We Need To Talk About Calculus 2
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- Опубликовано: 8 сен 2023
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We talk about Calculus 2 and why it's so hard. Also what can you do to do better in Calculus 2? Do you have advice for people? Please leave any comments below.
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"If Calculus is so good, how come there's no Calculus 2?" --Gottfried Wilhelm Leibniz
Personally, I got an "A" in Calc II, but an "A-" in Calc I and Calc III. Go figure!
@@LaughingManRasame so far, hoping to get that final A in calc 3 next semester
Calc 2 is hard because all the different ways to integrate makes it like playing chess. There's integrating by parts, trig sub, plain u-substitution, partial fraction decomposition, trig integrating by using trigonometric identities, and then having to _apply_ all of these in different contexts like Volumes of Revolution, Surfaces, Arc Length, and then their applications in Physics.
Total beginner here, so my question may be entirely naive.
However, is this different to factorizing polynomials, which can include factorizing by difference of squares, GCF, GCF with leading coefficient > 1, sum/differences of cubes etc?
@@christosbinos8467no sir, you’ll wish it was just that if you take it. And really Cal 2 isn’t hard, it’s that they make the questions really annoying and meticulous.
@@christosbinos8467I’m currently in Calc 2 at uni. The difference is you are mostly guessing when you have to apply this techniques. If you are good at calc 2 you have a good idea as to what to do, but if it doesn’t work you are guessing from there.
I got a D in Calc 1 in high school. Then, when I took Calc 1 again in college, my professor made the biggest difference. I plowed throw Calc 1, 2, and 3 with an A because of that one person who took the time to remind me about Algebra 1 and how Calc 1 was mostly Algebra 1 applied to more complicated looking “stuff” is what he called it. :)
It's what I've come to find so far. I'm someone who did not take any mathematics class in the last 2 years of high school. I've covered introductory set theory and attempted calculus, only to find I am missing basics like analytic trig, algebra 1 and functions.
Covering algebra 1 in particular has been a saving grace.
To use a video game metaphor.
Calc 1 is like a linear storymode game , where your character levels up as you defeat progressively harder bosses and by the end of the game your character is stronger than all the enemies.
Calc 3 is like a sequel to that game. You learn a new mechanic, thinking about functions with multiple variables, and the rest of the difficulty progression is similiar (until you get to stuff like stoke 's theorem which is kinda like DLC).
In contrast, Calc 2 feels like a game made by the same developer in a different genre; an open world game. The game difficulty isnt necessarily linear and it feels like you're going around collecting equipment, until you grind enough mobs to unlock the S-tier enchantment (the intuition necessary for being able to quickly pick the correct and efficient integration techniques and identies to solve any question in a timed exam). Until you get this enchantment, sometimes some of the early game areas even get harder , because you found some cool weapon in a different area (IBP) and you naively tried using it against enemies that are strong against it . And in the endgame, you end up getting introduced to different kinds of enemies as much faster pace. I've seen people try to cover everything from basic diff eqs, error correction, physics and probability applications, all the while trying to wrap up the power series section, during the few weeks between a third (or late second midterm) and the final .
beautiful way to put it
That's relatable
I’m in pre cal. You just eased all of my worries away with your explanation. I will reference this before every semester thanks.
@@yonkoh529 I'm glad to hear that. Good luck with calc !
Dark souls 2 is calc 2
Currently in calc 2 with a full time job, studying literally every free second I get, easily getting 4-6 hours in every day, still feels like I’m flying by the seat of my pants
good luck!!
I don’t mean to sound rude, but if you are studying 4-6 hours everyday for one class, you are going about it all wrong. Currently in calc 2, maintaining decent grades and I’m putting maybe 10 hours max into it per week.
Not everyone is the same as you Daniel. Not everyone learns the same. Some of us aren’t gifted in math but still worked our way up to where we’re at. And then you have different things that factor into the situation, maybe the teacher isn’t good at explaining certain things and topics? You never know.
@@LB_c_777 What makes you think I'm gifted in math. I had to take all high school level classes my first year of college. I am merely pointing out that OP might not be studying effectively if they are having to study 25+ hours in a single class. Though it is entirely dependent on the rigor of the class. Maybe their calc 2 course is just more difficult than mine.
Same
I think the reason Calc II is harder is because in Precalc we spend most of the time related more to Calc I. Too often series/sequences are given short shrift in Precalc so the stuff isn't so familiar. Other topics like partial fraction decomposition, polar coordinates are also not given the attention they need.
I’m a math major and I still cannot do partial fraction decomposition lol
That’s literally the only thing I cannot do in calc ii
I completely agree. The topics covered in calculus 2 are not focused on much so the it is difficult to have a good foundation before taking the class.
that's really intersting. i'm not much of a matg person but i loved calc I because it made all those years of pre calc finally click but afterwards i just kinda bombed. so i relate to maybe not being as prepared for calc II
True. Even then Precalculus is just a condensed one semester course trying to best prepare you for calculus, which is itself a 3 semester course.
