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

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

### Using Derivatives to Graph Functions, Part II

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
- Using Derivatives to Graph Functions, Part II 0:13
- Concave Up & Concave Down
- What Does This Mean in Terms of the Derivative?
- Point of Inflection
- Example I: Graph the Function 13:18
- Example II: Function x⁴ - 5x² 19:03
- Intervals of Increase & Decrease
- Local Maxes and Mins
- Intervals of Concavity & X-Values for the Points of Inflection
- Intervals of Concavity & Y-Values for the Points of Inflection
- Graphing the Function

### AP Calculus AB Online Prep Course

### Transcription: Using Derivatives to Graph Functions, Part II

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

*Today, we are going to continue our discussion of using the derivative to actually graph functions.*0005

*Let us jump right on in.*0011

*We have used the first derivative to talk about intervals of increase and decrease.*0016

*We have also used it to discuss local maxes and mins.*0021

*We have used the first derivative to find the intervals of increase and decrease,*0031

*where the actual function itself is increasing and decreasing, as well as local maxes/mins.*0047

*We know that if a function is from your perspective going from left to right, is decreasing then increasing,*0067

*we know the function is a local minimum.*0074

*If the derivative is increasing, it is positive, and then right after a certain point it becomes negative,*0079

*we know that it hits a local min and local max.*0085

*Now, let us go ahead and go a little further.*0090

*Let us look at a couple of segments of a couple of graphs.*0094

*Segments of the graph, these segments happen to be two increasing segments.*0103

*Let us look at two increasing segments because we do not know what is happening to the left or right, or what it is that I’m drawing.*0116

*Let us look at two increasing segments of two functions.*0131

*One, we have our axis like that and we have something that looks like this.*0140

*Over here, we have something that looks like this.*0148

*In both cases, it is increasing, it is increasing.*0156

*In other words, the derivative is going to be positive, the derivative is going to be positive.*0161

*But clearly, there is a difference between these two.*0166

*How do I differentiate, now is where we use the second derivative to differentiate between the two.*0168

*In both cases, f’(x) is greater than 0, how do we differentiate between the two?*0175

*One of them, let us call this one number 1 and let us call this one number 2.*0203

*Number 1 is called concave up, in other words, the concavity is facing upward.*0208

*Analogously, the other one is concave down.*0216

*In other words, the inside of the curve, the inside of the concave portion is facing down.*0221

*A nice definition, clearly, there is a tangent there, there is a tangent there, there is a tangent there.*0228

*These are all increasing, they are positive tangents.*0240

*Positive tangent, positive tangent, and positive tangent.*0243

*A nice definition of concavity is the following.*0249

*We fill it out here, not that we need it.*0259

*I know we know what concave up and concave down means.*0261

*These are pictorial motions, you know what concave is, it is the inside of the curve.*0264

*The inside of the bowl, as opposed to the outside of the bowl which is called a convex portion of a bowl.*0268

*A nice definition of concavity is as follows, just for the sake of having a definition.*0274

*Concave up, I will just use cu, the graph of the function, the curve, the graph was above its tangents.*0282

*Clearly, as you can see here, the tangents are all below the graph.*0302

*The graph never falls below the tangents.*0306

*Concave down is just the opposite.*0309

*The graph lies below its tangents.*0312

*In case you are not sure, in case it is really kind of subtle, draw a tangent and see whether the graph is actually below or above the tangent.*0320

*And that will tell you whether it is concave up or concave down.*0328

*Here we have a graph, let me go to red.*0331

*That is your graph, it lies below.*0335

*Let us go back to blue here.*0339

*It lies below these tangent lines, it is concave down.*0341

*Concavity can be very subtle.*0348

*Clearly, something like this is really obvious.*0349

*If you have something like this, it is obvious.*0351

*Something like this, it is obviously concave up.*0354

*You have something like this, ever so slight a curve, not so obvious.*0357

*This is where we are going to use our analytical techniques.*0363

*Now what does this mean in terms of the actual derivative?*0369

*What does this mean in terms of the derivative?*0377

*It means the following.*0393

*If f”(x) is greater than 0 over a given interval, I will call it just a over a given interval a, then f is concave up over that interval.*0395

