Concepts are more abstract than subjects before it, but everyone mind is different, what maybe easy to you may be harder to others, how you like calculus, I’m learning limits and continuity
@@BabyMessi_ That's like the end of Calculus II at most places (BC Calculus if you're in American High School). Calculus I ends a little bit after basic integration: it's actually really simple. Calculus II is when you get into the torturous stuff, like finding the volume of areas of rotation, Calculus with polar functions, and limits of summations. (and Calculus III (Multivariable) is when you question life, before finally having a bit of respite in Calculus IV (Differential)) At least, that's how it is at most places: some institutions may do things differently.
@@244295132171 Take Calculus, and you'll get what he's saying. Calculus concepts are extremely observable. Seeing the relationship between speed and velocity is so much easier than most Algebra concepts. This is all Calculus revolves around: using the slope of equations to figure stuff out. It's legit just finding slope, but on a larger scale with more moving parts.
Math tutor here, in my experience, it's not calculus that trips up students, the majority find the new concepts to be easy to understand. It's when calculus starts applying alg I + II, geometry, and trig that most students struggle with the course due to not remembering the material from previous years.
I’ve been thinking this the whole time I’ve been in college math but you put it a lot more eloquently! It’s really embarrassing to have to ask my Calc 2 professor how he factored something or about exponent rules, but then he’ll explain it and half the class goes “ohhhh yeahhh” 😂 Or like the other day when I was working on Taylor series and had to google how to add fractions.
People claim that math is hard because of perceiving it as arbitrary and that hurts retention. Until late pre-calc/early calc, I swear math curriculum goes out of its way to avoid explaining the interconnectedness of math subdisciplines. A pre-algebra student learning about slopes and exponents should be taught basic derivation because it is a tool that is not hard to use or understand that can dramatically increase their ability to reason through associated problems.
Answers to both problems here: Top problem: step 1. let u = √5x-2, making the problem way more simple (10 + 3u)/u = 3 + √10 step 2. multiply both sides by u: 10 + 3u = u(3 + √10) step 3. subtract 3u from both sides: 10 = u(√10) (the 3 on the right side of the equation in step 2 was technically 3u, thus allowing us to remove the 3 altogether) step 4. divide both sides by √10: √10 = u step 5. square both sides: 10 = u² step 6. replace u with the substitution made in step 1: 10 = 5x - 2 step 7. now this is just a simple algebra problem, in which x = 12/5 bottom problem: first off, integrating g(x) from 0 to y gives us the function (y⁹/9 - 126y). the entire function itself doesn't really matter in this case, the only important takeaway is that the leading degree of y is 9. since the nth derivative of a function lowers the degree of the leading exponent by n, the ninth derivative of this new function would be a constant (with x having a degree of zero). this means that any subsequent derivative has to equal 0, since the derivative of a constant is always zero. hence, the 10th derivative with respect to y is 0. *edited my answer to the bottom problem because i completely forgot about the integral. in this case, forgetting to integrate resulted in the same answer anyways so it was a lucky mistake
Ohh is THAT what the bottom problem was asking? Is that a calc 1 problem? I was so confused on the notation cuz I haven’t done calc in years (and I wasn’t very good at it)
Hey, i love your comment, but, i don't get the fourth step of the first problem. How does 10/sqrt(10) = sqrt(10) I kinda feel this is a dumb question 😅
@@formiga4106 its not a dumb question haha, but understanding the answer requires you to think of what a square root actually means. (√10)² = 10, because squaring a square root just cancels the square root. this means that (√10)(√10) = 10, or √10 = 10/√10 (dividing both sides by √10)
The study of pre-calculus involves the application of various mathematical formulas. Calculus, on the other hand, focuses on determining the slope of a curve and calculating the area under a curve.
Difficult = tedious when you’re a burnt out but intelligent high school senior. Doing busy work and long hard stupid questions for five hours every night is more difficult on your body and energy than putting extra thought into grasping more complex ideas. English 4 AP is probably easier for a lot of people than some random creative writing class because maybe the creative writing class is tedious and takes up too much time. As a chronically ill always tired person, anything tedious is super hard for me
For anyone curious about the top problem, the solution is 12/5: In[1]:= Solve[(10 + 3 Sqrt[5 x - 2])/Sqrt[5 x - 2] == 3 + Sqrt[10], x] Out[1]= {{x -> 12/5}}
And the bottom solution is 0: f(y)= y⁹/9 - 126y So derivating: f’(y) = y⁸ - 126 (one can actually start from here, if the antiderivative definition is used), f”(y) = 8y⁷ f”’(y) = (8•7)y⁶ So, the ninth and tenth derivatives are, respectively: f⁹(y) = (8•7•…•1)y⁰ = 8! (a constant) f¹⁰(y) = 0
@@EngagingOverImmersive Different learning speeds. Some intelligent students in Math might be poor in English, or Arts, Drama, Music, so we all have to support each others to get well-earned results!
I explained it to my calculus-phobic son like this: how do you figure out the area of a circle? Well, you start with a sqare inscribed in the circle, with corners touchin. Obsiously, the square is smaller. But now increase the number of sides: 6,12,100 ... The more sides, the closer you get. Next, you're filming a car accelerating on a drag strip, and you want to know how fast it's going at 1/8 mile. So you find the frame taken at the point. But you can't figure out the speed from one frame, so you take the next frame and measure how much it moved. That gets you the average speed between two frames. To get a better estimate, increase the frame rate. Keep increasing it until the change is too small to measure. In calculus language, the instananeous speed at any point is the limit of the difference in space (ds) with respect to the difference in time (dt) as dt approache zero. Similarly, the area of the circle is the limit of the sum (long s) of the areas of the incribed polygons, as base of their tangent sides approaches zero. Once you know the area, you can derive pi, but you can never know the area preciely, because pi has an infinte number of decimal digits. I learned calculus from Apostol's text, who introduces integral calculus before diffential calculus and was finicky about Newton's concept of an infinitesimal. My physics prof need us to know differential calculus from the get-go, so he explained as Newton did, defining an infinitesimal as "a number so small that you don't give a damn." An infinitesimal can be defined as the difference between 1 and 0.9999... You can teach math as abtract, dry and tedious, or with colour, getting to the abstraction from concrete examples. You could say that understanding of abstract concepts is the limit of the difference between examples and abstractions as the student's confusion approaches zero.
I disagree. Calculus 1 is far easier than pre-calculus. I mean, the rules for basic derivatives are a lot easier to remember than the quadratic formula as is. You could easily teach Calculus 1 before college algebra. It won't make as much sense, but people would still ace it far more often than college algebra. I still remember the first thing I was taught in calculus 1: "A line without a break in it is called a continuous line. If it has a break in it, it's called a non-continuous line." The professor even showed drawings and went into detail. Calculus 2, however, is a very cruel reminder for why some people have to choose between STEM subjects and good mental health.
Calculus is easier than pre-calculus however. This is because not only are derivatives and integrals easier to grasp fundamentally than things like polar equations, vectors, matrices and bearings; pre-calculus has you jumping from topic to topic often with no relation to one another whereas calculus is the building of one topic over another so you get better at both the last topic and the topic you're on right now. Also, I'd rather do the bottom problem since it's just power rule repeatedly, and since derivative of a constant is 0, answer is 0.
