Well, upthread, I thought Riemann summed it up pretty well. (har har) 
Wow, that would be horrible. I’ve never heard of a calc course leading off with the Fundamental Theorem of Calculus, and I can’t even imagine how it would be possible. Surely you have to introduce both derivatives and integrals before you can introduce the FTC, and you have to introduce limits before you can do either. When I took calculus, proving the FTC was the last thing we did in the class.
My take on it:
Prior to calculus, geometric proofs and solutions were only possible or practical for special-case idealized geometries; sphere, constant slope, triangles, etc. Calculus opened up math and physics to the irregular general case applications, which most of the real-world universe happens to be.
Having tutored calculus, I also hate the common traditional path that starts with the epsilon-delta proofs of limits. It’s weird arcane esoterica without context that students struggle through, and gets discarded and forgotten the instant that the next chapter comes along.
Ah, thanks!
ha ha ha ha ha!!!
Now this is what I like to call a damn good question.
All of the various math leading up to Calculus (Algebra, Trig, Geometry, etc.) was essentially about solving an equation. You could do things like add x to both sides, but it was still the same equation, really. The thing that’s different about Calculus is that you’re working one step above; changing one equation into another, and learning why and how they’re related. You need the mental model, and the notation, to talk about equations as units, as things to be manipulated. All that weird f(x) stuff finally comes in handy.
To me, that’s the big deal about Calculus.
Someone gave me some advice once that I should, if possible, take Calculus before Physics. So I did. When everyone else in my Physics class was trying to memorize five different version of basic kinematic equations, depending on what you knew and were trying to solve for, I already knew they were derivatives and I was all set.
My Calculus professor was excellent. I remember in the first week or so he drew a graph of an equation of a curve. Then he described what would happen if you had a camera with a zoom lens; and he drew a rectangle around part of the graph, and drew a larger version of that rectangle as if you’d zoomed in on that part of the curve. It had a line that was still curved, but not as much as it had been. So he zoomed in again. In a sense, that was the Fundamental Theorem, but presented pictorially. It worked for me.
As a liberal-arts major who never took or learned calculus, calculus is often held up to be this mythical, terribly difficult subject by non-math students, in Everest-like terms. And I cannot help but think that by treating calculus in these reverent, feared terms, that this *makes *calculus harder - that if you create this impression that calculus is fearsome and difficult, that it *will *be fearsome and difficult.
Any thoughts from our math majors?
This …
My understanding is that derivatives and integrals were around and known, what Newton and Leibniz did was connect the two, ∫ f’(x) = f(x) + C.
I’m not a math major but I have an engineering degree and I tutored math and physics for a bit in college. Is it a terrible and difficult subject? No. The difficulty with calculus comes from building the foundation to learn calculus.
You need to know algebra, arithmetic, some trigonometry, geometry helps, some analytical geometry helps. So that takes a few years to learn that mess generally in high school. And the way it’s taught leaves a lot to be desired. The way I learned these seemed pretty disjoint.
If you learn the important basics. What the different operations are. The order of operations. Learn and understand what a function and an inverse function are. Learn how to graph etc. You’ll be able to use calculus to solve problems. Will you understand it? Not necessarily. That takes time. But it isn’t extraordinarily hard. It’s just a bit more abstract than most people are comfortable with without putting in a proper amount of time.
My hardest part in tutoring was convincing those that I tutored that I couldn’t force understanding. I can show how but they need to work on internalizing the why. The why comes from working with the proofs and definitions and spending the time seeing how it works. That takes active learning. Pads of graph paper and pencil and actively recreating, with reference material, how this stuff is derived. That’s the trick. Working actively to learn it. Not just studying for a test or underlining crap in a guide. Work through the derivations and if you are unsure of a more fundamental concept, work through that as well. Make sure the foundation is strong.
Now most people do arithmetic for how many years? K-7? That’s a lot of time to learn stuff like 1/3 * 3/5 for example. How long do you have for algebra? A year or two? Yeah. No wonder people struggle.
