# How to improve accuracy when solving calculus questions

I find calculus to be a really interesting topic to study, and from what I've experienced it simply boils down to applying algebra to more complicated concepts. I understand calculus and can easily formulate proofs for myself as refreshers for things I don't quite remember.

However, when it comes to actually solving calculus problems, I really struggle in terms of accuracy. No matter what problem I approach, I always end up making stupid mistakes or miscalculations. For example, today I was doing a practice problem that involved applying integrals to a distance/velocity problem to find the total distance a particle traveled, given the s(t) function that represents position versus time. It took me three lengthy attempts to solve the problem before I got the correct answer, and EACH attempt paradoxically yielded three different answers (the last being the correct).

So the one solution I read in another post on Stack Exchange -- to take things slowly -- does not help, because when I solve calculus problems like a snail, I (mostly) do things correctly, but at the cost of time. This means that on timed exams, I may get more than half the questions correct, but I won't have enough time to finish the rest.

Others suggest practicing over and over to hone my skills so that I don't trip up and make these mistakes...but that doesn't help either. In fact, I've been practicing what I learned in my AP Calculus AB course for about a year now, and yet I still continue to frequently make miscalculations.

Again, what frustrates me is that I fully comprehend introductory calculus topics; it's not the application of calculus concepts or the use of formulas that gives me trouble, but rather it's maintaining accuracy while working quickly and efficiently.

Does anyone have suggestions on how I can alleviate my problem? I'm about to take a 2nd semester Calculus course in college when the Fall starts and I'm afraid that my grade will suffer if I continue to make these careless mistakes.

• If your primary concern is feeling rushed on timed exams, you should speak with your instructor. It is possible that your school has a formal student assistance program where you can officially take exams with extended time. Jun 27, 2016 at 20:58
• Good question. For written questions (not multiple choice), you want to write down every step methodologically and write neatly. This minimizes the risk of you making computational errors. Jun 27, 2016 at 21:08
• @AustinMohr Many times a students needs some sort documented learning disability, physical disability that interferes with test taking, and/or anxiety in order to qualify for extended time accomodations. Though the documentation required can be as simple as a doctor's note as far as I know. Jun 27, 2016 at 21:09
• You might also look at math.stackexchange.com/questions/1499262, math.stackexchange.com/questions/68279, and math.stackexchange.com/questions/1295552 for possible other helpful ideas. Jun 27, 2016 at 21:11
• @AleksandrH what sort of careless mistakes are you making? I got A/A+ in my calculi classes and still drop those darn negatives. I wouldnt sweat it too much. If you understand the material, and truly know all the forumulae, then you shouldn't have too much trouble. Calculus 2 is all about integration and antiderivatives. Calc 1 is undeniably the hardest of the three. Jun 28, 2016 at 1:37

Three concepts should always be a part of your mathematical problem-solving process.

1. Documentation. Write out each step carefully, using consistent and precise notation. Don't skip steps and don't be sloppy. Each step should be understandable and justifiable, as if you were explaining to a reader what you are doing.

2. Double-checking your computations. This means you should always go back and review your work. It doesn't mean that you just redo the same computations. Rather, you should look at your work critically, as if you are attempting to determine whether what you wrote is in fact correct.

3. Reasonableness. See if your answer makes sense. If the answer must be positive, is it positive? If it must have a particular unit of measurement, does it? Another aspect to this is to try to see if there is another way to obtain a solution. If so, try an alternative computation and compare the results.

The reality is that accuracy is not a talent, but a skill that is developed through persistence and good habits; it isn't something you can suddenly develop overnight. Accuracy is a result of experience.

• Hmm, I'll try this systematic approach next time. I think what's also part of the problem is that when I read a word problem and already know what formulas I'll need and what steps I need to take to arrive at the problem, oftentimes my brain works faster than my hand writes and I end up fumbling here and there. I guess I'll slow down for now and try to apply your step-by-step approach and see if I improve. Thank you! Jun 27, 2016 at 21:12
• This answer reminds me of Polya's "How To Solve It" framework. Jun 27, 2016 at 22:07
• ...and the OP could do worse than to read that timeless classic. @Aleksandr, do yourself a favor: read Pólya. Jun 28, 2016 at 1:20
• @Aleksandr, the one Austin said: How to Solve It. And if you have time to add to your reading list: Zeitz's The Art and Craft of Problem Solving. Jun 28, 2016 at 10:59
• @AleksandrH Your first comment on this answer is exactly what causes most students to make mistakes: their brain goes faster than their work. Writing down everything, including the things that seem obvious, is helpful because it helps you see any mistakes you've made and because it helps slow down your thinking. Jun 28, 2016 at 13:32

