– This is in section 2.8. And this is on differentiability. Alright. Differentiability. Of a function. Alright, differentiability of a function. We’re basically, up until now, we said “Does a limit exist?”

and we looked at the function to determine if a limit exists. We determined whether or not

the function was continuous. Now, we’re looking

whether or not a function is differentiable at a point. Alright, so that’s what we’re looking at. Is a function differentiable at a point. So I’ll start with this function. There we go. That’s some polynomial, everyone. This is just, I do want to point out, it’s a nice, smooth polynomial. Where is this function differentiable? The answer is, everywhere. For this polynomial here, you

don’t even need the equation. You can look at the image. So this is the X-axis, that’s the Y-axis. But this polynomial is differentiable everywhere. Everywhere. And so what do we mean by differentiable? Derivative of F exists everywhere. That’s what I mean. And we did two-sided

limits, that’s F of X. We did two-sided limits. Well, now you’re thinking about

how the slope from the left equals the slope from

the right at every point. And I can just pick a

point like right there. Is the slope, as it

approaches from the left side, matching the slope as it

approaches from the right side? Right there? Yeah. So, are the slopes equal? Coming from the left and from the right? Because slope of a tangent line

to a curve is a derivative. I’ll say it again: that’s

what a derivative is. It’s the slope of a

tangent line to a curve. – [Male Student] But one goes up and the other one comes down, doesn’t it? – That is no good. Very good! So that’s what we’re gonna focus on. I like how you think. That’s easy, it’s nice

and smooth everywhere. So that goes for all polynomials. They’re gonna be

differentiable everywhere. So, we’ve gotta focus on when

are they not differentiable. And you figured out one thing. You said, if you had something like this, something came up like this,

and some went like that. Right? Oh, thanks, Cher. You can just put it right on, right there. Thanks for doing that. She had about three hours of traffic. Hey, so you nailed it. So that’s what we’re gonna

focus on now, and alright, all polynomials, I guess they can be differentiable everywhere. Let’s focus on: where’s the function not differentiable? Then we’ll be good. So where are functions not differentiable? And you go: what’s differentiable mean? The derivative exists at the point. And you go: what’s a

derivative? (chuckles) We’ve gotta get all these words down. Derivative, we’re talking about the slope of a tangent line to a curve. And so again, before I erase this, like, even at that point right

there, does everyone agree? Like, the slope from that

side matches the slope from that side, right at that point, yeah? And that’ll exist

everywhere in this group. So, we get right to it. Alright. Differentiability of a function. Where is a function not differentiable? Or when? Should change where to when. Talk about the different occurrences. When is a function not differentiable? Alright. I wanna start out with yours. You did something like,

it came up like this, and it came up like that. Oh, their slopes are not

equal, at that point. So it wouldn’t be

differentiable at that point. So we’re going to start out with a C. Corners. C for corners. You got a corner on a graph. He nailed it, alright? If you got a graph, got

something like this, like a little little quirk. Like a V, somewhere. Yeah, at that point right

there, at any corner. At that location right there, let’s say there’s an X-value for this dot, the function is not

differentiable at that point. Do you all agree? And we go: why? Does the slope from the left of that dot match the slope from the right? No. So very good. I’m gonna put another word for C. – [Male Student] I have a question. Is there a way to prove that? – Let me finish this up, and

then I’ll get your question. I like this with corners,

but let’s also put “cusps”. I’m gonna show you what a cusp is. A cusp looks like this. It’s not really a V, but

it might come up like this, and then it comes up like

that, from that side. Now, and can you see, this

is really, really steep and positive, but this is really,

really steep and negative? And so that’s another one. Oh, go ahead. – [Male Student] Is there a

way to prove, mathematically? – Absolutely. Yeah. It’s just looking at, remember, ah, he does the math proof, so what you have to do is, remember the derivative definition? I’m gonna write it down. What’s the definition of derivative? It equals the limit, as H approaches zero, of F of X plus H, minus F of X, all over H, and you have to

do this from either side. So you have to do like

a plus or minus, here. Remember that, from the left

side and the right side? So, I’m gonna write this another way. Here’s another way to

look at F-prime of X. You can say F-prime of X is

the limit, as X approaches A, of F of X minus F of A, all over XYZ. Remember how we talked about this? I’ll hid this. Remember this was the slick, secret slope? And we add that and we always

get the slope of a tangent? So you would do this, but

then you do a left side and a right side. And that’s your mathematical proof. So that was a good question. Because could you do it? Man, like how do you do

this in an elegant way? Mathematics is interesting,

like, with notation. You have to do all that work, but you have to do a

negative for the left side, then a positive right side and

make sure their slopes match. Very good. Hey, so when C, corners, cusps. I got two more letters for you. The next one is: V. V for “vertical tangents”. V for “vertical tangents”. Alright. What do we mean

by vertical tangents? I’m gonna give an example of functions. So here’s my example

for vertical tangents: it’d be something that did this. The graph went up like this. Then it looped over like this. It actually changes concavity. Does anybody know that word, “concavity”? This is like, it went from concave up, to then the concave down. But right there, at that

one instant, at one instant, if you think about the slope

of a tangent line here, here, here, here, here, but right there, the slope of a tangent line to that curve would be totally vertical. What’s the slope of a line like that? It’s undefined, yeah. So that’s why we’re saying, oh, then the slope would be undefined. And that’s why we’re

