the interface plotting the tool palette data matters ... · tutortial plotting topic start section...

TEMATH 2.1 Tutorial The Interface Plotting The Tool Palette Data Matters

Matrix Calculations Solving Differential Eqns Evaluating Expressions The Visual Tools

Copyright 2000-2003 Adam O. Hausknecht and Robert E. Kowalczyk

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Tutorial: Interface

The brief tutorial in the following sections provides an introduction to the use of some of TEMATH's capabilities.

The Interface

TEMATH's interface contains four windows:

● the Graph window for plotting; ● the Work window for entering functions, equations, constants, . . . ;● the Domain & Range window for setting the graph viewing area and entering the values needed for

the tools on the tool palette;● the Report window for writing the results of TEMATH's numerical tools.

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Tutortial Plotting

Plotting

Enter and plot a function

Scale the y-axis

Change a plot's color

Find points of intersection

Plot a family of functions

Plot an implicitly defined curve

Plot an inequality

Plot an ellipse

Plot a function using constants

Plot a family of functions using an iterated constant

To plot the function y = x/(x^2 - 4) on the interval -4 x 4,

● Select New Function from the Work menu.● y1(x) = will appear as the first entry in the Work window.● Type the expression x/(x^2-4) into the first cell of the Work window. Press the

Enter or Return key.● Click in the Domain & Range window to make it active. Click in the cell to the left of “

x ” and type -4. Press Tab to move to the cell to the right of “ x ” and then type 4.

● Select Plot from the Graph menu. Note that TEMATH automatically scales the y-axis to fit the graph and displays the vertical asymptotes.

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Tutortial Plotting

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If you want to specify your own scaling for the y-axis, for example, -5 y 5,

● Click in the Domain & Range window to make it active. Click in the cell to the left of “ y ” and type -5. Press Tab to move to the cell to the right of “ y ” and then

type 5.● Click in the Autoscaling checkbox to turn off autoscaling.


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● Select Plot from the Graph menu.


To change the color of the plot,


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● Click in the Work window to make it active.● Click on the vertical line to the left of the function in the Work window. Each time

you click, a new color will be shown. TEMATH provides five different colors for plotting. Select the color that you want.



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Tutortial Plotting


To find the points of intersection of the curves y = x/(x^2 - 4) and y = x^2 - 4,

● Select New Function from the Work menu.● y2(x) = will appear as the first entry in the Work window.● Type the expression x^2-4 into the first cell of the Work window. Press the Enter or

Return key.● Select Overlay Plot from the Graph menu.


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Tutortial Plotting

● Click on the Curve Intersection tool on the tool palette in the Graph window. It is the ninth tool from the left and it shows the intersection of two curves. The tool's icon will become highlighted.

● Click in the Graph window near a point of intersection. The coordinates of the point of intersection will be written into the Report window and a point with displayed coordinates will be drawn at the position of the intersection of the two curves.

● Click near the second point of intersection.


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Tutortial Plotting


To plot the family of functions: y = x^2, y = x^2 + 1, and y = x^2 + 2,

● Select New Function from the Work menu.● Type the expression x^2 into the first cell of the Work window. Press the Enter or

Return key.● Select New Function from the Work menu.● Type the expression x^2 + 1 into the first cell of the Work window. Press the Enter

or Return key.● Select New Function from the Work menu.● Type the expression x^2 + 2 into the first cell of the Work window. Press the Enter

or Return key.● If you've plotted the previous two functions in this section, click on the checkmark


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Tutortial Plotting

to the left of those two functions in the Work window. The checkmarks will disappear and the two functions will be deactivated.

● Select Plot All from the Graph menu. All the checked (active) functions will be plotted in the Graph window.


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Tutortial Plotting


To plot the implicit equation x^2 + y^3 = 4 on -5 x 5 and -5 y 5,

● Select New Equation or Inequality from the Work menu.● Type the expression x^2 + y^3 = 4 into the first cell of the Work window. Press the

Enter or Return key.● Click in the Domain & Range window to make it active. Enter the domain -5 x 5

and the range -5 y 5.● Select Plot from the Graph menu.


