The Calculus Diaries - Jennifer Ouellette [19]
None of this, alas, helps Sean avoid an imaginary traffic ticket. He grudgingly admits defeat. The mark of all good scientists is the willingness to abandon a pet hypothesis if the experimental evidence contradicts it—but that doesn’t mean they have to be happy about it.
THE SUM OF ALL THINGS
Taking a derivative is pretty straightforward. Finding the integral is trickier. Conceptually, it’s just the flip side of the derivative: With the derivative, I can figure out my car’s speed based on how its position changes over time. With the integral, I should be able to determine how far we’ve traveled in the Prius based only on measurements of its speed at given locations along our high-tech highway.
Thanks to modern technology, I can just use the car’s odometer and built-in GPS system to find the answer. But what if the odometer is broken, the computer has malfunctioned, and we find ourselves stranded in the middle of nowhere, with no other cars in sight? These highfalutin hybrids with their onboard computers and hordes of sensors are pretty sensitive, after all.
Assuming our cell phones still work, we can call AAA, but we need to be able to tell them our precise position. There are no obvious landmarks. “Third tumbleweed on the left next to the giant boulder” isn’t going to narrow things down sufficiently. We know we haven’t passed Baker. Even if you missed the Mad Greek Cafe—despite the fact that it is gaily painted with the colors of the Greek flag and adorned with plaster replicas of naked Greek statues out front—you’d certainly notice Baker’s other main attraction: the World’s Tallest Thermometer. Baker is located at the 188-mile mark between our Los Angeles loft and the Luxor in Vegas. Let’s say that an hour before we got stranded, we stopped for coffee in Barstow, which is at the 110-mile mark. So I know we are somewhere between 110 miles and 188 miles from our home in Los Angeles.
Had our speed been perfectly constant, this would be a simple task, and we would have no need for calculus. Assuming a constant speed of 60 mph, for instance, and knowing that exactly one hour has elapsed since we left home, I can multiply our speed by the time and conclude that we have gone 60 miles. It’s probably a pretty good approximation. But that doesn’t reflect actual driving conditions; a car’s speed is constantly changing, even more so if there are spots of heavy traffic, and if my lead-footed spouse drives faster than 60 mph to make up for lost time whenever traffic clears.
The only concrete information I have about our velocity is from monitoring the speedometer. Fortunately that’s all I need to figure out how far we’ve traveled and thus pinpoint our location for AAA. The speedometer has displayed our speed at every instant along our journey; taken together, this gives me our velocity function. So I should know exactly how fast I was going at any given moment.
How do I take the variation in speed into account? I set boundaries around the correct answer to get a workable range for determining the distance. First, I do a series of calculations based on the slowest (starting) speed—in this case, at the point where we left Barstow—breaking that journey into smaller and smaller increments of time and adding up the pieces to arrive at a close approximation to the total distance traveled. But this will be an underestimate. So I also need to do the same labor-intensive process for the fastest speed the car was traveling over our entire one-hour journey. The resulting approximation will be an overestimate of how far we went, but at least I know that the correct distance is somewhere in between those two values. I then go through