AP Calculus AB More on Slopes of Curves

Hello, welcome back to www.educator.com, and welcome back to AP Calculus.0000

Today, we are going to continue our discussion about slopes of curves.0005

Let us get started.0010

Let me work in blue and see how that goes today.0020

We said that given f(x), some function of x, its derivative which we symbolize with f’(x),0025

we said that this derivative gives us the slope of the tangent line to f(x) or to the graph of f(x),0048

at a particular point, at a particular x.0067

It says at a particular point, xy.0077

We speak of the slope of the tangent line.0081

We can be a little loose, as we speak about the slope of the curve.0086

Essentially, it is the slope of the tangent line.0088

The slope of the curve is the slope of the tangent line.0091

We have something like this.0093

We got some graph like that, that is our f(x).0096

We pick some particular point that we are interested in, it is an x value.0105

The point itself is some xy value, the y value is just f(x).0110

There is a tangent line like that.0118

This is our tangent line.0121

f'(x), when we put a particular x value in, once we found what f’ is, as a function, f’ is the slope of this line.0128

This is our tangent line, this is the slope of the curve at a given point.0145

It is the derivative, it is the most important slope that we are interested in.0149

There is another slope that I'm going to introduce.0155

There is another slope along a curve that I want to introduce.0157

It is, this is we call it the tangent line.0182

The tangent line touches the curve at one point and one point only.0185

The slope that I'm going to introduce now, it is the slope of something called the secant line along a curve.0190

A secant line hits a curve at two points and two points only, generally two points only.0209

If I were to take this point and this point, and connect them, that is the secant line.0217

That is the difference.0226

Tangent line touches the graph at one point, secant line two points.0228

Let us go ahead and redraw this.0235

We have something like this and we have our function like that.0240

Let us go ahead and take that one point and that is another point.0247

Let us go ahead, that is our secant line.0252

If I were to take that point and draw a tangent line through it, that would be one tangent line.0258

Tangent line at this point, that would be another tangent line.0265

A secant line is the line that connects any two points on a curve, on a graph.0271

The slope of the secant line, any secant line, is actually very easy to find.0298

Because you have two points, you just do your normal slope formula Δ y/ Δ x, y2 - y1, y2 - y1/ x2 – x1.0304

If you have two points, that is it, something that you have been doing for years now.0313

The slope of the secant line is easy to find.0319

That is just, I will call it m sec, that is just Δ y/ Δ x.0334

In other words, y2 - y1/ x2 – x1.0342

We call the slope of the secant line the average slope of the function.0348

The average slope of the function between x1 and x2.0376

Here is our x1, here is our x2.0387

When we calculate the slope of the secant line, two points are necessary.0390

It is the average slope between x1 and x2.0395

You will hear the slope of secant line, we also call it the average slope.0400

I will go ahead and do it on the next page.0409

We call the slope of the tangent line the instantaneous slope at x because it only involves one point.0424

The instantaneous slope of the function at x, whatever x happens to be.0437

The derivative is the instantaneous slope.0453

The derivative is not the average slope.0457

The derivative is the instantaneous slope.0459

The derivative of f(x) which is f’(x).0471

I do not want to introduce more notation, that is not necessary.0484

The derivative of f(x), I will just say that, is the instantaneous slope.0488

Because we said the derivative is the slope of the tangent line.0500

Let us go ahead and draw something like that.0521

Let us draw a little piece of the curve like this.0528

Let us go ahead and take that is one point, that is another point, this is our x1, this is our x2.0532

Our secant line, I’m going to draw that way.0540

Let me work in blue now.0567

If I took several points between x1 and x2 along the curve and if I took tangent lines, tangent line,0569

You get the idea.0590

You see that the tangent line, in this particular case, as it moves up this curve, the tangent line is increasing.0596

If I were to calculate the instantaneous slopes0608

at various points between x1 and x2, if I average them, I would get the average slope.0628

That is essentially what it is.0655

Again, when we say average slope from x1 to x2, essentially,0656

what we are saying is that you are going to have a bunch of slopes, instantaneous slopes, along there.0662

But if we took an average of those slopes, it is going to end up being the same.0667

Again, what an average is, an average is just taking the ending and the beginning, and taking a mean,0671

the average value of the slopes in between that point.0678

That is all that is happening.0683

It is essentially the relationship between the average and the instantaneous ones in between.0684