So to answer the question is calculus hard? Not really. It’s just time consuming to master the fundamentals sufficiently to work with or understand calculus. Especially since a lot of times the course work preceding isn’t taught in a manner that is optimized to learn calculus. There is so much wasted time in Trig class for example going over useless identities in my opinion. Just look that junk up man. So you can get bogged down in years of unnecessary details.
Most of trig I’ve never used. Not even in other math classes. Algebra gets hard fast. Quite often the algebra can be harder to actually solve than calculus. At least with regards to the differentiation of functions. Integration isn’t as easy from an analytical point of view.
Short answer, if your mind can handle limits, functions, coordinate systems, and continuity and assuming you know algebra then you are more or less ready to learn calculus from a good teacher.
I tried to learn calculus but I started the class late and was behind everyone else and they kept learning faster and faster and I could never catch up.
My explanation of calculus to the crowd who hasn’t had it yet goes like this. How do you figure how far an object falls in a certain amount of time under constant acceleration? What’s the volume of a cone? What’s the potential energy when you stretch a given spring by a given amount?
To the pre-calculus folks, there’s a formula for all of these. If you know calculus, you don’t need a formula, you just figure it out. All those formulas came from people who know calculus. While you could memorize formulas for all the classes of things that you know, what happens when you come across something new?
Then on to differential equations: if a population grows every year by an amount proportional to its size, what does that growth look like? If you have a mass on a spring and you set it in motion, what’s that motion look like? What kind of function tells you each of these? That’s the extension of calculus called differential equations.
There appears to be a difference between coming up with a new idea that has never existed before and being taught something that someone else with real brains figured out.
Can you imagine how shocking the invention of the bow and arrow must have been?
psik
I think that you’re right that calculus (possibly since it is the highest level of math that non-mathematicians hear about) gets this unholy reputation as the ultimate in math.
Any subject can be hard for a person if they don’t have the mind for it. Philosophy seems like it would break my brain, but that’s because I have never had a reason to plod through all the cumbersome language. Practice makes perfect, though.
I took a year of calculus in college. Everyone had hyped it up to be this uber hard thing, and because I hadn’t taken it in high school, I was certain I was going to struggle with it. But no. I mean, did I have to do my problems every night? Yes. Did I have to drag my ass to all the evenings tutorial sessions and labs, just because some of the concepts didn’t come easy for me? Sure. Which only means calculus was no different from most of the core “weed out” courses I had to take. In fact, I preferred it over organic chemistry, which required way too much memorization for my pea brain.
If algebra, geometry, and trigonometry were subjects that you fared okay in, then you’d be able to do okay in an average calculus class.
(Our junior-high library had a “Calculus for 15-year olds” book or such which I read when I was 12. I’m afraid this was my typical pattern, so I barely noticed whether the curricular teachings were good or bad.)
As for the historical development of calculus, both Newton and Leibniz were awesome geniuses but calculus did not suddenly spring from them without precedent. Cavalieri and Fermat were among those who had developed differential calculus a few decades before; elementary forms of integral calculus was in use much earlier in India and the Islamic Empire, and of course by Archimedes. Here’s a 1912 edition of parts of Archimedes’ The Method, translated. (I’ve “opened” the book to the page with a figure similar to that in the Wiki article cited above.)
Identities, those details, are the very tools needed to understand and master the subject. Yeah, it’s a lot of rote memorization and I agree it’s likely without practice little will be retained years later, but that is the only time you are ever ‘doing’ Trig. The rest of the time a student is just watching.
Later, Differential Equations or Abstract Algebra are no different.
Education lets down students decades earlier making proficiency in these subjects near impossible later. ‘Forget the small stuff’ is just advice to disregard proficiency all together.
I have to make a point, practice is what people typically call talent and and interest is the drive people call “a mind for it”. While there is quite a bit of variation among individuals and some do have limitations, like my challenges with spelling and writing the main limitations for most people surround exposure and practice.
