Sarah's mother pushed back her chair and stretched out her arms. Sarah came in with coffee for both of them.
“So now I know how to differentiate polynomials. I still don't see any relation between all this and population growth."
Sarah sipped her coffee. Her mother always wanted the big picture, to know where she was going. Often Sarah was content just to do the work assigned and assume that at some point it would all make sense. However, her mother clearly wanted a response.
“Yeah, you need to learn how to find the derivatives of a lot more functions — square roots and sines and cosines and ... "
Sarah's mother looked a little discouraged. Sarah hurried to add, “But to understand the simplest population models, you only need to look at functions whose derivatives are multiples of the original function."
“What do you mean, `multiples of the original function'?"
“Like a function whose derivative is twice the original function."
“Hmm," her mother muttered. She paused with her cup in mid-air, frozen in thought. Sarah looked at her watch and realized that Stan would be by in a couple of minutes.
Her mother drank some coffee and set the cup down. “But we don't know any functions like that! Well, maybe the function that is always zero, but we can't use that to describe anything interesting. The linear functions have derivatives that are constants, and the derivative of kt2 is just 2kt. Maybe, higher powers ... . No, you always decrease the degree of a polynomial when you differentiate."
Sarah was impressed that her mother had picked up on the degree thing. She was pretty good at this. It was amazing how much smarter her parents appeared after she had been away at college for a year. But she needed to move this along or she was never going to be ready when Stan arrived.
“You need to look at the exponential functions."
“Isn't t2 an exponential function? There's a 2 in the exponent."
“Well, yeah, but what they mean by an exponential function is something like 2t or 3t or 10t."
“I've never looked at functions like that before. Why in the world would something like that turn up when we are talking about population growth?"
Sarah tried to remember — what had they looked at just before exponential functions. She went to the computer and clicked through the pages. Oh yeah, that stuff about interest rates.
“Here, look at this page. A bank account at fixed interest grows at a rate that is proportional to how much money you have in it, just like a population. Except it only grows each time they calculate interest. It's right there. I've got to get ready for Stan."
Sarah felt guilty as she hurried from the room before her mother could ask any more questions. Her mother still looked a little confused. However, in a few minutes Sarah had forgotten all about it.