Even though I've been much involved in the gourmet world lately, I haven't forgotten my calculus at all. I do problems every night. My current problem work has me drawing graphs, which I love, since it's something like art. I enjoy plotting out the curves on the coordinates by hand, though I also have graphic calculators on two of my computers. As I draw these, I experience a kind of kinetic feel as they move through their curves. It's like being in motion just to look at them. Some of these, of course, actually describe motion, such as the ubiquitous parabola, or the exponential curve of the speed of falling objects. But others are less speedy, and include their negative reflections, such as hyperbolas and the up-down and suddenly across curve of f(x) = x³. Some hyperbolas have a breezy sail-like quality, gliding over a grid-patterned sea. These are only simple curves; there are plenty others which yield all sorts of wild forms. But my favorite so far is the heroic tower produced by the equation f(x) = 1/x². Here is a plot of it from one of my graph-drawing programs:

You can see how the two halves of the inverse parabola curve out at the base of the "tower." There are buildings which do this in the "real world," especially William James Hall, the building that houses Harvard University's psychology department. I lived in Cambridge for 12 years (1976-1988) and for ten of those years, I passed by this building every day. Built in 1963, its white modernism is out of place on the red-brick Harvard campus and the building is much maligned by Harvard's architectural conoisseurs. It also had the misfortune to be designed by the hapless Minoru Yamasaki, who was the architect of the doomed World Trade Center in New York. But I loved William James Hall, not only for the way the yellow-orange winter sunlight made its white surface gleam like brilliant brass, but also because of that slight parabolic curve at the bottom, which is only somewhat evident in the website's picture. My Harvard years were math-free, and yet even then, I recognized that the base of Yamasaki's building was a parabola, or at least a section of one.

The problems have me drawing secants through these curves and finding the slope of that secant, and then through the process of limits, finding the slope of a tangent line at some point on that curve. Meanwhile I am holding on to the virtual railing, or the steering wheel, as my vehicle speeds around the curve of the graph. I feel the acceleration, just as if I were actually driving my Electron Car on that trajectory. But to find my instantaneous velocity while driving, all I have to do is look at the speedometer. Really? Even that is an approximation, derived from the turn rate of the transmission shaft (or front axle, for a front-wheel drive car like mine). And even more, since my car is an Electron, I will never know just how fast it is going, as long as I know where it is (or vice versa). For now, all is approximation.

Posted at 2:02 am | link