Calculus Sentence in English in textbook "Calculus", by Robert A. Adams

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The discussion revolves around a translation issue in the sixth edition of "Calculus" by Robert A. Adams, specifically in the second chapter on differentiation. The participant seeks clarification on the title of a section, suggesting it might be "Velocity and Acceleration," and requests the English version of a specific sentence from Example 2 regarding velocity continuity. There is confusion between the Mean Value Theorem and the Intermediate Value Theorem, with one participant noting that the terms can be easily mixed up due to their similar names. They explain that while the Mean Value Theorem deals with average values, the Intermediate Value Theorem pertains to the continuity of functions, emphasizing that every derivative has the intermediate value property. The conversation highlights the nuances in mathematical terminology and the importance of precise definitions in calculus.
mcastillo356
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Hi, PF

I've got a translation into spanish of the sixth edition of "Calculus", by Robert A. Adams. At the second chapter, "Differentiation", eleventh section, I would like to know: the title of the section (Could it be "Velocity and Acceleration"?); and a sentence of the Example 2 ("A point P moves along the x axis..."): at the solution of the question (d), I have something like "Velocity is continuous for all t, so, according to the Mean value theorem...". Wich would be the sentence in English?

Thanks!
 
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The velocity is continuous for all t so, by the Intermediate-Value Theorem, has a constant sign on the intervals between the points where it is 0.
 
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In my edition, it mentioned Mean Value Theorem, and it didn't make sense.

Greetings!
 
Yeah, it's probably easy to mix up 'Teorema del valor medio' and 'Teorema del valor intermedio'.
 
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The English word "mean" refers to an average, so the mean value theorem talks about the average value.
 
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actually every derivative, continuous or not, has the intermediate value property, and the proof uses the mean value theorem, or at least its equivalent version, rolle's theorem. ( a function that goes up and then down, must take the same value twice and hence must level off somewhere in between.) so many interpretations are possible.
 
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