Texas Instruments TI-89 user guide: Moving the Cursor in 3D

The Axes and Style formats are specific to the 3D graphing mode. Refer to “Changing the Axes and Style Formats” on page 165.

Setting the Graph Format

Exploring a Graph

As in function graphing, you can explore a graph by using the following tools. Any displayed coordinates are shown in rectangular or cylindrical form as set in the graph format. In 3D graphing, cylindrical coordinates are shown when you use use:

$F_1$ 9 — or — TI-89: \blacklozenge $I$ TI-92 Plus: \blacklozenge F to set Coordinates = POLAR:

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Moving the Cursor in 3D

When you move the cursor along a 3D surface, it may not be obvious why the cursor moves as it does. 3D graphs have two independent variables (x,y) instead of one, and the x and y axes have a different orientation than other graphing modes.

How to Move the Cursor

Note: You can move the cursor only within the x and y boundaries set by Window variables xmin, xmax, ymin, and ymax.

On a 3D surface, the cursor always follows along a grid wire.

Although the rules are straightforward, the actual cursor movement can be confusing unless you know the orientation of the axes.

In 2D graphing, the x and y axes always have the same orientation relative to the Graph screen.

In 3D graphing, x and y have a different orientation relative to the Graph screen. Also, you can rotate and/or elevate the viewing angle.

Tip: To show the axes and their labels from the Y= Editor, Window Editor, or Graph screen, use: TI-89: ◆ [1] TI-92 Plus: ◆ F

Simple Example of Moving the Cursor

The following graph shows a sloped plane that has the equation z1(x,y) = - (x + y) / 2. Suppose you want to trace around the displayed boundary.

When you press [F3], the trace cursor appears at the midpoint of the xy grid. Use the cursor pad to move the cursor to any edge.

Tip: By displaying and labeling the axes, you can more easily see the pattern in the cursor movement.

Tip: To move grid points closer together, you can increase Window variables xgrid and ygrid.

When the trace cursor is on an interior point in the displayed plane, the cursor moves from one grid point to the next along one of the grid wires. You cannot move diagonally across the grid.

Notice that the grid wires may not appear parallel to the axes.

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Example of the Cursor on a Hidden Surface

On more complex shapes, the cursor may appear as if it is not on a grid point. This is an optical illusion caused when the cursor is on a hidden surface.

For example, consider a saddle shape z1(x,y) = (x² – y² ) / 3. The following graph shows the view looking down the y axis.

eyeθ=90. eyφ=70. ezφ=0. xmin=-10. xmax=10. xgrid=14. ymin=-10. ymax=10. ygrid=14. zmin=-10. zmax=10. ncontour=5.

Now look at the same shape at 10° from the x axis (eyeθ = 10).

Tip: To cut away the front of the saddle in this example, set xmax=0 to show only negative x values.

zc:2.04082 xc:-2.85714 yc: -1.42857

You can move the cursor so that it does not appear to be on a grid point.

zc:2.04082 xc:-2.85714 yc: -1.42857

If you cut away the front side, you can see the cursor is actually on a grid point on the hidden back side.

Example of an “Off the Curve” Cursor

Although the cursor can move only along a grid wire, you will see many cases where the cursor does not appear to be on the 3D surface at all. This occurs when the z axis is too short to show z(x,y) for the corresponding x and y values.

For example, suppose you trace the paraboloid z(x,y) = x² + .5y² graphed with the indicated Window variables. You can easily move the cursor to a position such as:

Trace cursor

Valid trace coordinates

zc:27.551 xc:4.28571 yc: -4.28571

eyeθ=20. eyφ=45. ezφ=0. xmin=-5. xmax=5. xgrid=14. ymin=-5. ymax=5. ygrid=14. zmin=0. zmax=5. ncontour=5.

Tip: QuickCenter lets you center the viewing cube on the cursor's location. Simply press ENTER.

Although the cursor is actually tracing the paraboloid, it appears off the curve because the trace coordinates:

• xc and yc are within the viewing cube. — but —
• zc is outside the viewing cube.

When zc is outside the z boundary of the viewing cube, the cursor is physically displayed at zmin or zmax (although the screen shows the correct trace coordinates).

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Rotating and/or Elevating the Viewing Angle

In 3D graphing mode, the eyeθ and eyeφ Window variables let you set viewing angles that determine your line of sight. The eyeψ Window variable lets you rotate the graph around that line of sight.

