As I stated in the intruction of my project for EMT 670 students in Korean
high school are expected to solve complicated function problems without
help of any technology. So the knowledge about differential calculus is
as important as solving the problems themselves. Let us guess that a rational
function is given. Students can get a lot of information of the function
with the differentiated function. Students first will graph the simple figure
of the function, and usually mark the roots of function, if possibly calculated,
and some points where the differentiatial coefficients are zero. I think
every knowledge from the process should be understood conceptually by students.
"Why do they try to find the roots of a function?" and "What
does it mean for the students to have 0 as differential coefficients?".
I think the use of Algebra Xpressor will provide meaningful knowledge about
questions raised above by comparing a mother function and its differentiated
fuction visually in a scene. Next are an example of rational function .
The graph can be easily drawn by Algebra Xprossor, but, the main focus is
how to use it to make students understanding about relationship between
a function and its differential function . Let's look
at the function graph first.
There are three distinct parts of a graph. Students can draw an arbitrary
tangent line on a point of the graph. The slopes of all the tangent lines
of the graph will explain another function. Since the sines of slopes are
changing along the graph. Algebra Xpressor can draw the differentiated function
if students can calculate the derivative function of the mother function.
Following is the graph of the derivative function . One thing that
students have to notice is that the actual graph does not have a value when
x=2 and x=-2, but, Algebra Xpressor shows the two lines x=2 and x=-2. Students
will have a chance to discuss about the case of value when the denominator
of a rational function is zero.
Look at the following graphs and see if we can find significant information
Red is a function , and green is its differentiated function
. Look at the region with 4<x<10 of two graphs. What does green
say? If a student draw a tangent line on a point in 4<x<10, then the
tangent line will have positive slopes and the slopes will be increasing.
Green graph is showing the exact movement of the slopes of tangent lines.
What if I draw a tangent line with x=1? Green graphs says that the value is negative so the tangent line will be a linear function with a negative slope. Same kind of logic can keep going on for x<-3.
If you would have drawn the differentiated function first, then you could have understand that you would see three different parts of a function .