To plot the inequality 2x + 5y 4,

● Select New Equation or Inequality from the Work menu.● Type the expression 2x + 5y 4 into the first cell of the Work window (for , type

Option < or enter using a pop-up menu).● Press the Enter or Return key.


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Tutortial Plotting



To plot the ellipse (x-1)^2/(3^2) + (y+1)^2/(2^2) = 1,

● Select Conic — Ellipse from the Work menu.● Fill-in the template e#: ellipse(1,3,-1,2) in the first cell of the Work window. Press

the Enter or Return key.● Select Plot from the Graph menu.


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Tutortial Plotting


To plot a function using constants,

● Select New Constant from the Work menu.● c1 = will appear as the first entry in the Work window.● Delete c1 = and type the expression m = 3. Press the Enter or Return key.● Select New Constant from the Work menu.● c2 = will appear as the first entry in the Work window.● Delete c2 = and type the expression b = -2. Press the Enter or Return key.● Select New Function from the Work menu.● Type the expression m x + b into the first cell of the Work window. Be sure to type a

space between the m and the x. Press the Enter or Return key.


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To plot a family of functions using an iterated constant,


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● Select the cell containing b = -2 in the Work window.● Delete its contents and type the expression b = -2 to 2. Press the Enter or Return

key.

● Select Plot from the Graph menu. Five lines will be plotted in the Graph window for b = -2, -1, 0, 1, 2.

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Tutortial Plotting



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Tutorial Tool Palette

The Tool Palette

Getting Started

The Zoom Tool

The Root Finder Tool

To start, enter and plot the function y = x^3-x^2-16x-10 on the interval -5 x 5 by doing the following:

● Select New Function from the Work menu.● Type the expression x^3-x^2-16x-10 into the first cell of the Work window. Press the

Enter or Return key.● Enter the domain -5 x 5 and click on autoscaling.● Select Plot from the Graph menu.

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To zoom-in on the left-most negative root (zero) of the function,

● Click on the Zoom tool on the tool palette in the Graph window. It is the first tool on the left. The tool's icon will become highlighted.

● Click in the Graph window near the x, y coordinates (-5, 10) and hold down the mouse button. As you drag the mouse, a gray rectangle will be displayed. Move the cursor to a point near the x, y coordinates (-3, -10).

● Release the mouse button. The portion of the graph contained in the gray rectangle will be plotted using the entire Graph window. You can now visually estimate the values of the left-most negative root.


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To return to the original plot,

● Click on the thumb of the scroll bar on the right edge of the Graph and hold down the mouse button.


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● Drag the thumb to the top end of the scroll bar and release. The original “un-zoomed” plot will be redrawn.


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To approximate the roots more accurately,

● Click the Single Root Finder tool on the tool palette. It is the eighth tool from the left.

● Click in the Graph window near the leftmost root. The value of the root will be calculated and written into the Report window.

● Click in the Graph window near the rightmost negative root.


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Tutortial Data Matters

Data Matters: Entering, Plotting, and Fitting Data

Enter a data table using the keyboard

Plot data

Fit data with a least squares line

Enter a data table using the mouse

Generate a data table from a function

To enter a data table using the keyboard,

● Select New Data Table from the Work menu.

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● As an example, enter the following data by typing into the Data Table window:

-5 Enter -5 Enter-3 Enter -3.2 Enter-1 Enter -0.9 Enter1 Enter 1.2 Enter3 Enter 2.9 Enter5 Enter 5

Here "Enter" means to press the Enter key (not the Return key).


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● Click the close box of the Data Table window and click the Yes button in the Add Data Table window. The data table will be added to the Work window.


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To plot the data,



To fit the data with a least squares line,

● Select Least Squares — Line from the Tools menu.


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To enter a data table using the mouse,

● Plot any object in the Graph window or simply select Plot Axes from the Graph menu.● Click the Coordinate tool on the tool palette. It is the fourth tool from the left.

● Click in the Graph window at the position where you want to enter a data point. If a curve has been plotted and the selected style of the point has a line through it (see the icon tool's pop-up menu), then the x-coordinate will be used to draw a point on the curve, otherwise, the point will be placed at the position of the click. Enter as many data points as you want. You can also hold down the mouse button and drag to enter a sequence of points.


