Lagrange’ s Mean Value Theorem
Let f be a real function, continuous on the closed interval [ a, b ] and differentiable in the open interval( a, b). Then, there is at least one point c in the open interval( a, b), such that
Geometrically Any chord of the curve y = f( x), there is a point on the graph, where the tangent is parallel to this chord.
Remarks In the particular case, where f( a) = f( b). The expression [ f( b) – f( a)/( b – a)] becomes zero. Thus, when f( a) = f( b), f ‗( c) = 0 for some c in( a, b).
Thus, Rolle‘ s theorem becomes a particular case of the mean value theorem.
Approximations and Errors
1. Let y = f( x) be a given function. Let Ax denotes a small increment in Δx, corresponding which y increases by Δy. Then, for small increments, we assume that