*Remember, when we talk about intervals, we are talking about the x axis.*0429

*Let us say from 1 to 2, every single x value in there, when I put in the second derivative,*0433

*it happens to be positive then I know that in between 1 and 2, the graph is concave up over that interval.*0439

*This gives us a numerical way of measuring concavity.*0449

*In those cases where you have a slight very subtle curvature,*0452

*where you cannot really decide or you cannot read it off the graph.*0456

*Everything in math may have a geometric interpretation.*0460

*We use geometry to help us along with the pictures.*0463

*I mean, clearly, we are going to use all this to draw the graph of the functions.*0466

*The graph is important but the algebra, the math, the analytics, that is was paramount.*0470

*Number 2, if f”(x) is less than 0 over a given interval, then f is concave down over that interval.*0478

*This is how you do it, you have a function, you take the first derivative that tells you whether the graph is increasing or decreasing.*0514

*You take the second derivative and you check points to see where the second derivative is negative,*0521

*where the second derivative is positive.*0527

*Let us define something called a point of inflection.*0530

*It is any point where f is, first of all continuous there and the graph changes from concave up to concave down.*0540

*In other words, the second derivative goes from negative to positive or positive to negative.*0568

*The second derivative passes through 0.*0574

*Setting the second derivative equals 0 gives us an analytic way of actually finding these values of x*0576

*where there is a change from concave up to concave down.*0583

*In any point where f is continuous there, and the graph changes from concave up to concave down or concave down to concave up,*0587

*either way, where it makes the transition.*0604

*For a point of inflection, f” passes through 0, very important.*0611

*In other words, f” at a point of inflection is going to equal 0.*0632

*Let us see what is next here.*0642

*Let us go ahead and draw a version of this.*0646

*Here we have something like that and let us say we have a graph that looks like this.*0661

*This right here, there is a point right here.*0669

*That right there, from this point to this point over this interval, the graph is concave up.*0673

*The second derivative is going to be positive.*0682

*Over this interval, it is concave up which means that f” is greater than 0.*0686

*From here onward, it is concave down.*0692

*F”, if I take any x value in here, it is going to be less than 0.*0696

*At this point, which is a point of inflection, it changes from concave up to concave down.*0701

*In this particular case, this is a point of inflection.*0705

*This point happens to be a point of inflection, the x value.*0710

*If I want the y value, I put it into the original function to find the y value of where it actually flips.*0714

*At this point, f”(x) is going to equal 0.*0722

*This is a point of inflection, that is it.*0727

*That is your geometric explanation of what a point of inflection is.*0729

*Once again, over a certain interval, if the second derivative is greater than 0, the graph is concave up.*0734

*It could be very clear concave up or it can be very subtle.*0742

*If the second derivative over a certain interval is less than 0, it is concave down.*0747

*At a point of inflection, it is going to equal 0, provided it is actually continuous there.*0752

*Most of the functions that we are going to be dealing with are going to be continuous*0756

*but there are going to be some instances where we have to be a little extra careful.*0759

*We do not just take these rules and just apply them blindly.*0763

*We always want to step back a little bit to use other resources at our disposal,*0768

*to make sure we know exactly what is going on with the graph.*0772

*That is really what mathematics is all about, is every class that you take teaches you a little bit more of something,*0776

*a little bit more of something, gives you something else to put in your toolbox.*0781

*You are going to apply some or all of these tools to a particular situation, depending on how complex or how easy the situation is.*0784

*Let us do an example.*0796

*Here says sketch the graph of a function that satisfies the following conditions.*0801

*It is telling me that the f’(0) is equal to the f’(3), is equal to the f’(5), which is equal to 0.*0806

*The first derivative of these points is equal to 0.*0814

*That tells me that I’m hitting either a local max or a local min, or it is going to be not differentiable there.*0817

*It tells me that the first derivative is greater than 0, when x is less than 0.*0829

*Which means that when x is less than 0, the graph is actually rising, it is increasing.*0834

*It is also rising between 3 and 5, which means between 0 and 3, it is actually falling.*0841

*After 5, it is also falling.*0850

*It tells me that f’(x) is less than 0, the first derivative is less than 0 between 0 and 3.*0855

