Once we’ve identified the integral we need, we don’t have to resort to the tedious process of dividing up the area under the curve into tiny pieces and multiplying and adding ad nauseam. Instead, we just subtract the value of the integral at the end of the curve from the value at the beginning of the curve to get the answer. Let’s say we want to take the integral from 1 to 4 of the function x 4. We can rely on our little trick above to determine that an integral for x4 is Now we simply plug in the highest and lowest values for x in the range of interest (1 to 4) and subtract the results. Our answer: 1,023 divided by 5, or 204.6. This means that we have gone 204.6 miles between those two points—or that the area under the curve between point 1 and point 4 along the x axis is 204.6.
A physicist who blogs anonymously at Gravity and Levity describes physics and calculus at the high school level as a kind of game. “It was like a little logic puzzle where the rules of the game were given to you (usually on a formula sheet) and you were asked to use them cleverly to come up with a solution,” he says. “A friend of mine once put it succinctly: ‘Physics is all about finding out which variables you know and which variable you want, and then searching through your formula sheet for an equation that has all of those letters in it.’ That, more or less, was the physics game. You rearrange some symbols on a paper and you come up with an answer. Instant gratification.”
Some students take to the game quite naturally; others, like me, do not. But none of us will realize the full power of calculus until we move beyond treating it as a game and learn how to use it creatively to solve real-world problems.
YOUR MILEAGE MAY VARY
Even if we lose at the poker tables, I’ve gained something tangible from our weekend excursion: a valuable insight into the fundamentals of calculus. The derivative and integral are two different ways of looking at the same situation, namely, our Prius driving down a straight, level road. I can use the derivative to find our speed from our position and use the integral to figure out how far we’ve traveled based on our speed.
The speedometer and odometer in the Prius do these sorts of calculations all the time. It was quite ingenious of human beings to build these handy little devices whose primary purpose is to determine the exact information about speed and position that early mathematicians so meticulously calculated by hand. What is their secret? They have much more real-time data at their disposal. Both the speedometer and odometer are designed to collect every possible data point (for speed and distance, respectively) that it can in real time. The speedometer gives us a velocity function; the odometer gives us a position function. We can pretty much find out anything we need to know with this information, with no need to resort to calculus.
Speedometers measure the speed of a car by counting every single rotation of the tires. In older cars, they are mechanical, connected to a drive cable snaking its way from the transmission to the dashboard instrument cluster. The drive cable is basically a cluster of tightly wound coil springs wrapped around a center wire. When the wheels of a car turn, the gears in the transmission turn, and their rotational speed is sent down to the speedometer, where it can be measured and displayed.
The Prius speedometer is electronic (as is the odometer) and gets its rotational data from a vehicle speed sensor mounted to the crankshaft, rather than a drive cable. The sensor is little more than a toothed metal disk and a simple detector covering a coil that emits a magnetic field. The teeth interrupt the magnetic field as the disk rotates past the coil, creating a series of pulses, which are sent to the car’s computer via a single wire. The computer counts the magnetic pulses as each tooth of the metal disk passes by the coil. The real-time speed is displayed on the speedometer, so we can keep track of how fast we are traveling. The speedometer is linked to the digital odometer, so for every forty thousand pulses, the odometer adds one mile.
In fact, the Prius onboard computer goes even further: It combines the data on speed and distance with data collected from sensors monitoring gas usage to determine how many miles the car is traveling for each gallon of gas it consumes—both in real time, and on average over a given period. All this information is processed and presented in a colorful digital display that constitutes a real-time video game, showing how much gas you use at any given moment, and how your driving behavior can change that consumption for better or worse. Really, it’s a miracle that we Prius drivers manage to avoid plowing into ditches and rear-ending other cars all the time, given how distracting it is to have that constantly changing dynamic information on display before us.
Let the naysayers knock my plucky little hybrid if they must, but thanks to that real-time graphic display, I am now hyperaware of how much energy I consume when driving, and how much even tiny changes in driver behavior, type of terrain, or weather conditions can affect my overall mileage. By virtue of constant feedback on your fuel-efficiency performance, the Prius trains you to be a more energy-conscious driver. For instance, accelerate gradually, and you’ll use less energy than if you put pedal to the metal in a vain attempt to go from 0 to 60 in a few seconds.
