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Sequence Graphing

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Chapter 9: Sequence Graphing 9 Preview of Sequence Graphing 140 Overview of Steps in Graphing Sequences 141 Differences in Sequence and Function Graphing 142 Setting Axes for Time, Web, or Custom Plots 146 Using Web Plots 147 Using Custom Plots 150 Using a Sequence to Generate a Table 151 This chapter describes how to graph sequences on the TI-89 / TI-92 Plus. Before using this chapter, you should be familiar with Chapter 6: Basic Function Graphing. Sequences are evaluated only at consecutive integer values. The two general types of sequences are: ¦ Nonrecursive - The nth term in the sequence is a function of the independent variable n. Each term is independent of any other terms. In the following example sequence, you can calculate u(5) directly, without first calculating u(1) or any other previous term. u(n) = 2 ù n for n = 1, 2, 3, ... n is always a series of consecutive integers, starting at any positive integer or zero. u(n) = 2 ù n gives the sequence 2, 4, 6, 8, 10, ... Note: A recursive sequence can reference another sequence. For example, u2(n) = nñ +u1(nì 1). ¦ Recursive - The nth term is defined in relation to one or more previous terms, represented by u(nì 1), u(nì 2), etc. In addition to previous terms, a recursive sequence may also be defined in relation to n (such as u(n) = u(nì 1) + n). In the following example sequence, you cannot calculate u(5) without first calculating u(1), u(2), u(3), and u(4). The first term is undefined u(n) = 2 ù u(nì 1) for n = 1, 2, 3, ... since it has no previous term. So you must specify an initial value to use for the first term. Using an initial value of 1: u(n) = 2 ù u(nì 1) gives the sequence 1, 2, 4, 8, 16, ... The number of initial values you need to specify depends on how deep the recursion goes. For example, if each term is defined in relation to the previous two terms, you must specify initial values for the first two terms. Chapter 9: Sequence Graphing 139

