GILBERT STRANG: OK well,
here we're at the beginning. And that I think it's worth
thinking about what we know. Calculus. Differential equations is the
big application of calculus, so it's kind of interesting to
see what part of calculus, what information and what
ideas from calculus, actually get used in
differential equations. And I'm going to
show you what I see, and it's not everything
by any means, it's some basic ideas, but not
all the details you learned. So I'm not saying
forget all those, but just focus on what matters. OK. So the calculus you
need is my topic. And the first thing
is, you really do need to know basic derivatives. The derivative of x to
the n, the derivative of sine and cosine. Above all, the derivative of e
to the x, which is e to the x. The derivative of e to
the x is e to the x. That's the wonderful equation
that is solved by e to the x. Dy dt equals y. We'll have to do more with that. And then the inverse function
related to the exponential is the logarithm. With that special
derivative of 1/x. OK. But you know those. Secondly, out of those
few specific facts, you can create the derivatives
of an enormous array of functions using
the key rules. The derivative of f
plus g is the derivative of f plus the derivative of g. Derivative is a
linear operation. The product rule fg
prime plus gf prime. The quotient rule. Who can remember that? And above all, the chain rule. The derivative of this--
of that chain of functions, that composite function is the
derivative of f with respect to g times the derivative
of g with respect to x. That's really-- that
it's chains of functions that really blow open the
functions or we can deal with. OK. And then the
fundamental theorem. So the fundamental theorem
involves the derivative and the integral. And it says that one is the
inverse operation to the other. The derivative of the integral
of a function is this. Here is y and the
integral goes from 0 to x I don't care what
that dummy variable is. I can-- I'll change that
dummy variable to t. Whatever. I don't care. [? ET ?] to show
the dummy variable. The x is the limit
of integration. I won't discuss that
fundamental theorem, but it certainly is
fundamental and I'll use it. Maybe that's better. I'll use the fundamental
theorem right away. So-- but remember what it says. It says that if you take a
function, you integrate it, you take the derivative, you
get the function back again. OK can I apply
that to a really-- I see this as a key example
in differential equations. And let me show you the
function I have in mind. The function I have in
mind, I'll call it y, is the interval from 0 to t. So it's a function
of t then, time, It's the integral of this,
e to the t minus s. Some function. That's a remarkable
formula for the solution to a basic
differential equation. So with this, that
solves the equation dy dt equals y plus q of t. So when I see that equation
and we'll see it again and we'll derive this
formula, but now I want to just use the
fundamental theorem of calculus to check the formula. What as we created-- as we
derive the formula-- well it won't be wrong because
our derivation will be good. But also, it would
be nice, I just think if you plug that in,
to that differential equation it's solved. OK so I want to take
the derivative of that. That's my job. And that's why I do it here
because it uses all the rules. OK to take that
derivative, I notice the t is appearing there
in the usual place, and it's also
inside the integral. But this is a simple function. I can take e to the
t-- I'm going to take e to the t out of the--
outside the integral. e to the t. So I have a function t
times another function of t. I'm going to use
the product rule and show that the
derivative of that product is one term will be y and
the other term will be q. Can I just apply the product
rule to this function that I've pulled out of a
hat, but you'll see it again. OK so it's a product
of this times this. So the derivative dy dt
is-- the product rule says take the derivative
of [INAUDIBLE] that is e to the [INAUDIBLE]. Plus, the first thing times
the derivative of the second. Now I'm using the product rule. It just-- you have to notice
that e to the t came twice because it is there and
its derivative is the same. OK now, what's the
derivative of that? Fundamental theorem of calculus. We've integrated something, I
want to take its derivative, so I get that something. I get e to the minus tq of t. That's the fundamental theorem. Are you good with that? So let's just look
and see what we have. First term was exactly y. Exactly what is
above because when I took the derivative
of the first guy, the f it didn't change
it, so I still have y. What have I-- what
do I have here? E to the t times e to
the minus t is one. So e to the t cancels
e to the minus t and I'm left with q
of t Just what I want. So the two terms
from the product rule are the two terms in the
differential equation. I just think as you saw the
fundamental theorem was needed right there to find the
derivative of what's in that box, is what's
in those parentheses. I just like that the use
of the fundamental theorem. OK one more topic
of calculus we need. And here we go. So it involves the
tangent line to the graph. This tangent to the graph. So it's a straight line and what
we need is y of t plus delta t. That's taking any
function, maybe you'd rather I just
called the function f. A function at a point
a little beyond t, is approximately
the function at t plus the correction because
it-- plus a delta f, right? A delta f. And what's the delta
f approximately? It's approximately delta t
times the derivative at t. That-- there's a lot of
symbols on that line, but it expresses the most basic
fact of differential calculus. If I put that f of t on
this side with a minus sign, then I have delta f. If I divide by that delta
t, then the same rule is saying that this is
approximately df dt. That's a fundamental
idea of calculus, that the derivative
is quite close. At the point t-- the
derivative at the point t is close to delta f
divided by delta t. It changes over a
short time interval. OK so that's the tangent line
because it starts with that's the constant term. It's a function of delta
t and that's the slope. Just draw a picture. So I'm drawing a picture here. So let me draw a
graph of-- oh there's the graph of e to the t. So it starts up with slope 1. Let me give it a
little slope here. OK the tangent line,
and of course it comes down here Not below. So the tangent
line is that line. That's the tangent line. That's this approximation to f. And you see as I-- here
is t equals 0 let's say. And here's t equal delta t. And you see if I
take a big step, my line is far from the curve. And we want to get closer. So the way to get
closer is we have to take into
account the bending. The curve is bending. What derivative tells
us about bending? That is delta t squared
times the second derivative. One half. It turns out a one
half shows in there. So this is the term that
changes the tangent line, to a tangent parabola. It notices the
bending at that point. The second derivative
at that point. So it curves up. It doesn't follow it perfectly,
but as well-- much better than the other. So this is the line. Here is the parabola. And here is the function. The real one. OK. I won't review the theory there
that it pulls out that one half, but you could check it. Now finally, what if we
want to do even better? Well we need to take into
account the third derivative and then the fourth
derivative and so on, and if we get all
those derivatives then, all of them that means,
we will be at the function because that's a nice
function, e to the t. We can recreate that
function from knowing its height, its
slope, its bending and all the rest of the terms. So there's a whole lot
more-- Infinitely many terms. That one over two-- the
good way to think of one over two, one half, is one over
two factorial, two times one. Because this is one
over n factorial, times t to the
nth, pretty small, times the nth derivative
of the function. And keep going. That's called the Taylor
series named after Taylor. Kind of frightening at first. It's frightening because it's
got infinitely many terms. And the terms are getting
a little more comp-- For most functions,
you really don't want to compute the nth derivative. For e to the t, I don't mind
computing the nth derivative because it's still e to the
t, but usually that's-- this isn't so practical. [INAUDIBLE] very practical. Tangent parabola,
quite practical. Higher order terms, less--
much less practical. But the formula is
beautiful because you see the pattern, that's
really what mathematics is about patterns,
and here you're seeing the pattern in
the higher, higher terms. They all fit that pattern and
when you add up all the terms, if you have a nice function,
then the approximation becomes perfect and you
would have equality. So to end this lecture,
approximate to equal provided we have a nice function. And those are the best functions
of mathematics and exponential is of course one of them. OK that's calculus. Well, part of calculus. Thank you.
