Calculus 6.4

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6.4 Exponential Growth and Decay
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The number of bighorn sheep in a population
increases at a rate that is proportional to the
number of sheep 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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Modeling Growth
At the beginning of summer, the population of a
hive of hornets is growing at a rate proportional
to the population. From a population of 10 on
May 1, the number of hornets grows to 50 in
thirty days. If the growth continues to follow
the same model, how many days after May 1 will
the population reach 100?
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Continuously Compounded Interest
If money is invested in a fixed-interest account
where the interest is added to the account k
times per year, the amount present after t years
is
If the money is added back more frequently, you
will make a little more money.
The best you can do is if the interest is added
continuously.
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Of course, the bank does not employ some clerk to
continuously calculate your interest with an
adding machine.
We could calculate
but we wont learn how to find this limit until
chapter 8.
(The TI-89 can do it now if you would like to try
it.)
Since the interest is proportional to the amount
present, the equation becomes
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Radioactive Decay
The equation for the amount of a radioactive
element left after time t is
This allows the decay constant, k, to be positive.
The half-life is the time required for half the
material to decay.
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Half-life
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Using Carbon-14 Dating
Scientists who use carbon-14 dating use 5700
years for its half-life. Find the age of a
sample in which 10 of the radioactive nuclei
originally present have decayed.
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Newtons Law of Cooling
Espresso left in a cup will cool to the
temperature of the surrounding air. The rate of
cooling is proportional to the difference in
temperature between the liquid and the air.
(It is assumed that the air temperature is
constant.)
If we solve the differential equation
we get
p
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Newtons Law of Cooling
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Newtons Law of Cooling
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Using Newtons Law of Cooling
A hard boiled egg at 98ºC is put in a pan under
running 18ºC water to cool. After 5 minutes, the
eggs temperature is found to be 38ºC. How much
longer will it take the egg to reach 20ºC?
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