Lesson 1 - What Is A Derivative? (Calculus 1 Tutor)May 10, 2020
the slope of the line just barely touching the curve, it looks something like this, so when I say the slope of the tangent line I mean a line that just barely touches a point, okay, this is
whatthe slope would look like, so obviously. the slope changes as you go up the curve because you're accelerating okay that's why this problem is a little more challenging than the last one so I'm asking you
whatthe speed would be like just like we did last time v of Speed is a function of time. Well, I already told you that it was speeding up, so the speed must increase as I move forward in time.
Well, that means that when I start here I have a certain speed and it gets bigger and bigger. and bigger and I'm accelerating and I'm accelerating and that's why my speed is increasing up this would be like sitting in your car and constantly pressing the gas pedal down, down, down, down, you start at 10 miles per hour 20 miles per hour 30 miles per hour 40 miles per hour what are you doing are you accelerating that's what you're doing so your speed is increasing okay so the
derivativeof the position again gives you the speed in this case your slope here which one is this is the slope of the line tangent to the curve is very low, so your speed is low because the slope of this line tangent to the curve is the
derivativeof this and the derivative of this is the speed as you go down the curve. curve. the slope becomes larger, so the derivative becomes larger and its velocity is greater, the slope of this small tangent to the line on the curve here is even greater and since the slope of the tangent line is the derivative and the derivative is the velocity, the velocity increases So as the slope increases, the velocity increases, so you've taken a derivative in graphical media, so if you're going to write it down and you could say that v of t is twice t, let's say okay, in fact, this is it. the exact derivative here, but I haven't shown you how to calculate it yet in terms of functions, but this is the derivative of t squared is two t and this is a line with a certain slope, okay, so describe this equation here now.
I ask you what is the acceleration a of t. Well, we already said that in the initial problem statement you are accelerating correctly, so we know we are accelerating, so you have to ask yourself what the derivative of the velocity is. acceleration that's another way of saying what the slope of this curve is well the slope is constant the slope didn't change unlike this the slope is constant because this is a straight line so I'm going to go up here to 2 okay and I'm going to draw a line here with a constant slope which is a constant acceleration, which means as I go my acceleration I'm accelerating at a constant speed like this, I could be accelerating at 2 meters per second squared or something like that. constant acceleration.
I'm constantly increasing my speed by the same amount every time I move forward in time, okay, so this is a constant acceleration, in this case it's equal to 2, so you have a constant acceleration, so you've got an introduction. a basic introduction to derivatives the big picture to remember here is the derivative of something that is just a slope of the line tangent to the curve, you know, if it's just a line, it's just the slope, if you have something with curves, then at each point there will be a line tangent to that curve and the slope of that line is called the derivative, it just so happens that position, velocity and acceleration are very familiar things that we can talk about because you all drive their cars and they know what those terms are.
The terms mean, but you could take the derivative of any function. The function could be describing the change in pressure in a container. It could describe the trajectory of a spaceship. Anything like this. Okay, whatever function you write, you can take its derivative when you do that, you're taking the slope, you're looking at the slope, you're looking at how that line changes or how that curve changes with respect to time or with respect to something else that you might be looking, okay, great, let's go ahead and make this a little more concrete in terms of what your book will actually tell you.
I'm just going to write a few things here that I want you to remember that the derivative is, by definition, equal to the slope. of the tangent line to the curve add a point at a given point so that the slope can change as you go up and then your derivative changes, so to show you a little the way you write this I'm going to show you the symbols in
calculusthat we use to talk about derivatives remember that we said that the speed was the derivative of the position well you write that like this the speed is well it is equal to p prime of t remember that p of t is the position when you put this prime here, this means derivative, it is okay, so you have the position function, you take the derivative, which is the rate of change, the slope of that tangent line and what you get is called velocity. another way to write this is exactly equivalent ways to write it is the derivative of p of t with respect to t so you would say in words dp dt here you would say p prime of t in any case you are saying the same thing what you are saying is give me the slope whatever function you're talking about p of t a of your of t doesn't matter so in this case you're taking the derivative of p with respect to time because time is the dependent variable and the independent variable there, so this is the terminology here p prime of t dp dt you will see that both in your books mean the same thing, they are just different ways of writing it and, similarly, acceleration as a function of time is equal to v prime of t you take the derivative of the velocity, that's what this prime means and you get the acceleration and to write it this way you would say d v d t okay so the derivative of the velocity is equal to the acceleration now I'm going to show you something that might confuse you a little but I promise I'll make it clear that this is also equal to p double prime of t do you have any idea what that might mean? a small prime means first derivative it means I give you a function take its derivative which means look at the slope and then you will get a function which will be called derivative when you have two little ticks here which means do the derivative twice so take a derivative and you get some function and then you take the derivative again, you look at the slope again, so twice in total and you get that, so if you remember, we had our position graph i' We'll put p pos our velocity graph and then our acceleration graph and I said this was a velocity, it was the derivative of the position and then the acceleration was the derivative of the velocity, so that's what we're writing here, well, if you start with the position and go here the first derivative here is the second derivative well, you get to the acceleration so what we are saying is that the acceleration is the second derivative of the position one two it really is not not rocket science the first derivative is the velocity the second derivative of the position is the acceleration so the other way to write this in terms of that d business here is d squared v d t squared is just notation.
I'm just trying to teach you notation here what you're saying is I've taken the second derivative second derivative of the oh I have a typo here, the second derivative of position with respect to time and that's equal to the acceleration. Well, then it's just notation. Now the next thing I want to say is just to point things out and make them clear. Let's say you have a random curve that looks a little crazy and then there's a sharp point or something, well, clearly, if I want to see the slope of this curve as I go, it would be like the slope here, so the derivative would be Whatever the value of the slope is, which is a positive value if I'm looking at this right now, the slope looks like this, the slope of the line tangent to this curve, which is negative, a negative slope because it goes towards the backwards, to the right, so that's another slope value and so at each point you have a slope value based on the tangent line to this curve, okay, it's constantly changing as you go up and down, the slope is changing at this point. you have a cusp the derivative is not defined here the derivative is not defined basically the rule of thumb what you really need to remember is that to take the derivative you need to have a smooth function, you can't have a discontinuity or a sharp point or something like that or a function stepped, you have to have a smooth curve to be able to see that line tangent to it because when you think about it, how would you draw the line?
I mean, you have only one point here, would it be? here would be here I don't know so you can't really do that
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