Yes, multi variable calculus was the one time I handed in a mostly blank exam, it was so confusing to me.
6m later and I still have trouble doing vector analysis problems.
Like I can't for the love of me NOT apply the divergence theorem on non-closed sets ....
Not having a dedicated topology class/section before multivariable was BRUTAL :(
Calc 2 was easy for me. Calc 3 was wild on the other hand
I had the same experience. Loved calc 2 and passed with an A. Calc 3 was more of a struggle. Got by with a B.
Agree cal 3 was harder
Sounds about right. Calc 2 was my favorite math class. Calc 3 was traumatizing
Leaving this comment because I plan to come back to it after 4 weeks left of calc2. Needless to say I got into an accident and many other things that just wrecked me. But I have a very slim chance to pass and i'm going to try. Essentially I have to get at least a high C on the next 2 exams then a 70 on the final to pass. Been studying all say for series and im at the point where all the homework is done, the study guide is done and i'm reviewing for tomorrow. You are absolutely correct for the fact that sometimes life gets in the way of learning. Thank you . Also Love the series videos you make., they really helped me this chapter.
Hope you got better, and passed the course. Good luck!
I do enjoy the Stewart book. Has all the theorems, quite a bit of proofs, and lots of explanations. All kinds of exercises, even a few proofs in the exercises to make sure you understood. Finishing up Calc I and looking foward to Calc II next year 😁
The hardest part for me was the techniques of integration. I could never figure out the best technique to use, but once I learned that any problem could be done using trigonometric substitution, I became an expert at that and used it for everything.
BOOM!!!!!!! Shout that out to the world.
Any and all calculus is trig. Period.
x=cosine. y=sine. versine=negative cosine.
As the cosine goes to zero (must go to zero, no dx or dy) the versine must go to 1. That is the limit.
As long as a student never forgets this then they will do just fine in calculus.
@@williejohnson5172 any integral can be soluved by trigonometric substitution?? like even something that would require the power rule
Math sorcerer thank you for uploading this, Im taking calc 2 right now and its the first time I've really struggled in a math class in college. Its feels good to know its hard for other people too, even people who are really into math.
I'm old an retired guy learning Calculus on my own. Quite a coincidence that I just finished Calc I using Larson's textbook and in the last few days I've using Stewart's (odd problems only) as a review before starting Calc 2 next week.
Thanks and hello agin for this set of videos on calculus - I'm starting my math master's journey this year by auditing Cal 1, 2, 3 simulataneously now, and so far it went well and I agree that Cal 2's area/volume calculations are a challenge hurdle for many students.
Before finding your channel i used to like math, but you made me fall in love with mathematics, thank you math sorcerer!! ❤
I used to love maths back in high school when I got to university my love slowly decipated. But thanks to this guy and my recent lecturers my love is way bigger now. Maths is life
5:30 man that's so powerful and heart touching! Im sure he'll remember you for all his life
Calc II was intense (did it over eight weeks, summer '84), but it was also a lot of fun. You build a large toolkit, and spend the time hammering out a lot of interesting solutions to assigned problems. However, then Calc III is a breeze.
At least, that's how it was for me in 1984. I started college in '83 with Pre-calc, and ran from there.
Here in 2024 G
Great video! Solids of revolution was my favorite part of Calculus 2. That said, the setup and bounds of integration could be tricky for sure.
Currently in Calc 2, returning to college 13 years after dropping out. The reason it’s so hard for me is because of the amount of precalc and trig stuff that comes up that you need to be able to both see and do. I’m looking at problems, unsure where to go, and it’s like, oh, as the very first step you need to complete the square of the denominator or factor the difference of two cubes. Not necessarily hard, but I don’t remember and it so it doesn’t occur to me. I spend a lot of time struggling to solve something after missing a crucial early step. I get nowhere, get frustrated, and waste a lot of time.
Then it’s similar with the trig stuff. Something like sin(arcsec(5)) will come up and I’ll be drawing triangles and chanting Sohcahtoa because my trig stuff is super rusty and the rest of the class just did it and moved on. I’m pretty good at unit circle stuff but some of the identities just don’t pop out to me the way they need to.
Anyway, it could just be me, because I’m so far removed from the years of doing math on a regular basis, but to me it feels like you really need all your algebra, trig, and precalc to be super solid for Calc 2, and so instead of just learning whatever is in the current lesson, I also need to relearn several things on the fly as we go.
I also found infinite series to be incredibly cool and my favorite part of calculus. It is just feels so incredibly different from most of what we were doing up to that point and it felt like it was touching on a lot of deeper math concepts. I also thought it was incredible how we could determine convergence on an infinitely repeated operation.
