Conceptual question of integration and derivation

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Discussion Overview

This discussion revolves around the conceptual understanding of integration and differentiation in calculus. Participants explore the mechanics behind these operations, questioning how integration yields the area under a curve and how derivatives represent rates of change.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how the area under the curve of the function f(x) = 2x between the bounds [1,10] can be derived from F(10) - F(1), where F(x) = x^2.
  • The same participant seeks clarification on the mechanics of derivatives, specifically why multiplying the coefficient by the exponent yields the derivative function (2x) that indicates the rate of change.
  • Another participant suggests that understanding these concepts is beyond human comprehension, implying a reliance on faith in mathematical confirmation.
  • A third participant provides an explanation of the average rate of change and the limit process that leads to the derivative, as well as a visualization of integration using rectangles to approximate area under a curve.
  • There is a challenge to the dismissive tone regarding the understanding of calculus, with a suggestion to consult a calculus textbook for deeper insights.

Areas of Agreement / Disagreement

Participants express differing views on the nature of understanding calculus, with some emphasizing the need for rigorous study while others express skepticism about human comprehension of the underlying principles.

Contextual Notes

Some participants' statements reflect a lack of consensus on the fundamental understanding of calculus concepts, indicating that the discussion includes both exploratory reasoning and contested viewpoints.

BBRadiation
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Okay I have several questions that have been bothering me for some time now, so bear with me please.

  • How does integration work? Integrating the function f(x) between the bounds a,b finds the area under f(x) between a and b. But how does this work?If my function is f(x) = 2x and my bounds are [1,10], how come F(10) - F(1) equals the area under 2x from 1,10? (Where F(x) = x^2).
  • Furthermore, why does a derivative work? The derivative of the function f(x) yields a relationship between the rate of change in f(x) and a given input. But how does this actually work?

    Or put in a different way, if my function is 1x^2, why can I multiply the coefficient by the exponent to yield a function (2x) that tells me the rate at which the function will change?
  • What does it mean that time is being removed (divided in a derivative) or added (multiplied in an integral)?

Thanks for reading.
 
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BBRadiation said:
Okay I have several questions that ... how does this work?

Okay, God can tell you how it work. Human just try to compute and believe in conformation.
 
The best way to think of why it works is as follows:
the average rate of change on f(x) between x and x+h where h is some constant will be (f(x+h)-f(x))/h If we take the limit as h->0 for this expression then this calculates the instantaneous rate of change, or the derivative at x, since we are finding the rate of change between x and x+0, or x. The best way to visualise this is as a straight line between two points, A=(x,f(x)) and B=(x+h,f(x+h)), and then imagine B coming along the line closer and closer to A, the line will approach the tangent at point A, which is its rate of change at that moment.

For integration the same sort of idea can be used, this time imagine large rectangles approximating the area under a curve. As the width of these rectangles is lowered and a larger number of rectangles are used, the approximate area under the curve will become more accurate to the real area. If we take the limit as width->0 of the sum of the area of all these rectangles then you will get the area under the curve.
 
brahman said:
Okay, God can tell you how it work. Human just try to compute and believe in conformation.
That's an insult to mathematicians. Or are you saying mathematicians are not human?

On topic: these are questions that span a large portion of calculus, why don't you just take a look at a (good) calculus book (with proofs) and then ask more specific questions?
 
brahman said:
Okay, God can tell you how it work. Human just try to compute and believe in conformation.
Well, there are people, called mathematicians, who assit God in this work.
 

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