Section	Page
TI-89 / TI-92 Plus Guidebook	1
Front Matter	2
Important and US FCC Information	2
TI-89 Shortcut Keys	3
TI-92 Plus Shortcut Keys	4
Table of Contents	5
Chapter 1: Getting Started	5
Chapter 2: Operating the Calculator	5
Chapter 3: Symbolic Manipulation	6
Chapter 4: Constants and Measurement Units	6
Chapter 5: Additional Home Screen Topics	6
Chapter 6: Basic Function Graphing	6
Chapter 7: Parametric Graphing	6
Chapter 8: Polar Graphing	7
Chapter 9: Sequence Graphing	7
Chapter 10: 3D Graphing	7
Chapter 11: Differential Equation Graphing	7
Chapter 12: Additional Graphing Topics	8
Chapter 13: Tables	8
Chapter 14: Split Screens	8
Chapter 15: Data/ Matrix Editor	8
Chapter 16: Statistics and Data Plots	8
Chapter 17: Programming	9
Chapter 18: Text Editor	9
Chapter 19: Numeric Solver	9
Chapter 20: Number Bases	9
Chapter 21: Memory and Variable Management	10
Chapter 22: Linking and Upgrading	10
Chapter 23: Activities	10
Appendix A: Functions and Instructions	10
Appendix B: Reference Information	11
Appendix C: Service and Warranty Information	11
Appendix D: Programmer’s Guide	11
Flash Applications	12
Applications	12
Hardware/Software Requirements	12
Hardware Setup for the Computer	12
Installing a Flash Application from the CD- ROM	12
Running a Flash Application	12
Transferring a Flash Application from another TI-89 / TI-92 Plus	13
Backing up a Flash Application	13
Deleting a Flash Application	13
Keystroke Differences	14
What's New!	16
Introducing Advanced Mathematics Software Version 2.0	16
Language Localization	16
Improved User Interface	16
Upgradability with Flash ROM	17
Custom Menu	17
Chapter 1: Getting Started	18
Getting the TI-89 Ready to Use	19
Installing the AAA Batteries	19
Getting the TI-92 Plus Ready to Use	20
Installing the AA Batteries	20
Setting the Contrast and Selecting a Language	21
Turning the Unit on and Adjusting the Display Contrast	21
Languages on the TI- 89 / TI- 92 Plus	21
Important Information About the Language Process	21
Localizing the TI- 89 / TI- 92 Plus	22
About the Home Screen	23
Turning the TI- 89 / TI- 92 Plus Off	24
Performing Computations	25
Graphing a Function	28
Chapter 2: Operating the Calculator	30
Turning the TI-89 / TI-92 Plus On and Off	31
Turning the TI- 89 / TI- 92 Plus On	31
Turning the TI- 89 / TI- 92 Plus Off	31
APD (Automatic Power Down)	31
Batteries	31
Setting the Display Contrast	32
Adjusting the Display Contrast	32
When to Replace Batteries	32
Using the TI-92 Plus Cover as a Stand	32
The TI-89 Keyboard	33
Overview of Some Important Keys	33
Moving the Cursor	33
The TI-92 Plus Keyboard	34
Keyboard Areas	34
Cursor Pad	34
Modifier Keys	35
Modifier Keys	35
Examples of [2nd] and [diamond] Modifiers	35
Other Important Keys You Needto Be Familiar With	36
Entering Alphabetic Characters	38
Entering a Letter Character on the TI-89	38
Typing Alphabetic Characters on the TI- 89 / TI- 92 Plus	38
Automatic Alpha-Lock in TI- 89 Dialog Boxes	39
For Special Characters	39
Home Screen	40
Displaying the Home Screen	40
Parts of the Home Screen	40
History Area	40
Scrolling through the History Area	41
History Information on the Status Line	41
Modifying the History Area	41
Entering Numbers	42
Entering a Negative Number	42
Entering a Number in Scientific Notation	42
Entering Expressions and Instructions	43
Definitions	43
Implied Multiplication	43
Parentheses	44
Entering an Expression	44
Entering Multiple Expressions on a Line	44
If an Entry or Answer Is Too Long for One Line	45
Continuing a Calculation	45
Stopping a Calculation	45
Formats of Displayed Results	46
Pretty Print Mode	46
Exact/Approx Mode	46
Display Digits Mode	48
Exponential Format Mode	48
Editing an Expression in the Entry Line	49
Removing the Highlight from the Previous Entry	49
Moving the Cursor	49
Deleting a Character	49
Clearing the Entry Line	49
Inserting or Overtyping a Character	50
Replacing or Deleting Multiple Characters	50
Menus	51
Displaying a Menu	51
Selecting an Item from a Menu	51
Items Ending with > (Submenus)	52
Items Containing “. . .” (Dialog Boxes)	52
Canceling a Menu	52
Moving from One Toolbar Menu to Another	53
Example: Selecting a Menu Item	53
Using the Custom Menu	54
Turning the Custom Menu On and Off	54
Restoring the Default Custom Menu	54
Selecting an Application	55
From the APPLICATIONS Menu	55
From the Keyboard	56
Setting Modes	57
Checking Mode Settings	57
Changing Mode Settings	57
Overview of the Modes	58
Using the Clean Up Menu to Start a New Problem	60
Clean Up Toolbar Menu	60
Using the Catalog Dialog Box	61
Displaying the CATALOG	61
Selecting a Built-in Command from the CATALOG	61
Information about Parameters	62
Selecting a Flash Application Function	62
Selecting a User- Defined Function or Program	63
Storing and Recalling Variable Values	64
Rules for Variable Names	64
Data Types	64
Storing a Value in a Variable	65
Displaying a Variable	65
Using a Variable in an Expression	65
Recalling a Variable’s Value	65
Reusing a Previous Entry or the Last Answer	66
Reusing the Expression on the Entry Line	66
Recalling a Previous Entry	67
Recalling the Last Answer	68
Auto-Pasting an Entry or Answer from the History Area	69
Why Use Auto-Paste	69
Auto-Pasting an Entry or Answer	69
Status Line Indicators in the Display	70
Status Line Indicators	70
Finding the Software Version and ID Number	72
Displaying the “About” Screen	72
When Do You Need this Information?	72
Chapter 3: Symbolic Manipulation	74
Preview of Symbolic Manipulation	75
Using Undefined or Defined Variables	76
How Undefined and Defined Variables Are Treated	76
Determining If a Variable Is Undefined	76
Deleting a Defined Variable	77
