Variable use & Graphs

Math in ONE incorporates numerous independent variables defined for different applications. In the program, application descriptions explain the usage of independent variables. This program supports a real and complex independent variables. Independent variables x and y are used to represent real numbers. Using x and y in a function as real numbers, will not necessarily generate real results (e.g. sqrt(x) for x=-1). A complex number ' z ' is defined as z = x + i( y ). ' i ' is an imaginary number and i * i = -1.
(if x and y are not real, evaluation of the expression may produce unexpected results. It is the user's responsibility to ensure that the imaginary part of the expression indicated by ' i ' is a real value.)
Note: x + i( sqrt( cos( y )) for some values of ' y ' (e.g. y = pi) will produce unexpected results.

Generate Graphs:

(W) To generate a graph, follow the outline below :

1. Select graph type
2. Enter expression into the Input Field.
   1. In a real numbers expression with one independent variable, acceptable variables names are x, t, θ .
   2. In a real numbers expression with two real independent variable, acceptable variables names are x, y.
   3. In a complex numbers expression, z can be used as an independent variable (e.g. sin(z)) or a complex form is allowed f( x ) + i g( y )(e.g. sin( x ) + i (cos( y ))).
   Variables t and θ are intended for Parametric Equations and Polar Coordinates respectively, but x is acceptable as well. This program does not recognize any difference between x, t, and θ variable names. In Cartesian Coordinates, multiple number of expressions can be graphed at the same time. For Parametric Equations, a pair of two expressions must be defined in the designated memory. All expressions need to be added to Text Storage in the Input field.
3. Configure memory for selected graph type: Graph display operates in From-To mode. Data Entry To Memory window will help to configure memories for different graph types.

4. Y(x) window can be launched in Cartesian coordinate and Level Curves display. When a graph is displayed, right mouse click any place on the graph and select Open Y(x) window. A new window will open displaying a value of an independent variable and values of displayed functions that correlates to the position last clicked on the graph.
5. Zoom in and out: When graph is displayed, right mouse click on any place on the graph and select Zoom in or Zoom out. The graph size will be adjusted 50% based on zoom selection. In the Cartesian coordinate and the Level Curve display, the point on the graph where the mouse was clicked will be moved to the center of the disply area.

In a calculus mode, the display can be configured to include in a graph, the original expression, its derivatives 1st, 2nd, . . . and the integral of the expression for different initial conditions. For further details, refer to the object popup help (derivative check box) and application description (1st order differential equations).

(AG) Graphing Calculator

Videos:
1. Draw in Cartesian Coordinates
2. Derivatives in graphing display
3. Area calculation in graphing display

XY = Xmin, Xmax, Ymin, Ymax
1. drawCar( XY, Fn0 , Fn1, Fn2, ...) Draws functions Fn0, Fn1, Fn2, ... in Cartesian coordinates
• Xmin - calculation range from (e.g. -5 or m1)
• Xmax - calculation range to (e.g. 5 or m2)
• Ymin - display Y range from (e.g. -5 or m3)
• Ymax - display Y range to (e.g. 5 or m4)
• Fn0 - first function that is evaluated (e.g. ln(x) or F0)
• Fn1 - second function that is evaluated (e.g. exp(x)) or F1)
• Fn2 - third function that is evaluated (e.g. x) or F2)

drawCar(-5,5,-5,5,ln(x),exp(x),x)
In Cartesian coordinates, when one touches the screen at a point, it will turn on the display of the function x and y values at that point. A tap on the circle in bottom left corner will turn off the display.
2. drawPol( XY, Pix, Fn0 ) Draws function Fn0 in Polar coordinates.
• Xmin - calculation range from (e.g. 0 or m1)
• Xmax - calculation range to (e.g. 15 or m2)
• Ymin - display X & Y range from (e.g. -1 or m3)
• Ymax - display X & Y range to (e.g. 1 or m4)
• Pix - number of iterations (e.g. 700 or m5)
• Fn0 - function that is evaluated (e.g. cos(x) or F0)

drawPol(0,15,-1,1,700,cos(x))
3. drawPar( XY, Pix, Fn0, Fn1 ) Draws two functions as parametric equations.
• Xmin - calculation range from (e.g. 0 or m1)
• Xmax - calculation range to (e.g. 50 or m2)
• Ymin - display X & Y range from (e.g. -2 or m3)
• Ymax - display X & Y range to (e.g. 2 or m4)
• Pix - number of iterations (e.g. 1700 or m5)
• Fn0 - first function of the pair that is evaluated (e.g. sin(x) or F0)
• Fn1 - second function of the pair that is evaluated (e.g. cos(2*x) or F0)

drawPar(0,50,-2,2,1700,sin(x),cos(2*x))

Navigating in the Cartesian Coordinate's mode

In the Cartesian coordinate's mode, the functions that are displayed can be manipulated. The bottom buttons can be used to resize and move the graph of the functions. If you tap in the middle of the screen, the display will change to trace mode. By tapping on the '<--' or '-->' buttons, the display will show function values for a given x variable. When you select one or two functions, the button 'S' (setting) will be displayed in the top right hand corner. By pressing the 'S' button, a new window will pop up with the following options.