I think a way to visualize this is a ladder that doesn't have equally distanced steps going from algebra to precalc is a harder jump than going from precalc to calc so people will view calc as easier than precalc but for someone who only knows algebra they're equally hard
The hardest lower division math course I took was Calc 2, calc 3 was easy compared to calc 2 because it is basically calc 2 with nested variables, partial differentials, and vectors. After calc 2 every math course was fairly easy for me until my first semester of abstract algebra, and the last semester of real analysis.
I’m in calculus right now, and can’t understand derivatives to save the life of me. And I’m sure that’s a problem, because I can only imagine the derivatives will only get harder from here. Does anyone have any helpful tips or tricks that helped them pass their calculus class? Any help would be greatly appreciated
Oh man I struggled pretty badly back in AP Calc AB. I ended up using a free trial of Aleks to nearly perfect my math skills from about algebra up to pre calc. Perfecting those skills really helped me in that class. And then practice. Oh man please practice. Literally anything. Me and a friend would draw out random functions on a white board that we would make up on the spot and try to figure things was an integral possible? Critical points? Derivative? This is great and all but if this is Calc 1 in college you may not have much time. Something that helped me in college classes and college calc classes was like pre studying before a class/lecture. So if you see on the syllabus that you have chapter 3.4 Chain Rule on the agenda for the day then you can do a Google search of "Chain Rule Calculus 1" and try to either find summaries of it online, or short videos going over it. There's been a few times that I find a 50 minute lecture and I watch maybe the first 10-20 minutes of it. This is usually enough to prime myself for when class comes. Even now though we have things like chatgpt so you could always say smth like "Hey I'm a learning calc 1 student and I want an introduction in XYZ topic" so that's an idea XD also making friends in your class or in higher level classes if you can find some. That honestly made studying seem less stressful. Anywho I hope this can help you a little bit, enjoy the rest of your day!
I’m doing the Australia curriculum, and I’m only year 11, but something that helped me is relating the derivative of distance to speed. The rate of change of distance is the speed you are moving at, so that’s the derivative
@@jadenroach4832 i'm sorry if my comment sound condescending but y'all are overcomplicating a very simple concept ... all a derivative is that it is a FUNCTION that represents instantaneous change in one variable with respect to another variable .... if u plug in values of say the x coordinate into this function then u will get the rate of change of y with respect to x at the point on the graph corresponding to the x coordinate u plugged in just now ... this rate of change is geometrically the slope of the tangent at THAT particular point of the curve
anyone wondering, the solution to the bottom problem is 0, the problem is requesting for de 10th derivative of the f(y) function, and said function is of degree 9 at most, so, by the 9th derivative we will have a constant, and so the 10th will be 0
How I always measure difficulty: How long did it take me to actually understand the concept? How does it effect my stress levels when I'm doing it? How much work does it take after I know how to do it regarding number 2? So in other words calc 1 WAS easier for me because the precalculus class I took only went with what the book said. The teacher also had us practice 50 precalculus questions every night which just stressed the ever loving god out of me because I'd be like "Ah ok I get it" then get through like 5 questions in just to look at the rest of them to see 90% the same question with different numbers. Remember this was every day and also a terrible way to teach a topic. Reason why it is generally a terrible way to teach: If you work on one chapter only once the entire year then you may forget how to do the math, not every chapter teaches the same math. So by the time I'm on Chapter F, I have already potentially forgotten something from Chapter A but Chapter F also doesn't go over that topic in Chapter F-l and comes back up at Chapter M. Chapter M becomes hell. I watched half the class cheat durring the tests.
Uptil single variable calculus if you have done pretty good algebra and coordinate geometry and trigonometry then it might feel pretty easy and you will be able to explore the conceptul part of calculus more like solving tough problems but when it becomes multivariable shit starts to become tough af
I am much beyond both precalc and calculus. I do still believe Calculus is easier, much easier in fact. Calculus 1 concepts _are_ very easy, and honestly more digestible than precalc. A lot of precalc work implicitly assumes calculus in a weird way, in that calculus is later used to post-justify what you were taught in pre calc. As well as that, the step up is definitely smaller. Transitioning from pre calc to calculus is a much smaller step than from basic algebra to pre calculus.
calc 2 and calc 3 are slightly harder but theyre both easy in their own ways. its when you get to real analysis and mathematical reasoning (and depends on the prof) is where shit gets hard if not already in linear algebra and diff eq
When you talk about pre-algebra, precalc and calc 1,2 and 3, it seems like you are referring to well established concepts that should be understood by anyone with a background in science. But the words themselves looks like course names. Are math courses nationally defined in your nick of the universe?
The top problem took me 9 minutes because i forgot how to make the denominators equal and spent 8 minutes to figure that out. And the calculus problem took me 5 seconds because it is easy
I'm going to show it to my students in high school math honors. In order to get to calculus, they have to solve a lot of tedious stuff. There's no way around it. I just started teaching but I remember all this easier concepts at the tip of my fingers, decades later. Practice is all they need but it is so hard to convince them that one time when I was there in the seat, I had to do it 500 times before I dig into Riemann integrals 😂
Yup Calculus 2 was this for me. I love calculus 1, 3 ODE, PDE. The only class in the calculus family that I don’t like is calculus 2 or integral calculus.
Precalculus was not just tedious but also difficult to understand without the context given by calculus. Calculus would be easy to understand even without the context given by precalculus, because the concepts of integration and derivation are so easily understood, and they largely follow simple rules.
often times people are decribing the courses. Not the concepts. Precalc was harder for me mainly cause of other circumstances, but also the tests and such have more opportunities for mistakes that aren't not knowing the concepts.
The hardest thing about the calculus series (1/2/3/Diff Eq) is that it all builds off eachother, if you struggle to establish a good understanding of something it will then carry over to other sections, not to mention if your algebra and trig skills aren't that good you will struggle a lot like I did when I took calc 1 the first time.
There is easy vs hard, and there is simple vs complex. “Simple and complex” are descriptions of the nature of a problem and how many steps are required to complete it. “Easy and hard” describes our subjective experience in solving the problem. A simple problem may be hard to solve and a complex problem might be easy-depending on one’s abilities. I think that “tedious” describes the latter-a complex yet easily solvable problem.
For me, differential geometry is easier than whatever kind of euclidean geometry they're teaching at my highschool. Also, the problems are a lot more varied and more satisfying to figure out as you progress further into advanced maths.
Sorry, but I still strongly believe that, at least for me, Calculus 3 was easier than Calculus 2. Calculus 3 was mostly just stuff that is talked about in Calc1&2 but WRT 3D surfaces or even higher-dimensional surfaces. However, while it sounds complicated based on that description, working the problems out is actually very straightforward. At least for my class, Calc 2 was FAR more dense with content than Calc 3, and most of the mindnumbing integrals you might have to face in Calc 2 were nowhere to be found in Calc 3... ie the integrals in Calc 3 were just easier than those of Calc 2, because the focus is no longer on the integration strategies.
thats just because less people know calculus though. if we taught everyone how to solve algebraic problems like this and integral problems like this in middleschool, everyone would choose to solve the much Easier calculus problem.
