AZ-MATH Software - University of Arizona

A large collection of MSDOS and Macintosh software packages which can be used in the teaching of mathematics has been written at the University of Arizona. The latter link points to a page that contains information on the downloading of these packages and provides means for downloading the entire collection. To download individual files (in self-extracting format) use the anonymous ftp connection. These programs (in zip format together with a short abstract) are also available through the Mathematics Archives using the links below:

One of the problems we encounter while teaching calculus is that many students do not have the mathematical skills which are essential for success. This is true even though the prerequisite courses (College Algebra and Trig) have been passed. To address this problem, the Mathematics Department of the University of Arizona decided to do the following. Write a computer program which a student can use before starting calculus. This program would quiz the student on all sections of algebra and trig which are essential to calculus. The whole purpose of the program would be to hone the student's skills in those parts of algebra and trig which are essential for success in calculus. This program quizzes students in the following areas: Greek symbols, elementary trigonometry, elementary algebra, factoring, fractions, exponents, functions and equations, inequalities, curves, functions from formulas, and trigonometric identities.

ARE YOU READY FOR CALCULUS II ? Version 1.03
One of the problems we encounter while teaching Calculus II is that many students do not have the mathematical skills which are essential for success. This is true even though the prerequisite course (Calculus I) has been passed. To address this problem, the Mathematics Department of the University of Arizona decided to do the following. Write a computer program which a student can use before starting Calculus II. This program would quiz the student on all sections of Calculus I which are essential to Calculus II. In particular the quizzes cover trigonometric identities, logorithms, limits, differentiation and integration. The whole purpose of the program would be to hone the student's skills in those parts of calculus which are essential for success in Calculus II.

ARE YOU READY FOR CALCULUS III ? Version 1.04
One of the problems we encounter while teaching Calculus III is that many students do not have the mathematical skills which are essential for success. This is true even though the prerequisite courses (Calculus I, and II) have been passed. To address this problem, the Mathematics Department of the University of Arizona decided to do the following. Write a computer program which a student can use before starting Calculus III. This program would quiz the student on all sections of calculus I and II which are essential to Calculus III. In particular: functions and curves, logs and exponentials, differentiation, integration, and areas. The whole purpose of the program would be to hone the student's skills in those parts of calculus which are essential for success in Calculus III.

ARE YOU READY FOR COLLEGE ALGEBRA ? 1.02
One of the problems we encounter while teaching college algebra is that many students do not have the mathematical skills which are essential for success. This is true even though the prerequisite course (Intermediate Algebra) has been passed. To address this problem, the Mathematics Department of the University of Arizona decided to do the following. Write a computer program which a student can use before starting college algebra. This program would quiz the student on all sections of algebra which are essential to college algebra. The quizzes cover linear equations, exponents, functions, and factoring. The whole purpose of the program would be to hone the student's skills in those parts of algebra which are essential for success in college algebra.

ARE YOU READY FOR INTERMEDIATE ALGEBRA ? Version 1.0
One of the problems we encounter while teaching intermediate algebra is that many students do not have the mathematical skills which are essential for success. This is true even though the prerequisite course (Beginning Algebra) has been passed. To address this problem, the Mathematics Department of the University of Arizona decided to write a computer program which a student can use before starting intermediate algebra. This program would quiz the student on all sections of algebra which are essential to intermediate algebra. In other words, it would not review all of intermediate algebra, but would concentrate on those areas which are actually needed in intermediate algebra(i.e. decimals and exponents, numbers and expressions, factors and fractions, and linear equations).

ARE YOU READY FOR ORDINARY DIFFERENTIAL EQUATIONS (ODEs)? Version 1.05
One of the problems we encounter while teaching differential equations is that many students do not have the mathematical skills which are essential for success. This is true even though the prerequisite courses (Calculus I, II, and III) have been passed. To address this problem, the Mathematics Department of the University of Arizona decided to do the following. Write a computer program which a student can use before starting ODEs. This program would quiz the student on all sections of calculus which are essential to ODEs(i.e. logs and exponents, differentiation, integration, and power series). The whole purpose of the program would be to hone the student's skills in those parts of calculus which are essential for success in ODEs.

