Hp 48g Graphing Calculator Manual do Utilizador Página 238
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Plotting a Phase Plane for a Vector-Valued Solution
The HP 48 also lets you plot vector-valued equations and select which
vector value is plotted on which axis. As described earlier, another
way to write the second-order equation
y” = ai(i)i/' + ao(t)i/ + 5'(t)
Where w is
fw is
fw * w + c * g(t)
, and c is
0 1
ao(t) ai[t)
The initial condition y'{to) ~ Do and y{to) = yi can be written w''(to)
= [ yo yi
Example:
19
This is a vector-valued initial condition.
Plot the following equation for tc(l) given that ?/(()) = 0
and ?/'(0) = 0 (w(0) = [00]):
y" = W + Jyy + .5f + 1
where y(0) = 0 and y'(0) = 0 (dy(0) = [00 ]),
Step 1: Convert the equation into a first-order equation.
0 1
.5 .5
.5i -|- 1)
Step 2: Store the values in fiu and c.
Qcm acm
0
ciEQ
1
® SICDD -5 iSPCl -5 [ENTER 1 CD ®
© [STO) QQD 0 [SPCl 1 [enter) Q @ C
CEoD
19-12 Differential Equations
- HP 48G Series 1
- Regulatory Information 2
- WKSl HEWLETT 3
- HP Part No. 00048-90126 3
- Printed in Singapore 3
- Acknowledgements 4
- Edition History 4
- Contents 5
- Stack Command Menu 6
- 7. The Equation Writer 7
- Contents-4 8
- Contents-5 9
- Contents-6 10
- Contents-7 11
- Contents-8 12
- Contents-9 13
- Contents-10 14
- Organization of the Display 15
- {(.)> 16
- The Stack 17
- Organization of the Keyboard 18
- ^ ! j ■ 19
- To cancel a shift key: 20
- Cursor Keys 21
- Menus: Extending the Keyboard 22
- , Г. . . . . . . . . . j г 23
- To display the previous menu: 24
- To key in a simple number: 25
- 123.4 i 26
- To key in a single character: 27
- Keying in Special Characters 28
- 'abcdefghijkImno 29
- [SPC) between each element 30
- Using the Command Line 31
- Command Line Operations 32
- !r»i I ENTRY 33
- Viewing and Editing Objects 34
- Viewing or Editing an Object 35
- 2 Using the EDIT Menu 36
- EDIT Menu Operations 36
- №liEDIT ]: 36
- O ■ ^■ C'EL 36
- Making Calculations 37
- 3.7 (a(LN) 38
- 2401 fsp^ 4 39
- 3-4 Stack 40
- To clear the entire stack: 41
- The Interactive Stack 42
- Interactive-Stack Operations 43
- C' R P N 44
- 8mn3HamiS8a«iajiiaiinM3EMi5i 46
- Cormiiand/Description 47
- Input Output 47
- DEPTH Returns the 47
- DUPN Duplicates n 47
- 3-12 Stack 48
- Using the MODES Application 49
- Setting the Display Mode 50
- Mode Definition Annunciator 51
- 4-4 Modes 52
- [rzGzct)] 53
- Modes 4-5 53
- 4 Setting the Beep 54
- 1. Press 54
- Setting the Clock Display 54
- Setting the Fraction Mark 54
- To set the fraction mark: 55
- To use a flag command: 56
- MODES Submenus 57
- MODES Operations 58
- Memory 5-1 59
- PICT) and the 60
- Memory 5-3 61
- Where to Store Variables 62
- ^^neJECTS IN i HBHE } 63
- Creating New Variables 63
- 5-6 Memory 64
- (MEMORY) 65
- To edit a variable: 66
- 67
- (CANCEL)) 67
