Multivariable limits. It's like proving a negative to prove that they exist. You can hold x constant, and show that it gets to one number. Then you can hold y constant, and show that it gets to the same number. Then you can approach it along a generalized linear path, where y = k*x, and show that it still gets to the same number. That seems like it should be enough to prove the limit exists. But that still isn't enough, because someone could come up with an adversarial function and an adversarial path that relates y and x, where the limit ends up equaling something else entirely.
I feel like to really grasp the central idea behind calculus, it might be beneficial to find a way to intuitively explain what a limit is to a lay person. I don’t think just plotting a function with a hole in it is a good introduction, as it’s usually introduced. I’m yet to think of a good analogy that maps onto a common experience a person might have. In essence it feels like it comes from formal logic, there exists this “thing” that we can get closer to forever, and it’s an exact specific thing by the very nature of being able to always find a value(or set of values) closer to it. That seems to be what’s at the heart of integration, the integral symbol is hiding the implied limit of us approaching the area under the curve by summing more and more rectangles together.
@@zurabmelua7989 I appreciate your feedback! Yeah, that's totally valid. My goal was to keep the video to about 5-6 minutes and I could only cram in so much. But, what you just said seems like it'd be worth making another video on for explaining the idea of limits and infinitesimally small change in calculus for the lay person. I'll keep that in mind for future videos :)
Unfortunately intuition takes a while to get. Some people use repetition to etch intuition for a topic into their minds, this has too many flaws to list here. It is a "Shortcut" for intuition. A way to get to the end of the tunnel, when you dont know which way you are facing. The other is to understand it perfectly such that you have no doubts. This is a much more powerful method compared to repetition but it requires a lot more active thought and time put into the topic. It's powerful because you not only understand a topic easily, you also develop a wide range of different but interconnected skills with the topic in hand. What this video attempts to do is a mix of both while leaning heavily towards the latter, which is neither necessarily wrong nor is it correct but the second method of learning requires quite a lot of energy put in, which just isn't possible for a normal person within the span of a few minutes. They'd be better off learning from the absolute basics, rather than put 5 minutes here. Do not get me wrong. This is in no way a waste of time but the timeframe utilised is just too small to make a enough difference to justify putting this video over a more in depth one. The other issue is that calculus itself is a really massive branch of mathematics. Limits, Derivatives, Integration and continuity are all very important parts of calculus that are interconnected to each other. It is not possible to understand derivatives and continuity without derivatives and it is not possible to understand integration without derivatives.
I like your use of the term "stitch together". However, I think you spent too much time on non-essentials (all the business about units, area of triangle, etc) and yet tossed in terms like "limits of integration" with barely any explanation at all. Even a brief mention of where the integration symbol comes from (stylized "S" for "sum") would help. Likewise, the "dx" part. Your wife kept asking about "d". Again, you introduced a term without really explaining it (in simplest terms, anyway) - more of a hand-wave - and still didn't really answer her question. Wouldn't it have been simpler to go back to Riemann sum (aka rectangle rule) form, showing f(x) times delta(x)? For someone familiar with simple geometry, it should be fairly painless to go from making delta(x) smaller and smaller to get better approximation, to "now imagine if delta(x) was infinitesimally small - a single point", and there are "infinitely many" of them that we're now going to "stitch" together" which will give us not just a very good approximation, but an *exact* answer. THAT's calculus! :)
There’s so much more that could be covered but yet, I wanted to keep the video short. I’ll think about your points for a future video. And correction: That’s ONE part of calculus. There’s more to calculus than a Riemann sum.
@@NumberNinjaDave You'ree right. I should have been more specific: that's the calculus **part**. :) Thanks for taking this on-board and your kind reply. My wording can be a little too direct sometimes and I'm often taken as snide or snarky when I don't mean to be. Cheers!
