) constant of proportionality. We apply the method of
“separation of variables” described earlier in the section. Thus
dM/dt
M 2 = C

so that

dM/dt dt = Cdt.
M 2
Evaluating the integrals, we find that
1
− = Ct + D.
M
We have combined the constants from the two integrations. In summary,
1 .
M(t) =− Ct + D
For the problem to be realistic, we will require that C< 0 (so that M> 0 for
large values of t) and we see that the population decays like the reciprocal of a
linear function when t becomes large.
year the account has TEAMFLY
Re-calculate Example 6.32 using this new law of exponential decay.

6.5.4 COMPOUND INTEREST
Yet a third illustration of exponential growth is in the compounding of interest. If
principal P is put in the bank at p percent simple interest per year then after one

p
P · 1 +
100
dollars. [Here we assume, of course, that all interest is reinvested in the account.]
But if the interest is compounded n times during the year then the year is divided
into n equal pieces and at each time interval of length 1/n an interest payment of
percent p/n is added to the account. Each time this fraction of the interest is added
to the account, the money in the account is multiplied by

1 + p/n .
100
Since this is done n times during the year, the result at the end of the year is that
the account holds

P · 1 + p n (∗)
100n

®
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