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Texas Instruments TI-82 User Manual - Page 212

Fundamental Theorem of Calculus

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Fundamental Theorem of Calculus The TI.82 can graph functions that are defined by integrals or derivatives, using the functions fnInt( and nDeriv( from the MATH MATH menu. Problem 1 Demonstrate graphically that ‰x F(x) = 1àt dt = ln(x), x>0 and that 1 [ ‰ ] x Dx 1àt dt = 1àx 1 Procedure 1 1. Press z. Select Simul and the default MODE settings. Press o and turn off all functions. Press y [STAT PLOT] and turn off all stat plots. 2. Press p. Set the viewing WINDOW. Xmin = .01 Xmax = 10 Xscl = 1 Ymin = M1.5 Ymax = 2.5 Yscl = 1 3. Press o. Enter the numerical integral of 1àT and the mathematical integral of 1àX. Y1=fnInt(1àT,T,1,X) Y2=ln X 4. Press r. The busy indicator displays while the graph is being plotted. Use the cursor keys to compare the values of Y1 and Y2. 5. Press o. Turn off Y1 and Y2, and then enter the numerical derivative of the integral of 1àX and the function 1àX. Y3=nDeriv(Y1,X,X) Y4=1àX 6. Press r. The busy indicator displays while the graph is being plotted. Again, use the cursor keys to compare the values of the two graphed functions, Y3 and Y4. 14-18 Applications

