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## Contents of the MTOOL.DOC file

MATHTOOL

AN INTERACTIVE

MATHEMATICAL

FUNCTION INTERPRETER

USER'S GUIDE

VERSION 1.1

(C) COPYRIGHT 1988 - ALL RIGHTS RESERVED

by

MARTIN J. MAHER

MJM Engineering

P.O. BOX 2027

HAWTHORNE,CA 90251

TABLE OF CONTENTS

-----------------

ITEM PAGE

---- ----

TABLE OF CONTENTS .................................. 1

INTRODUCTION ...................................... 2

HISTORY ............................................ 3

USING MTOOL ........................................ 4

EDITING ............................................ 5

ENTERING A FUNCTION ................................ 6

MAIN MENU .......................................... 10

EVALUATION AT A SINGLE POINT ....................... 11

EVALUATION OVER A RANGE OF POINTS .................. 12

INTEGRATION ........................................ 13

EVALUATION AND INTEGRATION COMBINED ................ 14

DERIVATIVE ......................................... 15

SOLUTIONS .......................................... 16

GRAPHICS ........................................... 17

DIFFICULTIES ....................................... 19

LOOPING ............................................ 20

APPLICATIONS ....................................... 21

LEGALITIES ......................................... 25

SHAREWARE .......................................... 26

REGISTRATION FORM .................................. 27

1

INTRODUCTION

------------

Mtool is a function interpreter. The "M" in Mtool stands for

"Math." The program can handle virtually any well-behaved

mathematical function of a single variable. It is designed to be easy

to use, prompting the user at every step for whatever information it

needs. Once you have defined a function, Mtool gives you the

following capabilities:

1) Evaluate the function for any value of the independent

variable.

2) Evaluate the function over a range of values of the

independent variable.

3) Numerically integrate the function.

4) Evaluate the derivative of the function for a given

value of the independent variable.

5) Find solutions of the function over some range of

the independent variable.

6) Plot the function.

Mtool is written in Turbo Pascal, and all of its operations are

carried out completely in memory. Mtool requires a minimum of 256k

of memory. Since Mtool uses dynamic data structures, it can access

as much memory as is available. Testing shows, however, that even

very complicated functions rarely require more than about 50k of

memory. The plotting capability requires graphics capability. Since

Mtool is written in Turbo Pascal, version 4.0, it supports all common

PCompatible graphics standards - CGA, EGA, VGA, and Hercules.

2

HISTORY

-------

Mtool fills a need that I first perceived several years ago when

I was a student in engineering school. Often when I was reading some

text or other, I came across a complicated mathematical function or

formula. Normally the author of the text would make some statements

about the formula regarding its behavior or certain of its

properties. Usually at this point I thought that seeing a plot of

the function would make things much clearer. Most of the time I had

three alternatives - look in vain in the text for a plot; plot it

myself by hand, a job only a masochist could enjoy; or forget about

it. Naturally, alternative three was my usual choice.

Eventually, I became the owner of a Compaq Deskpro and a copy of

the Turbo Pascal compiler. I started writing a program that would

plot any function that I could type in. My purpose, at first, was

only to learn Turbo Pascal. Then I started adding some other things,

like integration and derivatives. The idea gradually grew that Mtool

might make a good shareware program, and so here it is.

The full power of Mtool is demonstrated later in this document

with the inclusion of script that you can use to generate some very

complicated results. The program is powerful enough that I wish I

would have had it when I was in engineering school, for it certainly

adds a fourth and most attractive alternative to the list above.

3

USING MTOOL

-----------

Using Mtool is easy. It is merely a matter of answering prompts

the program issues, and of making menu choices. First, the program

leads you through a series of steps which define your function.

Options such as the ability to define characters representing

constants and subfunctions add to the complexity of function that you

can define.

After your function is defined, Mtool presents you with its main

menu. Depending on your choice Mtool further prompts you for any

information it needs. Finally, after Mtool has completed your choice

of operations, it allows you the options of continuing to use the

same function, of defining a new function, of editing your function,

or of leaving the program.

