# The mean value theorem

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I need to write a paper for a calculus class, it asked me to use the mean value theorem. I have the formula for it, I have calculations, I just have no idea what any of those numbers or calculations actually mean for the function. I’m staring at the definition, and that definition sounds identical to at least 3 others.

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Has this topic not been discussed in class at all? It sounds like you have not seen MVT used before. Why do you need to write a paper on it?

The ratio (f(b) – f(a))/(b-a) for a < b is the slope of the line connecting the points (a,f(a)) and (b,f(b)). If you were graphing position f(t) against time t, then that ratio would be a change in distance divided by the corresponding change in time: the *average velocity* over the time interval [a,b]. In math and statistics, the word “mean” is a synonym for “average”. So the “mean value” in MVT refers to the left side of the equation you’re looking at.

The number f'(c) is the slope of the tangent line to y = f(x) at the point on the graph where x = c. If we were graphing position vs. time again, f'(c) would be the instantaneous velocity at time c.

Saying for a < b that we can write (f(b) – f(a))/(b-a) as f'(c) for a number c between a and b is saying the *average rate of change* of f(x) over the interval [a,b] equals the *instantaneous rate of change* of the function at some number (maybe more than one number, but at least one number) between a and b.

Graphically, this means if you slide the line through (a,f(a)) and (b,f(b)) perpendicular to its direction (so it maintains the same slope) then at some moment your line becomes a tangent line to the graph of y = f(x). At that point your original slope has become a tangent slope: (f(b) – f(a))/(b-a) has turned into f'(c) for some c between a and b: just look at the picture on the top of the Wikipedia page for MVT.

The importance of MVT in calculus is not in direct practical uses, but in the conceptual background: it *explains* mathematically many important properties of derivatives on intervals: why a function with derivative 0 on an interval is constant on the interval (which later in calculus explains the +C showing up in antiderivative formulas), why a function with positive derivative on an interval is increasing on the interval, and why a function with negative derivative on an interval is decreasing on the interval. The MVT also can be used to explain error estimates in Taylor polynomial approximations.