1. Drawing First or Second order Derivative of selected function. If two functions are selected, only the first one will be processed.
2. Points A,B,C selection - each point contains a 'x' value and two functions that are associated with that point (reference - 2a). If the functions do not intersect at the 'x' value, (point) 'A' represents a vertical segment of the line from point 'A1' on the first function to point 'A2' on the second function (see below 1st and 2nd area examples). Only one point can be processed at a time. When the 'S' button is pressed, the 'x' value of the pointer's (small dots) location is moved to the display box for the first not selected point. Also, the original two functions that were selected are associated with that point (reference - 2a). If only one function was selected, the x-axis is associated with that point and is represented as f0. At this time you have four options:
   • You can accept the 'x' value by pressing the button 'Set P'.
   • You can change the 'x' value by tapping the display box to display the soft keyboard. After the 'x' value is entered, the 'done', 'enter', or 'next' key on the keyboard must be pressed in order to save the number that was typed. Using the return key on your device will not save your entry.
   • You can press the 'Inters' button to find the 'x' value of the two functions intersecting or if one function is selected, to find the zero of that function.
   • You can press the 'Return' button without setting a point and start over with a new selection.
3. 'Return' button - press to exit the selection window.


4. 'Set P' button - press to set (save) a selected point. To reset points, return to the graphic display and press the 'cercal' button located at the bottom in a middle. This button is between left and right arrow buttons.
5. 'Inters' button - press to find the intersection of two functions or the zero of the function if only one function is selected. This option will not work with the derivative of a function.


6. 'Rem F' button - press to remove the selected function. None of the points can be selected when this option is use.
7. 'PERP' button - press to find the line perpendicular to the function at a specific point on the function. One point has to be selected first. This option will not work with the derivative of a function.
8. 'TANG' button - press to find the line tangent to the function at specific point on the function. One point has to be selected first. This option will not work with the derivative of a function.


9. 'LINE' button - press to find the line passing through two selected points on one or two functions. Two points have to be selected first.
10. 'AREA' button - press to find the area surrounded by one, two or three functions and two or three points. The derivative of a function cannot be included in this setup.

Note: point Ax < point Bx < point Cx



One function selected and two points
Point A = x1, functions selected - f1 = sin(x), f0 = x-axis
Point B = x2, functions selected - f1 = sin(x), f0 = x-axis


One function selected and three points
Point A = x1, functions selected - f1 = sin(x), f0 = x-axis
Point B = zero of sin(x), functions selected - f1 = sin(x), f0 = x-axis
Point C = x3, functions selected - f1 = sin(x), f0 = x-axis

When one function is used to calculate area and function is crossing x-axis, the crossing point has to be a 'B' point.


Two functions selected and two points
Point A = intersection of sin(x) and cos(x), functions selected - f1 = sin(x), f2 = cos(x)
Point B = intersection of sin(x) and cos(x), functions selected - f1 = sin(x), f2 = cos(x)


Two functions selected and three points
Point A = intersection of sin(x) and cos(x), functions selection - f1 = sin(x), f2 = cos(x)
Point B = intersection of sin(x) and cos(x), functions selected - f1 = sin(x), f2 = cos(x)
Point C = intersection of sin(x) and cos(x), functions selected - f1 = sin(x), f2 = cos(x)

When two functions cross each other in calculation area, the crossing point has to be a 'B' point.


Two functions selected and three points
Point A = zero of cos(x), functions selection - f1 = cos(x), f0 = x-axis
Point B = intersection of sin(x) and cos(x), functions selected - f1 = sin(x), f2 = cos(x)
Point C = zero of sin(x), functions selected - f1 = sin(x), f0 = x-axis


Three functions selected and three points
Point A = intersection of cos(x) and ax+b, functions selected - f2 = cos(x), f3 = ax+b
Point B = intersection of sin(x) and cos(x), functions selected - f1 = sin(x), f2 = cos(x)
Point C = intersection of sin(x) and ax+b, functions selected - f1 = sin(x), f3 = ax+b



Three functions selected and three points
Point A = intersection of sin(x) and cos(x), functions selected - f1 = sin(x) f2 = cos(x)
Point B = intersection of cos(x) and ax+b, functions selected - f2 = cos(x), f3 = ax+b
Point C = intersection of sin(x) and ax+b, functions selected - f1 = sin(x), f3 = ax+b




To view the set up process for the area calculation, follow the step by step example below:



Consider the following area that needs to be calculated.

Three functions selected and three points
Point A = 0.25, functions selected - f1 = exp(x), f2 = 1/x
Point B = 0.5, functions selected - f2 = 1/x, f3 = 0.5x+7/3
Point C = intersection of exp(x) and 0.5x+7/3, functions selected - f1 = exp(x), f3 = ax+b
1. From the graphing pull down menu, select the drawCar function.
2. In the IOD (Input Output Display) type: drawCar(0, 1.1, 0.7, 4.7, exp(x), 1/x, 0.5*x+7/3) and then press enter.
3. In the graphing window, tap any place in the middle of the screen. The display function's background color will change and the 'x' values in the functions will be replaced with the 'x' value of the point tapped on the screen.
4. Select exp(x) and 1/x functions by tapping on them. The background color will change again to indicate a selection.
5. Press the 'S' button in the top right hand corner to select the selection window. Note: To display the selection window only one or two functions have to be selected.
6. Tap on the point 'A' in the display box. The soft keyboard will pop up.
7. Overwrite the displayed number to 0.25 and press the 'enter', 'done' or 'next' button.
8. Press the 'Set P' button to set point 'A'.
9. Press the 'Return' button to get back to the graph display.
10. Select the 1/x and 0.5*x+7/3 functions and press the 'S' button to return to the Selection window.
11. Tap on the point B display box and enter 0.5 and then press the 'enter', 'done' or 'next' button.
12. Press the Set P button to set point B.
13. Press the 'Return' button to get back to the graph display.
14. Select the exp(x) and 0.5*x+7/3 functions and press the 'S' button to return to the Selection window.
15. Press the 'Inters' button to find the intersection between the exp(x) and 0.5*x+7/3 functions.
16. Press the 'Set P' button to set point C.
17. Press the 'Area' button to calculate the area predefined above.