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Texas Instruments TI-86 User Manual - Page 268

Reservoir Problem, Solving Bernoulli's equation for v

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256 Chapter 19: Applications Reservoir Problem On the TI-86, you can use parametric graphing animation to solve a problem. Consider a water reservoir with a height of 2 meters. You must install a small valve on the side of the reservoir such that water spraying from the open valve hits the ground as far away from the reservoir as possible. At what height should you install the valve to maximize the length of the water stream when the valve is wide open? Assume a full tank at time=0, no acceleration in the x direction, and no initial velocity in the y direction. Also, ignore valve-size and valve-type factors. Integrating the definition of acceleration in both the x and y directions twice yields the equations x=v0t and y=h0N(gt2)à2. Solving Bernoulli's equation for v0 and substituting into v0t results in this pair of parametric equations: xt=t‡(2g(2Nh0)) yt=h0N(gt2)à2 t = time in seconds h0 = height of the valve in meters g = the built-in acceleration of gravity constant When you graph these equations on the TI-86, the y-axis (x=0) is the side of the reservoir where the valve is to be installed. The x-axis (y=0) is the ground. Each plotted parametric equation represents the water stream when the valve is at each of several heights.

Section	Page
TI-86 Graphing Calculator Guidebook	1
Important and US FCC Information	2
Table of Contents	3
TI-86 Quick Start	3
Chapter 1: Operating the TI-86	3
Chapter 2: The CATALOG, Variables, and Characters	4
Chapter 3: Math, Calculus, and Test Operations	5
Chapter 4: Constants, Conversions, Bases, and Complex Numbers	5
Chapter 5: Function Graphing	6
Chapter 6: Graph Tools	6
Chapter 7: Tables	7
Chapter 8: Polar Graphing	7
Chapter 9: Parametric Graphing	8
Chapter 10: Differential Equation Graphing	8
Chapter 11: Lists	9
Chapter 12: Vectors	9
Chapter 13: Matrices	9
Chapter 14: Statistics	10
Chapter 15: Equation Solving	10
Chapter 16: Programming	11
Chapter 17: Memory Management	11
Chapter 18: The TI-86 Communication Link	11
Chapter 19: Applications	12
Chapter 20: A to Z Function and Instruction Reference	12
Appendix	12
Index	12
TI-86 Quick Start	13
Preparing to Use Your New TI-86	14
Installing the AAA Batteries	14
Turning On and Turning Off the TI-86	14
Adjusting the Contrast	14
Resetting All Memory and Defaults	15
Calculating on the Home Screen	15
Calculating the Sine of a Number	15
Storing the Last Answer to a Variable	15
Using a Variable in an Expression	16
Editing an Expression	16
Displaying a Complex Number as a Result	17
Using a List with a Function	17
Displaying the Integer Part of Real Numbers in a List	18
Removing (Exiting) a Menu	18
Finding the Square Root	19
Calculating Derivatives	19
Retrieving, Editing, and Re-evaluating the Previous Entry	20
Converting Degrees Fahrenheit to Degrees Celsius	20
Storing an Unevaluated Expression to an Equation Variable	21
Plotting Functions on the Graph Screen	21
Displaying and Entering Functions in the Equation Editor	21
Changing the Graph Style of a Function	22
Plotting a Function on the Graph Screen	23
Tracing a Function	23
Evaluating y for a Specific x Value (During a Trace)	24
Changing a Window Variable Value	24
Deselecting a Function	25
Zooming In on a Portion of the Graph Screen	26
Chapter 1: Operating the TI- 86	27
Installing or Replacing Batteries	28
When to Replace Batteries	28
