Sloppy Calculus
January 24, 2011 6:25 PM   Subscribe

Please help me evaluate integrals more accurately. Sloppy calculus is killing me.

So I'm taking a probability course and have to do a lot of integration of continuous probability distributions as part of the work. I get a lot of problems wrong because my calculus is sloppy, not because I set things up incorrectly.

Anyone have any general tips/tricks/strategies for evaluating integrals or general algebraic/calculus operations more accurately? Obviously just mowing through the problems isn't really working and I'm curious if anybody has any tips that have helped them, even if it's just as simple as "I write down every single step, no matter how small" or whatever.

Thanks a ton!
posted by PFL to Education (14 answers total) 5 users marked this as a favorite
Forming a study group/homework group and checking our work against each other is one of the only ways I could get my analytical math right every time. Also, do you have an equation solver, even a crude one like Matlab? Checking your work numerically is good for homework assignments, although it may not help you with tests.

And yes, writing down your work is important, and it also hopefully will slow you down a little bit.
posted by muddgirl at 6:36 PM on January 24, 2011

This won't help you so much at getting better, but will at least help you verify if your answer is correct or incorrect: You can use an integral solver to verify your answers. You can use the Wolfram Alpha or Online Integrator to symbolically integrate, and some of the more powerful calculators can help with this, also. Most graphic calculators at least have a numeric integrator, e.g. fnInt on TI calcs. You can use this to check your answer if you are calculating definite integrals.
posted by Herschel at 6:40 PM on January 24, 2011

even if it's just as simple as "I write down every single step, no matter how small"

Yeah, you'll probably have to do (at least) this. Until you get the hang of it, you won't really know exactly where your calculus is sloppy unless you write down every step.
posted by mhum at 6:48 PM on January 24, 2011

Are TI-89s allowed on tests? If so get one, once I got mine it helped me a ton, mostly because I never wanted to sit down and spend a lot of time working on integrals.
posted by DJWeezy at 6:49 PM on January 24, 2011

I sincerely believe there is such thing as "muscle memory" when it comes to solving equations. Let me explain what I mean by that. You say:

mowing through the problems isn't really working.

So stop mowing through problems! Do them slowly and deliberately. Now, when you check and answer and find you got it wrong, I bet you try to scan through it to find your mistake. Then you say "ah, ok, I see what I did wrong. Moving on". Don't do that anymore. Start it from the beginning, and do all the steps over again. This time you'll notice the mistake you made.

Whenever you get a problem wrong, do the WHOLE SOLUTION over. Keep doing it until it's perfect. Then start the next question. You have to be fastidious. As my creepy youth basketball coach used to say, "Practice doesn't make perfect; Perfect practice makes perfect".
posted by auto-correct at 7:06 PM on January 24, 2011 [2 favorites]

Even if you just have a ti-83/84, you can still solve a (single) integral on it pretty easily, which should help you check your mistakes. Message me if you'd like me to step by step it.
posted by kylej at 7:12 PM on January 24, 2011

What sorts of errors are you making? Are you making the same type of error consistently? In addition to auto-correct's great advice, take note of which errors you make. Becoming more conscious of your blind spots, as it were, may help you eliminate them.
posted by wiskunde at 7:14 PM on January 24, 2011

I am a mathematician. This is mathematical advice.

Write down all the steps.
posted by leahwrenn at 7:21 PM on January 24, 2011 [7 favorites]

Write down all the steps neatly.
posted by telegraph at 7:23 PM on January 24, 2011 [2 favorites]

Thanks all for the quick replies. Not making the same mistakes every time, it's just my brain moving faster than it should be and doing something stupid.

Graphing calculators aren't allowed, and I'd rather do what I can to just get better at it manually. Looks like moving slow is the thing to do here.
posted by PFL at 7:25 PM on January 24, 2011

I had a similar problem. My mind works fast and I understood the math, but I'd make mistakes. I bought one of those expensive mechanical pencils and some graph paper and I focused on being both thorough and neat. I would go so far as rewriting the formulas I was supposed to use at each step. I can't stress enough how much the neatness factor helped me though. Focusing on being neat made me slow down enough that I was able to catch my mistakes. It also allows you to more easily check your work.
posted by milarepa at 8:04 PM on January 24, 2011

In addition to writing down steps to avoid making mistakes, there are a number of ways to check to see if you've made any. Most generally, you can use the good ol' fundamental theorem of calculus. After you find an integral, differentiate it and make sure you get the original term. Another way to check if your answer is right is to try to make sure that your answer gives you correct values of trivial or already solved cases (typically when some parameter goes to 0 or infinity, sometimes 1). Also, if the problem can be cast in terms of units, make sure your solution has the units you expect.
posted by Schismatic at 3:06 AM on January 25, 2011 [1 favorite]

What I do is:

(1) solve the problem on a chalkboard
(2) arrange myself so that I can't see the chalkboard, then resolve the problem on paper.
(3) If the answers agree, I'm done. If not, figure out why they're different and what went wrong.

This has the benefits of auto-checking your work and increasing the number of times you've solved problems (even though they're the same). I learn almost exclusively through writing, rather than listening or reading, so this was essential to me in school.
posted by bessel functions seem unnecessarily complicated at 7:23 AM on January 25, 2011

For tips in terms of setting out the work, and what has worked for me (and others) over years of doing calculus:

1. State what is you hope to find at the beginning. "We aim to show/find ...."

2. Label your lines, 1), 2), 3) etc. Makes referring to lines far easier, and reinforce the idea of separate logical steps

3. Commentary version 1: Put justifications for each line such as in brackets to the right of each line. e.g. [rearranging (2)] or [putting (3) into (4)]. See how useful the previous suggestion becomes.

4. Commentary version 2: Use structure and headings, such as "To find K". Have some structure to it. At times I imagine it's a story with sentences, paragraphs, and chapters.

5. White space: For long derivations/proofs put logical gaps between sections. Makes following it far easier.

6. Make sure it's clear where the proof is going, what I term 'following the path of least resistance'.

7. Check the final step is there, and is what you intended to show/find (see step 1).

8. Read back through it, and think how might someone get lost, where and why. Can you justify the step/reasoning in words? If not, then that's probably where the mistake is.

Hope that's of some help.
posted by 92_elements at 8:50 AM on January 25, 2011

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