What exactly constitutes as an exact value?

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Discussion Overview

The discussion revolves around the concept of what constitutes an "exact value" in the context of calculus, particularly in relation to evaluating integrals and expressing results. Participants explore the implications of using decimal approximations versus closed form expressions in mathematical answers.

Discussion Character

  • Conceptual clarification, Debate/contested, Homework-related

Main Points Raised

  • One participant expresses uncertainty about the correct interpretation of "exact value" after receiving feedback on an exam regarding the perimeter of a region defined by a curve.
  • Another participant suggests that a closed form expression is what was required for the answer, implying that this is a standard expectation in such problems.
  • A third participant provides an example, stating that while "4π" is an exact answer for the circumference of a circle, its decimal representation is not considered exact.
  • The original poster reflects on their notation and feels justified in their approach, despite the feedback received from their teacher.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition of "exact value." There are competing views on whether decimal approximations can be considered exact and how to interpret the requirements of mathematical problems.

Contextual Notes

Limitations include the lack of specific problem details and the ambiguity in the definition of "exact value" as it pertains to mathematical expressions and evaluations.

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I'm not sure if this is the correct section to post in, but since it relates to calculus and this is more of a general concept than a single problem I figured here would be fine.

On a recent exam we were given a region bounded by a simple curve generated from a function that was bounded from the y-axis to a constant. We were asked to find the exact perimeter of the region. Solving the problem involved solving for the curve and the other two linear sides. Now for the curve I understand you do the integral of the square root of the derivative plus one. I did that, but when I evaluated on the calculator it came out to a very long irrational number. I felt that including a certain number of decimals would be an approximation, not an exact number. So I left it as the value of the other two sides plus the integral. The answer they wanted was anything with at least three decimals. I was not given points as my teacher felt that I did not properly evaluate the problem. Who is the right here?
Sorry if I'm being vague. We are asked not to post the problems/solutions so I was doing my best to be clear without actually giving the problem.
 
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My guess: there is a closed form expression (what you put into the calcuator) and that is what was asked for.
 
If you were give x^2+ y^2= 4 and asked to find the circumference, "4\pi would be an "exact" answer. "12.566370614359172953850573533118" would not be.
 
Okay, thank you to both of you. The answer could not be written out, so I still think I was correct in the notation I used. That being said, I don't make the decisions so I'll just have to live with things.
 

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