The next several sections of this document give a detailed

description of how to use Mtool, including many examples.

4

EDITING

-------

Mtool includes some simple editing tools as part of its user

interface. It shows, in reverse video, the length of string that can

be entered in reply to a prompt. For example, the main function

definition can hold as many as 73 characters, so the main function

prompt is followed by a reverse video bar 73 spaces long. Another

example is the independent variable definition which allows only one

character and so is followed by one reverse video space.

Mtool supports five editing keys. The keys and their functions

are:

Left arrow - moves cursor to left.

Right arrow - moves cursor to right as far as end of string.

Backspace - deletes a character and moves cursor to left.

End - moves cursor to end of string.

Home - deletes entire string.

A character is replaced by placing the cursor directly on it and

typing the desired character - that is, the editor works in

overstrike mode. Insert mode is not supported at this time.

Suppose that you had entered the string

cos98*x) + cos(7*x)

and wished to replace the "9" with a "(". Assuming that the cursor

is at the end of the string, hold down the left arrow key until the

cursor is on the "9", press "(", then press "End" to return the

cursor to the end of the string. One note of caution - if you press

the return key the string will be truncated at the cursor position.

Any characters to the right of the cursor will be lost, so pressing

the end key before the return key is important.

These simple editing keys apply at all Mtool prompts.

5

ENTERING A FUNCTION

-------------------

Mtool starts with a message from the author. After this

advertisement, Mtool issues its first prompt:

Independent variable:

Any lower case letter except for the first four are legal

responses to this prompt. The letters "a" through "d" are reserved

for use as subfunctions, which will be described below. If you try

to enter an answer that the program does not allow, it will issue a

warning, beep, and prompt again:

Independent variable:

Invalid choice - try again.

This will continue until Mtool gets an answer it likes, namely,

any lower case letter from "e" to "z".

Defining an independent variable does not mean that the other

letters from "e" to "z" are off limits. You can use any of them

anywhere in your function, and Mtool will consider them to be

constants and query you for their values - more on this later.

Next, Mtool asks for subfunctions. A subfunction is best

described with an example. Suppose that

f(x) = a + b

where a = cos(8*x) and b = cos(9*x).

Then a and b are subfunctions. Mtool allows you to define as many as

four subfunctions, each of which may contain as many as fifty

characters. Any operation that can be performed on a variable can be

performed on a subfunction, so in the example above we could have

f(x) = a*b, or f(x) = a/b, or f(x) = a^b, etc.

The program asks how many subfunctions you wish to define. The

appearance of the second screen depends on your answer to this

prompt. The number of subfunction lines which appear is the same as

your answer. An example of the first two screens with four

subfunction definitions and a main function definition appears on the

next page.

6

INITIALIZATION

Independent variable: x

How many subfunctions(min 0; max 4): 4

ENTER FUNCTION

f(x) =

where a =

b =

c =

d =

a = cos(3*x)

b = cos(5*x)

c = cos(7*x)

d = cos(9*x)

f(x) = a + b + c + d

Before the program accepts a subfunction definition, it makes a

first pass at parsing the expression. In this pass it is looking for

errors or constructions that it cannot deal with. Examples are

unbalanced parentheses or back-to-back operators. If the program

detects any errors, it will issue a warning message and prompt again

for the same subfunction. It will not allow you to enter a string of

more than fifty characters as a subfunction.

The next prompt after the subfunctions have been defined is the

main function prompt. This string may be as long as 73 characters.

Once again, the program will make a first pass at the string looking

for errors or bad constructions. It will also make sure that you

have not attempted to use an undefined subfunction character. If it

finds any problems, it will issue a message and prompt again for the

function definition.

Besides variables and constants, Mtool recognizes several

operators and many standard mathematical functions. A complete list,

with examples, follows on the next page.

