Introduction to Smootheology -- for Students

You have been doing calculus for a long time now. Why? What is this thing called a derivative and what business does it have occupying so much of your life?<br>In this talk, I will give an answer to this question, which will help make a connection between calculus and linear algebra that you probably have seen hints of in past classes, but never quite knew why. But then, after building up some intuition as to what derivatives are, I will smash this intuition with some strange examples. These strange examples, though, show up naturally in various areas of math, and mathematicians over the last century have invented various methods of dealing with them, and getting their derivatives back. I'll discuss some of these, and time-permitting how we are getting glimpses of something deeper hiding behind many of these methods.