saying that’s another place where a function is not differentiable. You go right here, at this

location in an instant, and we look at the slope of this line. The slope would be totally

vertical, and it’s undefined. So V for “vertical tangents”. I do wanna point out sometimes, if you’re just given an

image, it’s hard to tell. So don’t worry about that. If you’re unable to, maybe like, I can’t tell if it’s

really a vertical tangent. Maybe it’s just really steep. And I understand that. Got one more letter, J. I don’t know. Does JVC

still make equipment? Like, electronic equipment? Oh, good. So that’s what I use. I

like this acronym, JVC. Acronyms help us remember things. You know, like mnemonic devices. JVC. Where is that function not differentiable? J-V-C: V for “vertical tangents”, C for the “corner cusps” that you hit on. J stands for “jump discontinuities”. Jumps. And discontinuities. J for “jump” or “jumps”. I can give you some

example of those, everyone. What’s a jump discontinuity? Something like this. Ah, the graph went like this, stopped here, started here. That’s one example of

a jump discontinuity. See how it jumped? Right? Basically I’m talking discontinuity. Where it’s not continuous. Here’s another one. And when this is a jump discontinuity. Oh. Then it continues like

that, but there’s a hole. So I’m talking about

right at this location. The function is not differentiable here. Right here, the function

is not differentiable. Cool? Right there, right there,

there just can’t be any what? Jumps. So the acronym JVC, J for

“jump discontinuities”, V for “vertical tangents”,

C for “corners and cusps”, and we’re able to identify

exactly where functions are not differentiable. He went on to the algorithm. You said, what about math proof? If you wanted to really prove it, you could go and dive into

the function they give you and show this from the

left and right side. Cool? Alright, so, in my next

exercise, I’m gonna draw a graph. I’ll put some numbers down. I was wondering if you can

identify all the X-values where the function is not differentiable. Good news: you’ll see practice problems that I chose in the text book on this. But I’m just gonna draw a picture. I’ll go left to right. I guess I have to do it over here. If I do it over here, would it hit it? Or is it too far away? – [Camera Operator] No, that’s cool. – Awesome. I’ll do it right here. Alright, and I’ll leave that up. Alright. Here’s my graph. Alright, this graph does this. Let’s see, comes down like this, I’m gonna put a bunch of X-values down. (whooshing noises) Okay, that’s good. One, two, three, four, five, six, seven, is the X-axis. Negative one, negative

two, negative three, negative four, negative

five, negative six, negative seven, negative

eight, negative nine. Gotta make sure these line up, though. ‘Cause I know you all

are already identifying where this function is not differentiable. I’ll just call this F of X. I mean this does pass the vertical line test, right everyone? Let me make sure that’s just,

I’ll just put an arrow there. Maybe I can make this

look a little bit better. I wanted that to be smooth. Trying to make the thing kind of… There we go. That’ll work. That will work. At this location, right here, let me make sure that lines up. That’s it. X equal to negative five. Right there. Alright. Let’s see if we can do this. Where is this function not differentiable? Not differentiable? Question mark! Just list the X-values. As you can tell, I didn’t

put any Y-values down. Anybody got one? – [Female Student] X negative five. – X equal to negative five, right there! Boom. Corner or cusp, right? If you’re looking, is that

a corner, is it a cusp? It’s one of those at X

equal to negative five. I’ll but a comma because

there are more, right? What’s another one? – [Second Female Student] Negative two? – Negative two! Another corner or cusp. Any more? – [Multiple Students] Two, positive. – Two. Positive two. And maybe that’s it, but do you see one where, perhaps it is? Where? – [Second Female Student] Four. – Four. What’s it look like is

going on at X equal to four? (unintelligible responses) Yeah, but see, it’s like, again, I’d have to make a really good sketch. I was trying to show that this thing got really, really what? – [Male Student] Vertical. – Yup, totally vertical. Totally vertical, here. So, if I can improve that, make it look a little bit better. Make that smooth right there. So it does look like there,

when X equal to four, there, there’s a vertical tangent right here. So I’m gonna mark this thing out. Right there, there’s a vertical tangent. This was a jump, right? Jump discontinuity. This was a corner, this was a corner. Awesome. Everyone okay with that? Put a problem like this on the next test. Easy graph, yeah. No JVCs, right? What we’re saying is the function is not differentiable at JVCs. And you’ll laugh. A student who graduated ten years ago, on Facebook, wrote me a message, like within the last year,

and that was what he wrote. He’s like, “JVC: function

not differentiable”. It’s like, that’s what you remember? All that calculus. It’s crazy he remembered this. I was cracking up. It’s like, how do you remember that? Ugh. So I guess it worked. I guess it worked. We won’t do a proof for this problem, but I do want you to think about, I’m gonna put something on the board, I’m gonna put a piecewise function, and I’m gonna ask you about three things. Does the limit exist? Is the function continuous at the point? And is the function

differentiable at the point? Now, before I do that, I

think we should put a phrase with the words “differentiable”, “continuous”, and “limit”. We should make a little diagram for that. So how about this? Washington DC is lovely. Will that work? Washington DC is lovely. That’s not as cool as JVC, but DCL. What I’m saying is, if a function is differentiable at a point, that arrow means “then”, then the function has to be what? Continuous at the point, which also implies that the