How the Viewing Angle Is Measured

The viewing angle has three components:

• eyeθ — angle in degrees from the positive x axis.
• eyeφ — angle in degrees from the positive z axis.
• eyeψ — angle in degrees by which the graph is rotated counterclockwise around the line of sight set by eyeθ and eyeφ.

Note: When eyeψ=0, the z axis is vertical on the screen. When eyeψ=90, the z axis is rotated 90° counterclockwise and is horizontal.

In the Window Editor ( [icon] [WINDOW] ), always enter eyeθ, eyeφ, and eyeψ in degrees, regardless of the current angle mode.

Do not enter a ° symbol. For example, type 20, 70, and 0, not 20°, 70°, and 0°.

eyeθ=20. eyeφ=70. eyeψ=0. xmin=-10. xmax=10. xgrid=14. ymin=-10. ymax=10. ygrid=14. zmin=-10. zmax=10. ncontour=5.

Effect of Changing eyeθ

The view on the Graph screen is always oriented along the viewing angle. From this point of view, you can change eyeθ to rotate the viewing angle around the z axis.

Note: This example increments eyeθ by 30.

z1(x,y) = (x³y - y³x) / 390 In this example, eyeφ = 70

eyeθ = 20

eyeθ = 50

eyeθ = 80

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Effect of Changing eye\phi

Note: This example starts on the xy plane (eye\theta = 90) and decrements eye\phi by 20 to elevate the viewing angle.

By changing eye\phi, you can elevate your viewing angle above the xy plane. If 90 < eye\phi < 270, the viewing angle is below the xy plane.

z1(x,y) = (x³y - y³x) / 390 In this example, eye\theta = 20

Effect of Changing eye\psi

Note: During rotation, the axes expand or contract to fit the screen's width and height. This causes some distortion as shown in the example.

When eye\psi=0, the z axis runs the height of the screen.

When eye\psi=90, the z axis runs the width of the screen.

As the z axis rotates 90°, its range (-10 to 10 in this example) expands to almost twice its original length. Likewise, the x and y axes expand or contract.

The view on the Graph screen is always oriented along the viewing angles set by eye\theta and eye\phi. You can change eye\psi to rotate the graph around that line of sight.

z1(x,y)=(x³y-y³x) / 390 In this example, eye\theta=20 and eye\phi=70

From the Home Screen or a Program

The eye values are stored in the system variables eye\theta, eye\phi, and eye\psi. You can access or store to these variables as necessary.

TI-89: To type \phi or \psi, press [◆] [alpha] [F] or [◆] [alpha] [Y], respectively. You can also press [2nd] [CHAR] and use the Greek menu.

TI-92 Plus: To type \phi or \psi, press [2nd] G F or [2nd] G Y respectively. You can also press [2nd] [CHAR] and use the Greek menu.

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Animating a 3D Graph Interactively

After plotting any 3D graph, you can change the viewing angle interactively by using the cursor. Refer to the preview example on page 154.

The Viewing Orbit

Note: The viewing orbit affects the eye Window variables in differing amounts.

When using ◀ and ▶ to animate a graph, think of it as moving the viewing angle along its “viewing orbit” around the graph.

Moving along this orbit can cause the z axis to wobble slightly during the animation (as you can see in the preview example on page 154).

Animating the Graph

Note: If the graph is shown in expanded view, it returns to normal view automatically when you press a cursor key.

Tip: After animating the graph, you can stop and then re-start the animation in the same direction by pressing:

TI-89: [ENTER] or [alpha] [←]

TI-92 Plus: [ENTER] or space bar

Tip: During an animation, you can switch to the next graph format style by pressing:

TI-89: [I]

TI-92 Plus: F

Tip: To see a graphic that shows the eye angles, refer to page 162.

Animating a Series of Graph Pictures

You can also animate a graph by saving a series of graph pictures and then flipping (or cycling) through those pictures. Refer to “Animating a Series of Graph Pictures” in Chapter 12: Additional Graphing Topics. This method gives you more control over the Window variable values, particularly eyeψ (page 162), which rotates the graph.

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Changing the Axes and Style Formats

With its default settings, the TI-89 / TI-92 Plus displays hidden surfaces on a 3D graph but does not display the axes. However, you can change the graph format at any time.