BAYES' THEOREM 1.01
This program deals with Probabilities, Conditional Probabilities, Bayes' Theorem, and how they are used in Search and Rescue to look for a missing subject lost in a hostile environment.

COMPLEX NUMBERS
This program will evalute complex expressions, find the n th roots of a complex number (n < 9), and graphically display complex numbers. This program also has games involving addition, subtraction, complex conjugates, multiplication, division, and recipricals (all of complex numbers).

COMPOSITION OF FUNCTIONS
This slide show shows how f(g(x)) can be constructed graphically from the graphs of f(x) and g(x). It also shows how the chain rule
f(g(x))' = f'(g(x)) g'(x)
can be obtained graphically using this construction.

CONICS 1.01
This program plots conics (i.e. parabolas, ellipses, hyperbolas) as well as quadratic curves. The constants that usually occur in these formula can be changed at will and the curves redrawn. There are also demonstrations of the definitions of the parabola, hyperbola and ellipse as the locus of points satisfying certain conditions as, for example, the ellipse is the locus of points which sum of distances from these points to the foci is a constant.

CONVERT UNITS
This program converts between various type of units. Abbreviations are given as well as the dimensions of the various units.

FEUERBACH'S THEOREM 1.01
This program allows you to experiment with the size of a triangle and see the resulting incircles, excircles, circumcircles and nine-point circles. It is designed to give a pictorial representation of Feuerbach's Theorem which states that the nine-point circle of a triangle T passes through the following nine points: the midpoints of each side of T, the foot of each altitude of T and the Euler points of T. Furthermore, the nine-point circle of T is tangent to the inscribed circle of T and the three exscribed circles of T.

FINDPOLY 1.09
This program will answer most questions you have about a given polynomial except what it is. You must find the polynomial. The degree is less than 8, the coefficients are integers, and all the "action" is between -100 and 100. The program gives clues to the polynomial by giving the user information about the polynomial and its first and second derivitives(i.e. roots, graphs, and by evaluating these at a point.

FORTUNE 1.38
This program will graph up to two functions of x, and then allow you to "twiddle" any parameters a, b, c, contained in the functions. This program also evaluates the functions at specificpoints, gives a table of values, and has an animated plot of both secant and tangent lines. Sample functions and projects are provided.

FOURIER SERIES version 1.10
This program will graph the first 20 Fourier polynomials of y = f(x), after you enter f(x) and the period 2L. You can also enter the Fourier coefficients a(n) and b(n) (fast and accurate), or have the computer perform numerical integration to evaluate them (slow and approximate). The program also lets the user create even and odd functions.

HISTOGRAMS 1.01
This program will calculate the mean, median, and standard deviation of a data set. It will also generate a histogram, a bar chart, a box and whisker plot, and a stem and leaf plot.

IDENTIFY THE FUNCTION 1.08
This program will see whether you can identify polynomials, rationals, trig functions, exponentials, and logs. The program gives clues by graphing the function and by evaluating the function at a particular point. The computer will graph a function selected from its bank of functions. You are to identify that function.

IMPLICIT FUNCTIONS 1.05
This program tries to plot implicit functions of the form f(x,y) = c. Thus it can be used to plot implicit functions, defined by f(x,y) = 0, together with contour lines (level surfaces) of the function z = f(x,y).

INTERACTIVE DEMONSTRATIONS AND TEACHING AIDS 1.02
A collection of seven miscellaneous programs: Grid, Polar Grid (draws rectangular and polar grids resp., then these grids can be projected onto a screen), Trig 1, Trig 2, Trig 3 (graphically demonstrates the definitions of the trig functions), x^n sin(1/x) (animated display of tangent lines to these curves), and Polygons (draws regular polygons in a circle and calculates area and perimeter).

INTERACTIVE DEMONSTRATIONS AND TEACHING AIDS 2 Version 1.00
A collection of seven miscellaneous programs: Number Line, Temperature (elementary graphing), Venn Diagrams, Percent - Square, Percent - Circle (percentages), Galton Box, Shuffle (permutations).

INTERPOLATION 1.11
This program will plot a data set together with various interpolations, viz. piecewise linear, piecewise quadratic, Lagrange interpolation, and cubic splines. Sample data sets are provided.

