Slope, nice to put M in all this, slope M as the curve, this is the curve at that point. Now that's going to be a big deal for us. We're going to do literally dozens of these problems, right? For different techniques and so on.
So we're interested in finding the slope of the line at that one point. Now the problem is, our slope formula is what? Let's see if we see the problem.
Remember, rise over run, what is it? y2 minus y1 over what? Yes? Can you like, rephrase what's on...
Oh, sorry. Over here? Yes, I'll rephrase what it is.
A tangent line touches a curve once, at one point. The slope of the tangent line, where it touches it, is the slope of the curve. Same thing, both slopes are equal to each other. I'm going to reduce this because this is just way too big.
It's going to scale. Okay, there it is. We made that too much. Alright, so, everybody with me?
A secant line, so, with tangent line, let's mind we put secant, again, we're learning the language. Secant line touches the curve, not once, but... So a tangent line touches the curve one point, a secant line touches the curve, well, let's say two or more times. Let's just be safe.
Two or more times. At least twice. So here is the problem.
See if you see what the problem is. The problem is, up to this point, we have the, how do we find slopes? We take two points, right?
And we can use the slope formula and see what the slope between them is. Or we can visually... look and see what it is, right? Like go up three, over two, so the slope is three over two. Everybody following me?
Or down two, to the right, so it's negative two-thirds. Okay, that's what you're doing on your homework right now. That's the problem we did this morning. Everybody with me?
So that's the idea. Now the problem is if it only touches it one time, how do I get a slope? That's a problem. Does everybody follow me? That is the difference between them.
So we have to have a way to do this. So what we're going to do is, and again, just visually, what we're going to do in the next page is, okay, so we have the curve. again, pretend that's the same curve.
What we're going to do is we're going to find what? We're going to try to, let's say we want to slope there, right? That point.
We're going to take a secant line and try to go parallel with it. So we're trying to touch it twice. Now we have two points, don't we? And I'm going to get closer and closer to the point until it's almost indistinguishable.
So I'm zooming in on that one point until I get very, very close to that one point. Everybody follow me? That's what I'm trying to do. And the closer I get to it, the closer I get to the true slope. Because this only approximates the slope.
Yes? So we're going to choose two points. on the curve that are closer and closer to the point we want. That's the point we want to find right there.
So we're about to do it right now. I'm just telling you what I'm gonna do, but you're probably gonna understand it better because I'm gonna do an example. Or I can do a real example. Everybody ready?
This is what we're gonna do. do, let's do it. Like Nike said, what did Nike say?
There you go. That's what we're going to do. We're just going to do this thing. So I'm going to do example one in the Stuart book. It's a great example.
So you're going to be doing this many many times. Find the equation of a tangent line. By the way, what's the equation of a line?
Well, unless you like the other version. point form or the slope intercept form. I use that one.
You can use the other one. So when I say the equation of a tangent line, I mean what? y equals mx plus b? What do you have to find?
Two things to find. What separates one line from another line? has a different, well, the x and y are the same.
The slope is different, or the y-intercept is different, or they're both different. You're going to be doing this problem dozens of times, so let's get used to it now. Find the equation of the tangent line. They're talking about this.
You need to find a slope, you need to find an intercept. Period. End of story.
Every time. It doesn't matter what we're doing. We're doing derivatives.
Find the equation of the tangent line. Find the equation of the tangent line. Well, it says line, so that's the equation of the line.
So, we'll see it in a second. We haven't written the problem out yet. Line 2. The parabola.
This is interesting because when you think about this, y is equal to x squared, and they have to give you some point, right? I have a point of p, 1, 1. So this is your point p. I'm going to call it p. That's the coordinate.
The coordinates are what? x is 1 and y is 1. Okay, so the problem is, we said that what? We need the slope, but the techniques that we know, we need two points, right?
We need a slope for the equation of a line, but... Can everyone read my chicken scratch? If not, just say something, okay?
You'll get used to it. But we only have, what was the problem? What did I just say the problem was?
