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150 Chapter 9: Sequence Graphing
09SEQUEN.DOC TI-89/TI-92 Plus: Sequence Graphing (English) Susan Gullord Revised: 02/23/01 10:59 AM Printed: 02/23/01 2:14 PM Page 150 of 14
Using the predator-prey model in biology, determine the numbers of
rabbits and foxes that maintain population equilibrium in a certain
region.
R
= Number of rabbits
M
= Growth rate of rabbits if there are no foxes (use .05)
K
= Rate at which foxes can kill rabbits (use .001)
W
= Number of foxes
G
= Growth rate of foxes if there are rabbits (use .0002)
D
= Death rate of foxes if there are no rabbits (use .03)
R
n
=
R
n-1
(1 + M
ì
K W
n-1
)
W
n
=
W
n-1
(1 + G R
n-1
ì
D)
1. On the Y= Editor (
¥#
), define the sequences and initial values
for
R
n
and
W
n
.
u1(n) = u1(n
ì
1)
ù
(1 + .05
ì
.001
ù
u2(n
ì
1))
ui1 = 200
u2(n) = u2(n
ì
1)
ù
(1 + .0002
ù
u1(n
ì
1)
ì
.03)
ui2 = 50
2. Set
Axes = TIME
.
3. On the Window Editor
(
¥$
), set the Window
variables.
nmin=0. xmin=0. ymin=0.
nmax=400. xmax=400. ymax=300.
plotStrt=1. xscl=100. yscl=100.
plotStep=1.
4. Graph the sequence
(
¥%
).
5. On the Y= Editor, set
Axes = CUSTOM
,
X Axis = u1
, and
Y Axis = u2
.
6. In the Window Editor, change
the Window variables.
nmin=0. xmin=84. ymin=25.
nmax=400. xmax=237. ymax=75.
plotStrt=1. xscl=50. yscl=10.
plotStep=1.
7. Regraph the sequence.
Using Custom Plots
CUSTOM axes give you great flexibility in graphing sequences.
As shown in the following example, CUSTOM axes are
particularly effective for showing relationships between one
sequence and another.
Example: Predator-
Prey Model
Note: Assume there are
initially 200 rabbits and
50 foxes.
Note: Use
…
to individually
trace the number of rabbits
u1(n) and foxes u2(n) over
time (n).
Note: Use
…
to trace both
the number of rabbits (xc)
and foxes (yc) over the
cycle of 400 generations.
u(n)
n
u2(n)
u1(n)
u1(n)
u2(n)
- TI-92 Plus 1
- TI-89 Shortcut Keys 3
- TI-92 Plus Shortcut Keys 4
- Table of Contents 5
- Flash Applications 12
- 00_front.doc 13
- Keystroke Differences 14
- What’s New? 16
- Upgradability with 17
- Flash ROM 17
- Custom Menu 17
- Chapter 1: 18
- Getting Started 18
- Installing the AAA 19
- Batteries 19
- Installing the AA 20
- Note: English cannot be 21
- Note: Until you complete 22
- Select a Language dialog 22
- About the Home 23
- Turning the 24
- TI-89 / TI-92 Plus 24
- Performing Computations 25
- 01START DOC TI 26
- Graphing a Function 28
- Chapter 2: 30
- Operating the Calculator 30
- Setting the Display Contrast 32
- The TI-89 Keyboard 33
- Modifier Keys 35
- Other Important 36
- Keys You Need to 36
- Be Familiar With 36
- Important Keys 37
- (continued) 37
- Home Screen 40
- Note: For an example of 41
- Entering Numbers 42
- Note: You can also select 44
- by using 44
- (page 44) 44
- Note: When you scroll to 45
- is displayed at 45
- Formats of Displayed Results 46
- Exact/Approx 47
- Display Digits Mode 48
- Exponential Format 48
- Displaying a Menu 51
- Selecting an Item 51
- Note: Because of limited 52
- Using the Custom Menu 54
- Selecting an Application 55
- F1 F2 F3 F4 F5 56
- WINDOW GRAPH TBLSet TABLEY= 56
- Setting Modes 57
- ¦ Performs 60
- Clean Up Toolbar 60
- Using the Catalog Dialog Box 61
- Indicated command 62
- Brackets [ ] indicate 62
- Note: Use the VAR-LINK 63
- Flash applications. See 63
- Chapter 21 63
- The variable’s value 65
- Tip: Reexecuting an entry 66
- Recalling a Previous 67
- Note: Refer to ans() in 68
- Appendix A 68
- ¦ You can use 69
- Why Use Auto-Paste 69
- Auto-Pasting an 69
- Entry or Answer 69