function has to have a…? Limit existing at the point. The only thing is, can

you reverse this order? No. I wouldn’t always be necessarily true. So make sure you don’t reverse it. But, if a function is differentiable, then it must be what? Yeah, at that point. If a function is

differentiable at a point, then the function must be

continuous at the point. If a function is continuous

at a point, then the limit, the two-sided limit

must exist at the point. The only thing is, you can’t

do this, alright,? Look. Does C imply D always? No. Not always. I’m gonna

put a slash through it. Can anybody give me an

example of a function that is continuous but

it’s not differentiable? Watch. Someone will come up with one. To show the counter-example

of why we can’t say: C implies D. – [Male Student] Vertical tangent. – That’s a good one. I love that. He says a vertical tangent. You know, because hey, it’s continuous. Yeah, that was a good one. It’s continuous right there. That’s nice and continuous

but it’s not differentiable. There’s Martin. How about the activate valued X-graph? Like that. That was a good one. How about the Y-collapse

of valued X-graph? We all know that’s a V, right? It’s continuous right

there, but it’s not what? It’s not differentiable there. So we can’t say C implies D, right? But Washington DC is lovely. D implies C implies what? L. Always. So if that helps you in your

logic, I think that’s great. I do wanna point out you can reverse these and do a contrapositive

if you put the words not. For instance, that’s not true, but you can say this and that. Not C does imply not D. You can do that. You can do that. It’s

called a contrapositive. You can reverse it and put “nots”. It’s actually logic. Anybody ever take a logic course? Oh yeah, there we go. It’s logic, it’s the signing of logic. You can reverse it and put “nots” on it. If a function is not

continuous at a point, then it is…? Not differentiable at the point. Alright, here’s my problem. Let’s see how you do with this. Can I erase this? Yeah? Alright, JVCs, we got it. Alright, differentiability of a function. Here it is. F of X equals, I’ve got a piecewise, I made this up, while the other class was taking a quiz. Sitting there, brainstorming something. 3X minus one. If X is, greater or equal to two, comma, if X is less than two. Hey, you are all awesome, coming

out on this Thursday night, hustling through crazy

traffic and all that. – [Male Student] It’s ridiculous. – See? So, this problem won’t

be on the test Tuesday, but it will be one just

like this on the final. So I can give you that heads up, ’cause you made it all the way out here. Plus, the people watching the video. (laughter) They’re the ones who got

stuck and had to turn around. No, but yeah, I’ll put one up. You’ll see one of these on the final. It’s a nice final-type of question, dealing with conceptual

stuff, but alright. Alright, there’s the function. It’s a piecewise function, right? That’s a parabola, what’s this? Ah-ha! So you get parabola, straight line! But maybe it is

differentiable when they meet. It’d have to be smooth wouldn’t it? Right? Yeah. The slope from the left would have to match the slope from the right. You could design a

roller-coaster like that, right? Something came up like this, parabola, and then you’d pass the nice,

straight line right here. As long as you hit the slope

perfectly, you’re good. True? So that’s what we’re looking at. So here’s the question, you know? Which statement is true? The limit, as X approaches

two, of F of X exists. Alright? Letter B: F is continuous at X equals two. And letter C: F is differentiable. Differentiable means the

function’s derivative exists, at X equals two. And it can be more than one of these. All three of these could be true, only one of them could be true. None of them could be true. We’re just gonna check ’em all out, right? Almost like a true-false. Which statement is true?

Is this true or false? Is this true or is this false? Is this true or is this false? We’re gonna check it out. I’m gonna start with limit.

How do you check limit? How do you check limit? (muttered student response) Yeah, we could do that. I was thinking this: let’s check continuity, right? And if it’s continuous there,

then it has to have a what? – [Male Student] A limit. – Does that work? So, when I say let’s

check continuity, first, do we have to make a graph? No, because he had an idea,

last class, at lectures. What do we simply have to

do to check continuity? – [Male Student] Replace the two with X. Or, X for two and solve for even. – Love it. You know what we can do

for check continuity? We’ll start there, ’cause it’s so easy. Plug a two in, plug a two in. They gotta match. If they match (claps hands)

continuous at X equal two. If their Y-values don’t

match, it’s broken. Not continuous. – [Second Male Student]

What if if matches, but it’s not continuous? Like it (unintelligible)? – Well, if it matches,

it’s gotta be continuous. If these two Y-values match? Right? As long as I use that same thing. But now you’re onto it. You go, oh, it might

not be differentiable. There you go. And we’ll get to that.

We’ll do that afterwards. Very good. Very good. This is calculus, right? You got functions that are continuous but sometimes, they’re

not so differentiable. Alright, is it continuous there? Check. Very good. True, everyone, and we’re

gonna show why. Look. In this case, what’s two squared plus one? Five. What’s a three times two minus one? Five. Y-values equal. These two Y’s are the same. So, what does letter A have to be? True. You know, if the function’s

continuous at the point, then the function’s gotta have a limit exist in there, right? Yeah, it’s connected. So of course the limit

exists from both sides. I was just curious. We

don’t have to do a proof. But we do have to know

how to get derivatives. Who knows the derivative of this graph? AKA, here’s my hint, derivative