We only have one point, which we call P. And not two points to calculate the slope. I call myself abbreviating and calculate smoke. So we're going to choose. So what's our solution?
What did I say we're going to do? We're going to take two points that are close to that one point. So on the curve, think of that you're walking on the curve.
So you have a... So we have a point you're trying to get to, we're gonna pick. Well, why would you pick them way, way far away, right?
We don't care. We're trying to get close to that point and approximate the curve. Are you with me?
Now we'll see why it's an approximation. So we get closer and closer to choosing points, do a calculation, get. get the slope, get the slope, get the slope, get the slope, and get closer and closer until the real slope we're trying to find is very close and you can see the pattern, if you follow me.
That's kind of what we're gonna do. So you're gonna be able to see it right away. a couple problems like this.
So we're going to choose a point, and I'm just using their symbology. So P, what's after P? Q. So we'll call it, we'll choose a point X. And y nearby. You can call this, you know, we have x1, y1.
We can call this x2, y2, right? Okay. Nearby. Nearby the point 1, 1, we'll call this x1, y1.
So, in order to do this, what I just said, And call the slope, the slope between one of the two points, P and Q, right? So I'll just label them P and Q. This is the slope between P and Q.
So we know that a slope is, if you don't understand all the, the technique is what I want you to learn. So if you don't understand all the little details, all the labels, that's fine. So again, please tell me this is something everyone should know. What is the slope between two points?
Go, that's the point. Okay, slow between two points. Rise over run formula. You're good.
Y2, run, roll, run. Amen. Y2, run, roll, run. Yep, exactly.
y2-y1 over x2-x1. That's what we need. Now we're going to plug in what we just did.
So here's what we're going to do. We know that y is really what? What's the curve? So in other words, we call this... x1, y1, we call this x2, y2.
So this y2 is really what? The curve. y is equal to what? x squared?
So this is going to be x squared minus... So what is... Y1.
Everybody see how I labeled it? You see how it was easy once I labeled it, right? Label your points, please. Over. I just want to shove this at you and say where did this come from?
I'm showing you where it's from. Again, we're doing the concept learning. Concept. And then what is X2? X2 I called it what?
Just X, right? Minus. What's X2?
One. See how I created that? So what I've... say it again? Europe X twice.
My X... I didn't hear you. My X twice. Europe X twice.
Oh, well, where? You write it. Yeah, the formula is y is equal to x squared, right?
So I just plugged in x squared for it, because I don't want a different letter there. Are you with me? Now this is a function all in one variable.
Remember, we are what? Are we multivariable or single variable? Of course. No, no, that's calculus 3. We're a single variable course. We only deal with one variable at a time.
So that's one variable at a time. That would be a whole different problem if we left in another variable. Everybody follow me? So that's the idea. So that's where that came from.
So now we have a formula, right? So we created a general formula for any points on there. So we can go ahead now and plug in points, right? This formula works for anything on this curve. We just create it.
And I think they'll probably actually create it through the homework or practice. So that's the only thing we can't plug into here. Well, no, what would cause a domain problem?
Uh, maybe it's a zero? Yes. So x cannot be 1, right?
So you'd probably say, hey, why don't we just plug it right in? We can't. Because we created something that would go to an infinity, right?
I'm going to plug it in. I'm going to go to zero and block, but we can get very close to it. So here's the technique I want you to know. A very important technique. We're going to look at, I have a note here.
Show the full chart on page 68. I'm not sure we have time to do that. Let's see. There is, if we go to the book, I should have done this ahead of time, but here's the book. The problem is my download of the desktop, this thing keeps wiping itself every time. Okay, this doesn't have enough memory to open it.
Wonderful. Okay, what was that idea? What I want to do is I want to show you, if you want to scroll and use technology at the perfect time, if you want to go open up the Stewards book, there's a table there of values.
And we can, good, now whip out your calculator. We can actually do this ourselves, but I wanted to, that's how you practice it. So if you go to table on page, I believe it's 68 in the Stuart book. When I say Stuart book. Okay, we can recreate that, of course.
So here is the technique. You choose. Ever closer values to 1. So how can we approach 1?
