2 Rolle’s TheoremLet f be a function that satisfies the following three conditions:f is continuous on the closed interval [a,b] .f is differentiable on the open interval (a,b) .f(a) = f(b) .Then there is a number c in (a,b) such that f ′(c)=0.Examples on the board.
3 If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b:Mean Value Theorem for DerivativesThe Mean Value Theorem says that at some point in the interval, the actual slope equals the average slope.Note: The Mean Value Theorem only appliesover a closed interval.
4 An illustration of the Mean Value Theorem. Tangent parallel to chord.Slope of tangent:Slope of chord:
5 Corollaries of the Mean Value Theorem Corollary 1: If f ′(x)=0 for all x in an interval (a,b), then f is constant on (a,b) .Corollary 2: If f ′ (x) = g′ (x) for all x in an interval (a,b), then f - g is constant on (a,b) , that is f(x)= g(x) + c where c is a constant.(see the next slide for an illustration of Corollary 2)
6 These two functions have the same slope at any value of x. Functions with the same derivative differ by a constant.