Temporarily Overriding a Variable	77
Using Exact, Approximate, and Auto Modes	78
EXACT Setting	78
APPROXIMATE Setting	79
AUTO Setting	80
Automatic Simplification	81
Default Simplification Rules	81
How Long Is the Simplification Process?	82
Delayed Simplification for Certain Built-In Functions	83
Functions that Use Delayed Simplification	83
Substituting Values and Setting Constraints	84
Typing the “With” Operator	84
Substituting for a Variable	84
Substituting for a Simple Expression	84
Substituting Complex Values	84
Be Aware of the Limitations of Substitutions	85
Specifying Domain Constraints	86
Using Substitutions vs. Defining a Variable	86
Overview of the Algebra Menu	87
The Algebra Menu	87
Common Algebraic Operations	89
Adding or Dividing Polynomials	89
Factoring and Expanding Polynomials	89
Finding Prime Factors of a Number	89
Finding Partial Expansions	89
Solving an Equation	90
Solving a System of Linear Equations	90
Finding the Zeros of an Expression	91
Finding Proper Fractions and Common Denominators	91
Overview of the Calc Menu	92
The Calc Menu	92
Common Calculus Operations	93
Integrating and Differentiating	93
Finding a Limit	93
Finding a Taylor Polynomial	93
User-Defined Functions and Symbolic Manipulation	94
For Information about Creating a User- Defined Function	94
Undefined Functions	94
Single-Statement Functions	94
Multi-Statement vs. Single- Statement Functions	95
If You Get an Out-of-Memory Error	96
Freeing Up Memory	96
Simplifying Problems	96
Special Constants Used in Symbolic Manipulation	97
true, false	97
@n1...@n255	97
Infinity, e	97
undef	97
Chapter 4: Constants and Measurement Units	98
Preview of Constants and Measurement Units	99
Entering Constants or Units	100
From a Menu	100
From the Keyboard	100
Combining Multiple Units	101
Using Parentheses with Units in a Calculation	101
Converting from One Unit to Another	102
For All Units Except Temperature	102
For Temperature Values	103
For Temperature Ranges	103
Setting the Default Units for Displayed Results	104
If You’re Using the SI or ENG/ US System	104
Setting Custom Defaults	104
What is the NONE Default?	104
Creating Your Own User-Defined Units	105
Why Use Your Own Units?	105
Rules for User-Defined Unit Names	105
Defining a Unit	105
List of Pre-Defined Constants and Units	106
Defaults for SI and ENG/ US	106
Constants	106
Length	106
Area	106
Volume	107
Time	107
Velocity	107
Acceleration	107
Temperature	107
Luminous Intensity	107
Amount of Substance	107
Mass	107
Force	107
Energy	107
Power	107
Pressure	108
Viscosity, Kinematic	108
Viscosity, Dynamic	108
Frequency	108
Electric Current	108
Charge	108
Potential	108
Resistance	108
Conductance	108
Capacitance	108
Mag Field Strength	108
Mag Flux Density	108
Magnetic Flux	108
Inductance	108
Chapter 5: Additional Home Screen Topics	110
Saving the Home Screen Entries as a Text Editor Script	111
Saving the Entries in the History Area	111
Restoring the Saved Entries	111
Cutting, Copying, and Pasting Information	112
Auto-paste vs. Cut/ Copy/ Paste	112
Cutting or Copying Information to the Clipboard	112
Pasting Information from the Clipboard	113
Example: Copying and Pasting	113
Creating and Evaluating User-Defined Functions	114
Format of a Function	114
Creating a User-Defined Function	114
Creating a Multi-Statement Function	115
Evaluating a Function	115
Displaying and Editing a Function Definition	116
Using Folders to Store Independent Sets of Variables	117
Folders and Variables	117
Creating a Folder from the Home Screen	118
Creating a Folder from the VAR-LINK Screen	118
Setting the Current Folder from the Home Screen	118
Setting the Current Folder from the MODE Dialog Box	118
Using Variables in Different Folders	119
Deleting a Folder from the Home Screen	119
Deleting a Folder from the VAR-LINK Screen	119
If an Entry or Answer Is “Too Big”	120
If an Entry or Answer Is “Too Long”	120
If There Is not Enough Memory	120
Chapter 6: Basic Function Graphing	122
Preview of Basic Function Graphing	123
Overview of Steps in Graphing Functions	124
Graphing Functions	124
Exploring the Graph	124
Setting the Graph Mode	125
Graph Mode	125
Angle Mode	125
Checking the Status Line	125
Defining Functions for Graphing	126
Defining a New Function	126
Editing a Function	126
Clearing a Function	127
Shortcuts to Move the Cursor	127
From the Home Screen or a Program	127
Selecting Functions to Graph	128
Selecting or Deselecting Functions	128
From the Home Screen or a Program	128
Setting the Display Style for a Function	129
Displaying or Changing a Function’s Style	129
If You Use Above or Below Shading	129
From the Home Screen or a Program	129
Defining the Viewing Window	130
Displaying Window Variables in the Window Editor	130
Changing the Values	130
From the Home Screen or a Program	130
Changing the Graph Format	131
Displaying Graph Format Settings	131
Changing Settings	131
Graphing the Selected Functions	132
Displaying the Graph Screen	132
Interrupting Graphing	132
If You Need to Change the Viewing Window	132
Smart Graph	132
Displaying Coordinates with the Free-Moving Cursor	133
Free-Moving Cursor	133
Tracing a Function	134
Beginning a Trace	134
Moving along a Function	134
Moving from Function to Function	135
Automatic Panning	135
Using QuickCenter	135
Canceling Trace	135
Using Zooms to Explore a Graph	136
Overview of the Zoom Menu	136
Zooming In with a Zoom Box	137
Zooming In and Out on a Point	137
Changing Zoom Factors	138
Saving or Recalling a Viewing Window	138
Restoring the Standard Viewing Window	138
Using Math Tools to Analyze Functions	139
Overview of the Math Menu	139
