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Topic: [Apparently Not] New Insight on lim x->0 sin(x)/x (Read 10601 times) |
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william wu
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[Apparently Not] New Insight on lim x->0 sin(x)/x
« on: Sep 5th, 2003, 8:10pm » |
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Recently I learned that it's apparently not so clear that lim x->0 (sin(x))/x = 1. The standard answer for showing this is to simply use L'Hospital's rule. But using L'Hospital's rule here requires knowing that the derivative of sinx is cosx. However, this is circular, because in most calculus textbooks, the proof that the derivative of sinx is cosx depends on knowing that lim x->0 sinx/x = 1, which is the fact you started out trying to prove. See below where I try to prove that (d/dx)sinx = cosx from the basic definition of derivative:
(d/dx) sin x
= lim h[to]0 (sin(x+h)-sin(x))/h
= lim h[to]0 ((sin(x)cos(h) + cos(x)sin(h)) - sin(x))/h
= lim h[to]0 [ sin(x)(cos(h)-1)/h + cos(x)sin(h)/h ]
= [ sin(x) lim h[to]0 (cos(h)-1)/h ] + [ cos(x) lim h[to]0 (sin(h)/h) ]
So now if we can determine lim h[to]0(cos(h)-1)/h and h[to]0 sin(h)/h, we will be set. However, to determine h[to]0 sin(h)/h, we use L'Hospital. Hence we're running in circles.
To resolve this, we need a way of evaluating lim x->0 sin(x)/x which does not necessitate knowing that (d/dx)sin(x) = cos(x), because the latter statement is probably circular and depends on lim sin(x)/x.
A solution is detailed at the following link:
http://www.maths.abdn.ac.uk/~igc/tch/ma1002/diff/node27.html
It uses a cool sandwich argument and the fact that sin(x) is an odd function. I don't know to what mathematician this is credited to ... perhaps Newton?
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| « Last Edit: Sep 9th, 2003, 4:31am by william wu » |
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Sir Col
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #1 on: Sep 6th, 2003, 2:42am » |
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Couldn't this be resolved by using the power series expansion of sin(x)?
As sin(x) = x–x3/3!+x5/5!–...
d[sin(x)]/dx = 1–x2/2!+x4/4!–... = cos(x)
All we need to derive these power series is the derivation of ex=1+x+x2/2!+x3/3!+... and Euler's formula:
e[smiley=i.gif]x = cos(x)+[smiley=i.gif]sin(x)
e[smiley=i.gif]x = 1+x[smiley=i.gif]+(x[smiley=i.gif])2/2!+(x[smiley=i.gif])3/3!+...
e[smiley=i.gif]x = 1+x[smiley=i.gif]–x2/2!–x3[smiley=i.gif]/3!+x4/4!+x 5[smiley=i.gif]/5!–...
e[smiley=i.gif]x = (1–x2/2!+x4/4!–...) + [smiley=i.gif](x–x3[smiley=i.gif]/3!+x5[smiley=i.gif]/5!–...)
So, cos(x) = 1–x2/2!+x4/4!–..., and sin(x) = x–x3[smiley=i.gif]/3!+x5[smiley=i.gif]/5!–...
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| « Last Edit: Sep 6th, 2003, 2:58am by Sir Col » |
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william wu
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #2 on: Sep 6th, 2003, 3:12am » |
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Right, if you use that there's no problems. This is why I said (d/dx)sin(x) = cos(x) is probably circular ... there's actually ways to show this without depending on lim sin(x)/x. In the math books I learned calculus from though, I don't think I recall them showing (d/dx)sin(x) = cos(x) using power series. I think they succumbed to circularity.
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| « Last Edit: Sep 6th, 2003, 5:23am by william wu » |
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Sir Col
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #3 on: Sep 6th, 2003, 3:28am » |
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It's a fascinating thought that the limit, sin(x)/x, as x [to] 0, is not certain.
Since making the post, I've been pondering how we produce the power series for ex and Euler's formula. It's possible to produce the power series for ex by using MacLaurin series, but I'm not sure if it applies to complex numbers – we'd need our resident complex analysis guru to comment on this. Equally, I'm not sure if there is some circular reasoning in how we derive Euler's formula, using, at some point, the small angle approximation for sine.
