Reunion - Lesson 2 - The Infinite Power of the Infinite Magnifying Glass - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act I

2. The Infinite Power of the Infinite Magnifying Glass

2.6. Reunion

Let’s remind ourselves what we did in this chapter.

1.We noticed that curvy stuff is generally harder to deal with than straight stuff. However, we realized that if you zoom in on curvy stuff, it starts to look more and more straight. We noticed that if we could zoom in “infinitely far” (whatever that means), then any curvy thing would be exactly straight. That is, if only we had an infinite magnifying glass, we could potentially reduce curvy problems to straight problems. Rather than being sad that we didn’t have an infinite magnifying glass, we just pretended that we had one.

2.We used this idea of an infinite magnifying glass to define the steepness of curvy things. We did this by computing the steepness using two points that were “infinitely close to each other” (though we weren’t entirely sure what that meant).

3.We briefly discussed a few contraptions that people have invented over the years to allow them to avoid the idea of infinitely small numbers. We will occasionally use the contraption called a “limit,” but we’ll usually just use the idea of infinitely small numbers directly. We’ll get the same answers with both methods, so it’s okay to switch back and forth.

4.We discussed lots of different names and abbreviations that textbooks use to talk about these ideas. Textbooks usually call the steepness of M at x “the derivative of M at x.” Some common abbreviations for this idea are (i) the notation M′(x), which emphasizes that the derivative of M can be thought of as a machine in its own right, and (ii) the notation , which emphasizes that the derivative of M can be thought of as the “rise over run” between two points that are infinitely close to each other.

5.We proceeded to test our new idea on two non-curvy examples: constant machines and straight lines. We did this to make sure that our new idea gave sensible answers in simple cases where we already knew what to expect.

6.Then we tested our idea on some curvy machines, and eventually figured out how to use it on any machine that currently exists in our universe: the plus-times machines, or “polynomials.”

7.We discussed how the derivative often allows us to find the extremes of machines, namely, the places where they achieve their highest and lowest values. We explained why such extremes can usually be found by figuring out which points x make the derivative m′(x) equal to zero, and when this idea breaks down.