Exponential Growth and Decay

Greg Kelly, Hanford High School, Richland, Washington

Glacier National Park, MontanaPhoto by Vickie Kelly, 2004

Ex. Find the equation of the curve in the xy-plane thatpasses through the given point and whose tangent ata given point (x, y) has the given slope

2

slope = Point: (1, 1)3yx

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 kydt

dy kydt

1 dy k dty

1 dy k dty

ln y kt C

Rate of change is proportional to the amount present.

Divide both sides by y.

Integrate both sides.

1 dy k dty

ln y kt C

Integrate both sides.

Exponentiate both sides.

When multiplying like bases, add exponents. So added exponents can be written as multiplication.

ln y kt Ce e

C kty e e

ln y kt Ce e

C kty e e

Exponentiate both sides.

When multiplying like bases, add exponents. So added exponents can be written as multiplication.

C kty e e

kty Ae Since is a constant, let .Ce Ce A

C kty e e

kty Ae Since is a constant, let .Ce Ce A

At , .0t 0y y00

ky Ae

0y A

1

0kty y e This is the solution to our original initial

value problem.

0kty y eExponential Change:

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.