Math for Non-Geeks/ Mean value theorem

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The mean value theorem is one of the central theorems of differential calculus. It states (roughly speaking) that the slope of the secant between two different points of a differentiable function somewhere between these two points is assumed to be a derivative. Thus, the mean value theorem links the secant slope with the derivative of a function. Global properties, which can be expressed using the secant slope, can thus be traced back to properties of the derivative using the mean value theorem. In the section „Schrankensatz“ we will examine a useful application. Others then follow in the chapters constant functions, derivative and local extrema and L'Hospital's rule. The main theorem of differential and integral calculus is also based on the mean value theorem.

Motivation

File:Mittelwertsatz der Differentialrechnung - Erklärung und Beispiele.webm We've already viewed at Rolle's theorem. To repeat: Rolle's theorem states that for each continuous function $f:a,b]\to ℝ$, which is differentiable in $(a,b)$ and for which $f(a)=f(b)$ must give an argument $\xi \in (a,b)$, which satisfies $f^{\prime }(\xi )=0$:

Sketch of Rolle's theorem

How can we generalize this theorem for the case $f(a)\ne f(b)$? Does the derivative $f^{\prime }(\xi )$ for a $\xi \in (a,b)$ also have to have a certain value? First of all, it is noticeable that $f^{\prime }(\xi )$ does not necessarily have to be $0$:

figure 1 about the mean-value theorem

Let's reconsider what the situation was with the situation with Rolle's theorem. On the one hand, the slope of the tangent at the graph is $f$ in $(\xi ,f(\xi ))$ equal to $f^{\prime }(\xi )=0$. On the other hand, the slope of the secant through the two boundary points $(a,f(a))$ and $(b,f(b))$ from $f$ equal to $\frac{f(b)-f(a)}{b-a}=0$, since $f(a)=f(b)$ and thus $f(b)-f(a)=0$. The secant between the points $(a,f(a))$ and $(b,f(b))$ and the tangent in the point $(\xi ,f(\xi ))$ are thus parallel:

figure 2 about mean-value theorem

Be more general $f(a)\ne f(b)$. Consider the secant slope $\frac{f(b)-f(a)}{b-a}$ between the points $(a,f(a))$ and $(b,f(b))$. This is unequal to zero and corresponds to the mean slope of $f$ in the interval $[a,b]$. For example, if we consider the function as a position function of a car as a function of time, the average slope corresponds to the average speed $\bar{v}$ of the car in the time from $a$ to $b$.

If the car at the moment $a$ drives faster than $\bar{v}$ (meaning: The derivation $f^{\prime }(a)$ is greater than the secant slope $\frac{f(b)-f(a)}{b-a}$) so it is has to exist a moment $b$ at which it has driven more slowly than $\bar{v}$, otherwise it cannot reach the average speed $\bar{v}$. During an acceleration or braking process, the car takes on all speeds between the starting and final speeds and does not simply jump from the starting to the final speed (here we assume that the speed function is continuous). As the car was sometimes faster and sometimes slower than $\bar{v}$, there must exists a moment at which it has exactly the speed $\bar{v}$. Analogously we can argue, if the car at the moment $a$ drives more slowly than $\bar{v}$. For our $f$ function this means that there must actually be a $\xi \in (a,b)$ with $\bar{v}=f^{\prime }(\xi )=\frac{f(b)-f(a)}{b-a}$. This is the message of the mean value theorem.

Image 3 about the mean-value theorem

So there seems to be a $\xi \in (a,b)$ with $f^{\prime }(\xi )=\frac{f(b)-f(a)}{b-a}$. In the following we want to form this intuition into a theorem and prove it formally correctly. In our argumentation we have used, for example, that the derivation is continuous. Now the studied function does not have to be constantly differentiable. However, we will show in the proof that the mean value theorem is also fulfilled in this case.

Mean value theorem

The mean value theorem of differential calculus is a generalization of Rolle's theorem and reads as follows:

Equivalence of mean value theorem and Rolle's theorem

The mean value theorem and Rolle's theorem are actually equivalent. To show this, we have to deduce Rolle's theorem from the mean value theorem and prove the mean value theorem from Rolle's theorem. But we have already done the latter in the proof of this chapter, so that we only have to deduce Rolle's theorem from the mean value theorem.

