WEBVTT mathematics/ap-calculus-ab/hovasapian
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Hello, welcome to www.educator.com, welcome to the first lesson of AP Calculus AB.
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I thought what I would do is take about 5 or 10 minutes to give a nice overview of the course as a whole,
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so that we have a sense of where we are going.
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Then, we are going to launch right into the calculus proper.
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Let us get started and welcome.
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Calculus is basically going to be about two things, two processes.
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We are going to spend the first half learning how to do something called differentiating a function.
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We are going to spend the second half learning how to do something called integrating a function.
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That is essentially it.
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Each one of those tools is going to yield different applications.
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That is all we are going to be doing.
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Let me work in blue here.
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Given a function f(x), we will be spending our entire time doing two things to this function.
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The first thing is we are going to be differentiating it.
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It also called taking the derivative.
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Both of those terms, we will use interchangeably.
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The second thing we are going to be doing to this function,
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it is just going to occupy the second half of calculus, is we are going to be integrating it.
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These are the two major rivers of calculus.
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These are the two great rivers of calculus.
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The differential and the integral, two basic set of tools that allow us to solve certain types of problems with the differential calculus,
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certain other types of problems with the integral calculus.
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Essentially, what is going on is this.
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We have some function f(x).
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One of the things that we can do to it is differentiate it.
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When we differentiate a function, we are going to get another type of function.
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We are going to symbolize that with f’(x).
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The other thing that we can do to it is we can integrate this function.
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And then, when we integrate a function, we are going to get yet another type of function
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which we will symbolize normally with F(X).
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This is a new function and we call this function the derivatives.
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That is called the derivative because it is derived from the original function.
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This one is also a new function.
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We call this one the integral.
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That is essentially what calculus comes down to.
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It is finding ways to find derivatives of functions, applying it in certain cases, and then finding integrals of functions.
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Applying the integral in certain cases.
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Let us move on to the next page here.
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These two tools namely differentiation and integration are very powerful.
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It will allow us to solve the most extraordinary problems, that is what beautiful about the calculus.
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So far in your mathematical studies from elementary school, all the way through high school,
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there has been a pretty steady increase, in terms of the level of difficulty, the techniques that you develop.
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Some or more problems that you can solve.
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That is essentially what it is, it is just, you are making the class of problems that you can solve bigger and bigger.
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Now the calculus, it is a huge jump.
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It is not a little jump, it is not a stair step.
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You are going to be introduced to these tools, differentiation and integration.
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They are very powerful tools.
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It is going to take you up here, in terms of mathematical sophistication and the types of problems that you can solve.
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It is really quite extraordinary.
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Everything that we enjoy in the modern world, I do mean everything,
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It is worth writing down.
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I do mean everything.
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Let us capitalize this one.
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I do mean everything is made possible with calculus.
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It really is the single tool, these two things, differentiation and integration.
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They have allowed us to absolutely enjoy everything that we enjoy, computers, cell phones, cars, you name it.
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If there is anything in the modern world that you take for granted, that you enjoy, it is because of the calculus.
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Once again, we are going to spend the first half learning and applying differential calculus.
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And then, we are going to spend the second half, roughly, learning and applying the integral calculus.
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It is going to turn out that these two independent techniques, the differentiation and the integration,
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are in fact so deeply related, that we call this relationship the fundamental theorem of calculus.
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You will often see it as FTC.
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Differentiation and integration, they are reasonably independent techniques.
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There is no reason to believe that one is actually related to the other.
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They solve different set of problems.
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Yet, there is a very deep relationship that exists between them.
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This relationship is what we call the fundamental theorem of calculus.
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What is really interesting about the fundamental theorem of calculus,
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something that many of you who go on to higher mathematics will discover,
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is that this relationship that exists between differentiation integration is actually true, not just in one dimension.
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In other words the real line, which is what we have essentially been dealing with ever since elementary mathematics.
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We stayed on with the real numbers.
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It is true in any number of dimensions, dimensions 2, 3, 4, 5, 6, 15, 34, or 147.
