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# NewCalculus

last edited by 12 years, 10 months ago

An introduction to new features of GeoGebra 4.0 related to calculus.

This document highlights features of GeoGebra 4.0 that are both new and related to calculus. The full list of new features can be found in the release notes on the GeoGebra wiki. Several documents that look at an overview of features related to calculus can be found looking at the front page of this wiki.

I) Symbolic differentiation and integration –

• With release 4.0, values in the algebra window are rendered in LaTex, rather than in a text format. The results of symbolic differentiation and integration also use exact algebraic expressions rather than rational approximations.

• The underlying CAS system has been shifted from mathpiper to reduce, making it more robust. I find it can do most problems I would assign in a first year calculus sequence.

• With release 4.0, functions of two variables have been introduced. You can take partial derivatives of these functions.

• Partial fractions – While neither symbolic differentiation nor integration, it is used in a technique of integration covered in a standard course.

It is worthwhile to first look at a collection of functions in command view. Some were entered directly and are called free, and others are dependent on the free functions.

It is then worthwhile to compare to the regular view.

II) The commands Min, Max, Extrema, Root, and Roots

• The Min, Max, and Extrema commands find local extrema. You can either specify a starting point for a search or a range to look in. It is noteworthy that when given an interval, these commands find the appropriate extrema in the open interval.

• The Root command looks for a single root either from a starting point or in an interval. The Roots command finds all the roots in an interval. Since these commands use numerical methods, they may not find all roots in some instances.

Once again it is useful to look first at the command view, then at the value view.

III) Function Inspector tool – This is a new pop-up tool from the graphics view that gathers information about a function.

The interval view gives information about the function on a specified interval. The points view either gives information at a specified point, or for a grid of points with a specified step size. The line for Roots, either tells you there are not roots, or gives the value of the unique root, or tells you that there are multiple roots. Area is total area between the function and the x-axis while Integral is the usual signed area.

In the points view, the first button lets you make a chart of about 10 points with a specified step between them. As noted, you can add more columns of information. It is useful to be able to compare a difference quotient with a specified step size against the derivative.

IV) Riemann sums

While GeoGebra has been able to do a graphical or numeric evaluation of Riemann sums of a function in the past, the new version allows all the variations typically seen in a first year course.  (Sums that are upper, lower, lest, right, midpoint, or trapezoids.)

V) Functions of 2 variables –

As mentioned above, with release 4.0, GeoGebra can handle functions of 2 variables. It is worth noting that you can either evaluate using f(x,y) notations, or f(P), where P is a point. You can also plot implicit functions for polynomial equations. Implicit functions can be used to plot contours.

As noted above, GeoGebra 4.0 will evaluate partial derivatives. This allows for reasonable graphing of slope fields. Release 4.0 has also included a SolveODE command that plots numerical solutions to differential equations. (It uses Runge-Kutta 4.)

(The GeoGebra file for slope fields is at SlopeFields2A.ggb.)