Divine Heritage, Chapter 3, section 2.
Dr. Roper had set a fast pace, as Ms. Emory predicted he would. Already he was treating higher-order derivatives and their meaning during the first half of class, and presenting different methods for integration during the latter half. He'd given homework, some of which tried to confuse his students about which order of derivative to set equal to zero in order to find a local extreme of the next lower order. Other problems involved integration, which would have been devilishly convoluted for someone without experience in knowing when to use a trigonometric substitution, when to integrate by parts, and when to have a peek in The CRC Handbook of Standard Mathematical Tables and then reverse-engineer the logic behind an integral identity.
So far, my experience had enabled me to surf the class without having to exert myself much. I'd earned the gratitude of a few students one day by dropping by a study hall frequented by math and science majors, and correcting a few of my fellow college freshmen who had neglected to transform the differential dx to its new space, f(u) du, after making a substitution.
Yes, Brookstone College considered me a freshman, even though Brookstone GS called me a sixth-grader.
Dr. Roper had assigned a homework problem in which we were to find the analytic solution to an indefinite integral. The integral looked difficult, but it was not. You started with a trigonometric substitution, x equals the tangent of u, and you worked out the trigonometry until you obtained the transformed integral in its simplest form. Then you used integration by parts, grouped terms, applied a couple of trig relations, and did some factoring. But in the study room, when my older classmates asked me for help, I went to the blackboard and wrote:
∫ [ (7x Arctan x) / (1+x²)² ] dx
(A miracle occurs here.)
= (7/4) [ x + (x²−1) Arctan x ] / (x²+1) + K
The amusement that greeted my abbreviated demonstration was loud enough to bring Ms. van Peenen, a math teacher of Dutch extraction who treasured peace and quiet, out of her office to tell us students to hush. Then I had to get to my next class. I heard voices tapering in decrescendo behind me.
"How'd she do it?"
"Solved it in her head on the way down the hall."
"I know. I can barely chew gum and walk."
Physics 101 wasn't nearly as challenging. It was almost like high school physics, with a little calculus thrown in. Our hardest homework problem so far had been to derive the formula by which one would calculate the horizontal range of a projectile on a flat, airless world having a gravity field that did not vary with altitude, as a function of its initial velocity. I turned in the ridiculously easy homework assignments and tried to hide my boredom. When would the really good stuff begin?
I'd kept thinking about Vanessa Emory's words "not even you," as if she were Lois Lane reminding Superman that even he had his limitations, through my calculus and physics classes. Since first meeting her on the bus in Atlanta, Ms. Emory had shown an interest in me that had been very unusual for such a highly placed executive, not to mention someone whose aging father owned most of the school I attended. At first, I'd thought that she was shepherding me because my famous IQ-test results made me a feather in Brookstone School's cap, but lately I'd begun thinking that her interest was even greater than that could account for.