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PROFESSOR: Welcome back
to recitation.

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In this video I'd like us to
work on the following problem.

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We're going to let capital F of
x equal the integral from 0

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to x of little f of t dt.

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And I want us to find the
quadratic approximation for

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big F near x equals 0 in
terms of the little f.

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And then I want us to answer
this question.

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What assumptions do we have
to make about little f?

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You'll see we're making some
assumptions about little f.

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I want to know what those
assumptions are.

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So I'm going to give you a
little bit of time to work on

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this problem.

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And then I'll be back and I
will work on it with you.

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Welcome back.

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Hopefully you were able to make
some good headway with

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this problem.

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So again, what we're trying to
do, is we've defined capital

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F. And we want to find the
quadratic approximation for it

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near x equals 0 in terms
of little f.

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Knowing that capital F of x is
equal to the integral from 0

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to x of little f of t dt.

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And then at the end, we'll
talk about what are the

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assumptions we've made.

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So to start off, what I actually
want to do is remind

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us what is the quadratic
approximation for capital F,

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just in terms of itself?

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Of capital F?

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So let me come over to the right
and we'll write down

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what the quadratic
approximation is,

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near x equals 0.

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So, this gives us that f of x is
approximately f of 0 plus f

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prime of 0 times x plus
F double prime at 0

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over 2 times x squared.

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This is just the quadratic
approximation of capital F at

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x equals 0.

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That's what it is.

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So now all we need to do, I've
asked you to write it in terms

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of little f.

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So what we need to do is figure
out what each of these

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things actually is.

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What capital F of 0 is, what
capital F prime of 0 is, and

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what capital F double prime
of 0 is, in terms of the

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function little f.

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So let's see.

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Well first, let's figure out
what capital F of 0 is.

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Let's write down what it is.

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The integral from 0
to 0 of f of t dt.

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Well we know, when you take the
integral from a to a of a

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function, you should get 0.

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So capital F at 0 is just 0.

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So we have one piece.

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Now, let's figure out what
capital F prime is.

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Well, capital F prime of x, we
know using fundamental theorem

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of calculus--

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let's just write it down so
we can see it easily--

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we know this is just
little f of x.

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So if I wanted capital F prime
at 0, I just come over here.

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That's just going to
be little f at 0.

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So now, if we look back up at
what we have at the top, we

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filled in this part-- or we
know that part, that's 0--

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we know this part.

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So now all we need is capital
F double prime

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evaluated at x equals 0.

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And then we'll be done.

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Now, how are we going
to do this?

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Well we know what capital
F prime is.

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So let's come in, sort of
up here, and look again.

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We know what capital
F prime is.

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We know it's little f.

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So if I want to take a second
derivative, of capital F-- let

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me write that down here--

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that's going to be taking a
derivative of little f.

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Right?

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That'll be, that's d dx
of little f of x.

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So I can just write this
as little f prime of x.

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So again, let me show you
where that came from.

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We knew big F prime was
equal to little f.

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So when I take another
derivative of this, I have to

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take a derivative of this.

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So that's why capital
F double prime is

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equal to little f prime.

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So we're almost there.

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I think I'm about to run out of
room down here, so let me

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come over a little
bit further.

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So now, we just need to
evaluate this at 0.

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Well, that's just going
to be evaluating

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little f prime at 0.

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So now I have all the pieces.

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Let's come back in and fill
it in one more time.

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So what do I get?

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I get-- let me, now I
have all the pieces.

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Now to answer the question.

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Little f-- or sorry--

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Big F of x, its quadratic
approximation is going to be

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big F of 0, which
we said was 0.

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So let's come over and look.

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Here's big F of 0.

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I know that's 0.

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And then I'm going to add to
that big F prime at 0 times x.

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That's little f of 0 times x.

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Let's come back over
and do that.

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So we add to that little
f at 0 times x.

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And let's come back.

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One more thing we're
going to need.

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We need one more term.

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That's big F double prime at
0 over 2 times x squared.

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That I've left big F double
prime at 0 is

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little f prime at 0.

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So that's f prime at 0 over
2 times x squared.

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So we had all the pieces.

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We just had to put them in.

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This does not look like an
approximately by the way.

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Sorry.

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There we go.

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So what do we get?

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This expression above
is equal to--

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well why did I write
the 0 again?

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Sorry I was trying to
avoid writing the 0.

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It's equal to f of 0 times
x plus f prime of 0

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over 2 times x squared.

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So the quadratic approximation
of capital F in terms of

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little f is exactly this.

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Now what assumptions are we
making about little f?

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Well to apply the fundamental
theorem of calculus, you need

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little f is continuous.

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But we actually need
more than that.

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Because what have we done?

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We've actually taken a
derivative of little f.

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So we need that this function,
little f, is actually

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differentiable.

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So that is, that actually
includes, of course, that

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little f is continuous.

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So we don't even have to say
continuous in the assumptions.

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We just have to say, we're
assuming little f is

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differentiable, at
least near 0.

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So let's go back and just remind
ourselves what we did.

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I'm going to go all the way
over, remind ourselves of the

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problem and kind of bring
us back through.

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So we started off with this
definition of big F. And I

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wanted us to find a quadratic
approximation for it

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near x equals 0.

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And then we just mentioned,
what are the assumptions?

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So I reminded you, first,
of what is the quadratic

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approximation of big F in terms
of itself, in terms of

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big F. And we just
write it out.

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And then we had defined what is
this in terms of little f,

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what is this in terms of little
f, and what is this in

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terms of little f?

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And so what do we look at?

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Well, we evaluate f of 0.

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That's easy enough.

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We found big F prime in
terms of little f.

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That's fundamental theorem
of calculus.

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And then we evaluate
that at x equals 0.

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And then we know that if we
want to take a second

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derivative of this function, we
need to take a derivative

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of the little f.

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And so we get big F
double prime is

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actually little f prime.

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So we evaluate that at 0 to get
big F double prime at 0.

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And then we come in and
we just fill in

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everything we needed.

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And so ultimately, this is the
final answer of the quadratic

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approximation of big F
in terms of little f

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near x equals 0.

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So I think that's where
we'll stop.

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