Hello, I am Mr. Tarrou. Today in my Precalc class I covered something called the definition
of the difference quotient. Now, I am early in the textbook so it did not really explain
what this expression did. It was really more about practicing your use of the function
notation. But, because this is precalculus and I know what is coming up in the first
chapter of calculus which is derivatives and finding slopes of curved lines, I did not
go all this way with my introduction today, but I did make sure that those kids knew that
this was basically a very fancy form of slope. So, I would like to show you that today as
well. We have a couple of points on a straight line. Of course, hopefully we know that the
slope formula is equal to y sub 2 minus y sub 1 all over x sub 2 minus x sub 1. You
might know that as rise over run. Well, the difference in y is a vertical movement. A
difference in x is our horizontal movement. So..slope. Well, that is fine if you have
two completely random points and we are not in function notation. Let's remember that,
for a second, if I write y equals two x minus one, in function notation....that is not this
line by the way...in function notation that would be f of x equals two x minus one. You
know function notation just is a fancy way of writing the letter y. We do use it though
in upper level math incase we are using more than one equation and we want to name them
so we know what we are working on. So the expression of f of x, it is not f times x,
f(x) is a replacement for y. Ok, so that being said instead of, instead of labelling this
as (X1,Y1), I am going to take this and change it a little bit. I am going to take out the
y and replace it with f(X1). Now this is going to be f(X2). So, now my formula which is a
basic slope formula that we are used to, I am going to rewrite that with function notation
and have it look like this. Instead of Y2 we have f of x sub two minus f of x sub one
will replace y sub one just to emphasis again function notation. F of x is a replacement
for y. Now we have this formula. Ok. Well, this is a bunch of x's and y's and some function
notation, but this definition has the variable of h in it. Let me show you were that came
from. I am going to take this original diagram, and what they do and this is important for
derivatives if you are just starting calculus is, they have one fixed point and they measure
the other point based on a horizontal direction. That horizontal direction/distance is defined
by the letter h, for let's just say horizontal distance. So, now I have this fixed value
of x comma f of x, from there I am going to move the right a length of h. Well if I define
this as h, and we are going to go up to this point, I don't need this random x sub two
like some random x somewhere floating around. No. This x is going to be h units right of
this point, so I don't need another variable. I am going to take this out and we are going
to rename this point once again. It is going to be, if this is X1 and I have moved over
h units, now this point is X1 plus h. Ok, well what does that mean for the function
notation? I had just a second ago f of x sub 2. Well x sub 2 is not here any more. Now
it is x sub on plus h. Honestly, if I am only using one x I don't need that subscript any
more. Now the y value, if this is x plus h, this is the x. How do I find my y's? I plug
them back in to my equation. Which I am pointing like there is one there, but there is not.
Function notation will say that if you want your y value, f of whatever is fancy notation
for y, this y value if found by taking x+h and plugging it into an equation if we had
one. I skipped a numerical example because with Youtube I only have fifteen minutes.
So now I have a new set of labels for my two points. Well, if I am using these labels,
going back to Y2-Y1 over X2-X1, the generic formula for slope, now I have got this fancy
function version of slope where y came out and f of x went in its place. Well, now with
this new labeling with one point and a horizontal distance I have got this. Y sub two, so f(x+h)
minus, y sub one this is y sub two minus y sub one in the old form that you might be
familiar with, f(x). Oh, this is starting to look like this up here:) All over x sub
two which is x+h minus x sub one, or X2-X1 x+h minus x. If I clean this up, look the
x minus x, those are going to cancel out. Now we have what we started with. So, if your
book has defined this definition of a difference quotient, it is not some kind of like Whoo.
It is going to come back around chapter 11 maybe, when you start doing your calculus
introduction. It is basically a fancy version of slope. Let me make sure that you understand
completely that this is slope. I am going to do that by coming in here and saying, you
know what, I said a minute ago that equation that I wrote wasn't going to represent this
line. Let's go ahead and do that. Let's say that this line is equal to y equals two x
minus one. I said I wasn't going to use that, but let's go ahead. Because the y intercept
is below the x axis, let me get that for just a second, and the slope it is not drawn to
scale but let's say we are going up two and over one for slope. Now that means that this
line has a slope of two. Lines do not have a slope that change. So, if a line has a constant
slope everywhere when I use this example, I probably should have f(x) here, when I use
it with this definition that my precalculus book gave, it should come out to be an answer
of two if it does represent a fancy version of slope. Well, let's see. I am going to highlight
this in blue, because the notation with this stuff can be really kind of tough. Forget
the rest of this. F of x plus h, that means take function f and wherever there is an x
plug in x+h. So, I really should have used f(x) there, but f of x equals two x minus
one. So the function is two times something minus one. I am going to do this in blue to
match this. Two times something, 2x, minus one. Where the x used to be, x plus h goes
in its place. Now there is a minus sign, it is white and I am going to keep that white.
