We Need To Talk About Calculus 2
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- เผยแพร่เมื่อ 28 ก.ย. 2024
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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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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
"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
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
It was definitely my hardest class. I loved Calc 3 and Diff Equations - but just because I could use the tables to find most integrals. I graduated 20 years ago and have been a professional engineer for the last 10 years. Calculus 2 is very valuable though - it gives you problem solving skills needed for higher level engineering courses. And it helps you gain confidence to do well if you want to pursue professional licenses later on. I have a few textbooks - going to used Book stores is a godsend....I have the Stewart text. And yes, I still like to work problems to keep my problem solving skills up to snuff in general, even though I review plumbing engineering drawings for a living. Mathematics is the Queen of all STEM fields - the language of STEM and thus more invaluable than you will ever know until you get out of college. God Bless you Daniel! And yes - I got a C in Calculus II - it is the time in your life in college that can make things challenging.
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.
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 😁
5:30 man that's so powerful and heart touching! Im sure he'll remember you for all his life
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 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
I think it is hard because you have to be utterly familiar with the basics of calculus. If you really run the basics over and over, you have a firm base to stand on.
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.
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.
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 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.
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.
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.
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 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.
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.
The volume by discs and shells bit was Calc I when I took it. Chapter 5, Thomas and Finney, 5th Edition.
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.
Why Calculus 2 is hard for many people? 1. It is your first real college level math class. For starters, to understand Calculus 2, you need to understand Calculus 1 and Trigonometry/Pre-Calculus (the latter course is an advanced high school math class). If you struggled with either Calculus 1 or Pre-Calculus/Trigonometry, Calculus 2 will expose you. I did well in Honors Pre-Calculus/Trigonometry in high school (11th grade and I got a B+ in both semesters) but I somewhat struggled with Calculus 1 in Freshman year of College (I got a B- in that class). In addition, Calculus 2 requires you to think critically. If you do not have good critical thinking skills, prepare to get exposed by Calculus 2. I remember having trouble telling the difference between convergence vs divergence in improper integrals and sequences and series and needed to seek help from a tutor on campus. 2. The topics are mostly unrelated to each other. For instance, one moment you might be learning intro to differential equations. Another moment you might be learning the calculus of Sequences and Series or the calculus of Parametric Equations or integration by parts or improper integrals. 3. You got to do more than memorization to pass this class. You also have to think critically and apply some skills to pass this class. 4. If you have a hard Calculus 2 professor, you are probably screwed. Luckily for me, my Calculus 2 professor was the same professor I had for Calculus 1. I barely passed Calculus 2 though with a C on my first try. P.S. Also, I found Calculus 3 and Differential Equations to be harder than Calculus 2 because they required knowledge of Calculus 1 and 2 and some Linear Algebra. I found Linear Algebra to be harder than Calculus 1 but slightly easier than Calculus 2. I got a C+ in Linear Algebra the first time. Calculus 3 and Differential Equations, I got a D (both classes) the first try but passed on the second try (C on the second try). The point is Math in university can be challenging compared to high school. High school math was a joke compared to college level math.
please do all the calculus proofs in a step by step manner!
The Calculus can be hard to learn because in most books, there is precious little on how to actually solve the problems.
Very long on heavy duty theory, but very short on fully worked out examples.
Starting a summer semester for Calc II next week. Definitely very nervous for all the bad rep calc II gets
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.
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).
It's not the books. It's the calculus tradition, probably the American calculus tradition, because Europeans do math differently. The tradition includes curriculum content habits, managed and reinforced by schools teaching one calculus series for all students, engineering, math, business, and humanities. The result is that a multitude of topics are pressure-jammed into one course series. The situation is exacerbated by calculus book publishers who edit and revise to optimize their sales across many institutions. Publishers are reluctant to seriously revise books for fear of losing adoptions, and thus profits.
What is needed is a national calculus reform commission, something analogous to a constitutional convention, to completely re-envision the topics in their various environments, and construct subject matter and topic templates for various types and levels of curricula, in effect designing a new set of textbooks for various calculus-taught majors and learners.
Not only are engineering calculus, math calculus, and business calculus different in their needs and goals, but so are their attitudes toward the subject. And rightfully so. An architecture student likely does not need an over-the-head water-bucket dowsing of integration calculation methods for weeks on end, under time and exam pressure, clearly understanding that they are only skimming the top of the problems mountain, when other concerns are closer to the forefront of their learning and professional concerns.
Calculus II is terrible because there is too much stuff crammed into it, and too little time to consider, to learn, and to practice it to a comfortable level of competence. The course is more akin to an initiatory right-of-passage administered by sadistic sociopaths who have self-selected into departments of mathematics. If engineering people need lots of detailed calculus calculation rigor, let them create their own math courses in their own departments. If the humanities and arts people want to understand the beauty of math in the contexts of nature and inspired creations, let them create courses that present such to their students. Unfortunately, the calculus-containing world is much too stuck in a one-size-fits-all curriculum paradigm, and the world needs an organized reconsideration of the situation.
