# College Math Teaching

## December 1, 2012

### One challenge of teaching “brief calculus” (“business calculus”, “applied calculus”, etc.)

Today’s exam covered elementary integrals and partial derivatives; in our course we usually mention two variable functions and show how to calculate some “easy” partial derivatives.

So today’s exam saw a D/F student show up late (as usual); keep in mind this is an 8 am class (no class prior to it). He, as usual, got little or nothing correct. Of course we had the usual $\int \frac{1}{x^2} dx = ln(x^x) + C, \int^1_0 3e^{5x}dx = (15e^5 -15) + C$, etc.

But there was this too: note that we had barely discussed partial derivatives and how to calculate them “by the formula”. But I did give the following bonus question: “is it possible to have a function $f(x,y)$ where $f_x = x^3 + y^3$ and $f_y = 3xy$? Yes, this is a common question in multivariable calculus (e. g., “is this vector field conservative?”) but remember this is a “brief calculus” course.

A few students took the challenge; some computed $\int(x^3 + y^3)dx = \frac{x^4}{4}+ xy^3 + C, \int (3xy^2)dy = \frac{3}{2}xy^2+C$ and noted that the two functions cannot be made to match (I didn’t expect them to recognize that functions of one variable alone represents constants of integration). Some took the second partials and noted $f_{xy} = 3y^2, f_{yx} = 3y$ and that these don’t match. Again, this was NOT a problem that we practiced.

Another instance: given the ideal gas law $PV = nRT$ I challenged them to show $\frac{\partial P}{\partial V}\frac{\partial V}{\partial T}\frac{\partial T}{\partial P} = -1$ and someone got it!

Bottom line: in one course, we have some bright, interested students who enjoy thinking and we have some who either don’t or can’t. This makes teaching difficult; if one tries to “teach to the mean” one is teaching to the empty set. It is almost: either bore half the class, or blow away half the class.