Also like once you've done pre Calc you have a base level understanding making higher levels seem easier than doing pre Calc with no base understanding
I would counter that the jump from Algebra into Precalculus is a bigger one than from Precalculus to Calculus. Its the jump in difficulty that matters.
I will die on the hill that the amount of information and difficulty of adapting to new concepts in calc 2 is easier than applying those concepts in calc 3.
It's also about the difficulty curve. Going from algebra to pre-calc can often be very daunting and challenging, but going from pre-calc to full calculus, even if calculus is harder, might not be that big of a leap.
To solve the f10(y) problem, you integrate g(x) with respect to x first. You should get [x9/9 -126x] with boundaries y and 0 which equals y9/9 - 126y. So, f(y)= y9/9 -126y Next, the highest power of the function is 9. So, the ninth derivative is a constant, which means that the tenth derivative is 0. So, f10(y)=0 and the problem is solved
I find calculus easy and I say it's easy but that's only because I learnt it from my own research and the amazing guidance of 3b1b and other teachers. I'm definetly expecting some struggle when I actually get into a calculus class.
Some classes are also just harder at a higher or lower level. Sometimes you have courses that just have more difficult exams and homework. Calc 3 at my university was known for being harder than some significantly more advanced courses just for it's homework load.
The other consideration here is the difficulty jumps between levels of math. For many people, going from high school algebra/geometry to the trigonometry, college algebra, complex numbers, polar coordinates, etc. found in precalculus is a greater increase in difficulty than the jump from precalculus to the derivative rules and simple integration in calculus I. The material is still harder, and the concepts from precalc were necessary to understand the new material, but you might find yourself struggling less with Calc I, for this reason.
As a person who struggled with math, I could solve the bottom problem in my head, I would need to write the top problem down and do all the steps. Calculus, especially when it comes to polynomials is easier to learn than algebra. I think a large part of that is how math is taught, basic algebra could be taught with arithmetic. x+3=4 is simple enough that one who knows how to add and subtract can solve it if shown, same for x-4=1, 3+x=5, etc. It is just moving around numbers. If basic algebra was integrated into teaching arithmetic, by the time students got to algebra they would have an understanding of the basics and the harder stuff would be easier to learn.
To solve second task it's enough to understand, that we got max у⁹ in f(у) and therefore answer is zero. You don't have to be good at it to understand the gist of the problem in ten seconds. On the other hand, if we were to take the ninth derivative, we would have to tinker with it. It's just a matter of the task, it won't be difficult to make a task from the topic of the first task that can be solved in ten seconds and looks difficult at first glance
For anyone wondering, for the first problem, x =2.4 and in the second problem, the answer is 0. What I find funny is the fact that both problems are overcomplications of very simple things. The first problem is just 5x-2=10 while the second problem is zero.
Well, as someone that went into full youtube precalculus, calculus 1, 2 and 3. Here it is my PERSONAL opinion: precalculus was hell because I never had any idea on those things at the beginning, calculus 1 was acceptable, calculus 2 was HELL, calculus 3 was the easiest with triple derivatives and triple integrals, it seemed the easiest . still my personal opinion.
For the record, I say this, but what I mean when I do is something different than tedious vs hard. What I always say is, assuming that you understand the prerequisite concepts to the same degree, calculus is easier to learn than precalculus. Basically, I think it’s much harder to learn precalculus when you know only the required algebra and geometry stuff than it is to learn calculus when you know precalculus.
So now, correct me if im wrong, havent done any sort of calculus in a year or so, but cant you do f prime, get rid of the integral so f'(y)=g(x) plug in g(x) then take the derivative 9 more times? I know the answer is 0, but cantyou do that?
I just call it difficult or complicated instead of teedious or difficult. It's simple to open a heavy door but it's difficult while it's easy to solve a math problem but more complicated
I have a mentality that if I can't solve a problem instantly, it's too difficult for me. So I prefer difficult but straightforward problems to simple and tedious ones.
As for me personally, calculating math problems fills my head with negative emotions. I don't become emotionless. They are cold, hard, negative emotions that fill my head. I feel misery and I become enraged. Honestly, it has nothing to do with doing logic work, because I like doing logic work and I like juggling things in my head. I like puzzle games and I like learning concepts. It's more of tediousness that bothers me. Solving calculus is something you have to do while listening to music and dancing. You have to do it while occupying other areas of the brain. It's like how when I play a certain game so much (like fortnite especially), it becomes tedious in certain parts of the game, so I have to listen to a podcast while playing the game. Also, I don't think it's really only about tediousness. I am a very simple man. I girls. That's it. And when I am not thinking, I am preoccupied with a problem I am trying to solve whatever it may be. Sometimes, even just learning math concepts feels so painful unless I see it necessary to help me solve a grander problem that I take with personal important. It is like reading books like pride and prejudice. It's obviously not that the book cannot preoccupy me. It's that I don't care about that book. I really don't. It doesn't help me solve any of my problems. Obviously, for motivated students, that purpose is impressing their parents and acing the test right? So, if you find the school to be important, then you're gonna do well. If not, then no. One can certainly have that kind of mindset. I personally am a very competitive person because of my testosterone. It's no use if the school does not reward competitiveness or show how competitive you're. I'm going to tell you right now. There is no intelligent kid who will find that subject interesting because it stimulates intelligence because they don't know jack shit about what that subject even does. If any kid does, that kid is a fucking genius, and they are really fucking rare. They are either genius, or their parents are hyper fucking educated. That's probably why. Other than that, no kid in his right fucking mind will find school interesting. Learning math has degrees of depth. There is a point where you understand math just enough to know what it is basically about. Calculus is a study of change. Instantaneous change, accumulation of change. There is that, and there is the depth where you solve many different types of problems. It's like learning a language in a sense, but how do you find it interesting to learn language for sake of learning it? You don't know jack shit about what the problems are going to even do or used for (I mean there are some poorly made applied word problems using these concepts, but you still don't know jack shit about if they are still going to be used in the future, or you're learning about some archaic knowledge). Like I said above. Kids study hard because people told them they are important. It's never because of some kind of personal passion or some genius-like inspiration or curiosity. And in America, where testosterone is suppressed, I can't find motivation to do jack shit. The American education system does not breed true leaders. It rewards subservient cowards or privileged snobs. We must look toward the Korean education system. What about the army? Do you the army recruits highly competitive people? No. The West Point (although it is easier to get in than it is to get out) still looks for educated kids. And the army does not want high testosterone males. High testosterone males cause trouble. They want low testosterone males who only do what they are told and can wag their tail, wallowing in their own shit all happy, while being treated like shit. Who have more fear responses in their habits than competitive and brave responses. That's why they look for test scores and the education levels.
