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# A honeybee population starts with 100 bees and increases at a rate of $n'(t)$ bees per week. What does $\displaystyle 100 + \int^{15}_0 n'(t) \, dt$ represent?

## $n(15)=100+\int_{0}^{15} n^{\prime}(t) d t$

Integrals

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Okay when you look at this problem I would use the fundamental theorem of calculus. Okay. Uh to explain just this part. 0 to 15 yeah. Of N. Prime of T. D. Teeth. And the fundamental theorem of calculus says that that equals the anti derivative. It's just N. F. T. Because they undo each other and then you have the upper bound after minus the lower bound. Yeah. So what we're looking at at that piece is a, well actually I don't really care about that. Um What I'm gonna do is I'm going to add an of zero over. So what I'm deducing is that I'm writing this in green on purpose and zero plus because I'm adding it over here. The in a row from 0 to 15 of end prime of T. D. T. needs to be equal to and 15. So this I'm gonna put a circle I'm going to circle this and you might be sitting there and saying why are you already circling this is because instead of writing end of zero this is the number of bees at time zero. So no time has elapsed. It's the starting and so what I'm getting at starts with 100 is the problem didn't write and of zero here wrote 100 100 plus all of this. So that's equal to end 15. If you look at the units um what that formula gives you then is the amount of bees wow. Um Let's see because it's per week. Um That's what the 15 represents at 15 weeks. And this was a rate. So that's why this one is not a rate. So I mean circle is this is the interpretation.

Integrals

Integration

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