Loved the video. Addressed the uneasiness I've always felt from seeing trig functions defined using series (and conspicuously ignoring the geometry). +1 for the naive/natural/straightforward approach!
Hi! This is Swetha, founder of Melodies for Math. Thank you so much for these videos! I definitely feel like a lot of elements of calc are glossed over, and this video definitely helped me understand trig derivatives a lot better :)
It’s not just that we define x to be cos and y to be sin on the unit circle, that’s literally what happens if you apply opposite/adjacent over hypotenuse to the triangle formed in the unit circle since the radius is 1.
Hey Dan, great video. As someone is who is undergoing a crash refresh on Calculus and about to begin analysis, do you think it would be worthwhile to study all the "alternate/un-natural" definitions as a way to get exposed to some of the underlying ideas/tools of analysis? Or do you think they're just a waste of time? I'm about to begin reading your Fourier Analysis recommendation shortly, so thanks for that as well.
Of course for someone in the field it's important to know all the ways of looking at the subject. In these videos I'm just advocating an approach that is motivated and can actually establish all the important theorems without skipping steps. But once you know a bit about power series or differential equations, it's a good check if you can prove everything from a different starting point. That's great that you're getting into Fourier Analysis. Also, the best book I know for getting into real analysis is Bressoud's A Radical Approach to Real Analysis.
@@DanielRubin1 Highly appreciate the advice my man. I am about the same age as you, I stopped at multi-var calculus in college and I'm trying to pick it up from there. Keep making awesome videos, you have a unique perspective.
Yow! I mostly get upset with internet videos because they go too slowly. I do not have that complaint about this video. Good (and lots of) material covered. Great job!
Keisler used the definition that sin t and cos t are parameter functions for the circle with t = int^x_0 dq/sqrt(1-q^2) being the arclength. That certainly feels more natural than the spivak definition and its easy to prove everything associated with them. Though my understanding is that trig functions weren't that important for calculus until euler's complex exponentials, you just worked with the cartesian equation x^2+y^2 rather than a particular parametrisation. Otherwise, it was just empirical rules which one looked for in a table. Interestingly, Cauchy just asserts the sin x/x -> 1 by the inequality sin x
First of all, love the videos. And I agree on all the big picture stuff. I was aware of most of the issues you raise and share your criticisms of some of the unnatural approaches. There is some tension between rigor and pedagogical appropriateness. For the Spivak, you seem to be objecting to the pedagogy (I agree). But you don't make explicit claims that a certain approach is best pedagogically. But around @6:17 you critique the area approach taken by most modern intro calculus textbooks. I understand your objection (the argument rests on knowing the area of a circle), but this doesn't seem like too much to ask of the audience. Even if a better (more correct) proof of the area of a circle exists that the student is unaware of, it doesn't ruin the meaning or flow of ideas to prove the sinx/x limit using the circle area theorem. As far as I can tell from watching your later videos then, this approach is not circular (hence invalid), just... incomplete. Is that correct? To me this seems like a reasonable decision to make when teaching the sinx/x limit -- "As long as A=1/2*C*r is known, this arguments suffices."
Yes, if you're willing to accept the relationship between the area of a circle and its circumference, you can give the area-based inequalities to get the limit of sinx/x. But I do think it's worth pointing out that the proof of the area of a circle relies on the length inequalities that could be more directly used to find the limit of sinx/x.
That's the key issue. I am taking angle measure between two lines to be the (signed, counterclockwise) length of the subtended arc of the unit circle centered at the intersection.
you cannot use Archimedes as a proof since it requires defining a flat spacetime or a tangent space and introducing covariant derivatives. Also you need to prove E=mc2 so we are sure Pi is well behaved in all frames of reference. Just in case.
Fair question. This equation just expresses that the area of a sector is in the same proportion to the area of the whole circle as the length of the arc that it subtends makes to the whole perimeter of the circle. I did not give complete definitions of area (or arc-length) in this video, but I am making use of the properties that area is invariant under rotations and translations, and is additive under (countable) disjoint union, which you can take as axioms. The statement is then instantly clear for rational sectors of the circle, and you can add countably many such to get the statement for arbitrary sectors. Or you can show that the area of a sector is a linear function of the arc-length, since the difference area is independent of where the sector terminates (the differentiability can be established by the bounds in Episodes 2 and 3). In any case, the proof I give of the limit sin(x)/x in Ep 2 is all about lengths, not areas.
