AP Calculus AB Overview & Slopes of Curves

Hello, welcome to www.educator.com, welcome to the first lesson of AP Calculus AB.0000

I thought what I would do is take about 5 or 10 minutes to give a nice overview of the course as a whole,0007

so that we have a sense of where we are going.0012

Then, we are going to launch right into the calculus proper.0014

Let us get started and welcome.0016

Calculus is basically going to be about two things, two processes.0025

We are going to spend the first half learning how to do something called differentiating a function.0030

We are going to spend the second half learning how to do something called integrating a function.0033

That is essentially it.0037

Each one of those tools is going to yield different applications.0039

That is all we are going to be doing.0043

Let me work in blue here.0046

Given a function f(x), we will be spending our entire time doing two things to this function.0051

The first thing is we are going to be differentiating it.0087

It also called taking the derivative.0099

Both of those terms, we will use interchangeably.0106

The second thing we are going to be doing to this function,0110

it is just going to occupy the second half of calculus, is we are going to be integrating it.0114

These are the two major rivers of calculus.0123

These are the two great rivers of calculus.0128

The differential and the integral, two basic set of tools that allow us to solve certain types of problems with the differential calculus,0139

certain other types of problems with the integral calculus.0154

Essentially, what is going on is this.0158

We have some function f(x).0160

One of the things that we can do to it is differentiate it.0163

When we differentiate a function, we are going to get another type of function.0166

We are going to symbolize that with f’(x).0171

The other thing that we can do to it is we can integrate this function.0175

And then, when we integrate a function, we are going to get yet another type of function0179

which we will symbolize normally with F(X).0183

This is a new function and we call this function the derivatives.0187

That is called the derivative because it is derived from the original function.0195

This one is also a new function.0199

We call this one the integral.0206

That is essentially what calculus comes down to.0209

It is finding ways to find derivatives of functions, applying it in certain cases, and then finding integrals of functions.0212

Applying the integral in certain cases.0218

Let us move on to the next page here.0228

These two tools namely differentiation and integration are very powerful.0229

It will allow us to solve the most extraordinary problems, that is what beautiful about the calculus.0256

So far in your mathematical studies from elementary school, all the way through high school,0274

there has been a pretty steady increase, in terms of the level of difficulty, the techniques that you develop.0278

Some or more problems that you can solve.0284

That is essentially what it is, it is just, you are making the class of problems that you can solve bigger and bigger.0285

Now the calculus, it is a huge jump.0291

It is not a little jump, it is not a stair step.0294

You are going to be introduced to these tools, differentiation and integration.0296

They are very powerful tools.0300

It is going to take you up here, in terms of mathematical sophistication and the types of problems that you can solve.0301

It is really quite extraordinary.0308

Everything that we enjoy in the modern world, I do mean everything,0311

It is worth writing down.0331

I do mean everything.0332

Let us capitalize this one.0339

I do mean everything is made possible with calculus.0342

It really is the single tool, these two things, differentiation and integration.0358

They have allowed us to absolutely enjoy everything that we enjoy, computers, cell phones, cars, you name it.0364

If there is anything in the modern world that you take for granted, that you enjoy, it is because of the calculus.0372

Once again, we are going to spend the first half learning and applying differential calculus.0380

And then, we are going to spend the second half, roughly, learning and applying the integral calculus.0384

It is going to turn out that these two independent techniques, the differentiation and the integration,0391

are in fact so deeply related, that we call this relationship the fundamental theorem of calculus.0420

You will often see it as FTC.0461

Differentiation and integration, they are reasonably independent techniques.0464

There is no reason to believe that one is actually related to the other.0469

They solve different set of problems.0472

Yet, there is a very deep relationship that exists between them.0474

This relationship is what we call the fundamental theorem of calculus.0477

What is really interesting about the fundamental theorem of calculus,0481

something that many of you who go on to higher mathematics will discover,0483

is that this relationship that exists between differentiation integration is actually true, not just in one dimension.0487

In other words the real line, which is what we have essentially been dealing with ever since elementary mathematics.0495

We stayed on with the real numbers.0499

It is true in any number of dimensions, dimensions 2, 3, 4, 5, 6, 15, 34, or 147.0502

It is true in any number of dimensions.0508

It is a very profound relationship.0509

We will actually be getting to that, when we are getting to integration.0517

We will study the fundamental theorem of calculus.0521

Now the nice thing is, I can already tell you now what this relationship is.0523

You have an idea of where is that we are going.0526

I can tell you now what this relationship is.0532

Later, we will explore this relationship.0535

I can tell you now what this relationship is.0540

Again, there is nothing here, this is just overview.0547

Just giving you an idea of what it is that we are in for, so that you are not going into this blindly.0549

