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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: Note: k can be either positive or negative What happens in each of these instances? Initial Condition:

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**Law of Exponential Change**

Let’s solve this differential equation: Separate variables Integrate Exponentiate Laws of Logs/Exps

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**Law of Exponential Change**

Let’s 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 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 Interest can be compounded monthly (k = 12), weekly (k = 52), daily (k = 365), etc…

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**Compounding Interest Solution:**

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 1. k = – 0.5, y(0) = 200 Solution:**

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 2. y(0) = 60, y(10) = 30 Solution: or**

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 (a) (b)**

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 This is always the half-life**

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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**The sample is about 866.418 years old**

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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**There were 1250 bacteria initially**

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 (a) Find the element’s half-life. Half-life = days

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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 (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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Do Now How long would it take for an initial deposit of $1000 to grow into $1500 if you deposit it into an account that earns 4% interest compounded monthly?

Do Now How long would it take for an initial deposit of $1000 to grow into $1500 if you deposit it into an account that earns 4% interest compounded monthly?

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