ITERATE 1.18
For the study of discrete dynamical systems, there are eight major iteration routines available, called Numerical Iterates, Plot Iterates, Successive Iterates, Graphical Analysis, Attractors, Density Distribution Analysis, Initial Data Depen- dence, and Orbit Diagrams. Example functions and student projects provided.

LIMITS 1.08
This program will try to find the limits of f(x) as x goes to a, where a can be finite or infinite. Numerical demonstration in which the program starts one unit away from the the point a and then chooses another point in between the new point |a-1| and a. This can be repeated up to 30 iterations. Both right and left hand limits can be evaluated as well as two sided limits.

LINE INTEGRAL 1.06
This program will graph functions of the form y = f(x), functions of the form x = g(y), polar equations of the form r = r(t), where t is the angle, as well as parametric equations of the form x = x(t), y = y(t). Up to nine equations can be plotted. It also computes line integrals over these curves.

LINEAR ALGEBRA 3.17
Package to be used in first course of linear algebra or matrix algebra. Manipulation of matrices and vectors. This program computes the standard matrix operations(i.e. row operations, sum, difference, scalar multiples, matrix product, transpose, inverse, powers, determinants, linear equations, eigenvalues, reduced row echelon form, and similarity transformations. This program also deals with vector operations(i.e. sum, difference, scalar multiples, inner product, cross product, linear dependence, and the Gramm-Schmidt process. Sample homework assignments and projects are provided.

LINEAR SYSTEMS 1.05
This program allows you to solve graphically linear systems of equations in two unknowns. This program also gives numerical solutions, in fractional and decimal form, of the system of equations.

NUMERICAL INTEGRATION 1.17
This program will numerically integrate f(x) in nine different ways, e.g. by the left endpoint rule, the right endpoint rule, the midpoint rule, Riemann Sums, the trapezoidal rule, Simpson's rule, Romberg integration, Gaussian quadratures, and the Monte Carlo method.

ORDINARY LINEAR DIFFERENTIAL EQUATIONS 1.11
This program plots numerical solutions of 1st and 2nd order ordinary linear differential equations containing parameters a, b, and c. These parameters can then be "tuned" and the solution replotted. A user supplied solution function, and a user supplied power series, can also be plotted. Sample equations are provided.

POLAR EQUATIONS 1.11
This program will graph polar equations of the form r = r(t), where t is the angle, as well as parametric equations of the form x = x(t), y = y(t). Up to two equations can be plotted, and then you can "twiddle" any parameters a, b, c, contained in the functions r(t), x(t), y(t). Sample equations are provided.

ROOT FINDER 1.23
This program will find the roots of y = f(x) in four different ways, viz. by the bisection method, Newton's method, the secant method, and the method of false position. It also plots the graph of f(x), so a root can be found "by eye". This program will also evaluate f(x) at a point and give a table of values for f(x) over an interval. Sample equations are provided.

SEQUENCES AND SERIES 1.12
This program lets you create sequences a(n), and then shows the values of successive terms in the sequence, both numerically and graphically. It also computes: the partial sums of the sequence, ratio, difference, addition of sequences, subtraction of sequences, multiplication of sequences, and division of sequences. This program also lets you create a positive sequence and an nth root sequence. Sample sequences are provided.

SIMPLEX METHOD 1.0
This program will perform the Simplex method in three different ways, viz. having the computer show the answer, having the computer show the pivots, or having the user go through the step by step method.

SLIDE SHOW FOR FOURIER SERIES
This slide show consists of graphs of various functions together with some of their Fourier Series approximations:
• Triangular Wave
• Square Wave
• Saw Tooth Wave
• Cosine Expansion of Sine
• Interrupted Square Wave

SLIDE SHOW FOR FUNCTIONS
This slide show consists of graphs of various functions.
• sin(1/x)
• x sin(1/x)
• Continuous, but not differentiable
• x and 3x3
• sin x and 3sin x3
• sin x/x
• (1 - cos x)/x
• a^x and its derivative
• sin 2cx + sin 2cax

SLIDE SHOW FOR NEWTON'S METHOD
This slide show consists of graphs of various demonstrations of Newton's Method including some graphs showing how Newton's method fails to converge.

