Calculus: Acceleration and the Derivative

This lesson has not been reviewed.

Please purchase the lesson to review.

Practical Application of the Derivative
Position and Velocity
Acceleration and the Derivative Page [1 of 1]
Before we get going, I want to just take a look back and try to summarize, really quickly, with sort of big picture, with big ideas what's been going on. We've been spending an awful lot of time working at sort of the micro-level and the technical ends of things and getting our hands and fingernails really deep inside these calculations. And I want us now--I invite you now--to step back and just think about the big picture, because now, the big picture is really going to be the picture we're going to focus on. And the technical details of taking the derivative will now just be one small step in a large process of doing an actual application.
So, how did this all begin? Our mission was to find instantaneous velocity, and we remembered that velocity, a rate, equals the change in distance over the change in time. And the problem was that when you plug in instant, we get zero over zero--zero over zero. So, this was the problem that got us going.
So, how do we get around that? And well, we discovered after a lot of work that the derivative gives us the answer, because the derivative represents the instantaneous rate of change. So, by taking the derivative, we, in fact, can actually resolve the question of what is instantaneous rate of change, what is this velocity.
Then, we made an interesting observation. We saw that the instantaneous rate of change has a geometrical significance, and the geometrical significance is that, in fact, well, the derivative represents the slope of a tangent line--the line that just grazes the curve. And so, armed with that, we were able to then - to march off and take a look at really pretty curves--all sorts of exotic ones.
And if someone asked, "Well, what's the slope of the tangent line right here?" Well, we're now able to find that out. We just take the derivative and compute. What's the slope of the tangent line here? No worries. I just take the derivative. What about here? Not a problem, I just take a derivative and evaluate it there. So, armed with a derivative, we were able to find both the slopes of tangent lines, and the--simultaneously, simultaneously--instantaneous rates of change all arising from the zero over zero paradox and the study of movement.
Now, what we want to do next is actually first study this notion of movement and look a little bit more closely at the notion of velocity, and even now the more--fairly more--elaborate notion, which we haven't talked about, the notion of acceleration.
And what is acceleration? Well, acceleration is just an analog, basically, of the velocity. Velocity, remember, is the rate of change of position. How my position is changing, that is velocity. Well, acceleration, in some sense, is just sort of one notch up from that. Acceleration asks, "How is my velocity changing?" So, what are changes in velocity? Remember velocity is just changes in position. But, how is my velocity changing? That's called acceleration.
So, when you're driving your car and you jam on the brakes, and you feel the car pushing up against you as you slow down, you're feeling, actually, the change in velocity. You're actually feeling the acceleration.
So, we'll talk about acceleration and velocity, take a look at applications of that kind. And then we'll actually take a look at applications of this kind. You might think, "Well, gee whiz. This I can see applications of--movement and what not, but are there really applications of finding slopes of tangent lines?" The answer is yes. We're trying to evaluate certain complicated functions before, in fact, the advent of the computer. And, in fact, one might ask how do computers actually evaluate complicated functions? Turns out these techniques are actually related to that. So, we'll take a look at the inside of computers, in some sense, and see how calculus is even on the inside of computers.
Then we'll take this notion of looking at slopes of tangent lines to see how we can actually optimize things. You have a business. You want to maximize your profits. How do you maximize profit? Well, it turns out it's basically looking at a particular tangent line. So, in fact, we'll see a lot of applications to this, and a lot of applications to the notion of movement.
So really, when you think about it, it all comes down to looking the instantaneous change of things, and the derivative really captures the spirit of that. And we finally answered the conundrum and the paradox of the zero over zero problem to the notion of limit, and then building that to the derivative.
So really, it's all speed. It's all speed--the notion of going fast, the notion of traveling. This is exactly what we're up against. And how things move, we now see, is describable by the derivative. So, up next, we're going to take a look at the velocity and the acceleration. And, in fact, my last thing I'm going to do here before I leave, I'm going to get into the car, close the door, put on my seatbelt, and I'm going to smash into the zero over zero form.
And let me just tell you a little behind-the-scenes fact here. I was challenged by the staff that if I can knock down all the pieces, I actually get the massage bug. So, I'm really excited about this. So, let's see if we can do this. This would be really great. This is live, live video. Oh, please. Should I do it at an angle, you think? Oh, I don't know. I'll do it head on--uh--I don't know. Okay, here we go. Okay. Yes! I did it! Could we have the massage bug? Where's the massage bug. I did this! This is great! The power--this is now mine. The massage bug is mine. (electronic buzzing)
Obviously the derivative pays. We'll look at applications--a whole bunch of them--up next. I've got the massage bug.