BAM!!!
Mr. Tarrou In this video we are going to take the pre out of precalculus in our last chapter
and start looking at some of the core concepts you will learn in the first chapter of Calculus
if you take that next semester or next year.
Now in our textbook we have had a couple of examples in areas where we have talked about
limits.
We have talked about horizontal asymptotes and where did the graph or what values did
the graph approach, what y values did the graph settle down to or approach as x went
to infinity.
Maybe the horizontal asymptote is y=2, and the y values of the graph would come down
and approach two from the top and become infinitely close to two but never exactly equal to two.
That is a form of limits.
We also had a chapter, I think it was chapter two for my textbook, that actually introduced
limit notation which we are going to review in this chapter as we find limits both graphically
and numerically.
It is fairly simple material as far as the mechanics of the work in this introductory
chapter or section.
So, I am going to first do a couple of scenes here showing you how the idea of limits are
used in Calculus.
There is two core concepts that you are going to learn.
One is derivatives and the other is integration which you are going to learn later on in the
second half of your first year or first semester of Calculus.
You might have in your Precalculus book this idea of a Quotient Rule, the formula for the
quotient rule.
We reviewed it or were introduced to this formula when we were first learning or being
reminded of function notation and reminded that if it was f(2) you would take out the
x and replace it with 2.
If it was f(a+2), you would take all the x's out and replaced it with a+2.
They through in this formula to help you practice relearning or reminding yourself how that
function notation worked.
I did a video on this because I knew it would come up in chapter 11 on this quotient rule.
The reason why it was in your Calculus book is because... f of something is fancy notation
for y, so f(x+h) minus f(x) means that your are subtracting two y values.
Now what equation did we learn that had a subtraction of two y values in the numerator.
Y sub 2 minus Y sub 1 over X sub 2 minus X sub 1.
Yet this denominator is equal to h.
That is because there has been some simplification that has occurred.
How do you find this y value or this function value when you plug in.... oh, when you plug
in.
That must mean that x+h is acting as, what we learned in algebra 1 is my maybe X sub
1 or X sub 2.
Then to get this y value, this function value we plugged in x.
Now we sort of a fancy version of y sub 2 minus y sub 1 over x sub 2 minus x sub 1.
We know that is basically or is a version of slope.
So, you will be learning an idea soon called derivatives that is basically...
When you find a derivative in Calculus, you are finding a version of your original function
or equation that will allow you find slope at any point.
Here is a curved line.
Here is a fixed point on this curve.
You need two points to find slope.
Even the quotient rule said that we needed two points.
Maybe I will put a point here and we will call it (x+h,f(x+h)).
So we have a fixed point, another point based of this fixed point, and there is some kind
of unknown horizontal distance between those two points.
I am going to use those two points to estimate the slope at that point.
Well this yellow line is not really estimating the slope at this point very well.
So if I were to take the horizontal distance and reduce it maybe let the point, the movable
point go to here and shrink my distance of h, now using those two points to estimate
the slope at this point.
My estimation is still not great, but it is certainly better than yellow line.
Then if I were to move the point even closer I would get a better estimation of the slope
at that point (x,f(x)) using the second point of (x+h,f(x+h)).
What Calculus will allow you to do with this idea of limits in this section, is if you
let h the horizontal distance between your two points you are using to find slope...and
you do need two points...but if you let that horizontal distance approach zero...let it
approach zero...it is a limit as h approaches zero... then your going to get a better and
better estimation of the slope of this curved line.
Calculus will be the first time, and actually we are going to do this in our last chapter,
this will be the first time you will know how to find the slope of a curved line.
That is really really cool!!!
So if we allow h to become zero, because this denominator was originally h because x minus
x is zero, we cannot actually divide by zero.
So with Calculus and this idea of limits we are going to let that denominator approach
zero but not actually be zero.
Basically this denominator is .00000...00001.
This fraction is still defined.
That's why we are learning an introduction to limits in this discussion.
Because it is going to allow us with the old formula of slope that you learned in Algebra
1 along with this idea of a limit and allowing the distance between these two points to approach
zero, calculus is going to allow us to find something called a derivative.
This will allow us to find the slope of a curved line along almost any given point or
along almost any x value.
Let's do one more example of why need to learn limits and then learn the concept.
So on this board I have got some diagrams drawn.
Three curves.
I have a bunch of rectangles drawn in these curves.
Another idea you are going to learn, another key concept in Calculus you are going to learn
after you finish with the idea or concept of derivatives...finding first and second
derivatives, is you are going to then start trying to find areas under curves.
In Calculus 1 that will be two dimensional curves.
Well, this is...we would call this a right skewed distribution in Statistics.
I could have drawn any shape.
We don't have area formulas for every single unique curved shape.
So we do this with Calculus when we start introducing this concept of integrals.
We explain how we are going to find the areas of a bunch of small rectangles which are just
length times width, and then add the areas of all those rectangles together to estimate
the area under the curve.
Now, which of these drawings do your think is going to give us the best estimation of
the area underneath this curve.
I have just tried to draw these curves as equally as I could.
Clearly it is going to be the middle one with all the very small rectangles.
We have the least amount of error with the over and underestimations.
With this easier to see with this orange rectangle drawing.
When you draw rectangles to estimate the area of a curve, you are going to over estimate
some areas and underestimate others thus having an error.
Well in Calculus again with this idea of limits, we are going to learn with integration how
we can take the widths of these rectangles and shrink them to have widths approaching
zero.
So we are kind of going to be finding the area of an infinite number of rectangles to
get what is kind of almost or basically an infinitely accurate estimate of the area below
the curve.
These are concepts that are many months away, or at least many chapters away when you get
into Calculus.
I am not going to try and explain how to do an integral.
But this idea of limits allow w to approach zero, the widths of the rectangles to approach
but not exactly equal zero, but approach zero.
That idea will again allow us to do something called integration in Calculus.
This again shows the importance of why we need a good understanding of limits are in
the next couple of sections in our PreCalculus book where we find them numerically and graphically.
So let's get to our core concept, define, and do some examples.
BAM!!!