Be $f:a,b]\to ℝ$ a continuous function with $a<b$ and $(a,b)$ is differentiable. $f$ is therefore a function to which the mean value theorem can be applied. Furthermore, $f(a)=f(b)$ applies, so that all prerequisites of the set of role are given. According to the mean value theorem there is now a $\xi \in (a,b)$ with $f^{\prime }(\xi )=\frac{f(b)-f(a)}{b-a}=0$, because with $f(a)=f(b)$ is $f(a)-f(b)=0$. So there is actually a $\xi \in (a,b)$ with $f^{\prime }(\xi )=0$, which is exactly the statement of Rolle's theorem. Thus, the mean value theorem and Rolle's theorem are equivalent.

Application: Proof of inequalities

The mean value theorem can often be used to prove useful inequalities. The trick is to first apply the mean value theorem to an auxiliary function (often on one side of the inequation). Then we estimate the bounds to the expression $f^{\prime }(\xi )$ appropriately.

Logarithm inequalities

For all $x>1$ , we have the inequality $1-\frac{1}{x}<\ln (x)<x-1$.

Application: Lipschitz continuity of differentiable functions Template:Anchor

Visualization: Differentiable functions are Lipschitz continuous. The Lipschitz constant is given by the "maximum derivative value".

Practical example: Speed control with light

A photoelectric sensor for speed measurement

Some people who got caught speeding have come into contact with the mean value theorem. At least if it was by a Photoelectric sensor. Imagine driving a car on a country road. The maximum permitted speed is $v_{\max }=100\frac{\mathrm{k}\mathrm{m}}{\mathrm{h}}$ or 60 mph. The distance covered by your car is given by the differentiable position function $s$, which depends on the time $t$. The derivative of the position function at the time $t$ corresponds to the current speed, i.e. $s^{\prime }(t)=v(t)$. When measuring speed with light barriers, one passes through two light barriers which are placed at two fixed points $s_0$ and $s_1$. If you pass the two light barriers at the times $t_0$ and $t_1$, the average speed between these two measuring points is

Since the position function $s:[t_0,t_1]\to ℝ$ fulfils the conditions of the mean value theorem, there is a time $\tilde{t}\in (t_0,t_1)$ with

The average speed measured between the two barriers must therefore have been reached at least at one point of time. If now $\bar{v}>100+\text{tol}$, where $\text{tol}$ is a certain tolerance limit (usually 3%), you will get a speeding ticket! To avoid wrong measurements, in practice more than two light barriers are used and more than one measurement is carried out. But the principle remains the same. Another more recent technique for measuring speed is based on the Doppler effect and uses a Radar to determine speed.

Second mean value theorem

There is another version of the mean value theorem, which is called the second or also generalized mean value theorem. Therefore the "usual" mean value theorem is also called the "first mean value theorem". We will see that also the second mean value theorem follows from the Rolle's theorem. For the second version we need, besides our function $f$, another function $g$, which also fulfils the requirements of the mean value theorem. The second mean value theorem will be useful for deriving L'Hospital's rule.

Remarks

Remark 1: Obviously we get for $g(x)=x$ the first mean value theorem from the second. But we have concluded the second one from Rolle's theorem. Since the first mean value theorem and Rolle's theorem are equivalent, the second mean value theorem also follows from the first. The two mean value theorems are therefore equivalent.

Remark 2: If we omit the precondition $g^{\prime }(x)\ne 0$ for all $x\in (a,b)$, the second mean value theorem still applies in the form

Overview. Implications of mean value theorems

In the introduction, we already mentioned that several important results can be deduced from the mean value theorem.

• We have proved Lipschitz continuity of differentiable functions with limited derivation. This allows to show Lipschitz continuity of numerous functions.
• Another practical conclusion is the constant function theorem. This states that a function is constant if $f^{\prime }\equiv 0$ (the derivative is constantly zero). Thus we can derive the identity theorem of differential calculus. It says that two functions with identical derivative differ only by one constant. It is an essential part of the fundamental theorem of calculus . A further consequence of the criterion for constance is the characterisation of the exponential function via the differential equation $f^{\prime }=f$.
• Likewise, the mean value theorem serves for proving the monotonicity criterion for differentiable functions. This establishes a connection between the monotonicity of the function and the sign of the derivative function. More precisely, $f$ is monotonically increasing (or decreasing) exactly when $f^{\prime }\ge 0$ (or $f^{\prime }\le 0$). From this, one can derive a sufficient criterion for the existence of an extremum of a function at a point.
• From the second mean value theorem, L'Hospital's rule can be concluded. With their help, numerous limit values of quotients of two functions can be calculated, computing certain derivatives.

The points listed are summarised in the following overview diagram:

diagram about the consequences of the mean value theorem

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