Displaying the GRAPH FORMATS Dialog Box

From the Y= Editor, Window Editor, or Graph screen, press:

F1 9 — or — TI-89: [diamond] [1] TI-92 Plus: [diamond] F

• The dialog box shows the current graph format settings.
• To exit without making a change, press ESC.

To change any of these settings, use the same procedure that you use to change other types of dialog boxes, such as the MODE dialog box.

Examples of Axes Settings

To display the valid Axes settings, highlight the current setting and press [enter].

• AXES — Shows standard xyz axes.
• BOX — Shows 3-dimensional box axes.

The edges of the box are determined by the Window variables xmin, xmax, etc.

In many cases, the origin (0,0,0) is inside the box, not at a corner.

For example, if xmin = ymin = zmin = -10 and xmax = ymax = zmax = 10, the origin is at the center of the box.

Tip: Setting Labels = ON is helpful when you display either type of 3D axes.

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Examples of Style Settings

To display the valid Style settings, highlight the current setting and press ①.

1:WIRE FRAME 2:HIDDEN SURFACE 3:CONTOUR LEVELS 4:WIRE AND CONTOUR 5:IMPLICIT PLOT

Tip: WIRE FRAME is faster to graph and may be more convenient when you're experimenting with different shapes.

• WIRE FRAME — Shows the 3D shape as a transparent wire frame.
• HIDDEN SURFACES — Uses shading to differentiate the two sides of the 3D shape.

Later sections in this chapter describe CONTOUR LEVELS, WIRE AND CONTOUR (page 167) and IMPLICIT PLOT (page 171).

Be Aware of Possible Optical Illusions

The eye angles used to view a graph (eyeθ, eyeφ, and eyeψ Window variables) can result in optical illusions that cause you to lose perspective on a graph.

Typically, most optical illusions occur when the eye angles are in a negative quadrant of the coordinate system.

Optical illusions may be more noticeable with box axes. For example, it may not be immediately obvious which is the "front" of the box.

Looking down from above the xy plane Looking up from below the xy plane

Note: These examples show the graphs as displayed on the screen.

eyeθ = 20, eyeφ = 55, eyeψ = 0 eyeθ = 20, eyeφ = 120, eyeψ= 0

Note: These examples use artificial shading (which is not displayed on the screen) to show the front of the box.

To minimize the effect of optical illusions, use the GRAPH FORMATS dialog box to set Style = HIDDEN SURFACE.

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Contour Plots

In a contour plot, a line is drawn to connect adjacent points on the 3D graph that have the same z value. This section discusses the CONTOUR LEVELS and WIRE AND CONTOUR graph format styles.

Selecting the Graph Format Style

Tip: From the Graph screen, you can switch from one graph format style to the next (skipping IMPLICIT PLOT) by pressing: TI-89: \boxedF5 TI-92 Plus: F

Note: Pressing: TI-89: \boxedF5 TI-92 Plus: F to select CONTOUR LEVELS does not affect the viewing angle, view, or Labels format as it does if you use: TI-89: \boxedF4 \boxedF5 TI-92 Plus: \boxedF4 F

In 3D graphing mode, define an equation and graph it as you would any 3D equation, with the following exception. Display the GRAPH FORMATS dialog box by pressing [F1] 9 from the Y= Editor, Window editor, or Graph screen. Then set:

Style = CONTOUR LEVELS – or – Style = WIRE AND CONTOUR

• For CONTOUR LEVELS, only the contours are shown.
  • The viewing angle is set initially so that you are viewing the contours by looking down the z axis. You can change the viewing angle as necessary.
  • The graph is shown in expanded view. To switch between expanded and normal view, press \boxed×.
  • The Labels format is set to OFF automatically.
• For WIRE AND CONTOUR, the contours are drawn on a wire frame view. The viewing angle, view (expanded or normal), and Labels format retain their previous settings.

\begin{tabular}{lll} Style & z1(x,y)=(x$^3$ y – y$^3$ x) / 390 & z1(x,y)=x$^2$ +.5y$^2$ – 5 \ \hline & Looking down z axis & \ CONTOUR LEVELS & & \ & Using eye\theta=20, eye\phi=70, eye\psi=0 & \ CONTOUR LEVELS & & \ WIRE AND CONTOUR & & \ \end{tabular}

Note: These examples use the same x, y, and z Window variable values as a ZoomStd viewing cube. If you use ZoomStd, press Z to look down the z axis.