7

Operators

---------

Addition + 2 + 2 = 4

Subtraction - 3 - 2 = 1

Multiplication * 3 * 2 = 6

Division / 4 / 2 = 2

Exponential ^ 2 ^ 3 = 8

Mathematical Functions

----------------------

Absolute value abs(x)

Exponential exp(x)

Cosine cos(x)

Sine sin(x)

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)

The x in each math function represents the argument of the

function. It must be enclosed within parentheses. This argument can

be any legal Mtool expression. The arguments for trigonometric

functions and the values of inverse trigonometric fuctions are

assumed to be in radians. If you wish to use degrees, then include a

factor to convert from degrees to radians. This can be done by using

two constants, setting one equal to pi and the other equal to 180.

An example of how to do this is shown in the Main Menu section below.

Some examples follow. These examples show all the Mtool prompts

and all the necessary replies to define functions.

Example 1 - Fifth degree polynomial

Independent variable: x

How many subfunctions(min 0; max 4): 0

Function definition:

f(x) = x^5 + 4*x^4 - 7*x^3 + 10*x^2 + 2*x - 12

8

Example 2 - Trigonometric function with constants

Independent variable: x

How many subfunctions(min 0; max 4): 0

Function definition:

f(x) = cos(m*x) + cos(n*x)

(Later Mtool will ask for values for m and n.)

Example 3 - Function with one subfunction

Independent variable: x

How many subfunctions(min 0; max 4): 1

Subfunction definition:

a = 3*x^3 + 2*x^2

Function definition:

f(x) = cos(a)

Example 4 - Function with four subfunctions

Independent variable: t

How many subfunctions(min 0; max 4): 4

Subfunction definitions:

a = cos(t) + cos(3*t)

b = cos(5*t) + cos(7*t)

c = cos(9*t) + cos(11*t)

d = cos(13*t) + cos(15*t)

Function definition:

f(t) = a + b + c + d

9

MAIN MENU

---------

After the function has been defined, Mtool will echo the

function definition and list the main menu. If the function were the

one in Example 4 above, then Mtool would put this on the screen:

f(t) = a + b + c + d

where a = cos(t) + cos(3*t)

b = cos(5*t) + cos(7*t)

c = cos(9*t) + cos(11*t)

d = cos(13*t) + cos(15*t)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice:

Mtool expects you to answer with a number from 1 to 7 depending

on what you want it to do. After you make your choice Mtool will ask

you for any other information it needs. This information, and thus

Mtool's prompts, vary depending on your choice. The next several

sections will describe these prompts and possible responses.

10

EVALUATION AT A SINGLE POINT

----------------------------

The first choice in the menu causes Mtool to evaluate the

function for a single value of the independent variable. This means,

of course, that Mtool must query you for the value you wish to use.

Mtool will also query you for the values of any constants that may

have been included in your function. An example that includes

constants is shown below.

f(x) = cos(m*x) + cos(n*x)

What is the value of x: 0.2000

What is the value of n: 9.0000

m = 8.0000

n = 9.0000

Evaluating function

f( 0.2000 ) = -2.5640161698E-01

Press any key to continue.

Note that the arguments of the cosine terms in this function

have the unit radians. Also note that the spacing that appears on

the screen is accurately represented in this example. The prompt

asking for the value of m was issued but is not shown here because it

was overwritten by the prompt for n.

11

EVALUATION OVER A RANGE OF POINTS

---------------------------------

The second menu selection causes Mtool to evaluate the function

over a range of points. How this works should be clear from the

example below. Screen formats are not reproduced in this example.

f(x) = cos(p*x/n)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 2

What is the value of p: 3.141592654

n = 180.0000

Lower limit of x: 0.0000

Upper limit of x: 10.0000

Number of subdivisions(even for integration): 10

Evaluating function...

f( 0.0000000000E+00 ) = 9.9999999999E-01

f( 1.0000 ) = 9.9984769515E-01

f( 2.0000 ) = 9.9939082702E-01

f( 3.0000 ) = 9.9862953475E-01

f( 4.0000 ) = 9.9756405026E-01

f( 5.0000 ) = 9.9619469809E-01

f( 6.0000 ) = 9.9452189537E-01

f( 7.0000 ) = 9.9254615164E-01

f( 8.0000 ) = 9.9026806874E-01

f( 9.0000 ) = 9.8768834059E-01

f( 10.0000 ) = 9.8480775301E-01

Press any key to continue.