Yeah, sorry, let me restate. I mean that if you take a sector that comes from an arc that's a nice fraction of the perimeter of the circle, like 1/5, then it's clear that that sector is 1/5 the area of the whole circle, and the reason is I can appeal to two axioms of area, namely that if I rotate the sector around 1/5 of the way, it has the same area, and when I combine the two the area adds together. If you are willing to sum up infinitely many smaller and smaller sectors like that, they can approach any portion of the circle. The other way is to consider the area of a sector as a function of the arc, say counterclockwise from the positive x-axis for a circle centered at (0,0). To approximate the derivative, you take a difference quotient where the difference between the areas of sectors of angles \theta and \theta +h is the area of a tiny sliver sector of angle h. But the area of the sector of angle h doesn't depend on \theta, so the derivative (if it exists) is a constant, and so area is a linear function of angle, so the proportion holds.
The limit is 1, and the reason the proof is geometric is that the sine function was defined geometrically in terms of parameterizing the unit circle by arc-length. With that definition, which I think is the most natural approach to trig, you have to appeal to the geometry to get at the function's analytic properties. If you define the sine in terms of a Taylor series, then the limit is clear without geometry, but then you have to work to connect that series to the geometric meaning of sine.
Your videos would be so much easier to follow if you did not doubb yourself (which results in you talking about things you did not even write down on paper at that particular time you are talking about them) ... Unless you try to appeal to people that already know that stuff and/or are geniuses... It is math after all better to write stuff down in real time and talk maybe pause talking while finishing drawing that cyrcle etc **let that stuff sink in for the viewer**
Fair point. This series has been an experiment with this format of speeding up the writing to match normal speech. The thinking was that there was an audience for this material that would find it easier to watch a lecture like this than to watch a full-length lecture where the writing is in real time. I want to continue the series with this format, but I also want to fix the problems you mention since I know it is hard to follow when the speech doesn't match the writing or when the page is turned too quickly. This video (Episode 1) was my very first attempt at this lecture format, and I think I've gotten better at addressing those issues in later episodes.
I have wanted a reasonable discussion of sine since grade school-more than 20 years!!-thank you so much for getting into the details!
Thank you so much. I’m so close to crying from happiness right now.
Loved the video. Addressed the uneasiness I've always felt from seeing trig functions defined using series (and conspicuously ignoring the geometry). +1 for the naive/natural/straightforward approach!
Well said!
Hi! This is Swetha, founder of Melodies for Math. Thank you so much for these videos! I definitely feel like a lot of elements of calc are glossed over, and this video definitely helped me understand trig derivatives a lot better :)
Glad you like them!
It’s not just that we define x to be cos and y to be sin on the unit circle, that’s literally what happens if you apply opposite/adjacent over hypotenuse to the triangle formed in the unit circle since the radius is 1.
Again, thank you so much for providing these very smooth and informative lectures!
Hey Dan, great video. As someone is who is undergoing a crash refresh on Calculus and about to begin analysis, do you think it would be worthwhile to study all the "alternate/un-natural" definitions as a way to get exposed to some of the underlying ideas/tools of analysis? Or do you think they're just a waste of time? I'm about to begin reading your Fourier Analysis recommendation shortly, so thanks for that as well.
Of course for someone in the field it's important to know all the ways of looking at the subject. In these videos I'm just advocating an approach that is motivated and can actually establish all the important theorems without skipping steps. But once you know a bit about power series or differential equations, it's a good check if you can prove everything from a different starting point. That's great that you're getting into Fourier Analysis. Also, the best book I know for getting into real analysis is Bressoud's A Radical Approach to Real Analysis.
@@DanielRubin1 Highly appreciate the advice my man. I am about the same age as you, I stopped at multi-var calculus in college and I'm trying to pick it up from there. Keep making awesome videos, you have a unique perspective.
Yow! I mostly get upset with internet videos because they go too slowly. I do not have that complaint about this video. Good (and lots of) material covered. Great job!
Keisler used the definition that sin t and cos t are parameter functions for the circle with t = int^x_0 dq/sqrt(1-q^2) being the arclength. That certainly feels more natural than the spivak definition and its easy to prove everything associated with them. Though my understanding is that trig functions weren't that important for calculus until euler's complex exponentials, you just worked with the cartesian equation x^2+y^2 rather than a particular parametrisation. Otherwise, it was just empirical rules which one looked for in a table. Interestingly, Cauchy just asserts the sin x/x -> 1 by the inequality sin x
Is that a strum louvile problem with von Neumann boundary conditions?
