Poetry in Calculus

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JeffM

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DrPhil said:
The idea of "name changing" is not useful. The only function that does NOT "change its name" when differentiated is the exponential:

\(\displaystyle \displaystyle \frac{d}{dx}\left( e^x\right) = e^x\)

No other function besides the exponential is its own derivative.

The reason there is still a cotangent in the first example is because of the power law.

\(\displaystyle \displaystyle \frac{d}{dx}\left( u^n\right) = n\ u^{n-1}\ \frac{du}{dx}\)

The "logic" of trigonometric derivatives is that the derivative of the sine is the cosine, and the the derivative of the cosine is the negative of the sine. All other derivatives of trig functions follow from those two - which are the only ones I have memorized. There is symmetry and beauty in that truth. That's all I know on earth, and all I need to know.
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Hey Dr. Phil. What a place this is. No "Two Cultures" here. A little Keats is not a bad thing. In fact, a lot of Keats is not a bad thing.
 
JeffM said:
Hey Dr. Phil. What a place this is. No "Two Cultures" here. A little Keats is not a bad thing. In fact, a lot of Keats is not a bad thing.
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And a bit of Shelly and Swinburn...
 
Subhotosh Khan said:
And a bit of Shelly and Swinburn...
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I always feel that liking Swinburne proves that I am somewhat trivial.

"Red sprays of ruined springs" sounds wonderful with the "r"s and the "p"s and "pr"s, but I am still not sure it makes much sense. Nevertheless, I do find him wonderful. I get captured by the sound and worry not about the sense.
 
JeffM said:
I always feel that liking Swinburne proves that I am somewhat trivial.

"Red sprays of ruined springs" sounds wonderful with the "r"s and the "p"s and "pr"s, but I am still not sure it makes much sense. Nevertheless, I do find him wonderful. I get captured by the sound and worry not about the sense.
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But what images!!

Reading those lines, I remember watching the creek near our house - floating red azaleas covering the rocks and bumping around and getting stuck again (aftermath of torrential spring rain) - it just looked like "Red sprays of ruined springs".

I love the sound and the images I can impose on it - just like impressionistic paintings ..... hang the meaning of it....
 
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