means slope of tangent. Who knows the slope of a straight line? Slope of a straight line, MX plus B. I’m gonna write that, here. I’m hearing it. Isn’t that a straight line? MX plus B, straight line? Which number is the slope? – [Male Student] The three. – Three! Ah-ha! So what’s the slope from the left side? The slope from the left side is a what? Three. True? Alright, so I’m gonna write this down. Slope from left at X equal to two. I’m just putting words. You could do this, if you wanted to. You have to put a little

minus there, though. If you want to write the

math notation, everyone, like a math, you know,

like a mathematician, you’re like, I don’t wanna write words. You’d have to write all of

this, then put a two here, and then put in the minuses. But I’m calling out

the slope from the left is the value what? Three. Sure. Now how do we get the

slope from the right? I taught you this at

the end of last class. Remember the power rule? – [Multiple Students] F-prime and two. – Yeah! What’s the slope? How to get the derivative? Remember what you can do instead

of doing it the long way? Move the two in front, and then we have to

decrease the power by what? – [Multiple Students] One. – One. So let me write that down. That slope from the right, I’m gonna put a wall, what’s the slope, I mean

slope from the left, what’s the slope from the right? We’re saying its the derivative, so I’m gonna put that word down, you know, we’re doing our derivative from the right. Just like derivative from the left. Solve for the right. I’m gonna

use the power rule, though. Two in front, decrease the power by what? One. But, oh, and what’s the

derivative or slope of a constant? – [Male Student] Zero. – Zero. So this is it. But what number do I have to

plug in to see if it matches? – [Male Student] Two. – Yup. It’s always at that, you know,

right there at that value. So, I’m gonna put, everyone

would you just plug in X equals two at this expression. What’s two times two? – [Male Student] Four. – Are the slopes matching? So what’s the answer?

What’s going on with C? True or false? False. No. Is F differentiable? No. No, for this guy. Slope is three, slope is four. – [Male Student] How come

you took the derivative of X squared plus one,

but not the derivative of the bottom one? – Oh, you could do it. Do you wanna do it? ‘Cause I was thinking

we could think algebra, slope of a line is tangent,

slope of a tangent line. It’s just slope. ‘Cause remember, derivative means slope? It’s easy with straight lines, isn’t it? But, if you wanna do it,

what’s that power right there? – [Second Male Student] Nine P three. – Oh good, so it’d be one

times three, X to a zero. Now, I’m gonna write that. You get one times three

is three X to a zero. Well, what’s X to a zero, now? One. So, very good. You can do it every time. – [Male Student] Can we

just look at the other continuous points and see

whether it like, you know, goes in a linear pattern, or? – You could do that. Yeah, but the key is, whenever

we’re talking about that, for differentiable, for

being differentiable or not, we really have to focus on

that slope right at that point. We have to do the derivatives. Hey try one more, everyone. This time, just do letter C, okay? Just do letter C. True or false. One more for your notes. There you go. No, I like this better. Try that one. Is F differentiable at two? ‘Cause I used two again,

remember, there’s my cut-off two. That’s smooth everywhere

except where one about two. By the way, if you notice

something, everyone, I made sure it was already continuous. I wanna point that out. I made sure it was already

continuous. Can you tell? Two squared plus one is what? Five. What’s four times two minus three? Five. I made sure it was continuous

because D implies C equals L, but not C implies what? Not D. If it wasn’t continuous,

it’s not differentiable. So I didn’t want to, you know, I wanted to make sure I gave you something that was continuous. This one and this is already continuous. At X equal to two. I was just curious if you could figure out whether or not it’s differentiable. So. Well, this’ll give you

the same slope, right? What’s the slope from the left? Four. What’s the slope? Four. What do you think? Parabola meeting a straight line, but the slope is exactly what? Four, at that instant. From the left side, from the right side. A little… From the left. From the right. I’m sure mathematicians love

to just keep writing all that stuff, but I want

you to focus on, you know, it’s like, oh, I just wanna

look at slope, the derivatives. – [Male Student] Why is the

slope for the derivative from the right still equals four? Because last time you did it, negative 2X, and you multiply the two– – Oh, you’re right! So that

was the same thing, here. I wanna make sure, this is

from the right side, right? Alright, I’m gonna break all this. So this went with this, that’s true. That stayed. That stayed the same. This, now, this part came from here. But try it. What’s that derivative? Of that function? Well, that’s a straight line. So we know what the slope is. It’s four. But you could still do

this rule, the power rule. There’s a one right there. One times four, decrease it by one. You get 4X to a zero. But X to a zero is just what? One. Or, if you wanted to do that,

but that’s the long way. Yeah, okay. We did that once. We get it. (chuckles) Cool? Awesome. (unintelligible question) Yeah, I’m sorry. I never

put the answer down. Yes. Let me get that out of the way. Sitting on the video going, “False!” Well, this is what? This is true, the answer is yes. Yes it is. And what would this look like, here, when you have a parabola, then what? What goes on at two? It’d hit here, it would’ve hit here, and it would come up like this. So, here’s the picture. And then right at that

point, two comma five, you had it meet this what? This straight line, but

it would be nice and what? Smooth. Smooth. Slope left, slope right,

right there at that point. Right? It’s like someone’s designing a roller-coaster that came down and what? Just went off in a straight line. If you’ve ever seen things like this. Or water-slides like that. You know, the water-slide

comes down, then it’s like… But they make sure, they don’t give you a water-slide like this, do they? Eh, and then it’s like (laughs). True? Right. Right. I used to guard Nation City, everyone. This is back in the day. I don’t even know if you were born yet. 1987? And they’d just built a

