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Hi.

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Welcome to recitation.

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Last time in lecture we graphed
some trigonometric functions

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and some inverse
trigonometric functions.

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And there was a slight
error in one of the graphs

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that Professor Jerison did.

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So I just wanted to
talk a little bit

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about it and about
what the problem was

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and with the correction is.

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So the function in
question is the arctangent

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or the inverse tangent.

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And so I like to write arctan,
where Professor Jerison usually

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writes tan to the minus 1.

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But they just mean-- they're
just two different names

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for the same function.

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So the inverse
function of tangent.

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Now I've got a
graph set up here.

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And what I've graphed
are the lines,

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y equals x-- that's
this diagonal line--

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and the graph y equals tangent
of x-- so that's this curve--

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and here I've got one
of the asymptotes of y

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equals tangent x at pi over 2.

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Right?

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So as x approaches pi over 2
from the right, tangent of x

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shoots off to infinity getting
closer and closer to this line.

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And, you know, it does
something similar down here.

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And then of course,
it's a periodic function

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so there are many
copies of this.

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So one thing to
notice about this

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is that the tangent
comes in here.

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The graph y equals
tan x comes in

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and it is tangent to the line
y equals x at the origin.

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So the slope of tan x
is just its derivative.

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So we saw in an
earlier recitation

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that d over dx of tan x is
equal to secant squared of x.

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And so the derivative
at 0 is secant squared

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of 0, which is 1 over 1
squared, which is just 1.

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So the slope is 1.

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And in fact, a
stronger thing is true,

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which is that for positive x,
tangent of x is larger than x.

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So this falls away.

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So you can figure
that out, for example

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by looking at the difference
and higher derivatives

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if you wanted to.

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So the result of this, is that
the graph of the arctangent,

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that is what you
get when you reflect

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this graph across
the line y equals x,

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and because of the way
these graphs-- because

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of this property
that this graph has,

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that it lies above the line
y equals x for positive x,

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when you reflect it what you
get is something that lies just

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below the line y equals x.

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When you reflect this whole
picture, that the piece,

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this piece gets reflected
and comes entirely

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on the other side of that line.

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So the height here
will be pi over 2.

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That'll be the
horizontal asymptote.

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And it'll come below-- so
this is y equals arctan x.

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So it will come below that line.

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And similarly, over here it'll
come, it'll be the reflection,

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so it'll come above that line.

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And again it has an asymptote,
horizontal asymptote,

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at minus pi over 2.

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So the one feature
I want to point out

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is specifically these two curves
only intersect at the origin.

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So in the graph Professor
Jerison showed you,

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they looked more
like square root

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of x and x squared, which have
a later intersection point.

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But here, for x bigger
than 0, y equals

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tan x is always bigger than
x, which is always bigger

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than y equals arctan x.

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And then they come in and right
at the origin, their tangent

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to each other, they both
have derivative 1 here.

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And then for negative
x, and then they cross.

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And so arctan x is larger
than x is larger than tan x

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when x is less than 0.

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So that was all I wanted
to share with you,

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just this slightly
cleaner picture

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of the arctan of x that I get
by being able to put it up

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on the board ahead of time.

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So that's that.