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william wu
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #4 on: Sep 6th, 2003, 5:21am » |
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on Sep 6th, 2003, 3:28am, Sir Col wrote:| using, at some point, the small angle approximation for sine. |
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Speaking of this, my first response to the claim that lim x[to]0 sinx/x is uncertain was the following non-circular derivation of (d/dx)sinx = cosx:
(d/dx) sin x
= lim h[to]0 (sin(x+h)-sin(x))/h
= lim h[to]0 ((sin(x)cos(h) + cos(x)sin(h)) - sin(x))/h
= lim h[to]0 [ (sin(x)(1) + cos(x)h - sin(x))/h ]
= lim h[to]0 (cos(x)h/h)
= cos(x)
So I used two approximations: 1. cos(h)[to]1 as h[to]0, and 2. sin(h)[to]h as h[to]0 (small angle approximation). The guy I was talking to then asked me if there were any rigorous justifications for these approximations, to which I couldn't reply.
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| « Last Edit: Sep 6th, 2003, 5:24am by william wu » |
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Icarus
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #5 on: Sep 6th, 2003, 8:10am » |
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on Sep 6th, 2003, 3:28am, Sir Col wrote:It's a fascinating thought that the limit, sin(x)/x, as x [to] 0, is not certain.
Since making the post, I've been pondering how we produce the power series for ex and Euler's formula. It's possible to produce the power series for ex by using MacLaurin series, but I'm not sure if it applies to complex numbers – we'd need our resident complex analysis guru to comment on this. Equally, I'm not sure if there is some circular reasoning in how we derive Euler's formula, using, at some point, the small angle approximation for sine. |
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There are several of us who are more than familiar enough with complex analysis to answer these questions. I am sure that Pietro and Barukh are adequately versed. I strongly suspect that James, Towr, and SWF may be as well.
First of all, the derivation of limx[to]0 (sin x)/x =1 that William gave is standard - I'm somewhat surprised that you haven't seen it before, as has been given in every calculus book I have seen (except baby -erh, I mean - business calculus). It almost certainly predates Newton. It seems likely to me that a less well-defined version of it was known to the Arab originators of Trigonometry.
Since the concepts involved can be defined and proven without the use of the limit, the proof is rigorous, so there is no circular logic going on.
You could use the power series expansion to prove that lim (sin x)/x = 1, except that you need the derivative of sin x in order to prove that the power series actually is sin x. (You can prove it without the derivative, but only by means that are essentially the same, and still require the (sin x)/x limit.)
Concerning the justifications for the complex power series: They are justified because they represent the definition of these functions for complex values:
Let [smiley=f.gif] be a real function which is analytic (i.e. representable by a power series) in a neighborhood of the real number [smiley=a.gif]. We define [smiley=f.gif] for complex numbers near [smiley=a.gif] by allowing the variable in the Taylor series expansion of [smiley=f.gif] to take on complex values. The radius of convergence is the same for complex values as it is for real ones, so this extends [smiley=f.gif] to a disk centered at [smiley=a.gif]. If we choose a complex value [smiley=b.gif] near the edge of this disk, and take the Taylor series expansion of [smiley=f.gif] around [smiley=b.gif], we may (and commonly do) find that its radius of convergence is large enough to extend outside the original disk. This allows us to expand the definition of [smiley=f.gif] even farther. If we find another point near the boundary, we can repeat the process.
This method of extending the definition of [smiley=f.gif] is called "analytic continuation". In order for the resulting function to be well-defined, whenever the convergence disks of the taylor series around each of two different points overlaps, the value of [smiley=f.gif] for each of the two series must be the same. This is usually the case. The exception comes when [smiley=f.gif] has been extended in such a way that it encircles an "essential singularity". In this case, as we extend around the singularity, when we get back to where we started from, we discover that our new values for [smiley=f.gif] differ from the original ones by some constant amount. circling the singularity again will add the same constant amount a second time, etc. Such functions are called "multi-valued". The classic example is ln x, which picks up a constant 2[pi][smiley=i.gif] every time it circles the origin in a CCW direction.
Thus the expression [smiley=e.gif][smiley=supz.gif] = [sum] [smiley=z.gif][supn]/[smiley=n.gif]! is true for complex [smiley=z.gif] because it is the definition of [smiley=e.gif][smiley=supz.gif]. Euler's formula is an easy result of this definition.
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Sir Col
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #6 on: Sep 6th, 2003, 4:48pm » |
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on Sep 6th, 2003, 3:28am, Sir Col wrote:| we'd need our resident complex analysis guru to comment on this |
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Actually I was thinking of Kitty. 
