ࡱ> M 3bjbj== ,XWW.lXXX8,D LX(jllllll$  <4<<<j<j<.<jj ! Xjj0j<j< NOTE: This document was typed in TI-Interactive! and Exported to a WORD Document. The latest editions of TI-Interactive! support a programming capability that is very similar to that found on the TI-83, 89, and 92 calculators. It is a modified form of Basic programming. The most significant difference is that one can only write functions that return one object. Procedures, or prgm's, are not allowed. This means that some accommodations will need to be made when we graph functions that we define using the programming mode of the tool. We start our investigation of TII programming by first defining a function without using the programming mode. We simply click on the calculator icon or press -m and write f(x):= when(x <=2, x^2, =x+1). When we have entered the function, TII reformats the expression and the following appears.  We repeat the above exercise using TI-Interactive! Programming Logic instead of the built in "when" function. As before, click on the calculator icon. At the bottom of the calculator change the Input Mode from 'Auto Format' to 'Plain Text'. There are certain rules which must be followed carefully while entering a function. First of all do not use the Derive-type ':=' operator to define your function. (1) Use "define =func" (2) Always press -Enter at the end of each line of the function definition (3) Each line and new statement of the function definition must begin with two colons, :: (4) After typing ':: endfunc' press Enter this will complete the function definition. Pressing Enter instead of -Enter before this will cause an error message to be printed. Using these rules we enter our copy of f(x) which we call fp(x).  Let's compare the performance of the two functions for selected values of x  It appears that the two functions are the same. Let's check out their graphs. There are three ways to open a graph in TI-Interactive. The easiest is to click on the Graph icon. Other ways are to press -G or pull down the Insert menu and choose the "Graph, Y=" sequence. This last option is useful for doing X-Y function plots, parametric plots, and polar plots. Statistical plots are always an option with any choice of graph type. When we fill in the "y1(x):= " box with f(x), the following graph appears.  When we fill in the "y2(x):=" box with fp(x) we get an error message saying that a test could not evaluate to either true or false. This is a confusing message. I was fortunate enough to have a contact with the TI-Interactive! development team (The TI-Interactive help contact told me that they do not deal with programming questions.) who could help me. It seems that the variable x is considered to be an unbound variable within the programming logic mode. I need it to bind it to a real variable by typing the following for y2(x):= when(xdim(l1). You may prefer some other value.  We will test this plotting device on a simple sequence: seq(i3, i,-3, 3). The plot is hardly attractive, but it shows that the goal can be accomplished.  Let's try a sequence that is a little more substantial. Recall that the Syracuse function applied to an integer n is the number of steps until the sequence, s, starting with s(1)=n and the remainder of terms defined as  yields a value of 1. The following function returns the value for the Syracuse sequence starting a some positive integer, n.  We define a list consisting of the first 200 values of this sequence and plot it.   Exercise: 3. Extend the function listPlot1(x, l1) to a function listPlot2(x, l1, l2) where l1 is an ordered list of values of the independent variable and l2 is a list of the corresponding values of the dependent variable. If we prefer to deal with 2 X n matrices instead of two lists we can use the following funtion to attain the goal of the above exercise. Here we list the values of the independent variable in row 1 of the matrix and the values of the dependent variable in row 2.  As an example of applying this fundtion we draw the graph of ln(1+x) for 0 < x < 20.   Finally, if we wish to have a general 2-dimensional point plotting program where the plot may include vertical lines as well as horizontal and sloping lines, we need to go to a parametric plotting program. If the difference between the x-coordinate of the starting point and that of the ending point is the same, a vertical line will be plotted.  To illustrate this program we plot the matrix, house, and show the action of the matrix, a, as well as the tendency of repeated action of multiplying by a will tend to reduce the product to its dominant eigenvalue. The graphical inset shows plots of house, a*hous, and a^10*house.    Now that you have seen how to use the programming logic in TI-Interactive!, I hope that you will be able to use this very powerful, attractive, and inexpensive tool to meet your particular classroom needs. Appendix I Drawing Slope-Fields This is the project that started it all as far as my interest in programming in TI-Interactive! is concerned. I was disappointed that I could not use the TI-89/92+ facility for drawing slope fields and direction fields. I discovered programming and began by writing a function to give the value at x = xin of a line passing through the mid point of a box containing xin on the horizontal axis and centered at yin on the vertical axis. We assume that the region under consideration a < x < b and c < y < d is divided into n and m subintervals, respectively. The differential equation is given by the expression y / = s(x, y). We represent s as a string.  The following example shows how this function can be used to numerically evaluate the solution to a first order differential equation of the form: y / = x*y sin(x). Note that the x-interval [0,4] is divided into 8 subintervals and the y-interval [0, 5] is divided into 10.  We also want to be able to show a solution to the differential equation passing through a particular point in our region. For this purpose we define a Fourth Order Runge-Kutta routine that will give the solution at x = xx lying on a curve passing through the point (ix,iy).  Below we show the solution to the equation: y / = x*y sin(x) for x = .5 and having initial value y(1) = .2 .   INCLUDEPICTURE "C:\\My Documents\\TIIimagefile14238.gif" \* MERGEFORMATINET  However, the real goal is to duplicate the slope-field utility on the TI-89/92+ calculators. We want to draw the slope field and also enter the values for solutions for different sets of initial conditions. Unfortunately, this requires that we draw each line of the slope field individually and also enter the solution equation for each set of initial conditions. In order to draw the graph we had to enter a function of the form: when(x* jU jU5\ jU6]FST$%TZ[. ?      Q T X 1$7$8$H$^1$7$8$H$ $1$7$8$H$a$23X Y b l ij9XYd ehi U $1$7$8$H$a$1$7$8$H$UV\]RTX $1$7$8$H$a$1$7$8$H$78<>ABJKLMNOPVWZ[L M !"!`!a!!!!!!!!!!!!! " """##-$.$$$$$$$$$$$$$$$$$$$$ %%H* 5CJ\ jlU jjU j&hU j\U jWU jSU>* j|MU6] jEU jBUMX\^ !!!! 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