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Math Help - A problem about understanding the underlying conceptual issues of calculus.

  1. #1
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    A problem about understanding the underlying conceptual issues of calculus.

    I have a problem in understanding the underlying conceptual issues of calculus. I will use an example to illustrate my problem.

    One definition of the the Mean Value Theorem is "If f is a continuous function on [a, b] (closed set) which is differentiable in (a, b) (open set), then there exists a point x belongs to (a, b) such that f(b) - f(a) = (b-a) f'(x)".

    The expression that "if f is a continuous function on [a, b] which is differentiable in (a, b)" is also used in other mean value theorems as well as in other mathematical concepts.

    Can someone explain the underlying rationality of this expression? Why do we need this expression so that various mean value theorems as well as other mathematical concepts can hold?

    Thank you very much in advance XD
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by Real9999 View Post
    I have a problem in understanding the underlying conceptual issues of calculus. I will use an example to illustrate my problem.

    One definition of the the Mean Value Theorem is "If f is a continuous function on [a, b] (closed set) which is differentiable in (a, b) (open set), then there exists a point x belongs to (a, b) such that f(b) - f(a) = (b-a) f'(x)".

    The expression that "if f is a continuous function on [a, b] which is differentiable in (a, b)" is also used in other mean value theorems as well as in other mathematical concepts.

    Can someone explain the underlying rationality of this expression?

    Why do we need this expression so that various mean value theorems as well as other mathematical concepts can hold?
    You could examine the proof of the mean value theorem more closely to figure out where and how, exactly, that particular requirement, that f be continuous on [a,b], is used.

    But I'll tell you right here why that assumption is needed: it is needed in order to prove that the function f(x), or, rather the function f(x)-\left\{\frac{f(b)-f(a)}{b-a}(x-a)+f(b)\right\} assumes a largest and a smallest value somewhere on that interval [a,b] (aka. "Maximum Value Theorem").
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  3. #3
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    Isn't it clear that if you are going to talk about a property involving f'(c), then the derivative, f' must exist- that is, that f is differentiable? And, in order to be differentiable, f must be continuous?
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    Great! Thank you very much!
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  5. #5
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    Here's an example on why we need differentiability. Find c such that

    f'(c) = \dfrac{f(b) - f(a)}{b-a} where the interval is [-1,1] and the function f(x) =| x |.
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