Note: Do not confuse the contours with the grid lines. The contours are darker.

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How Are Z Values Determined?

You can set the ncontour Window variable ( [■] [WINDOW] ) to specify the number of contours that will be evenly distributed along the displayed range of z values, where:

$ increment = \fraczmax - zminncontour + 1 $

The z values for the contours are: zmin + increment zmin + 2(increment) zmin + 3(increment) ... zmin + ncontour(increment)

eyeθ=30. eyeφ=70. eyew=0. xmin=-10. xmax=10. xgrid=14. ymin=-10. ymax=10. ygrid=14. zmin=-10. zmax=10. ncontour=5.

The default is 5. You can set this to 0 through 20.

If ncontour=5 and you use the standard viewing window (zmin=-10 and zmax=10), the increment is 3.333. Five contours are drawn for z= -6.666, -3.333, 0, 3.333, and 6.666.

Note, however, that a contour is not drawn for a z value if the 3D graph is not defined at that z value.

Drawing a Contour for the Z Value of a Selected Point Interactively

If a contour graph is currently displayed, you can specify a point on the graph and draw a contour for the corresponding z value.

1. To display the Draw menu, press: TI-89: [2nd] [F6] TI-92 Plus: [F6]

2. Select 7:Draw Contour.
3. Either:
   • Type the point’s x value and press [ENTER], and then type the y value and press [ENTER]. – or –
   • Move the cursor to the applicable point. (The cursor moves along the grid lines.) Then press [ENTER].

For example, suppose the current graph is z1(x,y)=x² + .5y² – 5. If you specify x=2 and y=3, a contour is drawn for z=3.5.

Tip: Any existing contours remain on the graph. To remove the default contours, display the Window editor ([■] [WINDOW]) and set ncontour=0.

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Drawing Contours for Specified Z Values

From the Graph screen, display the Draw menu and then select 8:DrwCtour. The Home screen is displayed automatically with DrwCtour in the entry line. You can then specify one or more z values individually or generate a sequence of z values.

Some examples are:

Tip: To remove the default contours, use \boxed[WINDOW] and set ncontour=0.

DrwCtour 5 — Draws a contour for z=5. DrwCtour {1,2,3} — Draws contours for z=1, 2, and 3. DrwCtour seq(n,n, -10,10,2) — Draws contours for a sequence of z values from -10 through 10 in steps of 2 (-10, -8, -6, etc.).

The specified contours are drawn on the current 3D graph. (A contour is not drawn if the specified z value is outside the viewing cube or if the 3D graph is not defined at that z value.)

Notes about Contour Plots

For a contour plot:

• You can use the cursor keys (page 164) to animate the contour plot.
• You cannot trace (\boxedF3) the contours themselves. However, you can trace the wire frame as seen when Style=WIRE AND CONTOUR.
• It may take awhile to evaluate the equation initially.
• Because of possible long evaluation times, you first may want to experiment with your 3D equation by using Style=WIRE FRAME. The evaluation time is much shorter. Then, after you're sure you have the correct Window variable values, display the Graph Formats dialog box and set Style=CONTOUR LEVELS or WIRE AND CONTOUR.

TI-89: \boxed[◆][I] TI-92 Plus: \boxed[◆] F

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Example: Contours of a Complex Modulus Surface

The complex modulus surface given by z(a,b) = abs(f(a+bi)) shows all the complex zeros of any polynomial y=f(x).

Example

In this example, let f(x)=x³+1. By substituting the general complex form x+yi for x, you can express the complex surface equation as z(x,y)=abs((x+y*i)³+1).

1. Use [MODE] to set Graph=3D.
2. Press [♦][Y=], and define the equation: z1(x,y)=abs((x+y*i)^3+1)
3. Press [♦][WINDOW], and set the Window variables as shown.
4. Display the Graph Formats dialog box: TI-89: [♦][1] TI-92 Plus: [♦]F Turn on the axes, set Style = CONTOUR LEVELS, and return to the Window editor.
5. Press [♦][GRAPH] to graph the equation. It will take awhile to evaluate the graph; so be patient. When the graph is displayed, the complex modulus surface touches the xy plane at exactly the complex zeros of the polynomial: -1, 1/2 + \frac\sqrt32i, and 1/2 - \frac\sqrt32i

Note: For more accurate estimates, increase the xgrid and ygrid Window variables. However, this increases the graph evaluation time.

