WEBVTT
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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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Because this is one
over n factorial,
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times t to the
nth, pretty small,
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times the nth derivative
of the function.
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And keep going.
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That's called the Taylor
series named after Taylor.
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Kind of frightening at first.
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It's frightening because it's
got infinitely many terms.
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And the terms are getting
a little more comp--
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For most functions,
you really don't want
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to compute the nth derivative.
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For e to the t, I don't mind
computing the nth derivative
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because it's still e to the
t, but usually that's-- this
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isn't so practical.
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[INAUDIBLE] very practical.
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Tangent parabola,
quite practical.
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Higher order terms, less--
much less practical.
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But the formula is
beautiful because you
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see the pattern, that's
really what mathematics
00:14:02.340 --> 00:14:04.300
is about patterns,
and here you're
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seeing the pattern in
the higher, higher terms.
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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
00:14:18.280 --> 00:14:21.560
becomes perfect and you
would have equality.
00:14:21.560 --> 00:14:27.800
So to end this lecture,
approximate to equal provided
00:14:27.800 --> 00:14:30.080
we have a nice function.
00:14:30.080 --> 00:14:34.510
And those are the best functions
of mathematics and exponential
00:14:34.510 --> 00:14:36.010
is of course one of them.
00:14:36.010 --> 00:14:39.030
OK that's calculus.
00:14:39.030 --> 00:14:40.990
Well, part of calculus.
00:14:40.990 --> 00:14:42.790
Thank you.