Not sure every school, but at the tech school where I took Calc 1, 2, 3, and DE Calc 2 was treated as the weed out course for engineering transfer students and that was the justification for the difficulty. Organic Chem 2 is similarly used to weed out Pre med students in many places.
In Calc 2 right now, and I absolutely agree it’s the hardest. I’m not personally struggling, but every single minute in this class I’m learning a ton and my math knowledge and intuition is expanding. The amount of content is staggering and it’s not simple. I’m making vast leaps in my understanding and it’s *just* enough to keep up. If I wasn’t making consistent epiphanies about how this stuff works, I’d be falling behind like crazy.
I was the only one who managed the finish the entire first exam, it’s brutal
You got this!!! I barely passed but after you pass you will feel great!!!
I think for most people calc 2 is the first class where just plugging in a formula or algorithm doesnt automatically give you the right answer. You learn a lot of identities and technique , but getting the correct answer is contingent on developing the intuition to use the correct tools. And sadly , even if you are working towards a "correct" solution, having bad intuition can cause you to take a very lengthy route to get there, which is horrible for timed exams .
It begins as early as people incorrectly visualizing or inefficiently setting up integrals for solids of revolution, spikes exponentially once you get to integration techniques (for some of the integrals you can create entire trees of possible mistakes and lengthy routes you can take) and by the time people get to power series or error approximation it almost feels like a whole other class.
In contrast Calc 1 is intrinsically a lot more straightforward ( you derive derivatives, and derivative rules , get used to using them and then just learn a bunch of plug and play applications) .
Even calc 3 is better as a lot of it is just extending familiar material from calc 1 and 2 to multiple dimensions and generally ending with some applications (unless you go down the vector calculus stuff ). While the issues with calc 2 can still technically hurt you here, the amount of material seems to generally for professors fo test you on concepts introduced in calc 3 , rather than trying to trip you up with a tricky integral from the previous course.
Calc 3 tripped me up the most because I can't think in 3 dimensions/variables for the life of me
Calc 2 was regarded by the engineering students as one of the hardest courses. I loved techniques and applications of integration. I still love integrals. Series is what sucked for me. I got my lowest exam score on the series exam. Everything after series was very easy.
Series didn't click with me until after the fact. Similar to looking at an integral and thinking about all the techniques that could work, the same is similar with evaluating a series.
I took calc 2 as a four week course this past summer. I did well but I paid for it with my blood, sweat, and tears. It was by far my favorite math class.
4 weeks, wow!
Eight weeks, Summer '84. It was brutal (the pace), but fun.
I took linear algebra in 4 weeks. It was nuts, but fun. My instructor actually does research in undergrad math education, and her favorite to teach is apparently linear algebra
Same here, I actually had no problem with it and enjoyed it. Now in Calc 3 I’m already having more trouble than I did the whole semester of Calc 2. Same Professor for both as well. At this rate I’d say Calc 3 is much harder but maybe because it’s a new concept and it’ll get easier
Like the dumbbells in the background, looks really nice. I would however put them on some kind of "mini stand", because round weights on a flat surface is giving me sweat pearls just from looking at it..
I've taught BC Calc, AB Calc, and dual credit Calc 3 at the high school I teach at. It's interesting because for me, I always found Calc 3 the hardest as a student. I just struggled to visualize and conceptualize in 3D what I was calculating. Now that I'm teaching it and we have a full year with the course instead of just a semester, I have a much better grasp of it and want my students to understand the concepts before the calculations. Calc 2 was also my favorite - I loved infinite series also! I think what makes it hard for students is that infinite series is more than just using a formula or technique. It requires logic, reasoning, being willing to go back to the drawing board and try again when your first attempt fails, and really understanding the bigger picture of how functions relate to each other instead of just analyzing functions on their own (like you might do with an integral). It's way more pieces to the puzzle than taking a derivative, doing a u-sub, finding the flux across a surface, etc. But - it made me a way better problem solver and critical thinker! Still one of my favorite math classes I have ever taken and taught.
I love this guy and his videos. Great stuff.
thank you!
He's slowly forcing me back to math with remote care by his teachings... that's called a teacher
I think calc 2 is the most difficult because of the number of new topics introduced. Series and sequences are barely introduced in the precalc, and in calc 2 you go pretty hard on them. Polar coordinates are also introduced for the first time. Last, which I think is the most important, is that you are starting to learn how to set up the problem more than solving the problem itself. In calc 1, it was usually just taking the derivative or integral and you're good to go. In calc 2 you have to think a bit harder on how to set up the disk and shell method. Setting up the sequence is also difficult because you can have multiple right answers.
really good points here, thank you for this comment!!!