So you just have some sense of why we are doing what we are doing.0554

I can tell you now what this relationship is.0557

The relationship is, each of these processes namely differentiation and integration,0564

each of these processes, dif and int is the inverse of the other.0573

In other words, if I start with some function f(x), we said that I can differentiate it.0592

You know what I’m going to do actually here, I will go ahead and write dif down here.0619

I said that I could differentiate it to come up with a new function, the derivative.0627

If I want to go back, I integrate this function.0632

If I integrate, it will take me back to my original function.0638

Or if I integrate, it will give me some other function which we call the integral.0642

If I want to go back to the original, I just differentiate.0648

That is the relationship between the two.0653

Two entirely independent techniques, for the most part, because they solve entirely different set of problems.0654

Yet, the relationship is one is the inverse of the other.0661

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.0663

If you want to go back, you exponentiate this thing and it takes you back.0670

The logarithm and the exponential are inverse processes.0675

The sin and the inverse sin are inverse processes.0681

Cos, inverse cos, are inverse processes.0685

Differentiation or integration are inverse processes.0688

This is extraordinarily deep, extraordinarily beautiful.0691

Again, there is no reason to the world to believe that they are connected and yet they are.0694

Let us write that last part.0701

There is no obvious reason why they should be connected, at least cosmetically.0704

There is no reason to just sort of look and say that this is related to that.0711

There is no obvious reason why differentiation and integration should be related, but they are.0715

This relation has consequences that go further and deeper than you can imagine.0746

One of the beautiful things about mathematics is that, you will have different people0779

or perhaps the same person will investigate different areas of mathematics, to solve a certain type of problem.0783

When you take a look at this set of mathematics that you develop for this problem0792

and this set of mathematics that you develop for this problem,0796

when you realize that there is actually a connection between those two mathematics, they come together, you unify that.0799

That relationship that exists between the various areas of mathematics takes you to a deeper level,0806

a deeper understanding of reality, a deeper understanding of how the physical world works.0813

This is what we strive for, we strive for unification.0819

This is what makes it beautiful, things that should not be related, at least, as far as our intuition is concerned,0823

they end up being not only related but very deeply related.0829

The consequences of those relationships are profound.0833

Anyway, this is really beautiful stuff and it begins right here, with your first course in calculus.0838

Once again, welcome, and let us get started.0845

We are going to spend the first half talking about differentiation.0856

We are going to put integration on the shelf, for the time being.0859

We will come back to it later.0861

We are going to begin with taking derivatives, differentiation.0861

We begin with differentiation.0865

Actually, let me write something here.0880

Differentiation or also called taking the derivative.0883

I will just write it over here, taking the derivative.0890

In other words, starting with some f(x), performing the differential operation on it,0896

and ending up with a new f’(x), a new functions that is going to give us other information.0904

Either by the situation, or it is going to give us information about the original function, whatever it is.0910

But it is a new function that we have derived.0915

Two questions, the obvious questions.0920

Two questions, what does the derivative mean and how do we find it?0926

Given this, how do we find the derivative and once we have the derivative, what does the derivative mean?0933

What does it give us, what does it tell us, what problems does it solve?0939

Two questions, what does the derivative mean?0943

Two, how do we find it?0958

How do we find it given some f(x)?0961

How do we derive and get?0967

The answers are as follows.0971

The answers, the answer to number 1, what does a derivative mean?0974

The derivative is the slope of a function curve.0978

A function is just some curve that you draw on the xy plane at a given point on the curve.0999

We will explain what that means more, in just a second.1011

That is pretty much it.1015

A derivative is a slope of a curve at a particular point.1017

The strange thing is you have been doing derivatives for many years now.1024

If you have the function y = 3x + 4, you know that is the equation of the line.1027

The curve itself is a straight line but we call it a curve.1037

In general, it is a line in space.1040

What is the slope of this line, it is 3.1043

The function is 3x + 4, what is the derivative of that function?1046

The derivative is 3 because no matter where you are on that line, the slope is 3.1049

That is what you have been doing, you have been finding derivatives.1056

Now in calculus, we are not just going to find the slopes of straight lines.1058

We are going to find the slopes of curves.1062

What is that slope, the slope there, slope there.1066

The slope is going to change as you move along the curve, that all we are saying.1069

We are just giving you a fancy name and calling it the derivative, that is all.1074

Number 2, let us go back to blue here.1080

The answer to the question, how do you find it?1083

Here is how you find, f’(x), in order to find f’(x), here is what you do.1087

Limit as h goes to 0, you form f(x) + h, given whatever f is, you subtract from it the original f(x).1094

