[미적분] 미분과 적분의 관계에 대한 자료 (relationship between integration and differentiation)

5. The Relation between Integration and Differentiation.

Theorem 5.1. First Fundamental Theorem of Calculus. Let f be a function that is integrable on [a,x] for each x in [a,b]. Let c be such that a ≤ c ≤ b and define a new function A as follows:
A(x) = if a ≤ x ≤ b. Then the derivative A`(x) exists at each point x in the open interval (a,b) where f is continuous, and for such x we have A`(x) = f(x).

Theorem 5.2. Zero-Derivative Theorem. If f`(x) = 0 for each x in an open interval I, then f is constant on I.

Definition of Primitive Function. A function P is called a primitive(or an antiderivative) of a function f on an open interval I if the derivative of P is f, that is, if P`(x) = f(x) for all x in I.
- x-δ < t < x+δ, |f(t) - f(x)| < ε/2, .

Theorem 5.3. Second Fundamental Theorem of Calculus. Assume f is continuous on an open interval I, and let P be any primitive of f on I. Then, for each c and each x in I, we have
P(x) = P(c) + .
- A(x) = .
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