Turning On and Turning Off the TI-86	29
Adjusting the Display Contrast	29
The Home Screen	30
Displaying Entries and Answers	30
Entering Numbers	31
Entering Negative Numbers	31
Using Scientific or Engineering Notation	32
Entering Complex Numbers	32
Entering Other Characters	32
The 2nd Key	33
The ALPHA Key	33
ALPHA-lock and alpha-lock	33
Common Cursors	34
Cursor Direction Keys	35
Inserting, Deleting, and Clearing Characters	35
Entering Expressions and Instructions	36
Entering an Expression	36
Using Functions in Expressions	37
Using an Instruction	37
Entering Functions, Instructions, and Operators	37
Entering Consecutive Entries	38
The Busy Indicator	38
Interrupting a Calculation or Graph	38
Diagnosing an Error	39
Correcting an Error	39
Reusing Previous Entries and the Last Answer	40
Retrieving the Last Entry	40
Retrieving and Editing the Last Entry	40
Retrieving Previous Entries	40
Retrieving Multiple Entries	41
Clearing the ENTRY Storage Area	41
Retrieving the Last Answer	41
Using Ans Preceding a Function	42
Storing Results to a Variable	42
Using TI-86 Menus	43
Displaying a Menu	43
The Menu Keys	44
Selecting a Menu Item	44
Exiting (Removing) a Menu	45
Viewing and Changing Modes	46
Changing a Mode Setting	46
Chapter 2: CATALOG, Variables, and Characters	49
The CATALOG	50
Storing Data to Variables	51
Creating a Variable Name	51
Storing a Value to a Variable Name	52
Storing an Unevaluated Expression	52
Storing an Answer	53
Copying a Variable Value	53
Displaying a Variable Value	53
Recalling a Variable Value	54
Classifying Variables as Data Types	54
The CATLG-VARS (CATALOG-Variables) Menu	55
Selecting a Variable Name	56
The CUSTOM Menu	56
Entering CUSTOM Menu Items	56
Clearing CUSTOM Menu Items	57
Deleting a Variable from Memory	57
The CHAR (Character) Menu	57
The CHAR MISC (Miscellaneous) Menu	58
The CHAR GREEK Menu	58
The CHAR INTL (International) Menu	58
Adding a Modifier to a Vowel	58
Chapter 3: Calculus, and Test Operations	59
Keyboard Mathematical Functions	60
The MATH Menu	61
The MATH NUM (Number) Menu	61
The MATH PROB (Probability) Menu	62
The MATH ANGLE Menu	63
The MATH HYP (Hyperbolic) Menu	63
The MATH MISC (Miscellaneous) Menu	64
The Interpolate à Extrapolate Editor	65
The CALC (Calculus) Menu	66
The TEST (Relational) Menu	67
Using Tests in Expressions and Instructions	68
Chapter 4: Constants, Conversions, Bases, and Complex Numbers	69
Using Built-In and User-Created Constants	70
The CONS (Constants) Menu	70
The CONS BLTIN (Built-In Constants) Menu	70
Creating or Redefining a User-Created Constant	72
The Constant Editor Menu	72
Entering a Constant Name in an Expression	73
Converting Units of Measure	73
Converting a Unit of Measure	73
The CONV (Conversions) Menu	74
The CONV LNGTH (Length) Menu	75
The CONV AREA Menu	75
The CONV VOL (Volume) Menu	75
The CONV TIME Menu	75
The CONV TEMP (Temperature) Menu	75
The CONV MASS Menu	76
The CONV FORCE Menu	76
The CONV PRESS (Pressure) Menu	76
The CONV ENRGY (Energy) Menu	76
The CONV POWER Menu	76
The CONV SPEED Menu	76
Converting a Value Expressed as a Rate	77
Number Bases	77
Number Base Ranges	78
One’s and Two’s Complements	78
The (Number) BASE Menu	78
The	79
The BASE A-F (Hexadecimal Characters) Menu	79
Entering Hexadecimal Digits	79
The BASE TYPE Menu	79
The BASE CONV (Conversion) Menu	80
Converting Number Bases	80
The BASE BOOL (Boolean) Menu	80
Results of Boolean Operations	81
The BASE BIT Menu	81
Using Complex Numbers	82
Complex Results	82
Using a Complex Number in an Expression	83
The CPLX (Complex Number) Menu	83
Chapter 5: Function Graphing	85
