Which Came First? (historically)

[NSX]

Limits, derivatives, or integration?

 

I thought it was that exact order, but my friend said the order was:

 

Integration

Derivatives

limits

 

why?

 

He says "they just tend to find stuff first and then come up with the reasoning later

 

I'm stumped.

Anyone know?

[Dave]

I don't know whether you've ever found derivatives of functions by first principles, but if you did then you'd see that your order is correct.

 

The idea is to take two points on a curve, joining them up with a line. If you keep one point fixed, moving the other one closer to it, then as you move them closer and closer to each other, you can obtain the limit of the line joining the two points. Then when the seperation tends to zero, the gradient of your line tends to the gradient of the curve at that point.

 

Indeed, the derivative is defined using limits:

 

dy/dx = lim_h->0 [ (f(x+h) - f(x))/h ]

 

I'm pretty certain integration came afterwards, using the same kind of principles. Could be wrong about that though.

 

Hope this helps.

[NSX]

dave said in post #2 :

I don't know whether you've ever found derivatives of functions by first principles, but if you did then you'd see that your order is correct.

 

The idea is to take two points on a curve, joining them up with a line. If you keep one point fixed, moving the other one closer to it, then as you move them closer and closer to each other, you can obtain the limit of the line joining the two points. Then when the seperation tends to zero, the gradient of your line tends to the gradient of the curve at that point.

 

Indeed, the derivative is defined using limits:

 

dy/dx = lim_h->0 [ (f(x+h) - f(x))/h ]

 

I'm pretty certain integration came afterwards, using the same kind of principles. Could be wrong about that though.

 

Hope this helps.

 

THat's what I thought to.

 

Afterall, the def'n of the derivative is:

*

 

How would that be possible w/out limits?

 

*http://mathworld.wolfram.com/Derivative.html

[Dave]

It's not, is the short answer.

[NSX]

dave said in post #4 :

It's not, is the short answer.

 

To this: (?)

I thought it was that exact order, but my friend said the order was:

 

Integration

Derivatives

limits

[Dave]

NSX said in post #3 :

*

 

How would that be possible w/out limits?[/i]

 

It's not, is the short answer.

 

And that other order just can't be right.

[NSX]

Okay. I get ya know dave.

 

Although, would it be possible that they found areas under curves empirically?

 

ie., I think it was Archimedes that estimated through exhaustion the area of a cricle by alot of triangles.

 

Just like the Riemann sums (ignore the limit part for the sake of the argument). Someone just made alot of rectangles under a curve and saw the trend? Then made a model for the data, much like eq'ns are created to model empirical data?

 

After areas were done, they did the same for slopes?

 

Then found that you get closer and closer to a value, thus, creating the limit?