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● When you are finished entering your data points, select Generate Table from Points from the Tools menu. A data table containing the points that you entered will be added to the Work window.


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To generate a data table using the function f(x) = x^2 sin(x),

● Enter and plot the function f(x) = x^2 sin(x) over the interval -5 x 5.


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● Select Data Generator... from the Tools menu. Enter the number of points that you want to generate or simply click OK to generate 10 points.


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● Select Overlay Plot from the Graph menu to plot the data on top of the plotted curve.


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Tutorial Matrix Calculations

Matrix Calculations

The Matrix Calculator

Enter a matrix

The determinant

Store a matrix

Find the inverse of a matrix

Multipy two matrices

Perform an elementary row operation

Multipy a row by a scalar

Save a matrix in the Work window

To perform matrix calculations,

● Select Matrix Calculator from the Work menu. The Matrix Calculator window will be displayed.

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To enter the matrix

|1 3 2 | |4 1 5 | |8 6 7 |

● Double-click the Resize command in the matrix command list located in the lower left of the Matrix Calculator window.


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● Type 3, 3 into the command area located near the top of the Matrix Calculator window. Press the Enter key.

● Type 1 Enter 3 Enter 2 Enter 4 Enter 1 Enter 5 Enter 8 Enter 6 Enter 7

Here "Enter" means to press the Enter key and not the Return key which moves down columns!


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To find the determinant of the matrix,

● Double-click the Determinant command in the matrix command list. The value of the determinant will be written into the command area of the Matrix Calculator window.


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To save a backup copy of the matrix into memory location A,

● Click the Store button and then the A button.


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To find the inverse of the matrix,

● Double-click the Inverse command.


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To check the accuracy of the inverse matrix, multiply the inverse matrix by the backup of the original matrix. To do this,

● Click the multiplication * button, the A button, and the enter button.


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To perform the elementary row operation -4*R1+R2 -> R2 on the matrix stored in memory A,

● Click the Recall button and then the A button.


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● Select row 2 of the matrix by placing the cursor over the R2 label (the cursor should become a right-pointing arrow) and clicking the mouse button.

● Double-click the Elementary command.


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● Select row 1 of the matrix.

● Type -4 into the command area. Press the Enter key or click the enter button.


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To multiply row 2 by 1/(-11),

● Select row 2 of the matrix.

● Click the multiplication * button.


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● Type 1/(-11) into the command area. Press the Enter key or click the enter button.


To save the current active matrix to the Work window for future use,

● Select Add Matrix from the Work menu.


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Tutorial Solving Differential Equations

Solving Differential Equations

Enter and solve a differential equation

Erase plotted solutions

Draw a slope field

Draw a family of solutions

Solve a second-order differential equation

Plotting solutions in the phase plane

To solve the differential equation dy/dt = f(t, y) = -0.2y(y-3) subject to the initial condition y(0) = 1,

● Select Diff Eq from the Graph menu to use the Differential Equation Solver.

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● Select New Differential Eqn — First Order from the Work menu.

● Type -0.2y(y-3) into the first cell of the Work window. Press the Enter or Return key.

● Click in the Graph window to make it active and click the Axes button.


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● Click in the Domain & Range window to make it active and type 0 into the t-cell and 1 into the y-cell. Click the Enter button or press the Enter key.

● You can also click at any point in the Graph window to enter an initial condition and draw the corresponding solution curve. Try this.


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To erase the drawn solutions,

● Click the Axes button.


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To draw the slope field,

● Click the Slope Field button.


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To draw a family of solutions,

● Click on the Family of Solutions tool on the tool palette in the Graph window. It is the first tool on the right.

● Click in the Graph window at the initial t-value that you want. For example, click on the y-axis for an initial t-value of 0.


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The harmonic oscillator with damping is modeled by the second-order differential equation

y'' = -ks*y - kd*y'

where

y = the displacement of the free end of the spring from rest,

ks = the spring coefficient, and

kd = the damping coefficent.