Tip: When you animate the graph, the screen changes to normal view. Use [X] to toggle between normal and expanded views.

6. Press [F3], and move the trace cursor to the zero in the fourth quadrant. The coordinates let you estimate .428-.857i as the zero.
7. Press [ESC]. Then use the cursor keys to animate the graph and view it from different eye angles.

This example shows eyeθ=70, eyeφ=70, and eyeψ=0.

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Implicit Plots

An implicit plot is used primarily as a way to graph 2D implicit forms that cannot be graphed in function graphing mode. Technically, an implicit plot is a 3D contour plot with a single contour drawn for z=0 only.

Explicit and Implicit Forms

In 2D function graphing mode, equations have an explicit form y=f(x), where y is unique for each value of x.

Many equations, however, have an implicit form f(x,y)=g(x,y), where you cannot explicitly solve for y in terms of x or for x in terms of y.

y is not unique for each x, so you cannot graph this in function graphing mode.

By using implicit plots in 3D graphing mode, you can graph these implicit forms without solving for y or x.

Rearrange the implicit form as an equation set to zero.

$$f(x,y) - g(x,y)=0$$

In the Y= Editor, enter the non-zero side of the equation. This is valid because an implicit plot automatically sets the equation equal to zero.

$$z1(x,y)=f(x,y) - g(x,y)$$

For example, given the ellipse equation shown to the right, enter the implicit form in the Y= Editor.

If $x^2 + .5y^2 =30$, then $z1(x,y)=x^2 + .5y^2 - 30$.

Tip: You can also graph many implicit forms if you either:

• Express them as parametric equations. Refer to Chapter 7.
• Break them into separate, explicit functions. Refer to the preview example in Chapter 6.

Selecting the Graph Format Style

In 3D graphing mode, define an appropriate equation and graph it as you would any 3D equation, with the following exception. Display the GRAPH FORMATS dialog box from the Y= Editor, Window editor, or Graph screen.

TI-89: [icon] [icon] TI-92 Plus: [icon] F

Then set:

Style = IMPLICIT PLOT

Note: From the Graph screen, you can switch to the other graph format styles by pressing:

TI-89: [icon] TI-92 Plus: F

However, to return to IMPLICIT PLOT press:

TI-89: [icon] [icon] TI-92 Plus: [icon] F

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• The viewing angle is set initially so that you are viewing the plot by looking down the z axis. You can change the viewing angle as necessary.
• The plot is shown in expanded view. To switch between expanded and normal view, press \boxtimes.
• The Labels format is set to OFF automatically.

Note: These examples use the same x, y, and z Window variable values as a ZoomStd viewing cube. If you use ZoomStd, press Z to look down the z axis.

\begin{tabular}{lll} Style & $x^2 - y^2 = 4$ & \sin(x)+\cos(y)=e^(xy) \ & $z1(x,y)=x^2 - y^2 - 4$ & z1(x,y)=\sin(x)+\cos(y)-e^(xy) \ IMPLICIT PLOT & & \ \end{tabular}

Notes About Implicit Plots

For an implicit plot:

• The ncontour Window variable (page 168) has no affect. Only the z=0 contour is drawn, regardless of the value of ncontour. The displayed plot shows where the implicit form intersects the xy plane.
• You can use the cursor keys (page 164) to animate the plot.
• You cannot trace (\boxedF3) the implicit plot itself. However, you can trace the unseen wire frame graph of the 3D equation.
• It may take awhile to evaluate the equation initially.
• Because of possible long evaluation times, you first may want to experiment with your 3D equation by using Style=WIRE FRAME. The evaluation time is much shorter. Then, after you're sure you have the correct Window variable values, set Style=IMPLICIT PLOT. TI-89: \boxed\diamondsuit\boxedI TI-92 Plus: \boxed\diamondsuit\boxedF

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Example: Implicit Plot of a More Complicated Equation

You can use the IMPLICIT PLOT graph format style to plot and animate a complicated equation that cannot be graphed otherwise. Although it may take a long time to evaluate such a graph, the visual results can justify the time required.

Example Graph the equation \sin(x^4+y-x^3 y) = .1.