- Status Line 70
- Indicators 70
- Chapter 3: 74
- Symbolic Manipulation 74
- Automatic Simplification 81
- Functions that Use 83
- Simplification 83
- Tip: Use the solve function 85
- Specifying Domain 86
- Constraints 86
- Using Substitutions 86
- Note: For a complete 87
- Common Algebraic Operations 89
- Solving an Equation 90
- Solving a System of 90
- Linear Equations 90
- Overview of the Calc Menu 92
- Common Calculus Operations 93
- Define h(x)= 94
- (sin(t)/t,t,0,x) 94
- Multi-Statement vs 95
- Single-Statement 95
- Functions 95
- ˆ represents infinity 97
- Chapter 4: 98
- Entering Constants or Units 100
- Combining Multiple 101
- Using Parentheses 101
- Calculation 101
- For All Units Except 102
- Temperature 102
- For Temperature 103
- 04UNITS DOC TI 104
- Defaults for SI and 106
- Constants 106
- Chapter 5: 110
- Additional Home Screen Topics 110
- Saving the Entries 111
- Restoring the Saved 111
- Auto-paste vs 112
- Cut/Copy/Paste 112
- Cutting or Copying 112
- Information to the 112
- Clipboard 112
- Pasting Information 113
- Example: Copying 113
- Format of a 114
- Function 114
- Creating a User 114
- Defined Function 114
- Creating a Multi 115
- Statement Function 115
- Evaluating a 115
- Displaying and 116
- Editing a Function 116
- Definition 116
- Folders and 117
- Variables 117
- To cancel the menu or 118
- VAR-LINK 119
- Chapter 6: 122
- Basic Function Graphing 122
- ¦ Use the 124
- Graphing Functions 124
- Exploring the Graph 124
- Setting the Graph Mode 125
- POLAR graphing mode 126
- Selecting Functions to Graph 128
- Tip: To set 129
- as the style 129
- Defining the Viewing Window 130
- Changing the Graph Format 131
- indicator shows while 132
- ¦ Use a 133
- Free-Moving Cursor 133
- Tracing a Function 134
- Overview of the 136
- Zoom Menu 136
- Math Menu 139
- Tip: To erase a drawn 142
- (ReGraph) 142
- Chapter 7: 144
- Parametric Graphing 144
- PARAMETRIC 146
- Enter x and y components 147
- You can define 147
- Chapter 8: 150
- Polar Graphing 150
- Preview of Polar Graphing 151
- Window Variables 154
- Setting the Graph 154
- Chapter 9: 156
- Sequence Graphing 156
- Preview of Sequence Graphing 157
- Tip: You can also evaluate 158
- Simply enter the n value 158
- Selecting 160
- Sequences 160
- Selecting the 160
- Display Style 160
- Note: Both of these graphs 161
- ¦ To exit without making any 163
- Displaying the 163
- Dialog Box 163
- Using Web Plots 164
- Example: 165
- Convergence 165
- Divergence 165
- Oscillation 166
- Using Custom Plots 167
- Example: Fibonacci 168
- Sequence 168
- Chapter 10: 170
- 3D Graphing 170
- Preview of 3D Graphing 171
- You can define 3D 174
- Moving the Cursor in 3D 177
- The Viewing Orbit 181
- Animating the 181
- Animating a Series 181
- Tip: Setting Labels = ON is 182
- WIRE FRAME 183
- Contour Plots 184
- The default is 5. You can 185
- Tip: To remove the default 186
- Implicit Plots 188
- Chapter 11: 192
- Graphing 195
- Differential 195
- Equations 195
- Tip: You can use the 196
- Setting Axes 198
- Note: For information about 199
- Dimension 202
- Transforming an 203
- Equation into a 1st 203
- Order System 203
- Invalid Axes 204
- Undefined 206
- Predator-Prey Model 208
- Note: The fuzzy line on the 212
- Example of the deSolve( 213
- ) Function 213
- Fields=DIRFLD 215
- Fields=FLDOFF 216
- If You Use the Table 216
- Screen to View 216
- Chapter 12: 218
- Additional Graphing Topics 218
- for this 219
- Use the complete function 221
- Using the When 223
- Using a Multi 224
- Statement, User 224
- From the Home 224
- Screen or a Program 224