Notice, some of these slopes are less than this.0690

Up here, some of the slopes are actually steeper than here.0695

On average, it is going to be this slope, that is all that is going on here.0699

If I were to calculate the instantaneous slopes of various points between x1 and x2, and average them,0705

I would get the average slope in the interval x1, x2.0709

Sometimes this is useful, sometimes we want average.0724

Sometimes we only want to work with average.0728

More often than not want, we want to work with instantaneous.0731

It just depends.0736

Other times, what we want is the slope at a particular instant.0742

That is at a particular x, that is the instantaneous slope.0773

Sometimes we want the average slope, sometimes we want the instantaneous slope.0785

In other words, we want the slope of that line.0795

We want the derivative.0804

Let us go ahead and work an example here.0809

Our example is going to be a very important one.0813

It will come up a lot in the applied problems that you run across.0818

It is going to be distance, time, and velocity.0821

I think I will actually work in red here.0836

Let us say, a car starts from rest and accelerates.0839

Physically, you are all familiar with the intuition behind this.0853

You start at rest and you start to accelerate.0856

Its distance from where it started, which we will call the origin.0858

It is a very convenient place, we will just make that as x = 0.0874

The origin is a function of time.0879

Distance is a function of time.0887

In other words, what that means is that, at a certain time t,0890

its distance from the origin, in other words where it started, is some f(t).0916

t is the independent variable, the distance is the dependent variable.0933

f(t) is the distance that it has traveled.0939

That is it, that is all that is going on, that is all of this says.0944

We have a graph.0948

Basically, it is just going to be some, it is accelerating.0951

It is going to look something like that.0955

t is the independent variable.0960

Distance d is the dependent variable.0962

The distance is going to be expressed in meters and the time is going to be expressed in seconds.0965

We might make a little bit of table of values, t and d which is equal to f(t).0975

Let us say at time = 0, we are at the origin.0981

Let us say one second later, we are half a meter away.0986

2 seconds later, we are 2 meters away.0991

3 seconds later, we are 4.5 meters away.0993

4 seconds later, we are 8 meters away.0997

This is a tabular version of the function.1000

This is a graphical version of the function.1002

Let us say that our actual function of t which describes this is d = ½ t².1005

This is our function of t.1022

My question to you is, what is the average slope of d(t) between t = 4 and t = 8.1027

What I'm asking is, this is t right here.1051

Let us say this is 4 and let us say this is 8.1053

That is f(t), that is whatever that is.1058

This is our f(4), this is f(8).1061

The average slope is the slope of that line.1066

It is the slope of the secant line between them.1072

I need to find the xy point, the xy point, and I calculate the slope.1074

That is all I’m doing here, that is all this means.1078

What is the average slope here?1081

Let us go ahead and work this out.1085

Let us go ahead and redraw this.1087

We got something like that.1091

We have 4, we have 8.1093

There is that point, there is that point.1097

We are trying to find the slope of the secant line.1101

When t = 4, d(4), we said that d which is a function of t is equal to ½ t².1104

d(4) is equal to ½ × 4², that is equal to 8.1118

This point is the point 4,8.1127

At t = 8, d(8), in other words the y value, this is going to be ½ of 8².1134

It is going to be 32, the point is 8,32.1143

This point up here is 8,32.1147

Our average slope is equal to the change in y/ the change in x.1155

Or in this case, because this is time and this is distance, it is going to be Δ d/ Δ t.1163

Now x and y has generic variables.1172

Now that we have actually applied to the real world situation,1177

where the variables actually mean something, where the x variable is time, the y variable is distance.1180

Now it is Δ d/ Δ t.1185

It is going to be 32 – 8, distance 2 - distance 1 divided by 8 – 4.1188

The number I’m going to get is 6.1199

The average slope of this function between 4 and 8, this function is 6.1202

What about the unit?1212

This is a physical situation, there has to be some unit associated with this.1219

What is the unit, we have Δ d/ Δ t.1225

d is expressed in meters, you are dividing it by t which is expressed in seconds.1230

