newexpgrowth - Exponential growth and decay section 7.5...

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Unformatted text preview: Exponential growth and decay section 7.5 What do • the bacteria in a culture • the internet traffic • the human population, and • the capital in a bank have in common? They can all be represented by a mathematical function that grows exponentially. What do • the bacteria in a culture • the internet traffic • the human population, and • the capital in a bank have in common? A function y=f(t) has “exponential growth” if it grows at a rate proportional to its size: dy dt = ky for some k ! ( dy / dt ) y = k ! The relative growth rate of y is a constant. If the constant k is negative, we talk about “exponential decay” dy dt = ky We need to study the differential equation where k is a fixed constant. Let y=e 3t . y satisfies the differential equation dy/dt=3y. True False The most general solution of the differential equation dy/dt=3y is where C is any constant. A) C e 3t B) e 3t + C Question Solving the differential equation dy/dt= 6y. dy dt = 6 y ! dy = 6 ydt ! dy y = 6 dt ! ln| y | = 6 t + c ! | y | = e 6 t + c ! | y | = e c e 6 t ! y = ± e c e 6 t ! y = C e 6 t This is a new constant, call it C There are infinitely many solutions, one for each value of the constant C. Find y such that dy/dt= 6y and y(1)=3e 6 y ( t ) = C e 6 t ! y (1) = Ce 6 ....
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This note was uploaded on 12/07/2010 for the course MATH MATH 2B taught by Professor Famiglietti,c during the Fall '09 term at UC Irvine.

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newexpgrowth - Exponential growth and decay section 7.5...

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