I have to argue that Calc 3 is by far the most difficult math class for non engineering majors. I had no experience with vectors, determinants, or R^3 in general so I was already having to go over a hurdle in the first week learning about vectors.
I feel like I had a greater understanding while taking analysis than I did taking Calc 3. Maybe if I retook the course it would be a different story…
I think Calc 3 is easier if you’ve taken Linear Algebra beforehand
Calc 3 and Linear were terrible for me. Calc 2 was great
Waterloo used Stewart when I was there, I remember using that! I will be using Larson soon for Linear Algebra (another school). Look forward to it! Omg, totally relate to teaching time causing less content than an online course.
I too loved the topic on infinite series. Being able to find the right test always left me feeling smart for a while
I really like your mindset about classes and your critic about the guy in your story. I was going to transfer to a much better university and I was a straight A student before the semester before I transferred which was full of Cs and Ds. Thankfully though I did much better the next semester. Sometimes in life you have huge distractions and need to take mental break. It happens to every one of us.
Thank you very much SIR for the clear explanation . really helpful video
To me, the part that made Cal 2 so challenging was techniques of integration. Recognizing which techniques tu use, in different situations was one of the hardest things I had to learn. I worked hard and got an A- in the class, but it was an atrocious process.
Love it thanks!!
I took my first calculus later on life returning to school. I loved it but struggled and I wish I had done math sooner. I LOVED disk and shell and STRUGGLED with infinite series.
For me, Calc 2 was the hardest of the 3 because of all the convergence tests. I taught AP Calculus BC last year for the first time in my teaching career at the high school level and I had to go back and study the material like I was the student. I will be teaching it again after Christmas this year and feel much more confident than last year. I had fun relearning and teaching the material.
Same here. I have been relearning math recently because I forgot some of it. This morning, I was reading a Calculus book and learned something new. I previously self-taught myself BC Calculus many years ago. Now, I am trying to help my son. He completed Calculus 1 on his own this summer.
I can’t do partial fractions decomposition
My trig sub is rusty but I can do it lol
I have The Math Sorcerer's Calc 2 Udemy course (I have all his courses). There's so much Calc 2 in there. It's not the same as doing every problem in the book but chatgpt doesn't explain the answers the way The Sorcerer does. It's absolutey worth it. I had to walk away from my math studies to go do something else entirely so I've only done the first several hours of the Calc 2. So what happened is I discovered a whole area of pre-calc that I was weak in even though I passed Calc 1. This lead me down a Dover maths book rabbit whole...why was that a rabbit whole?- So here's what happened: In the Dover math books- the first chapter is usually the set theory that you need for the course BUT then the author always abandoned's it. For like 10 Dover books, this keeps happening. Now I'm learning algebraic topology so I can construct problems better. I don't care about memorizing trig functions to pass a calc 2 class. I care about fully understanding the question. There's so much theory and proofs in Linear Algebra such that if I study for fun over the next few years then I should be able to pick up Calc 2 as if I had just taken calc 1 instead of having took calc 1 20 years earlier. I'm "rolling sideways" for as long as it takes. I don't need Calc 2 alone, to land an immediate job to save the world. It's not in the cards for billions of people on Earth. So if you need Calc 2 for school so that you can go past Calc 2 for all that other stuff, The Math Sorcerer's Calc 2 course is worth it because, ChatGpt will get things wrong (on purpose, even) and can't explain it to you like your personal teacher or tutor. It's worth getting and Udemy courses are like $80 but then the next week they're $17- this happens every week. Do not be discouraged because it's expense. It's not expensive. Do not think you can just use ChatGPT. AI throttles your usage, Udemy doesn't.
I am similar in the mindset that Calc II was my favorite (I found it to be the most straightforward, of the three-semester sequence, and I also had a great professor for it).
Soooo True....the timing in the life is everything!!!!!!!!!!!!
I took calc 1 and 2 my senior year of high school online because of Covid. I went on to community college and took vector calc and linear algebra and did well in all of them. I then changed schools and my major and haven't done math seriously for 2 years. I would love to get back into it and hope to take diff eq some day but I don't know how much I have retained or lost.
I’m at the end of calc 2. Just finished with the last lecture. Two tests left. I feel like the hardest part was the sheer number of new things I learned. Trig sub was a bit tricky until you did a ton of problems. I initially did not get series at all until right before the end of the chapter. I really hated the application sections , ironically pursing an engineering degree. 😂 but I have really enjoyed a good portion of it. No regrets. Study hard , but my biggest suggestion to those that come here to watch this video because calc 2 is coming soon on your schedule; self study before your class if possible. Relearn these things : properties of natural logs , limits / L’Hopitals Rule , Trig identities, unit circle, convergence /divergence , and just learn every type of integration you can. Make hella Flashcards too , all the Flashcards. It’s hard but it’s possible and dare I say it even gets fun. Good luck.