I do dislike our current educational system bias to rote memorization and ‘fundamentals’ but my job would be so much easier of people weren’t so afraid of math.
As an example I do a lot of work with distributed system, due to the complexity and interactions it can be very difficult to diagnose issues with latency and/or to detect resource limitations.
While my colleagues resort to reading chicken bones and tea leaves I can leverage tensors, with mix ranks built from various sources and test for displacement by using tools like numpy that help remove the dead language soup of traditional formulas.
Using differential and tensor calculus in a non-euclidian space I have a ton of freedom and can detect displacement when someone whom only knows some algebra or geometry is stuck with the inability to differentiate between local velocity and displacement.
I have had several cases at work over the past few years, where knowing higher calculus allowed me to solve problems in hours that had been worked on for weeks with dozens of much smarter people in a war room setting. Unfortunately due to a general lack of higher math and particularly calculus I cannot share my work with coworkers and empower them to have the same tools.
I took on the personal goal of learning higher math and science on my own as documented by this post from 2011.
It only took dedicating a few hours a day, over several years to learn the math relating to General Relativity to the point where I grew past the limits of a few Professors whom had tutored me. We live in an amazing era, where basically the entirety of of human knowledge is accessible on everyones cell phone. There are also incredible individuals whom have a passion for teaching and offer far better instruction that was ever afforded to me growing up in a rural area with a poor population and schools. As an example mathbff on youtube are invaluable for re-learning calculus as an adult as remembering simple things like FOIL or quadratics rules can be the most challenging aspect.
Note that I do not think of myself as a genius, and even if my natural abilities in some areas are above the norm most people would never need to advance to the level of regge calculus and above. Although I would say that there is a particular freedom to move past the point where you are hampered by Euclidean geometry.
While a drive to know “why” things happen vs just “how” is probably the hugest asset in my career, and not all of that is dependent on math. Calculus, or the mathematical study of change is far more useful than statistics for me.
To be clear, I started learning to prep for finishing my degree, but as I have a very successful career, and I wanted to go to school for my own betterment I am not upset overshooting my goal but learning too much on my own.
Is my knowledge as deep or as broad as it would be with a formal degree? No…but the usefulness still continues to surprise me. Although having the math tools to do my job more effectively increased the frustration when coworkers cargo-cult and treat problems as god boxes but are unwilling to learn to tools that will help them.
But back to the OP, Geometry is the study of the shape, size, relative position of figures, and the properties of space. Calculus is the study of changes, and is not just limited to Euclidian geometry.
A few hours a day for several years to learn tensor calculus does sort of put you ahead of most in terms of drive though. I’ve been meaning to learn the subject after a very brief introduction to it in a deformation of solids text but I find it hard to be motivated. Out of curious what does some of your work look like?
To be clear, I was working through this book, I would stop and refresh, re-learn or learn any math I didn’t understand.
I no longer have LaTeX installed on my laptop after a migration from macports to homebrew and my python is pretty ugly as I have no need to clean it up for peer review.
If I find something that is readable and not a pile of python with an unhealthy dose of awk, sed and grep I will PM you the information.
As “deep learning” is quite popular it would probably be best to learn their notation. But I am probably not the best resource for someone whom wanted to learn tensor calculus outside of the applied level surrounding a study of GR.
Note that NumPy has good tensor support, and I hear R does too, but it’s pretty hard to find information on search engines when they won’t accept R as a search term.
http://docs.scipy.org/doc/numpy/reference/generated/numpy.tensordot.html
I would recommend getting comfortable with tensors before using numpy because you still need to know when use transposition and broadcasting.
That was not my experience. I did very well in all three of those in high school, plus physics and chem.
I took the intro calculus class my first year at university and just never got it. I just never understood the basic concepts. It frustrated me and my prof, who gave me a lot of personal time to try to help me understand. It just never happened.
Lowest mark I ever got in university, and ended any interest I had in sciences.