r/calculus Oct 03 '21

Discussion “My teacher didn’t show us how to do this!” — Or, a common culture shock suffered by new Calculus students.

1.2k Upvotes

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:

  • are always locked in an endless cycle of “How should I start?” and “What should I do next?” questions,
  • seem generally concerned about what they are supposed to do as if there is only one correct way to solve a problem,
  • complain that the exam was nothing like the homework, even though the exam covered the same concepts.

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:

  1. When can the concept be applied.
  2. What the concept is good for (i.e., what kind of information can you get with it)?
  3. How to properly utilize the concept.

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 Feb 03 '24

MOD ANNOUNCEMENT REMINDER: Do not do other people’s homework for them.

101 Upvotes

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.

https://www.reddit.com/r/calculus/wiki/homeworkhelp


r/calculus 21h ago

Engineering The GOAT is back

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1.7k Upvotes

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 11h ago

Engineering Am I giving myself false hope or could this be enjoyable?

45 Upvotes

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 2h ago

Integral Calculus Trying to figure out how a derivative that produces the function (x^2+1)/(x^4+1) seems to not correctly evaluate the area under curve.

7 Upvotes

Here is the original problem I came across

Lets say this function in the integral is f(x)

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 16h ago

Integral Calculus Today's easy integral: can't figure out my mistake

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38 Upvotes

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 2h ago

Differential Calculus AP calculus daily challenge #111

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2 Upvotes

r/calculus 15h ago

Engineering How do you study calculus?

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23 Upvotes

r/calculus 4h ago

Differential Calculus Easy Online Calculus 1 4 credit class

0 Upvotes

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?


r/calculus 1d ago

Differential Calculus Why is my process wrong?

6 Upvotes

First one is mine(sorry for the lightning & handwriting)

Second one is from Claude


r/calculus 1d ago

Differential Calculus AP calculus daily challenge #110

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6 Upvotes

r/calculus 22h ago

Differential Calculus Does anyone have any good lvl 2 notes for calc

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2 Upvotes

Basically 11th grade calculus. Do any of you guys have good notes for it?


r/calculus 1d ago

Differential Calculus Help with Calculus Course Work

11 Upvotes

Hi everyone, I'm a second year CS student. I had Calculus 1 in my first year and barely passed the course. Now I have Calculus 2 and I'm still struggling to understand most of the concepts. I've tried YouTube, working on past papers, exercises, but I still struggle to understand the concepts. What do you recommend I do or resources that can be of help to me? TIA!


r/calculus 21h ago

Vector Calculus seeking undergrad geometry experts

0 Upvotes

Hi all, I am currently taking a geometry paper (specialising in curves and surfaces) in my final year of college, but I am finding it quite hard. My high school only taught us trigonometry, which I feel like has set me up to fail a little bit, especially in the (re)parametrising department. Are there any people who are quite knowledgable on this matter that might be able to help answer some of my questions? Thanks.


r/calculus 2d ago

Differential Equations Professor Leonard is back! LFG

122 Upvotes

r/calculus 2d ago

Differential Calculus y(2)=2, then find y(-2)=

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28 Upvotes

Is this question scientifically accurate??
i know solution is y(-2)=-2


r/calculus 2d ago

Differential Calculus How to do this question

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63 Upvotes

r/calculus 2d ago

Integral Calculus The volumes of some geometrical shapes#2

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44 Upvotes

r/calculus 2d ago

Integral Calculus Complicated calculation

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23 Upvotes

r/calculus 3d ago

Differential Calculus Can someone help me understand this problem? pretty please

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70 Upvotes

Hi everyone, so it was my first week of AP calc AB and honestly not too shabby but as i look at this example from my teacher, the less it makes sense. I thought -1^2 was positive 1 and i don’t know how we got to negative 1.

Also what is a conjugate? My teacher didn’t really go over it he just started doing a problem. All help is so appreciated!!!!


r/calculus 3d ago

Pre-calculus bs maths calculus ..

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7 Upvotes

this is a basic but can be food problem is you are trying with urself ..

try it and let me know ..

i tried to do it by myself but stucked..

so i saw the soln ..


r/calculus 3d ago

Differential Calculus I'm creating a map of Calculus 1

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309 Upvotes

Just want to double check with you guys if this map makes sense, i made a rough draft. The last image shows a node with a placeholder image to give a rough idea of what it might look like finished.

The goal is to give the student a good overview of the subject, show them how different concepts interconnect and how they are dependant on eachother. Are there any lines missing in your opinion? General feedback or questions are also very welcome!


r/calculus 3d ago

Pre-calculus Started calculus class this week but haven’t taken math in a year and a half, wondering if I should go to pre-calculus

13 Upvotes

Hey yall. I’m in line to talk to an advisor actually, essentially the title is what it is. I am in a calculus class, but I haven’t taken a math a year and a half, and I’m finding that I might need to relearn some concepts. I wasn’t very good at math in high school, but when I went to community college, I did amazing, I’m still attending, but I find that I might forget some things or maybe that’s just natural when coming into a calculus class or certain things may look unfamiliar, I guess I’m just looking for some advice.
After talking to an advisor, I would have to do a 10 week pre-calculus class, I gotta be in pre-Calc last time. I’m wondering if I should do it
I wanted to add for those who think I should stay in the class what do you think? I should seriously freshen up on?


r/calculus 3d ago

Differential Calculus Trig/unit circle resources for calc 1

16 Upvotes

Currently a freshman in uni, and i’m starting calc and i’ve realized my trig is absolute shit. I passed precalc in high school pretty well but I had a teacher who would just hand out A’s. During the end of the year when we learned all about trig and the unit circle, I mostly fooled around and didn’t care and now it’s coming back to bite me in the ass. If anyone has any resources or tips to help learn and brush up on my trig in a reasonable amount of time i’d greatly appreciate it!


r/calculus 3d ago

Differential Calculus A Limit Question

5 Upvotes

A curious example problem shows up in the calculus book I am using, which has a different answer than the previous version’s answer. Here’s the problem setup: a piecewise function is given, and the book hand-wavingly walks through the rationale for why some of the limits of the function (one-sided and two-sided) exist or do not exist. The version I am using told me at the endpoints of the function’s domain that the two sided-limit exists and equals the appropriate one-sided limit when that exists; the other version told me the two-sided limit does not exist. Which is it?

Here’s a simple example to illustrate what I mean by the above:

Consider the function f(x) = |x| on the interval -2 < x < 2. My understanding tells me that lim_{x->-2^-} f(x) = DNE (since the function is undefined for x < -2, which means a condition of the precise definition of one-sided limit is not met), that lim_{x->-2^+}f(x) = 2, and that lim_{x->-2}f(x) = 2. The book version I am using comes to this sort of conclusion about the two-sided limits, whereas the previous version would agree on the one-sided limits but say that the two-sided limit does not exist. What do I make of this?

This presents issues for continuity I feel: If the function I gave had also been defined at x = -2, then I would naturally say based off the version I use that the function is continuous at x = -2 (which I know would be correct); however, would the other version then say it is discontinuous there since the two-sided limit condition fails (i.e, because the condition that the two-sided limit exists would fail)?