Obtuse Angles: Measures And Properties Explained

Mean Value Theorem

f'(c) = (f(b) – f(a))/ (b – a)find f'(x) find f(b)- f(a))/ (b – a)solve

The Mean Value Theorem (MVT) is an important theorem in calculus that connects the concept of a derivative and the slopes of tangent lines to the idea of an average rate of change. The theorem states that for a differentiable function f(x) on an interval [a, b], there exists at least one point c in (a, b) such that the slope of the tangent line at c is equal to the average rate of change of f(x) over [a, b]:

f'(c) = (f(b) – f(a))/(b – a)

In other words, the MVT guarantees that there is at least one point in the interval where the instantaneous rate of change of the function (represented by the derivative) is equal to its average rate of change over the entire interval.

The MVT has important applications in calculus, including optimization problems, approximation of functions, and proofs of other foundational theorems. It is also intimately connected to the concept of continuity, as the theorem requires that the function be continuous on [a, b] in order to apply.

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