- To move a variable: 68
- To purge variables: 68
- Using Variables; The VAR Menu 69
- (33 0PT I 0 (ENTER) 70
- Defining Variables 71
- Evaluating Variables 71
- '1: 72
- 'fl+B' 72
- : 'RDD£' 73
- 5 Special Memory Operations 74
- Memory Reset 75
- Memory Reset Prompt 76
- Memory 5-19 77
- 5-20 Memory 78
- InpRiit Foniìs and Lists 79
- (ENTER) or 81
- ENTER ) or 81
- [ENTER) or OK 82
- (ENTER) or UK 83
- « Program »■ 84
- Real number L 84
- EQ) and system flags are 85
- (ENTER) (3(iD (SCZD 86
- (ENTER) (3I2D 1 fS^ 5 86
- (enter) (3(Tg (ENTER) Q© 86
- SFIRST ONES 86
- U=M0+ adtD 87
- Organized 88
- Constructing Equations 89
- To include exponents: 90
- To include roots: 90
- To include powers of 10: 91
- To include units: 92
- E) 2(3X0 2(3 93
- 0 1 0(0X 93
- ©X(03E2 0X020 93
- 2 ©X @©Y (cos) 94
- (ZD 2 ©® 95
- 0 © 1 © 95
- To edit by backspacing: 96
- To edit the full equation: 96
- @0 0 ©) 10 (© 97
- 5. Press REF1... 98
- 7-12 The EquationWriter 98
- Description 100
- (i±JLi_D 100
- The HatrixWriter 101
- Entering Arrays 102
- [[ 2 -£ 0 ] 103
- 8-4 The MatrixWriter 104
- Editing Arrays 105
- To insert a column: 106
- To delete a column: 107
- To insert a row: 107
- To delete a row: 107
- 8-8 The MatrixWriter 108
- Graphic Objects 109
- The PICTURE Environment 110
- Using the Picture Editor 110
- Graphic Objects 9-3 111
- 9-4 Graphic Objects 112
- 113
- FTliCLEARl E 113
- 9-6 Graphic Objects 114
- Graphic Object Coordinates 115
- Graphic Object Commands 116
- Graphic Objects 9-9 117
- Key P r ogr ammable 118
- Unit Objects 119
- The UNITS Catalog Menu 120
- Creating a Unit Object 121
- ® S) HQ© O 6 @ F 122
- Converting Units 123
- To convert to any units: 124
- Calculating with Units 125
- 1: 703. l£5_ft''2/d 126
- 1: 8_ft^3^s''3 127
- Percentage 127
- 1 ^ ' 128
- 1 BHMBEMBBaHBB 128
- 25_“C/nin 129
- £5_TFmin 129
- 45_°FFpiin 129
- 10 @(U1I1D(EHD 130
- UNITS) mh; 131
- Unit Objects 10-13 131
- Creating User-Defined Units 132
- H is treated as a prefix 134
- Symbolic Constants 136
- S)( -»NUMf 137
- User-Defined Functions 138
- CD© z 3(PUM3 140
- © © 3 X dED 3 Y (spc) 141
- 9 (ENTER)18(ENTER)21 142
- (VAR) BOXR 142
- 1: .4285?1428571 142
- Functions of Real 143
- Complex 143
- Trigonometric Functions 144
- Hyperbolic Functions 145
- Probability Commands 146
- Test Statistics Commands 147
- (IvrnTl fN^ 148
- Real Number Functions 149
- Percent Functions 151
- Other Real-Number Functions 151
- Complex Numbers 153
- S)0> enter the 154
- Other Complex-Number Commands 155
- ¥eet®cs and 157
- [rz:eztt)] 158
- Entering 2D and 3D Vectors 159
- Vector Math Commands 160
- Vectors and Transforms 13-5 161
- 1: [ 178.937160532 162
- ¿111.148894255 ] 162
- 1: 78.8585649505 162
- Fast Fourier Transforms 163
- 13-8 Vectors and Transforms 164
- 6. Press (ENTER) 166
- Disassembling Matrices 168
