world events: July 2009 Archives

Rad Intro to Calculus

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Today is the 40th anniversary of the first manned lunar landing (1969) .. a historic event for techies .. cuz it was techies who put us there. And techies used » calculus to put a man on the moon (some 240,000 miles away).

MoonAt the heart of calculus lies the notion of » limits (a concept I'm quite familiar with, unfortunately).

Calculus was invented to solve (among other things) the problem associated with finding the instantaneous rate-of-change .. as visualized by the slope of a line tangential to any given point on a curve (of a graph).

To calculate a slope, you might recall (dust off them algebraic brain cells) we pick two representative points, find their difference and divide the » "rise by the run" .. the change-in-Y / change-in-X (.. commonly referred to as "delta-Y over delta-X").

As the change in the X coordinates (recall from basic Algebra) gets smaller and smaller, we get closer and closer to determining the slope (rate-of-change) at a particular point.

The problem however .. is that a point has no size, so the "change" or 'difference' (in the X coordinates) becomes zero. And dividing anything by zero is a major mathematical no-no. (Defined as "undefined" .. a mathematical black hole that will crash your computer.)

See t=13:00 here, and especiaaly t=13:30.


The concept of » limits was introduced to address this problem. Imagine standing in your living room, and walking half the distance to the furthest wall. Then walk half the distance again. And again & again.

Each time, you keep getting closer & closer. But .. you'll never actually reach the wall (cuz you keep going only half the distance). A hundred years from now, you'll be very, very close (to the wall), but still not quite there.

Getting closer & closer to the wall is analogous to decreasing the size of the difference between the two X coordinates along a curve plotted on a standard graph (which contains an X & Y axis). But the limit (drum-roll, please .. here it comes) is » the wall! .. even tho, in reality, you never actually get there.

That's why the notion of a limit represents a mathemetical "concept" (not reality). If you think about it, you can't really have an instantaneous rate-of-change (.. cuz nothing can change in an instant, cuz an instant contains no time). And the word 'rate' implies "per-unit-something." That 'something can be (and often is) » time.

That's also why the result is called/termed a 'derivative' .. cuz you can't get there with conventional mathematical manipulations. It's kinda like what that old farmer told me down South when I asked for directions » "Son, you can't get there from here." =)

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