slide on the boardwalk. So my guard stand was

right off the boardwalk. And it was the first year of this thing. But there’s not much room

on the boardwalk, is there? For a waterpark? I don’t know if this

waterpark is still there. So, they put a slide in. I am not lying to you. It was like, super steep slide like this. The catch-pool, I am not

lying to you, was that long. Let me fill it in with water. I mean, you all, the

first time I saw this, because worked in a water

park years before this, I’m like, ah, man, are

they asking for trouble. I mean, they’re asking

for lawsuits and stuff. I mean, this is what I saw. It was super steep and stuff. And anybody ever go on a water-slide, you get on your shoulder blades? You bounce like a rock, yeah, so. They’re like no, we’ll put a sign up that tells them that, oh, come on. Everyone that, the manager there, he said people were losing teeth. Yes, loose teeth. I mean, it came in just, boom! Right into the wall. They’d bounce up on the cement. Cut all up and stuff like that. So they started, you’re

gonna laugh at this, they kept it running and tried

to keep the business going. So it had the attendants,

they had to work there, they’d hold a raft at the end. They’d all stand with rafts. I’m true. True story. I’d watch ’em and just stand there. I was getting paid you know, like, back then we got paid

like, four bucks an hour. I was like, four bucks an

hour to just go like that. This is true. I mean, you saw it coming. It’s just dealing with,

you know, basic laws. Physics. It’s like, aw, yeah. Gonna let that go. Alright, I don’t wanna be like Freeman, just keep talking about

water-slides, right? Hey, you wanna know the

last thing we gotta do? 2.8? Is, we’re gonna get a function of F, and we’re gonna graph

the derivative of it. So now we’re gonna graph the derivatives. Graph the derivatives. Of F. This is cool. So, I’m gonna give you a function, F. You and I are gonna get an idea what the graph of the

derivative looks like. So let’s see. This’ll be my first one. How about… How about this guy? Graph it like this. Almost like one of these

Trig graphs, you know? I know this is almost gonna look like one of the trigonometric graphs. Like a sine or cosine. Y equals sine of X, Y equals cosine of X. Let’s call this F of X. And now, if you can, in your notes, I think it helps, if you have room, right underneath it, let’s graph F-prime. ‘Cause it really helps us stay organized. Right underneath this,

we’re gonna graph F-prime. I’ll put it in green. F-prime of X. We’re gonna graph that. We’ll just keep it nice and organized. There we go. X-axis,

Y-axis, X-axis, Y axis. And we’re gonna graph the derivative. This is fun. If you’re curious, what

I like to do on my exams, on these type, I like to do a matching. And the textbook has a matching exercise. I put if for practice, as one

of your practice problems. To try out? So it’s good practice. See

if you can get that one. In this case I’m now gonna

graph the derivative. It’s not hard. This is what I want you to do: would you focus on this graph right now, and tell me where the

slope of the graph is zero? I’m talking like this. If you drew a tangent line of that graph, it would just be flat. Can you give me some spots, I (audio cuts out) That’s the first thing

I want you to look for. Do you see one? – [Male Student] At the top of the– – Nice! He goes right at the top. So when I’m at the bump, right there, and I’m actually gonna

draw the slope of it like, slope zero, right? Now, what I want us to do? Right away, come down here. Go straight down. And I want you to put a dot right there. You know why? Because derivative means slope. And what’s the slope right there? Zero. Do you all follow me? I know a lot of you are copying. I’ll wait until you’re done copying, but that’s what’s going on, here, really. Hey, your F-prime is zero, here. F-prime means derivative. It’s zero. Slope is zero. F-prime is zero. M is zero. M means F-prime. F-prime means derivative. You go, what does this mean? This means slope of tangent. Drawn to F. Right? It’s the slope of a tangent

line drawn to the curve. So okay, here the slope

of a tangent line is zero. If I’m graphing the slope, then

it would be what right here? – [Male Student] Zero. – So far, so good? Yeah. Any other spots? – [Female Student] Also, the bottom. – Let’s do it. Right over here, hit these first. And go down. Right here. There it is. Go straight down. Whoosh! Why is it right on this X-axis? Because the slope is zero

right here. F-prime is zero. F-prime equals zero. M equals zero, there, right? Now, to finish out our, we’re

just making a rough sketch of this graph, by the way. To finish this thing out,

let’s now look in between. In between these spots, ’cause there’s no more

slopes of zero, true? Right? Cool. Let’s go over here. Would you just put a

tangent on that thing? And tell me if it’s

positive or it’s negative? So, just go on this curve,

just, how about right here? Is that slope positive or negative? – [Multiple Students] Positive. – So, I’m going to write F-prime pos, and then, I’m gonna come down here, I won’t even do the dotted lines anymore, I’ll just come right here, and I’m gonna put a

dot up here, somewhere. I’m just estimating. You go: where is it? Well, I’d have to know

exactly that slope, right? I’d have to actually measure the, I’d have to actually measure

the rise over the run, there. But I know it’s positive, right? So where are the positive values? Up here. Where are the negative values? Down here. So I’m just gonna put

a dot right here, like, how about right there? That work? Now, everyone, how about

in between these two? Slope at zero, what’s

going on right there? I’m gonna put a tangent line here on. What’s that slope? F-prime is negative! So, I’ll just come down

here at this location and put a dot down here somewhere. Where do I put it? Well, I just know it’s negative. I’d have to measure the

rise over the run, there. That’s a bad-looking tangent. We’ll make that look a little better. Tangent hits that thing

in one spot. Look at that. There we go. So that negative, and one more. How about over here? What’s the slope right here? Slope’s positive. And just put a dot right here. The last thing, everyone, you

bet it’ll be nice and smooth, just connect all the dots. What would this do? It looks like this would

come like this, right? It might loop over.