- Graphing a Family of Curves 225
- Using the Two-Graph Mode 226
- Note: You can display non 228
- (such as the Home screen) 228
- Drawing a Function 229
- Parametric, or Polar 229
- Equation 229
- Drawing the Inverse 229
- Tip: The text cursor 233
- Saving a Picture of 234
- Saving a Portion of 234
- Note: If a variable name is 235
- Note: Due to its complexity 236
- In two-graph mode 237
- Chapter 13: 238
- Preview of Tables 239
- Chapter 13: Tables 223 240
- Tip: To cancel a menu or 242
- Displaying an Automatic Table 243
- 228 Chapter 13: Tables 245
- TABLE SETUP 245
- Building a Manual (Ask) Table 246
- Note: If the independent 247
- Chapter 14: 248
- Split Screens 248
- Preview of Split Screens 249
- Note: In two-graph mode 250
- TOP-BOTTOM 253
- Chapter 15: 254
- Data/Matrix Editor 254
- Πmeans the cell is in a 255
- Data Variable 257
- Matrix Variable 257
- Creating a New 258
- Data, Matrix, or List 258
- Variable 258
- The Data/Matrix 260
- Editor Screen 260
- Entering or Editing 260
- Note: If you enter more than 261
- Changing the 262
- Cell Width 262
- Clearing a Column 262
- Using the Shift 267
- Using the CumSum 267
- Sorting Columns 268
- Chapter 16: 270
- Statistics and Data Plots 270
- Calculate 272
- FUNCTION 275
- The Calculate 276
- is shown instead of =, you 277
- Statistical Calculation Types 278
- Statistical Variables 280
- Defining a Statistical Plot 281
- Plot Type = Scatter 282
- Mark = Box 282
- Statistical Plot Types 283
- ¦ When you trace ( 284
- Histogram 284
- Showing the List of 285
- Stat Plots 285
- To Graph Plots and 285
- Y= Functions 285
- Type the CBL list 289
- Chapter 17: 292
- Programming 292
- Preview of Programming 293
- Running an Existing Program 295
- Starting a New 297
- Program or 297
- Note: By default, Variable 298
- Example of Passing 301
- Values to a Program 301
- Defines the subroutine 304
- Declares the subroutine 304
- Calls the subroutine 304
- Using Variables in a Program 305
- Circular Definition 306
- Variable-Related 306
- Commands 306
- :Local i 307
- :For i,0,5,1 307
- : Disp i 307
- ¦ Include 308
- To Perform 308
- Symbolic 308
- Calculations 308
- String Operations 309
- String Commands 310
- Conditional Tests 311
- Control Toolbar 314
- For...EndFor Loops 314
- While...EndWhile 315
- Note: The 316
- command 316
- Configuration 317
- Entering the 317
- SetMode Command 317
- I/O Toolbar Menu 318
- Input Commands 318
- Output Commands 319
- Graphical User 319
- Interface 319
- Creating a Custom Menu 320
- Restoring the 321
- Default Custom 321
- Note: For more information 322
- Chapter 12 323
- Drawing on the Graph Screen 324
- Drawing Lines and 325
- Expressions 325
- SendCalc 326
- Example 1 328
- Example 2 328
- Example 3 328
- Example 4 329
- Example 5 329
- Tip: If the program is not in 330
- Chapter 18: 332
- Text Editor 332
- Preview of Text Operations 333
- Note: By default 335
- Entering and Editing Text 336
- Thin cursor between 339
- Cursor highlights 339
- Entering Special Characters 341
- Note: To help you find the 342
- Note: Neither calculator 343
- For a List of All 344
- Special Characters 344
- Creating a Lab Report 347
- Printing the Report 348
- Chapter 19: 350
- Numeric Solver 350
- Preview of the Numeric Solver 351
- The argument does not have to 352
- Defining the Known Variables 354
- Finding the Solution 356
- Graphing the Solution 357
- Chapter 20: 360
- Number Bases 360
- Preview of Number Bases 361
- Domain error 362
- APPROXIMATE 363
- Boolean Operations 364
- Chapter 21: 366
- Notice that “ ( ” is 368
