Most calculus one classes begin with the study of limits but they don�t always go into
great level of detail as to why we don�t always talk about what the reason for studying
limits is it�s easier just to get into the course and start doing these things called
limits. It really kind of helps if you understand the reason behind studying limits and so you
take a big picture view like a calculus one course there�s really two main questions
that we are trying to answer in this class the first one is what�s the slope of a given
curve so if you have a curve here and you highlight a particular value let�s say like
right here we are interested in what the slope would be instantaneously at that point exactly
at this point where in previous math classes like your algebra classes you may have found
a slope of a line but that stayed constant usually you use the order m y equals mx plus
b but if you have a curve notice the slope will change from positive to negative to positive
depending on where you are at on the curve and number 2 the other big question is what�s
the area under a curve so again if you take a somewhat looking curve and you give it a
starting and an ending value we will be interested in the answer to what�s the area under this
curve from a to b we call this a we call this b and it turns out limits help answer both
of these questions and so that�s really the main idea now I�m not going to go through
a big discussion of what limits are and how they value with limits that will come in other
videos but let me just briefly explain how these 2 topics or these 2 questions tie in
to the study of limits so when you looking at a slope of the curve exactly at this point
here alright you will notice we have never learned how to find the slope at one point
normally we need 2 points to evaluate so what we do as we throw in an extra point a supplementary
point to where this is x and this point here will be either an x plus h in some text books
or x plus delta x in other text books but it�s the same thing h or delta x is just
representing some small numerical value for instance x might be 5 and x plus h might be
5 point 1 where the h will be like point 1 for instance so if we find the slope between
these 2 values at these 2 points we get something like this we get f over x plus h the y value
here and the second x value minus f of x all divided by x plus h minus x and if you are
wondering where I got that from that�s roughly like y2 minus y1 over x2 minus x1 that you
learned in your algebra class both of these are standing for what we will call slope it�s
simply rise over run it�s the vertical change over the horizontal change in distance. Now
if you look closely you will notice that the slope of the yellow line is not exactly the
slope of the pink line so we go and now take this approximation and make it better and
better and as we think about how you would make this better I think what you will wind
up deciding on if I can move that second dot closer to the first dot if I can slowly move
it down the line then each subsequent secant line slope will get better and better and
more exact to the slope of the tangent line that touches only at one point and so another
way to say that is if you take f of x plus h minus f of x all divided by x plus h minus
x we take this quantity this will get more accurate if the h term went away in other
words as the h goes to zero then x plus h will move closer to x now that seems simple
on the surface of it but notice h technically is not allowed to be zero but yet we know
that this yellow dot does need to move to the left to make this slope more and more
like the slope of the pink line which is our answer this is where limits comes in study
of limits we can take these limits to see what this ratio approaches not what it equals
exactly when h is zero but see what happens as the h gets closer to zero and what we should
get is the slope of the tangent line we were looking for so again I�m leaving out a lot
of details for as how to find limits but I�m just trying to in this video give some connections
as to why we would want to study limits. The second big question studied in calc one is
how do you find the area under a curve from one point to another we obviously for most
functions there�s no nice geometric shape it�s not a box it�s not a triangle it�s
not a trapezoid so usually if it has a smooth top we pretty much stuck so one idea will
be to break this up into intervals and then within each little subinterval we can create
rectangles one right after the other the reason I chose rectangles was because I actually
know the area of a rectangle its simply width times height and I can actually determine
both of those as pretty simple algebra determine the width of each little subinterval and the
height you notice simply touches the curve at certain x values now obviously here there�s
some error involved you see there�s some area missing or area that�s extra that we
need to account for it but basically if we can take a sum of you know some areas here
I just call these a sub I a and the I subinterval i equals one to n at where n is the number
rectangle that will give you an approximation to the area but now here�s the question
how could I make this approximation little bit better how could I make it more like the
exact area under the curve well the answer simple I just need more rectangles the more
rectangles I have the more accurate this approximation will be now ideally I don�t want 10 I don�t
want 100 I don�t want 1000 I would actually want to take a here we go a limit or see what
it approaches as the number of rectangles approaches infinity now can I ever get an
infinite number of rectangles no not really but I can at least see where that would converge
to I can see where that will lead to which will give me the exact area under the curve
so both of these major landmark topics heavily heavily rely on the knowledge of limits now
just to be honest when we actually get into the algebra it turns out lucky for us that
there are shorter ways to find the slope of a curve without using limits and there are
shorter ways to find the area under a curve without using limits but for a solid understanding
of these fundamental topics a understanding of limits is absolutely necessary so now that
you can have some motivation for learning the topics of limits you can move on and look
at the different ways to evaluate limits and what limits actually are and a little bit
more formal sense in later videos.