Difficulty is the amount of work put in, tediousness is basically just that. How advanced something is doesn't mean it's harder. Those students are right, brother
i think they are speaking in terms of relativity. For example, where i am from, gr12 calc and calc I in uni cover almost the exact same things, the difference being that uni calc obviously goes more in depth and it covers integral calc while high school does not. objectively speaking, calc I in university is harder, as it covers more topics in a shorter amount of time; but if you have already taken gr12 calc then you have already seen the majority of the content. so, relative to gr12 calc when you had to learn all of it for the first time, uni calc I is much easier because, for the most part, you already know/have a basis for what is being taught and may only need to remember a proof for something that you weren't taught in gr12. I think that is what people mean when they say "x is easier than y"
My guy... I remember EXTREME tediousness in one of my differential equations classes where we were utilizing massive series and transformations. Difficult AND tedious. Same with analysis (unless you got that intuition baby).
Algebra is about a lot of tricks to solve weird factorizations. You forget the tricks, you forget the class. Calculus is about concepts, only thing you gotta memorize is a table of derivatives and you are basically done. Difficulty is subjective, but complexity is absolute. Being good at arithmetic doesn't help much with algebra, but being good ar algebra helps a lot in calculus.
Again though, you confuse difficulty of problems with difficulty of course overall. With Pre-Calculus what makes it hard is that you switch continuously from very unrelated topics (like going from polar equations to vectors) which are many things that are completely unique whereas in calculus every topic builds on each other, not only making the next unit easier to understand, but further building on the previous units. Which is why Calculus is easier than Pre-Calculus
Calculus is actually easier though, not just faster. The concepts are easy to understand and remember and the operations are simple. It doesn't get hard again until differential equations
No, the "higher" math is often easier than the "lower" math in the sense that the "higher" math might be a shorter step. For example, getting your mind around the concept of an "unknown" might be much more difficult for some than, say, the idea of infinite sums of infinitesimals (i.e. integral calculus). Thus, it would be perfectly correct in this case to say that calculus is "easier than" algebra, because the core concepts of calculus are easier for an incoming calculus student to grasp than are the core concepts of algebra for an incoming algebra student.
nah calculus 1/2 problems are way easier to understand it's just rate of change and area and applications + series which is more applications. precalc has a lot more jargon you have to go through which makes final tests harder because there's more to study for calculus if i just understand the main concepts i can ace any test. but every higher math class relies on the base you set up for yourself prior so if you didnt really understand precalc and are in calc then calculus will seem harder than it really is.
yeah it's basically never going to be objectively easier because it depends on you knowing the skills from pre-calc, but the right language to use would be "i had an easier time" and like, yeah, for me, the hardest class was calc 1, second hardest was calc 3, third was precalc/trig. calc 2 wasn't nearly as hard for me as any of those, it clicked with me the same way algebra and geometry clicked with me. idk but measuring the area under a curve is a lot more intuitive to me than imagining the rate of change, at the time those concepts were introduced to me, but by the time i was learning the former, i had already learned the latter. differential equations was also an easier time than calc 1 for me, but it heavily depended upon the knowledge i got from precalc, trig, calc 1, calc 2, and linear algebra. upper division math courses were also way easier, but that's assuming that you already know the other stuff, because most of it is like figuring out how to take some complex problem and put it in matlab
so "easiness" might not be the right language to use, but like how much i personally struggled with each topic, calculus 1 was the first time in my life when i genuinely hit a wall, couldn't just absorb things like a sponge anymore, and had to start doing homework and studying and shit if i wanted to pass, and i was so naturally good at learning the elementary stuff that my study habits were absolute trash
If you pretend like integrals don't exist, for me precalc was harder than calculus. However, and I have no idea why, I just absolutely suck at integrals, and no matter how much practice I get with them I still need help choosing which strategy to use. The strategies themselves are easy for the most part (integration by parts 3 times in a row is everyone's favorite thing) but I never know how to set them up or which ones to use
well this isn't really a question about difficulty, because math isn't exactly a difficult subject per say. generally, calculus needs some level of pre-calc to work with, while pre-calc (obviously) doesn't need calculus. that means that more people can do pre-calc problems than people can calculus, but it's more a lack of knowledge rather than a skill requirement or anything of the sort. as a subject, calculus is generally easier to learn than pre-calc, though.
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Calculus concepts are far more down to earth. Understanding anything about derivatives was easier that idk, polar functions.
Concepts are more abstract than subjects before it, but everyone mind is different, what maybe easy to you may be harder to others, how you like calculus, I’m learning limits and continuity
You literally have to find derivatives of polar functions . It is hell
@@BabyMessi_ That's like the end of Calculus II at most places (BC Calculus if you're in American High School). Calculus I ends a little bit after basic integration: it's actually really simple. Calculus II is when you get into the torturous stuff, like finding the volume of areas of rotation, Calculus with polar functions, and limits of summations. (and Calculus III (Multivariable) is when you question life, before finally having a bit of respite in Calculus IV (Differential))
At least, that's how it is at most places: some institutions may do things differently.
Bro confused between Analysis and Calculus. 😂
@@244295132171 Take Calculus, and you'll get what he's saying. Calculus concepts are extremely observable. Seeing the relationship between speed and velocity is so much easier than most Algebra concepts.
This is all Calculus revolves around: using the slope of equations to figure stuff out. It's legit just finding slope, but on a larger scale with more moving parts.
Math tutor here, in my experience, it's not calculus that trips up students, the majority find the new concepts to be easy to understand. It's when calculus starts applying alg I + II, geometry, and trig that most students struggle with the course due to not remembering the material from previous years.
Yes.
I mostly blame this on how schooling is overall structured: too much emphasis being placed upon memorisation and regurgitation on tests.
This is my experience, too, both as a tutor and as a grad student teaching classes.
Bingo
I’ve been thinking this the whole time I’ve been in college math but you put it a lot more eloquently! It’s really embarrassing to have to ask my Calc 2 professor how he factored something or about exponent rules, but then he’ll explain it and half the class goes “ohhhh yeahhh” 😂 Or like the other day when I was working on Taylor series and had to google how to add fractions.
I still think calculus was easier to absorb and retain than precalculus
Calculus concepts makes sense
Pre calculus is a mix of everything
I did so well in pre calc
Calc killed me
@@AaronLacasse-z2f what does calc stands for?
@@lefishe7431 calculus
Calculus
For anyone curious about the bottom problem, the solution is 0
what does f^10 notation mean? the 10th derivative? or f(f(f(…))) with 10 f’s?
@@HanzCastroyearsago 10th derivative
It took me 11 seconds
I guess I'm not so good in calculus(
Lmaooo it’s not 0
You clearly have no idea what you're talking about if you can't figure out the answer for this question @@BenMartin-f5v
People claim that math is hard because of perceiving it as arbitrary and that hurts retention. Until late pre-calc/early calc, I swear math curriculum goes out of its way to avoid explaining the interconnectedness of math subdisciplines.
A pre-algebra student learning about slopes and exponents should be taught basic derivation because it is a tool that is not hard to use or understand that can dramatically increase their ability to reason through associated problems.
