Okay, hi everyone and welcome back. So let's start off with a little warm up question before we begin. Most time, people see this stuff, it's I don't know, a little alarming maybe, it's sort of one of the longest math problems that they've ever done. Usually takes up a good portion of the page and at the end of the day, remember we're finding a slope of a line, which was something that, well, until maybe recently was really easy as we just had a fraction y2 minus y1 over x2 minus x1. So remember what we're doing here with our goal. Put this over here on the side. Our goal now is you'll be given some function. Remember, any function is allowed, and a point, and some point P, let's say just x, y, and you want the slope of the line. So our goal is to find the slope of the tangent line. Find the slope of the tangent line at the point P, so this is our point P. So if I have some curve, whatever it is and I have some point how about that one? I want the slope of this line. And first, you find the slope because then if you want, you can then find the equation. You can then find the equation and this is such a simple question and theory like find the equation of a line. You were doing this for years. It's amazing how this kind of comes back and cycles back, but because were specifically focused on the tangent line which leads to a division by zero in the formula for slope, you need limits to get there. And that's sort of the beauty and the power of limits here in this difference quotient of what we're going to do. So let's do an example. So let's start with the function, the square root of x, and let's pick the point 1, 1. So we'll pick my point P to be 1, 1. So in your mind, you should think of the square root graph and 1, 1 is somewhere around here, nicely there. So then of course there's a tangent line to the curve, looks something, whoops missed, looks something like that, that in theory touches the line right at the point. So something, just make the circle bigger, in case you miss. Find the slope of this line, and because I only have one point, my traditional slope formula for a line won't work. But now we have limits, life's amazing. Okay, here we go. Write the formula, so limit as h goes to 0 of f of, now remember, it's like a plus h but a is our one in this example, a is always the x value, so think of that as a. A is always the x value that we're going to plug in. So instead of a plus h, it's 1 + h minus f of a is our number. I should have put in 1, all over h. So a is 1, f of 1. So f of 1 is clearly just one, square root of 1 is 1, so that number right here is going to be 1 in a second. Put the formula down, plug in your number a, just show us. Remember, you have to write the limit. If you leave off limits, you missed the whole point, so write limits every time. Do I have to write the limit every time? Yes, you have to write the limit every time, all right. So the function if I plug in 1 + h, it just says give me anything and I'll take it, square root. This becomes the square root of 1 plus h minus f of 1, minus square root of 1, that's just one, okay. All over h, okay. So here we are. Now remember that we want the limit as h goes to 0. What does this function want to do as h goes to 0? I cannot just plug in h, it has a big h in the denominator and I cannot do that. So I have to do something else. Do you remember what to do? This is a little tricky, but it's so, I don't know, so awesome, I love it. You have to multiply by the conjugate when you have a square root. This is a little trick to get around this, so the conjugate is the, change the sign outside of the square root. Change the sign outside of the square root, you have to do it to the top and bottom. Remember, this is a fancy way to write to the number 1, write the number 1. Numerator and denominator are the same, so I'm multiplying here by the number 1. When you do that, what do you get now? We're going to write the limit because I haven't actually taken the limit, but I don't want to write the limit, write the limit. Then we foil, multiply fractions across the top and across the bottom, square root of 1 + h times a square root of 1 + h is just 1 + h. When you foil conjugates, outside and inside cancel, so they cancel. So it really becomes first and then last. So last is just minus 1 times positive 1 is minus 1. All over h times, and then don't distribute this. This is separate. This is going to look kind of nasty, but that's okay. Maybe you could even see where this is going, stuff's going to cancel. So, why do I like this look? What happens? The 1s cancel. And then, you're just left with an h up top, and an h on the bottom. That's amazing. Now, your hs cancel. So, your 1s cancel, and your hs cancel. And remember, that h in the denominator was the problem. Was the big sort of stumbling block, that division by 0 problem, that was preventing me from plugging in the denominator. So, we have 1 upstairs. The only thing that's left is 1. When you cancel the hs, you get 1 upstairs. And the denominator, you just have the big 1 plus square root of 1+h, under the square root, plus 1. That is not under the square root. And now, even though there's an h downstairs, it's being added to something. So, I actually can plug this in. We're going to plug in h here, and you get 1 plus the square root of 1+0 +1. Notice I didn't write the limit, because I actually evaluated the limit now. I'm plugging in, so I'm taking the limit. I don't write the limit. You have to write the limit until you do this step. Clean this all up. This becomes 1 over 1+0 is 1. Square root of 1 over 1, it's like 1+1 over 1. I love when there's one of the crazy calculus problem turns into 1+1. So now it's 1/2. Positive 1/2, this is the slope of the line. This is the slope of this line. The tangent line to the curve, slope is 1/2. This is what you have to do to get this tangent line, and this is not easy. This is why it took a while to figure this out. This is usually the first step. You always need this. So, hopefully this makes some sense, and we'll do more of these as we go along. But, sometimes they ask not just for the slope, but they want the equation of the line. That's not an unreasonable thing to ask. So, if they want the equation of the line, remember, we can use point slope, y-y1=m(x-x1). And we have all these pieces of information. We have x and y, 1 and 1, and we have m, which is 1/2. So we'll just plug this in. So, we have y-1 equals 1/2(x-1). So, this is the equation of a line. You can clean this up if you want, but you could totally leave it like this, as well. So, know your point slope formula. So, there's two sort of things to know here. Know your point slope formula. So, this little definition. And, of course, know the limit definition for the slope of a tangent line. Okay, that's a good example. And, make sure you go over that and understand that. Next, we want to talk about velocities. And sort of relating these, why this thing is actually important. So, if we have an object, so if an object moves in a straight line. So, in a straight line. According to some function, so usually they use S, and this is a function of time. So think about a car on a highway. Maybe it speeds up, maybe it slows down, but you have an object moving. We call this function, where you are, this is your position function. Position function tells