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
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 start Calc II in a week and I’m terrified.. barely got through precalc and calc I
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.
Ended up with an A in Calc 2, I’m very happy about that
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.
@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
I think that the reason students have such difficulty in Calc 2 is that it is all chopped up.
In calc 1 you get the big ideas of derivatives and integrals with the focus on derivatives.
In cal 3, you extend derivatives and integrals to functions of two variables and finally
bring it all together with a chapter on vector calculus and various forms of Stokes Theorem.
There is no underlying theme in calc 2. There is a treasure of mathematics in calc 2 but
it is all in nickels and dimes.
I start Calculus 2 tomorrow! I've taken trig, pre-calc, and calc 1. All A's except for Calc 1, i ended with a B. Hopefully Calc 2 i will end with an A.
My son who is an electrical engineer gave me his Stewart Calculus book for self -studying calculus. Can you tell me where to begin in Cal 2 and where to end in Cal II in the Stewart book? Thanks In awe from Bamer in retirement!
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 larson's calculus textbook is my favorite as well, my school uses thomas' but i find larson's textbook much more understandable
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.
In calculus 1 you build up towards the fundamental theorem of calculus while in calculus 3 you build up towards it's generalizations (stokes theorem/divergence). At the end you put all the pieces together to understand the fundamental theorems.
But while calculus 2 is necessary it doesn't have the same clear goal. For example, you might spend some time doing integration/ differentiation rules for logs, exponents, or hyperbolic trig functions which kinda belongs to calculus 1. On the other hand in calculus 2 you might cover parametric equations which kinda belongs to calculus 3.
The only really calculus 2 content are the series. So I think the problem is that the class is all over the place.
Maybe people should take Calculus as a non degree student before they actually have to take the class for an actual degree. That way, they can see where their weaknesses with Calculus are and either auditing the class or taking it as a non degree student will give you a kind of rehearsal for when you have to take the class I bought my 2nd calculus book 2 nights ago by someone name Marvin Bittinger. I bought it mostly because the problems looked easier BTW: I'm watching the video titled You Can Learn Calculus in 1 video on this math channel. I'm about an hour into it and its a helpful video Its talking about limits at the moment. Limits really don't seem that hard.
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.
Mathing
What's weird is Calc 2 was the best I had ever performed in a Math class.
Derivatives, domain, range, and other changes in x and y graph.
Taking calc II rn with college apps on the horizon, we’re all going to be through this 🙌
I’m reminded of the “rule of 5 p’s”: Prior Preparation Prevents Poor Performance- Calc2 follows Calc1, simply a progression that builds upon prior-learned skills in algebra and trigonometry. If you’re struggling in Calc2, the solution is simple-review your algebra and trigonometry. All good calculus books include an thorough review of these subjects, usually covered in the first chapter. Unfortunately most math instructors skip the first chapter of the book due to time constraints and fail to remind students the importance of reviewing these essential topics if they want to succeed in learning Calculus.
Bottom line: Don’t let ANYONE tell you should expect to struggle in Calc2. With good preparation, you’ll do just fine.
I am reviewing calc 2 with friend who is re-taking it as well so I can stay sharp as a tutor and I remember now what makes it so difficult on my end: all the things you need to both remember and recall to use. There are many multi-step problems that require you to utilise calc 1, trig, geometry, algebra, and hairy arithmetic skills together. How many of us remember off the top of our heads the other way to write cos^2(x) or the antiderivative of sec(x)? What about simpler things like sec(-pi/6)? Forgot how to complete the square because you haven't needed it in a while? There are some integrals that require that. Partial fractions and polynomial long division? You never thought those were coming back, right? Conic sections? Yep. It's all here. Every mathematics topic you were exposed to prior to calc 2 is used somewhere in the course. OK maybe you can get away with not knowing how to multiply matrices or take determinants. ;)
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)
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.
How is it that you mentioned that the disk and washer stuff is harder than all the integration techniques covered in calc 2? While I do agree that the volume section is very tricky it definitely isn’t harder than solving for integrals. Solving integrals requires you to solve them using the 6000 techniques used and for you to use your mathematical creativity to approach them. Lacking this creativity will make you struggle. I think this section is hard as series.
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.
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 😂
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.
Now THAT is a microphone!
It's fun, and truly interesting, but it can be devastating to have to rely on truly understanding sine, cosine, etc, and new concepts such as infinite series either converging or diverging.
Likewise I focused on the proofs to review early calculus 2 concepts after some difficulty. I seriously recommend everyone, and I mean everyone, watch and study proofs. TH-cam is such a good resource in this regard.
