I like $x e^{-x}$

Or you could give them the graph of $f'$ (no formulas) and ask them to draw $f$, labeling all relevant features.

@TedShifrin I have another question where I give them a graph of $f'$ and ask them to describe features of $f$ (intervals of concavity, extrema, etc). Maybe giving them the derivatives wouldn't be so bad in a different question.

@TedShifrin Ha! I was just typing that. :D

I saw :)

Here's a variant of the one you used already. $f(x)=(6x^5-1)/x^3$.

$f(x^2-1)=x^4-2x^2+1$?

I also used to give ones with an oblique asymptote that the curve crossed.

That's not the point, @CFG.

Zero standard calculus students would even start that one.

Maybe something like $\sin(x)/x$ on $[-\pi, \pi]$? Though their trig is weak, too. :/

that would sinc them

@copper.hat Indeed it would.

I agree with copper.

Ugh... it is so hard to come up with interesting problems that are fair. :/

life is not fair

That's why giving them $f'$ and $f''$ with the algebra done helps.

For what grade?

You might still have to make them do a little algebra by finding the asymptote(s).

@TedShifrin Yeah, I might do that.

That's a good idea.

i like Ted's suggestion of $f'$ and find the relevant characteristics

@copper.hat Like I said, that question is already on the exam.

Here's one from my past. Ha ha. $$f(x)=\frac{x^4+5x^3+7x^2+3x+8}{(x+1)^3}.$$

maybe the speed of a rocket with different stages?

Maybe one graphing problem is enough. Two max/min problems always.

@XanderHenderson are you testing their conceptual knowledge or their computational ability?

@copper.hat Yes.

@copper.hat Both. And this particular exam question is on the part of the exam where both skills are emphasized, but computation is the more important part.

@TedShifrin: Xander says no rational but you gave rational example?

@C.F.G I didn't say "no" to a rational function.

I said "no" to a rational function with a lot of tedious algebra.

@XanderHenderson I don't want to give them a rational function

@C.F.G Yes, but you aren't the instructor of record.

You'll have your turn, eventually, maybe.

one of my profs (mathematical physics) used to give enough straightforward questions that allowed a pass (assuming minimal work) and then some tougher parts.

@copper.hat That is typically what I do, also.

another used to give out all the questions during the year (but there were hundreds of them).

@C.F.G The conversation moved on from there.

And Ted persuaded me that giving them the derivatives is not a total cop-out.

@copper.hat I kind of do that, in that I assign a lot of homework problems, and try to pull the "straighforward" question from that bank of problems.

@TedShifrin This is so diabolic

Thanks for the input, all.

Back to writing questions.

why is that diabolic? i think it is good, much of the point of calculus

because teachers often teach and solve simple examples but asks hard problems.:) (thought of most lazy students)

in secondary school spoon feeding is ok.

in my early ta days i found it strange that third level students in berkeley wanted to be spoon fed.

SOME Abstract algebra students still expect to be spoon-fed. Same with differential geometry.

very common. as an undergrad, the word on the street for a large number of math majors was, generally, look for classes taught by postdocs because they are more likely to spoonfeed. and if someone was a around for a bit the recommendations would personalize, so-and-so absolutely does spoon feed.

i think a few people managed to get degrees without ever having taken an upper division class from faculty.

complex analysis was one of the main courses where this advice had some bite. it was required by a couple of other majors (physics, i think? or at least some 'track' in physics) so they had to offer a lot of it. there was usually a postdoc one who would treat it as a calculus class with contour integrals instead of real variable integrals.

Hmm, I was not a Berkeley postdoc, but even as a grad student I taught tough classes, and I certainly did as a postdoc at MIT. That said, I think I graded more generously than I did subsequently in my years at UGA.

I think part of it is the culture: Spend as little time on teaching as possible. One way to attain that goal is to be undemanding and barely accessible.

yep

i went the other route and tried only to get actual faculty. i figured that's what i was paying for. even though obviously tenured faculty can be just as bad as postdocs.

i had one good postdoc and one really bad one.

