Has it ever occurred to you that the kinematic equations for uniformly accelerated one-dimensional motion are Taylor series expansions? If not, you are in good company. I didn't know this myself until a colleague pointed it out to me many years ago, and I was stunned to learn something new and wonderful about something so familiar. Accordingly, my first objective in this paper is to clearly present the not-widely-known Taylor series derivations of these basic equations to a population primed to deeply appreciate them: people, like me, who teach introductory physics. Following this, I use the Taylor series approach to derive a generalized one-dimensional expression for x(t) that includes the jerk and further kinematic time derivatives, which have importance in many real-world applications and in which there has been renewed pedagogical interest. I also outline teaching suggestions and provide student-accessible video derivations to support instructors who would like to incorporate Taylor series kinematics into their teaching, while identifying sequencing-related challenges. I close with the observation that the traditional second calculus course, which is largely free of sequencing issues, could be a great place to incorporate and leverage Taylor series kinematics, and I briefly outline an early-stage pilot collaboration to explore this possibility.

5 pages, 1 figure