You divide it by h and then you subject it to this process called taking the limit as h goes to 0.1106

We will be discussing what this means, how does one do this.1114

It is actually quite simple, just algebraically tedious.1119

How do we find it? We find it like this.1124

What does it mean, it means it is a slope.1126

We are going to start with number 1.1131

As far as number 2 is concerned, the how, I’m going to leave that for a future lesson.1133

For the next couple of lessons, I’m going to be talking about what the derivative is, slope of curves,1139

getting ourselves comfortable with the idea of a slope of a curve, as opposed to just the slope of a straight line.1145

And then, once we have a reasonably good sense, once we feel comfortable with that,1150

we will talk about how to find this so called f’(x), the derivative.1154

Let us go over here, I will stick with blue.1163

Let us start with number 1, in other words, the meaning, what does it mean?1170

The derivative f’(x) is a function which gives us the slope of the curve or graph.1194

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.1232

Here is what this means.1260

Let us take a look at let us say the sine functions.1262

Our function f(x), original function is sin(x), also y = sin(x).1271

We are going to be working in the xy coordinate system.1280

We are never going to be moving out of that.1282

Whenever you see f(x), you can just replace it with y, it is the same thing.1284

We know what the sine function looks like, it looks like this.1289

This is 0, this is going to be our π, this is going to be 2 π.1302

Over here at π/2, it is going to hit a value of 1.1306

Over here at 3π/ 2, it is going to hit a value of -1, standard sine function like that.1310

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.1322

What we call the tangent line.1335

The slope of that line is the slope of the curve, at that particular value of x.1338

At π/2, think of it as just some tangent line that is following the curve along.1343

You have that slope.1351

At this point over here, the slope is that.1353

At this point over here, the slope is that.1356

At this point over here, the slope is that.1359

This point over here, the slope is that.1362

Here, the slope is that.1365

You can see that the slope changes depending on where you are.1366

That is what f’(x) gives.1372

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.1374

When you find the derivative, it is going to be another function which we symbolize with f’(x).1382

The different values of x, it gives us some number, when you actually solve that function.1386

That number is the slope of the line that touches the graph, at that particular xy value.1393

As we can see for various values of x, the slope at that point, the slope at the point xy,1401

if this is x, this is the point xy, it is different.1439

We can see it geometrically, if that is the case.1447

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.1449

Once again, given f(x), we differentiate it and it gives us f’(x).1457

This thing, this tells us what the numerical value of the slope is.1465

The derivative itself is a function of x, we do not know.1488

We have to put in different values of x, to see what the slope is.1491

It tells us what the numerical value of the slope is for different values of x.1494

Let us go over here.1522

Let me draw a little bit of a curve, I will draw it like this.1524

I will take a point on that curve.1529

Again, we are looking for the slope of the line that just touches that curve at one point.1531

This is our f(x).1539

This is called our tangent line.1544

This point here is going to be xy, or x, if you prefer f(x), however you want to list it.1552

Once again, the line that touches a curve at a single point is called the tangent line.1564

It is kind of redundant, the tangent line of the curve at that point, I just said that.1606

We will just say, it is called the tangent line.1611

I think it is perfectly clear what we are talking about.1613

It is called the tangent line.1616

The slope of the tangent, it is the specific numerical value of the derivative of the function at that point.1619

It is the derivative at that point.1644

Again, the point itself is a point xy in the plane.1666

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.1674

It is the x value of the point that we put into f’(x).1697

Notice, it is a function of x not a function of y.1706

If we want to find the particular numerical value of the derivative, if we have f’(x), it is a function of x.1709

We are going to put the x value in there and it is going to spit out some number.1716

That number is the slope of that line.1720

It is the x value that you put in.1723

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.1724

Let us see what we have got, let us do some examples here.1754

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).1759

f(x) = sin(x) or y = sin(x).1772

I’m going to go ahead and tell you what the derivative is here.1777

Again, we will talk about how we got this later on.1780

It will turn out that f’(x) actually equals cos(x).1783

The original function is sin(x), its derivative is going to turn out to be cos(x).1798

When I put in different values of x into cos(x),1804

that will give you the numerical value of the slope of the tangent line touching the sine curve, at that point, like this.1807

I think I will work in red for this one, that will be nice, a little change of pace here.1818

Of course, we have that there.1828

Let us go ahead and draw our sine curve again.1830

We said we have 0, we have π, we have 2 π.1834

This is 1 and this is -1.1838

Now let us take the point 0.1841

When x = 0, let us find the y value.1845

We know y is 0 but let us actually find it.1849

f(0) is equal to sin(0) which = 0.1856

Yes, our point is going to be 0,0.1863

Again, sometimes we do not have a graph to work with, which is the reason I went through it analytically here.1866

