Arrrrrrrgh! I flipped my pencil around and scrubbed out a line of equations... for the fourth time. 'I hate math!' I growled to myself. That's when the eraser decided to shear off and roll away under the table while the rest of the pencil gouged into the page, tearing it. As thoughts of a fiery doom for my math book filled my head, I smoothed down the rip in my notebook, trying to salvage a half-hour's worth of limits. Why on earth Mrs. Mann, calculus demoness from Hades that she was, wouldn't let us use laptops or even calculators was beyond me, but it translated to a lot of work on paper.
I ducked under the table to grab my eraser, and wished fervently that Jubatus were here. I'm sure he'd know how to do this problem.
"Good afternoon --"
I just about jumped out the roof. "Aaaagh!"
It was the plant lady... um, Sue Carter, I think. She arched finely drawn eyebrows at me and commented dryly, "A bit high-strung today, are we?"
Jubatus had warned me about her, that I shouldn't trust her or talk to her about him. So I put on the most innocent face I could, and said, "Um, I haven't seen Jubatus around lately, if that's what you're going to ask." Then I noticed that I'd dropped the eraser when I'd jumped, and scowled, looking around for it. Again. "You wouldn't happen to have seen the eraser I just dropped, would you?" I stooped to look under the table again, keeping one ear towards her for a reply.
"It went off to your left," she said. "You're Hallan, right?"
I looked left to where she pointed, and found the little bugger next to the piano. "Yes, ma'am," I replied as I settled back at the booth, careful to thread my tail through the slot at the back of the seat rather than sit on it.
"I couldn't help but overhear that you're having some trouble with math." She gestured toward my much-rumpled notebook. "Would you like some help?"
Setting the eraser down out of the way where it wouldn't get bumped off the table and lost for a third time, I nodded and answered, "It's introductory calculus stuff..." I pushed the book over where she could see it. "If that's not a problem, then no, I wouldn't mind."
She "hmmm"ed for a moment, then slid into the seat across from me and reached across for my notebook, producing a pen of her own as she flipped to a clean page. "Differential calculus, at its most basic level, is simply a mathematical tool to study the ratio of change of events. As an example," she quickly drew a square graph and a straight line, "here we have a line with a constant rate of change, said rate being the value 'b' in y=bx+c. You've taken that already?"
She drew another graph, this time with a perfect circle as the line. "In this case the slope is not constant, but the rate of change of the slope is a constant as this is a circle. In most cases however, the rate of change of the slope changes, a simple case being the elliptical orbit of Brin station."
She drew another graph, this time with the line being an ellipse. I was actually following this!
"Back to Differential Calculus. Right now you seem to be wrestling with the concept of limits, and probably have no clue as to why you're being stuck with such a useless task. The key to mathematics is understanding the reason why things are done. Differential Calculus exists to determine the slope; and in the case of a straight line the slope is trivially obvious. With a curved line though the slope varies, and thus you want the slope at a single point of the line, at a particular set of values, or at a limit of those values for that line." She drew another graph and this time a curved line arcing upward.
I half nodded, one ear dropping slowly back. As she went on, she talked progressively faster...
"The slope of this straight line can be represented by (f(b)-f(a))/(b-a) where f(b) and f(a) are represented by using the values of b and a respectively for x."
"Okay..." Where was she going with this?
"Allow the point b to move closer and closer to point a." She drew a number of lines of increasing slope so that a and b were closer and closer together. "Eventually, you reach the point where a and b are equal, and the straight line is the tangent line to the graph of y=x^2 at the value of x=1. Thus the slope of the line any line at any point can be represented by the general equation..."
She turned the page and wrote an equation in large print: Lim(x->a) [(f(x)-f(a))/x-a)] My head started to spin.
"In this case let y=f(x) be a function, and suppose that a is in the domain of f. In other words, a is a valid value of x within the equation y=f(x). A line L containing (a, f(a)) is a tangent line to f at x=1 if the slope of line L is the limit. Thus, your problem is actually determining the slope of y=x^2 for the value of x=1, what we designated a, or in the way the problem was given to you, the limit as x approaches 1 for the equation (x^2-1)/(x-1).
"Um..." I half reached for my notebook, but she flipped to the next page and started jotting faster, her eyes locked to what she was doing.
"Now, lets go through a more complex and practical example. Let's say that Babylon is approaching Brin station along a parabolic curve that can be represented by..."
