>> [music] Calculus is all about rates. And one main example of rates is velocity.
Remember velocity is the derivative with respect to time of your position.
Velocity measures how quickly your position is changing.
Here's a particularly nice example that we can look at.
A car driving past you. So I've drawn a diagram of the situation.
I've placed you at the origin, so your coordinates are 0, 0.
And I've placed the car here at this red dot.
Its x-coordinate is v t. I'm thinking of v as the velocity of the
car, and t as the current time. And I placed this y coordinate at B so the
car is going to drive along the line y equals b at a constant velocity v.
Let's compute how far the car is from you. I've drawn a line segment between you and
the car. I don't know how long that line segment
is. Well, imagining invisible right triangle
here, I can use the Pythagorean theorem to measure the length of the hypothenus if I
know the lengths of the two legs. This horizontal leg has length vt, the
vertical leg has length b. The length of this hypotenuse, the length
of this line segment is the square root of the sum of the squares of the lengths of
the legs of the right triangle. How fast is the distance between you and
the car changing? So I don't know how fast this, the
distance between you and the car is changing.
So take a derivative with respect to t. So this is the derivative of the square
root of something. And the derivative of the square root
function is 1 over 2 square root. So I'm going to be evaluating the
derivative of the square root at the inside, which is vt squared plus b squared
times the derivative of the inside function.
So the derivative of bt squared plus b squared.
Now I'll just copy down the same thing, 1 over 2 square root, v t squared plus b
squared times. What's the derivative of this?
Well, it's the derivative of vt squared. So that's 2v squared t, plus the
derivative of b, which is 0. So now I could write this all together.
I could, for instance, cancel this 2 and this 2, and get that the derivative of the
distance between you and the car is v squared t over the square root of v t
squared plus b squared. Let's take a look at this with a graph.
So here's a graph of that function. Remember, that function is the derivative
of the distance between you and the car, so it's negative over here, because the
car's distance to you is going down, and it's positive over here, because the car's
distance to you is increasing. You can look at it here as a little tiny
model. Here's the car.
Here's you right? The car maybe starts over here and then it
drives past and how does the distance between you and the car change?
Well here the distance is getting less, less, less, less, less.
This is as close as the car gets to you. More, more, further, further, further away
and that's exactly what you see from this graph, right?
The derivative's negative here because the car is getting closer to you.
It's zero here when the car's as close as it ever gets to you and the derivative is
positive over here because the car is getting further away from you.
We can hear this in one of the explorations on the website.
Here I've got a car, that's driving past. And here in blue is you, the observer.
Let's look at what happens when I turn on the sound.
It's higher frequency and lower frequency. Higher, lower.
You're hearing the derivative.