The 151 quizzes are good. I caught almost everyone with the trusty old “differentiate a constant” wheeze… of course, but you wouldn’t’ve known it by these guys on the day. In fact—math teacher at play— is the “answer set” (sure as heck not the solution set!)—with appearing several times (once as ) and a variant (… slightly better) on one other value.
Now. If I’m going to go in claiming that anybody “should have known” something… then how come they should’ve? Let’s see. 11e^x, 13e^\pi, cos(x), 1/x^2, polynomial, radical. How come two of ‘em are alike? What’s different about ‘em? This is pre-math stuff here: mere “test-taking skills”. Okay. What else?
Am I gonna say everybody oughta be trying to imagine a graph? Like, every single time? Well, maybe it’d be a good idea… at least right in here at first… to at least consider it. The trig function everybody darn well oughta know is some wavy thing; the polynomial’s a cubic so again one has a pretty good idea. The radical and the rational function are a little more esoteric and I wouldn’t blame a Calc I student a bit for not thinking too hard about it if nothing pops up for free in instant recall; if it comes to thinking about it, maybe it’d be a good time to break out the grapher. As for the problems with e‘s in ‘em, I do feel that everybody oughta have a mental image of “exponential growth” (, say [with A_0 and K positive constants]): it sure is a simple doggone thing to sketch…
So then. With any luck, one then asks oneself, “what’s the graph of e-to-the-pi then, eh?” and arrives somehow at the necessary insight. In fact, if one realizes only that one does not yet know the answer, one might, again, think of using the graphing feature of the calculator (or, of course, think of estimating e^pi directly on the command line [“homescreen”… whatever]). Once you see that flat line… on screen or paper or in imagination… you’ve got it.
OK, true enough. Is that all you got? Well, I suppose a drill-instructor style teacher might favor repeating over and over until blue in the face “pi and e are constants“. Might be worth a try…
Am I supposed to be, like, hanging on every bloody word then? Well, yes, ideally, but hanging on every bloody symbol would be a darn good place to start. One is always led to ask, “what does this symbol mean in this context?”—indeed, this sounds suspiciously like a description of reading itself (and not just of reading “math”—calculations and such).
One is curious at this point as to whether they’d’ve done any better with y= e^3… but enough.
Because the good news is that there were five otherwise-perfect papers and several other solid high scores; in particular, the class redeemed itself very nicely with the “epsilon-delta” proof. Nonetheless, here are some trouble spots.
No “Let “. For this section, this line is mandatory. (One may of course “fix” epsilon or some other very slight variant.) I’m declaring by fiat that “all epsilon-delta proofs begin by making epsilon the name of a constant“.
Equal sign for implication.
No mention of .
““—right there in the calculation as if english words were appropriate in the middle of algebraic expressions. In the actual exercise, the grader is hoping to see . But hold on. I put the “mixed media” (algebra-and-plain-english) on the very blackboard myself! Ah, but it was in a marginal note about problems of this type generically… in particular problems (of this type—limits of linear functions) one will work with honest algebraic code. Should students be expected to just recognize slangy shorthand as such in its context while we’re preaching meticulous attention to detail in some other part of the work? Maybe. We can’t be explicit about everything.
The very inequalities I most want to see… in reverse order. Or in variant orders harder to classify… “no apparent order at all” having made at least one appearance.
Inequalities replaced by equations; reversed inequalities.
The thing is… at least some of the lack of clarity here ought to be considered my fault. In particular, of the calculations after fixing epsilon and writing delta as a function of epsilon… the calculations that we should be careful to present in the reverse order to the one we actually discover it in… I now feel one should explicitly say “(Then) The Following Are Equivalent” (and I’ll introduce the abbreviation TFAE for this situation in the next class). If we’re going to obsess over this definition… and we should… then it’s the logical structure that seems to present the biggest challenge; the equivalence of a certain set of inequalities is pretty close to the heart of the matter; we should be trying to spell this out as clearly as we know how.
I’ll probably have more to say about this…