Acting Out His Follies

I’ve just turned in grades; 24 years of classroom teaching over and done.

This blog now becomes a collection of my notes, for me; thanks to any readers it may have had along the way. With any luck, I’ll have done something interesting I’ll want to link back to…

I’m developing various Math Ed projects; there’ll probably be plenty of mathy stuff in vlorblog.wordpress.com for some time to come. Leave a comment. For heck sake. It’s like pulling teeth. Or trying to get a job.

In fact, if you’ve got one of those lying around, well there I am…

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More Of The Same

6. Find the arc length for the spiral whose polar equation is r = e^\theta over the interval \theta \in [0, 1].

The “key example”: the easiest problem of its type. Should this be on a final exam? Heck no, it should be assigned early on and then referred to often. Here it is now. The old “unchanged by differentiation” trick…

5. Find a formula for the slope of the tangent to the “epicycloid” whose parametric equations are x = 5cos(t) - cos(5t) and y=5sin(t)-sin(5t).

Routine; I passed over it in near-silence on the review day. Also… owen by the way… played ‘em ILMB and For John Henry.

4.Recall that the Taylor Series for the exponential function is e^x = \sum_{n=0}^\infty {{x^n}\over{n!}}. Use the 3^{rd} Taylor polynomial P_3 to obtain a (rational number) estimate for e^2 (write out the first four terms of the series; put x=2; simplify). Find a reasonable upper bound on the “error term” R_3 for this estimate.

The class that had this final didn’t “get” the control-the-error parts of the course; this one I didn’t even really try. You should… whoever you are… but if you’re my student, I’ll go ahead and admit there won’t be an “error term” problem on this quarter’s exam. Be able to write out a Taylor Series and use it in estimating a number; the rest, for us, is gravy.

More still to come I imagine.

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Fearsome Pit Of Trigonometric Doom

A precalc final review blog by Kate f(t) Nowak; an AP Calc blog with lots of handwritten problems (something I’d like to have done a long time ago).

The Man Behind The Curtain

The carny’s getting set up in this comment thread.

Last Year’s Final. Partly. Kind of.

I asked the 153 students to tell me a problem they’d like to see on the final exam (sort of as a challenge to other students)… the three that were named aloud today in the hearing of all present were p.806#41, p807#89, and p796#11.

I passed around last years exam and looked over quite a bit of that. To wit.

10. Find the derivative of the vector-valued function \vec{r}(t) = \sec(t)\vec{i} + \tan(t)\vec{j}.

The topic is barely introduced in this course; the whole message is “termwise differentiation works” (integration too of course… pesky constants emerge… you know the drill). That, and I get an excuse to bawl out the ones who botched “differentiate tan-x three times” in the middle of a Taylor Series calculation on the last exam.

9. Find the angle between \langle - \sqrt{3}, 1\rangle and \langle 3, \sqrt{3}\rangle.

In the actual document I had the \vec{i}, \vec{j} notation; in today’s whiteboard notes, I switched as I’ve done here. It looks cleaner and easier to understand to my eyes. Anyway, one has \vec{v}\cdot\vec{w} = |\vec{v}| |\vec{w}| \cos(\theta), where theta is of course the desired angle; the “absolute value” is (again of course) the square root of the sum of the squares of the co-ordinates of the vector in question (its “norm”, “magnitude”, “length”, or what have you… ). These last two were “gift” problems offered by way of apology for not having had time to do much more than introduce certain topics. In the case of #10; it’s built into the syllabus… in place of #9 one would prefer a problem that uses the “angle” property in service of some geometric problem.

8. Find the equation of the plane through (2,2,0), (1,0,1), and (0,1,1).

I love this problem. Three points in (x,y,z)-space not all on a line; find a linear equation. The exact natural generalization of a problem we drill diligently into hundreds of Math 102 students every quarter [two points in (x,y)-space; find the linear equation]. That’s just got to be interesting. Sure enough, the picture, together with both products (“dot” and “cross”), tell the whole story; forget the messy “formula”.
See clearly how to do it and just do it this way.

Gotta go. More later.

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Big Theorem In Topology

Caviar For The General

Wikipedia is good (as usual) on the binomial theorem; part of today’s lecture will overlap parts of this. Ideally, one would already have the “natural number exponent” case completely under control before exhibiting Newton’s mighty generalization. And yet calculus students have typically never studied this incredibly basic topic; somehow this has been shunted off to the Statistics arm of the curriculum (where, ideally, it becomes part of Exhibit A in the—necessarily handwavy—”Central Limit Theorem” [roughly, "everybody tries to be Normal"; to be any less rough I have to tell you about Pascal's Triangle, Binomial Coefficients, Bernoulli Trials and stuff like that—the Binomial Theorem ("easy" form)]; this work is typically omitted and its results handled by “technology” rather than student understanding).

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