For me there is just so much more stuff, like in one lecture in calc 1 we learned product and quotiet rule rule, or derivatives of logs
Calc 2 one of the lectures we learned the formulas and how to draw 6 3D shapes, or how to find the equations for the oscillating and normal plane of a vector function in the same class we learned curvature
Love your videos and definitely come to your channel for motivation and wisdom, but i got to know: is that where you keep weights? I saw that table wobble when you picked up Stewart's calculus and was slightly nervous for the rest if the video 😂
I was really fretting taking Calc 2 when I returned to College later in life. A bunch of my Comp Sci friends I know had to take it twice after failing it the first time. Luckily the professor I had was really good. What was great about his class is that we had a quiz every Monday which covered the previous two lecturers, and we had 4 midterms. This meant you had more chances to earn points and only have two midterms plus the final. He also assigned homework that if you could understand and complete, meant you could do well on the tests.
Calc 2 is brutal for complex integrals and other divergent test. The trig sub actually blown my mind as first impression in spring 2020 when pandemic starts.
I got your udemy and it will be useful as self teach for the 2024 or 2025 semester. I am self learn calc based physics mechanic for future course
Great! Amazing! I love this channel!!!! Great channel!
Hardest part of Calculus II for me: finding areas bounded by polar curves.
Way harder than Washer/Shell Method, integration techniques, Convergence/Divergence tests for series, etc.
I'm honestly surprised no one else talks about it, haha.
That section took me so many all-nighters to figure out and even still some problems trip me up.
Mainly finding the bounds of integration: for the longest time it wasn't clicking for me.
Again, I am genuinely surprised no one seems to talk about it.
Side note: I personally struggled way more with Calculus III than I did Calculus II.
Particularly with LaGrange Multipliers, change of variables in double integration, and Stokes Theorem.
I smashed Calc II this last summer. I was really getting into math right around that time, I also concurrently found you. Anyway, I'm a math and physics major now! Calc III, personally, seems more difficult. There's just more scaffolding and steps and nuance to arrive at the conclusions being taught. Also, it's all being taught faster.
calculus III is way more difficult, I do not know why this isn't the majority thought
you jump into calc II with disk ,washer, shell rotations about x-axis which WILL BE the most strenuous and monotonous use of the fundamental theory of calculus which also lends itself to error in the process of defining the integral.
My theory remains that there are a few factors involved. I was the worst in the second of the three maths I took. One thing I found out years later is that I must see the math, because I'm visually dominant. Another is the background should it be missing. I struggled with linear algebra because I never had vector calculus. Once I could see what was happening, I did get it. What still bothers me is infinite series and sequences, for I notice patterns with a certain ease, but I just can't seem to see the result. I can tell you the day of the week of any date without a calculator or switch from the archaic system to the metric or the pattern of zip codes, but somehow the infinites seem to elude me.
I enjoyed Calculus 2 more than any of the other calculus classes. It was taught by Dr. Ralph G. Sanger, the head of the Math Department at Kansas State University in 1965. We used Protter and Morey, College Calculus with Analytic Geometry for all three calculus classes. Dr. Sanger was a real renaissance man. He sang opera as well as loving mathematics. He let us use a cheat sheet for the final exam. We were allowed to write down anything we could fit on both sides of a single 8 1/2 x 11 sheet of paper. I was able to put all of the important formulas on that sheet by printing very small as I had taken mechanical drawing in high school. I finished the entire exam with only a glance or two at the cheat sheet. Dr. Sanger knew that the studying we did to prepare the cheat sheet would be an excellent review for the final exam. By the way, I still enjoy doing solids of revolution!
lol I’ve used both of these books! in high school AP Calc we used calculus by Larsen but at my college we use the Stewart book (also Early Transcendentals)
I took calc 2 my senior year of highschool and i remember liking it a lot. I loved infinite series ❤
im watching prof leonard's videos which are very good! disc and shell he did in calc 1. integration by parts stretched my brain a bit with its odd substitutions.
I bought that Larson book last month, it's great! I found the disc and area under the curve stuff quite aggravating.
First off, bravo man. You rounded up the tiniest percentage to let some student pass. There are a lot of professors out there that will refuses to do the tiniest curve. Seriously that warms my heart because there are certain classes people do so hard and try and having this one class just keep them from doing everything else it makes me really happy that they can keep pursuing and to be motivated. My school requires Calc 1 done before you take Physics/Mechanics and the professor said he is probably only going to have a couple calculus problems throughout the whole semester
But in my personal opion, Calc 2 is only hard because they did a 180 on us. every math class up until now, I was allowed to have notes and a calculator and my professor took it all away. And im not the only one many of my students cannot do the same and our algebra skills are reeeeeeally showing now. I dont do good because I never had to do 2/5(8)^5/2 on paper until now and we only get a couple days to practice it
I think I do fine setting up things like the disc/washer method but when I have to do evaluate the integral it is a disgusting mess. I bought your course and I appreciate you going through the whole integral process without a calculator!
