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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 15
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PROFESSOR: Today we're moving on from theoretical things from the mean value
theorem to the introduction to what's going to occupy us for the whole rest of the
course, which is integration. So, in order to introduce that subject, I need to
introduce for you a new notation, which is called differentials. I'm going to tell you
what a differential is, and we'll get used to using it over time. If you have a function
which is y = f ( x), then the differential of y is going to be denoted dy, and it's by
definition f' ( x ) dx. So here's the notation. And because y is really equal to f,
sometimes we also call it the differential of f. It's also called the differential of f.
That's the notation, and it's the same thing as what happens if you formally just take
this dx, act like it's a number and divide it into dy. So it means the same thing as
this statement here. And this is more or less the Leibniz, not Leibniz, interpretation
of derivatives. Of a derivative as a ratio of these so called differentials. It's a ratio of
what are known as infinitesimals.
Now, this is kind of a vague notion, this little bit here being an infinitesimal. It's sort
of like an infinitely small quantity. And Leibniz perfected the idea of dealing with
these intuitively. And subsequently, mathematicians use them all the time. They're
way more effective than the notation that Newton used. You might think that
notations are a small matter, but they allow you to think much faster, sometimes.
When you have the right names and the right symbols for everything. And in this
case it made it very big difference. Leibniz's notation was adopted on the Continent
and Newton dominated in Britain and, as a result, the British fell behind by one or
two hundred years in the development of calculus. It was really a serious matter. So
it's really well worth your while to get used to this idea of ratios. And it comes up all
over the place, both in this class and also in multivariable calculus. It's used in many
contexts.
So first of all, just to go a little bit easy. We'll illustrate it by its use in linear