We might as well learn this now. So if I have a graph, let's, you know what, I'm going to show you this right now. Before we have to do it.
So let's do some notation. Okay, so let's say I have a curve, and let's say I have a point I'm interested in. This point corresponds to what? An x value, correct? A vertical line?
And then a horizontal line. Is that too small? So that point has a x value here, right? And it has a y value there, correct? So what we're going to say is this.
As we know, we usually go from left to right. We walk in on the number line, we always go from smallest to largest, that's left to right. That's how we read, too.
So what I'm going to say is, I can approach this point. By the way, this is a one-dimensional object, right? The line, the number line, I can approach it from how many sides?
I can come, walk this way on the X and approach it from, what would that be considered? Left or right? So this is a left hand side approach. What if I come from this way?
This would be a what? You get it. So what we do is, if we come from the left side, we say it's, we're approaching, let's say this is what? Let's pretend it's negative 1 or not, but let's pretend it, remember I'm not doing this, I'm just doing an example, right?
This is separate from this. So let's say that it's negative 1. So we say that I'm approaching 1 from the negative side or the left side. Does everybody see that? That means you're, it's a left.
hand side approach. This is just definitions I'm doing here. So how do you think we're going to do an approach from the right side? Negative 1, we're approaching the number negative 1 from the plus side, the right side. This is a right hand side approach.
That's how we're going to do it. So in other words, we want to get close to 1. Now, why can't we choose 1? Because this blows up, right?
And it becomes a 0 on the bottom. Everybody see the problem? So I'm going to approach 1 from the left-hand side, and I'm going to approach Ron from the right-hand side. So in other words, I'm going to choose values that are less than 1, but very close to 1. And I'm going to choose values to the right of 1, in other words, just bigger than 1, but very close.
to say what I'm saying. So in other words, what are some values very close to one from the left side? What's just below? Well, for example, we're not doing negative anymore. Sorry, that's just the example.
Here we're doing one. So we're approaching one from the, what's just smaller than one? 0.9. How about 0.99?
How about 0.99999? Right? I mean, you can do 0.5 and stuff, but we don't want to waste time.
We want to get as close as we can so we can see a pattern. And then what's to the right? That's the left hand side. What about the right side? Let's just pick one.
The very first one. Okay, we have a dollar. 99 cents is close.
What's close on the other side? A little bit over. A dollar and a penny, right? 1.1, 1.01, 1.000.
Everybody with me? So that's how we approach it. We choose numbers, plug it into this formula, make a chart, and see if we can see a pattern.
That's the idea. Is that cool? Okay. So that's what I'm going to do. I'm going to choose values ever closer to 1. Can I put, remember what VHS?
means? Both. Instead of left hand side, right hand side?
Both hand side. Both hand side. Both hand side.
So in other words, I'm saying from the left and the right. Everyone with me so far? It's not bad, right? So, Do a definition?
Okay, so I guess we could do a definition. So we're going to say the... I haven't introduced limits yet, but let's go ahead and say limit.
So the limit is when you're approaching, either from the left side, that's called a what? Left-hand limit? left hand side limit, are you with me?
And I approach from the right side, that's called a what? A right hand side limit. That's just terms, that's just the name.
That's just the definition. So as we approach, as Q, the point Q approaches P, right? As we get closer and closer to one, one, this slope becomes the actual slope.
Actual slope of the line. That's what's going to happen. As we get closer and closer we find the real slope. This is the real slope. Real slope, right?
Actual slope. So that's what we're going to do. So let's go ahead and do it.
Let's create a little chart. By the way, how am I doing with the time? I have no idea.
It's going so fast. 18 after? Oh, 28 after. Alright, now here's what we're going to do.
Everybody have a calculator? Who has their calculator? Yes.
This is a rock and take me. So I'm going to have to choose whether I need to win from the both heads on. Right.
So we're going to do it right now. I promise you, after I write down a few numbers, I'll take this to the pad. Alright.
You with me? Okay. So let's choose some numbers. Now, here's what they choose. They chose...
