Calculus in Context
As I try and develop meaningful calculus problems I’ll post them here.
Functions with discontinuities
Delta function & Heaviside function: I don’t know why I waited so long to introduce these two incredibly simple and versatile functions in a calculus. Both feature discontinuities and both are ubiquitous in the sciences. I suspect that we don’t see the delta function in calculus books since it is properly a distribution, but I’ve only had a half dozen math students in my life, the rest have been scientists and engineers and they’ll almost certainly use these objects without the proper justification. These functions also provide a useful way to review transformations of function (a precalculus topic). That is, given a graph of a function what does the graph of look like? Introducing these early will allow you to use them often and to beef up the applications you can explore. These should be introduced along side some continuous versions which are messy to let the students see that though the delta function and step function are idealizations of system input they are much easier to work with than more realistic functions.
Learning from mistakes
A class of problems generated by mistakes.
(1) Consider the following error: . Find all solutions to this equation and show there are infinitely many. Reconcile this with the fact this is wrong and that there are essentially no solutions.
(2) Consider the following error: . Find all functions that satisfy this relation.
Have them linearize gravity at the surface of the earth. This is a great exercise as it requires little calculus but a lot of thinking through the steps. Also the error between the linearization and ‘actual’ gravity is really small.
Somewhere I’ve got a problem about a pendulum clock where the arm is metal (still assumed massless) but expands with temperature. How does this affect the period and then how does it result the time keeping of the clock?
Area between curves:
This is a good time to do calculations of work for simple heat engines. Various cycles like the Otto cycle and diesel cycle on a PV diagram allow straightforward integration but require some thinking about how to do it. It connects quickly with practical considerations without requiring too much overhead.
Integration by parts:
laplace transform. One of the classes I’m teaching this semester is an engineering calculus class where I can be sure all the students are in one of three engineering disciplines. I grow tired of just doing random examples to illustrate various tricks or scenarios involving integration by parts (or any other rule/trick). I introduced the Laplace transform earlier this week. Most of the entries are done via integration by parts anyways. Moreover I’ve realized that there’s a treasure trove of good calculus here. Laplace transforms are also improper integrals and the growth of functions (as compared with the exponential) must also be considered. If you introduced delta and step functions earlier you can play with these now as well.
This is also a good time to talk about analogies between mechanical and electrical systems. While you won’t expect the students derive the differential equations themselves, you can do that and they can solve various versions using the Laplace Transform.