fundamental theorem of calculus

The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse operations: if a continuous function is first integrated and then differentiated, the original function is ret..

If f(x) is a continuous function on the closed interval [a, b], and if Area function is defined ..

Let f(x) be a continuous function defined on an interval [a,b]. between the limits a and b. This statement is also known as 'fundamental theorem of calculus'. We call b, the upper limit of x and a, the lower limit. If in place of F(x) we take F(x)+c as the value of the..

First Fundamental Theorem of Integral Calculus Let f(x) be a continuous function on the closed interval [a, b]. Let the area function A(x) be defined ..

We have already defined, for a continuous function f(x) on a closed interval [a, b] as the area of the region bounded by the curve y = f(x), X-axis and x= a and x = b. In other words, area of the shaded region is a function of x. The function A(x) is shown in figure below. This area function A(x) i..

If x is a rational number, then the sum (e x ) of the exponential series..

Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then (a) f is increasing on [a, b] if f '(x) > 0 for each x (a, b) (b) f is decreasing on [a, b] if f '(x) < 0 for each x (a, b) This theorem can be proved by using Mean Value Theorem. We sh..

Let f be real valued function in [a,b] such that, f is continuous in [a,b]. f is differentiable in (a,b)...

Let f be a continuous function on an interval I = [a, b]. Then, f has the absolute maximum value and f attains it at least once in I. Also, f has the absolute minimum value and attains it at least once in ..

If a > 0, then prove that..