SLIDE SHOW FOR ORDINARY DIFFERENTIAL EQUATIONS
This slide show cosists of these various graphs of solutions of ordinary differential equations:
• One parameter family of curves
• The US Population and logistic growth
• The cooling of coffee
• Numerical Methods - Euler
• Numerical Methods - Runge Kutta 4
• Damped free vibrations
• Series solution
• Bessel Function

SLIDE SHOW FOR PARTIAL DIFFERENTIAL EQUATIONS 1
This slide show consists of graphs of numerical solutions of a particular partial differential equation, viz. the wave equation
```
U  +  U  = 0,
t     x
```
with an initial condition of a hump resolved with 10 points. This slide show contains: exact solution, central difference scheme, Lax-Friedrichs scheme, upwind, Lax-Wendroff, and downwind graphs.

SLIDE SHOW FOR PARTIAL DIFFERENTIAL EQUATIONS 2
This slide show consists of graphs of numerical solutions of a particular partial differential equation, viz. the wave equation
```
U  +  U  = 0,
t     x
```
with an initial condition of a step function. This slide show contains: the exact solution, central difference scheme, Lax-Friedrichs scheme, upwind, Lax-Wendroff, and downwind graphs.

SLIDE SHOW FOR PARTIAL DIFFERENTIAL EQUATIONS 3
This slide show consists of graphs of numerical solutions of a particular partial differential equation, viz. the wave equation
```
U  +  U  = 0,
t     x
```
with an initial condition of a hump "resolved" with 1 point. This slide contains the graphs of: the exact solution, central difference scheme, Lax-Friedrichs scheme, upwind, Lax-Wendroff scheme, and downwind.

SLIDE SHOW FOR VIBRATING STRING
This slide show shows how two travelling waves generate a stationary wave.
Travelling Wave
This shows three waves simultaneously. The top one travels to the left. The bottom one travels to the right. The middle one is the average of the top and bottom, and is stationary. In fact the top and bottom are the same function, viz. sin(x) + sin(2x), each translated by an amount c/6 per slide, while the domain is -4c < x < 4c. The stationary wave is thus the function sin(x)cos(a) + sin(2x)cos(2a), where a is a multiple of c/6.
Stationary Wave [0 : 2c]
This shows the stationary wave of (A) for 0 < x < 2c.
Stationary Wave [0 : c]
This shows the stationary wave of (A) for 0 < x < c.
In each of these three cases you have the choice of stepping through a sequence of slides one at a time, or having the computer do it continuously and automatically (either slowly or quickly).

SLIDE SHOW FOR TAYLOR SERIES
This slide show consists of graphs of various functions together with some of their Taylor polynomials about the origin.

SLIDE SHOW FOR TROUBLE WITH GRAPHS
This slide show consists of various graphs of the same function, seen over different domains. Frequently people are under the impression that if "enough points" of a function are plotted, and then joined, the resultant graph accurately portrays the original function. This is the principle used by all computer graphing packages. People are even questioning the wisdom of teaching curve sketching in calculus, because of such packages. The following slide show is designed to display some of the pitfalls of graphing by plotting points, and then joining them. In fact calculus is essential in deciding the accuracy of any graph.

SLOPES 1.17
This program will graph the slopes (direction fields) and the integral curves of dy/dx = f(x, y), where f(x, y) = F(x, y) / G(x,y). Sample functions are provided.

TAYLOR SERIES 1.05
This program will graph the first 20 Maclaurin polynomials (Taylor polynomials about x = 0) of y = f(x), after you supply f(x) and the Taylor coefficients, a[n], where a[n] is f'(x). Sample functions are provided.

TRUTH TABLES 1.01
This program displays Truth Tables. Expressions are constructed from the statements p, q, and r, and the four operations, "v" (or), "^" (and), "'" (not), and ">" (implies).

TWIDDLE 1.36
This program will graph the curve y=f(x), and allow you to "twiddle" any parameters a, b, c, contained in the function. You can also plot a single data set. Sample projects are provided.

TWO D MAPS 1.14
This program allows you to construct and experiment with 2 dimensional affine transformations. You can experiment with fractals, finding eigenvectors by eye, showing the effect of a map on a set of points, and showing the solutions of a set of two linear equations.