Mean Value Theorem
The Mean Value Theorem (often abbreviated MVT) is considered one of the most important results in real analysis. An elegant proof of the Fundamental Theorem of Calculus can be given using Mean Value Theorem.
Let be differentiable function on the interval . The Mean Value Theorem states there exists a point such that
Note that is the slope of the secant line which passes through the graph of at points and . Let . As is continuous on since it is the linear combinatoin of the continuous functions and , and that Since is differentiable on and , by Rolle's Theorem, there exists some such that . Thus as desired.
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