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Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limit
by jdmichal 9y ago
Is that really true before calculus? In my recollection, even in pre-calculus it was pretty easy to accurately tie constructs to physical representations. Limits is the first time I remember learning something that could not be generalized to a physical form, because it specifically dealt with what happened when a function "broke".
- hasenj 9y agoI find this sentiment somewhat surprising. Limits is incredibly easy to link to real physical things: movement, speed, acceleration. If speed is distance over time, what's the speed of a moving object at a very specific instant?
- Koshkin 9y ago"Ma'am, you were driving at ninety miles an hour." "But, sir, I have only traveled twenty miles!"
- jdmichal 9y agoAre you thinking of derivatives? I agree that those are relatively easy -- it's the rate of change for a function. Or, in geometric terms, it's the slope of the tangent line of the function's curve. Like I said, limits are used when a function "breaks", so they're naturally hard to visualize. I'll use my typical example for when I discuss teaching math by solving problems: Approximating the area under a curve using rectangles. The thinner your rectangles are, the more accurate your approximation is. So you narrow the rectangles, getting more and more accurate... Until you hit zero width, where the value of the function suddenly drops to zero... A discontinuity. A limit lets you find what the value would have been had that discontinuity not happened. So, there's a clear visualization of what the limit is letting you do in this particular case -- pretend like the rectangles of zero width don't have zero area -- but there's no clear generalization on how to visualize a limit.
- hasenj 9y agoWill, speed breaks when you consider a specific instant in thing because there is no time span nor a distance the be traveled.
- Koshkin 9y agoI think these things have already been discussed to death some time around the year 1700, when they were called something like "mathematical analysis of infinitely small magnitudes" or some such; the difference between an "infinitely small magnitude" and zero was already clearly understood. The notion of a "limit" came only later, in an attempt to put this stuff on a firm logical foundation and was (and still is) seen more of a formality than something aiding our intuition in this area.
- Koshkin 9y agoEven simple objects, like vectors, for example, can very quickly become so abstract (higher dimensions; complex vector spaces) that the physical intuition stops being useful and can even lead to mistakes. This is the point when new kinds of mathematical intuition becomes necessary, and building them is very difficult and takes time.
- jdmichal 9y agoI remember doing only simple (two-dimensional Euclidean) vectors before calculus -- length, scaling, addition, subtraction... Agreed that they obviously can get pretty hairy, but I don't think that side is exposed to people before calculus? Complex vector math was calculus III for myself.