CHAPTER 5 SECTION 5.6 DIFFERENTIAL EQUATIONS: GROWTH AND DECAY
Separation of Variables This strategy involves rewriting the eqn. so that each variable occurs on only one side of the eqn.
1. Solve the differential equation:
1. Solve the differential equation: Separate variables first!
Glacier National Park, Montana Photo by Vickie Kelly, 2004 Greg Kelly, Hanford High School, Richland, Washington
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.
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:
Rate of change is proportional to the amount present. Divide both sides by y. Integrate both sides.
Integrate both sides. Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication.
Exponentiate both sides. When multiplying like bases, add exponents. So added exponents can be written as multiplication. Since is a constant, let .
Since is a constant, let . At , . This is the solution to our original initial value problem.
Exponential Change: If the constant k is positive then the equation represents growth. If k is negative then the equation represents decay.
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?
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?
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?
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?
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.
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. 4. passes through the point (0,10). Find y.
5. Find the equation of the graph shown.
5. Find the equation of the graph shown.
1. Crystal Lake had a population of 18,000 in 1990 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.
1. Crystal Lake had a population of 18,000 in 1990 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.
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?
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?