It is very clear that this is going to be 0,0.1872

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 0.1876

We can see it graphically but we would not always have a graph.1884

f’(0) of f’ is cos(x), that is the cos(0).1889

What is the cos(0)?1897

The cos(0) = 1.1898

What that means is that the slope of the tangent line through 0,0 has a slope of 1.1901

What is that tangent line?1926

Let me extend this out a little further so it goes down that way.1930

My tangent line is the line that touches the graph at that point, that is my tangent line.1936

The slope of that line is given 1, because the derivative of sin x is cos x.1944

Let us find what the slope is at π/2.1952

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 1.1959

Therefore, I know that the point that I’m talking about is π/2 and 1.1972

I think I will do a little purple.1979

I know my tangent line is there, it is the line that touches the graph at that point.1985

That is the tangent line.1991

What is the slope of that tangent?1992

Graphically, just by looking at it geometrically, I know that the slope is 0.1994

We would not always have a graph, let us do it analytically.1999

Analytically, I know that the derivative is cos(x).2002

f’ at π/2, remember, we put in the x value, is equal to cos(π/2) that is equal to 0.2008

As you can see geometrically, analytically here, the slope of the tangent line through π/2, one has a slope of 0.2021

Let us do one more.2050

Let us take the point 7π/ 6, not quite so easy this time.2054

We will take x = 7π/ 6.2059

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.2064

It is going to be -1/2.2077

Our point is 7π/ 6 is our x value, -1/2 is our y value.2086

You are looking at it on the graph.2096

7π/ 6 is somewhere like right over here.2097

This point right here, that point is our point 7π/ 6, -1/2.2103

If this is going to be 1, that is probably going to be that way.2115

I have not drawn it that great but you get the idea.2119

Let us try this again, shall we, all this crazy writing.2127

This point over here, on the graph it is 7π/ 6 and -1/2, that is the coordinate of it.2132

What about f’, the derivative?2145

f’(7π/ 6) = cos(7π/ 6) because the derivative of sin x is cos x.2148

The cos(7π/ 6) is –√3/2.2160

The slope of the tangent line, that tangent line,2175

the slope of the tangent line through the point 7π/ 6, 1/2 is –√3/2.2181

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.2202

Numerically, analytically, we have to use the formula for the derivative, to find its actual numerical value.2207

The slope is a derivative.2215

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,2217

which is going to be another function of x.2223

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.2225

That is the derivative.2232

The derivative of 7π/ 6 – 1/2 of sin x is equal to -√3/2.2234

The derivative of the function is cos(x).2241

The derivative of the function, the numerical value.2245

Let us stick with red here.2255

It would be very nice to have a general procedure for finding the derivative of f(x).2258

We have a general procedure.2291

That general procedure says, let me write it a little bit more clearly here.2293

And then later on, we will be a little bit more messy.2302

The limit as h approaches 0 of f(x) + h - f(x)/ h, this is our general procedure.2306

It is a procedure that we are going to address in a later lesson, not right now.2316

I’m going to save the procedure for how to find the how, I'm going to save for another lesson.2322

For right now, I want to concentrate on the y.2327

What does it mean, we want to get a feeling for this.2331

I will start discussing this procedure in a future lesson.2339

The first thing we are going to do is, when we do this, first, we will discuss what this part means, what that means.2361

The second thing we will do, then, we address the whole thing.2382

If you saying to yourself, why does he keep writing this thing over and over again?2397

There is a reason for it, there is a pedagogical reason for it.2400

This is a very important thing.2403

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.2406

That is the reason I'm doing it.2412

It is not because I’m obsessive compulsive, over h.2414

When we actually discuss this in a future lesson, the how, I’m going to discuss what limits are first, how to find limits.2419

And then, we will go ahead and address how to take the limit of this particular quotient, which will give us the derivative.2426

For the next few lessons, we will continue with slopes of curves and what derivatives mean.2436

Once again, we want to become familiar with this idea of the slope of a curve.2475

We want to be able to handle a few things, in basic brute force way.2481

We want to know what is going on, how this idea of the slope,2487

how we are going to relate it to what we have done with slope before.2492

We want to get comfortable with it, before we start actually introducing calculus ideas.2495

That is what is going to occupy us, for the next probably three lessons.2500

We are going to spend a couple of lessons discussing what these things mean and2505

we are going to do a lesson on some example problems.2508

We will begin by discussing this idea of a limit of a function.2511

What does this limit as h approaches 0 mean.2516

With that, I will go ahead and stop this first lesson there.2519

Thank you again for joining us, I hope this turns out to be a wonderful experience for you.2522

Thank you and see you next time, take care.2527