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MATHTOOL AN INTERACTIVE MATHEMATICAL TOOL USER'S GUIDE VERSION 2.03 (C) COPYRIGHT 1990 - ALL RIGHTS RESERVED by MARTIN J. MAHER MJM ENGINEERING P.O. BOX 2027 HAWTHORNE, CA 90251 TABLE OF CONTENTS ----------------- 1 INTRODUCTION .............................................. 1 1.1 Description ........................................... 1 1.2 History ............................................... 1 1.3 Interface ............................................. 2 1.3.1 Menus ........................................... 2 1.3.2 Pick List ....................................... 3 1.3.3 Message Windows ................................. 3 1.3.4 Query Boxes ..................................... 3 1.3.5 Help ............................................ 4 2 MENU TREE ................................................. 5 3 MAIN MENU ................................................. 6 3.1 Files ................................................. 6 3.2 Define ................................................ 6 3.3 Eval .................................................. 6 3.4 differenTiate ......................................... 6 3.5 Integrate ............................................. 6 3.6 Solve ................................................. 6 3.7 Graph ................................................. 6 4 FILES MENU ................................................ 7 4.1 Load .................................................. 7 4.2 Write ................................................. 7 4.3 Directory ............................................. 7 4.4 Quit .................................................. 7 5 DEFINE MENU ............................................... 8 5.1 New ................................................... 8 5.2 Edit .................................................. 8 5.3 Delete ................................................ 8 5.4 Clear ................................................. 8 6 EVAL MENU ................................................. 9 6.1 Single ................................................ 9 6.2 Range ................................................. 9 7 DIFFERENTIATE ............................................. 10 8 INTEGRATE ................................................. 11 9 SOLVE ..................................................... 12 10 GRAPH MENU ................................................ 13 10.1 Single ............................................... 13 10.2 Multiple ............................................. 13 10.3 Types ................................................ 14 10.4 Scaling .............................................. 14 10.5 Data File ............................................ 14 11 DEFINING A FUNCTION ....................................... 15 11.1 Form ................................................. 15 11.2 Operators ............................................ 15 11.3 Math functions ....................................... 15 11.4 Sets of functions .................................... 16 11.5 Difficulties ......................................... 16 12 EXAMPLES ................................................... 18 13 LEGALITIES ................................................. 19 14 PRODUCT INFO ............................................... 20 15 ORDER FORM ................................................. 22 1 INTRODUCTION 1.1 Description MathTool is a program for learning and doing math. Anyone studying or using subjects such as algebra, analytic geometry, calculus, physics, or engineering will find MathTool to be very useful. Using Mathtool you can define functions using normal mathematical notation, then investigate them in various ways. It is well known that human beings learn best visually. With MathTool math becomes visual and this is a great aid to understanding. Using MathTool you can define any well-behaved mathematical function of a single variable. (For a detailed description of what makes a function well-behaved, see section 11.5, Difficulties.) MathTool is designed to be easy to use, using menus, context-sensitive help, and clear prompts at every step. Once you have defined a function, MathTool gives you the following capabilities: 1) Evaluate the function for any value or set of values of the independent variable. 2) Numerically integrate the function. 3) Evaluate the derivative of the function for a given value of the independent variable. 4) Find solutions of the function over some range of the independent variable. 5) Plot one or more functions. MathTool is written in Turbo Pascal 5.5. MathTool requires a minimum of 256k of memory. MathTool will use your math coprocessor if you have one. The number of functions you can define is limited only by memory. Testing shows that even very complicated functions rarely require more than about 50k of memory, so you should never have problems with insufficient memory. The plotting requires graphics capability. Plots can be printed on an IBM graphics printer or compatible. MathTool supports all common PC graphics standards - CGA, EGA, VGA, and Hercules. 