Basic Math for Math Majors
November 13, 2013 2:08 PM   Subscribe

Hello! I'm a first semester grad student in applied math. I am a non-traditional student in a variety of ways (liberal arts undergrad, working full-time), and while I'm finding that I intellectually understand the concepts I'm learning, execution is being made difficult by my lack of fluency with algebraic manipulation and calculus.

So, here's an excellent example. In a problem set this week, we were asked to find the expected value of this god-awful density function. (We had to integrate over 0-inf, solve for the constant c, then do another integration to find E(V).) (it's 2/(sqrt(pi)*sqrt(lambda)), if anyone cares).

This, no joke, took me 4 hours. At times I found myself just completely frozen about what to do next. I understand stuff like u-substitution and integration by parts, I'm just mindnumbingly slow at it, and I end up psyching myself out.

I need to get better and more automatic with this stuff ASAP. What's the best way to do that, relatively quickly? Are there problem banks somewhere I can use? I have some old books, is going through them helpful? Is there a cool iphone game or something?
posted by downing street memo to Education (16 answers total) 10 users marked this as a favorite
Best answer: You could get a AP Calc BC book and work through it.

People have done integration bees at various colleges that you could solve through:

posted by Maecenas at 2:16 PM on November 13, 2013 [2 favorites]

Best answer: Getting better and faster with this stuff comes from practice, and practice alone.
How long does it take your fellow students to complete these problems?
Is there a reason you aren't working on problem sets in groups, even just to compare answers?
When you asked your professor this question, what was their suggestion?
posted by oceanjesse at 2:17 PM on November 13, 2013 [2 favorites]

Wolfram Alpha will do step-by-step symbolic integration. I'm NOT suggesting that as a tool to do homework, but as a way to work through the proper steps. Once you know the steps, practice.
posted by supercres at 2:20 PM on November 13, 2013 [2 favorites]

Best answer: As well as the resources people are suggesting, consider setting up homework groups with other students. I took a year off between undergrad (physics major, liberal arts college) and physics grad school (major research university), and there were some moments in lecture that were absolutely horrifying, when the professor would write an equation on the board that hinged around some large half-familiar mathematical symbol. So I'm sitting there thinking "crap, I know it means something important when they draw a circle in the middle of the integration symbol, but what the heck IS that?" and feeling like the world's biggest idiot and/or imposter. I ended up with another small-college guy as my lecture buddy, and we'd sit together and ask each other the "stupid questions" and sometimes conclude that it was worth raising a hand and asking (which eventually gave us the reputation of asking really good questions).
It worked out that about 6 of us would do homework sets together, which really helped. As you seem to already know, there's no substitute for just DOING math, and lots of it - but doing it right is the thing you really learn from. There's not a huge amount of benefit difference between staring at it for 3 minutes, then saying "Hey Joe, how the heck do you do this?" versus staring at it for 30 minutes, thinking of 3 things that don't work, and getting really frustrated, before you finally find the key that lets you do it right. So practice, but don't assume you must practice alone, or that all the other students are finding this way easier than you are. If it gets out that you're doing integration drills, don't be surprised if there are people who'd like to work with you.
posted by aimedwander at 2:38 PM on November 13, 2013 [1 favorite]

An online resource is to read cross validated and to ask questions there when you get stuck. Make sure to mark your problem as homework and explain what you've tried and they shouldn't ruin the problem too much.
posted by esprit de l'escalier at 2:41 PM on November 13, 2013 [1 favorite]

Best answer: I've been there before -- actually, I think a lot of, if not all, mathematicians have been there before, so don't feel bad about it. This is the first of this kind of problem you've done in a long time, so it's naturally going to take you some extra time to bumble through the procedures involved. There's really no getting around that, and it's not something to feel bad about.

The suggestion above to use Wolfram Alpha to deconstruct the integrals is a good one, I think. But don't use them on the actual HW problems themselves. I'm going to assume you're being assigned problems from a text of some kind, so run through WA the integrals in the problems you weren't assigned, so you can get a feel for what's involved. Then try out the actual HW problems. And practice, too, but on simpler problems (perhaps from AP Calc, as suggested above), just so you can get the muscle memory of integration by parts down. When I was still in school, I came up with my own ritual to grid the u and dv and the progression from u->du and dv->v in a systematic way, which really helped me prime the integration part of my brain. If you do enough problems, you might see something similar emerge.

Honestly, pretty much every out-of-school-and-working-with-math person who I know, when faced with an integral, they just run it through Maple or Mathematica and are done with it. There are some contexts where I'd imagine your approach to handling integrals would need more nuance than this, so the decision as to whether you want to bone up on this material more deeply than what would be needed to do well in this course is going to hinge on your post-grad-school plans and goals. That's something to consider too.
posted by un petit cadeau at 2:43 PM on November 13, 2013

Best answer: I would actually not work in groups. At least, not the way that people usually do it - IME, it's too easy to get buoyed along by someone who really gets it, which is ok for your homework grades but will kill you on tests and in the future, because you will think you actually get it when you don't. I would work separately for as long as you can stand, and then check answers with each other mid-week as per here.

Honestly, I think you're not doing anything wrong. It's going to be hard for a while and the correct thing to do, I think, is to continue to struggle with this type of problem, mostly on your own. Here's one stackexchange question that seems like it might have some useful resources for finding batteries of integration problems.
posted by en forme de poire at 2:47 PM on November 13, 2013 [2 favorites]

this god-awful density function

By the way, that's the Maxwell-Boltzmann distribution.
posted by stebulus at 2:57 PM on November 13, 2013 [4 favorites]

Best answer: My god, that image gave me 1st semester statmech flashbacks. Pretty sure I spent four hours on that exact integral more than once. Good times. I endorse en forme de poire's approach, which is how studying in groups usually worked out--there'd be three or four of us in one room working in near silence, and then occasionally speaking up when we were at a dead end. This does mean that you need to be a little picky, so that you wind up with people with similar levels of skill and maturity--you don't want to end up holding someone else's hand, nor do you want someone who speeds through and leaves while you're still only a quarter done.