- Inserting Rows and Columns 169
- Extracting Rows and Columns 170
- Swapping Rows and Columns 170
- Characterizing Matrices 171
- Transforming Matrices 174
- (AB) of two matrices: 175
- Expressions 176
- OdD ©O J S30 ® o 177
- Transforming Complex Matrices 177
- Equations 178
- Operations 182
- H T R R 0 W R S W P 183
- MODES) FLFiG 54 184
- To recoiTstruct ; 186
- Binary Integers and Bases 187
- (MTH ) BFlSE fNXT) RC 188
- To enter a binary integer: 188
- (ENTER 1 188
- ED or 0 189
- Using Boolean Operators 190
- Manipulating Bits and Bytes 190
- 1 “ ^ H0iB£lC6h 191
- Calculating with Dates 193
- 1: 4.301993 194
- 1= 1565 194
- Calculating with Times 195
- T I CKS to start timing 196
- Calculating with Fractions 197
- ______________ 198
- ' 987-^1 £5' 198
- Lists and Sequences 199
- List Processing 200
- SKU) 3 fSPC) 4 fspci 5 200
- { 4 5 6 } 201
- { 4 10 20 }I 201
- 17-4 Lists and Sequences 202
- SKU) 1 dED 2 dED 3 dED 203
- List Manipulations 204
- Sequences 205
- 6. Execute SEQ 206
- Solving Equations 207
- ENTER FUNCTION TD_ SOLVE 208
- To interpret the comput ilt: 209
- 18-4 Solving Equations 210
- Solving Options 211
- 18-6 Solving Equations 212
- To enter jriiTient: 213
- ST (enter) 214
- (^^[SOLVEI 215
- To use the polynomial roc 216
- ■ finder: 216
- To find all roots ilynomial: 216
- 1. Press iT^fSOCTEl (A) ® U 217
- (enter) to duplicate it 218
- STACK) F 218
- SOLVE) :E 218
- Using the Finance Solver 219
- 1 ^ 2 I 3 4 ^ 5 ^ 220
- ^ 2 ^ 3 4 5 y 221
- 18-16 Solving Equations 222
- (EHD® ® 223
- PYR = 12; End mode 224
- 0.00 End 225
- To calculate amortization: 225
- 18-20 Solving Equations 226
- Differential Equations 227
- ¿final 228
- 19-2 Differential Equations 228
- Differential Equations 19-3 229
- —1000 *{y — sin{t)) + cos{t) 230
- T (ENTER) 231
- (^ (ENTER) 231
- (ENTER) 231
- (ENTER)1 (ENTER) 231
- OUDSKID 0 dED 1 232
- ^^*PLDT V'CTJ:Fa,V)^^M 233
- TOiaHOTHKiiiaitiBiligamBi 233
- 19-8 Differential Equations 234
- (0 2 (ENTER) 235
- 19-10 Differential Equations 236
- 0 (IE) © T (ENTER) (0 237
- Qcm acm 238
- SIGD •5®©te)i 239
- Integration 241
- The Numeric INTEGRATE Screen 242
- RAD), if 242
- H^iSYMBOLICl OK 243
- 1: .693147180555 244
- T-cos(e) 245
- Symbolic integration 247
- Symbolic 248
- Differentiation 249
- Y and dZ<Y> 251
- Implicit Differentiation 252
- TAYLOR POLYNOMIAL I 253
- The TAYLOR POLYNOMIAL Screen 253
- OK to open the ISOLATE H 255
- Press MODES) FLAG 256
- Showing Hidden Variables 257
- (»^fSYIVIBOLIC) 258
- Manipulating Subexpressions 259
- EQUATION) and key it in 259
- PICTURE IRI 259
- (ENTER 1, or fCANCEL) 260
- (ENTER) to see the new 261
- Distributing 263
- Distributing (continued) 264
- KB, and Kn-arie 268
- 'SIN(2*(X+n> 269
- Symbolic Integration Patterns 270
- Antiderivative 271
- Statistic^ Dñcí 273
- 275
