Friday, September 20, 2019

Calculus in your life

Mathematics is a language that is ideal for describing how the world works and building models that help understand the underlying processes and make predictions.

As a tutor, I heard a lot of, "I'll never use this. Why do I have to learn it?" and, honestly, I sympathized. I was helping high school students learn concepts that I was taught in college courses. Some intended to go on into technical or theoretical curricula in universities...but not all.

So, why advanced math in high school? Well, first, calculus, as difficult as it's made out to be, is neither advanced nor difficult. Once you understand it, it's no more difficult than any other math. It's a language. It has a vocabulary and it has rules for putting it's "words" together to make sense. The problem is that many calculus teachers don't understand it, at least not well enough to impart understanding to their students.

Further, calculus is the last tool you need to understand the world most people see in everyday life. The sequence goes like this:

Arithmetic gives you the tools to count and measure.

Algebra helps you solve problems in math.

Trigonometry let's you figure out the distance across a river without actually having to get your feet wet - that is, surveying.

Geometry is what you need to design structures that will stand up.

And, calculus...calculus is the tool you use to deal with change, because the two concepts
(only two) at the center of calculus, differentiation and integration, are what we have to deal with rates.

You see, a derivative (that's what you find when you differentiate) is just a slope. Every handyman knows what that is - it's rise over run. Measure a horizontal distance, that's the run, and then measure vertically up to the ramp,  or stairs, or roof, and that's the rise. Divide the rise by the run and that's the slope. Where it gets a little complicated is when the object you're measuring the slope of isn't a straight line. What if it's the trajectory of a bullet or a curved pipe?

The speed of a car, or any other rate, is a slope. The run is the time it takes for the car to go a number of miles. The number of miles is the rise. Rise over run….miles over hours….miles per hour. But cars rarely travel at a constant rate, so it's sometimes nice to know how fast a car is going at any particular time -that's called "instantaneous speed"and you need calculus to figure that out...or a speedometer, but a speedometer is an analog calculus calculator. It adds tiny chunks of speed (of the wheels) to come up with an estimate of current speed. That's the other concept of calculus….integration.

Integration adds tiny pieces of area, volume, rates, what have you, together to get a total. It does the impossible by adding together an infinite number of infinitesimally tiny chunks to get an infinitely accurate total.

For instance, if you knew exactly what the instantaneous velocity of a car is at every instant of a trip, you could add all those rises together to know exactly where the car is at any particular point in the trip.

But does anyone really add infinite quantities? That would take an infinite amount of time. But calculus provided ways of estimating integrals to any degree is accuracy. Those tools are called "numerical analysis".

Want to actually do the addition? Luckily, there is a way, because the integral is the reverse process of the derivative! If you know a derivative, you can just start there and go backwards to find the integral!

Some wag once said, "there are two things in calculus, differentiation and integration. Everything else is application."

So what do I think that everyone should know about calculus? I think they should know what it is and what it's good for. They should also know where to pick up the knowledge of how to use it in case they encounter a problem that requires it.

Carpenters, construction workers, and practical statisticians rarely, if ever use calculus, but the mathematical tools they use are based on calculus and someone had to work those out. If you want to make a career out of fiddling with numbers (that's what theoretical mathematicians and scientists do) or if you want to build the mathematical tools that other people use, then expect lots of calculus. And if you plan to follow me into next year, also expect some calculus.

I'm about to embark on an adventure into the hard sciences, astronomy and physics, and calculus is a big part of their language but, hopefully I can explain well enough for you to catch the passion for lifelong learning while I break open the hood and show you all the workings inside the world.

By the way, if you feel adventurous and want to study calculus or any other advanced math, MIT has some great online courses at

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