Here is the full transcript of Professor Jeff Heys’ TEDx Talk: __What is Calculus Used For?__ at TEDxBozeman conference. This event took place on March 23, 2012 at Bozeman, Montana. To learn more about the speaker, read the bio here.

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__Jeff Heys – Associate Professor__

So I’d like to talk to you about a question that I’m guessing most of you have never asked. But it’s a question that I asked a lot. I was an engineering student just up the road at Montana State University and I had to take a lot of Calculus courses.

I also had to take a lot of math courses in high school. And the question that I kept coming back to is: what is all this stuff I’m using or learning — what is all this Calculus used for? And I never really got a satisfactory answer until I went to graduate school in Boulder, Colorado.

And the reason why I was able to get an answer there is because Boulder had a very bizarre – well, maybe not bizarre, but a very interesting smoking ordinance at the time. This was the mid 90s and the smoking ordinance in Boulder was basically that if you wanted to have a smoking area in your restaurant, it had to be sealed off. It had to be completely walled off with a door that would open and close.

As you would expect, most restaurants didn’t really have a smoking area because the smoking area was too expensive to construct but a few did and they were very thick with smoke. In one of my classes, we were asked to develop a mathematical model that would allow someone to calculate how much smoke would come out of these smoking rooms every time the door was open.

Or similarly allow us to calculate if you were sitting at a table somewhere else in the restaurant, how much smoke would you be exposed to? This was really sort of a transformative assignment for me.

I realized that Calculus and Mathematical models were useful. I could use them to calculate something I was interested in. When I went into a restaurant and the waitress wanted to seat me at a table, I could determine whether or not I really wanted to sit at that table based on how far close it was to the smoking area.

Since that original mathematical model, I’ve spent 15 or more years developing mathematical models and I’ve sort of come to realize that they fit in to about three categories. The first category are models that predict the future. So these are probably the ones you’re most familiar with. If you know the current pitcher and pressure around the world, you can solve some fairly complicated Calculus equations and use that solution to predict the weather over the next day or week or years.

Similarly, if you own some stocks somebody wants to buy an option from you to purchase those stocks, you could solve the __Black-Scholes equation__. It’s another complicated calculus based equation that would allow you to predict what that stock price is going to do over the next couple months and allow you to calculate the price that you should charge for the option.

Well, those are difficult models. They have a lot of uncertainty in them. It’s really hard to predict the future. So those are the types of models I largely stay away from.

Another category of models are models that we developed to avoid doing experiments, because the experiments are really really expensive. So a few decades ago, when Boeing wanted to design a new aircraft, what they would do is they would go into the wind tunnel and they would try out a whole wide range of shapes, for the wings, for the fuselage, a whole wide range of shapes for the engines and they would see what was the most efficient. Well that was tremendously expensive.

Now what they do is they go into a computer, or go onto a computer and they design the aircraft on the computer ahead of time and they only test out a couple of the very best designs in the wind tunnel. And this saves them tens of millions — hundreds of millions of dollars.

Also, if you want to design a billion-dollar experiment, it’s probably a pretty good idea to develop a mathematical model ahead of time to see whether or not that’s money well-spent, sort of get an idea of whether or not that experiment is going to work. And these are difficult models, generally not the ones we create.

The category of models we create, that I’ve worked on for the most part, are models where the experiment that would give you the same information is unethical. So __Wired magazine__, a few months ago, had a very interesting article that talked about seven experiments where if we could do them they would teach us so much about human health and human behavior. But we can’t do them because they’re unethical experiments.

Well, what I’d like to sort of highlight for you today is that in some cases, there’s an alternative to the unethical experiment. So this was one of the first mathematical models I ever worked on. If you took a slice of your eye — don’t do that — but if you did, you would find some tissues in there without any blood vessels: the cornea, the lens, no blood vessels. Your body produces a fluid that circulates in the front part of your eye to provide those tissues with nutrients.

But sometimes that circulation of fluid gets messed up and that leads to __glaucoma__. Now we don’t really understand how all these different forms of glaucoma develop. So we have two options. Option number one is I can get some really thin pressure transducers, maybe a dozen of them and put them in your eye and I can measure all the forces, the fluid inside your eye experiences. Do we have any volunteers for option one? I can’t see you very well but I’m guessing there’s not a lot of hands up.

So I’ll give you a second option. We developed a mathematical model that could make those same predictions about how various forces impacted the eye and that model gave us insights into the development of a couple different forms of glaucoma.

Another interesting question is there’s a lot of inhalable drugs already on the market and more being developed all the time. For example, they’re developing inhalable chemotherapy drugs to maximize delivery to where you need it most, the tumor that’s in your lungs. But everybody’s airways are different. Your airways are different from a seven-year old, especially, right?

So what size particle and density of particles should you inhale for your unique airways? Again, there’s two options. Option number one is you can inhale some radioactive particles and using an X-ray we can see where they go. Probably not a good option, right?

Option number two is we can develop a mathematical model and we’re working on this right up the road at MSU that will predict where different size particles will be deposited, so we can determine for your unique airway geometries what’s best for you.

Another example: I met an echocardiologist once upon a time and learned that they have developed micro-bubbles that they can inject into the blood stream and using just a standard ultrasound they can actually visualize these bubbles moving with the blood in your heart. They can see what the blood flow in your heart looks like. Well, that’s nice.

But what the echocardiologist really wants to know is, is your heart healthy, how efficiently is your heart operating? And they can’t get that information just from looking at some bubbles moving around on an ultrasound screen. So what we are working on is combining this very valuable data with a mathematical model so that we can someday add — to this someday add to this ultrasound display a health gauge or an efficiency gauge, so the echocardiologist will be able to use this data more effectively in designing treatments.

So I’d like to close just with sort of a challenge and this is a challenge no matter what your age is. But especially if your age is 7 or 10, which happens to be the ages of my two children, I’d like to just challenge you to consider learning math, consider learning Calculus, because there are so many things out there that we still need to develop mathematical models for because the experiment is impossible. One we’re working on right now, for example, is we’re working on using higher temperatures hyperthermia to kill cancer cells. We need better models of that process simply because the experiments are too difficult. So please consider learning Calculus.

Thank you.