*It is decreasing from 0 and 3.*0864

*It is decreasing when it is greater than 5.*0866

*It tells me that the second derivative f” is positive between 2 and 4.*0870

*It is negative to the left of 0 or when x is less than 2, or x is greater than 4.*0881

*Let us see here, I think there might be a mistake in something here.*0896

*But let us go ahead and see if we can work this out.*0906

*Let us go ahead and draw it.*0908

*I think I have might mis-worded something over here, but let us see.*0912

*F’ at 0 and 3 and 5, I have 1, 2, 3, 4, and 5.*0918

*It is a 5, 4, 3, 2, 1, and this is our origin 0.*0926

*I’m going to hit a local max or local min at this point, this point, and this point.*0937

*But I do not know whether it is going to be local max or local min yet, so I have to look at these.*0945

*F’(x) is greater than 0 when x is less than 0.*0952

*F’(0) is less than 0 between 0 and 3.*0956

*The function is rising to the left of 0 and it is falling between 0 and 3.*0962

*Therefore, I actually hit a local max at 0.*0974

*It is rising over here and it is falling over here.*0981

*F’(x) is greater than 0, the function is rising between 3 and 5.*0992

*That means here, it actually hits a minimum.*0996

*It is rising here, that means this comes this way and that is confirmed with this right here.*1001

*The second derivative between 2 and 4, it is positive between 2 and 4, yes it is concave up.*1009

*And then, it is less than 0 when x is bigger than 4, it is concave down.*1020

*At 5, it looks like we are probably going to f’(x) is a greater than 0, when x is between 3 and 5.*1026

*It is rising between 3 and 5, and is less than 0 when x is greater than 5.*1040

*It hits another max over here and that is less than 0, for x greater than 5.*1044

*It drops off like that.*1054

*We have a local max, a local min, local max, and these corroborates.*1057

*Let us make sure what we have here.*1063

*F’(x) is greater than 0, when x is greater than 2 and less than 4.*1065

*Greater than 2 and less than 4, yes, concave up.*1070

*F’(x) is less than 0, when x is less than 2, yes, concave down.*1074

*That is our concave down right here.*1079

*This is concave down right here, or x is greater than 4.*1082

*Yes, greater than 4, there was no mistake, it was exactly as written.*1089

*Concave down, concave up.*1093

*It certainly satisfies these conditions.*1101

*You are just using the information given.*1105

*Once again, when f’ = 0, it is a critical point.*1108

*It may or may not be a local max or min.*1114

*In this case, workout to be local maxes and mins.*1116

*When f’, the first derivative is greater than 0, the function is increasing.*1120

*When the first derivative is less than 0, the function is decreasing, the original function.*1123

*When f” is greater than 0 over that interval, the graph is concave up.*1128

*When f’ is less than 0, the graph is concave down, the original function.*1134

*Let us do another example.*1143

*Now we can bring all of our analytics to there.*1145

*For the function x⁴ – 5x², discuss the intervals of increase and decrease,*1148

*local maxes and mins, points of inflection, the intervals of concavity.*1153

*And then, use this information to graph the function.*1157

*There is no law that says you have to do anything at a given order.*1161

*Generally, you are going to take the first derivative first, and then take the second derivative afterward.*1165

*You do not have to wait until the end to actually draw your graph.*1170

*You can draw it any time you want.*1172

*If you have enough information and you feel comfortable enough*1174

*to actually conclude the things that you can conclude from the mathematics given.*1176

*You may use all the information, you may use only some of the information.*1181

*You may use other information that you have, asymptotes, roots,*1184

*whatever it is that you need, whatever things that you have in your toolbox.*1189

*This first derivative and second derivative stuff, these are just two more tools in your toolbox.*1193

*Let us start with the first derivative.*1199

*Let us go back to blue here.*1201

*F’(x) that is going to equal of 4x³ - 10x.*1206

*We want to go ahead and set that equal to 0.*1216

*I’m going to factor out the x.*1218

*X × 4x² - 10 that is equal to 0.*1220

*I get x is equal to 0 and I get 4x² - 10 = 0.*1227

*That is going to be one of my possible points, one of my critical values.*1239

*I’m going to have 4x² that is equal to 10, x² = 10/4 which is equal to 5/2.*1246

