## The Fundamental Theorem of Calculus

### If f is continuous on [a,b], 1. g(x) = ∫ f(t) dt on [a,x] is continuous on [a,b]; and g'(x) = f(x)2. ∫ f(x) dx on [a,b] = F(b) – F(a)

The Fundamental Theorem of Calculus is a theorem that connects differentiation and integration. It has two parts: the first part relates the derivative of an antiderivative to the original function, while the second part provides a method for computing definite integrals.

Part One:

Suppose f is a continuous function on an interval [a,b] and let F be an antiderivative of f. Then, the derivative of F, denoted by F'(x), is equal to f(x), where x is any point in the interval [a,b]. This can be written as:

d/dx ∫a^x f(t)dt = f(x)

Part Two:

Suppose f is a continuous function on an interval [a,b], and let F(x) = ∫a^x f(t)dt be the antiderivative of f. Then, the definite integral of f from a to b can be computed using the formula:

∫a^b f(x)dx = F(b) – F(a)

This means that to evaluate a definite integral, we do not need to find an antiderivative explicitly. Instead, we can simply evaluate F at the two endpoints of the interval and subtract these values.

The Fundamental Theorem of Calculus is a powerful tool in calculus as it allows us to switch between differentiation and integration. It also provides a method for computing definite integrals without having to find an antiderivative explicitly.

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