The Mean Value Theorem

Section 6.2 The Mean Value Theorem

Continuous functions satisfy the Intermediate Value Theorem; well, differentiable functions also satisfy their own, nice, theorem, known as the “Mean Value Theorem” (MVT). This is what we explore in this section. While it may seem daunting at first, the statement of the MVT is in the end fairly obvious. However, it has fundamental consequences. For instance, it follows from the MVT that any function whose derivative vanishes over some interval must be constant over that interval. As a consequence, it follows that two antiderivatives of a given function can only differ by the addition of a constant: a statement that we have used repeatedly! Using the MVT, we can now understand why this statement must be true.

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Objectives

You should be able to:

Explain and illustrate the Mean Value Theorem.
Relate the Mean Value Theorem to the notions of average and instantaneous velocities.
Solve simple problems using the Mean Value Theorem.

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Subsection 6.2.1 Instructional video

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Subsection 6.2.2 Key concepts

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Concept 6.2.1. Mean Value Theorem (MVT).

Let $f$ be a function that is continuous on the closed interval $[a,b]$ and differentiable over the open interval $(a,b)\text{.}$ Then the Mean Value Theorem states that there is a number $c \in (a,b)$ such that

f^{'} (c) = \frac{f (b) - f (a)}{b - a} .

$\begin{equation*} f'(c) = \frac{f(b)-f(a)}{b-a}. \end{equation*}$

Geometrically, the Mean Value Theorem is saying that there must be a $c \in (a,b)$ such that the tangent line to $y=f(x)$ at $x=c$ is parallel to the secant line between $(a,f(a))$ and $(b,f(b))\text{.}$

It can also be understood as saying that there must be a $c \in (a,b)$ at which the instantaneous rate of change of $f$ is equal to its average rate of change between $a$ and $b\text{.}$

Note that as for the Intermediate Value Theorem, $c$ need not be unique; there may be more than one $c$ satisfying the statement of the Mean Value Theorem.

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Concept 6.2.2. Rolle's Theorem.

In the special case where $f(a) = f(b)$ (the secant line is horizontal), then the statement becomes that there must be a $c \in (a,b)$ such that $f'(c) = 0$ (the tangent line is horizontal), which is called Rolle's Theorem.

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Concept 6.2.3. Important consequences of the Mean Value Theorem.

The following statements are consequences of the Mean Value Theorem:

If $f'(x)=0$ for all $x$ in some interval $(a,b)\text{,}$ then $f$ must be constant on $(a,b)\text{.}$
If $f'(x) = g'(x)$ for all $x$ in some interval $(a,b)\text{,}$ then $f-g$ must be constant on $(a,b)\text{.}$ That is, $f(x) = g(x) + C$ for some constant $C\text{.}$

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MATH 144: Calculus for the Physical Sciences I