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This section delves into two foundational theorems in calculus: Rolle's Theorem and the Mean Value Theorem (MVT). Rolle's Theorem states that for a continuous function on a closed interval [a, b], which is also differentiable on the open interval (a, b) and where f(a) equals f(b), there exists at least one point c in (a, b) such that the derivative f'(c) equals zero. The MVT extends this concept, asserting that under similar conditions, there exists a point c in (a, b) where the instantaneous rate of change (derivative) equals the average rate of change over the interval.
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Section 3.2 The Mean Value Theorem
ROLLE’S THEOREM Theorem: Let f be a function that satisfies the following three hypotheses: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). 3. f (a) = f (b). Then there is a number c in (a, b) such that f′(c) = 0.
THE MEAN VALUE THEOREM (MVT) Theorem: Let f be a function that satisfies the following two hypotheses: 1. f is continuous on the closed interval [a, b]. 2. f is differentiable on the open interval (a, b). Then there is a number c in (a, b) such that or, equivalently,
TWO THEOREMS Theorem: If f′(x) = 0 for all x in an interval (a,b), then f is a constant on (a, b). Theorem: If f ′(x) = g′(x) for all x in an interval (a, b), then f – g is a constant on (a, b); that is f(x) = g(x) + c,where c is a constant.
