Eudox can handle six numerical commands and two symbolic commands.
Calculate the sum of f from xstart to xstop. You can use step to specify
the step from one x value and the next. This calculates the sum of 0+1+2+3+4+...+10.
This function works very similar to Sum. Product calculates the product
of f from xstart to xstop. You can use step to specify the step from one
x value and the next.
Root[] finds a zero to the function f between xstart and xstop. This finds the x value where sin(x) crosses the x-axis. We tell Eudox
to search for the root between x value 3 and x value 4. Root[] uses two algorithms to find the root. In the following example notice that after the two errors, the function
return a value. The errors don't mean that the returned value is false,
just that Newton's method couldn't find any root. Root obtained the answer
using the bisection method instead. Although the answer is correct in the above example, you may want to
get rid of the errors. To do this you should always plot the function. We notice that the interval was unnecessarily large and therefore we
narrow it down a bit, and at the same time decide which root we really
want. Assume that we wanted the root between 6 and 8,then this command
will solve the problem. The general way of eliminating errors in Root[] can be described as follow:
Intersection[] works very much as Root[]. The difference is that Intersection[] finds a intersection between two functions instead of one function and the x-axis. This finds the intersection between the function x^3-x and x^2 between
x value -1 and x value 0. Because Intersection[f0, f1, x, xstart, xstop] is almost identical to Root[f0-f1, x, xstart, xstop], the errors for Intersection[] is identical to the errors for Root[]. Look at the description of Root above to learn about the errors, and how to avoid them.
Integrate[] can only approximate the integral from xstart to xstop. Integrate can't find the indefinite integral of a function. This calculates the area restricted by the function and the x-axis from
zero to Pi. Here is the same integral but from zero to 2*Pi this time. Note the result zero. That is because the area from Pi to 2*Pi is below the x-axis and therefore has the value -2, and 2+ (-2) is zero so the result will be zero. This specifies the number of sample points to use when calculating Integrate[].
You can use this to determine how exact you want your result to be. The result is far from exact but it was much faster calculated. The standard value for SamplePoints is 10000. You can choose a higher value for SamplePoints then 10000 to get a more exact result. SamplePoints is an option for Integrate. You will learn more about options and how they work under the chapter Advanced Plotting.
Eudox has a very powerful derivative function. It can find the derivative for all elementary functions!
This calculate the derivative for y=x^2 Here are some derivatives. If you specify "xvalue" Eudox will use a numerical method to find the
rate at a specific x value. This find the rate at which the f increase/decrease
a x=4
Here are some simplified expressions. Simplify[] is not that powerful, it can't calculate for example addition. We hope that in later versions of Eudox this will be fixed. |