CHAPTER 5 SECTION 5.6 DIFFERENTIAL EQUATIONS: GROWTH AND DECAY

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CHAPTER 5SECTION 5.6DIFFERENTIAL
EQUATIONSGROWTH AND DECAY
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Separation of Variables

• This strategy involves rewriting the eqn. so
  that each variable occurs on only one side of the
  eqn.

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1. Solve the differential equation
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1. Solve the differential equation
Separate variables first!
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Glacier National Park, Montana Photo by Vickie
Kelly, 2004
Greg Kelly, Hanford High School, Richland,
Washington
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Thm. 5.16 Exponential Growth and Decay Model

• If y is a differentiable function of t such
  that y gt 0 and y ky, for some constant k,
  then
• y Cekt.
• C is the initial value of y.
• k is the proportionality constant.

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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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1. Radium has a half-life of 1620 years. If 1.5
grams is present after 1000 years and Radium
follows the law of exponential growth and decay,
how much is left after 10,000 years?
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1. Radium has a half-life of 1620 years. If 1.5
grams is present after 1000 years and Radium
follows the law of exponential growth and decay,
how much is left after 10,000 years?
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2. An initial investment of 10,000 takes 5 years
to double. If interest is compounded
continuously
a. What is the initial interest rate?
b. How much will be present after 10 years?
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2. An initial investment of 10,000 takes 5 years
to double. If interest is compounded
continuously
a. What is the initial interest rate?
b. How much will be present after 10 years?
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3.The rate of change of n with respect to t is
proportional to 100 t. Solve the differential
equation.
4. passes
through the point (0,10).
Find y.
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3.The rate of change of n with respect to t is
proportional to100 t. Solve the differential
equation.
4. passes
through the point (0,10).
Find y.
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5. Find the equation of the graph shown.
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5. Find the equation of the graph shown.
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1. Crystal Lake had a population of 18,000 in
1990. Its population in 2000 was 33,000. Find
the exponential growth model for Crystal Lake.
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1. Crystal Lake had a population of 18,000 in
1990. Its population in 2000 was 33,000. Find
the exponential growth model for Crystal Lake.
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2. The number of a certain type of Kellner
increases continuously at a rate proportional to
the number present.
a. If there are 10 present at a certain time
and 35 present 5 hours later, how many will there
be 12 hours after the initial time?
b. How long does it take the number of
Kellners to double?
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2. The number of a certain type of Kellner
increases continuously at a rate proportional to
the number present.
a. If there are 10 present at a certain time
and 35 present 5 hours later, how many will there
be 12 hours after the initial time?
b. How long does it take the number of
Kellners to double?