2 ms·

So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have

So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? Unfortunately, I don't have an informal notion of growth rate in my head :/

mac01021 7y ago
Yes, that's right. The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition. Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?
nabla9 7y ago
If you would learn this rigourously like mathematicians do, you learn it trough infinite series and limits. The derivate is defined as limit for the slope when the endpoints of the slope get closer and closer together () Lots of work, proofs are needed to show that this actually works. Common person or engineer only needs to accept/trust that point in a curve has well defined 'slope' and it's not an approximation.
Sniffnoy 7y ago
> So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.

mac01021 7y ago

Yes, that's right. The growth over any finite window, if you partition it into smaller windows, is the sum of the growth within each partition. Draw enough pictures and you'll develop the intuition that, if you keep partitioning smaller and smaller, you'll reach a point where the average growth rate across a partition is never going to change very much by subpartitioning further. If instantaneous growth rate is going to be defined at all, it has to be very close to the average rate over that tiny interval, no?

nabla9 7y ago

If you would learn this rigourously like mathematicians do, you learn it trough infinite series and limits. The derivate is defined as limit for the slope when the endpoints of the slope get closer and closer together () Lots of work, proofs are needed to show that this actually works. Common person or engineer only needs to accept/trust that point in a curve has well defined 'slope' and it's not an approximation.

Sniffnoy 7y ago

> So you're just drawing a line from start to end and calling that the velocity? That just averages the whole thing out, doesn't it? No. We're talking about instantaneous velocity. You know, the thing the speedometer displays. How fast is the car moving at any given moment? Like, a car doesn't need to be moving at constant speed for a speedometer to give meaningful information, right? Sometimes it is moving faster and sometimes it is moving slower. Sometimes it is moving at a rate such that if it stayed at that rate it would go 60 miles in an hour, and sometimes it is moving at a rate such that if it stayed at that rate it would go 30 miles in an hour. This is the informal notion of instantaneous velocity you should already have. Now the question becomes, how do we formalize this? Which is what the page is trying to answer.