Calculus courses usually start with limits, and the precise definition of a limit is usually included but barely. There are a number of problems with this.
First off, limits seem alien and "why are we doing this"? It would be better to start with the derivative, connect it to slopes and velocities, and tell a bit of the history (Newton attempting to understand gravity). Perhaps do some hand-waving about the limits, and promise to come back to them.
The second big problem is that the limit definition is never connected with the derivative definition. In fact, the f(x) in the limit definition is not the same function as the f(x) in the derivative definition. After almost 30 years of teaching calculus, I finally wrote something to explain how the two definitions interact. (I found nothing online about this, and I have never seen it in a textbook.)
I've written it up two ways. One is a bonus chapter for my (not-yet-published) book, Althea and the Mysteries of Calculus, in which there is discussion among the kids and 'Mom'. The other leaves out the characters: How the Definitions of Limit and Derivative Interact
Posts
The limit definition is where derivatives either click or stay magic - walking through it this carefully is exactly right. Nice progression from slopes to the formal difference quotient.
ReplyDeleteI use an approach like S. Kuhn, The American Mathematical Monthly 98(1) (1991), 40–44. First talk about continuous change (without limits, in terms of "as small as we please" etc.); then rates of change. Extend the average rate/velocity [f(t)-f(c)]/(t-c) for a fixed value of c to a function continuous at c. The value that plugs the hole at t=c is the instantaneous rate/velocity, that is, the value of the derivative. Continuity can be used to explain how f'(c)(t-c) is a good linear model of the change in value f(t)-f(c). The example f(t)=-gt^2/2, in whatever units, can completely analyzed, including the continuity of the average velocity. Plus the students can toss things up. (Slopes come later. Not sure why calculus teachers are so obsessed with slopes, while at the same time being obsessed with real-world applications. The linear model f'(c)(t-c) is a good setup for generalizing to the Jacobian in multivariable calculus.)
ReplyDelete