*Therefore, x is equal to + or – √5/2 which is approximately equal to 1.58.*1257

*I need to check my intervals of increase and decrease.*1268

*Now that I find my critical values, 0, positive, this is positive or negative, +1.58 and -1.58.*1271

*I have to draw my number line and I have to check points in those intervals to see where I have increase or decrease.*1282

*Let us go ahead and do that.*1287

*I have 0, I have +1.58 and I have -1.58.*1290

*We are checking f’, I have to check points in this interval, this interval, this interval, this interval, to see,*1298

*I’m going to put them back in the first derivative to see whether the first derivative is positive or negative, increasing or decreasing.*1306

*F’(x), we said is equal to x × 4x² – 10.*1314

*I’m going to try some points, I’m going to try a point here, here, here, and here.*1322

*Over here, I’m going to try -2.*1325

*When I put -2 into the first derivative, I do not really need to calculate the value.*1329

*I just need to know whether it is positive or negative.*1337

*-2 that means the x is negative.*1340

*When I put -2 in here, I’m going to end up with a positive number.*1343

*A positive × a negative is a negative.*1349

*F’ is negative which means that the function is decreasing on that interval.*1356

*I’m going to try the value here, I’m going to try -1.*1362

*When I put -1 in for x into the first derivative, this becomes negative, this becomes negative, it is positive.*1367

*The first derivative is positive in this interval which means the function is actually increasing.*1377

*I'm going to try one over here, I get negative.*1382

*1 is a positive number.*1391

*If I put 1 in here, I get a negative number.*1393

*I get a negative which means the function is decreasing on that interval.*1395

*I have 2 which is going to be positive.*1402

*If I put that there, 2 × 2 is 4, 4 - 10 is also positive.*1405

*It is positive which means it is going to be increasing along that interval.*1410

*The derivative = 0 at this point, this point, and this point.*1416

*Those are our critical points.*1419

*Our intervals of increase are this interval right here, let me go to red.*1423

*This interval right here and this interval right here.*1429

*Our intervals of increase are -1.58 to 0 union +1.58 to +infinity.*1433

*Our intervals of decrease are going to be that and that.*1447

*Our intervals of decrease are -infinity to -1.58 union 0 to 158, 0 to 1.58.*1458

*We have also taken care of local maxes and mins here.*1471

*We have decreasing/increasing, there is a local min right here.*1475

*Increasing/decreasing, there is a local max, here, somewhere.*1482

*There is a local min for x = 1.58.*1487

*I have taken care of that as well.*1496

*Let us go ahead and deal with those.*1496

*We also know that -1.58 is a local min.*1501

*We know that 0 is a local max and we know that +1.58 is a local min, as well.*1517

*Let us go ahead and find the y values of these things, just for the sake of it.*1534

*Let us find the y values of these critical points, these local maxes or mins.*1540

*We just want to know where they are exactly, y values of these points.*1549

*When I do f(-1.58), I get -6.25.*1559

*I hope my arithmetic is correct.*1566

*When I do f(0), that is going to equal 0.*1567

*When I do f(1.58), I get a -6.25 again.*1572

*We already have much of our graph.*1585

*We do not yet know where it passes through the x axis yet.*1604

*We do not know its roots yet.*1615

*We can get those just by setting the original function equal to 0 and solving by with whatever means we have our disposal.*1616

*In general for these graphing ones, whether you are doing them for calculus class or for the AP test itself,*1627

*you may not necessarily need to know where it passes the x axis.*1635

*What they want is a good solid graph showing increase/decrease things like that.*1639

*But not necessarily if the function is too complicated and*1645

*you do not have a calculator at your disposal or a computer algebra system at your disposal,*1648

*it might be very hard to find a particular root.*1652

*That is your secondary concern, where it passes through the axis.*1656

*Let us go ahead and draw what we have so far.*1664

*F(0) is 0.*1675

*Let us do 1, 2, 3, 4, 5, 6, 7.*1681

*Let us do 1, 2, 3, 1, 2, 3.*1686

*At -1.58, at -6.25.*1694

*1.58, -6.25, it is going to put us somewhere around there.*1699

*It is also the same over here, somewhere around there.*1703

*I know these are the local maxes and mins.*1706

*It was a local max here, it is a local min here, it is a local min here.*1709

*We have our graph, at least some of our graph.*1715

*Of course we want to finish off with the other information which is points of inflection, intervals of concavity.*1722

