## Friday, October 10, 2014

### The Fundamental Theorem of Calculus Suppose  f  is continuous on  [a,b]. 1)    If  y=∫xaf(t)dt, then  dydx=f(x). 2)  ∫baf(t)dt=F(b)−F(a), where  F  is an antiderivative of  f. The theorem can be applied in many instances. One of which is as shown, for finding a function that increases or is concaved upwards. The function  F  is increasing on the interval(s) where its derivative  f  is positive in value.  The function  F  is concave up on the interval(s) where its derivative  f  is increasing.      Fundamental theorem of calculus:

Swapping the bounds for definite integral