Defining a Graph	86
Setting the Graph Mode	86
The GRAPH Menu	87
Using the Equation Editor	88
The Equation Editor (GRAPH y(x)=) Menu	88
Defining a Function in the Equation Editor	89
Notes about Defining Function Equations	90
Selecting Graph Styles	91
Setting the Graph Style in the Equation Editor	92
Using Shading Patterns to Differentiate Functions	92
Viewing and Changing On à Off Status of Stat Plots	93
Setting the Window Variables	93
Displaying the Window Editor	94
Changing a Window Variable Value	94
Setting Graphing Accuracy with Delta x and Delta y	95
Setting the Graph Format	95
Displaying a Graph	97
Pausing or Stopping a Graph in Progress	97
Modifying a Drawn Graph	97
Graphing a Family of Curves	98
Smart Graph	98
Chapter 6: Graph Tools	99
Graph Tools on the TI-86	100
The GRAPH Menu	100
Using the Free-Moving Cursor	101
Graphing Accuracy	101
Tracing a Graph	102
Stopping and Resuming a Trace	103
Resizing the Graph Screen with ZOOM Operations	103
The GRAPH ZOOM Menu	103
Defining a Custom Zoom In	105
Setting Zoom Factors	105
Zooming In and Zooming Out on a Graph	105
Storing and Recalling Zoom-Window Variable Values	107
Using Interactive Math Functions	107
The GRAPH MATH Menu	107
Settings That Affect GRAPH MATH Operations	108
Using ROOT, FMIN, FMAX, or INFLC	109
Using Integral f(x), DIST, or ARC	110
Using dy/dx or TANLN	111
Using ISECT	112
Using YICPT	112
Evaluating a Function for a Specified x	113
Drawing on a Graph	113
Before Drawing on a Graph	114
Saving and Recalling Drawn Pictures	114
Clearing Drawn Pictures	115
The GRAPH DRAW Menu	115
Shading Areas of a Graph	116
Drawing a Line Segment	117
Drawing a Vertical or Horizontal Line	118
Drawing a Circle	118
Drawing a Function, Tangent Line, or Inverse Function	119
Drawing Freehand Points, Lines, and Curves	119
Placing Text on a Graph	120
Turning On or Turning Off Points	120
Chapter 7: Tables	122
TABLE Menu	122
The Table	122
Independent and Dependent Variables in the Table	123
Navigating the Table	123
The Table Menus	124
Setting Up the Table	125
Viewing and Editing Dependent Variable Equations	126
Clearing the Table	126
Chapter 8: Polar Graphing	127
Preview: Polar Graphing	128
Defining a Polar Graph	129
Setting Polar Graphing Mode	129
The GRAPH Menu	129
Displaying the Polar Equation Editor	130
Setting the Graph Screen Window Variables	130
Setting the Graph Format	131
Displaying the Graph	131
Using Graph Tools in Pol Graphing Mode	131
The Free-Moving Cursor	131
Tracing a Polar Equation	132
Moving the Trace Cursor to a theta Value	133
Using Zoom Operations	133
The GRAPH MATH Menu	134
Evaluating an Equation for a Specified theta	134
Drawing on a Polar Graph	134
Chapter 9: Parametric Graphing	135
Preview: Parametric Graphing	136
Defining a Parametric Graph	137
Setting Parametric Graphing Mode	138
The GRAPH Menu	138
Displaying the Parametric Equation Editor	138
Selecting and Deselecting a Parametric Equation	139
Deleting a Parametric Equation	139
Setting the Graph Screen Window Variables	139
Setting the Graph Format	140
Displaying the Graph	140
Using Graph Tools in Param Graphing Mode	140
The Free-Moving Cursor	140
Tracing a Parametric Function	140
Moving the Trace Cursor to a t Value	141
Using Zoom Operations	141
The GRAPH MATH Menu	142
Evaluating an Equation for a Specified t	142
Drawing on a Parametric Graph	142
Chapter 10: Differential Equation Graphing	143
Defining a Differential Equation Graph	144
Setting Differential Equation Graphing Mode	144
The GRAPH Menu	145
Setting the Graph Format	145