Several things should be noted here. First, the total range of

values of the independent variable is defined by the responses to the

lower and upper limit prompts. Second, the number of points at which

the function is evaluated is always one greater than the number of

subdivisions. Third, the independent variable increment is the total

range divided by the number of subdivisions.

This example also shows how conversion from radians to degrees

works. The evaluated values are the cosines of angles from zero to

ten degrees at one degree intervals.

12

INTEGRATION

-----------

Mtool uses Simpson's Rule to numerically integrate. Simpson's

Rule requires that the interval of integration be divided into an

even number of subintervals. The greater the number of subintervals,

the more accurate the answer. Here's an example of how this works in

Mtool. All Mtool output is present, but screen formats are not

reproduced.

f(x) = 6*x^2

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 3

Lower limit of x: 0.0000

Upper limit of x: 5.0000

Number of subdivisions(even for integration): 20

Evaluating integral...

From 0.0000000000E+00 to 5.0000000000E+00

integral = 2.5000000000E+02

The antiderivative of this function is 2*x^3, and this evaluated

over the interval zero to five gives 250.0, which agrees exactly with

Mtool's result.

13

INTEGRATION AND EVALUATION OVER A RANGE OF POINTS

-------------------------------------------------

This selection combines the two previous functions. The inputs

and outputs are the same as before, as the following example shows.

Once again, all output is present, but screen formats are not

preserved.

f(x) = 6*x^2

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 4

Lower limit of x: 0.0000

Upper limit of x: 5.0000

Number of subdivisions(even for integration): 10

Evaluating function and integral...

f( 0.0000000000E+00 ) = 0.0000000000E+00

f( 0.5000 ) = 1.5000000000E+00

f( 1.0000 ) = 6.0000000000E+00

f( 1.5000 ) = 1.3500000000E+01

f( 2.0000 ) = 2.4000000000E+01

f( 2.5000 ) = 3.7500000000E+01

f( 3.0000 ) = 5.4000000000E+01

f( 3.5000 ) = 7.3500000000E+01

f( 4.0000 ) = 9.6000000000E+01

f( 4.5000 ) = 1.2150000000E+02

f( 5.0000 ) = 1.5000000000E+02

From 0.0000000000E+00 to 5.0000000000E+00

integral = 2.5000000000E+02

Press any key to continue.

14

DERIVATIVE

----------

This selection evaluates the derivative of the function for some

particular value of the independent variable. It uses the standard

definition of the derivative as a limit. This definition can be

found in any elementary calculus text. Only one prompt is necessary

after the menu choice, the value of the independent variable. The

following example shows how it works.

f(x) = 6*x^2

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 5

Value of x : 2

Evaluating derivative....

fprime( 2.0000000000E+00) = 2.3997318793E+01

Mtool's result is returned as fprime. Of course, the derivative

of 6*x^2 is 12*x, and for x = 2, 12*x = 24. Mtool's result is close

to this, but not exactly correct. This is an unavoidable byproduct

of the differing natures of computation and mathematics, but the

accuracy of derivatives in Mtool will rarely if ever be less than

four places.

15

SOLUTIONS

---------

One of Mtool's most powerful functions is its ability to find

roots of equations. All it requires is a range of independent

variable values within which to look. The example below shows a case

in which eleven roots are found.

f(x) = cos(8*x) + cos(9*x)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 6

Enter range to search for solution.