It's not hard to pass pre calc, calc, etc, without ever using sine and cosine beyond their basic function (SOHCAHTOA). Learning how there isn't a way to do some integrals WITHOUT trig substitution was a bit frustrating and counter intuitive to me.
It is all the formulas, the techniques, substitutions,etc... the meat and potatoes of applied maths. That is 2. Calc 1 is theory, 3 extension of that theory.
In terms of self-study, it is wise to study two math courses simultaneously?
Should we study one math course at a time?
I respect you Mr teacher truly.
Do you know ow what I found out about cal 2 is disk washers and shells method to to pipe volume calculation
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?
I have calculus book by Briggs and Cochran
Calculus 2 was hard mainly because I was lacking on really understanding Calculus 1. Now I must say Real Analysis 2 is so hard, I think it’s because there are things in Real Analysis 1 I just don’t fully understand.
Great! Amazing! I love this channel!!!! Great channel!
for me calculus 2 was and is the easyast
I didn’t mind calc II. It was tough, but I learned a lot. For me, my living hell is currently prob and stats.
Xcellent piece
I just got an ad on the math sorcerer. Did you get the same thing as me?
I got an army add... looks like I'm going back to hello operator connect me number 9
Another thing to consider is that most college students taking calculus 1 already had the course in high school.
If you could choose only one book, which would it be?
yes, infinite series were my favorite too. innocent and still so playful. freaking blast.
Calculus 2 is hard because they throw differential mathematics very early into calculus 2 while adding more intergral formulas to overload the avarage student mind
Dude, 3ff you for loving calc two lol jk, its like the bootcamp of calc... keep it up bud, loving your content..... they raised us on Stewart but i aint no foreman lol......... our world is four monsters, God speed
I didn’t struggle in calc 2 because my prof wasn’t a lead head, he went out of his way to challenge us significantly, however it was done with measure & purpose, something many inexperienced instructors lack. He show us “why” this works, & why other operations don’t in different applications, always & forever, the teacher makes the difference. Hands down.
I sorta disagree. I’m in calc 3 right now and the concepts are harder to grasp and visualize.
Is calc 2 the same as the second half of AP Calc BC
Actually Calculus 2 is the equivalent of AP Calculus BC.
I think what makes calc 2 difficult is that sequences and series are taught at the end of the term and they require, at least to me, the most mental energy to grasp. In addition, I think that they are often very poorly motivated. Taylor series, for example, are an incredible tool with loads of practical applications, in numerical methods for example, but I think too often students don’t understand why they’re learning the topic. This is a double dose of badness.
bruh it wasn't even the integrals that fucked me up on my calc 2 test. It was series. I bombed that part. I stil don't get it.
For me, calc 2 was a very boring course, despite me being well in the tests. Basically the teacher went into every topic very very quickly just to cover the scheduled curriculum, not much theory explanations was covered. I felt it was like just in the purpose of memorize some tools to solve some exercises robotically, not really going into the beauty of math.
On the other hand, real analysis was a very tough course in my opinion, but it was very fun at the same time.
The course title is normally a give away in the US: calc is more plug and chug; analysis is proof based and rigour is expected. But your prof can choose to teach their own way for either course and the mileage does vary as a result.
dude sniffed the textbook LMAO
Is there a pure axiomatic calculus? You only have the axioms and a few rules, nothing else. You even don't know what a limit is.
Hey man. just wondering, are you familiar with the book “calculus a full course” by A Adams. If you could you maybe make a video comparing it to James Steward? Oh and nice gains btw.
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.
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.
@@msfttempur I can't lie, you probably have a bad professor if you didn't know which method to actually use for certain problems. I thought the class was bad just because of how many different methods there were and our final was cumulative. Got the A and I'll never have to memorize the taylor series for a function again. :D
I kinda love that its like real world stuff
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.
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
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
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…
Calc 3 and Linear were terrible for me. Calc 2 was great
I agree for some reason I had a harder time with Calc 3 because of the extra dimension where as Calc 2 was mostly about 2D problems. But from what ai remember, a lot of people in my class failed the class because the they failed the final which was mostly on series.
To be fair to those who struggle more with Calc 2, there is a lot to cover in that class and seeing series for the first time can be a challenge. It was a challenge for me too but I liked it and for me that was my favourite part in that class.
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!!!
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
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
@@magicaldragon4673 They might have meant that many integrals they had struggled with that required techniques like partial fraction decomposition or tricky u-subs were made easier by the alternative option of trig subs. I do feel like trig subs are overkill for some problems though.
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
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
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.
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.
Calculus 2 is realy hard but Calculus 1 is easy 💜
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 hope you passed the class. Life can be challenging and I hope you are in a better place now.