Derivatives Without Limits, Part 1

First off, welcome to my blog, and thanks for reading!

My purpose with this blog is to talk about the things that interest me as a teacher, which is mainly advanced mathematics and helping students understand it.

Today in calculus I got to teach one of my favorite lessons. This course is the first semester of calculus for my students. They spent the first two days of class learning how to sketch derivatives and antiderivatives of given functions from a picture. Along the way, we had to develop terminology like increasing/decreasing, concave up/down, local maximum/minimum, and inflection point. Everything was done at a descriptive level.

Then today I put up the graph of $y=x^3-3x$.

Cubic function

I asked my students where they thought the local minimum of the polynomial was. They did not need any justification. They just needed to eyeball it. They unanimously agreed that it was at $x=1$.

So, now the question is whether we can check our guess. I asked them what they knew about local minimums that we might try to verify. Drawing on the previous couple days of class, they quickly threw out that the function had to be flat at a local minimum, and so its the tangent line to the function at that point must be horizontal. So I asked what they thought the tangent line to the polynomial at $x=1$ was. If the tangent line is flat and passes through $(1,-2)$, then it must be $y=-2$, they told me. So I added $y=-2$ to the picture.

Cubic with Tangent Line

Now I ask them how we will know that this horizontal line is tangent to the parabola. They tell me that the polynomial cannot pass through the line because then the two curves would intersect twice. Therefore, the polynomial needs to bounce off of the line.

Where have we seen polynomials bouncing off of horizontal lines before? When graphing polynomials, students learn that a polynomial $P(x)$ bounces off the $x$-axis at $x=a$ if $P(x)$ has a factor of $(x-a)$ to an even power.

However, this approach does not immediately work with our problem. The main obstacle is that our polynomial seems to be bouncing off of $y=-2$ instead of the $x$-axis. Opening the floor to suggestions, I get a student to suggest translating everything up 2 units.

Modified Cubic

So now we have a modified polynomial, $y=x^3-3x+2$, which bounces off of the $x$-axis if and only if our original polynomial bounced off of $y=-2$. It is easy enough to factor: $x^3-3x+2=(x-1)^2(x+2)$. Sure enough, they bounce where we thought they did.

We can now conclude that our original polynomial was flat and therefore at a local minimum at $x=1$.

This technique is actually a lot more powerful than just verifying guesses of local extremums, though. We can actually adapt the method to compute the derivative of any polynomial at any point. But that is a topic for a later post.