"Of course Brin is also moving in an elliptical orbit, and it is rotating. We can ignore quantum variances because the speeds involved are low enough that Newtonian mechanics still holds. If we want to calculate the distance between one end of the rotating line that represents Brin, call that point a, and Babylon, we generate this equation with respect to time, given that Brin's rotational momentum..."
"Hold on a--"
She turned another page and kept writing. "For large distances, we can abstract Brin to a point, and represent the distance as a straight line, the hypotenuse of a right-angled triangle made up of the points a (Babylon), b (Brin) and c. If we ignore motion, this becomes simple trigonometry. You've taken Pythagoras?"
I looked on in dismay, giving up on trying to get her attention outside of a bullhorn to the ear. And I didn't feel like losing my voice roaring at her. I guess she assumed I nodded because she plowed right on.
"From Pythagoras we get (a-b)^2 = (b-c)^2+(c-a)^2. But, remember that both Babylon and Brin are moving. If we use a first level mathematical approximation of their orbits..."
More numbers appeared. And letters. She filled the page and moved on to the next one.
As she droned on, I looked around in a panic and noticed Jubatus just coming in. He must have seen the desperation on my face, because he walked over and carefully pulled the green math demon's hand away from the page. I heard a rustle of flipped pages and saw that my notebook was back open to the original problem. "Jubatus, how..?
He ignored me and turned to the plant lady and asked her, "Carter, have you forgotten what grade the kid's in? This is graduate-level stuff, maybe post-graduate!"
I sighed with relief as she turned away from my book towards Jubatus.
"This would have been no problem for me at his age," she answered him.
"Excuse me..." I tried to interject.
Jubatus answered her, "You were in college at age 15?"
Sue glared at Jubatus. "I will admit that I was a little advanced for my age, but I had thought that the North American educational system still had some quality left within it."
He shook his head in apparent disbelief and then turned back to me. At least he had heard me. "Okay... what you want is a series of successive approximations. Start out with what looks like a decent guess, run that thru the equation, and if that doesn't get the exact answer, try some other guesses. Say you start out with 1.9, and it turns out to be a little low; the next guess might be 1.95. And so on. You want to look for what the values are trending --"
I started scribbling notes but she didn't let him finish, "Here we have the modern American way. Keep them in the dark and make math a simple mechanical tool. So tell me Mr. Jubatus, Why'd you keep your injury secret?! I could have helped you."
I flinched away from her shout, which drew looks from every other patron in the bar. By this time, I'd given up on Carter entirely and focused all my attention on getting Jubatus to finish his explanation. Meanwhile, they continued their rapid back-and-forth sparring.
He turned away from me and answered, "Carter, if someone asks you what time it is, you don't give them the blueprints for a bloody atomic clock!"
The plant lady turned her level of sarcasm up, "And if they don't know why an atomic clock works, why not just tell them what time it is?"
"Exactly," he retorted, the fur along the back of his neck rising.
She turned to me and sharply stated, "The answer is 2," before turning back to Jubatus. "I was concerned about you! If you'd have died --"
Jubatus cut her off. "-- it would be the same thing that happens to everyone, sooner or later."
"Jubatus, could you repeat what you said..?"
She plowed into him with barely a pause for breath, "Do you have any idea what the cost to hum-"
Apparently Jubatus had heard me, "So. Successive approximations. Does that make sense to you?"
"I think so, but --"
Sue bulled her way back into the conversation, and Jubatus' twitching tailtip suggested he wasn't going to stand that for much longer. "Mr. Jubatus, you will listen to me now. If I was present you would have recovered your mind faster. You know it; I know it. Since you want to keep the subject sec --"
"Jubatus..." I had a sinking feeling as to whether they were ever going to stop arguing, but I held onto one slim hope: If I could just squeeze a few more clarifying sentences out of --
Jubatus turned back to her and snapped, "Enough. You want to discuss this, we do it in private. Back room. Now." Then he turned and stalked off, Sue hot on his tail.
"Jubatus!" I called,too late to do more than have it bounce off the door that slammed shut behind the pair.
"Well, that was helpful," I grumbled as I picked up my pencil. I thought she'd said the answer was 2, and Jubatus had said something about successive approximation... Whatever 'successive approximation' was. Sigh... Back to figuring this out the hard way... Open book to page 12, and let's go over this again...