Thanks for the explanation, Icarus, but I'm afraid that I must admit I don't follow. It's not your explanation – I'm sure that if I'm going to understand it from anyone, it'd be you – but it's that area of mathematics that I just can't connect with. It always sounds like mathematicians are using the argument: "We shall define... hence...", whenever there is something that cannot be proved, in the sense that I understand proof. For example, "We shall define the zeta function to have non-trivial zeroes iff the real part is 1/2, hence the Riemann hypothesis is true." I'm probably missing the subtlety of the higher mathematics, but to me, it's like smoke and mirrors.
*prepares to be barraged for speaking out against the sacred art of complex analysis* 
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Icarus
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #7 on: Sep 6th, 2003, 8:19pm » |
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Shouldn't be a barrage. But you have misunderstood.
We don't "define" when we can't prove. We define when we have nothing to talk about otherwise!
What does "ab" mean?
When b is a natural number it's definition is easy: b copies of a multiplied together. We start here and discover a couple properties: ab+c = abac and (ab)c = abc. By demanding that the first be preserved, we can figure out how to extend our definition to all integers for b. By demanding that the second be preserved, we extend our definition to all rational exponents.
What about irrational exponents? There is no way to define them until we start looking into calculus. Then we realize that irrational exponent values are uniquely determined by the requirement that exponentiation be continuous with respect to the exponent.
All of these steps are not things that were proven. There was nothing to prove at each and every one of them. The concept of 2-1 is meaningless until you define it. It is by definition, not proof, that we know 2-1 = 1/2. (You can use other definitions and prove this, but then you are just taking as definition a different property of exponents, which is proved when using this definition).
So how do you extend the definition of exponentiation to complex powers? The algebraic properties of exponents suggest nothing useful. Continuity would be satisfied by infinitely many possibilities. The first (and only) clue comes when we get to power series in calculus and you prove that
ex = [sum]n=0[supinfty] xn/n!
Here finally is an exponential property that has a unique suitable extension into the complex numbers using behaviors that are already defined.
Just as we extended exponentiation to negatives and rationals by preserving the equations above, we extend exponentiation to complex exponents by demanding that the power series for ex also hold for complex x.
Until we make this definition, demanding a proof of "eix = cos x + i sin x" is as reasonable as demanding a proof that "all gimpols are extarpolated". Both sentences are meaningless.
Fortunately, the preservation of power series expansions gives us a natural way to extend the definition of many functions defined originally only for real numbers to complex ones as well.
Thus the difference between "analytic continuation" and your Riemann conjecture example. The zeta function is already defined. If we change the definition, we have a different function, so the original problem, about the original function, still remains. In analytic continuation, there is no previous definition - this is why we are resorting to it. (By the way, the zeta function is defined first for real numbers and then extended to the complex plane by -- analytic continuation! This is part of why it is so hard to figure out where its zeros are.)
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Sir Col
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Here's a simple proof to show that lim [theta][to]0 sin[theta]/[theta]=1:
Consider the diagram and let the circle have radius, 1.
Area triangle OAB = sin[theta]/2
Area sector OAB = [theta]/2
Area triangle OAT = tan[theta]/2
Therefore, sin[theta]/2 < [theta]/2 < tan[theta]/2, giving sin[theta] < [theta] < tan[theta]
As 0< [theta] <[pi]/2, sin[theta] is positive, so dividing by sin[theta] gives, 1 < [theta]/sin[theta] < 1/cos[theta]
As [theta][to]0, cos[theta][to]1, so 1/cos[theta][to]1.
Hence, [theta]/sin[theta][to]1.
That is, sin[theta]/[theta][approx]1 for small [theta].
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Icarus
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #9 on: Sep 8th, 2003, 8:31pm » |
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Isn't that the same proof that William linked to?
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TenaliRaman
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #10 on: Sep 9th, 2003, 12:49am » |
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that's the same proof they taught me at my school!!!(Along with a small intro to the sandwich theorem)(i thought that was nothing new!!)
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william wu
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Re: New Insight on lim x->0 sin(x)/x = 1
« Reply #11 on: Sep 9th, 2003, 4:18am » |
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on Sep 6th, 2003, 8:10am, Icarus wrote:
First of all, the derivation of limx[to]0 (sin x)/x =1 that William gave is standard - I'm somewhat surprised that you haven't seen it before, as has been given in every calculus book I have seen (except baby -erh, I mean - business calculus). It almost certainly predates Newton. It seems likely to me that a less well-defined version of it was known to the Arab originators of Trigonometry.