1.2 History MathTool fills a need that I first perceived several years ago. At the time I was a student in engineering school. Often when I was reading some text or other, I would come across a complicated mathematical function or formula. Normally the author of the text would describe the function and some of its properties. Sometimes he might include a plot illustrating these properties. At this point I usually thought that seeing more plots of the function would make things much clearer, and the more complicated the function the more I would feel this way. This left me with three alternatives: look in other texts in the hope that I might find more information; plot it myself by hand, a job only a masochist could enjoy; forget about it. Naturally, alternative three was my usual choice. MathTool User's Guide - Page 1 Eventually, I became the owner of a Compaq Deskpro and a copy of Turbo Pascal. I started writing a program that would plot any function that I could type in. My purpose, at first, was to learn programming in general and Turbo Pascal in particular. Then I started adding some other things, like integration and derivatives. The idea gradually grew that this might make a good shareware program. That eventually led to version 1.0 of MathTool. Later, when the program got to version 1.11, I renamed it MTool. Now, for version 2.0, I am going back to its original name. MathTool 1.0 was powerful enough that I wished I would have had it when I was in engineering school. It certainly would have added a fourth and most attractive alternative to the list above. In MathTool 2.0 I have taken the basic features of the original program and expanded their functionality in numerous ways. Before you could only enter one function at a time. Now you can have an unlimited number of functions with names of your own choosing. Log-log and semi-log plots have been added. Perhaps most importantly, the program sports a much improved user interface, with context-sensitive help, menus, and file support so that complex function definitions can be saved for future use. 1.3 Interface The MathTool interface should be familiar to anyone with even a modest amount of experience with MS-DOS software. The primary interface consists of a menu system that uses the highly popular moving light-bar menus. Query boxes are used to get the answers to questions. A pick list of the currently defined functions is available for choosing a function. Pressing the <F1> key at any time bring context-sensitive help. 1.3.1 Menus The highest level of the MathTool interface consists of a set of menus. The main menu is horizontal and appears across the top of the screen. Secondary menus are vertical and appear as appropriate. All of the menus work in the same basic fashion. When the menu appears one of the menu items is highlighted by a light bar. This light bar is a different color on color monitors or inverse video on monochrome monitors. The user has two methods of making a selection. First, he can move the light bar using the arrow keys, left and right arrows for horizontal menus, up and down arrows for vertical menus. If the light bar is at one of the extremes of the menu and he pushes the arrow key that would take him beyond that extreme, the light bar moves to the other extreme. For example, if the light bar is on the top item on a vertical menu and the user presses the up-arrow key, the light bar will move to the bottom item on the menu. Once the light bar is on the proper menu item, the user can select it by pressing the <RETURN> key. MathTool User's Guide - Page 2 The second method is even simpler. Each menu item will have one character that is capitalized. Usually, but not always, this will be the first character. The user can make his selection simply by pressing the key that is capitalized, for example "f" for Files or "c" for sCaling. 1.3.2 Pick List A pick list is a dynamic menu. The pick list in MathTool contains a listing of all the currently defined functions. Choosing a function from the pick list is done by moving the light bar to highlight your choice, then pressing <RETURN>. The alternative method of typing a capitalized letter does not work for the pick list. 1.3.3 Message Windows Message Windows are used throughout MathTool to display results and messages. A Message Window is a window on the screen which may have a frame around it. The frame usually appears as either a single or double line. If you have a color monitor, the Message Window will appear in a different color than the background. Each Message Window has a specified size. Sometimes the message is too long to fit in the window. When this happens you will see the word "More" on either the top of the frame, the bottom of the frame, or both. This indicates that more of the message is available by scrolling up or down. You can scroll the Message Window one line at a time using the UpArrow and DownArrow keys, or one "screen" at a time using the PgUp and PgDn keys. You can jump directly to the top or bottom of the message using the Home and End keys. 1.3.4 Query Boxes A query box is a window that prompts the user for one or more replies. An example is the Solve Query Box where the user is asked for