# mvt - Mean Value Theorem One theorem of particular...

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M ean V alue T heorem One theorem of particular importance in calculus is the Mean Value Theorem. At first, I did not understand why it was so important. After taking more math classes, though, I came to a deeper appreciation of this theorem. The theorem is stated as follows: Mean Value Theorem If a function f is continuous on a closed interval [ a , b ] and differentiable on an open interval ( a , b ), then there exists c such that () fb fa f c ba = for a < c < b Before we look at the proof of the theorem, let’s understand what it is saying. In words, the theorem says that if a function f is continuous and differentiable (it looks like an ordinary curve), then between a and b there is a point where the slope of the tangent line to the curve is equal to the slope between the two points, a and b . A picture should help to clarify this concept a bit more. The red line in the figure to the left represents the line between the points a and b . a b c f H a L f H b L f H c L The blue line represents the tangent line at the point c . Notice that both slopes are the same.

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## This note was uploaded on 08/31/2011 for the course MATH 20A taught by Professor Staff during the Fall '08 term at UCSD.

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mvt - Mean Value Theorem One theorem of particular...

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