Doesn’t have to be perfect. It just comes down like this. There you go. By the way, this probably would kind

of loop over like this, if you really wanted to know this design. It’d probably come over like that. The only reason I know that

is because I could tell the slope was really steep and it was getting close to zero, there. It’d kinda fade in like that. Maybe more like… And remember, you’re

just doing a matching. Just doing a matching. And then you can continue. Cool. – [Male Student] Now because

a tangent at like, zero, zero, for instance, there is a negative, is that why you put it down below? – That’s right. And you notice, you go:

where should I put it? Listen, I’d have to go in here and measure the little change in Y over

the change in X in that. We don’t have to do that. We’re

just making a rough sketch. – [Male Student] But

you could just put it, like, right below it? – Right below it. That’s good enough. Or you could have stuck it, like, way down here, you know? ‘Cause you’re just getting a sense of it. And if you’re wondering, like, do you have to make a perfect,

perfect sketch for me, I do matching, I think it’s very fair. You know? It’s the best way to go. – [Male Student] So that’s what we’d have to do for every graph? – Yeah, I’d give you like, the book did this, too. I’d like the design. You know what I’d do? I’d

probably do four graphs, on the left, four graphs on the right. And I’d be like, can you match these four with their derivatives? And once you did like, a

couple of these points, you’ll figure out where it is. Okay, let me give you another one. Alright. And that’s what we’ll do. We’ll just do a bunch

of these until the end. If you wanna create one on your own, you can, and we’ll see what it looks like. We’ll look for strange, strange scenarios. Alright. Here’s my next one. Alright how about just this, then? How about this? Continuing here, stop

there, then swing up. Yeah, I mean, seriously. Just something like maybe,

a parabola or something. But I didn’t give you

the equation of this, so I don’t know if you knew what one. Maybe that’s Y equals X squared plus one. Or maybe it’s Y equal to

X to the fourth plus one. So we’re just gonna make a rough sketch of what we think the graph’s doing, right? In fact, I can even make this thing really kinda flatten out, here. Don’t worry, I’m just gonna make it, I’ll make it be more level, you know, right here, this came out. Then it’s like, alright. Hits the bottom, hits

the bottom. Slope zero. Because graphs like Y equal

to X to the fourth, everyone, do stuff like that. Graphs like Y equal to X to the fourth. They get really, really,

really low slopes right there and there, and then it

hits this bottom part. Hits its lowest point, there. Alright, where is the slope zero? Everyone can see it, do you agree? Right there, that F-prime is zero, so where do I put the dot? Where does this go? I mean, does it go right here? No, it goes where? Right there. Does everyone agree? Because that’s where slope is zero. This is where slope is positive, this is where slope is negative. True? So far, so good. – [Male Student] I have a question. – Sure. – [Male Student] Because

the bottom is kind a flat, doesn’t extend, does the– – Great question! And there was one instant

where it got totally, totally at tangent slope would be zero. But then it just barely

started increasing up this way. – [Male Student] But,

even though it’s like– – Yeah, it just had a

little, had a little lift. And it kept going up. Like a very, very what? Not a very high slope at all. And just right after it hit

that point, right there. So, I didn’t want you to

think it flattened out from like, here to here

and was a straight line. It came down, came down, came down, back. And just like a bravo, but

then it did get some lift. But there’s only one spot,

right there, where it did it. – [Male Student] If you have

like, X to the tenth power, what will happen? – X to the fifth? – [Male Student] No. X to the tenth power. – Oh, exactly. Feel free to

look at it in your calculator. It does that, just kind of gets flat. – [Male Student] Would it

still be a single point? – Yeah. It’d be just one point. Yeah. You know where it is? At X equal to zero. X to the tenth, X to the 12th. You bet. Anyway, I’ve got my dot right there now, what’s going on over here? See that? It’s negative. Does everyone agree? You don’t even have to mark it up. I’m just gonna go like this, negative, and then just come

down here and go like that. Negative. How about right there? Positive. F-prime positive, right there. So what’s this graph

doing? Something like what? I mean, maybe it’s just a straight line? Or maybe it’s just a curve

that has a little lift to it. This is just a rough sketch, but isn’t it doing something like this? It’s kinda sloppy looking,

but yeah, isn’t it doing this? Yup, you bet. Now, I wanna point something out. What if this was Y equal to X squared? Joe, would you try to do

the derivative on that? What would be the derivative

of Y equals X squared? Just so we can make connections, here? – [Female Student] 2X. – Y-prime would be a two in front, X decrease the power by one, right? Is that a straight line? Straight line, parabola. Does everyone see the connection, here? See how you could do this with any graph? Now what if this was, here we go, I’m gonna do yours, really high power. What if this was X to the fourth, maybe? Plus a, and look, it’s

shifted up, so plus a one. What’s that derivative? It would be a what? Four in front, X decrease it by one power. 4X to the third? What’s

