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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,