Lower bound: 0.0000

Upper bound: 4

Solutions between x = 0.0000000000E+00 and x = 4.0000000000E+00

At x = 3.8807909265E+00 f(x) = 9.1622496257E-09

At x = 3.5111917891E+00 f(x) = 7.7397999121E-10

At x = 3.1415927394E+00 f(x) = 9.0949470177E-13

At x = 2.7719935179E+00 f(x) = 6.3664629124E-12

At x = 2.4023943823E+00 f(x) = -7.7034201240E-10

At x = 2.0327952467E+00 f(x) = 2.2337189876E-09

At x = 1.6631961111E+00 f(x) = -4.2809915612E-09

At x = 1.2935969755E+00 f(x) = 6.7411747295E-09

At x = 9.2399783991E-01 f(x) = -9.4601091405E-09

At x = 5.5439870339E-01 f(x) = -3.0299815990E-09

At x = 1.8479956780E-01 f(x) = 1.0559233488E-09

Press any key to continue.

The criterion used by Mtool is that, if the absolute value of

f(x) is less than 1.0e-8, then x qualifies as a root. Two cautions

are necessary. First, for some functions such as very high-order

polynomials, Mtool may not have great enough precision to find a

small enough value of f(x). In these cases Mtool will write its best

estimate along with the value of f(x) so that you can judge whether

it is indeed a root. Second, it may rarely occur that Mtool might

miss a root. This may occur when two roots are so close together

with respect to the size of the range being searched that Mtool sees

only one of them. Reducing the size of the range will allow Mtool to

see both. Coordinating Mtool's solution finding and graphics

capabilities will help you deal with these possibilities.

16

GRAPHICS

--------

Probably the most useful of Mtool's functions is its ability to

quickly generate plots of functions. Certainly this is the one that

I find most useful - indeed, it was just to have this that I wrote

the program in the first place. Once again, getting Mtool to do its

thing is very simple. Two examples are shown below - one for

automatic scaling and one for user-defined scaling.

EXAMPLE 1

---------

f(x) = cos(8*x) + cos(9*x)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 7

Lower limit of x: 0.0000

Upper limit of x: 10.0000

Number of subdivisions(even for integration): 400

Automatic y-axis scaling(Y/N): y

EXAMPLE 2

---------

f(x) = cos(8*x) + cos(9*x)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 7

Lower limit of x: 0.0000

Upper limit of x: 10.0000

Number of subdivisions(even for integration): 400

Automatic y-axis scaling(Y/N): n

Minimum y-axis value: -2

Maximum y-axis value: 2

17

The one thing missing in each of these examples is, of course,

the plot. But this is simple for you to remedy - just run Mtool,

type in the values shown, and you'll have the plots in about a minute

or so for each.

The autoscaling works by finding the maximum and minimum values

the function takes over the range of the plot, and setting the y-axis

accordingly. The user-defined option allows you to set the scale of

the y-axis so that you can compare different curves.

Mtool supports all common graphics standards - CGA, EGA, VGA,

and Hercules. It automatically detects what graphics board you have

in your computer and adjusts itself. This capability is one of the

many features added in version 4.0 of Turbo Pascal and - free

advertising for Borland International - a good reason to upgrade if

you haven't already.

Finally, if you have a graphics printer you can get printouts of

your plots. Before running Mtool run the program Graphics included

with PC/MS-DOS. This enables the computer to do a dump of a graphics

screen to the printer. When you have a plot that you wish to print,

just push the PrintScreen key.

18

DIFFICULTIES

------------

In the introduction I mentioned that Mtool can handle virtually

any well-behaved function of a single variable. A definition of what

constitutes a well-behaved function for Mtool is difficult, although

certainly any function that fits the calculus definition of a

continous 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 Mtool,

especially in the graphics area.

First, it may happen that a function does not exist at some

point. An example would be a rational function at a point where its

denominator becomes zero. If Mtool attempts to evaluate the function

at this point, it will issue an error message. However, it may occur

that you ask for a plot of this function over a range that includes

such a point but because of the choice of endpoints and number of

subdivisions, Mtool does 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 from the plot. You would

only become aware of this discontinuity if Mtool tried to evaluate

the function at this point and issued a division by zero error

message.