Professors who refuse to budget even a quarter of a percent to grant a student who has clearly been trying their best all semester a passing grade are egotistical. Props to this guy for bumping the guys grade. Professors who refuse to do this refuse to see how they are also not perfect. Maybe one year they took 3 points off someone's wrong answer and another year they took off 5 points for the same mistake. Maybe the person who got 3 points off was able to pass, but the person who got 5 points off failed. Failing to bump someones a negligible amount so they can pass an arbitrary bar to consider their grade "passing" assumes they have always graded everything perfectly consistently, and I find it hard to believe (impossible, even) that that is the case.
Taking calc II rn with college apps on the horizon, we’re all going to be through this 🙌
My son just graduated with a Computer Science degree and he is now taking a Calculus 3 class. I'm going to share your videos with him.
I love this channel, and enjoy you sharing opinions on different books. If you do not mind, hope you could talk about calculus text by Salas. Some of my teachers love it, but also some criticize it a lot. Would like to hear your opinion. Thanks.
This was already nearly a decade since I first came into the U.S. higher education environment, and I audited and followed a Calc. 2 the entire semester for the first time. It is difficult in the sense that it teaches multiple integration techniques, but further many "results" involve (or are based about) trigonometric identities and trig. functions -- if I really think about it, along with the electrical engineering training I used to have, then these trig-related results are not that much useful; however, their existence makes failing to master calculus that much harder as long as a student genuinely studies through them. In short, they are things "to remember" which helps anchoring ourselves into the Calculus context and helps stretching our (short-term) memory span; having said so, there are other (alternative) ways to achieve this, too, such as studying social sciences, political sciences' applications that are blended with Calculus, or even just engineering.
I respect you Mr teacher truly.
Calc II was also my favorite and I did the best in it. Calc III on the other hand threw me onto the curb and ran me over a dozen times in a row. I was shocked when I found out that a lot of people saw III as easier.
Half way through Calc 2 and it’s awesome stuff!! 😅
I didn't think Calc 2 was too bad. Funny story: I was accompanied through the calculus sequence and differential equations with a wiz kid (his parents were from Germany if I recall correctly); the boy couldn't have been older than 10 or 11, but he routinely smashed the exams and always got one of the top scores. The only exam I know for a fact I beat him on was infinite series and sequences. I'm not sure why I took to that subject so well. I remember after first being introduced to them, I knew of their reputation as being a filter and causing countless students to drop their STEM degree or having to retake the class. So, after the first week they were introduced I grabbed a latte and spent that Friday afternoon, evening, and all into the night until the library closed writing the dozen or so theorems out over and over and redoing every homework assignment. I'd take breaks and walk around campus, read a passage from random books in the library, or work on another course, but I made infinite sequences and series my main target. I guess my advice for those struggling is to just recreate what I did. There's only a dozen or so theorems so write them out until you have them memorized. Try to write them out in set theoretic / symbolic logic notation. Identify the antecedent in each: what conditions must be met for that theorem to be appropriate to use? Really try to understand what they are saying, and understand why they work/make sense. Think of them as your tools; with enough practice, youll be able to look at any problem your professor throws at you and know which tool you need for the job.
Edit: students are typically introduced to set theory and higher order logic in their first proofs class, which is often Discrete Mathematics. So I suppose that is my second piece of advice: take Discrete as soon as possible in your education, and really take the material to heart. If you aren't a math major, see if Discrete can be taken to fulfill one of your Major specific electives (if it isn't one already). It is such a great and beneficial course
I also had the hardest time with Disk and Shell at first. Other than that most of Calculus 2 was pretty easy outside of a couple of the trickier area problems in polar coordinates. Series I can go through in my sleep with little trouble.
I actually found Calculus 3 hardest the first time because I hadn’t really learned linear algebra at the time. Now it’s not bad. Especially because I went on to get a lot better with linear algebra and now am fascinated by how the gradient is a linear transformation.
Disk and shell are a bit hard to grasp at first. For me it was infinite series and differential equations. I also took calc 3 before linear algebra, which made understanding things like the Jacobian incomprehensible to me.
The Dish/Washer/Cylindrical Shell Method is only tricky when the Axis of Revolution is a line other than the x-axis or the y-axis (e.g., x=-2, y=5, etc.).
Im math's major student and taking Calculus 2 right now this semester (Aug-Dec). I love Maths and also this Calculus stuff, my math's professors are brilliants. My University requires lab for Math (Except Cal 3 or DE), Physics, Chemistry and Biology. This thursday 14, is my first exam on this Cal 2. Wish me luck.
