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U0S1V03 Moving closer and closer.txt
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U0S1V03 Moving closer and closer.txt
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#
# File: content-mit-18-01-1x-captions/U0S1V03 Moving closer and closer.txt
#
# Captions for MITx 18.01.1x module [bANtYKLugsU]
#
# This file has 107 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Welcome.
Calculus is all about functions.
You probably know that a function f takes an input x
and gives an output f of x.
But in calculus, we're not concerned with just one input
and finding the output for that one input.
We want to consider a whole range of inputs.
So we would want to know what happens when
the input "moves" or "varies."
For instance, we could ask what happens as the input moves
really close.
Closer and closer to some point.
Let's say 1.
And to be even more specific, let's
say that x is moving towards 1 from the left.
So if this is a number line, and we've
got the point 1 right there, then x
could start here, and just move closer and closer and closer
towards 1, from the left.
We'll use this arrow notation to denote that x is getting
really, really close to 1.
But a warning, this does not mean that x will ever
actually equal 1.
We're only concerned with values of x that are near one.
OK.
Now that that's said, as x moves,
we know that the output f of x is also going to move.
And so the question that we can ask
is as x moves closer and closer to 1 from the left,
does f of x move closer and closer
to some value of its own?
Let's be concrete here.
And pick a particular function f.
I'm going to choose f of x to be the square root of 3 minus 5 x
plus x squared, plus x cubed, all over x minus 1.
Kind of a complicated function, but you'll
have to trust me that this is a good example.
And what we can do in order to see what's
happening to f as x approaches 1 from the left
is just select certain values of x
that are getting closer and closer to 1 from the left.
So over here on the number line, we
could start with x equals zero.
And then they get closer, we could try x equals 0.5.
Or even closer, maybe 0.9.
Even 0.99.
These sorts of values.
And we want to know, what's happening to the output?
So we can just plug these values into the function,
and see whether the output gets closer and closer to anything.
Now there are technically infinitely many values
of x that we could have chosen here.
But let's just start with these four.
Remember though that one value of x
that we will definitely not consider
is x equal to 1 itself.
In fact, this function isn't even defined at x equals 1.
We'd have a zero denominator.
It is, however, defined when x is approaching one,
and those are the values we're considering.
OK.
Well let's make a table with our chosen
inputs and the associated outputs,
and let's just calculate those outputs.
So when we plug in zero we'll get a square root of 3
on top divided by minus 1.
So minus square root of 3, which is roughly minus 1.73.
Next up is x equals 0.5.
I'm going to have to bust out the calculator here.
So we've got 3 minus 5 times 0.5 plus 0.5 squared
plus 0.5 cubed, and then we need the square root,
and then we need to divide by 0.5 minus 1.
So 0.5 negative.
So we get minus 1.87, roughly.
So back to our table.
We've got f of x moving from minus 1.73 to minus 1.87.
Well that's not really enough data
to tell if f is getting closer and closer to anything
in particular.
So let's take our next two values of x and plug those in.
I'm going to fast forward through the calculations.
You ready?
x equals 0.9.
All right?
That's approximately minus 1.97, and finally 0.99,
and we've got minus 1.997.
So as we go down this table, f of x
is getting really, really close to what looks like minus 2.
So we can say that as x approaches 1 from the left,
f of x approaches minus 2.
Now f of x might never actually equal to minus 2,
just as x never actually equals one,
but it gets really, really close.
And if it gets arbitrarily close,
meaning as close as we could possibly want,
then that's really all we'll care about.
What I would like you to do now is
to do this same exercise, except this time have x approach 1
from the right.
You might be surprised at what you find.
We'll talk afterwards.