In this sub unit I'll introduce the idea of the derivative,
which is the central idea of differential calculus.
If you've had calculus recently
or feel pretty comfortable with it
you can probably skip this video
since it's pretty standard material.
So, we'll start by thinking about rates of change.
So, as an example,
suppose you take a bicycle ride for 30 minutes
and at the end of those 30 minutes you've travelled 21 kilometres,
and you might want to know: on average how fast was I going?
what was my average speed during this bicycle ride?
Well we can calculate this as follows:
So, average speed - how fast you were going during your thirty minute bicycle ride
well, that's the change in position,
how far you've travelled, over the change in time,
and that means how much time has elapsed,
how long were you bicycling for.
In symbol, that's often written delta s over delta t,
'delta', this triangle thing, means 'change in'.
s here will stand for: change in position - how far you went,
and so you went 21 kilometres in 30 minutes,
and so that would be the .7,
and the units are kilometres per minute.
Now, this doesn't mean that at
every instant you were going at exactly .7 kilometres per minute
most likely, sometimes, you're were going faster and sometimes slower,
faster down a hill, slower up the hill, faster when you feel more energetic,
slower when you're tired,
so, this is an average. You weren't going this speed the entire time,
but on average, this was your speed,
but then you might ask another question: How fast were you going, say,
10 minutes, at exactly 10 minutes into your ride?
So, this is a different sort of question.
It's asking your speed - how fast you're going
at a particular instant, not over an entire 30 minutes.
So, this is a harder and more subtle question.
So, one way you might think to answer this is:
look at your average speed for a time interval close
to this 10 minute point.
So, let me write out how you might think about that.
So, you might say: the average speed
from t equals 9 to t equals 11 minutes,
so, same idea - it's an average speed, but now the time interval is smaller,
and notice that 10 minutes is right in between 9 and 11,
and then that would be the distance travelled in this time,
in this case, not during your entire journey,
but just the journey from t equals 9 to 11,
and you'll divide that by the time that has elapsed
and that would just be 2 minutes.
So, that could be,
and just to make up a number for the sake of concreteness,
maybe in those 2 minutes, from t equals 9 to t equals 11 in your journey
let's say you went 1.2 kilometres,
divide by 2 minutes,
and let's say that would be 0.6 kilometres per minute,
so, your average speed from
t equals 9 to 11, in this case, and I'm just making up these numbers so we have
something to talk about,
is a little bit less than your average speed.
You're going a little bit slower than average here.