the function he wishes to solve and for the lower and upper bounds of the search range. A query box presents each query in the form of a prompt followed by a blank field. You enter your reply in this field. After you have typed your reply you move to the next item in the query box by typing <RETURN>. Once you have left a query you cannot return to it. You can, however, at any point during the query box operation, exit by pressing the <ESC> key. Any entries you have made will have no effect. MathTool User's Guide - Page 3 While you are entering a response to a query you can use the following editing keys: Left arrow: Move left one space Right arrow: Move right one space Home: Move to beginning of entry End: Move to end of entry Backspace: Delete character left of the cursor Escape: Exit F1: Help F3: Pick List (where supported) F10: Restore original string without exiting Ctrl-Backspace: Delete entire string 1.3.5 Help Context sensitive help is available at all times in MathTool. Press F1 for help. MathTool User's Guide - Page 4 2. MENU TREE Main Menu | | +---------------------------------------------------------+ | | | | | | | Files Define Evaluate differenTiate Integrate Solve Graph Menu Menu Menu Query Query Query Menu | | | Box Box Box | | | | | | | +--- Single eval | | | | | | | +- Range eval | | +---- New Single ---| | | | | |--- Edit Multiple ---| | | | | |--- Delete Types---------------------| | | Menu | | +--- Clear | | | |- Linear | |--- Load | | | |- lOg-log | |--- Write | | | +- Semilog | |--- Directory | | sCaling ---------------| +--- Quit Menu | | | |-- Autoscale | | | +-- User defined | | | data File -----+ MathTool User's Guide - Page 5 3. MAIN MENU There are seven menu choices on the Main Menu. Some lead to other menus and some lead to query boxes. These choices are described in the following sections. 3.1 Files The Files command on the Main Menu causes the Files Menu to be displayed. The Files Menu contains commands for reading and writing files, for displaying a directory listing, and for quitting the program. The Files Menu is discussed in section 4. 3.2 Define The Define selection on the Main Menu causes the Define Menu to be displayed. The Define Menu contains commands for defining functions, editing existing function definitions, and undefining functions. The Define Menu is discussed in section 5. 3.3 Eval The Eval command on the Main Menu causes the Eval Menu to be displayed. The Eval Menu contains commands for evaluating functions either at a single point or over a range of points. The Eval Menu is discussed in section 6. 3.4 differenTiate The differenTiate selection on the Main Menu causes the Differentiate Query Box to appear. This query box asks a set of questions related to the numeric differentiation of a function. The Differentiate Query Box is discussed in section 7. 3.5 Integrate The Integrate command on the Main Menu causes the Integrate Query Box to appear. This query box asks a set of questions related to the numeric integration of a function. The Integrate Query Box is described in section 8. 3.6 Solve The Solve selection on the Main Menu causes the Solve Query Box to be displayed. This query box contains a set of questions related to finding solutions of a function. The Solve Query Box is described in section 9. 3.7 Graph The Graph selection on the Main Menu causes the Graph Menu to be displayed. The Graph Menu contains a set of selections related to plotting a function or functions. The Graph Menu is described in section 10. MathTool User's Guide - Page 6 4. FILES MENU 4.1 Load The Load selection allows you to load a file of function definitions into MathTool. When you make this choice, the program displays the Files Query Box. This query box asks only for the name of the file that you wish to load. This file must be an ASCII file. It can be created using the Write selection from the Files Menu after defining your functions, or by any editor which writes ASCII files. You can add comments since anything on a line after an exclamation point is a comment. A directory path before the name of the file is optional. If you enter just a filename, the program looks for the file in the directory from which you started MathTool. If you wish to load a file in a different directory you must include the path. The path specifier may be either an absolute path such as (C:\MATH\DEF\FILENAME.DAT) or a relative path such as (..\DEF\FILENAME.DAT). If the program cannot find the file, it issues an error message. 4.2 Write The Write selection causes MathTool to write all the current function definitions to an ASCII file. After you make this selection MathTool will display a query box which asks you for the name of the file. The default directory is the current directory, that is, the directory from which you started MathTool. You can write the file to another directory by preceding the filename with a path specifier. Once you have saved a definitions file you can load it during future MathTool sessions. 