1. ## Speech about calculus - please critique

I am not a professional, but I will give a 6 minute speech about calculus in one of my clubs. Please critique my speech and correct me if I'm getting the concepts wrong. Below is the speech:

Nothing in the Universe is constant. One day you could be in love but the next day you could be heartbroken. Children don’t stay young forever. Planets don’t stay in one place. Even the sun which holds Earth in place moves around its own orbit around the Milky Way. Everything changes, everything moves. Isaac Newton wanted to predict the arrival in the sky of a certain comet known as the Great Comet. How do you make predictions about a constantly moving object? It’s simple. You just need to create an entirely new branch of mathematics that is based on the concept of change. Calculus is all about change. It’s about the way things move and behave. There are so many things that calculus can do.

Let’s have an example to illustrate the kind of thinking that is involved in calculus. Open your cellphone’s calculators and try to divide the number two by zero. What do you get? (wait) You get an error. Why is it an error? What does that even mean? There’s more than meets the eye here. Now try dividing the number two by 0.1. You get 20. Divide it by 0.01 and you get 200. Divide by 0.001 and you get 2000. And so on and so forth. You could keep dividing by a smaller number forever and end up getting a higher number each time. As you can see, the number is changing. It is increasing. So what does this tell us about our little math error? It tells us that it is actually a limit that is equal to infinity, a number that can never be reached. You can never reach the answer no matter how close you got to it. At least we now have some sort of explanation as to why it’s an error. It is actually this very concept of a limit which is the foundation of all the formulas of calculus. Oftentimes you will find that the concept of limits and infinities are intertwined.

Let me extrapolate further. There are an infinite number of points on a line. Let’s say you have a three-meter horizontal line. A point can be one meter away from the left end. It can be 1.1 meters away from the left end. It can be 1.01 meters away from the left end. And so on. By this logic then there are also an infinite number of areas on a surface. In fact, there are an infinite number of volumes in any given region of space. Calculus provides a way to make precise calculations about objects in our three-dimensional world despite all of these infinities provided you know the way in which those objects are changing or moving. Calculus allows us to solve for the length, area, and volume of things. In fact, a lot of formulas in geometry are derived from calculus. Calculus allows us to calculate how an object’s dimensions and position changes as time flows. With calculus you can calculate the area of a plot of land, when a balloon will pop, when a glass will overflow, the center of gravity of an object, where a projectile will land, the coverage area of a cell site, the distance traversed by an airplane, etc.

Newton proved both the validity of calculus as well as his equations of gravity all in one stroke when he made the correct prediction for when the Great Comet arrived in the sky. The Universe may be infinite and always changing but now we could make precise calculations about things because we had a mathematics that can account for the change. It gave rise to the Newtonian model of physics which in turn became the foundation for Einstein’s theories of relativity. Mathematics officially became the chief language of science and hypotheses were proved by the correct predictions they could make.

Calculus is also used to solve optimization problems which are especially useful for business. You can collect a large amount of data on cost and profit, do a statistical interpolation on that set of data, and apply calculus to the interpolation in order to estimate the optimal amount of expenditures to yield the highest returns.

To sum it up, calculus is a branch of mathematics about change, it is foundational to everything in modern physics, and is even used for maximizing profit.

2. ## Re: Speech about calculus - please critique

I think your second paragraph would be better using a finite limit. ${\sin x \over x}$ as $x \to 0$ for example. Reasons: everybody "knows" that $\frac20 = \infty$; the limit is not "equal" to infinity; calculus doesn't require the infinite of the very large, but rather the infinite of the very small (the "infinitesimal", if you like).

This has an impact on your third paragraph which I would change to talk about measuring speed and the twin ideas of average speed and instantaneous speed.

3. ## Re: Speech about calculus - please critique

@Archie
It will be a nonmathematical audience who might not know what a sine is, what a function is, etc. Let alone the sinc function which you mention. I wanted to provide something that they could interact with during the speech, some cellphone calculators don't have a sine function,hence the chosen example. Also, I wanted to show something that could be made sense of only in the light of calculus and limits. I could still make an example of a function that doesn't involve a trigonometric expression, but if I introduce a finite limit to the audience, how do I show it's significance? I know the definite integral is about getting the sum of infinitesimals (Riemann sum) between an upper and lower limit. How do I show the significance there?

You are saying that it is better to focus on how calculus uses the infinite of the very small. I believe I am already going there with my third paragraph? I can extend the concept of "infinite number of points in a line", "infinite areas on a surface", "infinite volumes in a region", to the concept of the infinitesimal correct? And then say that finding the length, area, and volume of things was just summing all of the infinitesimals, something that you could not do in algebra.

Since I showed the infinitely large in my second paragraph, my third paragraph can talk about the infinitely small. And I can add a new one that talks about rate, average rate, and instantaneous rate.

4. ## Re: Speech about calculus - please critique

I think a finite limit is more representative of calculus. Much of advanced calculus is related to determining how we can avoid singularites, and physics abhorrs them.

Your third paragraph seems to allude to the infinitely small only to get to infinitely large collects of geometrical objects.

I would talk more about Reimann sums, instantaneous velocity or gradients, but you would probably want a whiteboard for any of those.

Interactivity is good.