MATH 191, Sections 1 and 3
Calculus I
Fall 2007
Section 3.8, Applications: exponential and logarithmic models
Let’s talk a bit more about one of the most important functions in all of math-
ematics, the
functions.
Such functions come up almost any time we’re considering a quantity that grows
at a rate proportional to its own value at any time. That is, if
y
is our quantity
of interest,
dy
dt
=
,
where
k
is some constant of proportionality.
An equation such as this one, relating a quantity and one or more of its deriva-
tives to one another, is called a
equation
.
In general such
equations are hard to solve (that is, to find a function that makes the equation
valid), but this one’s easy:
the function
y
=
will be a function
making the equation true, as you should check below:
Where does this kind of equation arise in applications? Let’s examine some...
Examples.
1.
Population growth.
If a population
P
(
t
) is assumed to grow at a rate
proportional at any time to the population itself, we have a
model
for the
population:
dP

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