*We are looking for points where the concavity changes from positive to negative.*1728

*We see here clearly this is concave up.*1734

*But there are some points around here where the concavity goes from concave up,*1736

*all of the sudden it is concave down, or from concave down to concave up.*1742

*It is going to go up and pass through here.*1747

*Let us go ahead and deal with those next.*1750

*Intervals of concavity, let us go back to blue.*1754

*Our points of inflection, we are going to take f” and we are going to set it equal to 0.*1769

*Solve for x, that is going to give us our points of inflection.*1781

*We are going to do the number line again.*1785

*We are going to check points to the left and to the right of these points of inflection,*1786

*to see where the second derivative is positive or negative, concave up, concave down, respectively.*1790

*Let us rewrite, we have our original function f(x) is equal to x⁴ – 5x².*1799

*We have f’(x) which is 4x³ - 10x, that makes our double prime.*1807

*3 × 4 that is going to be 12x² – 10.*1815

*We go ahead and we set that equal to 0.*1821

*We get 12x² is equal to 10.*1827

*We get x² is equal to 10/12 which equals 5/6.*1832

*Therefore, x is equal to positive or negative √5/6 which is approximately equal to 0.913, + or -.*1838

*Let us go ahead and do our little number line.*1852

*We have +0.913 and we have -0.913.*1854

*We are going to check points to the left in between and to the right, to see where it is concave up or down.*1862

*We already know the answer, using the first derivative, you are able to get 90% of our graph.*1869

*The only thing we do not really know is where it hits the x axis.*1875

*We know where it is concave up and concave down.*1879

*This procedure, this analytical procedure, gives us the actual numbers 0.913 to let us decide.*1882

*We know that we are talking about something which is going to be concave up, concave down, concave up.*1889

*Based on what we drew on the previous page.*1898

*We just confirmed that here.*1900

*This we are working with double prime.*1905

*The hardest part of these problems is going to be, once you get the values of x and*1906

*putting it back into the original function, the first derivative, the second derivative.*1912

*You just have to be very careful which function you put them in.*1916

*You are going to be dealing with 3 functions.*1920

*The function of the first and second derivative.*1922

*You have to make sure you know which one you are putting them in.*1924

*Here, we are taking x values and we are putting them into the second derivative to see what sign the second derivative is.*1927

*F”(x) is equal to 12x² – 10.*1934

*I’m going to pick a point over on this side.*1940

*I’m going to pick a point -1.*1945

*When I put -1 into this, I'm going to get a positive.*1947

*The second derivative, f” is positive here, this is concave up.*1954

*When I check something in this region, I’m going to go ahead and check 0.*1960

*I’m going to get -, 0 - 10 is negative.*1964

*Negative means it is concave down.*1968

*When I check +1 which is in this region over here, +1, I’m going to get positive which means it is concave up here.*1972

*It went ahead and corroborated.*1984

*Let us go ahead and re-graph our graph.*1989

*We have this and we have something that looked like this, something like that.*2003

*This point was 1.58, that is where it hit the local min.*2017

*This was -1.58, that hit a local min.*2023

*Let me draw this a little bit better.*2028

*It hit the local min, the points of inflection are right here.*2033

*These x values, that +0.913 and -0.913.*2038

*That is a point of inflection, that is the point of inflection,*2042

*it is where it changes from concave up, concave down, back to concave up, over that number line that we did in the previous page.*2047

*Let us go ahead and find the y values.*2058

*In other words, we found the x 0.913, -0.913.*2060

*Let us go ahead and find the y values for these points.*2064

*Let us find y values for the points of inflection.*2069

*Remember, very important, if you are looking for y values, these x values of the points of inflection,*2092

