The First Fundamental Theorem of Calculus: Understanding the Relationship Between Derivatives and Integrals

TRUE or FALSE: If f’ is continuous on [3,7], then the integral from 3 to 7 f'(x)dx = f(3) – f(7)

FALSE

FALSE.

The correct statement is that if f'(x) is continuous on [3, 7], then the integral from 3 to 7 of f'(x) dx is equal to f(7) – f(3) (NOTE: the order of subtraction is important; it should be f(7) – f(3) and not f(3) – f(7)). This is known as the First Fundamental Theorem of Calculus.

The First Fundamental Theorem of Calculus states that if a function f(x) is continuous on the interval [a, b] and F(x) is an antiderivative of f(x) on that interval, then the definite integral of f(x) from a to b is equal to F(b) – F(a). In other words, if f'(x) is continuous on [a, b], then the integral of f'(x) from a to b is equal to f(b) – f(a).

To apply the First Fundamental Theorem of Calculus, we need to know that f(x) is the antiderivative of f'(x) on the interval [3, 7]. If this condition holds, then we can say that the integral from 3 to 7 of f'(x) dx is equal to f(7) – f(3).

It is important to note that the continuity of f'(x) on [3, 7] is a necessary condition for the validity of the First Fundamental Theorem of Calculus, but it is not sufficient. We also need to establish that f(x) is an antiderivative of f'(x) on the interval [3, 7].

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