4.3 Directory The Directory selection allows you to view a directory listing. After you make this selection MathTool displays a query box which asks you to enter a mask. This mask can include paths and wild cards. The default, chosen by simply pressing <RETURN>, is a complete listing of the current directory, the directory from which you started MathTool. 4.4 Quit Choose Quit when you wish to exit from MathTool and return to the DOS prompt. MathTool User's Guide - Page 7 5. DEFINE MENU 5.1 New Choose New when you wish to define a new function. After you make this choice a Definition Query Box will appear. This query box has only one question. The prompt is "Function:". You should type in the complete definition of your function, for example, f(x) = cos(10*x) - sin(7*x). When you press <RETURN>, MathTool will read and parse your definition. You can now use MathTool to investigate the function. If the definition you enter is one that MathTool cannot parse, a Correction Window will appear. The Correction Window contains a message describing why MathTool could not parse the function and a line on which you can edit it. The function as you defined it will appear on the line. You can edit using the same editing keys described in section 1.3.4 on Query Boxes. As an example of how this might work, say you wished to enter the function "f(x) = x - 1", but you made a typo and typed "f(x) = x -- 1" instead. The correction box would have the message that MathTool cannot parse a function that has two operators in a row and ask you to edit your function. Once you complete the editing and press <RETURN>, MathTool will parse the corrected function. 5.2 Edit The Edit selection allows you to change the definition of an already defined function. When you make this choice a Definition Query Box appears. The first entry in this box asks you for the old function. You can specify the function by typing its name, f(x), or by typing its complete definition, f(x) = x - 1, or by pressing <F3> to use the Pick List. The second entry asks you for the new function. Type in the new definition as described above under New, section 5.1. After you complete this step the function will have its new definition. This can be very useful when you are trying to determine the effect of varying some parameter in a more complicated function. Define the parameter as a constant function then use the constant in the more complicated function. It is then very easy to change the value of the parameter. 5.3 Delete This selection allows you to delete a function definition. The Definition Query Box, like the one for New, only asks for one entry, the function you wish to delete. You can specify this function using either its name, f(x), its definition, f(x) = x - 1, or by pressing <F3> to use the Pick List. 5.4 Clear Choose Clear when you want to delete all of the defined functions. After you execute this command you will have an empty function list as if you had just started the program. MathTool User's Guide - Page 8 6. EVAL MENU 6.1 Single This command allows you to evaluate a function at a single point. When you choose this command an Evaluate Query Box appears. First it asks you for a function. You can specify the function in one of three ways: type its name, type its entire definition, or press <F3> to use the Pick List. The second question is the value of the independent variable at which you wish to evaluate the function. Enter the numerical value. Next MathTool will display two windows. The upper window will contain the complete definition of the function you chose, including subfunctions. If the function definition is so large that the window must go into scolling mode, then the second window will not appear until after you exit the first window by pressing either <ESC> or <RETURN>. The lower window will contain the value of the function for the specified value of the independent variable. 6.2 Range This selection allows you to evaluate a function over some range of values. After you choose this command MathTool will display an Evaluate Query Box. The first entry asks you to specify the function you wish to evaluate. You can do this in one of three ways: type its name, type its entire definition, or press <F3> to use the Pick List. The second and third entries ask for lower and upper bounds of the range over which you wish to evaluate the function. The fourth entry asks for the number of divisions within the range you wish. An example may make this more clear. Say your function was f(x) = x^2, the lower bound 0, the upper bound 10, and the number of divisions 10. The program will display the following eleven values: f(0) = 0.0000 f(1) = 1.0000 f(2) = 4.0000 f(3) = 9.0000 f(4) = 16.0000 f(5) = 25.0000 f(6) = 36.0000 f(7) = 49.0000 f(8) = 64.0000 f(9) = 81.0000 f(10) = 100.0000 MathTool User's Guide - Page 9 7. DIFFERENTIATE Choosing the differenTiate Main Menu command allows you to numerically evaluate the derivative of a function. After you choose this command, MathTool displays the Differentiation Query Box. The first entry in this query box asks you to specify which function you wish. You can do this in any of the standard three ways: type the function's name, type the function's entire definition, or press <F3> to use the Pick List. The second entry in the query box asks for the value of the