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.
Best Answer
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.