- Editing Statistical Data 276
- ) MFiTR CuL COL..+ 277
- = 31n.c — 4 278
- Generating Frequencies 280
- /ariabis; 282
- _2Ke_EYS _EKV _NE 283
- Plotting 285
- INDEP: X H-VIEW:-6.5 6.5 286
- _flUTDSCflLE V-VIEW;-3.1 3.2 286
- ENTER FUMCTIDNtS), TO PLDT 286
- __________ 286
- The Default PLOT Screen 286
- EQs field 287
- EQ” by pressing OK 287
- 22-4 Plotting 288
- Environment 289
- Using Zoom Operations 290
- Setting Zoom Defaults 291
- Selecting a Zoom 291
- 22-8 Plotting 292
- Analyzing Functions 293
- The PICTURE FCN Menu 294
- Plotting 22-11 295
- 22-12 Plotting 296
- Plotting 22-13 297
- Contents of the PPAR List 298
- Description Default 299
- 22-16 Plotting 300
- Plot Types 301
- The Function PLOT Form 301
- 23-2 Plot Types 302
- TRACE Mode 303
- Special Notes 303
- Polar Plots 304
- The Parametric PLOT Form 307
- Examples 309
- 1 = 6.5 or do their 309
- 23-10 Plot Types 310
- Differential Equation Plots 311
- Conic Plots 311
- The Conic PLOT Form 312
- The Conic PLOT OPTIONS Form 312
- EXAMPLES directory then: 313
- Truth Plots 314
- (aCELHD®@c 316
- Statistical Plots 317
- Histogram Plots 318
- Bar Plots 319
- Scatter Plots 320
- The Scatter PLOT Form 321
- The Scatter PLOT OPTIONS Form 321
- Sampling Grid 322
- Output Grid 322
- Slopefield/Ps-Contour 323
- The YSLICE View 324
- Slopefield Plots 325
- The Slopefield PLOT Form 326
- 8 steps 327
- 1 of the stack 327
- Wireframe Plots 328
- The Wireframe PLOT Form 329
- Pseudo-Contour Plots 330
- The Ps-Contour PLOT Form 331
- Y-Slice Plots 332
- The YSlice PLOT OPTIONS Form 333
- The Gridmap PLOT Form 334
- Parametric Surface Plots 335
- The Pr-Surface PLOT Form 336
- F" T II 337
- Press fVAR) EkFIM 338
- NXT) LHBEL. The names 339
- 24-2 Advanced Plot Options 340
- [PRG) FGCT F'DIM 341
- 24-4 Advanced Plot Options 342
- (PRG) P I C T P D I M 343
- Saving and Restoring Plots 344
- Advanced Plot Options 24-7 345
- SKMDSI E0 to restore EQ 346
- HP 48G Series Advanced 347
- User’s Reference 347
- Using the Solver 348
- Using the Menu Keys 349
- Actions for Solver Menu Keys 349
- Q _ rO rr I 350
- Viewing the Picture 351
- Operation Key 352
- S) Pil-L 353
- O H1- !-■ 353
- Defining a Set of Equations 354
- Xi = Vo + a ■ ti 355
- {X2 - xi) = a ■ {to - h) 355
- Checking Solutions 356
- The Equation Library 25-11 357
- Using the Constants Library 358
- 02 dielectric constant 3.9 359
- Minehunt Game 360
- User-Defined Units 361
- 1. Press f7» 363
- To change the date or time; 364
- To set an appointment alarm: 365
- 1 enter 15 365
- Responding to Alarms 366
- Viewing and Editing Alarms 367
- Pj^fMODESl F'LFlG CF 368
- MODES 1 FLFiG 368
- Printing 370
- To set up a serial printer: 371
- I DEJECT; 372
- The Wire PRINT Screen 373
- PRINT PRST 374
- Computer 375
- The Wire TRANSFER Screen 376
- Using Kermit 377
