r/calculus • u/Basic-Feedback2952 • 5h ago
Differential Calculus Limits + Continuity Note Sheet
A few days ago, I came across this notes sheet for limits and continuity and wanted to share it because I thought it could come in handy for anyone else.
r/calculus • u/random_anonymous_guy • Oct 03 '21
A common refrain I often hear from students who are new to Calculus when they seek out a tutor is that they have some homework problems that they do not know how to solve because their teacher/instructor/professor did not show them how to do it. Often times, I also see these students being overly dependent on memorizing solutions to examples they see in class in hopes that this is all they need to do to is repeat these solutions on their homework and exams. My best guess is that this is how they made it through high school algebra.
I also sense this sort of culture shock in students who:
Anybody who has seen my comments on /r/calculus over the last year or two may already know my thoughts on the topic, but they do bear repeating again once more in a pinned post. I post my thoughts again, in hopes they reach new Calculus students who come here for help on their homework, mainly due to the situation I am posting about.
Having a second job where I also tutor high school students in algebra, I often find that some algebra classes are set up so that students only need to memorize, memorize, memorize what the teacher does.
Then they get to Calculus, often in a college setting, and are smacked in the face with the reality that memorization alone is not going to get them through Calculus. This is because it is a common expectation among Calculus instructors and professors that students apply problem-solving skills.
How are we supposed to solve problems if we aren’t shown how to solve them?
That’s the entire point of solving problems. That you are supposed to figure it out for yourself. There are two kinds of math questions that appear on homework and exams: Exercises and problems.
What is the difference? An exercise is a question where the solution process is already known to the person answering the question. Your instructor shows you how to evaluate a limit of a rational function by factoring and cancelling factors. Then you are asked to do the same thing on the homework, probably several times, and then once again on your first midterm. This is a situation where memorizing what the instructor does in class is perfectly viable.
A problem, on the other hand, is a situation requiring you to devise a process to come to a solution, not just simply applying a process you have seen before. If you rely on someone to give/tell you a process to solve a problem, you aren’t solving a problem. You are simply implementing someone else’s solution.
This is one reason why instructors do not show you how to solve literally every problem you will encounter on the homework and exams. It’s not because your instructor is being lazy, it’s because you are expected to apply problem-solving skills. A second reason, of course, is that there are far too many different problem situations that require different processes (even if they differ by one minor difference), and so it is just plain impractical for an instructor to cover every single problem situation, not to mention it being impractical to try to memorize all of them.
My third personal reason, a reason I suspect is shared by many other instructors, is that I have an interest in assessing whether or not you understand Calculus concepts. Giving you an exam where you can get away with regurgitating what you saw in class does not do this. I would not be able to distinguish a student who understands Calculus concepts from one who is really good at memorizing solutions. No, memorizing a solution you see in class does not mean you understand the material. What does help me see whether or not you understand the material is if you are able to adapt to new situations.
So then how do I figure things out if I am not told how to solve a problem?
If you are one of these students, and you are seeing a tutor, or coming to /r/calculus for help, instead of focusing on trying to slog through your homework assignment, please use it as an opportunity to improve upon your problem-solving habits. As much I enjoy helping students, I would rather devote my energy helping them become more independent rather than them continuing to depend on help. Don’t just learn how to do your homework, learn how to be a more effective and independent problem-solver.
Discard the mindset that problem-solving is about doing what you think you should do. This is a rather defeating mindset when it comes to solving problems. Avoid the ”How should I start?” and “What should I do next?” The word “should” implies you are expecting to memorize yet another solution so that you can regurgitate it on the exam.
Instead, ask yourself, “What can I do?” And in answering this question, you will review what you already know, which includes any mathematical knowledge you bring into Calculus from previous math classes (*cough*algebra*cough*trigonometry*cough*). Take all those prerequisites seriously. Really. Either by mental recall, or by keeping your own notebook (maybe you even kept your notes from high school algebra), make sure you keep a grip on prerequisites. Because the more prerequisite knowledge you can recall, the more like you you are going to find an answer to “What can I do?”
Next, when it comes to learning new concepts in Calculus, you want to keep these three things in mind:
When reviewing what you know to solve a problem, you are looking for concepts that apply to the problem situation you are facing, whether at the beginning, or partway through (1). You may also have an idea which direction you want to take, so you would keep (2) in mind as well.
Sometimes, however, more than one concept applies, and failing to choose one based on (2), you may have to just try one anyways. Sometimes, you may have more than one way to apply a concept, and you are not sure what choice to make. Never be afraid to try something. Don’t be afraid of running into a dead end. This is the reality of problem-solving. A moment of realization happens when you simply try something without an expectation of a result.