Answers to both problems here:
Top problem:
step 1. let u = √5x-2, making the problem way more simple
(10 + 3u)/u = 3 + √10
step 2. multiply both sides by u: 10 + 3u = u(3 + √10)
step 3. subtract 3u from both sides: 10 = u(√10)
(the 3 on the right side of the equation in step 2 was technically 3u, thus allowing us to remove the 3 altogether)
step 4. divide both sides by √10: √10 = u
step 5. square both sides: 10 = u²
step 6. replace u with the substitution made in step 1: 10 = 5x - 2
step 7. now this is just a simple algebra problem, in which x = 12/5
bottom problem:
first off, integrating g(x) from 0 to y gives us the function (y⁹/9 - 126y). the entire function itself doesn't really matter in this case, the only important takeaway is that the leading degree of y is 9.
since the nth derivative of a function lowers the degree of the leading exponent by n, the ninth derivative of this new function would be a constant (with x having a degree of zero).
this means that any subsequent derivative has to equal 0, since the derivative of a constant is always zero. hence, the 10th derivative with respect to y is 0.
*edited my answer to the bottom problem because i completely forgot about the integral. in this case, forgetting to integrate resulted in the same answer anyways so it was a lucky mistake
The 9th derivative is constant. You need to integrate then derive
Ohh is THAT what the bottom problem was asking? Is that a calc 1 problem? I was so confused on the notation cuz I haven’t done calc in years (and I wasn’t very good at it)
Hey, i love your comment, but, i don't get the fourth step of the first problem.
How does 10/sqrt(10) = sqrt(10)
I kinda feel this is a dumb question 😅
@@formiga4106 its not a dumb question haha, but understanding the answer requires you to think of what a square root actually means. (√10)² = 10, because squaring a square root just cancels the square root. this means that (√10)(√10) = 10, or √10 = 10/√10 (dividing both sides by √10)
@@ace_nano thanks lol just fixed it 👍
If only all calc problems were 10th derivative of a power 8...
That's why I'm bad at arithmetic but did well on set theory and calculus
Arithmetics can be very very tedious
The study of pre-calculus involves the application of various mathematical formulas. Calculus, on the other hand, focuses on determining the slope of a curve and calculating the area under a curve.
Difficult = tedious when you’re a burnt out but intelligent high school senior. Doing busy work and long hard stupid questions for five hours every night is more difficult on your body and energy than putting extra thought into grasping more complex ideas. English 4 AP is probably easier for a lot of people than some random creative writing class because maybe the creative writing class is tedious and takes up too much time. As a chronically ill always tired person, anything tedious is super hard for me
Bro I am more convinced now that calc is easier than precalc I was stumbling for a sec on the precalc problem
The difficult part of calculus is all of the algebra and trig you need to sort out before you get to the calc step.
For anyone curious about the top problem, the solution is 12/5:
In[1]:= Solve[(10 + 3 Sqrt[5 x - 2])/Sqrt[5 x - 2] == 3 + Sqrt[10], x]
Out[1]= {{x -> 12/5}}
And the bottom solution is 0:
f(y)= y⁹/9 - 126y
So derivating:
f’(y) = y⁸ - 126
(one can actually start from here, if the antiderivative definition is used),
f”(y) = 8y⁷
f”’(y) = (8•7)y⁶
So, the ninth and tenth derivatives are, respectively:
f⁹(y) = (8•7•…•1)y⁰ = 8! (a constant)
f¹⁰(y) = 0
I got 2/5
12/5*
I've noticed that honors math classes are less tedious than regular math classes of the same grade and subject
The kids in the honors class typically need to do less to get the same understanding
That might also be because you can converse the concept of math than focus on the computation
yeah because the teacher doesnt have to baby the students as much as the regular classes
Huh, kinda wish we had those
@@EngagingOverImmersive Different learning speeds. Some intelligent students in Math might be poor in English, or Arts, Drama, Music, so we all have to support each others to get well-earned results!
I explained it to my calculus-phobic son like this: how do you figure out the area of a circle? Well, you start with a sqare inscribed in the circle, with corners touchin. Obsiously, the square is smaller. But now increase the number of sides: 6,12,100 ... The more sides, the closer you get. Next, you're filming a car accelerating on a drag strip, and you want to know how fast it's going at 1/8 mile. So you find the frame taken at the point. But you can't figure out the speed from one frame, so you take the next frame and measure how much it moved. That gets you the average speed between two frames. To get a better estimate, increase the frame rate. Keep increasing it until the change is too small to measure. In calculus language, the instananeous speed at any point is the limit of the difference in space (ds) with respect to the difference in time (dt) as dt approache zero. Similarly, the area of the circle is the limit of the sum (long s) of the areas of the incribed polygons, as base of their tangent sides approaches zero. Once you know the area, you can derive pi, but you can never know the area preciely, because pi has an infinte number of decimal digits. I learned calculus from Apostol's text, who introduces integral calculus before diffential calculus and was finicky about Newton's concept of an infinitesimal. My physics prof need us to know differential calculus from the get-go, so he explained as Newton did, defining an infinitesimal as "a number so small that you don't give a damn." An infinitesimal can be defined as the difference between 1 and 0.9999... You can teach math as abtract, dry and tedious, or with colour, getting to the abstraction from concrete examples. You could say that understanding of abstract concepts is the limit of the difference between examples and abstractions as the student's confusion approaches zero.
(10+3√(5x-2))/√(5x-2)=10/√(5x-2)+3=3+√(10)⇒10/√(5x-2)=√(10)=10/√(10)⇒5x-2=10⇒x=12/5
No one asked you to find the value of X😭🙏🙏
@@ilikespaceengine Bro it literally says "Solve For x" 😭🙏
@@ilikespaceenginestupid ahh
@@bacchadumII NAAWWWW
i definitely think calculus 2 is the most humbling math course most stem students will have to take
Learning calc is way easier than learning precalc for me for some reason. Probably because i have a very good calc proffesor
I disagree. Calculus 1 is far easier than pre-calculus. I mean, the rules for basic derivatives are a lot easier to remember than the quadratic formula as is. You could easily teach Calculus 1 before college algebra. It won't make as much sense, but people would still ace it far more often than college algebra.
I still remember the first thing I was taught in calculus 1: "A line without a break in it is called a continuous line. If it has a break in it, it's called a non-continuous line." The professor even showed drawings and went into detail.
Calculus 2, however, is a very cruel reminder for why some people have to choose between STEM subjects and good mental health.
Calculus is easier than pre-calculus however. This is because not only are derivatives and integrals easier to grasp fundamentally than things like polar equations, vectors, matrices and bearings; pre-calculus has you jumping from topic to topic often with no relation to one another whereas calculus is the building of one topic over another so you get better at both the last topic and the topic you're on right now. Also, I'd rather do the bottom problem since it's just power rule repeatedly, and since derivative of a constant is 0, answer is 0.
I think a way to visualize this is a ladder that doesn't have equally distanced steps going from algebra to precalc is a harder jump than going from precalc to calc so people will view calc as easier than precalc but for someone who only knows algebra they're equally hard
Amazing distinction to make, well designed video
Though I don't have experience in calculus yet, I want to know the solution and how to calculate the bottom expressions.
0
Integrating an 8th degree polynomial gives you a 9th degree polynomial. Differentiating a 9th degree polynomial 10 times gives you 0.
