 MATHPAK IV Help Index

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Below is the menu of Mathpak help.

Please click on one of the options for selection.

Commands

Calculus

Matrix

Miscellaneous


Calculus menu commands

The Calculus menu offers the following commands:

Chain Rule	Solves an equation using Chain Rule.
Derivative	Takes first and second deriivatives of a function.
Polynomial	Determines if given two(2) functions are intersecting each other.
Trapezoidal	Computes an area of a function using Trapezoidal 's rule of integration.
Simpson	Computes an area of a function using Simpson's rule of integration.


Matrix menu commands

The Matrix menu offers the following commands:

Crammer	Solves a system of equations using Crammer's Rule.
Gaussian	Reduces a matrix to the Row-Echolen form.
Inverse	Finds the inverse of a matrix.
Operations	Add, subtract, multiply two matrices.


Miscellaneous menu commands

The Miscellaneous menu offers the following commands:

Conversion	Converts a number from one base to another.
Quadratic	Solves a quadratic equation.
Trigonometry	Solves a trigonometry function.
Line_Equation	Finds the equation of a line given two(2) points.
Linear_Eq	Solves linear equations.
User_Defined	Solves user-defined functions.
GraphSetup	Changes default graph setup.


Chain Rule Help

Given an equation of the form:

 F(x) = a(b X^c+ d X ^e + f) ^g

Taking  the first derivative of the equation using Chain Rule:

F(x) = ag(b X^c + d X ^e + f) ^(g-1) (bc X^(c-1) +de X^(e-1))

To enter values for coefficients:

Press TAB key repeatedly until the blinking bar (Cursor) moves

to a desired coefficient.

 Coefficients that are left blank are considered as 0's.

To see the graphs of the function and its 1st derivative:

Click on the "PLOT" button.


First and Second Derivatives

 Given  an equation of the form:

ax^b + cx^d  +  ex^f + g

         F(x)  = --------------------------

 	hx^i+ jx^k+ lx^m + n

Where a,b,c,...n are entered by users.

If they are treated as 0's if they are not entered.

This is a general equation that gives users flexibility in entering

various forms of equations.

  For example,  to enter an equation of 1/X,

make g = 1 and h and i = 1.

  The derivative of the above equation is solved by using

     the devision rule:

f(x)	f(x) g'(x) - f'(x) g(x)

  F(x) = ---------     ----->     F'(x) =  ---------------------------

g(x)	 	g(x)^2


Number Conversion

Convert a number from one base to another.

Following bases are supported: Base 2, Base 8, Base 10, Base 16.

To enter a number in a base:

Press the TAB key repeatedly until the blinking bar (cursor) appears

on a desired base.

To clear all inputs:

Click on the "CLEAR" button.

To close the window:

Click on the "CANCEL" button.


Trigonometry Equation

 Given an equation of the form

F(x) = a[Sine/Cos/Tan(bx^c)]

Where a,b,c are entered by users.

If they are not entered, they will be treated as 0s.

 The equation is solved for Sine, Cosine, and Tan.

 Options for plotting

Users have options for plotting of any combination of 3 Trigonometry

equations.

 Click on the Check Box button to select plotting options.

 If the Check Box is checked, it means selected.


Crammer's Rule

  The matrix on the left is an unsolved system of equations.

  The matrix on the right is a solved system of equations.

To select a number of variables for an equation:

Click on a radio button that specifies a desired number of variables.

2-Variables means solving a system of equations of the forms:

ax1 + bx2 = c

dx1 + ex2 = f

The matrix representation is:

| nnn  nnn  nnn|

| nnn  nnn  nnn|

Determinants:

The number of determinants depends on the number of variables

in the functions.  Crammer's rule  solves a system of equations by

determinants.  The determinants for the above system of 2-variable are:

DET	| a  b |

| d  e |

DETX1    | b  c |      DETX2     | a  c |

| e  f |	   	| d  f |

X1 = DETX1/DET	X2 = DETX2/DET

There are 3 determinants for a system of 2-variable equations.

View matrices for determinants

Click on the MATRIX button.

There are 3 matrices for a system of 2-variable equations.

One matrix for Determinant

One matrix for Determinant of X1

One matrix for Determinant of X2


Reduce a matrix using Gausian Technique

To select a size for a matrix:

Click on Number of Row or Number of Columns,

a menu of predefined sizes is displayed.

Select a desired from the menu.

The Input Matrix and Result Matrix will change accordingly.

To enter data into the matrix:

Set the Cursor to one of the entries in the matrix by either pressing

the TAB key or click on the entry.

If any data is entered, the Result matrix will be initialized to 0's.

To view the solution:

There are 2 ways of viewing the solution.

Click on the Result button which will

display the Final solution on the Result matrtix.

 	Click on the Step button.

This will display the process of solving the matrix by steps.

Every click on the Step button will show the next step toward

the solution of the matrix.


Finding the inverse of a matrix

To select a size for the matrix

Click on the Down Arrow of the "Matrix Size", then select a desired size

from the menu.

You will notice the matrices instantly change accordingly.

To enter data into the matrix:

Set the Cursor to one of the entries in the matrix by either pressing

the TAB key or click on the entry.

If any data is entered, the Result matrix will be initialized to 0's.

To view the solution of the inverse

There are 2 ways of viewing the solution.

Click on the Result button which will

display the Final solution on the Result matrtix.

 	Click on the Step button.

This will display the process of finding the inverse of the matrix by

step.

Every click on the Step button will show the next step toward the

the solution of the matrix.


Find the equation of a line

Finding the equation of a line using 2 points,

point 1 (X1,Y1) and point2. (X2,Y2)

The equation of the form:	F(x) = mX + B

Where m = slope and B = Y-Intercept.

