The first part of the FTC states that the definite integral of the derivative of a function f(x) from x = a to x = b is equal to f(b) - f(a). The second part of the FTC states that the derivative of the definite integral of a function f(t) from an arbitrary value to x is the function f(x) (the same as the original function, with only the variable changed).The fundamental theorem of the calculus essentially defines the integral as the inverse operation of the derivative, analogous to multiplication being the inverse of division.
There are two versions of the mean value theorem (MVT) in calculus: one for derivatives and one for integrals. The MVT for derivatives states that for a continuous function f(x), there must be some point c in the interval [a,b] that has the same derivative value f'(c) as the secant line (f(b) - f(a)) / (b - a). The MVT for integrals states that for a continuous function f(x), there must be some point c in the interval [a,b] that has the same value as the average value of f(x) from a to b.
The derivative test theorems state that the first and second derivatives of a function provide information about the critical points of the function. Specifically, for a function f(x), the zeros of its first derivative correspond to the maximum and minimum points of the function (the sign test is used to distinguish maxima from minima). The zeros of the function's second derivative correspond to the points of inflection of the function (points where the concavity changes from positive to negative or vice versa).
The extreme value theorem states that in any interval [a,b] of a continuous function f(x), the function has both a local maximum and a local minimum on the interval. The local minimum and maximum are not necessarily the same as the function's global maximum and minimum (the absolute highest and lowest values of the function, respectively). The extreme value theorem is useful in the calculus of optimization (finding the most efficient or highest-yield value given a function or set of functions).