independent variable at which you wish to evaluate the derivative. Enter a numerical value here. Shortly MathTool will display two windows. The upper window will contain a complete definition of your chosen function. The lower window will contain the value of the derivative. MathTool User's Guide - Page 10 8. INTEGRATE The Main Menu Integrate command allows you to numerically evaluate the integral of a function. When you choose this command, MathTool displays the Integrate Query Box. The first question asks you to identify which function you wish to integrate. You can do this in any of the standard three ways: type the function's name, type the function's entire definition, or press <F3> to use the Pick List. The second and third questions ask for the upper and lower limits of the integral. The fourth question asks for the number of integration steps to use in evaluating the integral. A warning is in order about this fourth step. MathTool uses Simpson's Rule to do numerical integration. Simpson's Rule, described in any introductory college calculus text, requires the number of divisions of the range of the integral to be even. If you enter an odd number, MathTool will issue an error message. After you finish answering the queries, MathTool again displays two windows. The upper window contains the complete definition of the function you chose. The lower window contains the value of the integral. MathTool User's Guide - Page 11 9. SOLVE The Solve command on the Main Menu allows you to find the roots of a function. When you choose this command, MathTool displays the Solve Query Box. The first entry asks you to identify which function you wish to integrate. You can do this in any of the standard ways: type the function's name, type the function's entire definition, or press <F3> to use the Pick List. MathTool uses an algorithm that searches for roots within a range of values of the independent variable. The second and third entries ask you to define the lower and upper limits of this search range. After you finish the query box, MathTool once again displays two windows. The upper window contains the complete definition of the function you specified, including subfunctions. Since finding roots can take some time, MathTool first displays a message that it is working in the lower window. After it has finished its search, Mathtool displays the roots it found in the lower window. Each root is displayed in the form "f(a) = b" where f(x) is the specified function, a is the value of the independent variable, and b is the value of the function. Usually b will have a value on the order of 1E-8 or less. MathTool User's Guide - Page 12 10. GRAPH MENU 10.1 Single Make this selection from the Graph Menu when you wish to plot a function. The Graph Query Box will appear. This query box will ask you either four or six questions, depending on the type of scaling in effect. First it asks you to specify your function. You do this using any of the three standard methods - type the function's name, type the function's complete definition, or press <F3> to use the Pick List. The second and third entries are the lower and upper values of the independent variable. The fourth entry is the number of points to be evaluated. The number of points you should choose depends on several things. More points equals more processing equals more time. However, more points also means finer resolution of the plot. Generally speaking, the more complicated your function, the more points you need to plot. A straight line only needs two points to display all the information. Second and third order polynomials need fifty to one hundred points. Very complicated functions such as complex trigonometric functions will need 200 or more points. Going above about 600 points will buy you nothing, however, since the graphics window only has about 600 pixels in the horizontal dimension. If the scaling is AutoScale, then you will be finished with the query box after answering these four questions. If it is User-Specified, then you will be asked to specify the lower and upper bounds for the vertical dimension in the plot window. Once you've completed the query box the processing for your plot begins. While this is happening MathTool displays a message telling you that it is working. As soon as it finishes it displays the plot. While the plot is displayed you may print it by pressing the letter "p", either lower or upper case. I developed and tested the printing function using a Star Micronics SG-10 dot matrix printer. This printer is an IBM Graphics Printer clone, so printing should work on any printer which emulates the IBM. After printing finishes you are returned to the main menu. Pressing any key other than "p" also returns you to the main menu. 10.2 Multiple Make this selection from the Graphics Menu when you wish to plot more than one function on the same graph. After you make this choice a series of Graphics Query Boxes will appear. The first asks you how many functions you wish to plot. MathTool allows as many as five. The rest of the series consists of one query box for each function. Each of these will be identical to the one described above