Some of the tenured faculty were pretty pathetic, but some were wonderful.

i had two tenured faculty as instructors at berkeley undergrad that i would have considered really bad. i didn't recognize it much at the time but realized it later.

I won't name names here.

When I was associate head at UGA, I had the problem of assigning courses to the worst teachers in the department. How do you do that to minimize damage? It's almost like you want to reward them for incompetence by giving them nothing. Can't do that, obviously.

haha. at iowa they seemed to solve that problem by assigning them to courses that were not prerequisites to anything else. i don't know for sure.

had they been doing that, they would have made the same assignments.

Well, assigning them to business calculus students will engender a lot of complaints.

Give them advanced courses with few students? Um, no.

berkeley did that. or appeared to.

And now the board of regents in GA has de facto done away with tenure. That ought to fix everything.

yes, that's the ticket.

Okay... so... first semester of calculus. My students seem to understand the idea that acceleration is the derivative of velocity is the derivative of position, so they have seen (and solved) simple differential equations of the form $s''(t) = C$ (with some initial values). As a kind of "capstone" exam question is it totally unfair to ask them to solve something like $f'' = -k f$? (maybe with the hint "what functions do you know about which are the negative of their own second derivative?")

That maybe seems a little unfair...

Yes, I would not go there.

What about something non-constant for acceleration?

They probably all fall in the trap of saying the constant of integration is the initial value.

@TedShifrin Yeah, that was my next best thought---something something a rocket with constant thrust but decreasing mass something something acceleration is given by (something simple but increasing) something something

Or just a car braking with non-constant deceleration. Does he hit the barricade 180 ft down the road?

But if they've never done those for homework, that too is unfair.

They've seen initial value problems of the type f''(x) = (polynomial in x), so they should be able to solve an initial value problem like f''(x) = sqrt(x). Maybe I'll invent a story about that.

I'm not a physicist---are there any real world phenomena where f''(x) = k sqrt(x)?

t??

You mean $f''(t)$. Doesn't seem too physical.

I guess it could be a struggling engine that is warming up slowly.

I tend to think more about deceleration rather than acceleration, but don't worry too much about physical reality :D

A wizard casts fireball. You see a small, red bead emerge from the tip of the wizard's wand. The bead accelerates away from the wizard at $\sqrt{t}$ m/s^2, and explodes with a radius of 10 meters after 4 seconds. CAN YOU GET OUT OF THE WAY?!

If it is non-physical, a wizard did it!

@XanderHenderson how fast can I run and how far away from the wizard am I?

@robjohn Yes, those would be the initial values.

starts crawling away now

What is the standard movement speed in D&D? 30ft every 6 seconds?

I cast a detect professor spell...

You can "dash" at twice that speed.

d&d?

oh got it

Don't get carried away and make this impossibly complicated.

Dungeons & Dragons @copper

@TedShifrin Heh.

@TedShifrin Thanks!

Does Lucy catch Ethel as she speeds away with 2 minutes head start?

one thing that we were taught from an early stage was exam technique. scan the questions, pick off the easy ones, stage the approach, watch your time, take question points into consideration, etc.

I was never taught such things.

But I did often tell students to start with things they were confident of, because that didn't occur to them.

can you give them staged questions? a), b) are straightforward, c) needs a little thought?

i mean in primary & seconndary school

in primary mainly for secondary entrance exams

Multi-part questions are tricky to grade, but if you grade c) based on perhaps wrong answers to a) and b), they're ok.

Actually, that might not be a terrible problem. Acceleration is given, we can assume at the moment the fireball is cast, we are running away at (speed), and that we started (distance) from the wizard. Maybe exchange the fireball for some thing which doesn't have an area of effect.

Do your students appreciate or resent creativity, @Xander?

@XanderHenderson i had a question on the gre about baseball. i really had no clue.

@copper.hat This is a question for the capstone part of the exam. I usually try to write questions which I think are going to be mostly impossible for all but the top 10-20% of the class. Then I present them with two or three options and say "Do your best with whichever option you chose."