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After you said this, I wondered how I could have not ever seen the proof I posted until recently. I looked in my old freshman year calculus book (Stewart) and indeed, it's there. However, it was taught in first semester calculus at Berkeley, which I skipped thanks to AP credits from high school. And my high school teacher definitely never covered this, probably because my high school is ghetto (25% attend college, and the number going to four-year institutions is ridiculously small). This however, does not exclude the possibility that the proof was in my high school calculus book (which I no longer have), and I just didn't read it.
Looking back on all I've done in the past year with this riddles gig, and the way this thread turned out, I feel a need to confess my crimes, so, I'll tell you my little story.
Firstly, I completed all math requirements for the degree very early in my college career, and amazingly, I earned near straight As in all of them while usually reading only what was necessary to get the week's homework done. This was just immaturity on my part. Also, I was obsessed with video games. For some weeks, my necessary reading material consisted of only a few equations in a blue box! I reduced everything to pattern-matching and string substitution. In retrospect, it's kind of impressive, in a pathetic way. A long history of brainless problem-solving at the foundations might shed some light on the perhaps unexpected level of math-conceptual deficiency in some of my posts, especially the earliest posts.
In later years, when I became a more serious student, I realized a need to develop deep intuitions for material presented to me in class, because anything not treated with this level of consideration is simply forgotten. My algorithms courses allowed me to do this to my satisfaction; while algorithms can be very tricky and clever, their descriptions are relatively not difficult to read -- they're just step-by-step procedures. CLR (Introduction to Algorithms) is a very readable book. So I believe anyone who desired to completely comprehend what was going on could do so by simply dedicating enough time to the subject. In my electrical engineering courses however, I found a very different story. You could spend inordinate amounts of time absorbing the material given to you, but you could feel that many connections were still missing, because the course deliberately adopted a silent policy on so many things. For instance, when introduced to the inverse z-transform, I was told this is actually a contour integral in the complex plane. This sentence was immediately followed by a sentence along the lines of: but that's beyond us -- we'll just use our z-transform lookup tables. Many of my signal processing courses were essentially formula plug and play version 1.0. Lots of holes everywhere, from Lebesgue measures to conformal mappings.
Sadly, a story like this seems to be the norm as far as I can tell, and there's very little variance. I've long felt that my understanding of math is unacceptable for a person in my field of expertise, at my point in my education. Yet when I consider the peers who graduated with me, there's few whom I can clearly say are any better, and I dare say that most are far worse. If you asked the graduating engineering class at UC Berkeley -- possibly the second greatest engineering institution in the US -- to write down the delta-epsilon definition of a limit, or even to explain what a basis is, I would bet at least 50% would fail! Aside from such hypothetical scenarios, I could tell many real-world stories involving highly "accomplished" peers exhibiting frightening misconceptions or ignorances about math, but I won't risk making personal insults. To me this strikes of a full-scale educational disaster. Yes, we students must take responsibility for our own failings, but I think the "system" must carry some burden too. I am fascinated by how easy it can be for students to excel grade-wise in dozens of courses swarming with mathematical ideas, and yet emerge four years later with an understanding of math as shallow as what they started with. I apologize for this rant, but it's been weighing heavily on my mind recently.
Anyhow, it's only been in the past month or two that I decided to really hit the math books again, and truly understand why all the tools I've learned actually work. Proofs of powerful theorems, which I used to skip reading without a second thought, now are the most interesting things to me. My curiosity is highly motivated by reflection on the many frustrating engineering courses I've taken, which I believe were intentionally designed to hide deep understandings of why anything works, in the misguided interests of compressing content or saving time. But I'm also motivated by a desire to just understand WHAT THE HECK ARE YOU TRYING TO SAY!  In many books and papers I would like to understand, and sometimes in this forum and even my e-mail inbox, I am regularly amused by strange-looking pieces of communication that sound like they were spewed by robots. Not understanding what a world-famous mathematician is trying to tell me about problems on my own website is pretty damn embarrassing. So, with such large opportunity to wash away confusions, I'm finding math books very exciting. I'm awed and humbled everyday by the sheer size of the mathematical universe, and the unfathomable amount of cleverness floating around in the form of slick proofs. Indeed, it would have been nice to have been a math major. To think that once last year, I worried about consuming all the good puzzles left in this world faster than the world could produce them! They're out there in the millions --- we just need to convert them into understandable word problems for the masses!
