*Creating a curve in the geometry tab*

*Observing the link between dynamic geometry and graphic representation*

The purpose is to show how functional modelling helps to approach a calculus notion.

The goal is to study the differential quotient for a point on the graph of a cubic function.

First create the function *f(x) = 2 x^3-x* and the parameter *a* in the *function list*.

### Creating a curve in the geometry tab

Switch to the *geometry tab* and select **Create Object** / **Curves**.

Figure 125- Menu Create Object / Create Curve

The curves of the function displays.

Figure 126 - Displaying of the curve

### Studying a gradient

Then create the coordinate point A (*a*, f(*a*)). Note that A moves on the curve when you animate the parameter *a*.

Create M, a free point on the curve (**Create Object** / **Point** / **Free Point on Curve**). Note that M moves on the curve when you drag it with the mouse.

Create the calculation (yM-yA)/(xM-xA) corresponding to the differential quotient relative to A. Observe the values of the quotient when M is near A for various values of *a*.

Figure 127 - Calculation value and representation

Create the calculation (xM – xA) which you will use like preimage.

Create in the *x-value list*, the value zero (x1 = 0). Model the function *c1* -> *c2*. Then, clicking on **Auto**, the correct domain of definition appears. Validate with **OK** and click on **Exit**.

Figure 128 - Calculus "Derivative number"

In the *graphic tab*, you can graph the function g and ask the limit when x approaches 0 (xM approaches xA) which is the differentiated number of f in *a*.

- either when the parameter is instantiated

Figure 129 - Formulas with instantiated parameter

- either when the parameter is “formal”

Figure 130 Formulas with formal parameter

### Observing the link between dynamic geometry and graphic representation

In the *geometry tab*, you can display the geometric aspect of the problem and in the *graphic tab*, the curve of the quotient values. A moving in one tab leads to a moving in the other tab.

Figure 131 -Geometry and graphic tabs