Question

Why the name Mean Value has came for mean value theorems ?

Answer 1

excuse me?

Answer 2

Thats Rolle's mean value theorem

Answer 3

Yeah there's a general mean value theorem and a special one.

Answer 4

My doubt is why do they used the word mean value

Answer 5

Mean means that the value is equally far from 2 different values

Answer 6

But in this the value is not equally far from 2 different values it is just a value in between them

Answer 7

Which value?

Answer 8

The value of x where the slope of tangent drawn to the curve is equal to the slope of line joining the two end points of the interval

Answer 9

"mean" is a synonym for "average" in this case

Answer 10

could be called "average value" theorem, but sometimes average means different things. mean, median, mode, all called "average" the word "mean' is more precise

Answer 11

it is more clear when you see "mean value theorem" for integrals, more like adding up and dividing my number of numbers you have. but the word "average" still applies here.
if you drive from mile marker 40 on the highway to mile marker 150 in 2 hours, your "average" speed is
\[\frac{150-40}{2}=45\] and the "mean value theorem" tells you that at some point you must have actually been going 45 mph
like saying
\[\frac{f(b)-f(a)}{b-a}=f'(c)\] for some c in the interval (a,b)

Answer 12

well actually
\[\frac{150-40}{2}=55\] but you get the idea

Answer 13

I thought the average was calculated as the sum of everything, divided by the number of everything... not the difference divided by the number of everything

Answer 14

Ya thank you very much

Answer 15

i am taking "mean" to be "average speed" in this case

Answer 16

oh alright

Answer 17

@ agdgdgdwngo
the mean value theorem for integrals looks like what you wrote

Answer 18

Yeah satellite is right, since your velocity function is continuous.