# Important Theorems in Calculus

Calculus allows us to get analyses of a function’s behavior over the time. Calculus includes the derivatives and integrals. The basic theorem in calculus defines the relationship between derivatives and integrals. There are several theorems that define important features of the derivative and integral values of functions.

## Fundamental Theorem of Calculus

The first part of Fundamental Theorem of Calculus says that definite integral of the derivative of  a function f(x) from x=a to x=b is equal to f(b) – f(a). Second part of this theorem says that the derivative of the definite integral of a function f(t) from an arbitrary value to x is the function f(x) where the calculus defines the integral as the inverse operation of the derivative, analogous to multiplication that is inverse of division.

## Mean Value Theorem

Mean Value Theorem has two versions. One is for derivatives and one is for integrals. Mean Value Theorem for derivatives says 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). Mean Value Theorem for integrals says 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.

## Derivative Tests

Derivative test theorems says that the first and second derivatives of some function gives information about the critical points of that function. If we have function f(x) the zeros of its first derivative correspond to the maximum and minimum points of the function. The Zeros of the function’s second derivative correspond to the points of inflection of that function which are points where the concavity changes from positive to negative or conversely.

## Extreme Value Theorem

The extreme value theorem says that in any interval [a,b] of some continuous function f(x) the function has both a local maximum and a local minimum on the interval. Local minimum and maximum are not the same as the function’s global minimum and maximum. Extreme value theorem is useful in the calculus of optimization, where you find the most efficient or highest yield value which is given  a function or set of functions).