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Example of a limit

Suppose you know that the position of an object depends on time as tex2html_wrap_inline1384 (here I'm being naughty and forgetting about units for the moment). Lets calculate the instantaneous velocity at t = 1.

So in this case tex2html_wrap_inline1388 , and tex2html_wrap_inline1390 . We'll want to try different values of tex2html_wrap_inline1358 and verify that we do appear to converge to a sensible final answer.

Let's start with tex2html_wrap_inline1394 , then tex2html_wrap_inline1396 so from eq. 3.1 we have, tex2html_wrap_inline1360 over this time interval is

equation131

OK that's fine, but this is clearly not an infinitesimal interval. Let's shrink the interval by 1/2 so that tex2html_wrap_inline1402 . Then

equation135

If we shrink the interval tex2html_wrap_inline1376 even further, so that tex2html_wrap_inline1406 then going through the same steps gives tex2html_wrap_inline1408 . If we now try tex2html_wrap_inline1410 , then tex2html_wrap_inline1412 .

It looks pretty clear that as tex2html_wrap_inline1414 we're coming up with an instantaneous velocity of 2.

This is what you'd expect since the derivative of tex2html_wrap_inline1418 is 2t . Evaluating this at t=1, we get 2.



Joshua Deutsch
Mon Jan 6 00:05:26 PST 1997