The other thing--and this is more of a mental adaptation to make--is that you have to be willing to go down the rabbit holes. It's okay, even normal (in my experience), to have your first few approaches to an integral wind up in a dead end. Just make sure you have a buttload of scratch paper, and don't beat yourself up if what you tried turned out wrong. (Also, don't throw away the scratch paper until you're done with the problem, at least, and ideally keep it around until the problem set is complete. A lot of the time something that you think is totally wrong will actually have set up something--a substitution or convergence--correctly for later in the problem, as wacky as it sounds.)
posted by kagredon at 3:12 PM on November 13, 2013 [7 favorites]

Best answer: I used that integral once in a lecture to first-year calculus students, specifically to show them that some integrals are really tough. I let the students drive the solution process, mostly, and we spent 20 or 25 minutes going around in circles on it before I cut it short and moved on. (We had actually made some progress, but it was progress of a type that was hard for the students to see as progress.) So, solving it on your own in four hours seems... just fine, for a first encounter.

But yeah, practice. Get a first-year calculus textbook and do every problem. (Stewart would be fine for this, for example.) Eventually you'll be fast.
posted by stebulus at 4:01 PM on November 13, 2013

Best answer: Yea, this is pretty common. And the reality is that after you get through with your courses you probably won't spend that much time solving integrals. Which is to say, don't let your difficulty with this make you feel like you're not cut out for the field, because it absolutely doesn't. Some practical tips:

1. My mantra for these kinds of problems is "you won't solve it by staring at a blank piece of paper." Write something down, try some approach, even if you don't think it'll work or are not sure what you'd do after that. Breaking through the paralysis is key.

2. Integration drills are a good idea. I'd suggest starting with Stewart or an AP prep book and first doing the problems at the end of the sections. The nice thing about this is you get a block of problems that are all u-substitution problems or all integration by parts or whatever. The goal of this is to get yourself to see a bunch of problems and know what a u-substitution problem looks like versus what a integration by parts problem looks like. This isn't always obvious and you won't be perfect at it, but there are some common giveaways* you can learn to recognize. Then go and do the chapter exercises where they're all mixed in to test yourself.

2a. The other nice thing about drills is you'll end up memorizing some of the more common integrals. For example, in a grad class I think it's totally fine to go straight from \int log(x) to x logx - x without showing how to do the integral. Again, this is something that will just come with practice and doing this enough times. I bet after another semester you'll be able to do this in your sleep.

*eg, in the problem you posted, it had to be a u-substitution problem because there's pretty much nothing else you can do with e^(-x^2). On the other hand if it were e^-x you'd want to integrate by parts, because you can make the powers of x disappear by differentiating while e^-x is easy to integrate.
posted by matildatakesovertheworld at 4:55 PM on November 13, 2013 [2 favorites]

Best answer: IIRC, the key piece of that integral is actually the 0-inf limit, which is convergent for a bunch of weird integrals that otherwise resolve into complex functions. That's unfortunately one of the strange little tricks that rarely is touched on in introductory calculus, but becomes extremely relevant to solving certain systems in applied math/physical science (see also: using symmetry to try to reduce your function into something simpler.) You might look into if there are any relevant-looking Dover books--I can't recommend any mathematics-focused ones specifically, but I own several (on physical chemistry), and what I like about them is that they present a ton of practice problems, at several levels of complexity, which is not always the case with graduate-level texts.

(I did a search and was sort of surprised to not find many recreational mathematics books about integration. I don't know that I'd exactly want to make a hobby of it, but there's something uniquely satisfying about solving a particularly intractable integral.)
posted by kagredon at 6:25 PM on November 13, 2013 [1 favorite]

Best answer: I majored in math as an undergrad and went back to to techniques for integration a few years after graduation because I'd always felt shaky with it. I went though the problem sets in the Stewart book and solved every single one for which there was a solution, checking the answer and making sure I understood any errors before moving on to the next one. It was painstaking and took a few weeks but I learned a lot. The Stewart books had a companion book with the solutions worked out.

What really raised my game with it was intensifying my use of tactics to keep the information in each problem organized, techniques to break down a problem with a lot of pieces and solve the pieces separately, then put the solutions back together at the end without losing track and without switching a sign. I wound up sometimes using more than one sheet of computer paper for a single problem, and would orient the paper the landscape way instead of the portrait way a lot of the time.
posted by alphanerd at 6:40 PM on November 13, 2013 [1 favorite]

A couple of years ago I went through the Schaum's Outline of Calculus book, just doing all the problems in the chapters whose topics I was rusty in. It was really useful.
posted by dfan at 11:52 AM on November 14, 2013

And yeah, that is a particularly gnarly one as others have said. I remember trying to prove what the area under the normal curve was once and almost threw my computer across the room when I learned you were supposed to square the integral, change one of the variables, transform into polar co-ordinates, and then take the square root again. Oh, obvi! But at least now I have that particular trick in my toolbox for when I see "exp(-kx^2) dx" again.
posted by en forme de poire at 12:22 PM on November 14, 2013

I'm not sure you can manage this with working full time, but I found that teaching really solidified my math. If your school has walk-in tutoring you can ask to be hired for a few hours a week in the evening, or you could start private tutoring. You'll get paid for all the mind-numbing repetition and coming up with the answer for the off-the-wall questions that intro to calc. students ask will really sharpen your mind.
posted by anaelith at 7:48 PM on November 18, 2013

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