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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 1
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Professor: So, again welcome to 18.01. We're getting started today with what we're
calling Unit One, a highly imaginative title. And it's differentiation. So, let me first tell
you, briefly, what's in store in the next couple of weeks. The main topic today is
what is a derivative. And, we're going to look at this from several different points of
view, and the first one is the geometric interpretation. That's what we'll spend most
of today on. And then, we'll also talk about a physical interpretation of what a
derivative is.
And then there's going to be something else which I guess is maybe the reason why
Calculus is so fundamental, and why we always start with it in most science and
engineering schools, which is the importance of derivatives, of this, to all
measurements. So that means pretty much every place. That means in science, in
engineering, in economics, in political science, etc. Polling, lots of commercial
applications, just about everything.
Now, that's what we'll be getting started with, and then there's another thing that
we're gonna do in this unit, which is we're going to explain how to differentiate
anything. So, how to differentiate any function you know. And that's kind of a tall
order, but let me just give you an example. If you want to take the derivative - this
we'll see today is the notation for the derivative of something - of some messy
function like e ^ x arctanx. We'll work this out by the end of this unit.
All right? Anything you can think of, anything you can write down, we can
differentiate it. All right, so that's what we're gonna do, and today as I said, we're
gonna spend most of our time on this geometric interpretation. So let's begin with
that.
So here we go with the geometric interpretation of derivatives. And, what we're
going to do is just ask the geometric problem of finding the tangent line to some
graph of some function at some point. Which is to say (x0, y0). So that's the
problem that we're addressing here. Alright, so here's our problem, and now let me
show you the solution. So, well, let's graph the function. Here's it's graph. Here's