Second, consider the function

f(x) = sin(x)/x , x <> 0

f(x) = 0, x = 0

Discontinuous functions of this nature cannot be defined in Mtool.

Only one of the two parts, say the first, could be defined. Then

f(x) = sin(x)/x becomes a type one function.

Third, and most troublesome to Mtool, 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 near x = pi/2 because Mtool 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, and if you get plots that seem

strange, check your function to see if it is one of these types.

19

LOOPING

-------

After any main menu operation has been completed, Mtool offers

you the chance to use the same function again without reentering the

subfunction and function definitions. The menu that it presents

looks like this:

f(t) = a + b + c + d

where a = cos(t) + cos(3*t)

b = cos(5*t) + cos(7*t)

c = cos(9*t) + cos(11*t)

d = cos(13*t) + cos(15*t)

S)ame function

N)ew function

E)dit function

Q)uit

Enter choice:

Mtool displays your current function definition and its simple

menu. The choices are obvious.

If you do choose to reuse the same function and if the function

contains constants, then Mtool will ask if you wish to change their

values. The prompts look like this:

Value of m same( 8.0000000000E+00)..(Y/N): y

Value of n same( 9.0000000000E+00)..(Y/N): y

Mtool shows the current value of the constant and asks if you

wish to leave it unchanged. If you answer no for any constant, then

at the normal position following the main menu, Mtool will ask you

for the new value. Immediately after you have answered these

constant prompts, Mtool goes into the main menu.

20

APPLICATIONS

------------

This section includes two rather complicated applications to

give you a taste of the types of things you can do with Mtool. Both

applications relate directly to the earlier History section because

they are both specific cases where I had wished that I had a tool

like Mtool.

Taylor Series

-------------

A Taylor Series can be found for any function f that has

derivatives of all orders at some point a. If we let T(x) be the

Taylor Series then

T(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... +

fk(a)(x-a)^k/k! + ....

where fk denotes the kth derivative of f. The special case of the

Taylor Series about the point a = 0 is called the Maclaurin series.

These series are discussed at length in introductory calculus texts.

One application of these series is to approximate certain

functions. For example, the Maclaurin series for f(x) = e^x is given

by

1 + x + x^2/2 + x^3/3! + x^4/4! + ....

Given sufficient terms, this series converges to e^x for all x. My

questions were, how many terms are necessary, and how closely does

some number of terms, say 4 or 5 or 10, approach e^x, and over what

range of values of x does this approximation hold. If you are

familiar with Taylor and Maclaurin series, then you know that there

exists a formula for the remainder which gives the maximum error for

the series, but this will not give you the same feel for the accuracy

of the series approximation as a plot. For a plot of the error of a

twelve term Maclaurin series approximation to e^x, type the function

on the next page into Mtool.

21

f(x) = a + b + c + d - exp(x)

where a = 1 + x + x^2/2 + x^3/fac(3)

b = x^4/fac(4) + x^5/fac(5) + x^6/fac(6)

c = x^7/fac(7) + x^8/fac(8) + x^9/fac(9)

d = x^10/fac(10) + x^11/fac(11)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 7

Lower limit of x: -2.0000

Upper limit of x: 2.0000

Number of subdivisions(even for integration): 200

Automatic y-axis scaling(Y/N): y

In about the same amount of time that it takes you to enter this

function you'll have a plot showing how closely this series matches

e^x over the given range of x. Using the "same" response to the loop

menu, you will be able to play with different ranges of x and with

different numbers of points to plot. If you're like me, you will

quickly get a much better feel for the accuracy of this approximation

than you would ever get from a formula or two.

22

Fourier Series

--------------

The second application comes from one of my engineering courses.

