Exponential Growth and Decay

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Exponential Growth and Decay
Glacier National Park, Montana Photo by Vickie
Kelly, 2004
Greg Kelly, Hanford High School, Richland,
Washington
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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
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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
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Rate of change is proportional to the amount
present.
Divide both sides by y.
Integrate both sides.
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Integrate both sides.
Exponentiate both sides.
When multiplying like bases, add exponents. So
added exponents can be written as multiplication.
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Exponentiate both sides.
When multiplying like bases, add exponents. So
added exponents can be written as multiplication.
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Exponential Change
If the constant k is positive then the equation
represents growth. If k is negative then the
equation represents decay.
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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.