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Glacier National Park, Montana
Photo by Vickie Kelly, 2004
Exponential Growth
and Decay
Greg Kelly, Hanford High School, Richland, Washington
Ex. Find the equation of the curve in the xy-plane that
passes through the given point and whose tangent at
a given point (x, y) has the given slope
y2
slope =
3 x
Point: (1, 1)
The number of bighorn sheep in a population increases at
a rate that is proportional to the number of rabbits present
(at least for awhile.)
So does any population of living creatures. Other things
that increase or decrease at a rate proportional to the
amount present include radioactive material and money in
an interest-bearing account.
If the rate of change is proportional to the amount present,
the change can be modeled by:
dy
 ky
dt

dy
 ky
dt
1
dy  k dt
y
1
 y dy   k dt
Rate of change is proportional
to the amount present.
Divide both sides by y.
Integrate both sides.
ln y  kt  C

1
 y dy   k dt
Integrate both sides.
ln y  kt  C
ln y
e
kt C
e
y  e e
C
kt
Exponentiate both sides.
When multiplying like bases, add
exponents. So added exponents
can be written as multiplication.

ln y
e
kt C
e
y  e e
C
kt
Exponentiate both sides.
When multiplying like bases, add
exponents. So added exponents
can be written as multiplication.
y  e e
C kt
y  Ae
kt
Since
 eC
is a constant, let  e
C
A.

y  e e
C kt
y  Ae
kt
y0  Ae
Since
 eC
is a constant, let  e
C
A.
1
k 0
At
t  0 , y  y0
.
y0  A
y  y0 e
kt
This is the solution to our original initial
value problem.

Exponential Change:
y  y0 e
kt
If the constant k is positive then the equation
represents growth. If k is negative then the equation
represents decay.

Population Growth
In the spring, a bee population will grow according to an
exponential model. Suppose that the rate of growth of the
population is 30% per month.
a) Write a differential equation to model the population
growth of the bees.
b) If the population starts in January with 20,000 bees, use
your model to predict the population on June 1st.