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It is true in any number of dimensions.
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It is a very profound relationship.
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We will actually be getting to that, when we are getting to integration.
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We will study the fundamental theorem of calculus.
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Now the nice thing is, I can already tell you now what this relationship is.
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You have an idea of where is that we are going.
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I can tell you now what this relationship is.
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Later, we will explore this relationship.
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I can tell you now what this relationship is.
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Again, there is nothing here, this is just overview.
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Just giving you an idea of what it is that we are in for, so that you are not going into this blindly.
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So you just have some sense of why we are doing what we are doing.
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I can tell you now what this relationship is.
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The relationship is, each of these processes namely differentiation and integration,
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each of these processes, dif and int is the inverse of the other.
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In other words, if I start with some function f(x), we said that I can differentiate it.
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You know what I’m going to do actually here, I will go ahead and write dif down here.
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I said that I could differentiate it to come up with a new function, the derivative.
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If I want to go back, I integrate this function.
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If I integrate, it will take me back to my original function.
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Or if I integrate, it will give me some other function which we call the integral.
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If I want to go back to the original, I just differentiate.
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That is the relationship between the two.
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Two entirely independent techniques, for the most part, because they solve entirely different set of problems.
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Yet, the relationship is one is the inverse of the other.
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The same way that if you will take a number and if you take the logarithm of that number, you end up with some other number.
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If you want to go back, you exponentiate this thing and it takes you back.
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The logarithm and the exponential are inverse processes.
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The sin and the inverse sin are inverse processes.
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Cos, inverse cos, are inverse processes.
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Differentiation or integration are inverse processes.
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This is extraordinarily deep, extraordinarily beautiful.
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Again, there is no reason to the world to believe that they are connected and yet they are.
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Let us write that last part.
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There is no obvious reason why they should be connected, at least cosmetically.
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There is no reason to just sort of look and say that this is related to that.
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There is no obvious reason why differentiation and integration should be related, but they are.
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This relation has consequences that go further and deeper than you can imagine.
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One of the beautiful things about mathematics is that, you will have different people
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or perhaps the same person will investigate different areas of mathematics, to solve a certain type of problem.
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When you take a look at this set of mathematics that you develop for this problem
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and this set of mathematics that you develop for this problem,
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when you realize that there is actually a connection between those two mathematics, they come together, you unify that.
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That relationship that exists between the various areas of mathematics takes you to a deeper level,
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a deeper understanding of reality, a deeper understanding of how the physical world works.
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This is what we strive for, we strive for unification.
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This is what makes it beautiful, things that should not be related, at least, as far as our intuition is concerned,
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they end up being not only related but very deeply related.
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The consequences of those relationships are profound.
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Anyway, this is really beautiful stuff and it begins right here, with your first course in calculus.
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Once again, welcome, and let us get started.
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We are going to spend the first half talking about differentiation.
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We are going to put integration on the shelf, for the time being.
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We will come back to it later.
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We are going to begin with taking derivatives, differentiation.
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We begin with differentiation.
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Actually, let me write something here.
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Differentiation or also called taking the derivative.
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I will just write it over here, taking the derivative.
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In other words, starting with some f(x), performing the differential operation on it,
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and ending up with a new f’(x), a new functions that is going to give us other information.
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Either by the situation, or it is going to give us information about the original function, whatever it is.
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But it is a new function that we have derived.
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Two questions, the obvious questions.
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Two questions, what does the derivative mean and how do we find it?
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Given this, how do we find the derivative and once we have the derivative, what does the derivative mean?
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What does it give us, what does it tell us, what problems does it solve?
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Two questions, what does the derivative mean?
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Two, how do we find it?
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How do we find it given some f(x)?
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How do we derive and get?
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The answers are as follows.
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The answers, the answer to number 1, what does a derivative mean?
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The derivative is the slope of a function curve.
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A function is just some curve that you draw on the xy plane at a given point on the curve.
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We will explain what that means more, in just a second.
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That is pretty much it.
