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GILBERT STRANG: OK well,
here we're at the beginning.

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And that I think it's worth
thinking about what we know.

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Calculus.

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Differential equations is the
big application of calculus,

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so it's kind of interesting to
see what part of calculus, what

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information and what
ideas from calculus,

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actually get used in
differential equations.

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And I'm going to
show you what I see,

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and it's not everything
by any means,

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it's some basic ideas, but not
all the details you learned.

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So I'm not saying
forget all those,

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but just focus on what matters.

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OK.

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So the calculus you
need is my topic.

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And the first thing
is, you really do

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need to know basic derivatives.

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The derivative of x to
the n, the derivative

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of sine and cosine.

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Above all, the derivative of e
to the x, which is e to the x.

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The derivative of e to
the x is e to the x.

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That's the wonderful equation
that is solved by e to the x.

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Dy dt equals y.

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We'll have to do more with that.

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And then the inverse function
related to the exponential

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is the logarithm.

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With that special
derivative of 1/x.

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OK.

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But you know those.

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Secondly, out of those
few specific facts,

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you can create the derivatives
of an enormous array

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of functions using
the key rules.

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The derivative of f
plus g is the derivative

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of f plus the derivative of g.

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Derivative is a
linear operation.

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The product rule fg
prime plus gf prime.

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The quotient rule.

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Who can remember that?

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And above all, the chain rule.

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The derivative of this--
of that chain of functions,

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that composite function is the
derivative of f with respect

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to g times the derivative
of g with respect to x.

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That's really-- that
it's chains of functions

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that really blow open the
functions or we can deal with.

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OK.

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And then the
fundamental theorem.

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So the fundamental theorem
involves the derivative

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and the integral.

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And it says that one is the
inverse operation to the other.

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The derivative of the integral
of a function is this.

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Here is y and the
integral goes from 0

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to x I don't care what
that dummy variable is.

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I can-- I'll change that
dummy variable to t.

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Whatever.

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I don't care.

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[? ET ?] to show
the dummy variable.

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The x is the limit
of integration.

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I won't discuss that
fundamental theorem,

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but it certainly is
fundamental and I'll use it.

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Maybe that's better.

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I'll use the fundamental
theorem right away.

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So-- but remember what it says.

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It says that if you take a
function, you integrate it,

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you take the derivative, you
get the function back again.

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OK can I apply
that to a really--

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I see this as a key example
in differential equations.

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And let me show you the
function I have in mind.

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The function I have in
mind, I'll call it y,

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is the interval from 0 to t.

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So it's a function
of t then, time, It's

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the integral of this,
e to the t minus s.

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Some function.

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That's a remarkable
formula for the solution

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to a basic
differential equation.

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So with this, that
solves the equation dy

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dt equals y plus q of t.

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So when I see that equation
and we'll see it again

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and we'll derive this
formula, but now I

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want to just use the
fundamental theorem of calculus

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to check the formula.

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What as we created-- as we
derive the formula-- well

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it won't be wrong because
our derivation will be good.

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But also, it would
be nice, I just

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think if you plug that in,
to that differential equation

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it's solved.

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OK so I want to take
the derivative of that.

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That's my job.

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And that's why I do it here
because it uses all the rules.

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OK to take that
derivative, I notice

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the t is appearing there
in the usual place,

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and it's also
inside the integral.

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But this is a simple function.

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I can take e to the
t-- I'm going to take e

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to the t out of the--
outside the integral.

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e to the t.

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So I have a function t
times another function of t.

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I'm going to use
the product rule

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and show that the
derivative of that product

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is one term will be y and
the other term will be q.

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Can I just apply the product
rule to this function

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that I've pulled out of a
hat, but you'll see it again.

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OK so it's a product
of this times this.

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So the derivative dy dt
is-- the product rule

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says take the derivative
of [INAUDIBLE] that

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is e to the [INAUDIBLE].

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Plus, the first thing times
the derivative of the second.

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Now I'm using the product rule.

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It just-- you have to notice
that e to the t came twice

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because it is there and
its derivative is the same.

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OK now, what's the
derivative of that?

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Fundamental theorem of calculus.

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We've integrated something, I
want to take its derivative,

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so I get that something.

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I get e to the minus tq of t.

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That's the fundamental theorem.

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Are you good with that?

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So let's just look
and see what we have.

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First term was exactly y.

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Exactly what is
above because when

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I took the derivative
of the first guy,

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the f it didn't change
it, so I still have y.

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What have I-- what
do I have here?

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E to the t times e to
the minus t is one.

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So e to the t cancels
e to the minus t

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and I'm left with q
of t Just what I want.

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So the two terms
from the product rule

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are the two terms in the
differential equation.

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I just think as you saw the
fundamental theorem was needed

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right there to find the
derivative of what's

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in that box, is what's
in those parentheses.