The Slope is computed as follows:

         X2 - X1

m =  -------------

         Y2 - Y1

The Y-Intercept is computed as follows:

Y-Intercept = (-m * X1) / Y1

The Distance between 2 points is computed as shown below:

Distance = [ (Y2-Y1)2 + (X2-X1)2]1/2


Linear Equation

Given 2 linear equations of the form:

f(x) = mx + b and g(x) = ay + d

For F(X) = G(X), and let F(x) = y, we have

mx + b - d

   F(x) = ---------------

       a

Given 2 linear equations of the form above (F(x)),

the program will determine conditions of the equations according to

the initial input value.


Matrix Operations

   Add, Subtract and Multiply 2 matrices.

Select matrix operation

Click on the Addition or Subtraction or Multiplication radio buttton

to select the operation.

 Matrix Addition and Subtraction requires

the matrix A and Bhave the same size (ie, same number of rows

as well as number of columns).

  Matrix Multiplication requires the number of columns

of matrix A to be equal with the number of rows of matrix B.

 Otherwise, the matrices cannot be computed with the 	selected operation.

 To change the sizes of matrix A or matrix B

Click on the Down arrow next to the default size of the matrix.

Then select a size from the menu.


Polynomial Intersection

Determine whether 2 polynomials are intersected

at a given point.

  Given 2 equations of the form

F1(x) = ax^3 + bx^2 + cx + d

F2(x) = ax^3 + bx^2 + cx + d

Coefficients a,b,c,d are supposed to be enterred.

If they are left blanks, their values are assumed to be 0's.

 How it works

The initial value X is substituded in the equations to solve for F(x).

If F1(x) = F2(x) then, it means these 2 functions are intersected at

the given point of X.


Numerical Integration using Simpson's Rule

 Given an equation of the form:

ax^b + cx^d + ex^f + g

         F(x)  = --------------------------

 	hx^i + jx^k + lx^m + n

Where a,b,c,...n are entered by users.

If they are treated as 0's if they are not entered.

This is a general equation that gives users flexibility in entering

various forms of equations.

  For example, to enter an equation of 1/X,

make g = 1 and h and i = 1.

 Simpson's Algorithm approminating the graph of F(x).

Integration of F(x) from a to b:

b - a

------- [ f(X0) + 4f(X1) + 2f(X2) + 4f(X3] + ...+2f(Xn-2) + 4f(Xn1) + f(Xn)

  3n

 Variable descriptions

n	: Number of partitions between the limit a and b.

Xi	: X value on the partition.

F(xi)	: Y value at Xi.

m	: can be 1,2, or 4.  The coefficient of the Simpson's algorithm as

: shown above.

Solution:  is the final result of the Simpson's algorithm.

  To view a value at a partition

Click on the Down arrow of Xi and select a desired parition.

The value for selected partition is displayed as F(xi) or mF(xi).


Numerical Integration using Trapezoidal Rule

 Given an equation of the form:

ax^b + cx^d + ex^f + g

         F(x)  = --------------------------

 	hx^i + jx^k + lx^m + n

Where a,b,c,...n are entered by users.

If they are treated as 0's if they are not entered.

This is a general equation that gives users flexibility in entering

various forms of equations.

  For example, to enter an equation of 1/X,

make g = 1 and h and i = 1.

 Trapezoidal algorithm approminating the graph of F(x).

Integration of F(x) from a to b:

b - a

------- [ f(X0) + 2f(X1) + 2f(X2) + ...+2f(Xn-1) + f(Xn)

  2n

 Variable descriptions

n	: Number of partitions between the limit a and b.

Xi	: X value on the partition.

F(xi)	: Y value at Xi.

m	: can be 1 or 2,  The coefficient of the Trapezoidal algorithm as

: shown above.

Solution:  is the final result of the Trapezoidal algorithm.

 Error Estimate for the Trapezoidal rule

The Maximum error using the Trapezoidal rule is NOT greater than

M(b-a}^3/12n^2

Where M is a positive real number such that F"(x) < M for

all x in [a..b].

Note: F" is second derivative of F(x).

  To view a value at a partition

Click on the Down arrow of Xi and select a desired parition.

The value for selected partition is displayed as F(xi) or mF(xi).


User-Defined Function

 Given an equation of the form:

ax^b + cx^d + ex^f + g

         F(x)  = ----------------------------

 	hx^i + jx^k + lx^m + n

Where a,b,c,...n are entered by users.

If they are treated as 0's if they are not entered.

This is a general equation that gives users flexibility in entering

various forms of equations.

  For example, to enter an equation of 1/X,

make g = 1 and h and i = 1.


Graph Setup - default settings

  Scale Value for Zoom in and Zoom out:

Zoom in means making each unit in the function equal to

x number of units on the display device.

If the Scale value is 5, then each function unit is equivalent to

5 units on the display.

Zoom out is opposit of the Zoom out.

If the Scale value is 5, then each unit on the display is

equivalent to 5 units in the function.

  Scale Value for Draw Grid

The scale value of grid is based on the Scale Value for Zooming.

The Grid Scale = the Grid Scale Value times Zooming Scale Value.

If the Scale value for grid is 5 and the Scale Value for

zomming in is 10, the grid will be drawn for every 50 units on the display.


Quadratic Equation

 Given an equation of the form

ax^2 + bx + c = 0

Where a,b,c are entered by users

If a or b or c is not entered, then it will be treated as 0.

 The equation is sovled by using

 If (b^2 - 4ac) is less than 0, the X solutions will be imaginary 	because the square root of a nagative number is undefined.

  Otherwise, the equation has 2 solutions (X1, X2).