Name: 講座 2 | MIT 18.01 單變量微積分，2007 年秋季（Lec 2 | MIT 18.01 Single Variable Calculus, Fall 2007）
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Okay so I'd like to begin the second lecture by reminding you

and we also did a couple of computations.

We worked out that the derivative of 1 / x was -1 /

And we also computed the derivative of x to the nth

power for n = 1, 2, etc., and that turned out to be x,

So that's what we did last time, and today I

want to finish up with other points of view

almost the most important thing I'll be saying in the class.

But you'll have to think about it again when you start over

and start using calculus in the real world.

So again we're talking about what is a derivative

and this is just a continuation of last time.

So, as I said last time, we talked about geometric

interpretations, and today what we're gonna talk about

So remember we drew graphs of functions, y = f(x)

and we kept track of the change in x and here the change in y,

And then from this new point of view a rate of change,

keeping track of the rate of change of x and the rate

of change of y, it's the relative rate of change

we're interested in, and that's delta y / delta x and that

Usually we would think of that, if x were measuring time and so

the average and that's when this becomes a rate,

and the average is over the time interval delta x.

And then the limiting value is denoted dy/dx

and so this one is the average rate of change

that I'd like to discuss now and give you

Well, first of all, maybe some examples from physics here.

and then dq/dt is what's known as current.

A second example, which is probably the most tangible one,

is we could denote the letter s by distance

and then the rate of change is what we call speed.

and I just want to illustrate the second example

in a little bit more detail because I think

it's important to have some visceral sense of this notion

And I get to use the example of this very building to do that.

So let's illustrates this idea of rate of change

let's see here's the building, and here's the dot, that's

the beautiful grass out on this side of the building,

not that small when you're close to them, that

You know everything at MIT or a lot of things at MIT

And so we're going to use instead of 300 feet,

use 80 meters because that makes the numbers come out simply.

at 80 meters at time 0 and then the acceleration due to gravity

gives you this formula for h, this is the height.

So at time t = 0, we're up at the top, h is 80 meters,

And at time t = 4 you notice, 5 * 4^2 is 80.

Okay, so this notion of average change here,

so the average change, or the average speed here,

since that's-- over this time that it takes for the pumpkin

to drop is going to be the change in h divided

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講座 2 | MIT 18.01 單變量微積分，2007 年秋季（Lec 2 | MIT 18.01 Single Variable Calculus, Fall 2007）

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