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A derivative is a slope of a curve at a particular point.
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The strange thing is you have been doing derivatives for many years now.
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If you have the function y = 3x + 4, you know that is the equation of the line.
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The curve itself is a straight line but we call it a curve.
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In general, it is a line in space.
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What is the slope of this line, it is 3.
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The function is 3x + 4, what is the derivative of that function?
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The derivative is 3 because no matter where you are on that line, the slope is 3.
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That is what you have been doing, you have been finding derivatives.
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Now in calculus, we are not just going to find the slopes of straight lines.
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We are going to find the slopes of curves.
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What is that slope, the slope there, slope there.
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The slope is going to change as you move along the curve, that all we are saying.
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We are just giving you a fancy name and calling it the derivative, that is all.
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Number 2, let us go back to blue here.
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The answer to the question, how do you find it?
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Here is how you find, f’(x), in order to find f’(x), here is what you do.
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Limit as h goes to 0, you form f(x) + h, given whatever f is, you subtract from it the original f(x).
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You divide it by h and then you subject it to this process called taking the limit as h goes to 0.
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We will be discussing what this means, how does one do this.
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It is actually quite simple, just algebraically tedious.
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How do we find it? We find it like this.
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What does it mean, it means it is a slope.
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We are going to start with number 1.
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As far as number 2 is concerned, the how, I’m going to leave that for a future lesson.
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For the next couple of lessons, I’m going to be talking about what the derivative is, slope of curves,
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getting ourselves comfortable with the idea of a slope of a curve, as opposed to just the slope of a straight line.
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And then, once we have a reasonably good sense, once we feel comfortable with that,
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we will talk about how to find this so called f’(x), the derivative.
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Let us go over here, I will stick with blue.
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Let us start with number 1, in other words, the meaning, what does it mean?
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The derivative f’(x) is a function which gives us the slope of the curve or graph.
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You know what, maybe I will just call it a graph, which gives us the slope of the graph of the original function at various values of x.
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Here is what this means.
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Let us take a look at let us say the sine functions.
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Our function f(x), original function is sin(x), also y = sin(x).
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We are going to be working in the xy coordinate system.
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We are never going to be moving out of that.
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Whenever you see f(x), you can just replace it with y, it is the same thing.
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We know what the sine function looks like, it looks like this.
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This is 0, this is going to be our π, this is going to be 2 π.
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Over here at π/2, it is going to hit a value of 1.
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Over here at 3π/ 2, it is going to hit a value of -1, standard sine function like that.
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The slope of the curve is, basically, what you are doing is you are finding the line that touches the curve at a given point.
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What we call the tangent line.
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The slope of that line is the slope of the curve, at that particular value of x.
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At π/2, think of it as just some tangent line that is following the curve along.
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You have that slope.
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At this point over here, the slope is that.
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At this point over here, the slope is that.
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At this point over here, the slope is that.
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This point over here, the slope is that.
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Here, the slope is that.
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You can see that the slope changes depending on where you are.
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That is what f’(x) gives.
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F’(x), you have the original function, you do something to it, which we will talk about later, how to find the derivative of it.
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When you find the derivative, it is going to be another function which we symbolize with f’(x).
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The different values of x, it gives us some number, when you actually solve that function.
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That number is the slope of the line that touches the graph, at that particular xy value.
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As we can see for various values of x, the slope at that point, the slope at the point xy,
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if this is x, this is the point xy, it is different.
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We can see it geometrically, if that is the case.
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It is a straight line, for tangent line it is just going to be touching it at a certain point, the slope is going to change.
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Once again, given f(x), we differentiate it and it gives us f’(x).
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This thing, this tells us what the numerical value of the slope is.
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The derivative itself is a function of x, we do not know.
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We have to put in different values of x, to see what the slope is.
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It tells us what the numerical value of the slope is for different values of x.
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Let us go over here.
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Let me draw a little bit of a curve, I will draw it like this.
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I will take a point on that curve.
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Again, we are looking for the slope of the line that just touches that curve at one point.
