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# New Edition of Blyth Institute Journal

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From Jonathan Bartlett, one of our contributors and friends:

He writes:

In this issue we have:
The mathematics of tiling patterns
A methodological note on the transitivity of explanation
The decidability of the randomness of bitstrings
A discussion of whether information is a single, static quantity
A new, more student-friendly proof for the derivative of sin(x)
A discussion of an axiomatic moral system

the standard way usually involves some unproven (although intuitively true) lemmas. Wait a minute. What way is "standard"? Is there a non-standard way? Just to test my memory I wrote (e^ix - e^-ix)/2i and differentiated with respect to x using the ubiquitous rules for exponentials and got (e^ix + e^-ix)/2. The first expression is identical to sin(x) and the second is identical to cos(x) by the manipulation of Euler's formula. What the heck is unproven, non-standard or non-rigorous about any of the foregoing? Euler's formula? Really?groovamos
January 2, 2021
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Here's how I explain why lim (h-> 0) of (sin h)/h = 1, with h = a very small angle dx, although this may be hard to follow without a picture Imagine a point P on the unit circle for a small angle. The perpendicular from P = sin h, and h the arc length from P to the intersection with the x axis. Obviously h > sin h. However, as h gets smaller and smaller, the difference between the two gets less and less. As h gets infinitely small, the difference between the sin h and the arc h also gets infinitely small, so the ratio goes to 1. I understand this is not rigorous, but it is similar to your treating dx as a straight line, I think.Viola Lee
January 2, 2021
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Thanks, Johnny. I didn't see the paper was there to look at. I'm sorry if I come across as a contrarian, but that doesn't seem as student friendly as the standard way You say the standard way uses identifies that student wouldn't be expected to remember, but if you're learning about the derivatives of the trig functions, bring back some fundament trig identities would be something I would expect to do. But more importantly, treating the arc dx as a straight line makes the same type of intuitive "shortcut" as that that derives lim (sinx/x) as x -> 0 = 1 on the unit circle (which you mention in your footnote). Also, at least when I taught calculus, we introduced the derivatives of sine and cosine earlier than we did the implicit differentiation at (15). However, as you say, it's really not different in kind than the standard way, but it is some nice algebra that looks directly at the question of how does sin x change in respect to x.Viola Lee
January 2, 2021
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Viola - if you just scroll down on the link, the full paper is there (it's very short), it just requires a login to download it.johnnyb
January 2, 2021
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VL, I have never had a problem with that site. KFkairosfocus
January 2, 2021
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We all know that the derivative of sin is death! (Romans 6:23) Sorry, but I couldn't resist.Fasteddious
January 2, 2021
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Thanks, KF, but that requires that I make an account or sign in with Google, both of which might compromise my anonymity. I do see that he uses the unit circle and the Pythagorean theorem, which may be the way I used to "prove" that lim (h->0) (sin h)/h = 1 and lim (h->0) (1 -cos h)/h = 0. That wasn't a rigorous proof, however, but more of a motivation based on geometry. Maybe Johnny will reply here.Viola Lee
January 2, 2021
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January 2, 2021
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I'd be interested in the "new, more student-friendly proof for the derivative of sin(x)", as the standard way usually involves some unproven (although intuitively true) lemmas. I'm not going to buy the publication, but maybe Johnny would give a summary here.Viola Lee
January 2, 2021
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Thanks for the plug, Denyse!johnnyb
January 1, 2021
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