in the discussion of the Single menu choice. If the scaling mode is user-specified you will be asked to specify the MathTool User's Guide - Page 13 lower and upper bounds of the vertical axis in the first query box. You only need to specify these values once so these questions will not appear in the later query boxes. After you finish each query box, MathTool does the processing for that function and then displays the next query box. When all the processing is done, MathTool displays the plot. If you have a color graphics system, each curve will be in a different color. The plot header will have the main definition for each curve, but subfunction definitions will not be displayed. Once again, you can print the plot by pressing "p"; pressing any other key returns you to the main menu. 10.3 Types MathTool supports three types of plots: linear, log-log, and semilog. You use this menu to determine what type of plot MathTool will use. After you make your choice, MathTool returns you to the Graph Menu. 10.4 Scaling MathTool supports two kinds of scaling, AutoScale and User- Specified. In autoscaling mode, MathTool finds the maximum and minimum values of your function over the range to be plotted. These two values then become the maximum and minimum values on the vertical axis and thus determine the vertical scaling. In user-specified mode, you directly control the scaling of the vertical axis by specifying the maximum and minimum values. 10.5 Data File Choose the data File selection on the Graphics menu when you wish to plot the contents of a data file. The sample file data.dat included on the MathTool distribution disk shows the format a MathTool data file must have. Comments can be included on any line after an exclamation point. Data must be tab delimited. The program reads the file and plots the data. Only linear plots are available in this mode and no header will be displayed. MathTool User's Guide - Page 14 11. DEFINING A FUNCTION 11.1 Form MathTool allows you to define virtually any function of a single variable. The independent variable and the function name can each be as long as ten characters each. So you can have a function like "f(x) = cos(x)" or one like "pressure(depth) = AtmPressure + rho*g*depth". This function gives the pressure in a liquid, where AtmPressure is the atmospheric pressure, rho is the liquid's density, g is the gravitational acceleration, and depth is the depth below the surface. It is defined in the file Pressure.dat on the MathTool distribution disk. Two warnings are necessary. First, if you define a function or a variable whose name is longer than ten characters, MathTool is guaranteed to act erratically. Second, MathTool will become very confused if you define two functions with the same name. For a detailed discussion of potential traps and pitfalls in function definition, see Difficulties, section 11.5. 11.2 Operators The following operators can be used in MathTool function definitions. OPERATION OPERATOR EXAMPLE Addition + 2 + 2 = 4 Subtraction - 3 - 2 = 1 Multiplication * 3 * 2 = 6 Division / 4 / 2 = 2 Power ^ 2 ^ 3 = 8 11.3 Math functions MathTool provides an extensive library of mathematical functions that you can use in your functions. These are listed below with their syntax. FUNCTION EXAMPLE Absolute value abs(x) Exponential exp(x) Cosine cos(x) Sine sin(x) MathTool User's Guide - Page 15 Tangent tan(x) Inverse cosine acos(x) Inverse sine asin(x) Inverse tangent atan(x) Factorial fac(x) Hyperbolic sine sinh(x) Hyperbolic cosine cosh(x) Hyperbolic tangent tanh(x) Inverse hyperbolic sine asinh(x) Inverse hyperbolic cosine acosh(x) Inverse hyperbolic tangent atanh(x) Logarithm, base 10 log(x) Natural logarithm ln(x) Please note that all angles are in radians. 11.4 Sets of functions One of the strengths of MathTool is that you define interrelated sets of functions. A good example of this is provided in the file Sqwave.dat on the distribution disk. This file contains a Fourier series approximation of a square wave. You can see this very clearly if you load the file and plot it over the range -5 to 5 with 200 points. This example shows how you can build very complicated functions in a piecewise fashion. 11.5 Difficulties In the introduction I mentioned that MathTool can handle virtually any well-behaved function of a single variable. A definition of what constitutes a well-behaved function for MathTool is difficult, although certainly any function that fits the calculus definition of a continuous function would qualify as well-behaved. Functions that are not continuous can cause problems. Generally speaking, there are three ways that a function can fail to be continuous. All three can potentially cause problems for MathTool, especially in the graphics area. First, it may happen that a function does not exist at some point. One example might be a rational function at a point where the denominator becomes zero. If MathTool attempts to evaluate the function at this point, it will issue an error message. However, you may ask for a plot of this function over a range that includes such a point but, because of your choice of end points and number of divisions, MathTool