the derivative of that? Zero. So, and what’s 4X cubed look like? What’s a cubic graph look like? Yup, so it’d be more like,

eh, but it’d still do this. That’s what I mean when I

kinda went sloppy, here. Well, I’d have to have a real,

a little bit more knowledge about that graph. Good enough though, right? We just wanna get a sense of

how the derivative would look. Cool? Alright, another one. This one’s fun, here. We’ll make this thing another Trig graph. Another Trig graph. I like Trig graphs. Anyone know what the

graph “sine X” looks like? Of course you do. Right? It’s like a prerequisite skill. Now, if we just graph the sine of X curve, it goes like this, and goes like that. Right? Y equals sine of X. Okay. Does anybody Y

coda cosine looks like? I’ll put it over here. What’s cosine of X look like? I’m just gonna put it over here. Y equals cosine of X. Phase shift. Let me go like this, then

it comes down like this. You all agree? It’s got the phase shift? Very good. Isn’t that cosine of X? That cosine of zero is one,

what’s the sine of zero? Zero. I was wondering, Alan, can

you get the derivative? Make a rough sketch of the derivative? Alright. Where’s the slope zero? Boom! Right there, the slope is zero. What about here? Boom! Slope is zero. What’s the slope in the middle? That slope right there? Positive or negative? – [Multiple Students] Positive. – By the way, I should say that, if someone was sitting

here the whole time going: I don’t know how it’s

positive or negative, yeah. Lines that go up from

left to right, positive. Lines that go down from

left to right, negative. I should say that. There could be someone here who goes: oh, I didn’t know that. So that’s a positive slope. So I just put that, what? Up in the air? There we go. Over here, looks like

it was what? Negative? And right there? Negative? Alright. It’s gonna be smooth. And guess what, everyone? Guess what the derivative of sine of X is? Take a guess. You got it. Then we’re just gonna

make this connection. Guess what? If Y is sine of X, guess

what its derivative is? Just something, you can

really kind of see it. If Y equals sine of X, the derivative of Y equals sine of X is cosine. And so now, there’s a

whole section in the book for Trig derivatives, so don’t worry about Trig derivatives,

yet, just sometimes we discover these things as we go along. You’re like, we didn’t prove it, but we’re pretty, pretty sure. We can see, oh my gosh, you know? These Trig functions, they

both behave very similarly, but there’s a phase shift going on. That’s what’s going on. – [Male Student] If Y

is cosine of X is, ah, Y-prime times X… – Oh, I like this. If Y equals cosine of X. He’s investigating this, now. See, I love when you do this stuff. You can play around with it. What do you think it is? I’m curious. – [Male Student] It would

be Y-prime, sine of X. – Sine of X? – [Male Student] Yeah. Sine of X. Play around with it. You’re close. – [Second Male Student]

Negative sine of X? – Oh, who nailed that? Who was that? That guy who just said it? Hey, if you played with

it enough, you would tell. You’d be like, oh, it’s

the complete opposite and you’d have flipped it. You know what makes a graph flipped over? Just a what in front? The negative. You, I’m serious, sometimes

it’s fun investigating. You’re like, can I figure this out? And then, so, if you play

around with these graphs, you can see the derivative

of cosine is actually a negative sine of X. Hey, don’t worry, we’ll get, ah, remember my JVC thing for jumps, vertical? I’ll give you something that helps remember sines and cosines. I’ll just give you a

little, when we get to that, and that’s after the next test, I’ll just give you something like this. And you’ll just go,

derivatives go that way. The derivative sine of X is? The derivative of Y

equal to cosine of X is? Yep. You can just write it down. (laughing) Hey, here comes my next one! This one, everyone, I’m

stealing from the textbook. But I changed a little bit,

so it’s not the same one, but I thought it was very interesting. Is there anything that could trick you? I’m serious, like, there’s no way you’re gonna be stumped on this. Yeah, so let’s look at something strange. Sometimes the strangest

ones are the easy ones. Like a straight line. Does anybody know about

straight lines and their slopes? The don’t change. – [Male Student] Is it a zero? – Well, if it was like this,

the slope would be zero, right? But, what do we know about the slope in terms of if it’s a straight

line like this or this? The slope remains the same. The slope is constant

over the entire portion of that line, right? Forever and ever and ever. True? That’s different than

these parabolas and stuff. The slope changes. That’s why you have

the Calc course, right? Change in slope, but if

it’s a straight line, the slope is always

staying constant, true? So, I’m gonna just put a graph here, here’s the one I made up. It’s not the same as the

book. It’s a little different. I just thought I’d do a… There you go. Let’s say there’s some

function doing this. So this is strange. And I don’t see these nice,

nice smooth hills anymore. So how do you deal with

this one, weird case? Well, they’re straight lines. And the slopes always

stay the same, right? So how about, that’s what we’ll look for. And I don’t see any hills, do you? So what I’m gonna tell you right now, there’s nowhere where the slope is zero. You go, isn’t the slope there, zero? Oh no. The slope does not what? The slope does not exist there. It’s a corner. Everyone agree? So, we’ll talk about

what we’ll do with this. If the slope doesn’t exist

here, that’s a corner. Does the slope exist there? No. So, we’ll figure out how

are we gonna represent that, if it doesn’t exist? Well, a lot of you already know. But for this thing, all I

