Editing 2042: Rolle's Theorem
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==Explanation== | ==Explanation== | ||
+ | {{incomplete|Go a little bit more into the explanation.Explain the museum reference. Do NOT delete this tag too soon.}} | ||
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In mathematics, a {{w|differentiable function}} is a function that is "smooth" everywhere, without any sudden breaks or pointy "kinks" or similar. The derivative of such a function is a new function that represents the "slope" or "rate of change" of the original. The function in this comic curves up from point (a) until a point above (c), smoothly turns around, and then curves down from (c) to (b). As a result, the derivative of this function is positive from (a) to (c), and then is negative from (c) to (b). At (c) itself, the function is "flat": the more one zooms in, the more horizontal it looks. The function is moving neither up nor down, so the derivative is neither positive nor negative, but zero. This is what ''f'(c) = 0'' means, as ''f''' is a common notation for the derivative of the function ''f'' in {{w|differential calculus}}. | In mathematics, a {{w|differentiable function}} is a function that is "smooth" everywhere, without any sudden breaks or pointy "kinks" or similar. The derivative of such a function is a new function that represents the "slope" or "rate of change" of the original. The function in this comic curves up from point (a) until a point above (c), smoothly turns around, and then curves down from (c) to (b). As a result, the derivative of this function is positive from (a) to (c), and then is negative from (c) to (b). At (c) itself, the function is "flat": the more one zooms in, the more horizontal it looks. The function is moving neither up nor down, so the derivative is neither positive nor negative, but zero. This is what ''f'(c) = 0'' means, as ''f''' is a common notation for the derivative of the function ''f'' in {{w|differential calculus}}. | ||
A {{w|theorem}} in mathematics is a statement that has been ''proven'' from former accepted statements, like other theorems or {{w|axiom}}s. This comic references {{w|Rolle's theorem}}. The theorem essentially states that, if a smoothly changing function has the same output at two different inputs, then it must have one or more turning points in between, as the derivative is zero at each one. As a special case, should the function remain flat between the two inputs, then its derivative is actually zero for every point between the inputs. To [[Randall]], this is obvious. However, the proof of this theorem is not as obvious as the result. | A {{w|theorem}} in mathematics is a statement that has been ''proven'' from former accepted statements, like other theorems or {{w|axiom}}s. This comic references {{w|Rolle's theorem}}. The theorem essentially states that, if a smoothly changing function has the same output at two different inputs, then it must have one or more turning points in between, as the derivative is zero at each one. As a special case, should the function remain flat between the two inputs, then its derivative is actually zero for every point between the inputs. To [[Randall]], this is obvious. However, the proof of this theorem is not as obvious as the result. | ||
− | + | In the title text, Randall mentions a line together with a ''coplanar'' circle. This simply means that both those two-dimensional objects must lay in the same plane in a higher, three-or-more-dimensional space. And by this means, every line drawn through the center of a circle is just a diameter which divides it into two equal parts. It is interesting to note that this theorem: even if it is trivial, {{w|Proclus}} says that the first man who proved it was {{w|Thales of Miletus|Thales}}. Auctioning of {{w|naming rights}}, also noted in the title text, refers to the practice of naming entertainment venues for companies which pay for the privilege, such as any of the three {{w|Red Bull Arena}}s or {{w|Quicken Loans Arena}}. The naming of mathematical theorems (along with lemmas, equations, laws, methods, etc.) is [http://www.maa.org/external_archive/devlin/devlin_09_05.html not always straightforward] and {{w|List of misnamed theorems|often results in misleading names}}. | |
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− | In the title text, Randall mentions a line together with a ''coplanar'' circle. This simply means that both those two-dimensional objects must lay in the same plane in a higher, three-or-more-dimensional space. And by this means, every line drawn through the center of a circle is just a diameter which divides it into two equal parts. | ||
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==Transcript== | ==Transcript== | ||
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:Rolle's theorem states that any real, differentiable function that has the same value at two different points must have at least one "stationary point" between them where the slope is zero. | :Rolle's theorem states that any real, differentiable function that has the same value at two different points must have at least one "stationary point" between them where the slope is zero. | ||
− | :[The graph shows a sine like curve in blue intersecting the x-axis at points "a" and "b" marked in red while in the middle a point "c" has a vertical dashed green line to the apex and on top also in green f'(c)=0 is drawn with a horizontal | + | :[The graph shows a sine like curve in blue intersecting the x-axis at points "a" and "b" marked in red while in the middle a point "c" has a vertical dashed green line to the apex and on top also in green f'(c)=0 is drawn with a horizontal line.] |
:[Caption below the frame:] | :[Caption below the frame:] | ||
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[[Category:Comics with color]] | [[Category:Comics with color]] | ||
[[Category:Line graphs]] | [[Category:Line graphs]] | ||
− | [[Category: | + | [[Category:Math]] |
[[Category:Wikipedia]] | [[Category:Wikipedia]] |