Sarah Allen - Fairy Tale Calculus: The Derivative

upon a time, on the edge of a deep forest, there lived two small trolls. Their names were Herbert and Erbert and they each lived in their own small burrows. Along the edge of the forest ran a wide dirt road, where many travelers would pass on their way to market. In between Herbert and Erbert lived a much larger, meaner creature. It had no name, and was so quick that no one had ever gotten a very close look at it. It was an ancient, secretive creature, and it hated people who were always rushing around.

When it heard people outside its burrow speeding along the path too quickly it would leap from its burrow and gobble them up. The villagers complained to the kings men, who hunted the beast, but to no avail. The tried to lure it out, smoke it out, and dig it up, but the creature was too wise and quick for them. Then the King tried to move the road, but on one side was the forest, which was filled with even worse beasts, and on the other side was a thick bog, which would support no road. Herbert and Erbert had come out of their burrows to watch the efforts to destroy their neighbor, and at last Herbert spoke up. Excuse me, he said.

The king and his men turned to the little troll. What if my friend Erbert and I just keep watch? We can measure the speeds of the travelers and tell them to slow down if they need to. The king thought this was a very good idea, and he commended the two trolls for their bravery and service to the realm. He also said he would pay them each one gold coin every market day that no one was eaten. He also gave them each a large brass pocket watch. On the first market day, they each stood proudly outside their burrows, with their pocket watch chains glinting on their chests.

When a traveler would come from Erberts side, Herbert would run to the middle, just past where the beast was, so that they could time the passage of the traveler. They each stood one hundred meters from the beast. If a traveler took less than 20 seconds to get to the beast from where one of the trolls stood, then the beast would eat them. The first person to come along was a farmer, pulling a heavy load of pumpkins. Herbert ran to the middle when he saw him coming, and started his pocket watch just as the man passed Erbert. The farmer trudged at a slow and steady pace and took a full one hundred seconds to cross the gap.

Herbert nodded politely to him as he passed. Next came a woman, hurrying in the other direction, pulling her daughter by the hand. Erbert ran to the middle and started his watch as she bustled by Herbert. Just before she reached the beasts lair, though, the child tripped. The woman sighed, picked her up, and slowed down. When she passed Erbert she was going more slowly than the farmer had, but because she had been going so quickly before, it had taken her less than 20 seconds to cross the distance.

Still, Erbert wasnt sure and he warned her to slow down, anyway. She looked at him like he was stupid and continued at her slow pace past the beasts lair, and was not eaten. The third to come was a very old man. He hobbled most of the way so slowly that Herbert was barely even looking at his watch anymore. Just before the man reached Herbert, though, he giggled. Then he pulled off his beard and dropped his cane.

Too slow! The boy yelled and took off, running towards the market. Just as he passed Herbert, though, a dark blur engulfed him. There was a crunch and a snap, and then the boy and the blur were gone. In that first day, ten people were eaten, which was less than usual, but still not desirable, and definitely not enough to earn them their gold. That evening, Herbert and Erbert met to discuss the problem. If people changed their speeds in the one hundred meters, then they still might be going too fast when they reached the beast, even though they took more than twenty seconds to cross the total distance.

They decided that they would each stand only fifty meters from the beasts lair, and make sure that people took no less than ten seconds. The next day only seven people were killed, which was better but still not ideal. So they moved so that they were only ten meters away. Now they could smell the beast. It smelled like dust, and smoke, and cinnamon. That day only one man was killed, but it was still too many, and Herbert and Erbert, who were sitting together at the side of the road, had begun to lose hope.

Its no use, said Erbert. No matter how close we get we still dont know exactly how fast theyre going right when they pass the beast. Just at that moment, two old men, who happened to be passing just in front of them, both stopped in their tracks and turned to face the two worried trolls. I am the great and powerful Leibniz, and I can give you what you seek, said the first. I am the wise and fearsome Newton, and I can give you what you seek, said the second, at exactly the same time. Herbert and Erbert looked at the two wizards, not sure which one to answer first, but wanting to appear as polite as possible to each of them.

They didnt know much about wizards, but they did know it was important to be polite to them. Luckily for the trolls, the wizards spoke again first. What you need, is the limit, said Newton. Yes, said Leibniz, narrowing his eyes at Newton. You must seek out the limit. Where can we find this limit? asked Erbert.

It is infinitely far away, said Leibniz. And infinitely tiny, said Newton. Or you can just make h go to zero, said Leibniz. Newton glared at him. H, like in my name? So wed both just be named Erbert? How is that- er, how would that help? You must occupy exactly the same spot. And measure exactly no time.

Erbert glanced at Herbert. They began to edge away slowly. Or, I can just give you this, they each said, holding out the exact same talisman: a small emerald green stone. Erbert and Herbert looked at each other warily as the two wizards jostled with each other, each trying to hold his magical talisman out farther than the others. Herbert and Erbert knew they could never protect the villagers completely with what they were doing, but these two wizards did not seem completely trustworthy. At last, they shrugged their shoulders, reached out and each took a talisman.

Nothing happened. Touch them together, said the wizards. Erbert and Herbert reached out and touched the stones together. There was a pop, and a cloud of smoke, and when the smoke cleared there stood one troll with two arms, two legs, two pocket watches, and two heads. The wizards, who had gone back to arguing with one another stopped when they saw the expressions on the trolls faces. Huh, said one.

Oops, said the other. Well, glad that worked out, said the first. Wed better be going now, said the second. More people to help and all that. No need to thank us! They were already almost over the next hill and around the bend. Then they hitched up their robes and ran away, all knobby-knees and elbows.

The talismans were gone. They searched the grass and forest late into the night, but could find no trace of them. Unable to decide whose burrow to go to, they slept that night at the roadside, and awoke cold and damp with dew. When the first travelers woke the trolls as they passed, out of habit they got up and began to time them. They found to their amazement that what the wizards had said was true. They were timing on their pocket watches infinitely small times, and of course the distance between them was infinitely small, and what they came out with was speed.