f of x, well f of x plus h...that is just what I wrote, two times something minus one
from down here. Now it is minus, I am going to need a set of parenthesis if I am going
to get my signs right, f of x again my function two x minus one. I am not taking x out and
replacing it with anything. There is no fanciness in here, just f(x) minus one. All of this
over h. So, the first part of the numerator is this blue. This is f(x+h) minus go back
to your original function and just copy it down all over h. We are going to clean this
up and hopefully we get two as an answer. Otherwise I have messed up. Multiply the two
through. I am going to distribute the negative through that is in front of these parenthesis.
We get 2x plus 2h minus one. This negative will through the parenthesis with distribution
like the two. Negative one time two, negative one times negative one is positive one all
over h. Let's see. Two x and negative two x cancel out. Nice! Two h, oops that does
not cancel with anything, let's get that back. The ones do, there is a negative one and a
positive one. We are left with, just to make sure that you see it, 2h which I almost cancelled
out, over h. What is 2h over h? Two. I started with a function that I knew had a slope of
two. I applied it or passed it through the definition of difference quotient which I
said was a fancy way of saying slope or defining slope and indeed once everything cleared out
I got an answer of two. Now, your precalculus book will have a more complicated version
of this where maybe it is a quadratic formula. If you have a curve line the slope is constantly
changing, so you will have more than just a regular two. It might be like 3x-4 plus
h. But, I am close to running out of time so thank you very much:) Have a great day!
Hey, I am back. I just checked my time. I had a spare five minutes. Let me see if I
can finish another example for you on that difference of quotient definition. Let's take
a look at this equation. And let's see if we can finish this in the allotted time and
check out this slope. So, we are going to run this function through this difference
quotient and see what happens. So we have, again I am going to highlight this in blue
on the first part, f of x plus h. So I am going to write this and everywhere there is
an x I am putting in x+h. Something squared minus four times that same something plus
one, that is this part of the definition. Now minus the original function, so again
I am going to write in x squared minus four x plus one. All of this is over h. I am going
to have to write quickly if I am going to finish this example, so I am going to be kind
of quiet and just show you the steps of this. x+h squared is going to be x squared plus
2xh plus h squared. Take the negative and distribute it. Negative four x and another
negative four h plus one. Take the negative that is in front of this parenthesis, if you
do not wrap that second half of the numerator in parenthesis you will have sign errors that
are going to cause you problems. Negative x squared. Negative and a negative is positive
four x, and a negative and a negative is minus one. All of that is over h. WHOO. Man, I have
a lot of stuff going on here. If you are not really neat with your work, it is extremely
easy to make a mistake that you will keep going and going and not catch with most algebra
problems. We have a positive x squared and a negative x squared. We have a positive h
squared and oops, that was the only h squared that was in there so that is a little mistake.
At any rate. We have got a negative four x and a positive four x. We have, what else,
we have one and negative one. 2xh, we have got a 4h, we have a h squared, and we have
an h in the denominator. Nothing else is going to cancel out. Let's see what happens here.
I am going to put the h squared first, then have 2xh, and we have got negative four h,
and that is all over h. ok, so now what? Some stuff will cancel out. All three of these
terms all have a factor of h in them, and we have a denominator of h. This will clean
up. This will reduce some. All of these have an h. How many can I take out? One. This has
two to give up, but I can't take two out if these only have one to give. So, I am going
to show a set of parenthesis. Factor out that h. H squared, if I take one out that leaves
me with one. 2xh, again I just took that one out that it has to give up, so I am only left
with 2x. Minus 4h, I just pulled out the h leaving negative four. All of this is over
h. Those h's cancel out and what do we have? h plus two x minus four. If you are in a Precalc
book and you are at the beginning of the year, that is it, this is your answer. If you are
at the end of precalculus, you are going to find out what it is exactly that h does, where
it goes, how this expression will be able to tell you the slope of a curve anywhere
which is extremely cool. If you are in Calculus, well this is old news. Thank you very much.
BAM!!!