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This is our f(x).
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This is called our tangent line.
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This point here is going to be xy, or x, if you prefer f(x), however you want to list it.
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Once again, the line that touches a curve at a single point is called the tangent line.
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It is kind of redundant, the tangent line of the curve at that point, I just said that.
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We will just say, it is called the tangent line.
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I think it is perfectly clear what we are talking about.
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It is called the tangent line.
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The slope of the tangent, it is the specific numerical value of the derivative of the function at that point.
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It is the derivative at that point.
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Again, the point itself is a point xy in the plane.
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When we find f’(x), let us say we have some f(x), we find f’(x), that is the derivative, it is the x value of the point.
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It is the x value of the point that we put into f’(x).
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Notice, it is a function of x not a function of y.
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If we want to find the particular numerical value of the derivative, if we have f’(x), it is a function of x.
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We are going to put the x value in there and it is going to spit out some number.
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That number is the slope of that line.
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It is the x value that you put in.
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When we find f’(x), it is the x value of the point that we put into f(x), in order to get a numerical value for the slope.
00:29:13.800 --> 00:29:19.200
Let us see what we have got, let us do some examples here.
00:29:19.200 --> 00:29:32.400
Some examples, I think what we will do is we will let f(x), let us go ahead and take the same function y = sin(x).
00:29:32.400 --> 00:29:37.500
f(x) = sin(x) or y = sin(x).
00:29:37.500 --> 00:29:40.200
I’m going to go ahead and tell you what the derivative is here.
00:29:40.200 --> 00:29:42.900
Again, we will talk about how we got this later on.
00:29:42.900 --> 00:29:58.400
It will turn out that f’(x) actually equals cos(x).
00:29:58.400 --> 00:30:04.000
The original function is sin(x), its derivative is going to turn out to be cos(x).
00:30:04.000 --> 00:30:07.600
When I put in different values of x into cos(x),
00:30:07.600 --> 00:30:17.700
that will give you the numerical value of the slope of the tangent line touching the sine curve, at that point, like this.
00:30:17.700 --> 00:30:27.800
I think I will work in red for this one, that will be nice, a little change of pace here.
00:30:27.800 --> 00:30:30.500
Of course, we have that there.
00:30:30.500 --> 00:30:34.400
Let us go ahead and draw our sine curve again.
00:30:34.400 --> 00:30:38.000
We said we have 0, we have π, we have 2 π.
00:30:38.000 --> 00:30:41.300
This is 1 and this is -1.
00:30:41.300 --> 00:30:44.800
Now let us take the point 0.
00:30:44.800 --> 00:30:48.700
When x = 0, let us find the y value.
00:30:48.700 --> 00:30:56.200
We know y is 0 but let us actually find it.
00:30:56.200 --> 00:31:03.000
f(0) is equal to sin(0) which = 0.
00:31:03.000 --> 00:31:06.000
Yes, our point is going to be 0,0.
00:31:06.000 --> 00:31:11.700
Again, sometimes we do not have a graph to work with, which is the reason I went through it analytically here.
00:31:11.700 --> 00:31:15.900
It is very clear that this is going to be 0,0.
00:31:15.900 --> 00:31:24.600
It is going to be very clear that this point up here is going to be π/2, 1, 3π/ 2, -1, π 0, 2π 0.
00:31:24.600 --> 00:31:28.800
We can see it graphically but we would not always have a graph.
00:31:28.800 --> 00:31:37.000
f’(0) of f’ is cos(x), that is the cos(0).
00:31:37.000 --> 00:31:38.500
What is the cos(0)?
00:31:38.500 --> 00:31:41.400
The cos(0) = 1.
00:31:41.400 --> 00:32:06.500
What that means is that the slope of the tangent line through 0,0 has a slope of 1.
00:32:06.500 --> 00:32:10.600
What is that tangent line?
00:32:10.600 --> 00:32:16.600
Let me extend this out a little further so it goes down that way.
00:32:16.600 --> 00:32:24.100
My tangent line is the line that touches the graph at that point, that is my tangent line.
