Scary Stories to Tell in the Dark of an animal clinic

Hi there! It's currently the 24th of October, and Howl-o-ween is literally next week! Right now our dog-tors are closing down the store so, while they're at it, let's talk about something really, really, really scary...the FALL of Dr. Doggssuck's clinic (because ours is much more pawso...I mean professional).

What Happened?

A long, long time ago (just like two weeks), there was this clinic that lost paw-pularity. It happened within a matter of days, months even, but why? Well, as discussed last time, we gained the popularity for being an overall good clinic, but how did that affect Dr. Doggssuck's clinic?

Here is that graph from last time, showing us the rate of patients per month, and how it has declined:

This data was recorded from month 1 (a) all the way until month 16 (b). That is a whole 16 months of this guy's continuous downfall! That's more than a whole year! Over one WHOLE year, twelve plus four, that's SIXTEEN whole months of this, oh my gosh!

This graph can be represented by the following derivative:

Although previously we used an apostrophe, or a prime marking, to indicate a derivative, we will use a lowercase 'f' in this scenario because we gotta find...

The ANTI - Derivative

Really all the antiderivative happens to be is just the original function, however in some cases more precise. This helps us find the exact area under the curve of our function��f, rather than an approximation with some errors as seen with Riemann's Sum.

Below, I will show you how to get the antiderivative. After all, I'm the one telling you this scary story, so all you gotta do is sit there really quickly (or stand, maybe grab a snack, this might take me a little bit).

Here is our antiderivative. Notice how the integral at the top has 1 and 16, this representing our a and b from earlier. The antiderivative we just found isn't quite complete yet, it is but now we gotta find...

The Exact Area

Now that we have our antiderivative, we have to use the bounds from the graph. In this scenario, we will be using the 16-month time frame that was measured and use it to calculate our exact area under the curve using the Fundamental Theorem of Calculus (FTC).

From here, I'll use my bounds and plug them into the antiderivative, and then subtract.

What this shows us is that Dr. Doggssuck's clinic had 74 patients before it closed down. Still not that bad, but, when compared to earlier times, it doesn't look that good.

Here is a visual of how the graph looks now with exact area:

Looking back at what we previously got, it's about the same as our lower estimate when using Riemann's Sum, but it's better to use the exact area to get exactly what we are looking for. For example, if we were measuring a diluted solution but approximated how much concentration the solution should have, we would get improper measurements and results - which is probably scarier than Dr. Doggssuck's story.

Anyways, it sounds like the doctors are almost done, I'm gonna go show them the scary downfall of Dr. Doggssuck's clinic.