1. Use MODE to set Graph=3D.
2. Press [Y=], and define the equation: z1(x,y)=\sin(x^4+y-x^3y)-.1
3. Press [WINDOW], and set the Window variables as shown.
4. Press: TI-89: [F5] TI-92 Plus: [F5] F Turn on the axes, set Style = IMPLICIT PLOT, and return to the Window editor.
5. Press [GRAPH] to graph the equation. It will take awhile to evaluate the graph; so be patient.
6. Use the cursor keys to animate the graph and view it from different eye angles.

Note: For more detail, increase the xgrid and ygrid Window variables. However, this increases the graph evaluation time.

Tip: When you animate the graph, the screen changes to normal view. Press [X] to switch between normal and expanded views.

eye\theta=-90. eye\phi=0. eye\psi=0. xmin=-10. xmax=10. xgrid=14. ymin=-10. ymax=10. ygrid=14. zmin=-10. zmax=10. ncontour=5.

The graph shows where \sin(x^4+y-x^3y) = .1

In expanded view, this example shows eye\theta=-127.85, eye\phi=52.86, and eye\psi=-18.26.

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Differential Equation Graphing

11

• Preview of Differential Equation Graphing ........ 176
• Overview of Steps in Graphing Differential Equations ........ 178
• Differences in Diff Equations and Function Graphing ........ 179
• Setting the Initial Conditions ........ 184
• Defining a System for Higher-Order Equations ........ 186
• Example of a 2nd-Order Equation ........ 187
• Example of a 3rd-Order Equation ........ 189
• Setting Axes for Time or Custom Plots ........ 190
• Example of Time and Custom Axes ........ 191
• Example Comparison of RK and Euler ........ 193
• Example of the deSolve( ) Function ........ 196
• Troubleshooting with the Fields Graph Format ........ 197

Note: A differential equation is:

• 1st-order when only 1st-order derivatives appear.
• Ordinary when all the derivatives are with respect to the same independent variable.

This chapter describes how to solve differential equations graphically on the TI-89 / TI-92 Plus. Before using this chapter, you should be familiar with Chapter 6: Basic Function Graphing.

The TI-89 / TI-92 Plus solves 1st-order systems of ordinary differential equations. For example:
y' = .001 y * (100 - y)

or coupled 1st-order differential equations such as:
y1' = - y1 + 0.1 * y1 * y2
y2' = 3 * y2 - y1 * y2

You can solve higher-order equations by defining them as a system of 1st-order equations. For example:
y" + y = sin(t) can be defined as y1' = y2
y2' = - y1 + sin(t)

By setting appropriate initial conditions, you can graph a particular solution curve of a differential equation.

You can also graph a slope or direction field that helps you visualize the behavior of the entire family of solution curves.

For graphing, the TI-89 / TI-92 Plus uses numerical methods that approximate the true solutions. The deSolve() function lets you solve some differential equations symbolically. This chapter introduces deSolve(). Refer to Appendix A for more details.

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Preview of Differential Equation Graphing

Graph the solution to the logistic 1st-order differential equation y' = .001y* (100 – y). Start by drawing only the slope field. Then enter initial conditions in the Y= Editor and interactively from the Graph screen.

\begin{tabular}{|p{0.25\textwidth}|p{0.15\textwidth}|p{0.15\textwidth}|p{0.35\textwidth}|} \hline \textbf{Steps} & \textbf{TI-89 Keystrokes} & \textbf{TI-92 Plus Keystrokes} & \textbf{Display} \ \hline

