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>> [music] Calculus is all about rates.
And one main example of rates is velocity.

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Remember velocity is the derivative with
respect to time of your position.

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Velocity measures how quickly your
position is changing.

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Here's a particularly nice example that we
can look at.

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A car driving past you.
So I've drawn a diagram of the situation.

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I've placed you at the origin, so your
coordinates are 0, 0.

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And I've placed the car here at this red
dot.

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Its x-coordinate is v t.
I'm thinking of v as the velocity of the

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car, and t as the current time.
And I placed this y coordinate at B so the

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car is going to drive along the line y
equals b at a constant velocity v.

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Let's compute how far the car is from you.
I've drawn a line segment between you and

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the car.
I don't know how long that line segment

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is.
Well, imagining invisible right triangle

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here, I can use the Pythagorean theorem to
measure the length of the hypothenus if I

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know the lengths of the two legs.
This horizontal leg has length vt, the

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vertical leg has length b.
The length of this hypotenuse, the length

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of this line segment is the square root of
the sum of the squares of the lengths of

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the legs of the right triangle.
How fast is the distance between you and

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the car changing?
So I don't know how fast this, the

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distance between you and the car is
changing.

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So take a derivative with respect to t.
So this is the derivative of the square

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root of something.
And the derivative of the square root

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function is 1 over 2 square root.
So I'm going to be evaluating the

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derivative of the square root at the
inside, which is vt squared plus b squared

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times the derivative of the inside
function.

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So the derivative of bt squared plus b
squared.

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Now I'll just copy down the same thing, 1
over 2 square root, v t squared plus b

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squared times.
What's the derivative of this?

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Well, it's the derivative of vt squared.
So that's 2v squared t, plus the

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derivative of b, which is 0.
So now I could write this all together.

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I could, for instance, cancel this 2 and
this 2, and get that the derivative of the

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distance between you and the car is v
squared t over the square root of v t

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squared plus b squared.
Let's take a look at this with a graph.

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So here's a graph of that function.
Remember, that function is the derivative

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of the distance between you and the car,
so it's negative over here, because the

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car's distance to you is going down, and
it's positive over here, because the car's

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distance to you is increasing.
You can look at it here as a little tiny

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model.
Here's the car.

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Here's you right?
The car maybe starts over here and then it

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drives past and how does the distance
between you and the car change?

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Well here the distance is getting less,
less, less, less, less.

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This is as close as the car gets to you.
More, more, further, further, further away

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and that's exactly what you see from this
graph, right?

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The derivative's negative here because the
car is getting closer to you.

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It's zero here when the car's as close as
it ever gets to you and the derivative is

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positive over here because the car is
getting further away from you.

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We can hear this in one of the
explorations on the website.

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Here I've got a car, that's driving past.
And here in blue is you, the observer.

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Let's look at what happens when I turn on
the sound.

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It's higher frequency and lower frequency.
Higher, lower.

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You're hearing the derivative.