Hi.
Welcome to recitation.
Last time in lecture we graphed some trigonometric
functions and some inverse trigonometric functions.
And there was a slight error in one of the graphs that
Professor Jerison did.
So I just wanted to talk a little bit about it and about
what the problem was and with the correction is.
So the function in question is the arctangent
or the inverse tangent.
And so I like to write arctan, where Professor Jerison
usually writes tan to the minus 1.
But they just meet, they're just two different names for
the same function.
So the inverse function of tangent.
Now I've got a graph set up here.
And what I've graphed are the lines, y equals x--
that's this diagonal line-- and the graph y
equals tangent of x--
so that's this curve--
and here I've got one of the asymptotes of y equals tangent
x at pi over 2.
Right?
So as x approaches pi over 2 from the right, tangent of x
shoots off to infinity getting closer and
closer to this line.
And, you know, it does something similar down here.
And then of course, it's a periodic function so there are
many copies of this.
So one thing to notice about this is that the
tangent comes in here.
The graph y equals tan x comes in and it is tangent to the
line y equals x at the origin.
So the slope of tan x is just its derivative.
So we saw in an earlier recitation that d over dx of
tan x is equal to secant squared of x.
And so the derivative at 0 is secant squared of 0, which is
1/1 squared, which is just 1.
So the slope is 1.
And in fact, another, a stronger thing is true, which
is that for positive x, tangent of x is larger than x.
So this falls away.
So you can figure that out, for example by looking at the
difference and higher derivatives if you wanted to.
So the result of this, is that the graph of the arctangent,
that is what you get when you reflect this graph across the
line y equals x, and because of the way these graphs,
because of this property that this graph has, that it lies
above the line y equals x for positive x, when you reflect
it what you get is something that lies just below the line
y equals x.
When you reflect this whole picture, that the piece, this
piece gets reflected and comes entirely on the other
side of that line.
So the height here will be pi over 2.
That'll be the horizontal asymptote.
And it'll come below--
so this is y equals arctan x.
So it will come below that line.
And similarly, over here it'll come, it'll be the reflection,
so it'll come above that line.
And again is has an asymptote, horizontal asymptote at
minus pi over 2.
So the one feature I want to point out is specifically
these two curves only intersect at the origin.
So in the graph Professor Jerison showed you, they
looked more like square root of x and x squared, which have
a later intersection point.
But here, for x bigger than 0, y equals tan x is always
bigger than x, which is always bigger than y equals arctan x.
And then they come in and right at the origin, their
tangent to each other, they both have derivative 1 here.
And then for negative x, and then they cross.
And so arctan x is larger than x is larger than tan x when x
is less than 0.
So that was all I wanted to share with you, just this
slightly cleaner picture of the arctan of x that I get by
being able to put it up on the board ahead of time.
So that's that.