1. Display the MODE dialog box. For Graph mode, select DIFF EQUATIONS. & \boxedMODE \rightarrow 6 \boxedENTER & \boxedMODE \rightarrow 6 \boxedENTER & \ \hline
2. Display and clear the Y= Editor. Then define the 1st-order differential equation: y1'(t)=.001y1*(100–y1) Press × to enter the * shown above. Do not use implied multiplication between the variable and parentheses. If you do, it is treated as a function call. Leave the initial condition yi blank. & \blacklozenge [Y=] \boxedF1 8 \boxedENTER \boxedENTER .001 Y1 × ( 100 - Y1 ) \boxedENTER & \blacklozenge [Y=] \boxedF1 8 \boxedENTER \boxedENTER .001 Y1 × ( 100 - Y1 ) \boxedENTER & \ & & & Important: With y1' selected, the TI-89 / TI-92 Plus will graph the y1 solution curve, not the derivative y1'. \ \hline
3. Display the GRAPH FORMATS dialog box. Then set Axes = ON, Labels = ON, Solution Method = RK, and Fields = SLPFLD. Important: To graph one differential equation, Fields must be set to SLPFLD or FLDOFF. If Fields=DIRFLD an error occurs when you graph. & \blacklozenge [I] \rightarrow 2 \rightarrow 2 \rightarrow 1 \rightarrow 1 \boxedENTER & \blacklozenge F \rightarrow 2 \rightarrow 2 \rightarrow 1 \rightarrow 1 \boxedENTER & \ \hline
4. Display the Window Editor, and set the Window variables as shown to the right. & \blacklozenge [WINDOW] 0 \rightarrow 10 \rightarrow .1 \rightarrow 0 \rightarrow 10 \rightarrow 110 \rightarrow 10 \rightarrow 10 \rightarrow 120 \rightarrow 10 \rightarrow 0 \rightarrow .001 \rightarrow 20 & \blacklozenge [WINDOW] 0 \rightarrow 10 \rightarrow .1 \rightarrow 0 \rightarrow 10 \rightarrow 110 \rightarrow 10 \rightarrow 10 \rightarrow 120 \rightarrow 10 \rightarrow 0 \rightarrow .00 \rightarrow 20 & t0=0. tmax=10, tstep=1 tplot=0, xmin=-10. xmax=10. xscl=10, ymin=-10. ymax=10. yscl=10. nCurves=0. diftol=.001 fldres=20. \ \hline
5. Display the Graph screen. Because you did not specify an initial condition, only the slope field is drawn (as specified by Fields=SLPFLD in the GRAPH FORMATS dialog box). & \blacklozenge [GRAPH] & \blacklozenge [GRAPH] & \ \hline \end{tabular}

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Chapter 11: Differential Equation Graphing 177

Overview of Steps in Graphing Differential Equations

To graph differential equations, use the same general steps used for y(x) functions as described in Chapter 6: Basic Function Graphing. Any differences are described on the following pages.

Graphing Differential Equations

Tip: To turn off any stat data plots, press F5 or use F4 to deselect them. Refer to Chapter 16.

Note: The Fields format is critical, depending on the order of the equation (page 197).

Note: Valid Axes settings depend on the Fields format (pages 190 and 197).

Note: Depending on the Solution Method and Fields formats, different Window variables are displayed.

Tip: F2 Zoom also changes the viewing window.

Set Graph mode ([MODE]) to DIFF EQUATIONS. Also set Angle mode, if necessary.

Define equations and, optionally, initial conditions on Y= Editor ([\blacklozenge] [Y=]).

Select ([F4]) which defined equations to graph.

Set the display style for an equation. TI-89: [2nd] [F6] TI-92 Plus: [F6]

Set the graph format. Solution Method and Fields are unique to differential equations. [F1] 9 — or — TI-89: [\blacklozenge] [1] TI-92 Plus: [\blacklozenge] F

Set the axes as applicable, depending on the Fields format. TI-89: [2nd] [F7] TI-92 Plus: [F7]

Define the viewing window (\blacklozenge [WINDOW]).

Graph the equations (\blacklozenge [GRAPH]).

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Differences in Diff Equations and Function Graphing

This chapter assumes that you already know how to graph y(x) functions as described in Chapter 6: Basic Function Graphing. This section describes the differences.

Setting the Graph Mode

Use [MODE] to set Graph = DIFF EQUATIONS before you define differential equations or set Window variables. The Y= Editor and the Window Editor let you enter information for the current Graph mode setting only.

Defining Differential Equations on the Y= Editor

Use t0 to specify when initial conditions occur. You can also set t0 in the Window Editor.

Use yi to specify one or more initial conditions for the corresponding differential equation.

You can define differential equations y1'(t) through y99'(t).

Tip: You can use the Define command from the Home screen to define functions and equations.

When entering equations in the Y= Editor, do not use y(t) formats to refer to results. For example:

Enter: y1' = .001y1*(100-y1) Not: y1' = .001y1(t)*(100-y1(t))

Do not use implied multiplication between a variable and parenthetical expression. If you do, it is treated as a function call.

Only 1st-order equations can be entered in the Y= Editor. To graph 2nd- or higher-order equations, you must enter them as a system of 1st-order equations. For information, refer to page 186.

For detailed information about setting initial conditions, refer to page 184.

