Why Do We Know What We Know?

In my AP Calculus BC class today I presented an activity from Christopher Danielson that I found through the MTBoS Search Engine

You should check out Christopher’s post. I asked the questions he proposed, but I had a new first one. I asked my students to discuss in their small groups what assumptions they were making about this function pictured here. I heard some good stuff. They talked about continuity, they talked about it being a polynomial function, hopefully one with an even highest power. They talked about the fact that circle C could not have a root since it is entirely above the x-axis (although one student raised the question of complex roots and this prompted a conversation about use of the word root versus calling them x-intercepts), they talked about the minimum number of critical values. In general, just some great recall. At our school, BC is a second year Calculus class so we were talking about ideas from last October/November. This led me to raise a question that I was a bit worried about. Earlier, we had made mention of the fact that polynomial functions with an odd highest power have all real numbers as their range. Sure, we know this. But why? Do we really have an idea why this is true? I was worried that this was too vague a question. I was worried that they would waive it away. We know this is true, Mr. Dardy. Why talk about this? Instead, we got some great GREAT conversations. I was told to think about limits, to think about derivatives. I jokingly asked if we should think about area between curves or optimization or some other time honored Calculus ideas. I was told to consider the limit as x grows without bound both for a positive leading coefficient and for a negative one. We discussed how all terms in the polynomial eventually become insignificant compared to the highest powered term. We talked about the derivative being even powered and what we know about those graphs. Man, I was just so pleased that they were willing to travel down this sort of hazy questioning path with me and reinforce what they know and WHY they know it. I say this every year, but this class absolutely spoils me.


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