will not attempt to evaluate the function at that point. A plot will appear, but the fact that the function does not exist at some point will not be apparent. You would only become aware of this discontinuity if MathTool tried to evaluate the function at the actual point of discontinuity and issued a division by zero error message. MathTool User's Guide - Page 16 As an aside, you may at some point wish to investigate a function that has one or more discontinuities. The discussion above points out a strategy you can use with these types of functions. Just be sure to define your endpoints and number of subdivisions so as to avoid the discontinuity. Second, consider the function f(x) = sin(x)/x , x <> 0 f(x) = 0, x = 0 Note: The symbol "<>" is notation for inequality. Discontinuous functions of this nature cannot be defined in MathTool. Only one of the two parts, say the first, could be defined. Then f(x) = sin(x)/x becomes the type of function discussed above - it has a discontinuity at x = 0. To see how this can cause problems, run MathTool and define f(x) = sin(x)/x. Try to evaluate it first at x = 0.0 and then at x = 1.0e-8. When x = 0 you hit the discontinuity and MathTool gives you the wrong answer. When x = 1.0e-8, no problems. For a graphic demonstration of this problem, plot this function from -30 to 30 twice - the first time with 200 points, the second time with 201 points. Third, and most troublesome to MathTool, are functions that are not defined at some point, but do have limits that vary depending on the direction of approach to the point. An example of this type of function is tan(x). As x approaches pi/2 from below, tan(x) goes to infinity. As x approaches pi/2 from above, tan(x) goes to negative infinity. If you plot tan(x) from x = 1 to x = 2, you will see a nearly vertical line at x = pi/2 because MathTool connects the points on either side of pi/2. The upshot of all this is, if you are dealing with non- continuous functions, be careful. If you get plots that seem strange, check your function to see if it is one of these types. Some other potential problems bear mentioning as well. When you are plotting functions in log-log or semilog modes, you should make sure to avoid zero or negative numbers since their logarithms are undefined. If you define a function or formula that represents a physical relationship such as the formula in Pressure.dat, you should be careful to use consistent units. This is entirely up to you because MathTool knows nothing about units. MathTool User's Guide - Page 17 12. EXAMPLES MathTool comes with several files that contain sets of functions ranging from very simple to very complicated. Each file is described below. Abc.dat contains a simple straight line and illustrates how to define and use constant functions. Cosines.dat contains one of my favorite test functions. I used this file extensively while I was developing MathTool to check out the graphics and solving capabilities. This file shows how to build a complicated function from simpler functions. If you want to duplicate one of my tests, try solving f(x) = cos(8*x) + cos(9*x) over the range 0.0 to 4.0. See how long it takes MathTool to find the eleven(!) roots in this range. Poly.dat contains a set of arbitrary polynomials. This example also shows how to build very complex functions from simpler ones. Exp.dat contains two main functions. The function f(x) is the Taylor series approximation for exp(x). This Taylor series can be made to approximate exp(x) to any degree of closeness by adding more terms to the series. The second function, g(x), is the difference between the Taylor series approximation and the actual exponential. Playing around with functions like this will quickly give you a good idea of how useful a learning tool MathTool can be. Pressure.dat contains the formula for determining the pressure in a liquid at some distance below the surface, where rho is the density of the liquid, AtmPress is the atmospheric pressure at the surface, g is the acceleration of gravity, and depth is the distance. Data.dat contains a sample data file that can be plotted using the "data File" option on the Graphics Menu. This file shows how a data file needs to be formatted for MathTool to read it. Sqwave.dat is the most ambitious of the examples included with MathTool. This example is the Fourier Series approximation of a square wave. Fourier was a nineteenth century French mathematician. He proved that any periodic function can be expressed as the sum of an infinite series of sinusoidal functions. The square wave approximation demonstrates this very clearly. This is another great example of how useful MathTool can be for a student - it's very illuminating to add terms to the series one by one and see how quickly the series starts to approximate a square wave. MathTool User's Guide - Page 18 13. LEGALITIES MathTool is shareware, also known as user-supported software. The diskette with the program and the user's guide can be freely copied and shared. The idea of shareware is that if the user finds the program worthwhile then he can, at his own discretion, support the author by sending him a contribution. MathTool is NOT a public domain program. It is copyright (c) 1990 by Martin J. Maher, and the author reserves all rights. In particular, he reserves the right to distribute MathTool, associated libraries, and all source code and documentation for profit. There is no guarantee that MathTool will work correctly in all situations, and in no event will the author be liable for any damages arising from the use or misuse of MathTool. User groups, bulletin boards, clubs, and shareware distributors are authorized to distribute MathTool under the following conditions: 1. No charge is made for the software or the documentation. A nominal fee may be charged to cover copying and distribution costs. 