Also, traveling at a steady speed, even in heavy traffic, is better than jerkily starting and stopping, because every time you restart after a full stop, you have to overcome the car’s inertia all over again. I try to leave a bit of extra distance between my car and the vehicle just ahead, so I can coast a little rather than brake suddenly. Under the best conditions, the difference can be as significant as getting 75 miles per gallon versus 25 mpg. I reflect on that whenever I feel frustration at Los Angeles’ notoriously congested freeways. I might be inching along at a snail’s pace, but I reap the benefit by averaging many more miles per gallon, even if it takes longer to reach my destination. Collectively, these practices have become known as hypermiling.
Even traveling at a steady speed, in general, the faster you go, the more energy it takes to maintain that speed because of increased air resistance (drag). The engine has to work constantly to overcome the resulting drag and thus consumes more fuel. It’s tough to correctly calculate the drag coefficient for anything but the simplest of shapes, but in general, at high speeds, the drag force increases as the square of the velocity. In plain English, this means that if you’re traveling at 100 mph, you’ll experience four times the drag force you’d experience if you were traveling at 50 mph.16
Small increments in improved fuel efficiency can add up significantly over time. So driving just at (or slightly under) the speed limit can result in considerable energy savings in the long term. Back in 1974, the federal government instituted a 55 mph speed limit on highways, not because it was safer17 but because it conserved fuel at a time when oil was scarce. Similarly, driving uphill uses more energy than coasting downhill—any avid bicyclist could tell you that—as does driving into a strong headwind. Certain driving conditions are beyond one’s control. Don’t even get me started on what a ten-hour drive from Salt Lake City to Los Angeles in gusting crosswinds through a mountain pass did to my average miles per gallon.
Why doesn’t everyone ditch their current gas-guzzling cars for a Prius or similar hybrid? The answer might surprise you. It turns out that many of us assume that saving gas (and therefore money) corresponds linearly with miles per gallon. But according to a June 20, 2008, article in Science by Richard Larrick and Jack Soll at Duke University’s Fuqua School of Business, the gas used per mile is actually inversely proportional to miles per gallon. They call this the mpg illusion.
Let’s say you own two cars: one with a 34 mpg rating, like Sean’s old Toyota Corolla, and another with an 18 mpg rating, like my father’s beat-up Chevy pickup. Should you replace the 34 mpg Corolla with a pricey 50 mpg hybrid, or the 18 mpg pickup for a cheaper 28 mpg nonhybrid vehicle, in order to achieve optimal savings? You want to optimize those gas savings to recoup your initial capital investment as quickly as possible. Run the numbers, and it becomes apparent that replacing a 34 mpg car with a hybrid that gets 50 mpg will save you 94.1 gallons of gas per 10,000 miles; in c
ontrast, replacing the 18 mpg truck with a 28 mpg vehicle will save you a whopping 198.4 gallons per 10,000 miles.
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 embedded in the road on which they are traveling. In this way, they can relay critical information, such as whether there’s been an accident up ahead, and also serve as anonymous data collectors.
Traffic jams are a bit like the process of freezing. On a sparsely populated highway the cars are far apart and can move freely at whatever speed they choose while maneuvering between lanes—much like the movement of molecules in a gas. In heavier traffic, the “car molecules” are more densely packed, with less room to maneuver, so cars move at slower average speeds and traffic behaves more like a liquid. If the car molecules become too densely packed, their speed is reduced and their range of movement is restricted to such an extent, they can crystallize into a solid, akin to that critical temperature/ pressure point at which water turns into ice.
It’s a useful analogy, but the reality is a bit more complicated. A physicist named Boris Kerner has analyzed data collected from several years of traffic monitored along German highways and found that traffic tends to follow the rules of self-organization. His model breaks down traffic into three basic categories: freely flowing, jammed (a solid state), and a bizarre intermediate state he calls synchronized flow, in which densely packed car molecules move in unison, like members of a marching band. When all the cars are traveling at roughly the same average speed because of the vehicle density on the roadway, they become highly dependent on one another, or “highly correlated.”
