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Okay so I'd like to begin the second lecture by reminding you

what we did last time.

So last time, we defined the derivative

as the slope of a tangent line.

So that was our geometric point of view

and we also did a couple of computations.

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

x^2.

And we also computed the derivative of x to the nth

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

I'm sorry, nx^(n-1).

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

want to finish up with other points of view

on what a derivative is.

So this is extremely important, it's

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

is rate of change as an interpretation

of the derivative.

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,

let's say.

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

has another interpretation.

This is the average change.

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

and this one is the instantaneous rate.

Okay, so that's the point of view

that I'd like to discuss now and give you

just a couple of examples.

So, let's see.

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

So q is usually the name for a charge,

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

So that's one physical example.

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.

So those are the two typical examples

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

of instantaneous speed.

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

Probably you know, or maybe you don't,

that on Halloween there's an event that

takes place in this building or really

from the top of this building which

is called the pumpkin drop.

So let's illustrates this idea of rate of change

with the pumpkin drop.

So what happens is, this building-- well

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

the beautiful grass out on this side of the building,

and then there's some people up here

and very small objects, well they're

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

get dumped over the side there.

And they fall down.

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

are physics experiments.

That's the pumpkin drop.

So roughly speaking, the building

is about 300 feet high, we're down here

on the first usable floor.

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

just for convenience purposes we'll

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

So we have the height which starts out

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,

the units here are meters.

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

I picked these numbers conveniently so

that we're down at the bottom.

Okay, so this notion of average change here,

so the average change, or the average speed here,

maybe we'll call it the average speed,

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

to drop is going to be the change in h divided

by the change in t.

Which starts out at, what does it start out as?

It starts out as 80, right?

And it ends at 0.

So actually we have to do it backwards.

We have to take 0 - 80 because the first value is

the final position and the second value

is the initial position.

And that's divided by 4 - 0; times 4 seconds

minus times 0 seconds.

And so that of course is -20 meters per second.

So the average speed of this guy is 20 meters a second.

Now, so why did I pick this example?

Because, of course, the average, although interesting,

is not really what anybody cares about who

actually goes to the event.

All we really care about is the instantaneous speed

when it hits the pavement and so that's can

be calculated at the bottom.

So what's the instantaneous speed?

That's the derivative, or maybe to be

consistent with the notation I've been using so far,

that's d/dt of h.

All right?

So that's d/dt of h.

Now remember we have formulas for these things.

We can differentiate this function now.

We did that yesterday.

So we're gonna take the rate of change and if you take a look

at it, it's just the rate of change of 80 is 0,

minus the rate change for this -5t^2, that's minus 10t.

So that's using the fact that d/dt of 80 is equal to 0

and d/dt of t^2 is equal to 2t.

The special case...

Well I'm cheating here, but there's

a special case that's obvious.

I didn't throw it in over here.

The case n = 2 is that second case there.

But the case n = 0 also works.

Because that's constants.

The derivative of a constant is 0.

And then the factor n there's 0 and that's consistent.

And actually if you look at the formula above it

you'll see that it's the case of n = -1.

So we'll get a larger pattern soon enough with the powers.

Okay anyway.

Back over here we have our rate of change

and this is what it is.

And at the bottom, at that point of impact,

we have t = 4 and so h', which is the derivative,

is equal to -40 meters per second.

So twice as fast as the average speed here,

and if you need to convert that, that's about 90 miles an hour.

Which is why the police are there at midnight on Halloween

to make sure you're all safe and also why when you come

you have to be prepared to clean up afterwards.

So anyway that's what happens, it's 90 miles an hour.

It's actually the buildings a little taller,

there's air resistance and I'm sure you

can do a much more thorough study of this example.

All right so now I want to give you a couple of more examples

because time and these kinds of parameters and variables

are not the only ones that are important for calculus.

If it were only this kind of physics that was involved,

then this would be a much more specialized subject than it is.

And so I want to give you a couple of examples that don't

involve time as a variable.

So the third example I'll give here

is-- The letter T often denotes temperature,

and then dT/dx would be what is known as the temperature

gradient.

Which we really care about a lot when

we're predicting the weather because it's that temperature

difference that causes air flows and causes things to change.

And then there's another theme which

is throughout the sciences and engineering which

I'm going to talk about under the heading of sensitivity

of measurements.

So let me explain this.

I don't want to belabor it because I just

am doing this in order to introduce you

to the ideas on your problem set which

are the first case of this.

So on problem set one you have an example

which is based on a simplified model of GPS,

sort of the Flat Earth Model.

And in that situation, well, if the Earth is flat

it's just a horizontal line like this.

And then you have a satellite, which is over here, preferably

above the earth, and the satellite or the system

knows exactly where the point directly below the satellite

is.

So this point is treated as known.

And I'm sitting here with my little GPS device

and I want to know where I am.

And the way I locate where I am is

I communicate with this satellite by radio signals

and I can measure this distance here which is called h.

And then system will compute this horizontal distance which

is L. So in other words what is measured,

so h measured by radios, radio waves and a clock,

or various clocks.

And then L is deduced from h.

And what's critical in all of these systems

is that you don't know h exactly.

There's an error in h which will denote delta h.

There's some degree of uncertainty.

The main uncertainty in GPS is from the ionosphere.

But there are lots of corrections

that are made of all kinds.

And also if you're inside a building

it's a problem to measure it.

But it's an extremely important issue,

as I'll explain in a second.

So the idea is we then get at delta

L is estimated by considering this ratio delta L/delta

h which is going to be approximately

the same as the derivative of L with respect to h.

So this is the thing that's easy because of course it's

calculus.

Calculus is the easy part and that

allows us to deduce something about the real world that's

close by over here.

So the reason why you should care about this quite a bit

is that it's used all the time to land airplanes.

So you really do care that they actually

know to within a few feet or even closer where your plane is

and how high up it is and so forth.

All right.

So that's it for the general introduction

of what a derivative is.

I'm sure you'll be getting used to this

in a lot of different contexts throughout the course.

And now we have to get back down to some rigorous details.

Okay, everybody happy with what we've got so far?

Yeah?

Student: How did you get the equation for height?

Professor: Ah good question.

The question was how did I get this equation for height?

I just made it up because it's the formula from physics

that you will learn when you take 8.01 and, in fact,

it has to do with the fact that this is the speed if you

differentiate another time you get

acceleration and acceleration due to gravity

is 10 meters per second.