TWSB: Back on the Chain Gang
This week’s science-related blog has to do with CALCULUS because calculus is awesome and because we actually just got tested on using the Chain Rule on Wednesday. The Physicist over at AskAMathematician.com responded to this question awhile back: “Is there an intuitive proof for the chain rule?” Which, when you think about it, is a really interesting question. We’re taught in calculus that when we’ve got a function “embedded” within another function [e.g., f(g(x))], when we take the derivative of that function, we take the derivative of the “outside function” f’(x) and then multiply it by the derivative of the “inside function” g’(x) to get f’(g(x))g’(x).
But why the heck do we need to do that?
As the Physicist very elegantly points out, it all has to do with slope. When you multiply a function by some amount, you squish it up by that same amount. Using their same example, say you’ve got a function f(x) and it’s “squished” function f(2x). These two functions are the same when x=6 and x=3 respectively, but that’s not the case for their slopes. The squished function 2x will have a steeper one by two. When you take the derivative of this function, then, you have to re-multiply it by two to deal with the squishing.
So how does this work out in general? If you replace the 2x with a more complicated function g(x), you get f(g(x)), and the slope of the whole thing depends, then, on g(x). So when finding the slope (taking the derivative) of f(g(x)), you have to re-multiply it by the slope of g(x) (or the derivative of g(x)) to deal with whatever g(x) is doing to the whole of f(g(x)).
And there’s your chain rule!
Take a look at the actual article. They’ve got pictures and actually have the derivatives and functions written out like I can’t do here.
VERY FREAKING COOL, PEOPLE.
This Week’s Science Blog: Taking You to a Higher Dimension
Okay. So this week’s science blog is really, really awesome, but I think if I try to summarize it and put it in my own words here it’ll lose a lot. So I’ll just copy down a few highlights. This is another question answered by the Physicist at AskAMathematician.com: What would life be like in higher dimensions?
Seriously. Really cool answer.
Highlights:
- In 4 or more dimensions orbits are always unstable, and in 1 dimension the idea of an orbit doesn’t even make sense.
- f you set off a firecracker in 3, 5, 7, etc. dimensions, then you’ll see and hear the explosion for a moment, and that’s it. If you set of a firecracker in 4, 6, 8, etc. dimensions, then you’ll see and hear the explosion intensely for a moment, but will continue to see and hear it for a while…it may not even be possible to understand people when they speak.
- Which elements are stable, and the nature of chemical bonds between them, would be completely rearranged.
- Every element after helium would adopt weird new properties, and the periodic table would be longer left-right and shorter up-down.
TWSB: Well, it certainly would make the cartographer’s job easier…
My favorite TWSBs involve hypothetical situations. Today, the hypothetical situation in question is as follows: what would earth be like if it were a cube instead of spherical?
Quick summary of coolest points:
– oceans and atmosphere would “puddle” into the centers of the six sides of the cube.
– life would be isolated to the perimeters of these ocean puddles, and it is very likely that each of the six regions would be completely isolated and each region would pretty much be its own biosphere.
– huge seasonal temperature fluctuations due to—again—the isolation of the atmospheres.
– sunrise and sunset would be sudden, not the gradual niceness we see on regular earth.
How cool, eh?
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