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

[mcastillo356]

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!

 

[Dragon27]

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.

 

[mcastillo356]

In my edition, it mentioned Mean Value Theorem, and it didn't make sense.

Greetings!

 

[Dragon27]

Yeah, it's probably easy to mix up 'Teorema del valor medio' and 'Teorema del valor intermedio'.

 

[Dr.D]

The English word "mean" refers to an average, so the mean value theorem talks about the average value.

 

[mathwonk]

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.