Your unit is meters per second.1241

Meters per second is the unit of velocity.1243

This is the unit of velocity.1255

Whenever you are dealing with a situation where time is on the x axis, distance is on the y axis, to your slope,1258

which is gotten by taking some change in x over some change in y.1266

It is always going to be meters per second.1271

The numerical value is a numerical value, the unit that it represents1273

is going to be the dependent variable divided by the independent variable.1277

In this case, meters per second.1283

When you have distance as a function of time, the slope is a velocity.1284

The average slope is average velocity.1291

Between 4 seconds and 8 seconds, your average velocity is 6 m/s.1295

If I ask you to find the tangent curve, the number that you get for that, the derivative of x = 5,1300

that is going to be the instantaneous velocity.1307

In other words, at 5.5 seconds, if I look at the speedometer, that is how fast my speedometer is going.1310

That is how fast my car is going at that moment.1316

In this case, you are going to be faster, a second later.1318

You are going to be slower, a second before that.1322

On average between 4 and 8, you are going 6 m/s.1324

Let us move on here.1335

What we have calculated is the following.1336

Between 4 seconds and 8 seconds, after the cars starts moving, the average velocity is 6 m/s.1354

Between 4 and 8, on average, it is moving in 6 m/s.1381

If 4 seconds is going to be less than 6, at 8 seconds, it is going to be more than 6.1384

But on average, in that time interval, it is going to be 6.1390

I know I’m repeating myself a lot, I hope you will forgive me.1401

Again, if this stuff is something that you already know, you are more than welcome to move on.1404

This is the 4 seconds, this is the 8 seconds.1414

On average, what we have calculated is the average velocity.1422

Notice, the instantaneous velocities, the lines, the slopes of the lines are increasing.1427

The velocity is increasing.1438

You know that already, you are accelerating.1440

On average, the average slope, average velocity is 6.1442

At any given point along this curve, if I were to take a tangent line,1447

that would give me the instantaneous velocity at that point in time.1452

During the time between 4 seconds and 8 seconds,1461

our instantaneous velocities at various t values are different.1480

Our instantaneous velocities, our instantaneous slopes, they are different, they change.1499

The slope of the tangent line.1505

The instantaneous slopes are changing, as you proceeded along the curve.1510

The average of all of these instantaneous slopes is the average slope.1546

That is all that is happening here.1564

Let us slow down a little bit, shall we?1578

What if I said find me the instantaneous velocity at t = 5.5 seconds.1581

What would you do, 5.5 seconds?1599

Now what we do is we would have to find f’(t).1609

Our original function is f(t).1622

We want the instantaneous slope, we want the instantaneous velocity.1624

That is a derivative, we need to find the derivative function f’(t).1628

And then, we need to put 5.5 in for t and solve.1633

We would have to find f’(t), then plug in 5.5 for t.1639

In other words, we are looking for f(5.5).1653

Again, no worries, we will get there.1657

Right now, we are discussing the why.1658

Later on, we will discuss the how.1660

If you want, I can do it for you right now, just real quickly, just so you have a little bit of a sense of what it is that is coming.1665

We said that f(t) is ½ t².1672

When I differentiate ½ t², what I’m going to end up actually getting is t.1679

f’(t) is actually going to equal t.1687

Therefore, f’ at 5.51, my instantaneous velocity is going to be 5.5 m/s.1691

That is all that is going on.1700

If I want an average, average is easy.1702

I just two points and I take the average.1704

If I want an instantaneous, I have to find the derivative.1707

In this particular case, the derivative of ½ t² happens to be t.1709

Again, you do not know that yet, you are not supposed to know where that came from.1715

I just threw it out for you just so that you can actually see it.1719

If I needed to do it, that is how I would do it.1722

Recap, our secant line, this is our average slope.1734

Our tangent line, this is our instantaneous slope also known as the derivative at that point.1747

Let us interpret what we mean by the slope.1784

Again, this might be something that you already understand, in which case,1789

you are more than welcome to skip it or it might be nice just to do it.1791

It is totally up to you.1795

Let us investigate, let us interpret slope.1796

A slope is a rate of change, our c.1811

A rate of change is this, it is the change that the dependent variable1824

which y experiences for every increment of one change in the dependent variable.1847

That is what a rate of change is.1876

A slope is a rate of change, the slope is dy/dx.1879

Dx is the independent variable, dy.1884

x is the independent variable, y is the dependent variable.1887

If I change x, if Δ x, if I change it by one unit, one increment, how much does y change?1890