Finding y(x) at a Specified Point	140
Finding a Zero, Minimum, or Maximum within an Interval	140
Finding the Intersection of Two Functions within an Interval	140
Finding the Derivative (Slope) at a Point	141
Finding the Numerical Integral over an Interval	141
Finding an Inflection Point within an Interval	141
Finding the Distance between Two Points	142
Drawing a Tangent Line	142
Finding an Arc Length	142
Shading the Area between a Function and the X Axis	143
Shading the Area between Two Functions within an Interval	143
Chapter 7: Parametric Graphing	144
Preview of Parametric Graphing	145
Overview of Steps in Graphing Parametric Equations	146
Graphing Parametric Equations	146
Exploring the Graph	146
Differences in Parametric and Function Graphing	147
Setting the Graph Mode	147
Defining Parametric Equations on the Y= Editor	147
Selecting Parametric Equations	148
Selecting the Display Style	148
Window Variables	148
Exploring a Graph	149
Chapter 8: Polar Graphing	150
Preview of Polar Graphing	151
Overview of Steps in Graphing Polar Equations	152
Graphing Polar Equations	152
Exploring the Graph	152
Differences in Polar and Function Graphing	153
Setting the Graph Mode	153
Defining Polar Equations on the Y= Editor	153
Selecting the Display Style	153
Window Variables	154
Setting the Graph Format	154
Exploring a Graph	155
Chapter 9: Sequence Graphing	156
Preview of Sequence Graphing	157
Overview of Steps in Graphing Sequences	158
Graphing Sequences	158
Exploring the Graph	158
Differences in Sequence and Function Graphing	159
Setting the Graph Mode	159
Defining Sequences on the Y= Editor	159
Selecting Sequences	160
Selecting the Display Style	160
Window Variables	160
Changing the Graph Format	161
Exploring a Graph	162
Setting Axes for Time, Web, or Custom Plots	163
Displaying the AXES Dialog Box	163
Using Web Plots	164
Valid Functions for Web Plots	164
When You Display the Graph Screen	164
Drawing the Web	164
Example: Convergence	165
Example: Divergence	165
Example: Oscillation	166
Using Custom Plots	167
Example: Predator-Prey Model	167
Using a Sequence to Generate a Table	168
Example: Fibonacci Sequence	168
Chapter 10: 3D Graphing	170
Preview of 3D Graphing	171
Overview of Steps in Graphing 3D Equations	173
Graphing 3D Equations	173
Exploring the Graph	173
Differences in 3D and Function Graphing	174
Setting the Graph Mode	174
Defining 3D Equations on the Y= Editor	174
Selecting the Display Style	174
Window Variables	175
Setting the Graph Format	176
Exploring a Graph	176
Moving the Cursor in 3D	177
How to Move the Cursor	177
Simple Example of Moving the Cursor	177
Example of the Cursor on a Hidden Surface	178
Example of an “Off the Curve” Cursor	178
Rotating and/or Elevating the Viewing Angle	179
How the Viewing Angle Is Measured	179
Effect of Changing eye theta	179
Effect of Changing eye phi	180
Effect of Changing eye psi	180
From the Home Screen or a Program	180
Animating a 3D Graph Interactively	181
The Viewing Orbit	181
Animating the Graph	181
Animating a Series of Graph Pictures	181
Changing the Axes and Style Formats	182
Displaying the GRAPH FORMATS Dialog Box	182
Examples of Axes Settings	182
Examples of Style Settings	183
Be Aware of Possible Optical Illusions	183
Contour Plots	184
Selecting the Graph Format Style	184
How Are Z Values Determined?	185
Drawing a Contour for the Z Value of a Selected Point Interactively	185
Drawing Contours for Specified Z Values	186
Notes about Contour Plots	186
Example: Contours of a Complex Modulus Surface	187
Implicit Plots	188
Explicit and Implicit Forms	188
Selecting the Graph Format Style	188
Notes About Implicit Plots	189
Example: Implicit Plot of a More Complicated Equation	190
Chapter 11: Differential Equation Graphing	192
Preview of Differential Equation Graphing	193
Overview of Steps in Graphing Differential Equations	195
Graphing Differential Equations	195
Differences in Diff Equations and Function Graphing	196
Setting the Graph Mode	196
Defining Differential Equations on the Y= Editor	196
Selecting Differential Equations	196
Selecting the Display Style	196
Setting Graph Formats	197
Setting Axes	198
Window Variables	198
The fldpic System Variable	200
Exploring a Graph	200
Setting the Initial Conditions	201
Entering Initial Conditions in the Y= Editor	201
If You Do Not Enter an Initial Condition in the Y= Editor	201
Selecting an Initial Condition Interactively from the Graph Screen	202
Note about Tracing a Solution Curve	202
Defining a System for Higher-Order Equations	203
Transforming an Equation into a 1st-Order System	203
Example of a 2nd-Order Equation	204
Example of a 3rd-Order Equation	206
Setting Axes for Time or Custom Plots	207
Displaying the AXES Dialog Box	207
Example of Time and Custom Axes	208
Predator-Prey Model	208
Example Comparison of RK and Euler	210
Example of the deSolve( ) Function	213
Troubleshooting with the Fields Graph Format	214
Setting the Fields Graph Format	214
What Order Equation Are You Graphing?	214
Fields=SLPFLD	214
Fields=DIRFLD	215
Fields=FLDOFF	216
If You Use the Table Screen to View Differential Equations	216
Chapter 12: Additional Graphing Topics	218
Preview of Additional Graphing Topics	219
Collecting Data Points from a Graph	220
Collecting the Points	220
Notes about SysData Variable	220
Graphing a Function Defined on the Home Screen	221
What Is the “Native” Independent Variable?	221
Copying from the Home Screen to the Y= Editor	221
Graphing Directly from the Home Screen	222
Clearing the Graph Screen	222
Extra Benefits of User- Defined Functions	222
Graphing a Piecewise Defined Function	223
Using the When Function	223