Selecting Differential Equations

You can use [F4] to select a differential equation, but not its initial condition.

Important: Selecting y1' will graph the y1 solution curve, not the derivative y1', depending on the axis setting.

Selecting the Display Style

With the Style menu, only the Line, Dot, Square, Thick, Animate, and Path styles are available. Dot and Square mark only those discrete values (in tstep increments) at which a differential equation is plotted.

TI-89: [2nd] [F6] TI-92 Plus: [F6]

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Setting Graph Formats

From the Y= Editor, Window Editor, or Graph screen, press:

F1 9

— or —

TI-89: ♦ 1

TI-92 Plus: ♦ F

The formats affected by differential equations are:

Important: The Fields graph format is critical in successfully graphing differential equations. Refer to “Troubleshooting with the Fields Graph Format” on page 197.

Tip: If you press ENTER while a slope or direction field is being drawn, the graph pauses after the field is drawn but before the solutions are plotted. Press ENTER again to continue.

Tip: To cancel graphing, press ON.

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Setting Axes

In the Y= Editor, Axes may or may not be available, depending on the current graph format.

If it is available, you can select the axes that are used to graph the differential equations. For more information, refer to page 190.

TI-89: [2nd] [F7]
TI-92 Plus: [F7]

Window Variables

Differential equation graphs use the following Window variables. Depending on the Solution Method and Fields graph formats, not all of these variables are listed in the Window Editor ([♦] [WINDOW]) at the same time.

Note: If tmax < t0, tstep must be negative.

Note: If Fields=SLPFLD, tplot is ignored and is assumed to be the same as t0.

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Window Variables (Continued)

Note: For information about how the Fields graph format affects whether ncurves is used, refer to page 184.

Standard values (set when you select 6:ZoomStd from the [F2] Zoom toolbar menu) are:

t0 = 0. xmin = -1. ymin = -10. ncurves = 0. tmax = 10. xmax = 10. ymax = 10. diftol = .001 tstep = .1 xscl = 1. yscl = 1. Estep = 1. tplot = 0. fldres = 14. dtime = 0.

You may need to change the standard values for the t variables to ensure that sufficient points are plotted.

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The fldpic System Variable

When a slope or direction field is drawn, a picture of the field is stored automatically to a system variable named fldpic. If you perform an operation that regraphs the plotted equations but does not affect the field, the TI-89 / TI-92 Plus reuses the picture in fldpic instead of having to redraw the field. This can speed up the regraphing time significantly.

fldpic is deleted automatically when you exit the differential equation graphing mode or when you display a graph with Fields = FLDOFF.

Exploring a Graph

As in function graphing, you can explore a graph by using the following tools. Any displayed coordinates are shown in rectangular or polar form as set in the graph format.

\begin{tabular}{ll} \textbf{Tool} & \textbf{For Differential Equation Graphs:} \ Free-Moving Cursor & Works just as it does for function graphs. \ [F2] Zoom & Works just as it does for function graphs. \ & \bullet Only x (xmin, xmax, xscl) and y (ymin, ymax, yscl) Window variables are affected. \ & \bullet The t Window variables (t0, tmax, tstep, tplot) are not affected unless you select 6:ZoomStd (which sets all Window variables to their standard values). \ Tip: During a trace, you can move the cursor to a particular point by typing a value for t and pressing [ENTER]. & [F3] Trace Lets you move the cursor along the curve one tstep at a time. To move approximately ten plotted points at a time, press [2nd] \blacktriangleright or [2nd] \blacktriangleleft. \ & If you enter initial conditions in the Y= Editor or let the ncurves Window variable plot curves automatically, you can trace the curves. If you use: \ & TI-89: [2nd] [F8] \ & TI-92 Plus: [F8] \ & IC from the Graph screen to select initial conditions interactively, you cannot trace the curves. \ Tip: You can use QuickCenter at any time during a trace, even if the cursor is still on the screen. & QuickCenter applies to all directions. If you move the cursor off the screen (top or bottom, left or right), press [ENTER] to center the viewing window on the cursor location. Use \ominus or \oplus to view results on all plotted curves. \ [F5] Math & Only 1:Value is available. \ & \bullet With TIME axes, the y(t) solution value (represented by yc) is displayed for a specified t value. \ & \bullet With CUSTOM axes, the values that correspond to x and y depend on the axes you choose. \ \end{tabular}

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