Like knowing that a baseball diamond is a square?

i mention it because you don't want a question to be distracting in a meaningless sense.

@TedShifrin Some of them like it, others hate it.

There are very few with no opinion.

@TedShifrin It isn't a square! It is a diamond! Geez...

I used to put a few students' names into problems from time to time.

@TedShifrin yes, or that a strike was, in fact, the opposite

Ah.

Depends on your viewpoint :P

i know, but it caught me completely unawares

I know. You were playing cricket.

you know, two countries separated by a common language

not even, rounders was the closest

When I taught probability, I had to start by explaining the basics of a deck of cards. There were plenty who had no idea.

And I had to explain that one divided 52 into 4 sets of 13 when one plays bridge, etc.

and that only once or twice. we have hurling youtube.com/watch?v=fgEMvRrOCRI, handball youtube.com/watch?v=rtGRUO0VdcA and a few others...

Handball, squash, etc., have little to do with football, baseball, etc.

hurling is a field sport, just pointing out that my reference sports had little overlap

what about curling?

one book that is entertaing from an educational perspective is diaconis' "magical mathematics"

curling :-) not quite. hurling is a tad faster

like lacross but with a hard ash stick

curling is on ice, right?

yup, sorta boule on ice

Oh, another twist on these kinds of problems might be to have them find the acceleration. "In order to be safety certified, a car must be able to go from 60 mph to a complete stop in (distance). If we assume that the car decelerates at a constant rate once the brakes are applied, what does that rate need to be?"

(with appropriate word smithing...)

i guess i would stay away from complicated setups?

unless you can get someone to proof the questions beforehand?

Yeah, if you haven't assigned homework of that ilk or done a problem in class and warned them it would be on the final, ... stick with things like what they've done.

@TedShifrin Again, the "capstone" questions are intentionally not things which they have done before.

OK, I didn't know this was "capstone." There's like one of those or two for the A/B students?

OK, make x a function of y.

really go in there and rip all of their brain wiring out

@TedShifrin Yes, that is exactly the idea.

Someone at UGA before my time used to ask his students for $dx/df$. I would have shot said instructor.

These are the two or three questions which are supposed to distinguish A students from B students.

OK, two or three out of 15?

I just worry that the C/D/F students might get 0 on them.

@TedShifrin 3 out of 30.

Whoa. 30 questions!!

can't make a capstone without stoning some caps.

@TedShifrin I don't worry about that---I expect that.

Most are very snappy, then.

i took a theoretical cs class from a manuel blum in berkeley and on the final he had a few questions that were ambiguous. he would not clarify during the exam so i wrote answers under both interpretations.

@TedShifrin to be fair, the first 24 are meant to be completed in less than 45 minutes.

(for a three hour exam)

"meant to be"

Less than 2 minutes a question.

So no multi-part "find the derivative/antiderivative" questions. They're each single questions?

@TedShifrin Because of the nature of the testing environment, they will have 60 minutes to complete the first 24. Then the remaining questions will be completed at a different time, with a limit of 2 hours.

my point being that it is hard to write unambiguous, challenging (as opposed to difficult) questions.

@TedShifrin Yes.

Oh, I see.

Typical questions in the first 24 are "compute (d/dx) x^2".

What's the difference between challenging and difficult, @copper?

@TedShifrin I suppose one needs more creativity?

really? typical? what about nested chain rule and product rule?

or "evaluate the indefinite integral $\int \cos(\theta) \,\mathrm{d}\theta$."

Oh, shoot, sorry, that is the first 21 questions.

Those are clearly trivial. But surely the 2-hour exam is for the word problems and curve sketching, so where are the less routine derivatives and antiderivatives?

21 super snappy questions, 6 intermediate questions (to distinguish C from B) (that is where the curve sketching stuff comes in), 3 hard questions.

i mean you can take a given question and make it difficult without adding any intellectual value, whereas a challenging one adds value. i realise this is a difference without distinction

Where are the two max/min problems (one closed interval, one not)?

do they get graph paper to sketch on?

One might be intermediate, one might be hard, I guess.