As a closing remark, I'm here to teach what little I can, but I'm also very much here to learn as much as possible, from the many people in this community so much more versed in problem-solving than me, people for whom I have nothing but respect and appreciation for. So if absolutely anything is wrong with what I write, regardless of the fact that I'm the administrator who spent lots of time to make all this or whatever blah blah blah, you should still rip it to shreds with zero hesitation. I will not get upset at all! Rather, I'll be upset if you don't rip it up!
4:27 AM 9/9/2003
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| « Last Edit: Sep 9th, 2003, 7:59am by william wu » |
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BNC
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #12 on: Sep 9th, 2003, 5:28am » |
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I don't intend to rip anything you say. In fact, in an ideal world, I would agree with you totaly.
BUT! I teach engineering courses at the local university, and I can tell you that I, too, skip important "background" issues. It's a matter of given time per course, and required essetials to teach at that time. I just don't have the time to cover all I would like to -- I have to choose!
The soluiton I found, is skipping as little as I can, hand out xiroxed explainations of as much of the rest as I can, and give references to the rest. And then, I just sit and hope my students will actually read that stuff. But I know they (well, at least the large majority of them) won't do it. I'm usually pleased if thay do the required chores themselvs. And it's not a bad university (some of our engineering researchers are acknowledged world-wide as leaders in their fields).
So, basically, as I see it, it comes down to the individula student. If the student want, a world of wander and knowledge awaits him (her). But the last if should actually be iff...
My 2c
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TenaliRaman
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #13 on: Sep 9th, 2003, 6:38am » |
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As a student of a engineering course, i must say that william spoke my heart out there.But then i do consider BNC's situation as well .... "too much to teach and too little time" ... maybe there is no way to strike a balance here (considering the amount of things that needs to be taught these days).Hence i stpped expecting wonderful things from my profs.
So i accept BNC's argument that a student needs to find his way out.I also consider this as a chance to feel the charm of independence (independent of others and dependent on only books).Well you may call this optimism.
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william wu
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #14 on: Sep 9th, 2003, 7:54am » |
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on Sep 9th, 2003, 5:28am, BNC wrote:I don't intend to rip anything you say. In fact, in an ideal world, I would agree with you totaly.
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Thanks for your thoughts. Regarding "ripping", my slang made me pretty vague about what I meant. I wasn't referring to my opinions on the educational process, although you're more than welcome to offer your thoughts about that too, as you have. I just mean to say that whenever I write something technical (as many of these puzzles deal with math) that isn't exactly correct, I'd like to be notified. Basically, all errors should be corrected. Maybe this is already happening to the best extent reasonably possible. However I simply wonder sometimes whether people are holding back because they're afraid I'll get annoyed by nitpicky corrections.
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towr
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #15 on: Sep 9th, 2003, 9:07am » |
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My take on university usually leads me to not read the books. In part that's because I simply don't like reading books, I like being lectured (but only by people who know their stuff, so not about anything ethical  ). Basicly I only use books to look up things I need, they're external memory, just like pen and paper (or computer) form external work-memory.
As long as I understand where to find the pieces of a puzzle, and how to fit them together I've been taught the most important things. Remembering definitions etc is only usefull if you actually use them a lot. It may be nice to know that z-transforms can be transformed back using a contour integral, but in most practical situation lookup tables beat doing it that way.
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Ragna
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #16 on: Sep 9th, 2003, 3:14pm » |
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Here's another viewpoint about our educational system today:
While Wu argues that many students nowadays are not sufficiently grounded in the fundamentals, and I tend to agree with that statement, I would argue that the problem stems more from the lack of a student's thinking ability than from having to remember the formalisms and details involved.
I know that last sentence was kind of confusing, so I'll take the epsilon delta limit definition as an example. Obviously a professor in analysis should be able to write down the definition without thought. Probably we would expect a professor in any sort of mathematics to be able to write down the definition with minimal thought. However, speaking for myself only, I tutored freshman calculus for 6 semesters in the dormitories, and while I knew the definition cold when I was working, at this point it'd still take me a few minutes to write down the formal definition of something as fundamental as a limit. On the other hand, I could explain in words and pictures quite easily what the epsilon delta definition means, and why it's important. As a computer science student, I expect that this is really all I'll need on a regular basis, and if it turns out I DO need the real definition, I can look it up.
