Last Quarter’s Final

1. Let f(x) = {1\over{x^2}}. Compute f'(x) using the definition (the limit of a difference quotient).

2. Differentiate (find D_x(f)):
a. f(x) = 5^{3x}

b. f(x) = \log_2(x^3)

c. f(x) = \arctan({1\over x})

3. Find formulas for y' using a. logarithmic differentiation and b. the “Quotient Rule”, where = {{(2x+1)^2}\over x}.

4. Evaluate the limits (where defined; “DNE” for any that are not):
a. \lim_{x\rightarrow 3}{{x-3}\over{|x-3|}}
b. \lim_{x\rightarrow \infty} {{3x^2 + 5x - 11}\over{2x^2 + 7}}
c. \lim_{x\rightarrow 0} {{e^x-1}\over{e^{2x}-1}}

5. Find the equation of the line tangent to g(x) = ln(x) at x = 1.

6. Use implicit differentiation to find {{dy}\over{dx}} given that y = e^{xy} + x^2.

7. Find an algebraic formula for cos(arctan(x)) using a sketch of a right triangle.

8. Compute the second derivative, {{d^2 y}\over{dx^2}}, where y = \sin(x) + xe^x.

9. A balloon rises at a rate of 10 feet per second from a point on the ground 100 feet from an observer. Find the rate of change of the angle of elevation of the balloon from the observer when the balloon is 100 feet off the ground.

10. Find (both co-ordinates of) the local extrema and inflection point for the cubic function y = x^3 - 9x - 27. Sketch the curve.

11. Estimate \root3\of{123} using a tangent line approximation (hint: \root3\of{125} = 5).

12. Find the antiderivatives.
a. \int {1\over{x^2 +1}}dx
b. \int x^3 dx
c. \int dx

Exercises. Answers go in comments.
Marry, sir. Find all the typos.
Moreover. Convince a skeptic that problem 1 or one much like it is worth learning how to do (and belongs on a Calc I final) even if they know f'(x) = {{-2}\over{x^3}}. Hint: even a god-damn computer knows that; this is a course for college credit.
Secondarily. Explain why one does not go on, “Harvard Calculus” style, to require discussions about graphs (without calculations)… I thought this was a course for college credit. For extra credit, justify the lack of an “epsilon-delta” proof on this test.
Sixth and lastly. Were notations for differentiation ({{dy}\over{dx}}, f'(x), D_x (f)) just tossed around at random to confuse everyone? Or what? Justify your conclusions.
Thirdly. How in the name of the nameless name is it even remotely possible that after using TeX for seventeen freaking years, Vlorbik doesn’t know how to typeset those limits properly? For that matter, why didn’t he just scan last year’s test and upload the S.O.B.? How can one survive without being able to work the most basic tools?
Explain why, despite having been present on two or more occasions when Vlorbik explained how to do the tangent line problem on the calculator with no calculus whatsoever, students will blank out on problem 5. No, that’s easy. Explain why they think we made up a whole new symbol for the natural log when we really meant the reciprocal function… now that’s a mystery. And why should it be hard to recognize a constant function, for hecksake? What’s up with these people who haven’t learned calculus yet? Why don’t they get it?

And to conclude. Any questions?

Buy Conditionally

The 151 quizzes are good. I caught almost everyone with the trusty old “differentiate a constant” wheeze… y = 11e^\pi \Rightarrow {{dy}\over{dx}}= 0 of course, but you wouldn’t’ve known it by these guys on the day. In fact—math teacher at play— \{11, \pi 11e^{\pi-1}, 11e^\pi, 11\root7\of{e^{22}}, e^\pi, 0\} is the “answer set” (sure as heck not the solution set!)—with 11e^\pi appearing several times (once as {{11}\over{e^{-\pi}}}) and a variant (11\pi e^{(\pi - 1)}… 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” (y = A_0 e^{Kx}, 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.

\bulletNo “Let \epsilon \rangle 0“. 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“.

\bulletEqual sign for implication.

\bulletNo mention of \delta.

\bullet\delta = {\epsilon\over{|slope|}}“—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 \epsilon\over5. 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.

\bulletThe 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.

\bulletInequalities 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…

I Can’t Deny/This Feeling That I…

Deep Down Inside at y of x. Random Dylan links from last week. Oh. The exams are great; mostly A’s and B’s. Homeworks too. Lots of gaps to plug of course…

MT@P

Well I tried to differentiate a constant
But I never got a whole lot outta that
That’s ’cause differentiation gives the slope
And the slope is nothing if the line is flat.
If a function only has a single value
Then its dee-dee-ex is zero where defined
And conversely if why-prime is always zero
Then why’s the same for all ex in the line.

You can sort of fit it to “Red River Valley” if you like. Not that I ever did. Till now. Alas.

We Are Everywhere

A Calc I blog by what appears to be very talented and hardworking student somewhere. Newly discovered by me. I’ll be back.

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Proudly Burned Toast

So the Calc I class are taking their Exam #1. As a show of good faith I just took my Exam #1 by loudly naming each student without looking down at the desk for my notes. Eighteen names isn’t that much of a stunt but they get to directly see me working on learning skills I’m not particularly good at and for me right now that’s a win.

The first one’s hit the desk; one small mistake. So I say it out loud. “This is the one to beat.” Here’s a few more. Wow. They’re getting it. They actually pay me for this. Gee but it’s satisfying.

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The Sky’s The Limit (At Infinity)

So today I finally steeled myself to ask for an\epsilon-\delta proof as homework (that’s “epsilon-delta”, for all you non-Calculus people; a famous sticking point in the introductory course). And I’ll bet at least a few of ’em’ll be substantially correct, too… but have to admit I won’t bet much

Anyway, I’ve got a pretty good feeling about it and Thursday we’ll see how it actually went. Best-case scenario: several students confident enough to work out a problem like “Prove using the definition that the limit at 13 of 2x – 7 is 19” (I had this as a formula but the computer ate it and I’m impatient), at the board (very best case: such a student leads a discussion of the problem… this is rare but I’ve seen it done and it sure makes me proud [so I’m saying it]).

This class has stepped up with blackboard work quite impressively, with students at the front of the room calculating derivatives using the definition of “derivative”—i.e., “the hard way” (we will soon develop rather a large collection of techniques for avoiding these calculations; they are nevertheless considered essential material for a first course in Calculus [and rightly so in my view]). One of the themes in my work as a learner of The Art (of teaching math) is “the less I talk the better I like it” (which is, alas, not entirely true… but I’ve found it a nice slogan to express my firm conviction that the student often learns best when the student is moving the pencil [or chalk] and talking about it…); I’m pretty excited about the presentations I’ve seen out of this group (hence the guarded optimism of my first paragraph).

I haven’t even begun what I was planning on saying. But now I want to see if it’ll post. Here goes.

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