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I just like that the use
of the fundamental theorem.

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OK one more topic
of calculus we need.

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And here we go.

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So it involves the
tangent line to the graph.

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This tangent to the graph.

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So it's a straight line and what
we need is y of t plus delta t.

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That's taking any
function, maybe

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you'd rather I just
called the function f.

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A function at a point
a little beyond t,

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is approximately
the function at t

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plus the correction because
it-- plus a delta f, right?

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A delta f.

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And what's the delta
f approximately?

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It's approximately delta t
times the derivative at t.

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That-- there's a lot of
symbols on that line,

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but it expresses the most basic
fact of differential calculus.

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If I put that f of t on
this side with a minus sign,

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then I have delta f.

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If I divide by that delta
t, then the same rule

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is saying that this is
approximately df dt.

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That's a fundamental
idea of calculus,

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that the derivative
is quite close.

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At the point t-- the
derivative at the point t

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is close to delta f
divided by delta t.

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It changes over a
short time interval.

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OK so that's the tangent line
because it starts with that's

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the constant term.

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It's a function of delta
t and that's the slope.

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Just draw a picture.

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So I'm drawing a picture here.

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So let me draw a
graph of-- oh there's

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the graph of e to the t.

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So it starts up with slope 1.

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Let me give it a
little slope here.

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OK the tangent line,
and of course it

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comes down here Not below.

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So the tangent
line is that line.

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That's the tangent line.

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That's this approximation to f.

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And you see as I-- here
is t equals 0 let's say.

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And here's t equal delta t.

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And you see if I
take a big step,

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my line is far from the curve.

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And we want to get closer.

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So the way to get
closer is we have

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to take into
account the bending.

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The curve is bending.

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What derivative tells
us about bending?

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That is delta t squared
times the second derivative.

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One half.

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It turns out a one
half shows in there.

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So this is the term that
changes the tangent line,

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to a tangent parabola.

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It notices the
bending at that point.

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The second derivative
at that point.

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So it curves up.

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It doesn't follow it perfectly,
but as well-- much better

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than the other.

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So this is the line.

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Here is the parabola.

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And here is the function.

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The real one.

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OK.

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I won't review the theory there
that it pulls out that one

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half, but you could check it.

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Now finally, what if we
want to do even better?

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Well we need to take into
account the third derivative

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and then the fourth
derivative and so on,

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and if we get all
those derivatives then,

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all of them that means,
we will be at the function

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because that's a nice
function, e to the t.

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We can recreate that
function from knowing

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its height, its
slope, its bending

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and all the rest of the terms.

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So there's a whole lot
more-- Infinitely many terms.

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That one over two-- the
good way to think of one

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over two, one half, is one over
two factorial, two times one.

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00:12:55,830 --> 00:12:59,650
Because this is one
over n factorial,

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00:12:59,650 --> 00:13:04,170
times t to the
nth, pretty small,

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00:13:04,170 --> 00:13:07,175
times the nth derivative
of the function.

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And keep going.

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00:13:13,020 --> 00:13:19,310
That's called the Taylor
series named after Taylor.

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00:13:19,310 --> 00:13:25,550
Kind of frightening at first.

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00:13:25,550 --> 00:13:28,910
It's frightening because it's
got infinitely many terms.

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00:13:28,910 --> 00:13:31,731
And the terms are getting
a little more comp--

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00:13:31,731 --> 00:13:33,740
For most functions,
you really don't want

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to compute the nth derivative.

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00:13:35,680 --> 00:13:39,100
For e to the t, I don't mind
computing the nth derivative

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00:13:39,100 --> 00:13:44,790
because it's still e to the
t, but usually that's-- this

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isn't so practical.

220
00:13:46,950 --> 00:13:48,570
[INAUDIBLE] very practical.

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00:13:48,570 --> 00:13:51,380
Tangent parabola,
quite practical.

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00:13:51,380 --> 00:13:55,150
Higher order terms, less--
much less practical.

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00:13:55,150 --> 00:13:59,210
But the formula is
beautiful because you

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see the pattern, that's
really what mathematics

225
00:14:02,340 --> 00:14:04,300
is about patterns,
and here you're

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00:14:04,300 --> 00:14:08,880
seeing the pattern in
the higher, higher terms.

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00:14:08,880 --> 00:14:14,050
They all fit that pattern and
when you add up all the terms,

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if you have a nice function,
then the approximation

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00:14:18,280 --> 00:14:21,560
becomes perfect and you
would have equality.

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00:14:21,560 --> 00:14:27,800
So to end this lecture,
approximate to equal provided

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we have a nice function.

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And those are the best functions
of mathematics and exponential

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00:14:34,510 --> 00:14:36,010
is of course one of them.

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OK that's calculus.

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00:14:39,030 --> 00:14:40,990
Well, part of calculus.

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Thank you.