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Exponential Growth and Decay

Exponential Growth and Decay. Dr. Dillon Calculus II SPSU Fall 1999. Today’s Goals. Identify growth and decay problems Learn to solve growth and decay problems. Which is which?. “Growth’’ refers to exponential growth. “Decay” refers to exponential decay. Recognize the Problem.

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Exponential Growth and Decay

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  1. Exponential Growth and Decay Dr. Dillon Calculus II SPSU Fall 1999

  2. Today’s Goals • Identify growth and decay problems • Learn to solve growth and decay problems

  3. Which is which? “Growth’’ refers to exponential growth “Decay” refers to exponential decay

  4. Recognize the Problem • Key words • population • radioactive decay • Newton’s Law of Cooling • One differential equation says it all

  5. A differential equation? • Information about the derivative of a function. • To solve a differential equation, find the function. • We want y, a function of x, which satisfies

  6. Technicalities • The TI-89 solves differential equations. • This one is easy to solve by hand. • One diff. eq. underlies every growth & decay problem. • We only have to solve it once.

  7. The Solution

  8. Are we there yet? No. So far, we just have an outline of the type of problem we want to solve.

  9. Example A population of bacteria doubles in twenty minutes. How long will it take to triple in size?

  10. Where is the diff. eq.? If P is the size of any population at time t then P grows at a rate proportional to itself, i.e.

  11. Thus we know... where P(t) is the size of the population at time t

  12. It’s always true... • that C is the value of the function (in this case P) when the variable (in this case t) is zero • that k is a feature of the situation at hand (in this case, the bacteria in your petri dish)

  13. Showing that C=P(0) which gives us…

  14. A Note about Notation

  15. Always ask... • What do we know? • What are we looking for?

  16. Find k

  17. Finally... Find t when P=3P(0) as follows

  18. The Moral A diff eq underlies every problem The solution is always of the form k is different in every problem Work with what you know to find what you seek.

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