About the Mean Value Theorem.

Dr. Wei-Chi Yang

Radford University

Radford, VA 24142

www.radford.edu/~wyang

The Cauchy Mean Value Theorem.

Suppose the function and are continuous and that their restrictions to are differentiable. Moreover, assume that for all in Then there is a point $x_{0}$ in at which MATH

$\QTR{bf}{Remark:}$ This is not intuitive at all, we should see how this can be made eaiser to understand from geometric point of view.

Cauchy Mean Value Theorem, Inequality Version. If functions and are continuous on and differentiable on and if is strictly monotone, then implies
Mean Value Theorem, Inequality Version.implies
- If a car is going at a speed between and units (say miles per hour) in hours ( ) then the car has traveled between and miles in hours.
A Physical Application: If car is going at a speed which is between and times the speed of car then car will have traveled between and times of the distance of car [R. Vencil Skarda of Brigham Young University]

Example.

Let and we may pick to be any function that satisfies the conditions of the theorem. Note that one easy way to find is to graph its derivative so $y=g\prime (x)$ does not touch -axis. Say, , then $g(x)=-\cos (x)+2x$ .

We first sketch the graphs of and in as follows:
graphics/CauchyMeanValueweb__47.png

If and satisfy the conditions of Cauchy Mean Value, can we find so that satisies the Rolle's Theorem

One way to interpret the Cauchy Mean Value Theorm is that if we are given a strictly monotone function on (since on and if is arbitrary differentiable function on we can find a function such that satisfies the Rolle's Theorem. [We may define ]
Another way to interpret the Cauchy Mean Value Theorem is that if the function is defined to be $f(x)+m\ast g(x),$ where and are continuous on and differentiable in and is strictly monotone. Then by compressing or stretching and adds it to we can find a number (which can be ) and $x_{0}$ in so that

One question here: Is the number is unique for

Note that if then implies that for some $x_{0}$ in If we use the Example 2 above, we see and and assume We have and
We link to Maple. Note that if we follow the curve where and intersect, then we will get $F^{\prime }(x)=0.$ [Or precisely, ] Note that if we follow the curve the changes of with respect to direction is ]
- The intersection:
- For example, let's take
- For example, let's take
- Now, let Then

Express as a composite of and $g^{-1}.$

Another way to interpret Cauchy Mean Value Theorem: Let : $X\rightarrow Y$ and $g:X\rightarrow Y$ . The functions and are continuous on , and and are differentiable on the interiors of . If the function is strictly monotone (WLOG, we assume is strictly increasing here), then we define where $x=g^{-1}(y).$ Therefore, we apply the regular Mean Value Theorem on $F=f\circ g^{-1},$ which says that there is a between such that
- Example: Let and $g(x)=\sqrt{x+2}$ so $f:R\rightarrow R$ and and If Let's take Then it follows from the regular Mean Value Theorem on that there is a $x_{0}$ such that
- To find the $x_{0}$ satisfying we can use CAS to solve this, which yields,
- We can also find a such that we find is approximately

Remark: One draw back of writing as $f\circ g^{-1}$ can be shown as follows: Note that if we take the same and replace by $-\cos x+2x,$ then is strictly increasing, but we may not be able to find a formula for $g^{-1}$ so we can not define this way.

Parametric Form

Consider a curve of the form $r=f(\theta ).$ If the curve is a smooth curve in the interval , it follows from MVT that we can find $\theta \in (a,b)$ so that

MATH

This implies that integrating over $\theta$ on both sides, we get MATH

If we define we will see and we may apply the Rolle's theorem on . This gives us a glimpse how we prove the Cauchy Mean Value Theorem.

Example

We consider $r=\theta ,$ and we write

MATH We have MATH Since the curve $r=\theta$ is a smooth curve, it follows from the Mean Value Theorem that there exists a $\theta$ such that MATH If we pick we can solve Solution is $\allowbreak$ from ClassPad.