For more information, please see full course syllabus of AP Calculus AB

For more information, please see full course syllabus of AP Calculus AB

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### More on Slopes of Curves

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Slope of the Secant Line along a Curve 0:12
- Slope of the Tangent Line to f(x) at a Particlar Point
- Slope of the Secant Line along a Curve
- Instantaneous Slope 6:51
- Instantaneous Slope
- Example: Distance, Time, Velocity
- Instantaneous Slope and Average Slope
- Slope & Rate of Change 29:55
- Slope & Rate of Change
- Example: Slope = 2
- Example: Slope = 4/3
- Example: Slope = 4 (m/s)
- Example: Density = Mass / Volume
- Average Slope, Average Rate of Change, Instantaneous Slope, and Instantaneous Rate of Change

### AP Calculus AB Online Prep Course

### Transcription: 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 slopes *0608

*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 represents*1273

*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 variable *1824

*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 consistent *2185

*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 that *2609

*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

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