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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 16
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PROFESSOR: And this last little bit is something which is not yet on the Web. But,
anyway, when I was walking out of the room last time, I noticed that I'd written
down the wrong formula for c1 - c1. There's a misprint, there's a minus sign that's
wrong. I claimed last time that c1 - c2 was + 1/2. But, actually, it's - 1/2. If you go
through the calculation that we did with the antiderivative of sine x cosine x, we get
these two possible answers. And if they're to be equal, then if we just subtract them
we get c1 - c2 + 1/2 = 0. So c1 - c2 = 1/2. So, those are all of the correction. Again,
everything here will be on the Web. But just wanted to make it all clear to you.
So here we are. This is our last day of the second unit, Applications of
Differentiation. And I have one of the most fun topics to introduce to you. Which is
differential equations. Now, we have a whole course on differential equations, which
is called 18.03. And so we're only going to do just a little bit. But I'm going to teach
you one technique. Which fits in precisely with what we've been doing already. Which
is differentials. The first and simplest kind of differential equation dy/dx = some
function, f (x). Now, that's a perfectly good differential equation. And we already
discussed last time that the solution; that is, the function y, is going to be the
antiderivative, or the integral, of x. Now, for the purposes of today, we're going to
consider this problem to be solved. That is, you can always do this. You can always
take antiderivatives. And for our purposes now, that is for now, we only have one
technique to find antiderivatives. And that's called substitution.
It has a very small variant, which we called advanced guessing. And that works just
as well. And that's basically all that you'll ever need to do. As a practical matter,
these are the ones you'll face for now. Ones that you can actually see what the
answer is, or you'll have to make a substitution. Now, the first tricky example, or the
first maybe interesting example of a differential equation, which I'll call Example 2, is