In a proof I have encountered, this step is used:
|sinA-sinB| < |A-B|
But it was taken as a given and not proved itself. Why is this so? Am I missing something very obvious?
No, I do not know that theorem. I know derivatives and understand the basics of integrations (well, some of the concepts like the Riemann sum). But, is there no other way to prove that the relation I posted is always true?
Note that y=x is tangent to the sin x curve at origin. is a line parallel to y=x and passing through S=(A,sinA).
I have considered only first quadrant.
Note that PQRS is a parallelogram. ALSO:
So we get B-A>sinB-sinA.
Furthermore, doesn't this prove that |sin A - sin B| |A-B| rather than |sinA - sinB| < |A-B| ?
Beside the mean theorem, if is differentiable in the interval and continuous in then there exists a number wherefore ... (see Plato's post) . To answer your questions, is indeed the correct notation and the reason you'll get is because .
Important to understand the mean value theorem is to look at the graphic interpretation of it, that will help you more to understand this theorem.