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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.