- CHCujS OK to browse the 378
- TRANSFER form 378
- I-- I N 1 H 378
- Choosing and Using File Names 379
- Backing Up HP 48 Memory 380
- Sending Kermit Commands 381
- Using XMODEM 382
- Using Other Serial Protocols 383
- Ports and Piiyg-^n Cards 389
- Card Slot 1 390
- Card Slot 2 390
- Using Backup Objects 391
- To ewaluate a tjackyp object: 392
- To delete a backup object: 392
- Backing Up All Memory 393
- Using Libraries 394
- Write contents 399
- Read-only setting 400
- To remove a plug-in card: 402
- LIBRARY) 403
- LIBRARY) MERG . If 404
- ProgicaimiiitcRij the iiiP 407
- The Contents of a Program 408
- Programming the HP 48 29-3 409
- Calculations in a Program 410
- Structured Programming 410
- To enter a program: 411
- CD (©(hold) VOL (release) 413
- 1- 268.082573106 414
- Programming the HP 48 29-9 415
- Using Programming Structures 416
- CASE...END 417
- Loop Structures 418
- STF loop-clause NEKl 418
- STEP ... s 418
- S) FOF:; . Its syntax is 419
- DO...UNTIL...END 420
- WHILE...REPEAT...END 420
- Error Trapping Structures 421
- Using Local Variables 422
- Evaluating Local Names 423
- ■tiefining procedure: 424
- Directory 425
- HP 48G/GX 426
- Programming the HP 48 29-21 427
- Customizing Menus 429
- “MOLUtlE" CST } 430
- 1: .001075_m"3 430
- .0010?5_P1'"3 430
- 1: "VOLUME" 430
- ^THG l! MOLUME: .0010?5_r^3 431
- "MOLUME" CST } 431
- Customizing the Keyboard 432
- To actiwate User mode: 433
- KEYS STUK 434
- Disabling User Keys 435
- KEYS RCLK 0 DELK ST OK 436
- Answers to Common Questions 437
- Q: What does “object” mean? 438
- Environmental Limits 440
- When to Replace Batteries 440
- Changing Batteries 441
- 1 to turn the calculator on 443
- Nail grip 444
- Testing Calculator Operation 445
- Self-Test 446
- Keyboard Test 447
- Port RAM Test 448
- IR Loop-Back Test 449
- Serial Loop-Back Test 450
- Limited One-Year Warranty 451
- ■rrop Messages 455
- Sicursio^ 456
- .1 i i t- Г Г U j-‘ !.• e и 457
- (ENTER), or STR^ due 458
- ri-B i G JiU D -B X- •“! 459
- 0 g 01. i I..! 0 459
- Message Meaning 461
- V e B u f f e r 462
- Error Messages B-9 463
- B-10 Error Messages 464
- Menus C-1 465
- System Flags 467
- System Flags (continued) 468
- D-2 System Flags 468
- System Flags D-3 469
- D-4 System Flags 470
- System Flags D-5 471
- D-6 System Flags 472
- Table of Units E-1 473
- HP 48 Units (continued) 474
- E-2 Table of Units 474
- Table of Units E-3 475
- St (Stere) 476
- Table of Byilt-ln Equations 477
- Energy (31,36) 478
- Operatic"r h 481
- 1 Solve finance... HPiUK 482
- Operation Index G-3 483
- (x) containing 484
- Operation Index G-5 485
- G-6 Operation Index 486
- 1 EDIT CIRCL 487
- G-8 Operation Index 488
- ) MFiTR MflKE CON 489
- G-10 Operation Index 490
- Operation Index G-11 491
- G-12 Operation Index 492
- Operation Index G-13 493
- G-14 Operation Index 494