I generally only put one min/max problem on exams, usually in the middle section.

To me, max/min is the whole point of the course. A calculator can do most of the rest.

But I realize my viewpoint is not universal.

@TedShifrin Indeed. Optimization is important, but there are other things in the class, too.

i like the 'crossing a ploughed field' problems.

it was something non obvious that had a practical implication (for me :-)).

Word problems and writing skills are important for science/engineering students universally.

@copper You mean the running along the shore versus rowing in the water?

paddy has to cross an unploughed field, and a ploughed field, and then cut some turf, and then tarmac a drive, and then hit his brother with a coal shovel. please find the path minimizing the time it takes him to do this.

The six intermediate questions are, roughly, (1) min / max, (2) sketch a curve, (3) evaluate a derivative as the limit of a difference quotient, (4) some kind of related rates problem, (5) a slightly tricky integral evaluation, and (6) a slightly tricky differentiation, usually with nested chain and product rules.

Simmons had a fascinating problem that I had never seen anywhere else. If you do the motoring boat versus running on the shore, it's clear that if the boat's speed exceeds the running speed, you should boat all the way. What's the smallest speed of the boat for which that's still true?

@TedShifrin yes, same class of problems. i used to do orienteering as a sport, so had plenty of opportunity to experiment

@leslietownes well, optimal design of driveway to maximise initial appearance and minimise the time before it needs more work is a complicated problem....

OK, @Xander. I guess my exams tended to be a bit more on intermediate tending to a bit harder.

But I recognize that with your clientele your balance is way more appropriate.

copper, that's going to be the 'capstone' problem in my class if i ever teach again.

@TedShifrin That sounds like an application of Snell's Law of Refraction

It would say that the ratio of velocities is the sine of the angle from the perpendicular to the shore

It's a variant, yes, @robjohn. But you have to account for boundary situations.

It really is a question about when the optimal point stays at the endpoint.

how far downstream and how wide is the stream?

It just surprised me, because I initially intuited that as soon as the boat speed dropped below the running speed, one should have to do some running.

Well, those are obviously parameters.

I last taught this material back in 2007 or so, so I would have to recreate the whole thing.

According to Snell, the ratio of speeds should be $\frac{\text{boat speed}}{\text{running speed}}=\frac{\text{downstream}}{\sqrt{\text{downstream}^2+\text{width}^2}}$

i like it when problems challenge daily 'intuition'.

But that assumes an interior critical point.

i like problems that force people to also look for extrema at the boundaries as well. so they don't get complacent.

:59846921 That is the ratio of speeds that will make it so that the boat should be taken the whole way

Yes, precisely, @copper. Especially with my calc theory students I made a huge deal out of that.

Oh, that's the minimal boat speed, @robjohn.

Snell's Law is essentially a solution of this problem, but applied to light paths.

@TedShifrin you would be surprised (or maybe not) by the number of engineering students who fail to check reality.

@TedShifrin yes

Yes, but for Snell, we have refraction no matter what.

there's always the non smooth version of queen dido's problem...

personally i think convex should be taught much earlier :-)

@TedShifrin yes, but light takes the fastest path, and the path bends because the light is slower in the refractive medium

@copper I remember being very impressed by a calc theory student back in the 80s who worked a differential equation problem (something standard about exchanging bills at the bank) and complained that his answer contradicted common sense. I praised him for that!

mostly joking there

LOL, yes, @robjohn, I know that :P

@TedShifrin i have always been amazed how a little appropriate praise is such an enabler

So he had in fact messed up the set-up slightly, but, yes, for a freshman to check that things make sense is very good indeed. I still see this guy from time to time on FB.

@robjohn So, yes, your solution is exactly the boundary condition for the critical point to move to the endpoint. Most people I gave this question to did not have the right intuition.

OK, time for lunch. Bye, y'all.

:-) enjoy... acme baguette, rosemary ham and cotswold cheese for me

@TedShifrin I knew this only because I proved Snell's Law many years ago when I was studying optics.

oh man, acme.

its even better, colusa market a little boutique supermarket near kensington circle sometimes carries acme, saving me the trip to cedar & the wait with grumpy wealthy folk who find lines distasteful.

