How to Study Calculus

As of April 2025, I’ve been a TA for 6 first-year calculus courses at UBC. Students often ask me, “How do I succeed in this class?” (or “How do I pass this class?”). Here’s my advice.

1: Don’t Forget the Basics

With the fast pace of math courses, its easy to assume that you don’t have time to review previous material. But I think this is a terrible mindset. Calculus exams are unforgiving, and you will do poorly if you don’t have a solid understanding of pre-calculus. I highly encourage everyone who thinks they have gaps in their foundational knowledge to review the following:

• Exponent laws
• Logarithm rules
• Finding intercepts and asymptotes
• Function composition and inverse functions
• Trigonometric functions and the unit circle
• Graphing basic functions like \(e^{x}\), \(x^3\), and \(\sin (x)\)

It shouldn’t take you more than a couple hours to go over these concepts, and doing so and it will make the rest of your semester so much easier.

2: Make a Spreadsheet

Keep a list of questions you struggled with and write down what you learned from them. I use a spreadsheet that looks like this:

Color-code the questions: green if you understand everything, yellow if you’re a little confused, or red if you’re completely lost. Only include the questions you get wrong. This technique is a lifesaver during exam season, providing a personalized list of challenging questions for you to study from. Additionally, reading through the “Notes” column before an exam helps recall the techniques you’ve learned throughout the course.

3: Put the Calculator Away

If you came from the Canadian high school system, you were probably allowed a calculator for your pre-calculus classes. However, calculators are not allowed in the first-year calculus courses at UBC (and most upper-year math courses). Some students live in denial of this policy and continue using a calculator whenever possible throughout the semester. Unsurprisingly, these students later lose marks on exams due to computation errors. To minimize the chances of these silly mistakes, I think its best to avoid using a calculator while doing practice problems¹. Mental math isn’t fun, but it is necessary for success in calculus.

4: Embrace the Struggle

If you don’t have an exam soon, it’s okay to struggle with practice questions. You often learn more this way than by asking a friend or going to office hours. Trying new ideas and getting things wrong deepens your understanding of the material. When you finally get the right answer, you’ll understand not just how to solve the problem, but why the solution works. Spending an hour on a difficult problem can be more productive than doing ten easier problems in the same period. Of course, if you’ve exhausted every idea you have, it’s time to get help.

5: Practice Being Fast

If you have an exam soon, you should practice doing problems quickly. I’ve found that math exams have more restrictive time limits than exams in other departments, especially on midterms. The all-too-common experience of “I remembered how to solve the problem after the test!” is ultimately a problem of being too slow. Find out how long the test is, find out how many questions there are, and ensure you can work fast enough. The mathematics department posts tons of practice finals on their website. Ideally, you should try doing at least one of these practice finals under exam conditions.

6: Advice for Writing Tests

Ensure you are well-rested and well-fed. Bring drinks/food/gum to the exam if it helps you. Do not study within two hours of the test – do something relaxing instead. Do not write tests while drunk/high/hungover. If you are going to use stimulants, take a reasonable amount. Yes, I know this is trivial stuff. And yes, there are actually people who don’t do these things.

Okay, onto the more interesting advice.

If you can’t figure out how to do a problem in ~3 minutes, skip it and come back to it later. You’d be shocked at how good your subconscious is at coming up with ideas while you’re working on the other problems. Additionally, leaving the hardest problems to the end will help prevent you from ending up in a time crunch.

When checking your work, begin by making sanity checks. This means asking questions like “Does my answer have the right sign?” or “Does my answer seem unreasonably small or large?”. You can also do these as you work. Certain types of problems can be solved in reverse. For example, if a question asks you to evaluate an integral, check that the derivative of your answer is the original function. If your solution looks correct both forwards and backwards, it is much more likely to be right. Other types of problems have multiple viable solutions. If you see an alternative way of solving a problem, try working through it. If you get the same answer using two different methods, your answer is much more likely to be right.

Conclusion

First-year calculus courses at UBC are challenging for everyone, even math majors. I know someone who has taken over fifteen math courses at UBC and still says the MATH 101 final was the hardest exam they’ve ever written. Luckily, there are many resources available for students. Sign up for Piazza, go to office hours, and discuss the material with your friends. Success in these classes is not impossible—thousands of students score above 80% in MATH 1XX every year. I believe that you can be one of them. Good luck!