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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 10
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PROF. JERISON: So, we're ready to begin Lecture 10, and what I'm going to begin
with is by finishing up some things from last time. We'll talk about approximations,
and I want to fill in a number of comments and get you a little bit more oriented in
the point of view that I'm trying to express about approximations. So, first of all, I
want to remind you of the actual applied example that I wrote down last time. So
that was this business here. There was something from special relativity. And the
approximation that we used was the linear approximation, with a - 1/2 power that
comes out to be t( 1 1/2 v^2 / C^2).
I want to reiterate why this is a useful way of thinking of things. And why this is that
this comes up in real life. Why this is maybe more important than everything that
I've taught you about technically so far. So, first of all, what this is telling us is the
change in t / t, if you do the arithmetic here and subtract t that's using the change in
t is t' - t here. If you work that out, this is approximately the same as 1/2 (v^2 /
C^2). So what is this saying? This is saying that if you have this satellite, which is
going at speed v, and little c is the speed of light, then the change in the watch down
here on earth, relative to the time on the satellite, is going to be proportional to this
ratio here. So, physically, this makes sense. This is time divided by time. And this is
velocity squared divided by velocity squared. So, in each case, the units divide out.
So this is a dimensionless quantity. And this is a dimensionless quantity.
And the only point here that we're trying to make is just this notion of
proportionality. So I want to write this down. Just, in summary. So the error fraction,
if you like, which is sort of the number of significant digits that we have in our
measurement, is proportional, in this case, to this quantity. It happens to be
proportional to this quantity here. And the factor is, happens to be, 1/2. So these
proportionality factors are what we're looking for. Their rates of change. Their rates
of change of something with respect to something else.
Now, on your homework, you have something rather similar to this. So in Problem,