When cars are highly correlated, a tiny perturbation will send little ripples of slowdowns through the entire chain of cars behind the offending vehicle. What happens if the law-flouting driver in the Audi ahead of you decides to text his girlfriend and then has to brake too suddenly when he looks up and realizes he’s about to rear-end the BMW just ahead? That makes you brake too suddenly, and the person behind you, and so on.
That’s one reason why traffic jams are so common at freeway entrance and exit ramps, or—like on the I-15—when lanes are closed due to road construction (or a major accident). A state of steady synchronized flow, punctuated by these tiny ripple effects, can persist indefinitely, but the balance is delicate and highly unstable. If the volume of cars continues to increase, the density also continues to increase, and eventually you get a “pinch effect”: that frustrating stop-and-go phenomenon we are experiencing on the road to Vegas, in which you escape one narrow traffic jam only to encounter another a little farther down the road, until they all converge into a single wide jam. Traffic comes to a standstill.
Given world enough and time, even the worst traffic jams eventually unsnarl. We finally break free of the construction zone, and Sean gleefully accelerates to full speed. Zzyzx Road can eat our dust. Soon we’re happily chowing down on gyros and falafel at the Mad Greek Cafe, bagging some tasty pistachio baklava for the road, as well as some Alien Beef Jerky from the tiny store decked out in UFO paraphernalia down the street. (Baker’s a pretty colorful town.) An hour or so later, our hunger sated, we are cruising down the infamous Las Vegas Strip toward the Luxor Hotel, where Lady Luck—and the calculus of probability—will determine our fortunes in the casinos.
3
Casino Royale
The theory of probabilities is at bottom nothing but common sense reduced to calculus.
—PIERRE-SIMON DE LAPLACE
Legendary Vegas gambler Nick the Greek (aka Nicholas Andrea Dandolos) won and lost over $500 million in his lifetime by his own estimation, moving from rags to riches and back again countless times. Along the way, he met pretty much everyone, from Al Capone and Bugsy Siegel to the Marx Brothers, Ava Gardner, and John F. Kennedy. Every celebrity who visited Vegas wanted to meet the last of the true high rollers. So when the world’s most famous physicist, Albert Einstein, came to town for a symposium, naturally he sought out Nick the Greek, while indulging in a spot of gambling himself at the craps and roulette tables.18 Realizing that his gambling cronies would have no idea who Einstein was, Nick the Greek simply introduced him as “Little Al from Princeton—controls a lot of the action around Jersey.”
Sean is tickled when I tell him this (possibly apocryphal) story. Any serious gambler should have a cool nickname, he declares, and promptly dubs himself S-Money for the duration of our stay. We normally focus on poker when in Vegas, but this time, we’re interested in learning craps, b
ecause it is a natural fit for discussing the calculus of probability. Much of probability theory emerged from attempts to analyze games of chance, particularly those involving the throwing of dice, sticks, or bones. There is even a theorem known as the craps principle, dealing specifically with event probabilities under repeated trials. And what better way to explore that principle than to hit the craps tables in a bona fide casino?
There are many online guides to playing craps, some with in-depth analysis of all the related probabilities, but these tend to be dense and jargon-heavy. There are also online computer craps games where you can practice placing bets and rolling virtual dice without risking any actual money. But sooner or later, you have to step up and put your wallet on the line. Craps doesn’t really begin to make sense until you get your hands dirty and play the game in a real-world setting—like Las Vegas.
Craps is a raucous, fast-moving game—there is one roll of the dice every twenty seconds or so—and this pace can be intimidating for the average newbie still struggling to grasp the rules. So we have opted to take one of the daily introductory classes offered by the New York, New York casino. Our instructor is a dapper man, slight of build, with tidy salt-and-pepper hair, wire-rimmed glasses, and a wry sense of humor, whom I dub Dominic. Dominic has been a dealer for thirty years and is happy to share not just the rules of craps but colorful anecdotes from Las Vegas history.19
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 6
