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**CHAPTER 5 SECTION 5.6 DIFFERENTIAL EQUATIONS: GROWTH 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 > 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. Since is a constant, let

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Since is a constant, let At , This is the solution to our original initial value problem.

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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**

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**

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**

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**

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. 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**

3.The rate of change of n with respect to t is proportional to100 – t. Solve the differential equation. 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**

1. Crystal Lake had a population of 18,000 in 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**

1. Crystal Lake had a population of 18,000 in 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?

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