For example, x to be, I don't know why they're so far away, if I put a 2 in there, I don't know, you don't have to start this far away. So matter of fact, don't even write this down. Here's the chart from the table.
They put in a 2, plug it into the formula. What's 2 squared minus... What's 2 minus 1? What's 3 over 1?
The slope is 3. See how they did it? That's the actual slope or like the approximate slope? Isn't 2 pretty far away from 1? but I'm trying to get close to one, right?
I don't know why they started at two, that's pretty arbitrary. Let's get closer. They did 1.5 and let's save some time.
If you do it yourself, it's 2.5. If you do a 1.1, that's getting closer, right? That's a dollar and 10 cents.
So obviously, which side are we coming from? Left side or right side? The right side.
Coming from the right side, the plus side. Remember we call it the plus side? So that gives you a...
I can't even read what I wrote down, but I think it says 2.1. So it gives me a 2.1. So let's go ahead and get even closer.
Put this in your calculator. How about 1.01? Put that into the formula. So it's going to be 1.01 squared minus 1 over 1.01 minus 1. Where's my calculator?
Help us out. Let me get it. So you square that. You see, you don't want to do this by hand, right? Please, for the love of God, bring your calculator.
It is your tool. Use it. Use it. So we get this ratio. By the way, please practice on your calculator.
That's the... One of the first quizzes we're going to do is something like this. I'm going to have you do this.
So please make sure you know how to use your calculator. 2.01. So we get 2.01, which we'll put in here. Now again, in math, we look for the patterns, am I right? Can I get a witness?
Patterns. Pattern recognition. Very, very important.
So let's go ahead, and by the way, we're approaching from the what? One from the plus sign, right? We'll call it like that.
Just so you know. So let's get closer. Sure. Take a penny and cut it and cut it into 10 parts. So it's 1.001.
Everybody with me? Plug that into there, what do you get? You're probably saying, well, I don't want to do that. I want to do 1.000000.
Whatever you want, right? If you really want to see it, you can go even smaller. You notice that the bottom number is what?
It's always going to be positive, right? Because we're always slightly above it. So it's going to be positive. Just make note of that for later on. We'll do a technique just...
And the top... It can't be negative when we go below it. It can't be negative because we're above it. On the other side, we'll be below it. Yeah, we'll be less than.
Alright, so, but it's going to balance out top and bottom. So what do we get when we do that? Anyone? OK.
Do we see a pattern? It seems like we're approaching the input. The closer x gets to 1, the output seems to be approaching what value?
We have to do it to find a limit. You must, write this down, you must approach from both ends. So you know you have a limit. If you come from the left side and you're going towards the same number as you're going to the right side, the limit exists and it's equal to that number. Is everybody following?
So left side and right side. So now we have to finish the table by doing another one. Yes? Oh yeah, so a limit, don't worry, because we haven't even done limits yet, I'm sneaking ahead. So for a limit to exist, left hand limit, that means you're pushing with side.
The left side, well, left side, or a minus side, are you with me? Has to be equal to the right hand approach, which is from the plus side. Everybody get it? They have to be going towards the same number.
If they're not, the limit doesn't exist. In other words, the left hand side might be going to 5, the right hand side approach might be going to 3. That's not the same number, the limit doesn't exist, period. I just want to get that in your subconscious.
The limit doesn't exist. Exactly. There we go.
The limit has your... Well, we'll see all that good stuff later on. I just want you to learn it.
Now we're going to approach here. We're going to go from the what? 1 minus... So... So now we're going to go...
What? Let's go... I don't know. I'm going to do the reverse of what they did.
I don't know. They went to 0 because that's easy. My favorite number, right? Why is it my favorite number? My bank account balance.
You can ask me why is that. So this is going to be what? 0 squared minus 1 over 0 minus 1, which gives me what?
What does that give me? Take it times the opposite. Plug in 0. Let's get to the formula.
Remember the formula? x squared minus 1, right? Which is? Okay, let's plug in what, so in other words, we're approaching it from the other side, right?
We're coming up to it, like from the left side. Try 0.5. Now, you don't have to go 0.5, that's fine.
