Sarah was on the phone talking to Charlene, who was having trouble with her boy friend. This was nothing new — Charlene was always having trouble with her boy friend. Sarah saw her mother come in the room; she frowned and sighed heavily. Sarah listened to Charlene's complaint a little longer, then broke in: "Well Charlene, Richard may have the sensitivity of a pumpkin at times. Still, he did come through last month on your birthday. Look, can I call you back? I need to discuss something with Mom."
Sarah's mother sat down on the sofa holding a printout of graphs and arrows.
Sarah joined her. "Looks
like the calculus is getting you down."
"Well, Sarah, things were going fine until I got to these slope fields. What are all these little lines?"
"Oh, those. They're directions," Sarah replied.
"What do you mean, they're directions?"
"That bothered me too. I finally figured out that you should think of them as arrows pointing from left to right."
"All right. So they are arrows. What does that tell me?" Sarah's mother still did not seem satisfied.
"One of the TA's told me to think of it like this. Suppose you are a little person riding around in a little car on the t,P-plane. Where you start is determined by the initial condition. Once you start, where you go is completely determined by the differential equation. Whenever you come to a point in the plane that is the start of one of the arrows, you will be headed in the direction of the arrow. If you put your finger down here, then you can pretty much see what the graph of the solution has to be."
Sarah traced out a curve with the tip of her finger. "Oh, like this." Sarah's mother traced out another curve. "I think I see. So that is why the differential equation together with the starting value determines a unique function graph. Now tell me who a TA is."
"A teaching assistant. Often they run the labs."
Sarah's mother looked up from the book. "Oh, isn't that what you said
Stan was?"
"No, Mother. Stan was a lab monitor. He is just a year older than I am; the TA's are either advanced undergraduates or graduate students."
"Yes, he does look rather young."