Displaying the Differential Equation Editor	146
Setting the Graph Screen Window Variables	147
Setting the Initial Conditions	148
Setting the Axes	149
Differential Equation Graphing Tips	149
The Built-In Variable fldPic	150
Displaying the Graph	150
Entering and Solving Differential Equations	151
Graphing in SlpFld Format	151
Transforming an Equation into a First-Order System	152
Graphing in DirFld Format	153
Graphing a System of Equations in FldOff Format	154
Solving a Differential Equation for a Specified Value	156
Using Graph Tools in DifEq Graphing Mode	156
The Free-Moving Cursor	156
Tracing a Differential Equation	156
Moving the Trace Cursor to a t Value	157
Drawing on a Differential Equation Graph	157
Drawing an Equation and Storing Solutions to Lists	157
Using ZOOM Operations	159
Drawing Solutions Interactively with EXPLR	160
Evaluating Differential Equations for a Specified t	162
Chapter 11: Lists	163
Lists on the TI-86	164
The LIST Menu	164
The LIST NAMES Menu	165
Creating, Storing, and Displaying Lists	165
Entering a List Directly in an Expression	165
Creating a List Name by Storing a List	166
Displaying List Elements Stored to a List Name	166
Displaying or Using a Single List Element	167
Storing a New Value to a List Element	167
Complex List Elements	168
The List Editor	168
The List Editor Menu	168
Creating a List Name in the Unnamed Column	169
Inserting a List Name into the List Editor	169
Displaying and Editing a List Element	170
Deleting Elements from a List	170
Removing a List from the List Editor	170
Using List Operations	171
The LIST OPS (Operations) Menu	171
Using Mathematical Functions with Lists	173
Attaching a Formula to a List Name	174
Comparing an Attached List with a Regular List	175
Using the List Editor to Attach a Formula	175
Using the List Editor With Attached-Formula Lists	176
Executing and Displaying Attached Formulas	176
Handling Errors Related to Attached Formulas	177
Detaching a Formula from a List Name	178
Editing an Element of a Attached-Formula List	178
Chapter 12: Vectors	179
Vectors on the TI-86	180
Creating, Storing, and Displaying Vectors	181
The VECTR (Vector) Menu	181
The VECTR NAMES Menu	181
Creating a Vector in the Vector Editor	181
The Vector Editor Menu	182
Creating a Vector on the Home Screen	182
Creating a Complex Vector	183
Displaying a Vector	183
Using a Vector in an Expression	184
Editing Vector Dimension and Elements	184
The VECTR MATH Menu	185
The VECTR OPS (Operations) Menu	185
The VECTR CPLX (Complex) Menu	187
Using Mathematical Functions with Vectors	188
Chapter 13: Matrices	189
Matrices on the TI-86	190
Creating, Storing, and Displaying Matrices	190
The MATRX (Matrix) Menu	190
The MATRX NAMES Menu	190
Creating a Matrix in the Matrix Editor	190
The Matrix Editor Menu	191
Creating a Matrix on the Home Screen	192
Creating a Complex Matrix	192
Displaying Matrix Elements, Rows, and Submatrices	193
Using a Matrix in an Expression	193
Editing Matrices in the Matrix Editor	194
Editing Matrices on the Home Screen	194
The MATRX MATH Menu	195
The MATRX OPS (Operations) Menu	196
The MATRX CPLX (Complex ) Menu	197
Using Mathematical Functions with Matrices	197
Chapter 14: Statistics	199
Statistical Analysis on the TI-86	200
Setting Up a Statistical Analysis	200
The STAT (Statistics) Menu	200
Entering Statistical Data	201
The LIST NAMES Menu	201
The STAT CALC (Calculations) Menu	201
Automatic Regression Equation Storage	203
Results of a Statistical Analysis	204
The STAT VARS (Statistical Variables) Menu	204
Plotting Statistical Data	206
The STAT PLOT Status Screen	206