That is what a rate of change is, that is what the slope actually tells me.1897

Here is what this means.1903

Let us go to red, why not.1918

I know that slope is equal to Δ y/ Δ x.1921

The change in y, the rate of change is the change in y for every unit change in x.1932

Unit change in x, when you see the word unit, it means 1.1952

Unit change in x means a change by an increment of 1.1961

The word unit is equivalent to 1.1984

When we say the unit change, that means you are changing the variable by 1 unit, from 1 to 2, 2 to 3, 3 to 4, 4 to 5, not 1 to 7.1987

Examples, let us say I calculated the slope equal to 2.1996

This means that dy/dx is equal to 2.2010

This is the same as 2/1.2018

Again, sometimes you end you with whole numbers, 2.6, 5.2.2020

It is still dy/dx, it is a slope, it is a rate of change.2025

There are is still some number and some number.2028

There are still a dependent variable and independent variable.2030

It is better to write it this way.2033

Now we understand, what this says is that if I change x by 1, y changes by 2.2035

That is what this means, it is a rate of change.2056

It is the change that the y variable experiences for every unit change in the x variable.2060

It is a change that the dependent variable experiences for every change of 1,2065

every unit change in the independent variable.2072

Another example, let us say that we are given or that we calculated a slope of 4/3.2075

This means that Δ y/ Δ x = 4/3.2084

This is the same as 4/3/ 1.2091

That is really what is going on.2099

This says, if I change x by 1, then y changes by 4/3.2100

This last one could also be expressed exactly like your thinking.2123

Δ y/ Δ x = 4/3.2135

Excuse me, can also be expressed as, for every change in x by 3, y changes by 4.2141

That is fine, you are welcome to think about it that way.2170

But notice that the definition of the rate of change is, for every unit change in the x value.2173

Unit means 1, it is the personal thing.2180

I will just say that, but thinking about it as 4/3/ 1 or 4/3 to 1 is consistent2185

with the unit change in the x variable or the independent variable.2208

Let us use purple and see how nice that is.2228

Every slope is a rate of change.2233

Now when we assign the x and y variables, two quantities in a physical world,2247

like we did in the problem with time and distance, we always get some numerical value like 4/3.2259

We get the physical unit for the y axis.2292

In other words, distance and velocity example.2302

Distance by this unit, we mean physical unit, not unit 1, meters, seconds,2305

cubic centimeters, kilograms, miles, hours, whatever it is.2311

The unit for the y axis over the unit for the x axis.2315

Often expressed as the numerical value, whatever the numerical value of the slope is.2328

We have unit for y per unit for x.2335

Anytime you see something per something, it is a slope, it is a rate of change.2345

That is what is happening here.2350

Let us go ahead and do an example.2352

The example was, if you see 4 m/s, here, I automatically know that meters is my y axis, second,2356

the denominator is my x axis.2367

Second is my independent variable, x axis.2370

Meters is my y axis, this is time, independent variable.2375

Meters is distance, it is my dependent variable.2380

Here, distance is some function of time.2384

This is a rate of change, this is a rate of change.2389

It is a roc, it is a rate of change.2396

This is 4 m/s.2400

This is saying, this is the same as 4m/ 1s, that means for every 1 second that passes,2408

I'm going to be traveling 4 meters.2415

That is it, that is what is going on here.2418

For every 1 second that passes, I move 4 meters.2420

For every 1 second that passes, I move 4 meters.2423

This is a rate of change.2426

Another example, we know or maybe we do not, if you have taken chemistry then you know.2432

If not, maybe you did it in physics, you would also know it from physics.2445

But if not, it really does not matter.2449

Because again, these are general ideas, it does not matter what the units actually are.2452

All we have to know is it is going to be something per something.2456

We know that density = mass divided by volume.2460

Anything divided by something is the top thing per the bottom thing.2467

That is it always, that is what division is.2470

It something per something, the numerator per the denominator.2473

If I had something like a density, I measure the density of 8.6g/ cm³.2478

That is telling me that this is 8.6g/ 1 cm³ because it is always per unit change in the x variable.2488

This is a function, I know that the x value, the x axis is expressed in cubic centimeters.2500

It is the denominator.2507

I know that the y value is expressed in grams.2508

This is a mass, this is a volume.2513

The function here, it is, mass is a function of volume.2519

The function might look like anything.2530

It might be straight, it might go that way, might go this way.2531

But anywhere along there, if I calculate a slope, either a slope of that, the slope of that,2535

or the slope of the straight line which is constant, that is a rate of change.2541

