Surprisingly, math can be used to measure many things in any field you pursue. With this type of function below, we can use this to measure growing amounts of information, let's give an example:
Dr. Eyelovedogs and his assistant, Dr. Melovecats, started up a vet clinic in a small part of San Antonio in 2000. When they opened, they originally had 41 patients because of Dr. Eyelovedogs' great care when he did an internship program under a different vet. After their successful start, they only continued to grow, represented in the model above, where x represents time (years) and f(x) represents total patients at their clinic. That said, we'll find the derivative first, then critical points, and finally inflection points, where we will ultimately create a graph to show to Dr. Eyelovedogs and Dr. Melovecats that represents their overall growth.
The Derivative
When we find the derivative of this problem, we'll utilize the chain rule in this scenario.
Critical Points
In order to find our critical points, we'll need to set our derivative equal to 0.
Inflection Points
When finding inflection points, we wanna make sure to find the SECOND derivative, then solve for 0. In this case, we'll utilize the quotient rule.
Graphing the Function
Given the function we have will gradually increase to infinity, let's go ahead and restrict it. We'll consider 2000 to be our year zero and restrict the graph to 25 years. This should prevent us from reading anything before the clinic opened but also anything too far into the future.
This naked model shows restrictions up until the year 2025, but since it's currently 2023, let's find out the total patients they could have in 2024. We'll plug in this number into the original function but make sure that we're not doing 2024 and just 24 (we're not accounting for 2,024 years in the future that's too far).
When we plot this we'll make sure to mark on the graph for the dog-tors so it's easier for them. But looking back to the graph, it looks like their clinic had a steeper incline in 2002 than now in 2023. If we plug in the values, 2 and 23, into our derivative, we should be able to find the rates and then plot it onto our graph to see the difference.
When graphing the derivative, it will show the rates of the function. Here, the derivative, or the blue line, starts to decreases as the years go by, telling us the clinic's patients per year have slowed down. However, when looking at the graphed function, by 2024, they're expected to have about 102 total patients that go to their clinic, which is still a high number! Good luck to Dr. Eyelovedogs and Dr. Melovecats because that is a LOT of animals! Time to show them our findings.