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.
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.
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