What’s been the most frustrating part of learning calculus? 😢
Multivariable limits. It's like proving a negative to prove that they exist. You can hold x constant, and show that it gets to one number. Then you can hold y constant, and show that it gets to the same number. Then you can approach it along a generalized linear path, where y = k*x, and show that it still gets to the same number. That seems like it should be enough to prove the limit exists. But that still isn't enough, because someone could come up with an adversarial function and an adversarial path that relates y and x, where the limit ends up equaling something else entirely.
What I learned from this video: 5 minutes is not enough time to understand calculus.
How could I have explained it better
I feel like to really grasp the central idea behind calculus, it might be beneficial to find a way to intuitively explain what a limit is to a lay person. I don’t think just plotting a function with a hole in it is a good introduction, as it’s usually introduced. I’m yet to think of a good analogy that maps onto a common experience a person might have. In essence it feels like it comes from formal logic, there exists this “thing” that we can get closer to forever, and it’s an exact specific thing by the very nature of being able to always find a value(or set of values) closer to it. That seems to be what’s at the heart of integration, the integral symbol is hiding the implied limit of us approaching the area under the curve by summing more and more rectangles together.
@@zurabmelua7989 I appreciate your feedback! Yeah, that's totally valid. My goal was to keep the video to about 5-6 minutes and I could only cram in so much. But, what you just said seems like it'd be worth making another video on for explaining the idea of limits and infinitesimally small change in calculus for the lay person. I'll keep that in mind for future videos :)
I think you did not show how the area is the number of cows. Maybe trying it out by examples would help convince a beginner.
Did you watch the whole video?
Nice video!
Thank you!
Unfortunately intuition takes a while to get. Some people use repetition to etch intuition for a topic into their minds, this has too many flaws to list here. It is a "Shortcut" for intuition. A way to get to the end of the tunnel, when you dont know which way you are facing.
The other is to understand it perfectly such that you have no doubts. This is a much more powerful method compared to repetition but it requires a lot more active thought and time put into the topic. It's powerful because you not only understand a topic easily, you also develop a wide range of different but interconnected skills with the topic in hand.
What this video attempts to do is a mix of both while leaning heavily towards the latter, which is neither necessarily wrong nor is it correct but the second method of learning requires quite a lot of energy put in, which just isn't possible for a normal person within the span of a few minutes. They'd be better off learning from the absolute basics, rather than put 5 minutes here. Do not get me wrong. This is in no way a waste of time but the timeframe utilised is just too small to make a enough difference to justify putting this video over a more in depth one.
The other issue is that calculus itself is a really massive branch of mathematics. Limits, Derivatives, Integration and continuity are all very important parts of calculus that are interconnected to each other. It is not possible to understand derivatives and continuity without derivatives and it is not possible to understand integration without derivatives.
Yeah but still i don't think my wife would understand 😥
😭
I like your use of the term "stitch together". However, I think you spent too much time on non-essentials (all the business about units, area of triangle, etc) and yet tossed in terms like "limits of integration" with barely any explanation at all. Even a brief mention of where the integration symbol comes from (stylized "S" for "sum") would help. Likewise, the "dx" part. Your wife kept asking about "d". Again, you introduced a term without really explaining it (in simplest terms, anyway) - more of a hand-wave - and still didn't really answer her question. Wouldn't it have been simpler to go back to Riemann sum (aka rectangle rule) form, showing f(x) times delta(x)? For someone familiar with simple geometry, it should be fairly painless to go from making delta(x) smaller and smaller to get better approximation, to "now imagine if delta(x) was infinitesimally small - a single point", and there are "infinitely many" of them that we're now going to "stitch" together" which will give us not just a very good approximation, but an *exact* answer. THAT's calculus! :)
There’s so much more that could be covered but yet, I wanted to keep the video short. I’ll think about your points for a future video.
And correction: That’s ONE part of calculus. There’s more to calculus than a Riemann sum.
@@NumberNinjaDave You'ree right. I should have been more specific: that's the calculus **part**. :)
Thanks for taking this on-board and your kind reply. My wording can be a little too direct sometimes and I'm often taken as snide or snarky when I don't mean to be.
Cheers!