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A bacteria population is 4000 at time $ t = 0 $ and its rate of growth is $ 1000 \cdot 2^t $ bacteria per hour after $ t $ hours. What is the population after one hour?
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Calculus 1 / AB
Indefinite Integrals and the Net Change Theorem
University of Michigan - Ann Arbor
University of Nottingham
Idaho State University
In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.
In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.
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Okay here in this problem we're looking at population growth uh T. Represents hours. So when T equals zero the population is 4000. T. Equals zero. Means in the very beginning. Before we even started counting time uh T. Equals one. Would mean one hour has passed. T equals zero means we're at the very beginning no time has passed. So if the population is 4008 equals zero that means we were starting with a population of 4000. Now the rate of growth is given by this expression 1000 times two to the T. Power where T. Is the number of hours. And we want to find a population uh the entire population after one hour. Well after one hour uh We are going to have a population that we started with. But also we're going to have uh more population more bacteria. Because we're going to experience some growth. Because after the end of one hour where T. Is going to be one we're going to have some growth in uh number of bacteria. So the population after one hour Is going to be the original 4000 that we started with the original population plus uh the growth in the population. The growth in the population that takes place during the first hour is going to be given by this expression. So 4000 the original population plus uh the increase in the population as it grew during that first hour. So we have to do 1000 times two. And since we're looking for the population after one hour that means T. Is going to be one. Only one hour has elapsed. So to to the T. Will be two to the first. So this will be the entire population. 4000 is the population we started with 1000 times two to the first power. Is the growth in the population during the first hour. Now to to the first power is just too times 1000 will give you 2000 Plus. The original 4000 is a total of 6000. So what is the total population at the end of one hour? 6000 4000 That we started with plus 2 2000 that the bacteria grew by during that first hour. So we start with 4000 Uh bacteria. And then during the first hour they grew 2000 more bacteria. For a total of 6000 bacteria.
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