- indicates the 368
- Checking and Resetting Memory 370
- Flash ROM free on 371
- Which Applications 376
- Can You Use? 376
- Procedure 376
- From the VAR-LINK 378
- Archive free 380
- RAM free 380
- Chapter 22: 382
- Linking Two Units 383
- If you select: What happens: 385
- Note: The sending unit may 386
- Certificate 387
- Overview of 388
- The “Chat” Program 388
- Transferring 392
- Product Software 392
- (Base Code) 392
- TI-89 TI-89 TI-89 393
- Error Messages 394
- ID Lists and Group 395
- Certificates 395
- Collecting ID Lists 395
- Transmitting the ID 396
- List to a Computer 396
- Clearing the ID List 396
- Chapter 23: 400
- Activities 400
- Maximum Length of 401
- Pole in Hallway 401
- Hint: Use the auto-paste 402
- and press 402
- Performing 403
- Computations to 403
- Derive the Quadratic 403
- Note: This is only one of the 404
- Exploring a Matrix 405
- Exploring cos(x) = sin(x) 406
- Exploring a 3D 407
- Graph of the 407
- Surface Area of a 407
- Parallelepiped 407
- Finding the 408
- Minimum Surface 408
- Area Analytically 408
- Running a Tutorial 409
- 2:Clear split 410
- Decomposing a 411
- Rational Function 411
- Note: Be sure the Graph 412
- Filtering Data by 413
- Categories 413
- Note: Set up several box 414
- Note: Only Plot 1 through 415
- Plot 5 should be selected 415
- Hint: Press 417
- Optional Exercise 418
- Visualizing Complex 419
- Note: Calculating and 420
- Finding the Interest 421
- Rate of an Annuity 421
- Finding the Future 421
- Value of an Annuity 421
- TI-GRAPH 422
- Finding Factors 423
- Sampling-without 424
- Replacement 424
- Sampling without 424
- Appendix A: 426
- Functions and Instructions 426
- Quick-Find Locator 427
- Matrices 428
- Statistics 429
- ClrGraph 437
- CyclePic 446
- 8:Clear Home 448
- (creates the folder 451
- (deletes the folder 451
- DispHome 455
- DrawFunc 456
- DrawParm 456
- DropDown 457
- DrwCtour 458
- InputStr 475
- LineHorz 478
- LineVert 479
- {0 2 3 485
- PlotsOff 497
- PowerReg 498
- 8.124… 9.601… 11.078… 502
- 0. .904… 1.809… 502
- 0. 0. 0 502
- RandSeed 505
- SendChat 511
- Flash App 513
- Alternate Language 513
- ShowStat 517
- Unarchiv 533
- ZoomData 539
- ZoomPrev 541
- ZoomTrig 542
- , ', " 553
- Appendix B: 558
- Reference Information 558
- Note: For detailed 567
- Vector Format 569
- Pretty Print 569
- Split Screen 569
- Unit System 571
- Custom Units 571
- Language 571
- Disp char(127) 572
- TI-89 Key Codes 573
- ' 39 574
- " 34 8241 q 113 574
- Table 2: Arrow Keys 574
- Table 3: Greek Letters 575
- Key Modifier 576
- Table 6: Greek Letters 579
- Table 7: Tilde Letters 579
- Table 8: Caret Letters 579
- Table 9: Umlaut Letters 579
- Entering Complex Numbers 580
- Accuracy Information 583
- Order of Evaluation 585
- Parentheses 585
- Brackets, and 585
- Indirection 586
- Post Operators 586
- Exponentiation 586
- Negation 586
- Constraint ( 586
- Regression Formulas 587
- ¦ At each of the five points 589
- Algorithm 589
- Runge-Kutta Method 590
- Appendix C: 592
- Battery Information 593
- Lithium battery 594
- AAA alkaline 594
- Slide to open 595
- AA alkaline batteries 595
- In Case of Difficulty 596
- Home Page: 597
- education.ti.com 597
- Warranty Information 598
- Australia & New 599
- Zealand Customers 599
- All Other Customers 599
- Appendix D: 600
- Programmer’s Guide 600
- A (continued) 609
- C (continued) 610
- D (continued) 611
- E (continued) 612
- F (continued) 613
- G (continued) 614
- M (continued) 616
- P (continued) 618
- S (continued) 620
- U (continued) 622
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