The STAT PLOT Menu	207
Setting Up a Stat Plot	207
Turning On and Turning Off a Stat Plot	207
The PLOT TYPE Menu (Selecting a Plot Type)	208
Plot Type Characteristics	208
The STAT DRAW Menu	211
Forecasting a Statistical Data Value	211
Chapter 15: Equation Solving	213
Preview: The Equation Solver	214
Entering an Equation in the Equation-Entry Editor	215
Setting Up the Interactive-Solver Editor	216
Entering Variable Values	216
Controlling the Solution with Bounds and a Guess	216
Editing the Equation	217
The Solver Menu	218
Solving for the Unknown Variable	218
Graphing the Solution	219
Solver Graph Tools	219
The Solver ZOOM Menu	220
The Simultaneous Equation Solver	220
Entering Equations to Solve Simultaneously	220
Storing Equation Coefficients and Results to Variables	222
The Polynomial Root-Finder	223
Entering and Solving a Polynomial	223
Storing a Polynomial Coefficient or Root to a Variable	224
Chapter 16: Programming	225
Writing a Program on the TI-86	226
The PRGM Menu	226
Creating a Program in the Program Editor	226
The Program Editor Menu	227
The PRGM I à O (Input à Output) Menu	227
The TI-86 Key Code Diagram	229
The PRGM CTL Menu	230
Entering a Command Line	232
Menus and Screens in the Program Editor	232
Running a Program	233
Breaking (Interrupting) a Program	234
Working with Programs	235
Managing Memory and Deleting a Program	235
Editing a Program	235
Calling a Program from Another Program	236
Copying a Program to Another Program Name	237
Using and Deleting Variables within a Single Program	237
Running an Assembly Language Program	237
Entering and Storing a String	238
The STRNG (String) Menu	239
Creating a String	239
Chapter 17: Memory Management	241
Checking Available Memory	242
The MEM (Memory) Menu	242
Checking Memory Usage	242
Deleting Items from Memory	243
The MEM DELET (Delete) Menu	243
Resetting the TI-86	244
The MEM RESET (Reset) Menu	244
ClrEnt (Clear Entry)	244
Chapter 18: The TI-86 Communication Link	245
TI-86 Linking Options	246
Linking Two TI-86s	246
Linking a TI-86 and a TI-85	246
Linking a TI-86 and a CBL 2/CBL or CBR System	246
Linking a TI-86 and a PC or Macintosh	247
Downloading Programs from the Internet	247
Connecting the TI-86 to Another Device	247
The LINK Menu	248
Selecting Data to Send	248
The LINK SEND Menu	248
Initiating a Memory Backup	249
Selecting Variables to Send	249
The SEND WIND (Window Variables) Screen	250
Sending Variables to a TI-85	251
The LINK SND85 (Send Data to TI-85) Menu	251
Preparing the Receiving Device	252
Transmitting Data	252
Receiving Transmitted Data	252
Repeating Transmission to Several Devices	254
Error Conditions	254
Insufficient Memory in Receiving Unit	254
Chapter 19: Applications	255
Using Math Operations with Matrices	256
Finding the Area between Curves	257
The Fundamental Theorem of Calculus	258
Electrical Circuits	260
Program: Taylor Series	262
Characteristic Polynomial and Eigenvalues	264
Convergence of the Power Series	266
Reservoir Problem	268
Predator-Prey Model	270
Program: Sierpinski Triangle	272
Chapter 20: A to Z Function and Instruction Reference	273
Quick-Find Locator	274
Graphing	274
Lists	274
Mathematics, Algebra, and Calculus	275
Matrices	276
Programming	276
Statistics	276
Strings	277
Vectors	277
Alphabetical Listing of Operations	278
A	279
abs	279
Addition: +	279
and	280
angle	281
Ans	281
arc(	281
Asm(	281
AsmComp(	282
AsmPrgm	282
Assignment =	282
aug(	282
Axes(	283
AxesOff	283
AxesOn	283
B	283
b (integer b)	283
Bin	284
Box	284
C	285
Circl(	285
ClDrw	285
CILCD	285
ClrEnt	285
ClTbl	285