Furthermore, when learning new concepts, and your teacher shows examples applying these new concepts, resist the urge to try to memorize the entire solution. The entire point of an example is to showcase a new concept, not to give you another solution to memorize.
If you can put an end to your “What should I do?” questions and instead ask “Should I try XYZ concept/tool?” that is an improvement, but even better is to try it out anyway. You don’t need anybody’s permission, not even your instructor’s, to try something out. Try it, and if you are not sure if you did it correctly, or if you went in the right direction, then we are still here and can give you feedback on your attempt.
Other miscellaneous study advice:
Don’t wait until the last minute to get a start on your homework that you have a whole week to work on. Furthermore, s p a c e o u t your studying. Chip away a little bit at your homework each night instead of trying to get it done all in one sitting. That way, the concepts stay consistently fresh in your mind instead of having to remember what your teacher taught you a week ago.
If you are lost or confused, please do your best to try to explain how it is you are lost or confused. Just throwing up your hands and saying “I’m lost” without any further clarification is useless to anybody who is attempting to help you because we need to know what it is you do know. We need to know where your understanding ends and confusion begins. Ultimately, any new instruction you receive must be tied to knowledge you already have.
Sometimes, when learning a new concept, it may be a good idea to separate mastering the new concept from using the concept to solve a problem. A favorite example of mine is integration by substitution. Often times, I find students learning how to perform a substitution at the same time as when they are attempting to use substitution to evaluate an integral. I personally think it is better to first learn how to perform substitution first, including all the nuances involved, before worrying about whether or not you are choosing the right substitution to solve an integral. Spend some time just practicing substitution for its own sake. The same applies to other concepts. Practice concepts so that you can learn how to do it correctly before you start using it to solve problems.
Finally, in a teacher-student relationship, both the student and the teacher have responsibilities. The teacher has the responsibility to teach, but the student also has the responsibility to learn, and mutual cooperation is absolutely necessary. The teacher is not there to do all of the work. You are now in college (or an AP class in high school) and now need to put more effort into your learning than you have previously made.
(Thanks to /u/You_dont_care_anyway for some suggestions.)
r/calculus • u/random_anonymous_guy • Feb 03 '24
Due to an increase of commenters working out homework problems for other people and posting their answers, effective immediately, violations of this subreddit rule will result in a temporary ban, with continued violations resulting in longer or permanent bans.
This also applies to providing a procedure (whether complete or a substantial portion) to follow, or by showing an example whose solution differs only in a trivial way.
r/calculus • u/Basic-Feedback2952 • 5h ago
A few days ago, I came across this notes sheet for limits and continuity and wanted to share it because I thought it could come in handy for anyone else.
r/calculus • u/PeoniePetrichor • 7h ago
I don’t know the English term for it, but I essentially did the highest tier of maths a few years ago and took a break from school. I averaged nearly 100s for my STEM classes.
I’m starting again, and have to do Differential Calculus (calculus 1) and, I’m honestly extremely anxious about it. I can’t remember ANYTHING. I had to go through severe brain fog these last few years because of depression and meds.
I was told cal 1 is usually just a refresher of everything you learned prior, and cal 2 (mechanics) is when it gets actually hard. But, I’m debating dropping the course and giving up on doing anything engineering-adjacent. I was great at maths before, aka YEARS ago. But, now I feel like I genuinely won’t pass the class. It’s only been one day.
I really need some advice.
r/calculus • u/FlamingHotBoof • 2h ago
Hello everyone! I am posting this evening because I am going back to college after five years for mechanical engineering and need to prepare myself for calculus 2 next semester. I am already enrolled this semester and am taking two simpler classes. Ideally, I would like to master Calc 1 and at the very least get a head start on Calc 2 so that I can hit the ground running next semester.
For reference, I am familiar with the material, as I have passed Calc 1 in the past and attempted Calc 2 more than once. I planned on following along with Professor Leonard's Calc 1 videos, study enough to pass practice final exams, and continue on to his Calc 2 lectures. What I'm struggling with is how to divide up the work to be finished by the end of the year, and creating a schedule to go along with it.
Does anyone have any resources that might be able to share on how to learn Calc 1 and Calc 2 by the end of January and how to structure that schedule? I'm currently only enrolled in two classes and should be able to spare time each day to focus on learning this material. I cannot fail whatsoever since I was accepted back this semester on stipulations.
-- I have a copy of Calculus 9E- Early Transcendentals - James Stewart by the way as well.