Are we not integrating 10 times though? I’m probably too tired for this
@@meraldlag4336f(10) means differentiate f 10 times.
@@meraldlag4336 no, just differentiating 10 times
The hardest lower division math course I took was Calc 2, calc 3 was easy compared to calc 2 because it is basically calc 2 with nested variables, partial differentials, and vectors. After calc 2 every math course was fairly easy for me until my first semester of abstract algebra, and the last semester of real analysis.
I’m in calculus right now, and can’t understand derivatives to save the life of me. And I’m sure that’s a problem, because I can only imagine the derivatives will only get harder from here. Does anyone have any helpful tips or tricks that helped them pass their calculus class? Any help would be greatly appreciated
Oh man I struggled pretty badly back in AP Calc AB. I ended up using a free trial of Aleks to nearly perfect my math skills from about algebra up to pre calc. Perfecting those skills really helped me in that class. And then practice. Oh man please practice. Literally anything. Me and a friend would draw out random functions on a white board that we would make up on the spot and try to figure things was an integral possible? Critical points? Derivative? This is great and all but if this is Calc 1 in college you may not have much time.
Something that helped me in college classes and college calc classes was like pre studying before a class/lecture. So if you see on the syllabus that you have chapter 3.4 Chain Rule on the agenda for the day then you can do a Google search of "Chain Rule Calculus 1" and try to either find summaries of it online, or short videos going over it. There's been a few times that I find a 50 minute lecture and I watch maybe the first 10-20 minutes of it. This is usually enough to prime myself for when class comes.
Even now though we have things like chatgpt so you could always say smth like "Hey I'm a learning calc 1 student and I want an introduction in XYZ topic" so that's an idea XD also making friends in your class or in higher level classes if you can find some. That honestly made studying seem less stressful.
Anywho I hope this can help you a little bit, enjoy the rest of your day!
I’m doing the Australia curriculum, and I’m only year 11, but something that helped me is relating the derivative of distance to speed. The rate of change of distance is the speed you are moving at, so that’s the derivative
It’s just the rate of change.
@@jadenroach4832 i'm sorry if my comment sound condescending but y'all are overcomplicating a very simple concept ... all a derivative is that it is a FUNCTION that represents instantaneous change in one variable with respect to another variable .... if u plug in values of say the x coordinate into this function then u will get the rate of change of y with respect to x at the point on the graph corresponding to the x coordinate u plugged in just now ... this rate of change is geometrically the slope of the tangent at THAT particular point of the curve
if it helps then try watching differentiation courses from indian youtube channels that teach in english ... u'll master this topic in no time
My teacher literally said calc was easier than it
anyone wondering, the solution to the bottom problem is 0, the problem is requesting for de 10th derivative of the f(y) function, and said function is of degree 9 at most, so, by the 9th derivative we will have a constant, and so the 10th will be 0
Just started taking math classes in university. I want tedious back and difficult gone.
How I always measure difficulty:
How long did it take me to actually understand the concept?
How does it effect my stress levels when I'm doing it?
How much work does it take after I know how to do it regarding number 2?
So in other words calc 1 WAS easier for me because the precalculus class I took only went with what the book said. The teacher also had us practice 50 precalculus questions every night which just stressed the ever loving god out of me because I'd be like "Ah ok I get it" then get through like 5 questions in just to look at the rest of them to see 90% the same question with different numbers.
Remember this was every day and also a terrible way to teach a topic.
Reason why it is generally a terrible way to teach: If you work on one chapter only once the entire year then you may forget how to do the math, not every chapter teaches the same math.
So by the time I'm on Chapter F, I have already potentially forgotten something from Chapter A but Chapter F also doesn't go over that topic in Chapter F-l and comes back up at Chapter M.
Chapter M becomes hell. I watched half the class cheat durring the tests.
Uptil single variable calculus if you have done pretty good algebra and coordinate geometry and trigonometry then it might feel pretty easy and you will be able to explore the conceptul part of calculus more like solving tough problems but when it becomes multivariable shit starts to become tough af
And then you hit an abstract math class like linear algebra and calculus seemed easy
I disagree multivariate was the easier parts of calc 2 with a few added quirks.
@@citrus4419isn't linear algebra easy though?
I am much beyond both precalc and calculus. I do still believe Calculus is easier, much easier in fact.
Calculus 1 concepts _are_ very easy, and honestly more digestible than precalc. A lot of precalc work implicitly assumes calculus in a weird way, in that calculus is later used to post-justify what you were taught in pre calc.
As well as that, the step up is definitely smaller. Transitioning from pre calc to calculus is a much smaller step than from basic algebra to pre calculus.
Would you give some examples?
calc 2 and calc 3 are slightly harder but theyre both easy in their own ways. its when you get to real analysis and mathematical reasoning (and depends on the prof) is where shit gets hard if not already in linear algebra and diff eq
take it back diff eq was challenging but if u remembered how to do all of them youre solid
When you talk about pre-algebra, precalc and calc 1,2 and 3, it seems like you are referring to well established concepts that should be understood by anyone with a background in science. But the words themselves looks like course names. Are math courses nationally defined in your nick of the universe?
The top problem took me 9 minutes because i forgot how to make the denominators equal and spent 8 minutes to figure that out. And the calculus problem took me 5 seconds because it is easy
The concepts in introductory calculus are much easier to absorb and digest than the ones in precalc
I'm going to show it to my students in high school math honors. In order to get to calculus, they have to solve a lot of tedious stuff. There's no way around it. I just started teaching but I remember all this easier concepts at the tip of my fingers, decades later. Practice is all they need but it is so hard to convince them that one time when I was there in the seat, I had to do it 500 times before I dig into Riemann integrals 😂
So hear me out; What if they just had one bad precalc teacher and then an amazing calculus teacher. From school to school, it’s different.
Yup Calculus 2 was this for me. I love calculus 1, 3 ODE, PDE. The only class in the calculus family that I don’t like is calculus 2 or integral calculus.
Precalculus was not just tedious but also difficult to understand without the context given by calculus. Calculus would be easy to understand even without the context given by precalculus, because the concepts of integration and derivation are so easily understood, and they largely follow simple rules.
often times people are decribing the courses. Not the concepts. Precalc was harder for me mainly cause of other circumstances, but also the tests and such have more opportunities for mistakes that aren't not knowing the concepts.
x² =16² - 250 - 6 = 0
The hardest thing about the calculus series (1/2/3/Diff Eq) is that it all builds off eachother, if you struggle to establish a good understanding of something it will then carry over to other sections, not to mention if your algebra and trig skills aren't that good you will struggle a lot like I did when I took calc 1 the first time.
Difficult and tedious are two different different things, but they are both opposites of easy
There is easy vs hard, and there is simple vs complex.
“Simple and complex” are descriptions of the nature of a problem and how many steps are required to complete it. “Easy and hard” describes our subjective experience in solving the problem. A simple problem may be hard to solve and a complex problem might be easy-depending on one’s abilities.
I think that “tedious” describes the latter-a complex yet easily solvable problem.
