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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,