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Stewart6e3_8 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 3.8, Applications: exponential and logarithmic models Let’s talk a bit more about one of the most important functions in all of math- ematics, the functions. Such functions come up almost any time we’re considering a quantity that grows at a rate proportional to its own value at any time. That is, if y is our quantity of interest, dy dt = , where k is some constant of proportionality. An equation such as this one, relating a quantity and one or more of its deriva- tives to one another, is called a equation . In general such equations are hard to solve (that is, to find a function that makes the equation valid), but this one’s easy: the function y = will be a function making the equation true, as you should check below: Where does this kind of equation arise in applications? Let’s examine some... Examples. 1. Population growth. If a population P ( t ) is assumed to grow at a rate proportional at any time to the population itself, we have a model for the population: dP
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