Calculus Theorems

casillas's version from 2016-11-02 16:00

Section

Question Answer
Intermediate Value TheoremIf f is continuous on [a,b] and L is any number between f(a) and f(b), then there is a number c that is between a and b so that f(c)=L.
Root Location TheoremIf f is a continuous function on [a,b] and if f(a) and f(b) have opposite signs, then f(c)=0 for at least one number c on the open interval (a,b).
Extreme Value TheoremIf f is continuous on a closed and bounded interval [a,b] then f has both and absolute maximum and an absolute minimum. Absolute extrema of a continuous function on a closed interval exist.
Critical Number TheoremIf a continuous function f has a relative extremum at x=c, then either f ' (c)=0 or f ' (c) does not exist. Absolute extrema can only occur at critical points or at the endpoints a,b.
Rolle's TheoremLet f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If f(a) = f(b), then there is at least one number c in (a,b) such that f ' (c)=0.
Mean Value TheoremIf f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a least one number c in (a,b) such that [(f(b)-f(a))/(b-a)]=f ' (c)
Zero-Derivative TheoremIf f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) with f ' (x) = 0 for all x in (a,b), then f is constant on [a,b]
Constant Difference TheoremSuppose f and g are continous on the closed interval [a,b] and differentiable on the open interval (a,b). If f ' (x) = g ' (x) for all x in (a,b), then there i a constant c so that f(x)=g(x)+c for all x on [a,b]