How to improve accuracy when solving calculus questions
Three concepts should always be a part of your mathematical problem-solving process.
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.
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.
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.
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!
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.