The other day, I was watching one of the College Board Daily Videos for AP Calculus AB, and I came across something that prompted me to share it on here. The teacher in the video used a rumor spreading as an example of how we can use instantaneous rate of change. The instantaneous rate of change, for those who don't know, is the rate of change at any given point. Mathematically, it is equivalent to the derivative of the graph's function evaluated at that specific point.
We often face problems in math classes that ask us to graph things, such as the test scores of boys versus girls in a class or the amount of annual income in a company over time, but I never really thought about how a rumor spreading could be an example of calculus. In the problem given to me, the spread of a rumor is modeled by the function
where R(t) is the number of people that have heard the rumor and t is the number of days since the rumor started for 0 < t < 30. This interval, 0 < t < 30, simply means this function starts at 0 days and ends at 30 days. In other words, this is the function of a rumor spreading from its formation to the end of a 30-day period of its circulation.
Then, the graph of this function, R, was graphed with the line tangent to it at t=4. When I say R has a tangent line at t=4, I mean that there is a straight line that comes into contact with R only at t=4, and it was graphed with R.
Here, R is in black, and its tangent line is in purple.
The problem goes on to ask a multiple-choice question about which answer would give the best estimate for the instantaneous rate of change of R at t=4, or the instantaneous rate of change for the rumor spreading at 4 days.
The choices were as follows:
A) R(4)
B) The slope of the line joining (1, R(1)) and (4, R(4))
C) The slope of the line joining (4, R(4)) and (8, R(8))
D) The slope of the line joining (3.8, R(3.8)) and (4.2, R(4.2))
The best estimate for instantaneous rate of change of R at t=4 has to have an interval that includes t=4 but is also as close as possible to t=4.
If you're having trouble understanding what I mean, a good way to think about it is like this:
If an analog clock read that it was 10:10 AM, and you didn't have your glasses on, you wouldn't be able to see what time it was very well. Let's pretend you know it is past 10:00 AM, and you can see a fuzzy outline of the minute hand. You can see that it is somewhere past 10:00 AM, but it is no past 10:15 AM. You conclude that it is somewhere from 10:00 to 10:15 AM, but you want to know a more precise time in order to not run late for your doctor's appointment. You move as close as possible to the clock, but you still can't see the clock with 100% accuracy. This time, you conclude that it is between 10:09 AM and 10:12 AM, and this is the best estimate you will be able to attain because you can't get any closer to the clock.
This is the same concept for the instantaneous rate of change. We want to get as close as possible to t=4, so choice D is the answer. An interval from 3.8 to 4.2 is closer to 4 than an interval of 4 to 8 (Choice C) or 1 to 4 (Choice B). We know that A is incorrect right off the bat because it is a point on the graph, not a rate of change.
Besides teaching some calculus basics, the point I'm trying to bring home here is that even calculus is relevant to everyday situations. Rumors spread every day, but now you know that can be modeled using calculus. Calculus happens in real life every day, whether or not we model it with graphs, tangent lines, and derivatives.
Instantaneous rates of change have endless applications, but some to think about, which my Daily AP Video brought to my attention, are:
The Learning Curve
Heat Condition
Population (Growth & Density)
Chemical Reactions
Electric Current
Temperature
Growth & Decay
Rates In/Out
Flowing Liquid/Gas
Acceleration
Velocity & Speed
Marginal Cost (Prediction of Future Cost)
I have actually predicted future cost myself this past summer as part of Columbia University's Online Summer Immersion Program. I programmed in Python with a group to create a linear regression model that predicted CLI, or credit line increase. The basic foundations for the algorithms I used were based in calculus and statistics, but I was solving a business problem from the perspective of a bank or credit card company: provide appropriate credit lines to each customer based off of previous data.
Here are some slides from our final presentation below, which include some of our code:
Calculus is everywhere. Like the clock example from before, you just have to have the right prescription to see it.
Thank you for reading, and I hope you stop by soon for more upcoming articles!
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