Match case Limit results 1 per page

Chapter 9: Sequence Graphing

139

Sequence Graphing

Preview of Sequence Graphing

............................................................

140

Overview of Steps in Graphing Sequences

.........................................

141

Differences in Sequence and Function Graphing

..............................

142

Setting Axes for Time, Web, or Custom Plots

....................................

146

Using Web Plots

......................................................................................

147

Using Custom Plots

................................................................................

150

Using a Sequence to Generate a Table

................................................

151

This chapter describes how to graph sequences on the

89 / TI

92 Plus

. Before using this chapter, you should be familiar

with Chapter 6: Basic Function Graphing.

Sequences are evaluated only at consecutive integer values. The

two general types of sequences are:

Nonrecursive

— The

th term in the sequence is a function

of the independent variable

Each term is independent of any other terms. In the following

example sequence, you can calculate

u(5)

directly, without

first calculating

u(1)

or any other previous term.

u(n) = 2

for

n = 1, 2, 3, ...

u(n) = 2

gives the sequence

2, 4, 6, 8, 10, ...

Recursive

— The

th term is defined in relation to one or

more previous terms, represented by

u(n

, etc. In

addition to previous terms, a recursive sequence may also be

defined in relation to

(such as

u(n) = u(n

1) + n

In the following example sequence, you cannot calculate

u(5)

without first calculating

u(1)

u(2)

u(3)

, and

u(4)

u(n) = 2

u(n

for

n = 1, 2, 3, ...

Using an initial value of 1:

u(n) = 2

u(n

gives the sequence

1, 2, 4, 8, 16, ...

The number of initial values you need to specify depends on

how deep the recursion goes. For example, if each term is

defined in relation to the previous two terms, you must specify

initial values for the first two terms.

Note:

A recursive sequence

can reference another

sequence. For example,

u2(n) = n

+u1(n

1).

n is always a series of

consecutive integers,

starting at any positive

integer or zero.

The first term is undefined

since it has no previous term.

So you must specify an initial

value to use for the first term.