Another analogy: Most of us who drive cars cannot explain how a car works. I would say most of the people driving cars do not know the different stages of the car combustion engine. However, that doesn't keep us from driving cars. Most of us have a TV. It wasn't until my antenna broke, and I was trying to construct a new one, that I realized I had no idea how long an 'optimal' antenna should be. That doesn't keep me from enjoying TV. Similarly, the business and biology students who are obligated to take calculus will never have to deal with epsilon delta definitions or proving the fundamental theorem of calculus. That is as it should be. (An aside: When I was tutoring calculus one of the hardest things to teach was the 'business' calculus class because it's very difficult to teach people how to generalize their solutions when they don't understand the fundamentals). However, it is clear that they should realize that the 'integral' of a curve is the area under the curve, and they should be able to compute integrals, or at the least, use a computer to do so (yes, I realize this is not quite true when the curve goes below the x-axis). Economics students generally do not have to take multivariable calculus, but they know how to do lagrangian multipliers, since they often have to maximize multivariate functions. Do they understand the theory behind them? I would guess no, but they do understand how to apply it as a tool, nor do I think this hampers them, as they would not be called upon to generalize the use of lagrangians. And in any case, if they do need to generalize, they would call in an expert.
Of course, there are always people who would like as full an understanding of what they're doing as possible (I personally am one of those kinds of people), and don't mind taking the time to work through all the symbolisms and formalisms, and that is fine. But then there are also the people who see math as simply a tool for doing things, and not for solving new puzzles and problems, and they are not necessarily at much of a disadvantage in what they do.
I suspect also that a disproportionate number of readers of these forums may be of people in the latter category, since these puzzles tend to deal with problems people wouldn't normally see in their day to day work.
And finally, as a parting shot, I would argue that a significant portion of the failure of our educational system is a result of our social problems today (I guess this only applies to US readers). It's hard to concentrate on education when there children have only one parent because of divorce, etc, when gangs form a child's surrogate family, and when teachers are grossly underpaid and schools grossly underfunded. However that is a topic for a different thread and a different day.
Wow I wish I could have written essays like these back in my college days.
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Icarus
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #17 on: Sep 9th, 2003, 6:55pm » |
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I can agree with Ragna somewhat, but not entirely. It's true that it would be ridiculous to expect everyone to be a master of mathematics. It's also true that I am not a master, nor strive to be one, in every aspect of my own life.
On the other hand, while I don't know all the intricacies of how a car operates, I do understand the basic principles. I also understand how this computer I am sitting at works at all levels, though I certainly couldn't sit down and design one. I may not know the optimal length for a TV aerial, but I know how to figure it out by experiment. I understand the basic concepts, and know where to find the formulas necessary for calculating it as well.
This knowledge is usually not of much worth in my day to day dealings (some of it is - after all, I work in technical field), but it does help me when faced with abnormal situations to know how to approach them - how to feel them out. It gives me insights that those who are more ignorant of such things miss.
The gripping hand is, while it is not particularly helpful to stuff your memory with formulas, definitions, etc. that you are likely never to see again, it is good to become aquainted with the basic ideas that underpin those equations; to understand first what a limit is about, and then why the definition of the limit is as it is, even if you don't remember the actual statement. The principles that the formulas reveal can be an important guide in dealing with many situations.
A study I heard about long ago, and can only talk about in the broadest terms, involved reseaching business people to find what factors in their background & education had the greatest effect on their success. The study looked for correlations that would best predict the future success of those entering the business world. Of all the factors studied, the one with the greatest correlation was calculus. Those with heavier exposure to (and presumably success with) calculus tended to be more successful that than those without it. And this factor outweighed every other educational and background consideration.
I used to tell my Business Calculus students this. They met the news with disbelief, because they had all talked to the Business professors, and the upperclassmen (and women - can't think of a gender-neutral equivalent that doesn't make me cringe) and had been told that after they got out of Business Calculus, they would never see any of that stuff again. "But", I told them, "go out into the real world - ask these successful business people how their calculus knowledge has helped them to succeed - what will they tell you?" The answer is: "Nah - I never used that stuff again!"
