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The process of letting delta t get smaller and smaller
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so that one comes up with an instantaneous rate of change
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can also be viewed graphically.
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This may give you another way to picture this process
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and, more generally, to think about what
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differential calculus is all about.
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So, as a starting point suppose we have a curve -
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some function that may be looks like this,
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and this curve could represent the growth of some process -
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who knows, it doesn't matter - it's just a curve,
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and then somebody asked you to find the slope of the curve -
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that would be like the instantaneous velocity,
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or instantaneous speed, or growth rate.
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You might say, well I don't know how to find the slope of a curve -
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a slope is a property of a straight line
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and the curve isn't a straight line
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but aha, then you might get an idea.
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You look at the curve - I'm looking at it now,
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well it looks like a curve - I can see it bending
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however, if I look really, really close -
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I zoom in until my eye is almost against it,
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it looks like a straight line
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and I know how to find the slope of a straight line,
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so, to find the slope at a point,
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I just zoom in until it looks sufficiently line-like
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and then I find the slope - no big deal.
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So this idea - that, if you take a curve
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and you zoom in on it, that it looks like a line.
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and you can calculate the slope of the line,
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that's the one big idea
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behind all of differential calculus.
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So, in the first term of a calculus sequence
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one typically spends most of the time
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defining this a lot more carefully,
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and then exploring the consequences of it.
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Here's another way to illustrate the fact
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that zooming in on a curve
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makes it look like a straight line.
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So, here's a curved function,
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viewed from this distance,
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it definitely does not look straight,
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and now imagine zooming in -
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so I'm moving the function
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closer and closer to the camera
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and you should see that the curve starts to
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look more and more like a straight line.
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If it doesn't look sufficiently straight you could zoom in more.
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I'm limited because eventually I'm going to hit
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the lens cover of the camera and I can't zoom in any more
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but if you zoom in more and more, and more and more,
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the line looks straighter and straighter and straighter,
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and you can try this at home -
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draw a curve on a piece of paper, zoom in
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and you'll see that it really does look like a straight line,
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so differential calculus works.