I'm a little sure, but plug in 0.5 and see what you get. Is it important to use your calculator in this class? Yes it is.
What does that give you? It gives you 1.5. I can't take it anymore.
I'm so excited. Alright, so let's go a little closer. Let's go dot dot dot. Let's go to 0.199 cents. So we're getting pretty close to one from the lower side, the left side, right?
The left side. Remember, one is here, my left, right. The right one. As you face it in front of you, number one, we're approaching it from both sides.
Plug that in, you get 0.99 squared minus 1 over 0.99 minus 1. What do you get? Any one? Actually, I forgot to write it down, so I don't know what it is. I think it's 1.99. 1.99?
Yeah. Can I get a witness? No, I meant if you found it. We trust you. We trust you, that's nice.
Alright, let's go even closer. What's closer to 99 cents? How about 9.99? 9, 9, 9, 9, whatever, however many 9s you want. Go with 6 9s, go with them.
In other words, get very accurate. Use larger or smaller numbers if you're not sure of the pattern. But I think we see a pattern here.
If I plug this in, I get what? You know what I meant to do? I'm sorry, I meant to write this here, then write that. I meant to do it that order. Sorry about that.
Because I want you to see it going. Yeah, let me do it. I got so excited I forgot what I was going to do.
So I'm just making this up. Dot, dot, dot. 0.99. I get 1.99. 0.999.
Can you see that? So I'll just erase this here. There's more of that. Is that cool?
So, do we have enough information to see what the slope is? What do we think the slope is? We think that, according to this chart, this chart is telling us something.
It's telling us... the limit as we approach one. So if you don't see a plus or minus, what does this mean?
Again, this is notation. This means we're approaching one from what? If you don't see a plus or minus, it says both hand sides.
So we just approach it from the left. We just approach it from the right. Matter of fact, let's write that down. So again, I'm kind of teaching you ahead of time, but I think it's cool.
So this is telling you right here. At the limit, here's the language of math. Remember, math is a language.
A lot of symbolism in it. This says, as I approach one, from which side? Plus or minus? Uh, right?
Yes. You're both right. Positive and... Right?
It's the same thing. As I approach this value of x squared, the function x squared minus 1 over x minus 1, I appear to be going to what value? What do you guys think?
So if we went really, really far, do you think it would be what? 2.000000? So let's take a guess. What do you think it would be? 2. Yes.
Exactly 2.000000. Good. Because look at the pattern.
I'm getting closer and closer to 2. The input is getting closer and closer to 1 from the right side. The output is getting closer and closer to 2. So I'm saying the limit, this is plus. No, I'm talking about the right connotation.
Limit as x arrow. So this says the limit, L-I-M is short for limit. As x approaches 1, from the what side? What's the plus side, folks?
The positive or right side. So the right side of it. So you look at the number, it's on the right side.
Look at the number, the left side of it is the negative. Okay? And here, this is telling me that as I, what, limit? As I approach 1 from the left side, or the negative side, of this function x squared minus 1 over x minus 1, what do I seem to be approaching? 1.99, what if I put in 0.99999999, I'd probably get 1.99999999.
Notice they match up, right? It's 3 there, 3 there. So it tells me it's going to what? So I'm coming from the left side going to 2, coming from the right side going to 2. What would the limit be as you approach 1?
So with all this information, this is telling me I can only do it by approaching both sides. The limit as x goes to 1 of x squared minus 1 over x minus 1 is what? I can say it's 2. By the way, this is equal to what? What's that?
The slope of the line we're trying to find, the tangent line, right? M is the slope of the TGT line, tangent line. So guess what, folks, what did we just do? We just found the slope. We're halfway done the problem.
Remember we're trying to find the equation? What two things we're always going to find of the equation of a line? What two things are different? That's it. We just found half of it.
Matter of fact, we just found the... the most difficult part. See what we had to do?
We had to do it. This is called a numerical technique. We had to do a numerical technique. By the way, I'm going to build this up.
