﻿ Casyopee - Curve and derivative

What they say:   I used Casyopée to help students solve optimisation problems. It allowed me to present students very open tasks. Students explored and found results that they had to prove afterward. Casyopée also helped students reuse basic strategies for solving.
What they say:   Casyopée is faster and more convenient than a calculator.... We have the geometric and algebraic side of the problem at the same time. It is easier to see how a function "reacts." It's useful and interesting.
What they say:   Casyopée makes it easy to calculate a derivative, to factor, to calculate zeros... and have a graph of the function next to it in the same window. It allows on a geometric problem to be able to establish variables that can then be used to study the problem by way of functions...
What they say:   Casyopee is a powerful application that can prove useful to both students and teachers It allows you to use various exploration and modeling tools, with the purpose of studying or teaching mathematical functions.
What they say:   Casyopee comes with lots of features. One of these features is the help provided for proving a function. There is also a feature for writing HTML reports that include the mathematical functions. Casyopee is guaranteed to improve the mathematical knowledge of its users.
What they say:   Besides the concept of number, the concept of function is the most important one in mathematics
What they say:   The notion of function is present in all scientific disciplines, and also in everyday life. Our experience as a teacher shows every day that it is a problem for many students. Situations with Casyopée can also be used outside of a technological environment and everyone will be able to reflect on her professional practice.

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

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

Creation date : 06/10/2014 - 16h17
Category : - help