I agree, parts of calc2 are hard, but my calc3 added in a little laplace transforms at the end and I found that difficult.
The limit theorem was my favorite, once you understood that your golden. My prof even put it on the test. It is central to calc. Not to be confused with the central limit theorem, which is statistics.
The only calculus book available in my country was Thomas' Calculus. I hope that one would be nearly as good as those you showed in the video. I'm currently honing my skills in precalc.
Starting a summer semester for Calc II next week. Definitely very nervous for all the bad rep calc II gets
To be honest, for me calc 3 was the hardest.
I would rate them, from easiest to hardest by: Calc 1 -> Calc 4 -> Calc 2 -> Calc 3.
By the way, Calc 4 for us was Vector Analysis.
I took it in a month long course during the summer a long time ago - got a decent mark. Used Stewart of course. I ended up never taking Calc III because I thought it was going to be impossible. No one ever told me that math would get easier ever again.
Reminds me of my uncle, who was a tenured math professor. In his own undergrad days, the only math course he didn't get an "A" in was Calculus 2. Guess not much has changed since the 70s!
Derivatives, domain, range, and other changes in x and y graph.
The disc/shell, are the rigorous entry route to Stokes and Gauss. Later you generalize Gauss to Green's by making a source point & field point different. Every problem with a nonzero source is an impulse (delta function) in simplest form. As for infinite series, physics folk lop of the O(n^?) terms at 2nd or 3rd.
Reminds me of 1st sem quantum mechanics. You start out with all wave functions living in 1-D (Cartesian flatland, 🤣)...later on, you have to go to 3-D and model hydrogen (& bigger atoms) in spherical coordinates.
I think its because there's a lot to remember that you have to use in creative ways which is unlike Calc 1 in my opinion. Also, if you don't put in the work to solve a good amount of questions then you wont be able to form the pattern recognition needed to bypass having to think creatively, which I don't think many people want to do.
where I come from
Calculus I - Review of analytic geometry. Differentiation of elementary, trigonometric, exponential, and logarithmic functions. Applications of the derivative. Integration. Fundamental Theorem of Calculus.
Calculus II - Inverse trigonometric functions. Techniques of integration. Improper integrals. Applications of the definite integral. Introduction to differential equations.
Calculus III - Sequences and series, convergence tests, and Taylor series. Curves, tangent vectors, and arc length. Applications of partial differentiation. Polar, cylindrical, and spherical coordinates. Multiple integration.
Calculus IV - Vector calculus. Line and surface integrals. The divergence, Green's, and Stokes' theorems. Differential forms.
I did these 4 classes, 1 per term, while I was working as an electronics technician, back in the 80s. I also did 2 linear algebra classes, and differential equations. Someone convinced me to take Boundary Value Problems instead of Partial Differential Equations, and that is where I bombed out, and never got around to taking any more math classes.
cylindrical shells, 3d shape revolution, they made me do that in cal 1 hence, Cal-1 ended up being the hardest cal course for me.
please do all the calculus proofs in a step by step manner!
Why doesn't he ever mention the Simmons book? I started it but put it on hold to review Bostock and Chandler which I studied as a kid. I wonder if Simmons is the one to go back to after (I chose it because it aligns with the OCW courses).
I scraped through Calc II, but to be honest I should have held off and retaken it the year after with a better skill set. I really wasn't ready enough for the final. I'm actually going through the parts of my calc two regularly even though I'm out of college. On my return I want to have this particular subject handled.
You know the author of of the dover book 'Essential calculus with applications' Richard Silverman wrote another calculus book named 'Modern Calculus and Analytic Geometry' (also a dover book). It is said to be proof based and just as rigorous as Spivak, but unlike spivak it covers analytic geometry and more topics (almost 1200 pages)You should check it out!
Also Daniel Vellerman author of 'How to Prove It' wrote a calculus book 'Calculus: A Rigorous First Course' which is rigorous but also readable. Both have solutions and answers to the odd numbers excersises.
Xcellent piece
what books are best for brushing up calc 1/pre calc skills before getting into calc 2? i'm starting calc 2 now and i know my knowledge is lacking and in need of practice
I studied Calc II at uni (long ago) and barely passed, now I use differential eqs everyday and even teach how to model with them.
I really enjoy math and calculus now, but hated the way we were force-fed stuff in Calc II without even having the time to comprehend it. Is that really the best way of organizing a calculus course?
A topics that are hard, Infinite sequence, Infinite series. Numerical Integration. Improper Integrals. Even introduction to Fourier series can be pretty confusing and taxing especially when it comes to piecewise functions. Introduction to 1st order Differential Equations are easy. The funny thing is 2nd order Differential equations can be tougher and more tricky in Calculus 2 than if you're taking it up in course in Linear Algebra and Differential Equations. In fact they've got a nice technique that surprisingly simplifies 2nd order Differential equations by converting them into 1st order Differential Equations then converting it back ti 2nd order to complete the solution. I didn't even know this technique existed until I first took up Linear Algebra and Differential Equations course.