According to Fourier analysis, a square wave can be approximated by

the series

f(t) = 1/2 + (2/pi)cos(t) - (2/3*pi)cos(3t) + (2/5*pi)cos(5t) - ....

The greater the number of terms the nearer the series approaches a

square wave. Usually in a text there will one and only one graph

that shows this. It would be nice to play around with this and see

how close series of different lengths actually do come to a square

wave. With Mtool this is possible. The following script shows how

to get a plot of the series when it has nine terms:

f(t) = a + b + c + d

where a = 1/2 + 2*cos(t)/p - 2*cos(3*t)/(3*p)

b = 2*cos(5*t)/(5*p) - 2*cos(7*t)/(7*p)

c = 2*cos(9*t)/(9*p) - 2*cos(11*t)/(11*p)

d = 2*cos(13*t)/(13*p) - 2*cos(15*t)/(15*p)

1) Evaluate at a single point.

2) Evaluate over a range of points.

3) Integrate.

4) Both 2 and 3.

5) Derivative.

6) Solve.

7) Graph.

Enter choice: 7

What is the value of p: 3.141592654

Lower limit of t: 0.0000

Upper limit of t: 10.0000

Number of subdivisions(even for integration): 200

Automatic y-axis scaling(Y/N): y

Once again, after a wait of about a minute or so, you'll be able

to see just how close this series approximates a square wave. If you

want you can zoom in on any part of the curve simply by choosing

"same" on the loop menu and resetting the range over which to plot.

23

LEGALITIES

----------

Mtool is shareware, also known as user-supported software. The

diskette with program and 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.

Mtool is NOT a public domain program. It is copyright (c) 1988

by Martin J. Maher, and the author retains all rights. In

particular, he retains the right to distribute this program and all

source code and documentation for profit. There is no guarantee that

Mtool 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

this program.

User groups, bulletin boards, clubs, and shareware distributors

are authorized to distribute Mtool under the following conditions:

1. No charge is made for the software or the documentation. A

nominal distribution 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 Mtool is available to 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. PC-DOS, CGA, EGA, and VGA

are trademarks of International Business Machines, Inc. Hercules is

a trademark of Hercules Computer Technologies, Inc.

24

SHAREWARE

---------

My hopes are that you will find this program useful and that,

of course, you will decide to become a registered user of Mtool by

filling out the registration form on the last page and mailing it to

me along with a check for $10.00. In addition to the warm glow of

virtue, registration will bring you the following benefits:

1) The most recent version of the program.

2) The most recent version of the program that supports the

80x87 math coprocessor chips. This version runs

significantly faster.

3) Notification of future major upgrades.

4) Reduced prices for future major upgrades.

The source code for Mtool is available to registered users. The

purchase price is $10.00, so that the total for both registration and

source code is $20.00.

Above I mention future major upgrades. These upgrades may

include, but are not limited to, logarithmic and semi-logarithmic

plots, file support, expanded function definition capabilities, and

improved user interfaces. Finally, if you have any comments,

complaints, bug reports, or suggestions I would very much appreciate

hearing from you.

25

REGISTRATION FORM

-----------------

You may register your copy of Mtool by filling out the following

form and mailing it with a check for $10.00 to the address below.

You will promptly receive a diskette with the most recent version of

Mtool, a version of Mtool that supports the 80x87 math coprocessor,

and the most recent version of the user's guide. You will also be

placed on our mailing list so that you may be notified of future

versions, and you will qualify for discounted prices on those future

versions. If your check is for $20.00, all Mtool source code will be

included on the diskette.

Thank you for your support.

Mail to:

Martin J. Maher

MJM Engineering

P.O. Box 2027

Hawthorne, Ca. 90251

Name_________________________________________________________

Address______________________________________________________

City/State___________________________________________________

Zip Code _________________________

How did you first learn about Mtool, or where did you receive

your first copy of Mtool?

___________________________________________________________________

___________________________________________________________________

Comments or suggestions

_____________________________________________________________________

_____________________________________________________________________

26

December 27, 2017
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