FYI -- I ran across this really cool video about how to do differential geometry in *finite* fields. Suppose you have a polynomial in a finite field. You can have a "tangent" line, a "tangent" conic, a "tangent" cubic, etc. https://www.youtube.com/watch?v=2yGuKIz2wfE DiffGeom7: Differential geometry with finite fields njwildberger https://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus! In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture. After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics. One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made. This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.