know is the slope is constant. Just tell me, right here, is

the slope positive or negative? Negative. So, if the slope’s negative here, along this entire portion. Okay. Then I’ll just do this. Then I’m drawing a flat line. How do I know it’s flat? Why not, oh, it’s just

somewhere it’s negative, but why do I know it’s going flat? – [Male Student] Slope remains constant. – That’s right, it’s a constant. Let’s make up a number,

so people can see this. Sometimes that helps. So, we’ll make up a slope here. What’s that slope? M is negative two, right? What’s the slope right there? M is negative two. What’s the slope right here? M is negative two. It’s not changing. It’s

always negative two, right? The slope is always a negative two. The slope is always negative two, until it gets to this corner. We’re not gonna put that corner. An open what? An open circle. – [Male Student] And why is that? – Wait, the slope is

always a negative two. – [Male Student] No, I

mean the open corner. – Oh, remember? Because

derivatives do not exist at J’s and V’s and C’s. At jumps, jumps is

kinda, vertical tangents, and at corners. So if it’s not differentiable there, the derivative now does not exist there, then I’m putting nothing there. Do you agree? Nothing. Does not exist. Hey, ah, what about here? – [Students] Positive. – Yeah, you wanna make up a number? What’s the slope of this line? One? You might say one. Slope two, whatever.

I’ll just use slope two. M is two, M is two, M is

two, slope is positive. So from here to here,

I’m just gonna do what? Draw from there, and then what? How about for the right-hand part? Can you predict it?

What’s it do right now? It goes back to what? There you go. That’s the weird one. Everything else is basically,

basically the same. How about… I’m serious, I can’t get too,

too much pleasure in these ’cause all you gotta do

is follow the process. Slope zero, all that stuff. I’ll try to make a crazy one, here. Alright. Came up. And came down. It came up, it came down, it came up. I’ll get it up a little bit. Get it up a little bit. Nice and smooth, isn’t it? Now, just try to get

a sense of that curve. What’s the function doing, right? It’s periodic. That thing’s periodic. I bet you its derivative is also what? Periodic. Great. It’s got a phase. Yeah, outstanding. I mean, you can get a sense

of just making up as we go. Now the question is, later in this course, are we ever going to go backwards? Yeah. (laughing) Like, if we have F-prime, can we go up? We’ll just reverse this, that’s all. Reverse the thinking, the whole lot. Look on this X-axis, you

know it has slope zero. That’s all it is. That’s all we’ll do. Hey, I’m gonna nail

these plots right here. Where’s the slope zero? I wanted to make that nice and smooth. (mimics tiny explosions) Slope zero. One more. Just a little longer. Now then, I’m just going

to do positive, negative, positive, negative, positive, negative, positive, negative, right? Slope right there, positive or negative? Positive. Slope here? Negative. Slope here? Positive. Slope here? Negative. Then get back to zero. Alright, looks like another

one of these Trig derivatives. Looks like it shifted a little

bit, but that’s alright. I think I’ve tried to

brainstorm every fun case. I saw one that did this. Just try this one out. Try this dude. It wasn’t in the textbook. I just saw this online. I was like, hey yeah, that’s a neat graph. What if it did something like this? Almost like the, anybody take statistics? A little bit similar to the bell curve. It just kinda went… This steep, and then, it was just something like that. We can talk more about it. It looks like there’s only one spot where there’s slope zero. Do you all agree? But, do you all agree, as

it kept going like this, it looked like the slope

was getting almost what? Yeah, it was getting very

close to being a zero. And over here, it looked like

the slope was leveling out and approaching a slope of what? Zero, I’m gonna use the

word “approaching” there. I don’t know if it really got flat, right? It looks like it kept going down, there. Very close to what? Zero. Alright, so where is the slope zero? You see it? Right there? Alright. Hit it. Hit it, everyone. Slope zero. That means, right here. Got it. How about the slope right there? Positive. Positive. That’ll be like, right there. That’ll work. Positive. How about the slope right here? Negative. I like to put little

tangents just so you see it. That’s negative. That

means it goes down here. But one last thing. Let’s make a half-decent graph here. What about right there? I can see this slope is

positive, but it’s really what? It’s positive, but it’s close to what? Positive, close to zero. Positive, close to zero. How about right here? Like, right above zero, but positive. Do you agree? Yeah? Does that work for you? Like, right there, I’m

like, it’s got a little, a slight positive slope to it. But it’s almost, it’s almost zero. Right? Like a road with a very low incline. And how about right here? Right there, that slope right there. – [Students] Negative. Slightly negative. – Yes! So how about, like, right here? So yeah. But that’s like

a bell-shaped curve. But this derivative would

do something like this. And I’m just gonna kinda

let it lie right there. It looks like it was getting really close, but we need more of that

curve to see what it did. And by the way, it probably

would just keep doing it. Just keep approaching zero. Like that. Cool? That’s it. I’ve hit every type of

scenario you could think of. You all really should crush the test. Rest of the class is review. I think George is gonna leave. Anyone else wants to leave, you can. Beat traffic and stuff, but stick around if you

have any questions for me. I’ll be glad to do it with you. I got nowhere to go. – [Camera Operator] Do you want

me to keep the tape rolling? – No, you can cut it off.

Thanks for posting!

Thank you so much! I came into class very late because of transit issues, and when I came into class the professor had just finished going over this chapter. Thanks again!