Any help would be greatly appreciated! Thank you! :D
r/calculus • u/Bitter_Ad469 • 9h ago
I'm currently taking Calculus AB (but self-studying for BC exam this year), and I'm genuinely enjoying it. I push off doing reading for English, and my statistics homework just to do more calculus worksheets. I even listen to videos teaching calculus. took statistics thinking it would be easy (it is), but I absolutely hate it.
What type of person prefers statistics vs calculus?
Quick question - What math should I do after this? I'm a junior in hs and I want to do Cognitive Science.
r/calculus • u/PeoniePetrichor • 10h ago
Hey guys! I’m continuing my journey with STEM by starting again with Cal 1. I’m debating whether or not it would be better to write my notes, exercises,homework etc on an iPad, instead of having to handwrite everything on paper like I’ve always done. I’ve noticed a lot of students have either a computer, or an iPad.
Is it quicker? Better? More organized?
r/calculus • u/Apprehensive_Cut2880 • 7h ago
r/calculus • u/CodeLiving • 3h ago
Like the title says, I am looking for solutions manual or just solutions to Tom Apostol's Calculus.
r/calculus • u/TheNextOne8797 • 4h ago
r/calculus • u/x_xwolf • 4h ago
r/calculus • u/ReplacementFresh3915 • 7h ago
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r/calculus • u/SadSuggestion7222 • 18h ago
My teacher provided us with this note about the POINT OF INFLECTION. It's so confusing. So, I asked AI about it and now, I am more confused. Can anyone explain this concept? And how do I find it?
r/calculus • u/SuperToast05 • 1d ago
After 3 years the GOAT of teaching mathematics has returned, I hope this guy has also changed lives like he changed mine. I was awful at math in hs and had to avoid engineering because of it but after 2 years in a major I hated I decided to give engineering a try. He is the one and only reason I passed all my classes and I owe him a debt that can never be repaid. I hope he changes/changed your life like he did mine. (He is 2000x better than the OCHEM tutor)
r/calculus • u/Efficient-Test-2494 • 17h ago
Hi everyone,
I’m starting Cambridge International A Level Further Mathematics and will mostly be self-studying because my school doesn’t have a Further Maths teacher.
Does anyone know where I can find useful PDFs, textbooks, notes, or other free resources for CIE Further Mathematics? I’d especially appreciate resources for understanding the topics and practising questions.
Thanks! 🙏
r/calculus • u/whoop_chicken • 1d ago
Here is the original problem I came across

When I evaluated the indefinite integral and confirmed with an integral calculator I got the function listed below.

I saw someone solving the problem online a different way and got the resulting function:

Here are all the equations graphed out:

What I am confused by is that deriving the H(x) function does give the equation in the integral: f(x) = (x^2+1)/(x^4+1). If H(x) derives to this f(x) function does that not mean it can used to evaluate the area under the curve of f(x). f(x) does correctly give the slopes of H(x) but evaluating H(x) gives incorrect areas. For example evaluating H(x) from x=-1 to x=1 gives an area of 0 while using the F(x) function gives ~2.22. The F(x) function seems to give the correct areas under the curve of f(x). I am just confused how the derivative of H(x) gives the parent function under the integral but cannot be used to correctly evaluate the area of the derived function.
r/calculus • u/JakeD51 • 1d ago
I'm only in calc 1, gotten up to squeeze theorems, but I genuinely have not enjoyed math this much in YEARS. The professor is quite blunt personally, but his video notes(online class) are wonderful and extremely detailed. Everything makes complete sense and I really have no sense of questioning anything. Am I being hopeful only to crash and burn when it gets to more intense Calculus? Early in the semester but I actually enjoyed Calculus more than any of my other courses so far.
r/calculus • u/greenvented • 15h ago
Was going through a textbook for practice problems and this one is throwing me off. I have no idea whether I simplified wrong, because im pretty sure the integration process is correct
r/calculus • u/AnalogRFIC_Wizard • 1d ago
Hey everyone.
Regarding today's easy integral. I can't figure out where my mistake is but I believe it is something silly.
I am basically starting with the substitution of u = ln(x^2026)
Algebraic manipulation leads me to 2025/2026 times the integral of sin(u) * e^-u. I then use integration by parts twice to get the second part of my expression.
Can anyone help me spot the mistake?
Thank you!
EDIT: oh I forgot to adjust the integration limits, that must be it. Still leaving the post here I guess.
EDIT2: exactly so with the substitution the lower integration limit becomes 0 and so the answer is 2025/2026 * 1/2 = 2025/4052
r/calculus • u/Due_Yogurtcloset4421 • 1d ago
I can’t seem to pass my university’s in-person Calculus I course, so I’m looking for an easier online 4-credit Calc I course I can transfer in. Has anyone taken one with manageable exams that they were able to pass? What university?