Two questions about the Total Change Theorem

[Skepticuss]

1) Just exactly what totally changes?

2) Why is it called the "Total Change Theorem", not the ""Net Change Theorem"?

Thx

 

[Cuddles]

What is the Total Change Theorem?

 

[andyandy]

The Total Change Theorem
The total change theorem is an adaptation of the second part of the Fundamental Theorem of Calculus. The Total Change Theorem states: the integral of a rate of change is equal to the total change.

If we know that the function f(x) is the derivative of some function F(x), then the definite integral of f(x) from a to b is equal to the change in the function F(x) from a to b.

so if you had a function that described the motion of a snooker ball - moving along a straight line at a certain velocity - then by integrating that function, and substituting values, one could find the displacement of the snooker ball in a given time frame. So you could know how far it had moved......

the website http://www.nipissingu.ca/calculus/tutorials/integrals.html
talks you through it.....

and gives an example to follow through here....
javascript:solution('examples/integrals/total_change/total_change.html');

 

[Skepticuss]

What is the Total Change Theorem?

I'm new here. If your question is indicative of the candle-power here (or lack of cp), then I am getting very skeptical that there is much enlightment to be found here. I had hoped to be able to converse with skeptics of great depth as well as breadth.

 

[boooeee]

I googled it. "Total Change Theorem" in quotes only gets 286 hits, which leads me to believe that the terminology is not widespread (I had never heard of it, not that that counts for much).

It seems to be another name for the second part of the fundamental theorem of calculus: "The integral of a rate of change is equal to the total change".

In reference to the questions in the OP, what is changing is whatever you took your derivative of. For example, if you integrate velocity over time, you get the change in distance, because velocity measures the rate of change in distance.

In regards to the second question, I agree that Net Change Theorem would be more appropriate, but mathematics is filled with inaccurate naming conventions (e.g. Fermat's Last "Theorem"). It is more important that everybody use the same name for the same theorem, whether it be "Total Change Theorem" , "Net Change Theorem", or "Duane".

 

[Skepticuss]

Andyandy and Boooeee, thanks a lot for your very helpful responses!

 

[andyandy]

I'm new here. If your question is indicative of the candle-power here (or lack of cp), then I am getting very skeptical that there is much enlightment to be found here. I had hoped to be able to converse with skeptics of great depth as well as breadth.

no need to be rude.....i think you'll find Cuddles is rather well thought of.


edit

let's keep things civil, and welcome to the forum

 

[boooeee]

no need to be rude.....i think you'll find Cuddles is rather well thought of.


edit

let's keep things civil, and welcome to the forum

Seconded. "Total Change Theorem" doesn't seem to be a very commonplace designation, so I think Cuddles asked a fair question. Maybe there's a "Total Change Theorem" in topology too.

 

[Skepticuss]

In regards to the second question, I agree that Net Change Theorem would be more appropriate, but mathematics is filled with inaccurate naming conventions (e.g. Fermat's Last "Theorem"). It is more important that everybody use the same name for the same theorem, whether it be "Total Change Theorem" , "Net Change Theorem", or "Duane".

Your point perfectly understandable and yet...

Each successive instance of the rate of change f(x) and derivative F'(x) associated with x's continuous "journey" (from x = a to x = b) is a partial change.

And the sum of all of those "partial changes" is a "total" change in the sense that any sum is a total.

What seems to argue for the term "net" instead of the term "total" is that a and b and F(a) and F(b) are not necessarily the first and last instances of the independent and dependent variables of the functions in question.

So the term "total" seems wrong, and the term "net" seems better.

But perhaps "total" is not so much wrong as seeming not right until we contrast "total" with "partial" instead of using another standard of scrutiny?

And considering how much math is taught by light-weights, maybe the name of that theorem fell out of favor because the average Calculus teacher got tired of wrestling with this "Why 'total' " question!

 

[Blert]

...And the sum of all of those "partial changes" is a "total" change in the sense that any sum is a total.

What seems to argue for the term "net" instead of the term "total" is that a and b and F(a) and F(b) are not necessarily the first and last instances of the independent and dependent variables of the functions in question.

So the term "total" seems wrong, and the term "net" seems better.

But perhaps "total" is not so much wrong as seeming not right until we contrast "total" with "partial" instead of using another standard of scrutiny?

...

Hmmm... I teach this stuff regularly, and usually look for several sources, and this is the first I've heard of this name; my ignorance seems boundless...

"Total Change" most likely refers to the total change in the function on the interval [a,b]---note that the "Mean Value Theorem" (whichever version you like) only calculates the "Mean Value" on a given interval.

I agree, though, "Net Change" is probably more accurate.

 

[Skepticuss]

Hmmm... I teach this stuff regularly, and usually look for several sources, and this is the first I've heard of this name; my ignorance seems boundless...

"Total Change" most likely refers to the total change in the function on the interval [a,b]---note that the "Mean Value Theorem" (whichever version you like) only calculates the "Mean Value" on a given interval.

I agree, though, "Net Change" is probably more accurate.

An "alter-ego"/close buddy of mine recently gave me this as a way of looking at the Total Change Theorem. While perhaps not totally mathematical, I think it is a good approximation. What do you think?

Suppose that you have a regular trigon (AKA, triangle) F with three vertices and three sides, with each side seen as a vector originating from its associated vertex. I.e., each side seen as having a specific length -- and direction.

Per the trigon, draw a straight line and mark off three equal lengths. And then at each mark, indicate one of the trigon's successive vectors.

My alter-ego tells me that this is analogous to the TCT's f. And that the trigon with its vectorial sides is analogous to the TCT's F. And that as the sides/vectors of the polygon increase from three to infinity, the successive approximations of the above f's and F's approach the TCT's real f and F.

If he is correct, the above seems to me to be a cool way of showing not only how f is related to F and F', but also how you can have a total change in displacement of O!

 

[dogjones]

I'm new here. If your question is indicative of the candle-power here (or lack of cp), then I am getting very skeptical that there is much enlightment to be found here. I had hoped to be able to converse with skeptics of great depth as well as breadth.

So, didya see the game last night?