Tutorial The Visual Tools

The Visual Tools

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.

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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.
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.
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.
Click in the Domain & Range window and type 2 into the "n =" cell. This resets the number of rectangles (trapezoids, parabolas) to 2.
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. 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.
Select New Function from the Work menu and type 3.5x(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 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 dynamically displayed in the Domain & Range window.

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