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EXPONENTIAL GROWTH AND DECAY Section 6.4a

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Law of Exponential Change Suppose we are interested in a quantity that increases or decreases at a rate proportional to the amount present… Can you think of any examples??? If we also know the initial amount of, we can model this situation with the following initial value problem: Differential Equation: Initial Condition: Note: k can be either positive or negative W What happens in each of these instances?

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Law of Exponential Change Lets solve this differential equation: Separate variables Integrate Exponentiate Laws of Logs/Exps

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Law of Exponential Change Lets solve this differential equation: Def. of Abs. Value Let A = + e C – Apply the Initial Cond. Solution:

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Law of Exponential Change If y changes at a rate proportional to the amount present (dy/dt = ky) and y = y when t = 0, then 0 where k > 0 represents growth and k < 0 represents decay. The number k is the rate constant of the equation.

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Compounding Interest Suppose that A dollars are invested at a fixed annual interest rate r. If interest is added to the account k times a year, the amount of money present after t years is 0 Interest can be compounded monthly (k = 12), weekly (k = 52), daily (k = 365), etc…

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Compounding Interest What if we compound interest continuously at a rate proportional to the amount in the account? We have another initial value problem!!! Differential Equation: Initial Condition: Look familiar??? Solution: Interest paid according to this formula is compounded continuously. The number r is the continuous interest rate.

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Radioactivity Radioactive Decay – the process of a radioactive substance emitting some of its mass as it changes forms. Important Point: It has been shown that the rate at which a radioactive substance decays is approximately proportional to the number of radioactive nuclei present… So we can use our familiar equation!!! Half-Life – the time required for half of the radioactive nuclei present in a sample to decay.

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Guided Practice Find the solution to the differential equation dy/dt = ky, k a constant, that satisfies the given conditions. 1. k = – 0.5, y(0) = 200 Solution:

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Guided Practice Find the solution to the differential equation dy/dt = ky, k a constant, that satisfies the given conditions. 2. y(0) = 60, y(10) = 30 Solution: or

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Guided Practice Suppose you deposit $800 in an account that pays 6.3% annual interest. How much will you have 8 years later if the interest is (a) compounded continuously? (b) compounded quarterly? (a) (b)

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Guided Practice Find the half-life of a radioactive substance with the given decay equation, and show that the half-life depends only on k. Need to solve: This is always the half-life of a radioactive substance with rate constant k (k > 0)!!!

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Guided Practice Scientists who do carbon-14 dating use 5700 years for its half- life. Find the age of the sample in which 10% of the radioactive nuclei originally present have decayed. Half-Life = The sample is about years old

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Guided Practice A colony of bacteria is increasing exponentially with time. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000 bacteria. How many bacteria were present initially? There were 1250 bacteria initially

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Guided Practice The number of radioactive atoms remaining after t days in a sample of polonium-210 that starts with y radioactive atoms is 0 (a) Find the elements half-life. days Half-life =

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Guided Practice The number of radioactive atoms remaining after t days in a sample of polonium-210 that starts with y radioactive atoms is 0 (b) Your sample is no longer useful after 95% of the initial radioactive atoms have disintegrated. For about how many days after the sample arrives will you be able to use the sample? days

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