00:32:24.100 --> 00:32:32.400
The slope of that line is given 1, because the derivative of sin x is cos x.
00:32:32.400 --> 00:32:38.800
Let us find what the slope is at π/2.
00:32:38.800 --> 00:32:51.900
When x is equal to π/2, the f value, the y value, the f(π/2) which is equal to sin(π/2), that is equal to 1.
00:32:51.900 --> 00:32:59.400
Therefore, I know that the point that I’m talking about is π/2 and 1.
00:32:59.400 --> 00:33:04.800
I think I will do a little purple.
00:33:04.800 --> 00:33:11.100
I know my tangent line is there, it is the line that touches the graph at that point.
00:33:11.100 --> 00:33:12.300
That is the tangent line.
00:33:12.300 --> 00:33:14.400
What is the slope of that tangent?
00:33:14.400 --> 00:33:18.900
Graphically, just by looking at it geometrically, I know that the slope is 0.
00:33:18.900 --> 00:33:22.200
We would not always have a graph, let us do it analytically.
00:33:22.200 --> 00:33:28.200
Analytically, I know that the derivative is cos(x).
00:33:28.200 --> 00:33:41.600
f’ at π/2, remember, we put in the x value, is equal to cos(π/2) that is equal to 0.
00:33:41.600 --> 00:34:10.000
As you can see geometrically, analytically here, the slope of the tangent line through π/2, one has a slope of 0.
00:34:10.000 --> 00:34:14.100
Let us do one more.
00:34:14.100 --> 00:34:18.900
Let us take the point 7π/ 6, not quite so easy this time.
00:34:18.900 --> 00:34:24.600
We will take x = 7π/ 6.
00:34:24.600 --> 00:34:37.400
f(7π/ 6), f is sin, sin (7π/ 6), we are just trying to find the y value first, to find out where the point is.
00:34:37.400 --> 00:34:45.700
It is going to be -1/2.
00:34:45.700 --> 00:34:56.200
Our point is 7π/ 6 is our x value, -1/2 is our y value.
00:34:56.200 --> 00:34:57.300
You are looking at it on the graph.
00:34:57.300 --> 00:35:03.100
7π/ 6 is somewhere like right over here.
00:35:03.100 --> 00:35:15.400
This point right here, that point is our point 7π/ 6, -1/2.
00:35:15.400 --> 00:35:19.000
If this is going to be 1, that is probably going to be that way.
00:35:19.000 --> 00:35:26.800
I have not drawn it that great but you get the idea.
00:35:26.800 --> 00:35:31.900
Let us try this again, shall we, all this crazy writing.
00:35:31.900 --> 00:35:45.000
This point over here, on the graph it is 7π/ 6 and -1/2, that is the coordinate of it.
00:35:45.000 --> 00:35:48.000
What about f’, the derivative?
00:35:48.000 --> 00:35:59.900
f’(7π/ 6) = cos(7π/ 6) because the derivative of sin x is cos x.
00:35:59.900 --> 00:36:15.000
The cos(7π/ 6) is –√3/2.
00:36:15.000 --> 00:36:21.200
The slope of the tangent line, that tangent line,
00:36:21.200 --> 00:36:42.000
the slope of the tangent line through the point 7π/ 6, 1/2 is –√3/2.
00:36:42.000 --> 00:36:47.400
Geometrically, we can see that it is going to have to be a negative slope because it is going from top left to bottom right.
00:36:47.400 --> 00:36:54.800
Numerically, analytically, we have to use the formula for the derivative, to find its actual numerical value.
00:36:54.800 --> 00:36:57.500
The slope is a derivative.
00:36:57.500 --> 00:37:02.900
When we say find the derivative of a function, we are saying do whatever you need to do to find the derivative of the function,
00:37:02.900 --> 00:37:04.700
which is going to be another function of x.
00:37:04.700 --> 00:37:11.700
And then, put in the x value of whatever point on the curve you want, that will give you the slope of the tangent line.