?? How is it that calculus has the greatest correlation with business success, when - with rare exception - business executives make no use of it ?? The reason, I believe, is not that calculus is a useful tool for everyday business, but because calculus is a very useful tool for education!. By learning the subtleties of logical thought that go into a good development of calculus, the student picks up more than just the theory itself. He or she also learns how to approach problems in a logical fashion. How to look at information from different viewpoints than the obvious ones. That not only is the value important, but so is how it changes, and how to interpret that change. All of these are skills that business people use all the time, without realizing that they were mostly developed by studying calculus.
(By the way, I am not denying the important applications of calculus and higher mathematics to the business world. Many procedures used daily in financial markets are the result of some very high powered mathematical research. But business people learn and apply the results of that research without learning the mathematics behind it. That's why the students are all told they won't see calculus again. Calculus underlies what they study, but they are never taught that part of it. And as William has confessed, and as I was guilty of too, the natural tendency is to do the minimum needed to get by, rather than really dig into anything outside your main interest - even if it undergirds it. It was my job to convince them that just learning the results was not enough.)
It is not necessary to learn the mathematics of any particular arena in detail. In fact, it's generally detrimental - you would need to spend so much time on the mathematics, you would not be able to comprehend the application. But it is better to know the principles involved. Knowing how to use z-transform tables may give you what you need for right now, but if you don't have any comprehension of what z-transforms are and how they work, then you are going to be at a complete loss when you find a situation that isn't in the table.
So why don't these courses emphasize the theory more? Two reasons, one of which BNC has amply stated: time. It is not just that there is not enough time in the classes to cover it in sufficient detail, but overall, people have a limited amount of time for everything. At some point, devoting more time on one important subject means taking time away from something else that may be even more valuable.
The other reason is that the students simply are not ready to comprehend all the details. We learn in layers. Our first introduction to a subject barely covers the surface, though we feel like we have been dragged through the depths. It is only later, after the information has had a chance to "sink in", that we have the understanding to look deeper. The latest posts to the 0.999... thread are from a student who has taken Calc 2, and states that he believes he has a pretty good grasp of the subject. Yet his posts reveal several miscomprehensions he has. I don't blame him. When I finished Calc 2, I also was very impressed with my own understanding of the subject. It was two more years before I learned just how bad my understanding was.
A rather trivial example: Once in a physics class, the professor had integrated a function and evaluated the constant to come up with a final function that was never zero. I asked him after class how this could possibly be, since the integral was the area under the function (-below yada yada yada) between the value of x and some fixed point x0. Clearly any integral must be zero at some point x0. The constant C merely depended on where this x0 is located. His answer forced me to finally confront the difference between the integral and the anti-derivative.)
If the teacher delves too deeply, he loses his audience, and then he is no longer teaching and they are no longer learning. I learned this in front of the class, not when I was sitting in it. I am truly embarrassed to recall my first attempt at teaching. Alas for the poor students, it was Trigonometry, and I have always had a love for all those trigonometric identities, and how you could figure out "exact values" for so many points. Which is more helpful to the student: cos 15o [approx] 0.966, available from any calculator, or cos 15o = ([sqrt]6 + [sqrt]2)/4, available by trig identities? The answer is obvious now, but it wasn't where I put my teaching efforts then!
Alright, I'll stop rambling and shut up now. 
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towr
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #18 on: Sep 10th, 2003, 12:10am » |
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on Sep 9th, 2003, 6:55pm, Icarus wrote:| Of all the factors studied, the one with the greatest correlation was calculus. Those with heavier exposure to (and presumably success with) calculus tended to be more successful that than those without it. And this factor outweighed every other educational and background consideration. |
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Ah, the perils of statistics.. I think it's just a common cause problem. I don't think that calculus makes them better businessmen, but that what makes them better businessmen also leads them to a larger interest in calculus (and probably other similar interests as well).
I'm sure that in some sense it's simply related to ambition, why settle for doing the bare minimum you need to, when you can also try to excel. And I also think calculus gives more status and opportunity then f.i. geography.
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Wikipedia, Google, Mathworld, Integer sequence DB
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Icarus
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Re: [Apparently Not] New Insight on lim x->0 sin(x
« Reply #19 on: Sep 10th, 2003, 3:09pm » |
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Possibly - but don't tell those bus. calc. students that!
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I wonder: Which is larger
When their digits are reversed? " - Anonymous
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