@TedShifrin the fact that students don't do that more often is rather dismaying

i'm in a joycean punctuation free mood

@Semiclassical savvy recruiting firms should identify such folk early on with a view to later exploitation

acme began delivering in the north bay in the early 90s. it was paradise. i think my mom can still get it in sonoma.

i realise its cultural, but for me nothing beats freshly baked bread

we'd go to get it early, sometimes it was still warm on the truck.

we lived behind la farine on college ave which has great baguettes. their other bread offerings are more standard.

when i spent some time in haikou, there was a bakery a few blocks away from my in-laws place, my father in law & myself would get up early to get a fresh loaf

la fairine was good

is there something other than baguettes?

oh yeah, the pan epis

they have batards and round things and various seasonal stuff.

for me, baguettes only. the standard, the rustic, or the sourdough.

oooohhhh, excusez moi, pain d'epi (said with jaw in painful dislocation)

i'm kidding. its hard to beat fresh bread of any kind, even the mass produced stuff tastes great just out of the oven

my latest comment: "Try doing some truly marginal work and Google <not relevant>. Then look at the first Wikipedia entry. Really, come on"

at least you didn't include a 'let me google that for you' link

@robjohn Fixing the MathJax issue on the Activity page seems to be very low on the priority list.

@RandomVariable there are a lot of items to be fixed, and the layout of the MathJax is a cosmetic problem. Not that it isn't important, but they usually focus on the data coherence first.

They fixed part of one of the problems I noted. There is a problem that would only be seen by mods, so I don't think that meta.SE is the place to report it.

Okay... the analysis / synthesis questions for the calculus final are done. I still need to write the recall questions, and then I have to deal with precalc, but I think that I am on pace to be ready on Monday.

It might be time to start drinking...

i've been drinking since 9am.

green tea is addictive

I been drinking co$-\phi$ since 8am

What could happen if you were to embed $M_i$ for $i=1,2,3,4$ into $\Bbb R^2$ where $i$ denotes the quadrant of $\Bbb R^2.$ Where $M_i \simeq \Bbb R^2$?

@robjohn It's a bit more than just cosmetic. Why are there so many issues? Was it not beta tested?

Xander, I'm going for a walk, but I'll meet you for martinis shortly :)

green tea is apparently good from a periodontal perspective

i have bad genes, from a periodontal perspective. i hope the two are balancing out

@leslietownes I've been drinking coffee since 5 this morning.

And I have had two cans of hipster water.

I think it is time to switch to whisky.

anyway you're in the desert, shouldn't it be peyote?

@leslietownes That isn't drunk.

@RandomVariable Have you visited the New responsive Activity page

Or maybe it is. I have no idea.

you could visit the future and talk to your ancestors about related rates and capstone problems

take enough of it and maybe you get a chance to become an ancestor

The last problem I wrote defined the symmetric derivative, and asked them to compute the symmetric derivative of the absolute value function at zero.

@robjohn Yes. I mentioned it the other day.

I don't really know the reason. I don't know their QA process. Whatever, it seems to be lacking.

Hi there, do you know any math software that cleverly simplify trigonometric identities symbolically? I have rotation matrices multiplications and I don't want to do them manually.

@CroCo wolfram

Mathematica can do that to some extent if you tell it to.

shouldn't positivity of b^2 - 4c be enough? the difference between the roots is a multiple of sqrt(b^2 - 4c), so nonzero, in that case.

You need $c<0$ to get positive roots!

oh, i missed 'positive'

yes.

But if $c<0$, how can $\sqrt{b^2-4c} < -b$

That!s not what you mean, is it?

Doesn't $c<0$ only guarantee distinct real roots? Not positive roots?

Isn’t $b^2-4c > b^2$?

real roots is given by b^2 - 4c nonnegative, and distinct real roots is given by b^2 - 4c positive. positivity then brings into consideration the rest of the thing from the quadratic formula. focus on the smaller root.