- Operation Index G-15 495
- G-16 Operation Index 496
- Operation Index G-17 497
- G-18 Operation Index 498
- Operation Index G-19 499
- G-20 Operation Index 500
- 1 (NXTI CM PL IM 501
- 1 (NXT 1 3 D V P H R 501
- G-22 Operation Index 502
- Operation Index G-23 503
- ] 1- I 1 ¿1.1 N E 504
- 3 R U R R S U 505
- 3 RULES n 505
- 3 R U ^ R R 3 n 505
- Operation index G-27 507
- G-28 Operation Index 508
- Operation Index G-31 511
- 2/) of a given type 512
- 1 PK I N \ KkTKM PK I pH 513
- 1 fNXT ) PURG 514
- 1 H V HI K b 514
- Operation Index G-35 515
- FLfiG ÍNXT) RCLF 516
- DfiTfl SDFiT 516
- PROB RDZ 516
- G-36 Operation Index 516
- PPliR RE.S 517
- 3D fNXTl 517
- VpfIR iNXT) RESET 517
- Operation Index G-37 517
- SOLVE) D I FFE Rl<FE 518
- D I F F E K RF i 518
- G-38 Operation Index 518
- BASE (NXT) BIT RR 519
- Operation Index G-39 519
- 2), a possible step 520
- PROC iNXT) SEQ 522
- I T H fNXT) S I H V 522
- G-42 Operation Index 522
- C; j fj p p' 523
- NLIKM oNKh 523
- 1 BfiSE fNXT) E:IT SR 524
- 1 FMT STD 524
- G-46 Operation Index 526
- Operation Index G-47 527
- r u t h 529
- Operation Index G-49 529
- PROB fNXin UTPF 531
- PROB (NXTl UTPN 531
- Operation Index G-51 531
- L- ri N M i i„ E w H j. L. t 532
- Operation Index G-53 533
- Operation Index G-55 535
- G-56 Operation Index 536
- Operation Index G-57 537
- G-58 Operation Index 538
- Commands 539
- ^checksum 541
- •’^XCOl •^ycol “ 542
- •^correlation 542
- •^covariance 542
- ■ ’^^Darcy 542
- >^n ew 543
- '’index 543
- { -i^start -^end } 543
- ■ ^sta rt 543
- ■ ^e n d 543
- 'symb' Xy 545
- 'symb' —^ 545
- 'symb' 546
- ,symb2)' 546
- ^increment 547
- { global } 548
- ' 0/1 550
- "ALL" ^ 551
- ‘name‘ ^ X 552
- "message" 552
- " string.^" 553
- 'NOT symb' 553
- for the first character 553
- [[^11 ■ ■ ■ n ]] Xll 554
- "prompt" 557
- X_ un/tj^ 558
- \epeat > 559
- •^m e n u 559
- 2 [[ matrix ]]3 560
- { list } 561
- { guesses } 561
- ■X LvI 4 LvI 3 562
- ■ ’^sdevl 563
- ■ ^sdev2 ■ ■ ■ ^sdevm ] 563
- ■ ’^density 563
- 'symb^ ' 567
- 'symb2 ' 567
- { global 568
- TVM variable' 568
- Object Type Numbers 568
- 'name' 569
- ■^port ■ 569
- Xfactor 569
- 'name' —► 571
- •’‘’left •’‘right 572
- 'ZFACTOR(sym£)j^ .symb^Y 573
- •’‘'low 574
- ■ ’‘’high 574
- 2 ' 575
- ^ Level 1 576
- 1 ... X]^ 1 ■ • m ]] 577
- Special characters 579
- L J for matrices, 2-6, 8-1 586
- Index-10 588
- Index-11 589
- Index-12 590
- Index-13 591
- Index-15 593
- Index-16 594
- Index-17 595
- Index-18 596
- Index-19 597
- Index-20 598
- Index-21 599
- Index-22 600
- PRG annunciator, 1-3, 29-5 601
- RiZ annunciator, 12-11, 13-2 602
- Index-24 602
- Index-25 603
- Index-26 604
- Index-27 605
- Index-28 606
- Index-29 607
- 10-2, 10-6 608
- Index-31 609
- Index-32 610
- Contacting Hewlett-Packard 611
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