00:37:11.700 --> 00:37:14.400
That is the derivative.
00:37:14.400 --> 00:37:20.700
The derivative of 7π/ 6 – 1/2 of sin x is equal to -√3/2.
00:37:20.700 --> 00:37:25.200
The derivative of the function is cos(x).
00:37:25.200 --> 00:37:35.300
The derivative of the function, the numerical value.
00:37:35.300 --> 00:37:38.600
Let us stick with red here.
00:37:38.600 --> 00:38:10.700
It would be very nice to have a general procedure for finding the derivative of f(x).
00:38:10.700 --> 00:38:13.400
We have a general procedure.
00:38:13.400 --> 00:38:22.400
That general procedure says, let me write it a little bit more clearly here.
00:38:22.400 --> 00:38:26.000
And then later on, we will be a little bit more messy.
00:38:26.000 --> 00:38:36.400
The limit as h approaches 0 of f(x) + h - f(x)/ h, this is our general procedure.
00:38:36.400 --> 00:38:42.100
It is a procedure that we are going to address in a later lesson, not right now.
00:38:42.100 --> 00:38:47.000
I’m going to save the procedure for how to find the how, I'm going to save for another lesson.
00:38:47.000 --> 00:38:51.200
For right now, I want to concentrate on the y.
00:38:51.200 --> 00:38:59.200
What does it mean, we want to get a feeling for this.
00:38:59.200 --> 00:39:20.900
I will start discussing this procedure in a future lesson.
00:39:20.900 --> 00:39:42.200
The first thing we are going to do is, when we do this, first, we will discuss what this part means, what that means.
00:39:42.200 --> 00:39:56.700
The second thing we will do, then, we address the whole thing.
00:39:56.700 --> 00:40:00.600
If you saying to yourself, why does he keep writing this thing over and over again?
00:40:00.600 --> 00:40:02.900
There is a reason for it, there is a pedagogical reason for it.
00:40:02.900 --> 00:40:06.400
This is a very important thing.
00:40:06.400 --> 00:40:12.100
I’m writing it over and over again so that by the time you actually do see it, it will be a sort of like you have seen it before.
00:40:12.100 --> 00:40:14.500
That is the reason I'm doing it.
00:40:14.500 --> 00:40:19.600
It is not because I’m obsessive compulsive, over h.
00:40:19.600 --> 00:40:26.400
When we actually discuss this in a future lesson, the how, I’m going to discuss what limits are first, how to find limits.
00:40:26.400 --> 00:40:36.600
And then, we will go ahead and address how to take the limit of this particular quotient, which will give us the derivative.
00:40:36.600 --> 00:41:15.300
For the next few lessons, we will continue with slopes of curves and what derivatives mean.
00:41:15.300 --> 00:41:21.500
Once again, we want to become familiar with this idea of the slope of a curve.
00:41:21.500 --> 00:41:27.500
We want to be able to handle a few things, in basic brute force way.
00:41:27.500 --> 00:41:32.600
We want to know what is going on, how this idea of the slope,
00:41:32.600 --> 00:41:35.600
how we are going to relate it to what we have done with slope before.
00:41:35.600 --> 00:41:40.100
We want to get comfortable with it, before we start actually introducing calculus ideas.
00:41:40.100 --> 00:41:44.900
That is what is going to occupy us, for the next probably three lessons.
00:41:44.900 --> 00:41:48.200
We are going to spend a couple of lessons discussing what these things mean and
00:41:48.200 --> 00:41:51.200
we are going to do a lesson on some example problems.
00:41:51.200 --> 00:41:55.800
We will begin by discussing this idea of a limit of a function.
00:41:55.800 --> 00:41:59.400
What does this limit as h approaches 0 mean.
00:41:59.400 --> 00:42:02.300
With that, I will go ahead and stop this first lesson there.
00:42:02.300 --> 00:42:06.700
Thank you again for joining us, I hope this turns out to be a wonderful experience for you.
00:42:06.700 --> 00:42:08.000
Thank you and see you next time, take care.