I mean the 10th derivative of a 9th degree polynomial is probably easier to solve than stressing about an algebraic mistake on the first question
For me, differential geometry is easier than whatever kind of euclidean geometry they're teaching at my highschool. Also, the problems are a lot more varied and more satisfying to figure out as you progress further into advanced maths.
Sorry, but I still strongly believe that, at least for me, Calculus 3 was easier than Calculus 2. Calculus 3 was mostly just stuff that is talked about in Calc1&2 but WRT 3D surfaces or even higher-dimensional surfaces. However, while it sounds complicated based on that description, working the problems out is actually very straightforward. At least for my class, Calc 2 was FAR more dense with content than Calc 3, and most of the mindnumbing integrals you might have to face in Calc 2 were nowhere to be found in Calc 3... ie the integrals in Calc 3 were just easier than those of Calc 2, because the focus is no longer on the integration strategies.
Precal was actually harder for me because at that time I was a senior and didn’t care about school as much as I do now
thats just because less people know calculus though. if we taught everyone how to solve algebraic problems like this and integral problems like this in middleschool, everyone would choose to solve the much Easier calculus problem.
Also like once you've done pre Calc you have a base level understanding making higher levels seem easier than doing pre Calc with no base understanding
I would counter that the jump from Algebra into Precalculus is a bigger one than from Precalculus to Calculus. Its the jump in difficulty that matters.
Bruh i need a video dedicated to derivatives.
I will die on the hill that the amount of information and difficulty of adapting to new concepts in calc 2 is easier than applying those concepts in calc 3.
10/sqrt(5x-2) = sqrt(10)
10 = sqrt(10) sqrt(5x-2)
a = 10, b = 5x-2
10 = sqrt(a) sqrt(b)
100 = a × b
10 = b
5x-2 = 10
5x = 12
x = 12/5 = 24/10 = 2.4
It's also about the difficulty curve. Going from algebra to pre-calc can often be very daunting and challenging, but going from pre-calc to full calculus, even if calculus is harder, might not be that big of a leap.
for anyone wondering, the second equation is solved with newton's binomial
No + it isn't even an equation 😭
To solve the f10(y) problem, you integrate g(x) with respect to x first. You should get [x9/9 -126x] with boundaries y and 0 which equals y9/9 - 126y.
So, f(y)= y9/9 -126y
Next, the highest power of the function is 9. So, the ninth derivative is a constant, which means that the tenth derivative is 0. So, f10(y)=0 and the problem is solved
Cal student here. Bottom problem looks way easier once you understand integrals.
Tedious does equal difficult.
Math is NP hard.
I find calculus easy and I say it's easy but that's only because I learnt it from my own research and the amazing guidance of 3b1b and other teachers. I'm definetly expecting some struggle when I actually get into a calculus class.
Some classes are also just harder at a higher or lower level. Sometimes you have courses that just have more difficult exams and homework. Calc 3 at my university was known for being harder than some significantly more advanced courses just for it's homework load.
The other consideration here is the difficulty jumps between levels of math. For many people, going from high school algebra/geometry to the trigonometry, college algebra, complex numbers, polar coordinates, etc. found in precalculus is a greater increase in difficulty than the jump from precalculus to the derivative rules and simple integration in calculus I. The material is still harder, and the concepts from precalc were necessary to understand the new material, but you might find yourself struggling less with Calc I, for this reason.
As a person who struggled with math, I could solve the bottom problem in my head, I would need to write the top problem down and do all the steps. Calculus, especially when it comes to polynomials is easier to learn than algebra.
I think a large part of that is how math is taught, basic algebra could be taught with arithmetic. x+3=4 is simple enough that one who knows how to add and subtract can solve it if shown, same for x-4=1, 3+x=5, etc. It is just moving around numbers. If basic algebra was integrated into teaching arithmetic, by the time students got to algebra they would have an understanding of the basics and the harder stuff would be easier to learn.
As someone who can solve both equations
I agree, I prefer the second one
To solve second task it's enough to understand, that we got max у⁹ in f(у) and therefore answer is zero. You don't have to be good at it to understand the gist of the problem in ten seconds. On the other hand, if we were to take the ninth derivative, we would have to tinker with it. It's just a matter of the task, it won't be difficult to make a task from the topic of the first task that can be solved in ten seconds and looks difficult at first glance
For anyone wondering, for the first problem, x =2.4 and in the second problem, the answer is 0. What I find funny is the fact that both problems are overcomplications of very simple things. The first problem is just 5x-2=10 while the second problem is zero.
I feel that "difficulty" just means making the problems harder by making people think harder instead of smarter
Well, as someone that went into full youtube precalculus, calculus 1, 2 and 3. Here it is my PERSONAL opinion: precalculus was hell because I never had any idea on those things at the beginning, calculus 1 was acceptable, calculus 2 was HELL, calculus 3 was the easiest with triple derivatives and triple integrals, it seemed the easiest . still my personal opinion.
For the record, I say this, but what I mean when I do is something different than tedious vs hard.
What I always say is, assuming that you understand the prerequisite concepts to the same degree, calculus is easier to learn than precalculus.
Basically, I think it’s much harder to learn precalculus when you know only the required algebra and geometry stuff than it is to learn calculus when you know precalculus.
I can confirm the bottom problem took me like 5 sec no problem I mean i would definitely prefer the bottom problem in this case
intro calculus is more tedious but easier than precalc; the real hard thing is not intro calculus but analysis
1)12/5 2)0
So now, correct me if im wrong, havent done any sort of calculus in a year or so, but cant you do f prime, get rid of the integral so f'(y)=g(x) plug in g(x) then take the derivative 9 more times? I know the answer is 0, but cantyou do that?
I just call it difficult or complicated instead of teedious or difficult. It's simple to open a heavy door but it's difficult while it's easy to solve a math problem but more complicated
You could also resolve the first in even less than 10 seconds though.
I have a mentality that if I can't solve a problem instantly, it's too difficult for me. So I prefer difficult but straightforward problems to simple and tedious ones.
No, calculus was definitely easier to understand than trigonometry. It’s easier and less tedious.
As for me personally, calculating math problems fills my head with negative emotions.
I don't become emotionless. They are cold, hard, negative emotions that fill my head.
I feel misery and I become enraged.
Honestly, it has nothing to do with doing logic work, because I like doing logic work and I like juggling things in my head. I like puzzle games and I like learning concepts. It's more of tediousness that bothers me.
Solving calculus is something you have to do while listening to music and dancing. You have to do it while occupying other areas of the brain.
It's like how when I play a certain game so much (like fortnite especially), it becomes tedious in certain parts of the game, so I have to listen to a podcast while playing the game.
Also, I don't think it's really only about tediousness. I am a very simple man. I girls. That's it. And when I am not thinking, I am preoccupied with a problem I am trying to solve whatever it may be. Sometimes, even just learning math concepts feels so painful unless I see it necessary to help me solve a grander problem that I take with personal important. It is like reading books like pride and prejudice. It's obviously not that the book cannot preoccupy me. It's that I don't care about that book. I really don't. It doesn't help me solve any of my problems.