I think that for me Calc 2 was hard because I did Calc 1 online due to covid and so I never learned the material well as we weren't tested nor taught enough . later, in college, I took an exam to skip Calc 1/2 because I alr did the classes. here, I realized that Calc 2 was actually very easy😂. the real problem is with Calc 1 not giving the students enough practice, and Calc 2 having too much material. I've seen at some colleges that Calc 2 is split into continuous and discrete, and Calc 3 is split into multivariable and vector calculus. I think the real issue is the lack of practice and time for the students to cover all the material.
Hey man, can you make a video on the best mobile games/apps to learn math? I’m trying to get more math work in but as a 11th grade student I got a lot of work in my weekly schedule and need to find something effective that takes little time from my schedule.
Ended up with an A in Calc 2, I’m very happy about that
I took Calc 2, 40 years ago and yes, I still have my 2ND EDITION Larson & Hostetler with the solutions book.. 🙂
Sounds like you were integrated to become the intergrand, innergrand.... I mean the answer to the test
I just pulled my copy of Larson from high school off the shelf, and discovered it's the 1979 first edition. Wish I had the solutions book though!
Calc 2 was the hardest math class for me, it felt so fast to me, I never took AP Calc BC so I felt left behind quick but I got a C that semester
My foundations were very weak, I had to retread a lot of algebra to get the integral methods down so that was a massive time sink.
But Calc 2 was the first class I had to LEARN, every class before then I could be SHOWN the problem solve and just remember it.
Calc 3 was a step down in difficulty because you're extending ideas, not learning new ones
Graduate Polish student here. I think that Calc II problems require something beyond symbolic integration, i.e. you might need to know if it's better to approach a volume problem via means of change to polar/cylindrical or spherical coordinates because if you didn't do so you would end up calculating really messy square root integrals. That is, for an effective solution you sometimes have to know upfront. Next, maybe some lack of intuition with respect to integrating along something more than x axis. An integral across some area or volume, although it is more of Calc III as far as I remember. Figuring out region over which an integral is evaluated is not always easy, but it is easy to make some silly mistake and end up with a bizarre result. Plus, when you manage to pass all the exams one after another and your brain has not cooled down, it might be harder for you to do more of the same, but "slightly" different. Maybe Calc III is easier for some, because it is different enough and the brain likes that diversity more, so to speak.
As someone who has taught and tutored all the calc series for about 8 years now, in my opinion, calc 2 can be harder for many students, as you said.
For the students that DO think calculus 2 is harder, it comes down to two things.
1. The problems become more independently of the math. You have to realize how you can apply calculus to the problem to solve it. For example the dx or dy in an integral when doing volumes, or Work or fluid force actually means something. It is more than “the variable of integration.
2. As far as the integration goes, i think calc 2 exposes students weaker algebra/trigonometry skills. Calc 1 does this too, but I believe all the algebra manipulation that is done in calc 2 really requires that you know your algebra rules and trig identities.
@6:13 , that is true. I have taken calculus 2 in 2019 and i barely pass it, because i had full job while i was studying.I decided to take again and got B
It should have been the hardest. I had the same professor for the first and third courses in the sequence. It was a night class. The second instructor took it easy on us. Except for a diversion into real analysts to show us the proof for the normal curve. He was consulting for a hedge fund at the time, so maybe it was relevant to his work and where we were in the course. The first prof seemed disappointed that we were missing knowledge. I wiped out in numerical methods (sat behind my calculus tutor…) and pretty much quit at that point.
Calc 2 is indeed hard! it took me over 500 integrals to solve as practice to ace my exams and the series somewhat between 75-150 series problems of all types to ace the series questions, other than that the rest of the problems revolves around some application of integrals and series. The last chapters, polar coordinates and finding the area were doable but were rushed by the end of semester. So basically as a tip for students who wanna ace calculus is to shoot for mastery, most of the students who did bad in this class, they studied in a way that it is not aiming for mastery( studying to get by, or thinking if I know the formula I could survive). So study for mastery, and you'll ace calc 2 and any math class you're in. Aim to be a master of the subject you're learning. Good luck!
You sir are a national treasure.
The only thing I thought was hard about Calc 2 was infinite series and testing for convergence. Everything else I felt I had a good precalc and calc 1 foundation.
That was almost 30 years ago. Now I teach calculus and physics, and infinite series still feels like the hardest part, becauae I don't use any of it amywhere else - except Taylor series, of course.
Do you know ow what I found out about cal 2 is disk washers and shells method to to pipe volume calculation