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 (FTC) is a crucial concept in calculus that establishes the relationship between differentiation and integration. The theorem states that:
If a function f(x) is continuous over an interval [a, b], and F(x) is an anti-derivative (or indefinite integral) of f(x), then the definite integral of f(x) from a to b (written as ∫[a,b]f(x)dx) is equal to the difference between the values of F(x) at the endpoints of the interval, i.e., F(b) – F(a).
In simpler terms, the FTC provides us with a method to evaluate definite integrals using anti-derivatives. Essentially, the theorem tells us that the area under the curve of a function (i.e., the definite integral) can be found by calculating the difference in the values of an anti-derivative of the function at the endpoints of the integral.
The FTC is split into two parts – the first part relates the derivative of the integral of a function to the function itself, while the second part establishes the relationship between the definite integral of a function and the anti-derivative of the same function.
The FTC provides an immensely powerful tool for solving problems in calculus, and many other branches of mathematics and science. It allows us to link concepts such as instantaneous rate of change, accumulation of values, and area under the curve in a unified way, and helps us to better understand the behaviour of complex mathematical functions.
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