The Calculus Diaries - Jennifer Ouellette [25]
That means you’re much better off replacing the lower mpg vehicle (the Chevy pickup) with a cheaper alternative to the Prius to get the biggest cost savings. This seems counterintuitive. After all, you’re getting a 16 mpg improvement in the first example, and only a 10 mpg improvement in the second. But if you put this data into graph form, you can clearly see the gas used per mile is inversely proportional to miles per gallon.
There is a steeper slope at lower mpg ratings and gradually diminishing returns as one moves up the graph to increasingly higher mpg ratings. So even such seemingly simple numbers can be deceptive, particularly since most of us are sadly deficient in our grasp of basic mathematical concepts. And in this case, our ignorance could prove costly.
That’s why I resist the occasional twinge of Prius envy when I read about the 2010 Prius with the solar-powered sunroof and even better mileage. Based on the above calculations, it simply isn’t cost-effective to replace my 2007 model with the newer model; it would take much longer to recoup that capital investment. I’m better off just driving my existing Prius into the ground.
MODEL BEHAVIOR
If digital speedometers and odometers do a better job than manual calculations of speed and distance, why do we still need calculus at all? Calculus is a vital part of almost every field of science because it enables scientists to construct mathematical models to study complicated real-world systems—including traffic patterns. Like the computer dashboard displays in the Prius, mathematical models are visual representations of abstract concepts, with the added advantage of enabling scientists to make useful real-world predictions.
Admittedly, not all mathematical modeling has a practical application. Topologists, for example, are interested in studying imaginary multidimensional shapes that simply couldn’t exist in our four-dimensional space-time. But much of the appeal of mathematical modeling for less exalted minds lies in how it can help make predictions about how a system is likely to behave, so we can make better, more informed decisions—such as whether to stay on a clogged freeway and wait out the congestion or try to find an alternate route to avoid any more potential slowdowns up the road. (The latter is not an option on I-15. There is no alternate route.)
The more data points you have to work with, the more accurate your models will be. Ideally you would like a continuous stream of real-time data rather than a collection of discrete data points. That’s why state and federal agencies spend about $750 million each year on traffic monitoring to gather better data in hopes of building better predictive models of traffic flow. For instance, several state transportation agencies—Maryland, Virginia, Missouri, and Georgia—are experimenting with software that uses radio signals from drivers’ cell phones as tracking devices to monitor traffic patterns. The phones just need to be turned on; the agencies swear they are not monitoring actual conversations.
“Listening posts” are placed throughout a designated region; they are capable of detecting but not sending radio signals. A post will pick up a cell-phone signal and time-stamp the signal’s arrival. By analyzing how long it takes the radio wave to reach the listening post from the cell phone, a computer can calculate almost precisely where that phone is located on the highway. You need data from three such listening posts to determine a two-dimensional position of a given cell phone user. Adding radio tags along the highways to time when vehicles pass between given points can determine the car’s location and speed. Berkeley, California, has a test-bed project dubbed Smart Cars and Smart Roads, whereby participating cars are equipped with wireless technology to pick up signals transmitted from sensors