To solve this second-order differential equation with

kd = 0.5, ks = 2, y(0) = 3, and y'(0) = 0,

● Select New Differential Eqn — Second Order from the Work menu.


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● Type -2y-0.5y' into the first cell of the Work window. Press the Enter or Return key.

● Click in the Graph window to make it active and click the Axes button.


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● If necessary, click on the Solution Curve tool on the tool palette in the Graph window. It is the second tool from the right.

● Click in the Domain & Range window to make it active and type 0 into the t-cell, 3 into the y-cell, and 0 into the y'-cell. Click the Enter button or press the Enter key.


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● You can also click at any point in the Graph window to enter an initial value and draw the corresponding solution curve (be sure that a value has been entered into the y'-cell). Try this.


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To draw the solutions of the harmonic oscillator in the phase plane,

● Click on the Style menu in the Graph window and select y' vs. y.


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● Click in the Graph window to draw a solution.


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Tutorial Evaluating Expressions

Evaluating Expressions

Enter and evaluate an expression

Evaluate a sum

Evaluate a polynomial

Compute the number of combinations

Evaluate functions in the Work window

To evaluate the expression cos(1.23)exp(-0.1)+ (2.3^2-1.7),

● Select Calculators — Expression Calculator... from the Work menu. The Expression Calculator window will open.

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● Type cos(1.23)exp(-0.1)+sqrt(2.3^2-1.7) into the Expression Calculator window. Press the Enter key. Pressing the Return key will only move the cursor to the next line and not evaluate the expression. Alternatively, you can use the mouse to click the Evaluate button in the lower left corner of the Expression Calculator window to evaluate the expression. The value of the expression, 2.19716033417, will be written on the next line.



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Note: When you press the Enter key, the entire expression on the line containing the flashing cursor will be evaluated. To evaluate only part of the expression, select the part of the expression that you want by dragging over it and then press the Enter key.

To find the sum 1 - 1/2 + 1/3 - 1/4 + ... + ((-1)^101) 1/100,

● Scroll the function list on the left side of the window until you find (i,1,n,f(...,i...)). Double-click on (i,1,n,f(...,i...)). The template () will be written into the window's work area. Finish typing the expression (i,1,100,(-1)^(i+1) 1/i) and press the Enter key.


To find P(3.21), where P(x) = 2 - 3x + 6x^2 - 4x^3 - 0.2x^4,

● Double-click the poly(x,a0,a1,...,aN) function in the function list. Finish typing the expression poly(3.21, 2, -3, 6, -4, -0.2) and press the Enter key.


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To compute the number of combinations ,

● Double-click the comb(n,m) function in the function list. Finish typing the expression comb(10, 3) and press the Enter key.



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You can also use the Expression Calculator to evaluate functions that have been entered into the Work window.

For example, if y1(x) = 3 cos(2x+1) has been entered into the Work window, you can evaluate this function at x = 0.23 by

● Typing y1(0.23) into the Expression Calculator window and pressing the Enter key.




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Tutorial The Visual Tools

The Visual Tools

The Dynamic Tangent Tool

The Integration Tool

The Dynamic Integration Tool

The Fixed-Point Iteration Tool

The Parametric Tracker

The Dynamic Tangent Tool

The Dynamic Tangent tool shows a moving tangent along a curve and it also plots the slope of the moving tangent. The following example demonstrates this visual tool:

● Select New Function from the Work menu and type sin(x) into the first cell of the Work window. Press the Enter or Return key.

● Enter the domain - x and range -2 y 2 into the Domain & Range window. The symbol can be entered by pressing the Option and P keys (don't press the shift key).

● Click off autoscaling.● Select Plot from the Work menu.● Click the Tangent tool on the tool palette. It is the seventh tool from the left. Click

on the tool's pop-up menu and select Dynamic.

● Click the Go button in the Graph window and watch the show.

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<● Click the Step button to move the tangent one step at a time.● To change the speed of the dynamic tangent, select Dynamic Plotting Speed... from

the Options menu.


