What is the Mean Value Theorem and its applications in calculus?

[moham_87]

again, I've another question i wish it is the last.
it is about "Mean Value Theorem",

* if u and v are any real numbers, then, prove that:
|sin(u)-sin(v)|<=|u-v|

* Prove that [(1+h)^(1/2)] < [1+(h/2)], for h>0

* Suppose that f'(x)=g'(x)+x for every (x) in some interval (I), how different can the function (f) and (g) be

I don't know from where to start and i would like you to know that I'm in exams' days, and that's not assignment

thank you a lot for your efforts

 

[PrudensOptimus]

Originally posted by moham_87
again, I've another question i wish it is the last.
it is about "Mean Value Theorem",

* if u and v are any real numbers, then, prove that:
|sin(u)-sin(v)|<=|u-v|

* Prove that [(1+h)^(1/2)] < [1+(h/2)], for h>0

* Suppose that f'(x)=g'(x)+x for every (x) in some interval (I), how different can the function (f) and (g) be

I don't know from where to start and i would like you to know that I'm in exams' days, and that's not assignment

thank you a lot for your efforts

loll... how old are you kid?

 

[HallsofIvy]

Prudens Optimus, why "lol"? These seem like reasonable questions to me.

moham_87, since you say that these are about the "mean value theorem", how about using that?

Mean Value Theorem: "If f is continuous on [a,b] and differentiable on (a,b) then there exist c in [a,b] such that
f'(c)= (f(b)- f(a))/(b-a)."

In the first problem, f(x)= sin(x). What is f'(x)? What is the largest possible value of f'(x)?

In the second problem, f(x)= (1+h²)^1/2. What is f'(x)? What is the largest possible value of f'(x)?

In the third problem, if f'(x)=g'(x)+x , what is f(x)?