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Exponential Growth and Decay. Glacier National Park, Montana Photo by Vickie Kelly, 2004. 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 - PowerPoint PPT Presentation
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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.