The Integration Tool

To approximate the area under a curve using rectangles, trapezoids, or parabolas (Simpson's rule) and to see a visual representation of the approximate area, use the Integration tool. The following example demonstrates how to use this tool:

● Select New Function from the Work menu and type x cos(x)+10+cos(x^2) into the first cell of the Work window. Press the Enter or Return key.

● Enter the domain 0 x 5 and range 0 y 15 into the Domain & Range window. If autoscaling is not off, click it off.

● Select Plot from the Work menu.● Click the Integration tool on the tool palette. It is the eleventh tool from the left.


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● Position the cursor over the Method menu in the Graph window, press and hold down the mouse button, and select Left End Point from the pop-up menu. This sets the approximate integration method to be the left-end-point rule using rectangles.

● Click the Draw button (it is the middle button) in the Graph window. Ten (the default value) rectangles will be drawn in the Graph window.


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● To double the number of rectangles, click the Up-arrow button. Continue to click the Up-arrow button until the area of the rectangles provides a good visual approximation to the area under the curve.


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● Click in the Domain & Range window and type 2 into the "n =" cell. This resets the number of rectangles (trapezoids, parabolas) to 2.


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● Select Trapezoid from the Method menu in the Graph window.

● Click the Draw button in the Graph window. Two trapezoids will be drawn in the Graph window.

● Continue to click the Up-arrow button until the area of the trapezoids provides a good visual approximation to the area under the curve.


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Dynamic Integration Tool

The Dynamic Integration tool dynamically plots the "area so far" function and shades-in the corresponding area under the curve. This tool can be used to gain some visual intuition about functions defined as integrals. For example, consider the sine-integral, that is,

Si(x) = sin(t)/t dt

To investigate this function,

● Select New Function from the Work menu and type sin(x)/x into the first cell of the Work window. Press the Enter or Return key.

● Enter the domain 0 x 15 and range -1 y 2 into the Domain & Range window.


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If autoscaling is not off, click it off.● Select Plot from the Work menu. ● Click on the Integration tool's pop-up menu and select Dynamic.

● Click the Go button in the Graph window and watch the show.


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Fixed-Point Iteration Tool

The fixed-point iteration tool visually demonstrates the fixed-point iteration process. This tool can be used to investigate "orbits" as follows:

● Select New Function from the Work menu and type 3.2x(1-x) into the first cell of the Work window.

● Enter the domain 0 x 1 and range 0 y 1 into the Domain & Range window. If autoscaling is not off, click it off.

● Select Plot from the Work menu.● Click the Fixed-Point Iteration tool on the tool palette. It is the first tool on the right.● Click in the Graph window near x = 0.6.● Click the Go button. Notice that the iterations cycle between two different x-values.

Click the Stop button to stop the iterations, otherwise, they will continue for 150 iterations.


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● Select New Function from the Work menu and type 3.5x(1-x) into the first cell of


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the Work window.● Select Plot from the Work menu.● Click in the Graph window near x = 0.6.● Click the Go button. This time, the iterations cycle between four different x-values.

● Select New Function from the Work menu and type 4x(1-x) into the first cell of the Work window.

● Select Plot from the Work menu.● Click in the Graph window near x = 0.6.● Click the Go button. This time, the iterations do not cycle.


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Parametric Tracker

The Parametric Tracker is used to visualize paths along parametric curves. For example, the throwing of a ball can be parameterized by the following equations:

x(t) = 30cos( /4)t, and y(t) = 2 + 30sin( /4)t - (9.8/2)t^2

To visualized this ball-throwing experiment,

● Select Parametric from the Graph menu.● Select New Function from the Work menu and type [30cos( /4)t, 2+30sin( /4)t-

(9.8/2)t^2] into the first cell of the Work window.● Enter the intervals 0 t 5, 0 x 95 and 0 y 95 into the Domain & Range

window. If autoscaling is not off, click it off.● Select Plot from the Work menu.● Click the Parametric Tracker tool on the tool palette. It is the sixth tool from the left.● Position the arrow cursor over the Parametric Tracker's slide on the top right side of

the Graph window. Press and hold down the mouse button. Drag down and watch the ball fly! Notice that the x- and y-coordinates of the position of the ball are


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dynamically displayed in the Domain & Range window.

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