Question

Mean Value Theorem - need help understanding this theorem

Answer 1

Mean Value Theorem
If f is continuous on the closed interval[a,b] and differentiable on the open interval (a,b), then there is at least one point c in (a,b) such that
\[\frac{f(b)-f(a)}{b-a}=f \prime(c)\]

Answer 2

http://en.wikipedia.org/wiki/Mean_value_theorem

Answer 3

Its pretty much an extension to Rolle's theorem, its a more generalized form.

Answer 4

f(x)=3sin2x [0, pi/4]

Answer 5

This theorem can be understood intuitively by applying it to motion: If a car travels one hundred miles in one hour, then its average speed during that time was 100 miles per hour. To get at that average speed, the car either has to go at a constant 100 miles per hour during that whole time, or, if it goes slower at one moment, it has to go faster at another moment as well (and vice versa), in order to still end up with an average of 100 miles per hour. Therefore, the Mean Value Theorem tells us that at some point during the journey, the car must have been traveling at exactly 100 miles per hour; that is, it was traveling at its average speed.

Answer 6

So set it up.\[f'(x) = \frac {f (\frac {\pi}{4}) - f(0)}{\frac {\pi}{4} - 0}\]

Answer 7

f(0)=0

Answer 8

f(pi/4)=3

Answer 9

(3-0)/(pi/4)

Answer 10

12/pi ?

Answer 11

Yes.

Answer 12

So that is what f' equals to.

Answer 13

cool thanks,
I guess I could verify it by taking the derivative

Answer 14

You have to solve for x now after setting the derivative equal to that.\[6 \cos (2x) = \frac {12}{\pi}\]

Answer 15

lets see divide both sides by 6

Answer 16

You'll need a calculator, not very nice numbers :P

Answer 17

cos(2x)=2/pi

Answer 18

yes, sometimes that happens. Unfortunately my professor doesn't allow calculators at all but keeps assigning problems that require the use of a calculator

Answer 19

lol, then you leave your answer as\[x = \frac {1}{2} \cos^{-1} (\frac {2}{\pi})\]

Answer 20

Anyways, do you know what this theorem is really saying?

Answer 21

yes another version of the derivative at a slope

Answer 22

Sorta... but no :P

Its saying that there exist a point on the graph of f(x) on the interal [a, b] whose instantaneous rate of change is equal to the average rate of change on the whole interval.

Answer 23

Well, when you take calc you'll appreciate it, don't worry ;)

Answer 24

Ah, ok, its nice that your helping your relative :)
I wish I could explore everything there is to math, but there are so much other subjects I have to learn as well ;)