Exponential Growth and Decay Mr. Peltier. Exponential Growth and Decay dy/dt = ky If y is a differentiable function of t, such that y > 0 and dy/dt =

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Presentation transcript:

Exponential Growth and Decay Mr. Peltier

Exponential Growth and Decay dy/dt = ky If y is a differentiable function of t, such that y > 0 and dy/dt = ky for some constant k, then: y = Ce kt P o k P o is called the initial value of P(t) and k is called the constant of proportionality. You get a growth equation when k > 0 and a decay equation when k < 0. A = Pe rt P(t) = P o e kt

Exponential Growth and Decay EX: A sample contains 1 gram of radium (half life = 1620 years). How much radium will remain after 1000 years? P(t) = P o e kt P(t) = 1e k(1000) P(t) = 1e ( )(1000) g P(t) = g

Exponential Growth and Decay EX 2: A population of Peltierians grows at an exponential rate. If there are 100 Peltierians after the 2 nd day, and 300 after the 4 th day, how many Peltierians were there to start with? P(t) = P o e kt 100= P o e 2k 300= P o e 4k

Exponential Growth and Decay EX 2 Continued:

Exponential Growth and Decay EX 2 Continued: Um, what were we doing? How many Peltierians were there to begin with? Oh yeah! 33 Peltierians So there were 33 Peltierians at the start!

Exponential Growth and Decay Find the solution to y’ = 7y satisfying y(0) = 8 The solutions to y’ = 7y are the functions y(t) = Ce 7t, where C is the initial value C = y(0)

Exponential Growth and Decay Other useful bits of knowledge, given that P(t) = P o e kt

Assignment Pages 369 Problems 1, 2, 7-15, 17, 18