2. Recipients are to be notified of the shareware concept and should be encouraged to support it. 3. The program and documentation are not to be modified in any way. 4. The source code for MathTool is available for purchase by registered users. They may alter it for their own use but may not distribute it for profit. Turbo Pascal is a trademark of Borland International, Inc. Compaq and Compaq Deskpro are trademarks of Compaq Computer Corp. MS-DOS is a trademark of Microsoft, Inc. IBM, PC-DOS, CGA, EGA, and VGA are trademarks of International Business Machines, Inc. Hercules is a trademark of Hercules Computer Technologies, Inc. MathTool User's Guide - Page 19 14. PRODUCT INFO As stated above in the Legalities section, MathTool is shareware. I hope that you find MathTool to be a useful and even enjoyable piece of software. If you do, I hope that you will register your copy. I've devoted a lot of time and effort to developing MathTool, and, quite frankly, I would like to get paid. Ok, fair enough, you say, but what's in it for me? When you register you will receive the latest version of MathTool, if it is different than the version you register. When future upgrades are released they will be available to you at reduced prices. My policy with version two has been to offer a free upgrade to every registered user of version one. While I cannot promise that this policy will apply in the future, I can promise that your savings will be considerable. Finally, a modest amount of technical support is available for registered users. I can be reached either by U.S. mail or on CompuServe as 72040,2404. I will try to answer your questions or help if you have problems. There's one other thing I would like you to do if you like this program - make copies and give them to your friends and associates. Not only are you allowed to do this with shareware, but I encourage you strongly to do it with mine. Other products will soon be available from MJM Engineering. MathTool incorporates the object-oriented programming technology introduced by Borland International in version 5.5 of Turbo Pascal. Most of the functionality in MathTool is based on objects contained in two extensive libraries, and these two libraries will soon be available either in object form, actually TPU files, or source-code form. Both will come with programmer's guides and example source code. Both libraries should be available in early 1991. All MathTool registered users will be informed of their release and, as another benefit of registration, will be able to purchase the libraries for reduced fees. Finally, all the source code for MathTool is available. This option includes complete source code for the entire program, including source code for both libraries and both programming guides when they become available. A handy order form with prices is included on the next page. One note - if you register MathTool without using the order form, please include a note telling me the version number of your copy of the program. This will help me determine if I need to send you an updated version. MathTool User's Guide - Page 20 15. ORDER FORM Name_________________________________________________________ Address______________________________________________________ City/State/Zip_______________________________________________ Item Price Quantity Subtotal MathTool registration 20.00 ___ ______ Version 2.03 Interface object library* 20.00 ___ ______ with source 40.00 ___ ______ Function object library* 20.00 ___ ______ with source 40.00 ___ ______ Note: requires interface library Both libraries* 35.00 ___ ______ with source 70.00 ___ ______ MathTool registration with complete source code for MathTool and both libraries 80.00 ___ ______ TOTAL ______ * Libraries available first quarter 1991. Shipping and handling and any applicable sales taxes are included in the these prices. If you are ordering from outside the United States, please send your funds as a money order drawn in U.S. dollars. Thank you for your support. Mail to: Martin J. Maher MJM Engineering P.O. Box 2027 Hawthorne, CA. 90251 How did you learn about MathTool, how did you get your copy? ________________________________________________________________ ________________________________________________________________ Comments or suggestions _______________________________________________________________ _______________________________________________________________ MathTool User's Guide - Page 21