Why Use a Dummy Variable in the Fundamental Theorem of Calculus?

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    Calculus Theorem
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Discussion Overview

The discussion revolves around the use of a dummy variable in the context of the Fundamental Theorem of Calculus (FTC). Participants explore the implications of using different variables in integrals and derivatives, focusing on notation and the necessity of distinguishing between the variable of integration and the limit of integration.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question why the dummy variable t is used instead of x, suggesting it may be a matter of tradition and clarity in notation.
  • Others argue that using the same variable for both the limit and the integrand leads to confusion and is not mathematically permissible.
  • A few participants propose that the dummy variable is essential for maintaining clarity in the relationship between the variable of integration and the limits.
  • One participant illustrates the potential confusion by comparing sums with dummy variables to integrals, emphasizing the importance of distinct variables.
  • There are discussions about the implications of using a constant in place of a variable in integrals, with some arguing that it renders the notation meaningless.
  • Participants also explore the relationship between indefinite and definite integrals, noting that they represent different concepts and should be expressed accordingly.

Areas of Agreement / Disagreement

Participants express differing views on the necessity and implications of using a dummy variable. While some agree on its importance for clarity, others challenge the need for it, leading to an unresolved discussion with multiple competing perspectives.

Contextual Notes

Some participants highlight limitations in notation and the potential for misunderstanding when variables are not clearly distinguished. There are also references to the need for proper hypotheses when stating theorems, indicating that the discussion is nuanced and context-dependent.

Who May Find This Useful

This discussion may be of interest to students and educators in mathematics, particularly those exploring calculus concepts, notation, and the Fundamental Theorem of Calculus.

  • #61
cyrusabdollahi said:
What the bleep??...

i think it was in empire strikes back when obi-wan tells luke not to go try save leia/han/chewie/etc & that it's a trap, he should wait until his training is done before he faces darth vader, & yoda says "yes! to obi-wan you listen!"

edit: start at the beginning. can you tell us what is wrong (if anything) with writing
\sum_{n=0}^n f(n)
 
Last edited:
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  • #62
Sure, your index starts at n=0, and goes to n. That would be like going from 0 to 0. n only ever takes on one single value.
 
  • #63
how do you fix it & explain why it works the correct way
 
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  • #64
I love all of you! No, honestly, I am sorry for being a pain in your rear. I am thankful for all your help! (to mathwonk)
 
  • #65
Well, to make it work you would have to change the n on top of the sigma to some other value. Then n can increment from zero to, let's say r. ( if we change that n on top of sigma to r.)

\sum_{n=0}^r f(n) But I am not sure about the n inside the f(n), would it always remain at zero, or would it change? I think it would change. It would increment until it reaches the value of r and stops.
 
  • #66
the d3 thing. the thing is that what the person who first poted it was trying to get across was something silly. ok? no, if g is any function then the symbol dg(x) is the same as g'(x)dx. right? but the original use of this was not to refer to 3 as a constant function but to simply a number. it is moot what the intention was. I *can* give it meaning, but that isnt' necessarily what was intended. maths isn't abuot some set of things that exist and come with notation already.
 
  • #67
^^ stop confusing cyrus :cry:

cyrusabdollahi said:
Well, to make it work you would have to change the n on top of the sigma to some other value. Then n can increment from zero to, let's say r. ( if we change that n on top of sigma to r.)

\sum_{n=0}^r f(n) But I am not sure about the n inside the f(n), would it always remain at zero, or would it change? I think it would change. It would increment until it reaches the value of r and stops.

yeah, & it's similar with \int_{t=a}^{x} g(t)dt

just like you can give \sum_{n=0}^r f(n) a name like F(r), we can give \int_{t=0}^{x} g(t)dt a name like G(x), where G'(x) = g(x). the 'dummy variable' t 'increments' from 0 until it reaches the 'value' x.
 

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