It is telling me that for every cubic centimeter the I increase in volume,2547

my density of my system is going to increase by 8.6 grams.2551

That it is a rate of change, it is a slope, that is what is happening.2557

Independent variable, dependent variable.2566

Independent variable, denominator, dependent variable, up there.2569

The slope is going to be the dependent variable divided by the independent variable.2572

This gives me the numerical value, this gives me the actual unit.2578

When I say something like I'm traveling 50 miles per hour, that means for every hour that driving, I'm moving 50 miles.2582

It is a rate of change, it is the slope of some function.2589

The function that it is a slope of is time, distance.2594

Distance in miles, time in hours.2599

The rate of change, the slope miles per hour.2602

Density is grams per cubic centimeter.2605

That is what is happening here.2608

Anytime you see something per something, you automatically know that2609

there is some function of the numerator unit of the denominator unit.2614

y is going to be a function of x.2621

Here distance is a function of time.2623

Density is a rate of change, density is a change in mass per change in volume.2627

It might be constant, it might not be constant.2634

Let us go back to red here.2638

Every numerical value and physical unit for y, per unit, per x,2643

I’m going to write it differently.2665

I’m just going to say something per something is a rate of change.2670

Anytime you see something per something, it is a rate of change.2683

What that means is that it is a slope, it is the slope of some function.2686

It is the slope of the graph of some function between the two something.2707

The bottom something being the x axis, the independent variable.2730

The numerator, the top something being the y variable, the dependent variable.2737

When we saw grams per cubic centimeter, here we saw 8.6g/ cm³.2747

We automatically know that there is a function grams up here, cubic centimeter here.2758

Denominator, independent variable.2769

Numerator of the unit, dependent variable.2772

I know that grams is some function of cubic centimeters.2775

Or more generally, mass is going to be a function of volume.2784

This represents a slope because it is a sum y value/ the x value.2790

It is a slope, it is going to be the slope somewhere along that.2797

It is a rate of change.2806

It is going to be the derivative.2807

If we are taking an instantaneous slope, it is going to be the derivative.2810

If we take an average, it is just going to be an average slope.2812

It represents a slope, that is what is going on here.2816

Anytime you see something per something.2819

If you saw kilometers per minute, I know that there is now some function.2821

Where minute is on the x axis, kilometer is on the y axis.2829

Kilometers per minute of some function, this is going to the slope of.2833

That is what is happening, I hope that make sense.2839

I’m sorry if I deliver the point.2842

Let us see, where am I now?2848

Slope, volume, it is correct.2853

Let us go ahead and finish this off here.2866

Let me go back to purple because I like it, it is very nice.2869

I have got some f(x), y = f(x).2874

We got some point on here, tangent line, that is a secant line.2888

I have two slopes I can form.2904

What is happening, I’m losing my mind here.2919

I have two slopes that I can form.2922

The average slope, the instantaneous slope.2926

The average slope between two points.2928

The instantaneous slope at a particular point.2930

A slope is a rate of change, I have two rates of change that I can form.2935

I have the average rate of change or the average slope.2944

I will say average slope or average rate of change.2951

This is going to be Δ y/ Δ x, your two points.2960

If this is x2 y2, this is x1 y1.2967

I form y2 – y1/ x2 – x1, that is my average slope.2974

That is my average rate of change and whatever it is that I happen to be discussing.2980

The average rate of change of the distance.2983

The average rate of change of mass.2988

In other words, the average velocity, the average density.2990

Or I can form the instantaneous slope which is the instantaneous rate of change.2993

This is f’(x) which we will discuss later.3006

I have spent a couple of lectures actually presenting some of the material.3015

Now let us go ahead and actually start solving some problems.3018

The next lesson is going to be example problems of these concepts that we have been discussing,3021

so that we can become more familiar with what is going on.3026

There are not going to be a lot of example problems in the next lesson, I think I only have like three of them.3029

But we are going to be going through them in detail.3035

In the process, I’m going to be discussing other things, ways of handling,3038

how to find instantaneous slopes, how to find other things geometrically.3042

By all means, take a look at these example problems, this is very important.3047

Thank you so much for joining us here at www.educator.com.3050

We will see you next time, bye.3052