*they go back to the original function.*2101

*Do not worry, we all make mistake like we all did.*2115

*That is the process, the original function, in other words f(x).*2118

*We have f(-0.913) = -3.4 and f(0.913) = -3.4.*2133

*Our points of inflection are -0.913, -3.4, and 0.913, -3.4.*2157

*Let us go ahead and write down the intervals of concavity, up and down.*2194

*Sorry about that, I do not think I actually wrote them down formally.*2200

*Intervals of concavity, it is going to be concave up from -infinity, we said to -0.913 union 0.913 to +infinity.*2203

*It is concave down from -0.913 all the way to +0.913.*2224

*Just to write them down formally.*2233

*That takes care of that, the last thing we should do is actually find out where it crosses the x axis, just for good measure.*2237

*The last thing we should do is find where f(x), the original function, crosses the x axis.*2246

*In other words, the roots, the x axis.*2266

*There is no guarantee that the function is actually going to cross the x axis.*2273

*It could not cross at all, in which case you are going to set f(x) equal to 0 and you are going to get no solution.*2276

*F(x), the original function is equal to x⁴ - 5x², we set that equal to 0.*2284

*I’m going to go ahead and factor out the x² and I'm going to get x² – 5 and that is equal to 0.*2293

*I get x² = 0 and I get x² - 5 = 0.*2302

*Therefore, it crosses x = 0.*2308

*It is a double root, it actually touches, we know that already from the graph.*2312

*It touches the graph there, it does not cross.*2315

*X² = 5, therefore, x = + or -√5 which is equal to + or -, roughly, approximately equal to 2.24.*2320

*There you go, final graph.*2333

*I will go ahead and put a point here, put a point there.*2340

*They cross there, cross there.*2346

*This is 2.240, this point is -2.240.*2359

*This point is a point of inflection, this point is of course 0,0.*2372

*This point is 0.913, -3.4.*2379

*This point is -0.913, -3.4.*2387

*This point, the local min, that is -1.58 and -6.25.*2393

*If I’m not mistaken, if I remember properly.*2400

*This one is 1.58 and -6.25.*2402

*Let us see what a better version of the graph looks like.*2408

*Local max, local min, local min.*2416

*Where it crosses the x axis, where it crosses the x axis, those are two roots.*2421

*Points of inflection are here and here.*2426

*Concave up until this point, this interval all the way to -0.913, it is concave up then it is concave down until it hits +0.913.*2431

*When it flips over and becomes concave up again, all the way towards infinity.*2443

*They are you have it.*2449

*The first derivative will tell us whether we have a local max or min.*2455

*It does not tell us increasing/decreasing.*2480

*It also tells us when it passes from increasing to decreasing or decreasing to increasing, we are going to hit a local max or min.*2481

*The first derivative is all we need for local max or min.*2487

*There is also a second derivative test, if we want it.*2493

*There is also a second derivative test for local maxes and mins.*2496

*It is generally unnecessary but we will go ahead and lay it out.*2517

*Let f”(x) be continuous near a point a.*2524

*To the left and right of it, it is going to be continuous.*2540

*If f’(a) is equal to 0 and f” at a is greater than 0, f’(a) is equal to 0.*2543

*It is either a local max or min.*2565

*F” if a is greater than 0, which means it is concave up, it is a local min.*2567

*Then f has a local min at a, and the opposite.*2577

*If f’(a) = 0, f” at a is less than 0, which means concave down.*2588

*Then, f has a local max at the point a.*2605

*What you are looking at is this.*2618

*You know this already, that point, that point.*2626

*Here f’ is equal to 0, f” greater than 0, concave up, local min.*2629

*Here you have got f’ is equal to 0, f” less than 0.*2645

*This is concave down, this is your local max.*2652

*That is it, second derivative test, in case you want to use it.*2657

*You really do not need it but it appears in the calculus books, we are going to go ahead and give it to you.*2659

*I’m going to go ahead and stop the lesson here.*2669

*Do not worry that there is only been a couple of examples in this lesson and in the previous lesson.*2672

*This was mostly just theory, something to get your feet wet.*2677

*The next couple of lessons are going to be nothing but example problems, using what we just did in these last two lessons.*2680

*Again, the next set of lessons is going to be just example problems.*2687

*We are going to do plenty of these, no worries.*2692

*Thank you so much for joining us here at www.educator.com.*2695

*We will see you next time, bye.*2697

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