Bruh. This is phenomenal. Ty ty
First of all, love the videos. And I agree on all the big picture stuff. I was aware of most of the issues you raise and share your criticisms of some of the unnatural approaches. There is some tension between rigor and pedagogical appropriateness. For the Spivak, you seem to be objecting to the pedagogy (I agree). But you don't make explicit claims that a certain approach is best pedagogically.
But around @6:17 you critique the area approach taken by most modern intro calculus textbooks. I understand your objection (the argument rests on knowing the area of a circle), but this doesn't seem like too much to ask of the audience. Even if a better (more correct) proof of the area of a circle exists that the student is unaware of, it doesn't ruin the meaning or flow of ideas to prove the sinx/x limit using the circle area theorem. As far as I can tell from watching your later videos then, this approach is not circular (hence invalid), just... incomplete. Is that correct? To me this seems like a reasonable decision to make when teaching the sinx/x limit -- "As long as A=1/2*C*r is known, this arguments suffices."
Yes, if you're willing to accept the relationship between the area of a circle and its circumference, you can give the area-based inequalities to get the limit of sinx/x. But I do think it's worth pointing out that the proof of the area of a circle relies on the length inequalities that could be more directly used to find the limit of sinx/x.
How are you defining the angle that two lines meet at?
That's the key issue. I am taking angle measure between two lines to be the (signed, counterclockwise) length of the subtended arc of the unit circle centered at the intersection.
you cannot use Archimedes as a proof since it requires defining a flat spacetime or a tangent space and introducing covariant derivatives. Also you need to prove E=mc2 so we are sure Pi is well behaved in all frames of reference. Just in case.
Why is Area OBC/Area Circle = h/Circumference ?
Fair question. This equation just expresses that the area of a sector is in the same proportion to the area of the whole circle as the length of the arc that it subtends makes to the whole perimeter of the circle. I did not give complete definitions of area (or arc-length) in this video, but I am making use of the properties that area is invariant under rotations and translations, and is additive under (countable) disjoint union, which you can take as axioms. The statement is then instantly clear for rational sectors of the circle, and you can add countably many such to get the statement for arbitrary sectors. Or you can show that the area of a sector is a linear function of the arc-length, since the difference area is independent of where the sector terminates (the differentiability can be established by the bounds in Episodes 2 and 3). In any case, the proof I give of the limit sin(x)/x in Ep 2 is all about lengths, not areas.
@@DanielRubin1 Can you explain that to me like I'm 5?
Yeah, sorry, let me restate. I mean that if you take a sector that comes from an arc that's a nice fraction of the perimeter of the circle, like 1/5, then it's clear that that sector is 1/5 the area of the whole circle, and the reason is I can appeal to two axioms of area, namely that if I rotate the sector around 1/5 of the way, it has the same area, and when I combine the two the area adds together. If you are willing to sum up infinitely many smaller and smaller sectors like that, they can approach any portion of the circle.
The other way is to consider the area of a sector as a function of the arc, say counterclockwise from the positive x-axis for a circle centered at (0,0). To approximate the derivative, you take a difference quotient where the difference between the areas of sectors of angles \theta and \theta +h is the area of a tiny sliver sector of angle h. But the area of the sector of angle h doesn't depend on \theta, so the derivative (if it exists) is a constant, and so area is a linear function of angle, so the proportion holds.
Is there a proof that no non-geometric proof exists of lim x->0+ {sin x/x} -> 1 ?
The limit is 1, and the reason the proof is geometric is that the sine function was defined geometrically in terms of parameterizing the unit circle by arc-length. With that definition, which I think is the most natural approach to trig, you have to appeal to the geometry to get at the function's analytic properties. If you define the sine in terms of a Taylor series, then the limit is clear without geometry, but then you have to work to connect that series to the geometric meaning of sine.
@@DanielRubin1 oopsie on the 1 Vs 0 limit typo. Fixed
Your videos would be so much easier to follow if you did not doubb yourself (which results in you talking about things you did not even write down on paper at that particular time you are talking about them) ...
Unless you try to appeal to people that already know that stuff and/or are geniuses... It is math after all better to write stuff down in real time and talk maybe pause talking while finishing drawing that cyrcle etc **let that stuff sink in for the viewer**
Fair point. This series has been an experiment with this format of speeding up the writing to match normal speech. The thinking was that there was an audience for this material that would find it easier to watch a lecture like this than to watch a full-length lecture where the writing is in real time. I want to continue the series with this format, but I also want to fix the problems you mention since I know it is hard to follow when the speech doesn't match the writing or when the page is turned too quickly. This video (Episode 1) was my very first attempt at this lecture format, and I think I've gotten better at addressing those issues in later episodes.
The old Bressoud copycat video.