us where you are. And then, if I take the average velocity. So, if I take the average velocity, remember, from physics, here velocity is change of distance over the change of time, all right? So, it starts to look like a slope formula like d2-d1. So, you can do all this. So, everything is the same, but instead of doing the traditional formula for slope, we do our difference quotient. So we say, well, let's pick a point a. Let's pick a point a, and we'll do f(a+)-f(a), all over h. So, we write it in our way, that's getting set up for the slope of the tangent line. Now there's no limit here. There's no limit. a is some fixed point. So, without the limit, and if you have some point, there's your curve. If I have a, look, I keep calling it x, but they call it a. And then I have some other point, and we call it a+h over h represents the distance between them. Then, this expression f(a+h)-f(a) becomes the change of y over the change of x. We know this. This is the secant line. This becomes the slope of the secant line. This is the slope. This is good old days dy/dx. The slope through two points is called the secant line. When we take a limit as h goes to 0, that'll get us towards there. So, in physics, this expression, which you probably knew already, or at least have seen at some point, is just a slope of the position function is the average velocity. So what do we do now with limits? So, As h goes to 0, which is what we do, we take a limit as h goes to 0. These are initially two different points become one and the secant line becomes the tangent line. So the slope of the tangent line, the slope of the tangent line, this thing that we're after. The thing that we just solved and found a couple, is the instantaneous. Instantaneous rate of change. So this is sort of how fast am I moving right now, instantaneous velocity is another way to say this instantaneous, because it occurs at one point. So at this instant what is going on instantaneous velocity? So there's a difference, and this is important between the average velocity which is given me two points was the average life, so I drive from New York to Baltimore. What was my average velocity between the two? Or I'm on the Jersey Turnpike? And what is my velocity right now? Those are two separate questions. They both give you different pieces of information. You can get a ticket for one of them if you're going too fast. And we want our instantaneous velocity and usually just as a friendly reminder, we usually do velocity in meters per second. Although there are some problems, of course we do miles per hours. So when you start to get into physics, just be careful, you gotta watch out for your units. And so we could do these problems just as an example, what you're goinna see when the physics side of things kicks in. So they'll give you displacement function. So it's all the same stuff, just think of it as word problems. We have a displacement function, or maybe they'll call it a position function. Or maybe there's a particle moving blah, blah, blah, you know the drill and then they'll give you some function 1 over t squared. And if they say, find the average velocity, let's say part A, find the average velocity. Find the average velocity, so I need to give you 2 points. So let's say over a time interval, over let's say, time yeah, let's go 1 to 2. Right, so some particles following this path. This is with position and I want to they stopped at 1 and I stopped at 2. What does that mean? It's a lot of words. It sounds fancy, but it's pretty straightforward where looking for the average velocity formula. This is just the change in the distance over the change in time. Distance for whatever reason is S over the change in time. This is just slope change y over change of x, rise over run. So I plug in 1 over 1 squared. Yeah, so we gotta plug in so it's S(1). I will see S(2)- S(1) over 2- 1. This should feel just find this whole problem, but of course it's in a word problem and there are always a little trickier. So what does this one-forth- 1 over 1 so minus three-fourth some like that. This thing was decelerating. So average velocity, is old news, we know how to handle it. If they had asked for not be able to fit in here, but find the instantaneous velocity. And we won't do this one at the moment, but just to talk about it. Find instant velocity at t =1. Now we're going to go ahead and do the limit problem for that, so that's a longer question and we can do that later. All right, so these are the kind of velocity questions you're going to get. Average velocity versus instantaneous velocity, know the difference and will do some more problems later on. But let's keep moving on for now, because I want to define the meat and potatoes of this class. And this is derivatives, so you may have heard this word before and not know what it is, but now you now you'll know what it is. So this is a definition. If I have a function f and will say the derivative on, so the derivative of some function f(x) at x = a. So some point in the domain will denoted by prime of x, is defined as. Here comes f prime of x equals the limit as h goes to 0 of f(a) + h- f(a) all over h. I'm going to put one little caveat in here, if the limit exist. Not every limit exists, and so since the derivatives are, Not every function will have a derivative at a point and we'll see some of those in a second. But this formula. So what do we just say? Basically, I talked about this over the tangent line. This formula for the slope of the tangent line is exactly so the derivative is so important. It goes by different names, so you also see this slope of the tangent line slope of the tangent. This is what we've been doing. So the tangent line, and then if you're the physics person, this is instantaneous rate of change. That's sort of the generic term for it, and you can also call it instantaneous velocity. If the function you're starting off with Is the position function velocity. Every sort of scientific discipline needs and uses the derivative. If you're in economics, they call it like marginal, so we talked about the cost function. It becomes marginal cost if you talk about the revenue function, you talk about the derivative, its marginal revenue. So each discipline sort of has this idea of finding the slope of the tangent line of instantaneous rate of change. How fast is something changing right now has come up and it's taken on different names. So one of the things you have to get comfortable with is like you're always going to be doing this question. You're finding derivatives, but they may not actually use the word derivative. [INAUDIBLE] will say [INAUDIBLE], of the tangent line physics find instantaneous velocity economics will say find the marginal cost or something equivalent. So I'm going to try to mix it up and give it to you. Present the question in different ways, but the end of the day we're all going to go ahead and find this derivative. This limit as H goes zero of F of A plus H minus F of a over H, so there is the formula where we will need to memorize this eventually. And hopefully by doing lots of lots of examples we will memorize it that way, so let this one sink in. We'll stop here and then will pick up with some extra examples on the next one. Okay, so see you on the next slide. Keep this example and definition handy. See you next time.
