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Hi.

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Welcome to recitation.

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Last time in lecture we graphed
some trigonometric

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functions and some inverse
trigonometric functions.

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And there was a slight error
in one of the graphs that

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Professor Jerison did.

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So I just wanted to talk a
little bit about it and about

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what the problem was 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

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usually writes tan
to the minus 1.

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But they just meet, they're just
two different names for

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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, y equals x--

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that's this diagonal line--
and the graph y

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equals tangent of x--

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so that's this curve--

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

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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

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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 so there are

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many copies of this.

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So one thing to notice about
this is that the

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tangent comes in here.

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The graph y equals tan x comes
in and it is tangent to the

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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 that d over dx of

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tan x is equal to secant
squared of x.

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And so the derivative at 0 is
secant squared of 0, which is

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1/1 squared, which is just 1.

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So the slope is 1.

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And in fact, another, a stronger
thing is true, which

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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 by looking at the

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difference and higher
derivatives 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 this graph across the

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line y equals x, and because
of the way these graphs,

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because of this property that
this graph has, that it lies

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above the line y equals x for
positive x, when you reflect

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it what you get is something
that lies just below the line

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y equals x.

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When you reflect this whole
picture, that the piece, this

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piece gets reflected and comes
entirely on the other

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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--

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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 is has an asymptote,
horizontal asymptote at

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minus pi over 2.

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So the one feature I want to
point out is specifically

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these two curves only intersect
at the origin.

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So in the graph Professor
Jerison showed you, they

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looked more like square root of
x and x squared, which have

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a later intersection point.

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But here, for x bigger than 0,
y equals tan x is always

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bigger than x, which is always
bigger than y equals arctan x.

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

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tangent 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 when x

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is less than 0.

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So that was all I wanted to
share with you, just this

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slightly cleaner picture of the
arctan of x that I get by

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being able to put it up on
the board ahead of time.

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So that's that.

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