We're going to find more and more techniques to be able to find this very, very important idea. Are you with me? So here we're going to say, okay, so this means so far I have y is what?
What's the equation? Put the 2 in. 2x plus b.
So I'm halfway done. So far I have... Now, second part, by the way, I do this every single time, right? So, this is just our first one, I'm going very much detailed into it. Now we have to find, again, we're trying to answer the question here, example one.
Find B, how do we find B? Well, wouldn't it be cool if we knew of a point on this curve? Is there a point on the curve that we know?
One comma one, yeah. One comma, yeah, I know what you meant. One comma one, right? They gave it to us. It gave us the whole problem was we need two points to find a curve.
But we did something about it. You can see right. And use this limit idea. So it's kind of cool. I'm just getting chills up my spine, but that's fine.
So it's kind of cool that they had to do this so many years ago. I'm 300 years. So we're going to plug in the points.
In other words, we're going to plug in the point that they gave us. To find what? What do we need to find?
We need to find B. Sorry. We're going to plug in this to find B into what we have.
Are you with me? So I'm trying to write things out more and more so students can come back and look at it. Especially if they're not even here, I don't know what's going on.
This is not a good day to miss, but very few days are good days to miss. Anyway, let's plug it in. So what's the source value called? What's the second one called? Okay, we need to know.
Just kidding. Alright, plug this into here. By the way, they're both the same number, right?
So you can't make a mistake. We're going to plug it into here and solve for B. Every time we're going to do this.
Now, be careful. What's this? Let's say this is one and this is two.
See, students, what mistake they might make, they might plug a one there and a two there. So be very careful. Make a note of that. X goes to X, which is on the right side. Y goes to on the left side.
Okay, don't, just because you see a one first, plug it in there first. I mean, this case works, so then go to one, one. So that's, I'm trying to come up with, so in other words, 1 is equal to 2 times 1 plus b.
So 1 is equal to 2 times 1 is 2 plus b. How do we solve this? Help me out? Yes.
Subtract 2, subtract 2. So this tells me that v is what? Negative 1. So I have the slope, right? I have the slope. Let's put it together. What is the solution?
What's the equation of the tangent line to that curve at that point? It is, drumroll please, or just put it in. It's what? Fill it in. y equals 2x.
You got it. That's not bad. That's the tangent line.
That wasn't bad. That just didn't require more than a rhythm tape, right? That was pretty much the rhythm tape we were using. Tangent line to the curve at that point. To x squared at the point.
So this would be the answer to the first question? That's the answer to the second question. But what we're going to repeat is we're going to use this technique over and over again. Okay, now I'll try. How long should we be able to do it?
How fast should we be able to do it? It depends on how much you practice. So I'm going to give you one in class. And make sure you can use exponentials, by the way, because it can be any function. It doesn't have to be a simple quadratic like this.
It can be an exponential function, a log function. It's okay. You can plug it in your calculator if you know how to plug it in your calculator. So please, for the love of God, practice on your calculator.
That's really going to help you. How many people time do we have? Really? But anyway, I went into it in detail.
Now, we're going to look at the second part of it, which is velocity, which is almost the same thing. We're going to use very, very similar techniques. So let's just talk about that. Now, it's in the homework, but I know that you'll see it once in a while, but the tangent problem is the one you'll see a lot.
So, The velocity problem is very simple. Now, the problem is also, it's the same problem, actually. Does everybody remember taking physics? Remember Newton's laws? It requires that what?
You know the initial velocity, does this sound familiar? And the final velocity? No?
Okay. So it involves initial velocity. Yes, yes.
All along that stuff. So you need the initial velocity and the final velocity. Well, that's two points, right? Same problem. If something is traveling, I have to know two points.
I have to know what it's, the time is here and here to be able to find the velocity, right? So I need two points. You see the same problem? What if I only have one point?
How do I find the velocity? I'm gonna use the same technique. I'm gonna use a secant line on the curve and get closer and closer and closer. So I'm gonna choose closer and closer times and look for the same pattern. Isn't that kinda cool?
