Readers are presented with three specific technology based activities that address the following mathematical topics: the mean value theorem, the Cauchy mean value theorem, the inverse image of parametric curves, and the global minimum of total squared distances among non-intersecting curves or surfaces.

We demonstrate below another example of a traditionally static core, Mean Value or Cauchy Mean Value Theorem, done differently.

Example 1 About the proofs for the Mean Value or Cauchy Mean Value Theorems.

We can prove the Cauchy Mean Value Theorem (CMV) in a similar manner.

we obtain the conclusion of Cauchy Mean Value Theorem (see [15] for details).

In this paper, we presented three technology based mathematical conceptual activities, as part of our dynamic core, designed to improve teachers' mathematical knowledge for teaching the mean value theorem, the Cauchy mean value theorem, the inverse image of parametric curves, and the global minimum of total squared distances among non-intersecting curves.

There are two video clips which give some geometric insights of how we prove Mean Value Theorem and Cauchy Mean Value Theorem, they are located respectively at

Next, we give a geometric description of how the Cauchy Mean-Value is stated and shed some light on how we can arrive at the function to which Rolle's Theorem is applied to yield the Cauchy Mean Value Theorem holds.

3 Geometric Interpretation of Cauchy Mean Value Theorem

We use similar approach mentioned above to demonstrate how geometric interpretations of the Cauchy Mean Value Theorem can be explored with the help of DGS and CAS.

then we will obtain the conclusion of Cauchy Mean Value Theorem (see [1]).

We use the following example to give motivations for the conclusion and the proof of Cauchy Mean Value Theorem.