A few suggestions:

• Look at the places where you made mistakes. Is there a pattern to what kinds of mistakes you're making? "Careless mistakes" is a bit broad, and perhaps a bit unfair, because you may be being overly harsh on yourself. It could be that there is a gap or two in one of the earlier math courses that you took, and it's only coming to light now that it's assumed that you know it.
• Learn as best as you can from the mistakes you've made. This is good advice in general, not just for calculus.
• Consider whether something else is getting in the way. Some people are wired differently which makes going through the motions of a math exam under time pressure a bit difficult. If you feel that your exam grades aren't reflecting what you know, then there may be accommodations that can be made.
• Post your questions here in the same detail that you've done this one. It's likely that you'll get good responses that can pinpoint where things went wrong in your work.

Good luck!

• Thanks for your reply! So the problem was as such: "A particle moves in a straight line so that its position (in feet from a starting point) after t minutes is given by s(t) = 6t^2 - t^3. Find the displacement of the particle and the total distance traveled on the time interval 0<= t <= 5". The displacement part was pretty straightforward; for the distance, I used a definite integral of | v(t) | from 0 to 5, where v(t) = 12t - 3t^2. And then I set v(t) = 0, found where it's negative, and split the integral in two. My mistakes were mainly during the application of the Evaluation Theorem. Jun 27, 2016 at 20:57
• And it wasn't so much as improperly setting up the F(b) - F(a) evaluation as it was screwing up on the numbers when I was plugging them in. I honestly don't know how I managed to get 3 different answers. Jun 27, 2016 at 20:58
• Maybe it's just a matter of double-checking your numbers, then. But again, if this happens a lot then there may be underlying reasons.
– John
Jun 27, 2016 at 21:03
• Yeah, I try to double-check my work when I can, but it seems I can't break the habit of checking my work by assuming that what I've already written down is correct. Jun 27, 2016 at 21:10

Take comfort in the fact that real mathematics is not done under timed conditions like the examinations. When I was an undergraduate, I too found the introductory calculus and linear algebra courses one of the hardest, simply because I could not do computations as fast as other people. But mathematics is ultimately about theorems and proofs, not computation (that's now all doable by computers anyway). I then went into pure mathematics where almost all the higher-level courses involved mostly proofs and little computation.

• Anyway so, no, your grade may not suffer at all if you're like me (and you certainly seem so). Just try your best not to make careless mistakes, but don't fret too much over them. Jun 28, 2016 at 11:19
• Thanks! It's a shame that higher level math classes can't move slower and allow students to really appreciate the concepts they teach instead of just shoving them down their throats and saying "Hey, look, here's the stuff you need to learn. Okay, got it? Good. Now take a timed exam." Jun 29, 2016 at 0:00
• @AleksandrH: Yes, but "higher-level" is relative. Right now the classes you're taking may seem higher-level but the content is mostly devoid of rigorous proof. Proving identities is usually nothing more than algebraic manipulation, and the closest you'll get to real proofs is in Euclidean geometry. So if you like proving theorems in Euclidean geometry rigorously (no reliance on intuition!), then I think you'll have no problems past the first year of undergraduate mathematics. Jun 29, 2016 at 0:55

One technique is to do the problem in two very different ways.

If the methods are different enough, it's unlikely that you'll repeat the same mistake both ways, so comparing the answers gives you a way to check. And if there's a difference, you can often use what you learned from one method to validate your intermediate results from the other and find out exactly where the mistake is.

I have a couple of suggestions that may make sense for some people here.

1. Try to find the pattern of your mistakes. You can even write your mistakes down to get a bright image of what your mistakes are. After a while, you will find out on which part you make lots of mistakes. Thus, when you are dealing with problems, you will be wary of not redoing your mistakes.

2. Read the question carefully, word by word. Sometimes you go a long way to find a suitable answer to a question while the question, in fact, has wanted you to get another aspect of the problem. So, answer what question has asked you, not anything else.

3. Be doubtful about math questions as if they are your enemies wanting to deceive you. That is to say, when you calculated your answer, check it again, before choosing it as the final answer. Do not trust questions, some of them seem really easy that you can find the answer in second, however, they use tricky methods to deceive you to reach the wrong answer.