So it's the same technique. It just. involves a formula. Like for example, I guess we'll just do the example. So a ball, matter of fact, this is a good one for you to do, and we'll talk about it tomorrow.
A ball is dropped. Drop 450 meters. Again, we're not doing physics here.
We're just trying to use our same technique to find what's going on. Above the ground, find the velocity. I don't expect you to be able to do this because this is a physics problem.
I don't think physics is high school. If you didn't, that's fine. After 5 seconds.
The point is, we want to use the same technique we just did. The distance, the formula, the distance fallen is given by the formula. I'm just doing a little summary of what they gave us. It is S is your precision. It's the distance fallen S, right?
The distance fallen is... s, it's a function of what variable? t for time is equal to 4.9 t squared. In other words, don't worry, they're going to give you some meters.
They're going to give you the formula, right? We're not expecting you to know physics. Are you with me?
But isn't this another curve? It's actually a squared curve, so it's similar to what we just did, isn't it? Except it's time. Everybody follow me? So this is very, very similar.
So here, So here's the problem. We're given... How many points?
It says at what? Five seconds, right? How many points is five seconds? One. No, I mean one instant of time.
We're given one instant of time. You know, five seconds. So t is equal to exactly 5 seconds, right?
So, we, I'm just writing the problem out, we have no, we have no interval, so we don't have 2 points. Is everybody with me? We don't have 2 points. So we're going to approximate, this is what we're going to do, we're going to approximate, I can spell it, even my abbreviations can, approximate by Computing the average velocity.
Are you with me? And by the way, the average velocity looks exactly like the slope formula. I'll show it to you in a second. Over one-tenths of a second intervals.
Why did they choose 1 tenth? Because we're trying to get what? As close as possible.
We could have chose 1 over 100, like 0.1, 0.01, 0.001. Does that sound familiar? That's exactly what we just did. That's why I want to do this right now while it's fresh in your mind. 1 tenth second interval from t equals 5 to t equals 5.1.
My chicken's cracking. I'm running a little faster. It's getting worse.
Sorry about that. You've got to get this in. So here's what we're going to do.
So average velocity. Okay, this is... This is average velocity.
I'm just going to write this out. The average velocity is what? The change of position over the change of time.
Everybody with me? The change of position would be what? Don't we have this formula? We're going to plug this in at what value? 5.1 minus x5.
We're just going to plug this in because we're going to use this as a... So we're going to do 4.9, we're going to plug in 5.1 squared. This is so 4.95 squared over, now they didn't do this one.
So in other words, this is what? S2 minus S1 over T2 minus T1. Is that exactly where it comes? That's exactly where it comes from.
So the S is actually a formula that we plug in, right? S is this formula. So we're going to plug it in. Sure.
How do we know what S2 and Q2 is? S2 would be the, we're saying, the future value. So 5.1 is larger.
It's always the larger number. The larger minus the smaller. Because if you do it the other way, you get a negative time. Right? Everybody with me?
Over, what's the beginning time? We're going to say 0. minus 0.1 right that's the two times times 0 is the beginning at 0 right this is y2 minus x2 minus x1 right So, oh sorry, minus 0, let's do that again. Because at 5, that's our 0, at 5.1, that's that.
So we get what? We get 4.9, 5.1 squared, minus 4.9, so we write this again, 5 squared, over 0.1, which comes out to be 49.49 squared per second. So.
What they want you to do is, you can go ahead and look at the, on that page. They're going to have you do a chart, and they'll choose instead of 5.1, instead of 1.1, 2.01. It should get more and more accurate, and you should approach the actual value.
So I believe the, the value. I'm just going to approach the actual value of 10. I can't remember, whatever it was. But you see this is the same 10-minute, that's why I wanted to show it to you.
Because you'll have velocity problems on there. Everybody, let's start a little bit at a different time. Robert, we're going to be in the late Lucini. All right, so practice these problems.
So I'm going to be, what's it called? The tangent velocity problem right there. It's right in your homework. Thanks, guys.
Go ahead. Slide up there. Oh, yeah, you can pull these around.
No, I see it. No, I see it. I see it.
Are you good, Lucini?