 Alright everyone let's look at one last example of this net change theorem that is to say well let's apply the fundamental theorem of calculus to scientific applications here and for this last example here suppose we have a bacteria population of 4,000 at the start of some experiment time equals zero and we'll be measuring time in hours here and 4,000 is representing 4,000 bacteria and the bacteria are growing incredibly fast they have a growth rate of 1,000 times 2 to the t that seems really really fast right they're growing 1,000 times 2t bacteria per hour and that's and then so actually increases by the hour as well so this is our population if population is n of t this is our derivative 1,000 times 2t like so so if we want to change what you know what's the population after one hour to answer that question we're trying to integrate from zero to one so there's 4,000 at the initial there's 4,000 at the initial quantity there and then we want to see how so I should say like how much is there after one hour what we're trying to find right now is this is going to be the increase this is the increase after one hour we're not trying to figure out how many there are at one hour yet we're trying to figure out what's the increase here so increase after an hour here so we're going to integrate from zero to one the derivative in prime t dt from what we saw here that we get 1,000 times 2 to the t dt because this is an integral we can factor out the 1,000 just put it out in front integrate from zero to one 2t dt so we need to find a function whose derivative is 2t that is how we calculate antiderivatives now we know for a fact that the derivative of say like e to the x that's equal to e to the x and that tells us that the antiderivative of e to the x dt that's equal to e to the x plus a constant but things are a little bit different when you work like base 2 if we took the derivative of 2 to the t with respect to t here we'd actually get the natural log of 2 times 2 to the t there's this tariff that you have to pay for using the wrong base and so if we reverse this process if we want to reverse this process we actually get that the integral of 2 to the t dt this is going to equal 2 to the t but divided by the natural log the natural log of 2 plus this constant right here and so that's the principle that we want to use in this exercise here the antiderivative of 2 to the t uh this is going to give us a natural log of 2 in the denominator then we get 2 to the t um we want to go from zero to one plugging in one is going to give us a 2 so we get 2,000 over the natural log of 2 and then we subtract from that t to the zero right well t the zero is not zero two the zero is actually one so you get 1,000 over the natural log of 2 right there all right and so then subtracting those because they both have a common denominator of natural log of 2 you end up with 1,000 over the natural log of 2 and we could estimate what that value is if we so wanted to so this right here is this right here represents the net change this is the amount of increase that happened in that first hour well what what does that say for us we were trying to figure out what the population was exactly at one hour that was the question we had to do and the population after one hour that's going to be your initial population which was in of zero plus the increase how much did increase from zero to one hour and so we know it started off with 4,000 bacteria and then we just added 1,000 over the natural log of 2 and that is approximately i can write that approximation right here that's approximately 5,442 0.7 we'll round to the nearest bacterium so we'll say that's approximately 5,443 that's a good enough estimate here you can't have half of a bacteria or anything like that bacterium so that's gonna be our final answer right there and i want to mention here that this idea of using the net change is really what we've already were doing at the end of chapter four as we start top on antiderivatives we were basically doing this process already just in a slightly different perspective notice what we saw before is that our population in of t it was going to equal the integral of in prime of t dt if we know the antiderivative we can calculate it and so in this situation if we're integrating was it 1,000 t to the 2 to the t there dt we end up with 1,000 over the natural log of 2 to the t plus a constant right we had to know what that constant was and to determine what the constant would be we would look at some initial value in of 0 which would tell us we had 1,000 over the natural log of 2 times 2 to the 0 plus a constant right here which that first part would just become 1,000 over the natural log of 2 plus a constant but then we we would realize oh yeah that's what if you plug 0 into the function that's what you get but we also know it's equal to 4,000 right here and so plugging these things in here we can recreate this expression that we had before so I want you to be aware that if you're looking for just the net change like in this case just the increase you can integrate the definite integral and I'll give you the net change which was that this 1,000 over the natural log of 2 we saw before but if you're actually trying to figure out what is a specific value at in one well you can do what we did here right you can take you can calculate the increase from some from some known value in of 0 but that actually matches up with this initial value problem we were doing before with anti-derivatives so kind of further solidifying this idea that the fundamental calculus tells us that derivatives and integrals are inverse operations this this net change process we can talk about in section 5.4 in these lectures connects the idea that this net change is really capturing this idea of anti-differentiation that we're doing before thus validating once again the fundamental theorem of calculus and that brings us to the end of section 5.4 about net change which really was just an opportunity to look at story problems involving integrals right I appreciate you watching the videos here if you like them that would be great post some comments if you have any questions I'll be happy to answer them and as usual if you want to see some more of these videos feel free to subscribe so you can get notifications there are more information about this channel in the future I will see you next time everyone keep on calculating bye