Obviously, for motivated students, that purpose is impressing their parents and acing the test right? So, if you find the school to be important, then you're gonna do well. If not, then no. One can certainly have that kind of mindset. I personally am a very competitive person because of my testosterone. It's no use if the school does not reward competitiveness or show how competitive you're.
I'm going to tell you right now. There is no intelligent kid who will find that subject interesting because it stimulates intelligence because they don't know jack shit about what that subject even does. If any kid does, that kid is a fucking genius, and they are really fucking rare. They are either genius, or their parents are hyper fucking educated. That's probably why. Other than that, no kid in his right fucking mind will find school interesting.
Learning math has degrees of depth. There is a point where you understand math just enough to know what it is basically about. Calculus is a study of change. Instantaneous change, accumulation of change. There is that, and there is the depth where you solve many different types of problems. It's like learning a language in a sense, but how do you find it interesting to learn language for sake of learning it? You don't know jack shit about what the problems are going to even do or used for (I mean there are some poorly made applied word problems using these concepts, but you still don't know jack shit about if they are still going to be used in the future, or you're learning about some archaic knowledge). Like I said above. Kids study hard because people told them they are important. It's never because of some kind of personal passion or some genius-like inspiration or curiosity. And in America, where testosterone is suppressed, I can't find motivation to do jack shit.
The American education system does not breed true leaders. It rewards subservient cowards or privileged snobs. We must look toward the Korean education system.
What about the army? Do you the army recruits highly competitive people? No. The West Point (although it is easier to get in than it is to get out) still looks for educated kids. And the army does not want high testosterone males. High testosterone males cause trouble. They want low testosterone males who only do what they are told and can wag their tail, wallowing in their own shit all happy, while being treated like shit. Who have more fear responses in their habits than competitive and brave responses. That's why they look for test scores and the education levels.
Before I took calc 3 people were saying to me that Calc 3 was the easiest calculus class. How incredibly wrong they were.
Difficulty is the amount of work put in, tediousness is basically just that. How advanced something is doesn't mean it's harder. Those students are right, brother
i think they are speaking in terms of relativity. For example, where i am from, gr12 calc and calc I in uni cover almost the exact same things, the difference being that uni calc obviously goes more in depth and it covers integral calc while high school does not. objectively speaking, calc I in university is harder, as it covers more topics in a shorter amount of time; but if you have already taken gr12 calc then you have already seen the majority of the content. so, relative to gr12 calc when you had to learn all of it for the first time, uni calc I is much easier because, for the most part, you already know/have a basis for what is being taught and may only need to remember a proof for something that you weren't taught in gr12. I think that is what people mean when they say "x is easier than y"
No high schooler is solving that top algebra equation in a minute lmao.
The top one actually took me 50 seconds
Cal1 up to intro to integrals was easier than precalc. Calc 2 from techniques of integration to second order differential equations is way harder
My guy... I remember EXTREME tediousness in one of my differential equations classes where we were utilizing massive series and transformations. Difficult AND tedious. Same with analysis (unless you got that intuition baby).
Thank you ! Needed to be said
Algebra is about a lot of tricks to solve weird factorizations. You forget the tricks, you forget the class. Calculus is about concepts, only thing you gotta memorize is a table of derivatives and you are basically done.
Difficulty is subjective, but complexity is absolute. Being good at arithmetic doesn't help much with algebra, but being good ar algebra helps a lot in calculus.
I would much prefer that Calculus question ngl
the algebra problem took me 15 minutes and i didn’t even get it right lmaoo i wonder why i’m failing calculus
Again though, you confuse difficulty of problems with difficulty of course overall. With Pre-Calculus what makes it hard is that you switch continuously from very unrelated topics (like going from polar equations to vectors) which are many things that are completely unique whereas in calculus every topic builds on each other, not only making the next unit easier to understand, but further building on the previous units. Which is why Calculus is easier than Pre-Calculus
Me where Stats 1 had Normal Distribution, but Stats 2 only had Poison Distribution
So much fucking simpler!
I forgot how to solve the top problem but I know how to solve the bottom problem with ease haha
Pre calc feels like a barrage of disconnected and random new concepts
Calculus is actually easier though, not just faster. The concepts are easy to understand and remember and the operations are simple. It doesn't get hard again until differential equations
When I took AP calculus in high school, the only questions I got wrong were the ones where I made a mistake in the algebra
No, the "higher" math is often easier than the "lower" math in the sense that the "higher" math might be a shorter step. For example, getting your mind around the concept of an "unknown" might be much more difficult for some than, say, the idea of infinite sums of infinitesimals (i.e. integral calculus). Thus, it would be perfectly correct in this case to say that calculus is "easier than" algebra, because the core concepts of calculus are easier for an incoming calculus student to grasp than are the core concepts of algebra for an incoming algebra student.
Got it. Whoever designed school had a burning hatred for ADHDers.
nah calculus 1/2 problems are way easier to understand it's just rate of change and area and applications + series which is more applications.
precalc has a lot more jargon you have to go through which makes final tests harder because there's more to study for
calculus if i just understand the main concepts i can ace any test.
but every higher math class relies on the base you set up for yourself prior so if you didnt really understand precalc and are in calc then calculus will seem harder than it really is.
yeah it's basically never going to be objectively easier because it depends on you knowing the skills from pre-calc, but the right language to use would be "i had an easier time"
and like, yeah, for me, the hardest class was calc 1, second hardest was calc 3, third was precalc/trig.
calc 2 wasn't nearly as hard for me as any of those, it clicked with me the same way algebra and geometry clicked with me. idk but measuring the area under a curve is a lot more intuitive to me than imagining the rate of change, at the time those concepts were introduced to me, but by the time i was learning the former, i had already learned the latter.
differential equations was also an easier time than calc 1 for me, but it heavily depended upon the knowledge i got from precalc, trig, calc 1, calc 2, and linear algebra.
upper division math courses were also way easier, but that's assuming that you already know the other stuff, because most of it is like figuring out how to take some complex problem and put it in matlab
so "easiness" might not be the right language to use, but like how much i personally struggled with each topic, calculus 1 was the first time in my life when i genuinely hit a wall, couldn't just absorb things like a sponge anymore, and had to start doing homework and studying and shit if i wanted to pass, and i was so naturally good at learning the elementary stuff that my study habits were absolute trash
If you pretend like integrals don't exist, for me precalc was harder than calculus. However, and I have no idea why, I just absolutely suck at integrals, and no matter how much practice I get with them I still need help choosing which strategy to use. The strategies themselves are easy for the most part (integration by parts 3 times in a row is everyone's favorite thing) but I never know how to set them up or which ones to use
Nice
Calculus based physics is a lot easier than physics for students who don't know calculus.
I genuinely don't remember what happens in pre-calc classes.
But in all of this, it proves how simple calculas it
well this isn't really a question about difficulty, because math isn't exactly a difficult subject per say. generally, calculus needs some level of pre-calc to work with, while pre-calc (obviously) doesn't need calculus. that means that more people can do pre-calc problems than people can calculus, but it's more a lack of knowledge rather than a skill requirement or anything of the sort. as a subject, calculus is generally easier to learn than pre-calc, though.
As one of those students. Nothing before calculus made sense to me. Yet calculus solved everything for me.