Section	Page
TI-82 Graping Calculator Guidebook	1
Important & US FCC Information	2
Table of Contents	3
Getting Started: Do This First!	3
Chapter 1: Operating the TI.82	3
Chapter 2: Math, Angle, and Test Operations	4
Chapter 3: Function Graphing	4
Chapter 4: Parametric Graphing	4
Chapter 5: Polar Graphing	4
Chapter 6: Sequence Graphing	5
Chapter 7: Tables	5
Chapter 8: DRAW Operations	5
Chapter 9: Split Screen	5
Chapter 10: Matrices	6
Chapter 11: Lists	6
Chapter 12: Statistics	6
Chapter 13: Programming	6
Chapter 14: Applications	7
Chapter 15: Memory Management	7
Chapter 16: Communication Link	7
Appendix A: Tables	7
Appendix B: Reference Information	7
Index	7
Using this Guidebook Effectively	8
Structure of the Guidebook	8
Page-Design Conventions	8
Information-Mapping Conventions	9
Reference Aids	9
Glossary	10
Getting Started: Do This First!	11
TI-82 Menus	12
Displaying a Menu	12
Moving from One Menu to Another	12
Selecting an Item from a Menu	12
Leaving without Making a Selection	12
First Steps	13
Entering a Calculation: Compound Interest	14
Defining a Function: Box with Lid	16
Defining a Table of Values	17
Zooming In on the Table	18
Changing the Viewing WINDOW	20
Displaying and Tracing the Graph	21
Zooming on the Graph	22
Finding the Calculated Maximum	23
Other Features	24
Graphing	24
Sequences	24
Tables	24
Matrices	24
Lists	24
Statistics	24
Programming	24
Split Screen	24
Chapter 1: Operating the TI-82	25
Turning the TI-82 On and Off	26
Turning the Calculator On	26
Turning the Calculator Off	26
APD™ (Automatic Power Down™)	26
Batteries	26
Setting the Display Contrast	27
Adjusting the Display Contrast	27
When to Replace Batteries	27
The Display	28
Home Screen	28
Displaying Entries and Answers	28
Returning to the Home Screen	28
Display Cursors	29
Busy Indicator	29
Entering Expressions and Instructions	30
Expressions	30
Entering an Expression	30
Multiple Entries on a Line	30
Entering a Number in Scientific Notation	31
Functions	31
Instructions	31
Interrupting a Calculation	31
TI-82 Edit Keys	32
Setting Modes	33
Checking MODE Settings	33
Changing MODE Settings	33
Leaving the MODE Screen	33
Setting a MODE from a Program	33
TI-82 Modes	34
Normal, Sci, Eng	34
Float, Fix	34
Radian, Degree	35
Func, Par, Pol, Seq	35
Connected, Dot	35
Sequential, Simul	35
FullScreen, Split	35
Variable Names	36
Variables and Defined Items	36
Storing and Recalling Variable Values	37
Storing Values in a Variable	37
Displaying a Variable Value	37
RCL (Recall)	37
Last Entry	38
Using Last Entry	38
Multiple Entries on a Line	38
Reexecuting the Previous Entry	39
Accessing a Previous Entry	39
Last Answer	40
Using Ans in an Expression	40
Continuing an Expression	40
Storing Answers	40
TI-82 Menus	41
Moving from One Menu to Another	41
Selecting an Item from a Menu	41
Leaving a Menu without Making a Selection	41
VARS and Y-VARS Menus	43
VARS Menu	43
Y-VARS Menu	43
Accessing a Name from a VARS or Y-VARS Menu	43
EOS™ (Equation Operating System)	44
Order of Evaluation	44
Implied Multiplication	45
Parentheses	45
Negation	45
Error Conditions	46
Diagnosing an Error	46
Correcting an Error	46
Chapter 2: Math, Angle, and Test Operations	47
Getting Started: Lottery Chances	48
Keyboard Math Operations	49
Using Lists with Functions	49
+ (Add), – (Subtract), * (Multiply), / (Divide)	49
Trig Functions	49
^ (Power), ² (Square), (Square Root)	49
-1 (Inverse)	49
log, 10^, ln	50
e^	50
- (Negation)	50
abs	50
(Pi)	50
MATH MATH Operations	51
MATH MATH Menu	51
>Frac	51
>Dec	51
³ (Cube)	52
(Cube Root)	52
xth (Root)	52
fMin(, fMax(	52
nDeriv(	53
fnInt(	53
solve(	54
Controlling the Solution for solve(	54
MATH NUM (Number) Operations	55
MATH NUM Menu	55
round(	55
iPart	55
fPart	55
int	56
min(, max(	56
MATH HYP (Hyperbolic) Operations	57
MATH HYP Menu	57
sinh, cosh, tanh	57
sinh^ -1, cos^-1, tanh^-1	57
MATH PRB (Probability) Operations	58
MATH PRB Menu	58
rand	58
nPr	58
nCr	58
! (Factorial)	58
ANGLE Operations	59
ANGLE Menu	59
°( Degree)	59
' (DMS Entry Notation)	59
r (Radians)	59
R>Pr(, R>Pq(, P>Rx(, P>Ry(	60
TEST TEST (Relational) Operations	61
TEST TEST Menu	61
=, ne, >, ge>, <. le<	61
Using Tests	61
TEST LOGIC (Boolean) Operations	62
TEST LOGIC Menu	62
Boolean Operators	62
and, or, xor	62
not	62
Using Boolean Operations	62
Chapter 3: Function Graphing	63
Getting Started: Graphing a Circle	64
Defining a Graph	65
Steps in Defining a Graph	65
Exploring a Graph	65
Saving a Graph	65
Setting Graph Modes	66
Checking and Changing Graphing Modes	66
Setting Modes from a Program	66
Defining Functions in the Y= List	67
Displaying the Functions in the Y= List	67
Defining a New Function	67
Editing a Function	68
Clearing a Function	68
Defining Functions from the Home Screen or a Program	68
Evaluating Y= Functions in Expressions	68
Selecting Functions	69
Turning a Function “On” or “Off”	69
Leaving the Y= Edit Screen	69
Selecting Functions from the Home Screen or a Program	69
Defining the Viewing	70
TI-82 Viewing WINDOW	70
Checking the Viewing WINDOW	70
Changing a WINDOW Variable Value	70
Leaving the WINDOW Edit Screen	71
Storing to a WINDOW Variable from the Home Screen or a Program	71
Delta X and Delta Y	71
Setting WINDOW FORMAT	72
Checking WINDOW FORMAT	72
Changing WINDOW FORMAT	72
RectGC, PolarGC	72
CoordOn, CoordOff	72
GridOff, GridOn	72
AxesOn, AxesOff	72
LabelOff, LabelOn	72
Displaying a Graph	73
Displaying a New Graph	73
Pausing a Graph	73
Smart Graph	73
Graphing a Family of Curves	74
Exploring a Graph with the Free-Moving Cursor	75
Free-Moving Cursor	75
Graphing Accuracy	75
Exploring a Graph with TRACE	76
Beginning a Trace	76
Moving along a Function	76
Panning to the Left or Right	76
QuickZoom	76
Moving from Function to Function	77
Leaving TRACE	77
Using TRACE in a Program	77
Exploring a Graph with ZOOM	78
ZOOM Menu	78
ZBox	78
Zoom In, Zoom Out	79
Zoom Out	79
Leaving Zoom In or Zoom Out	79
ZDecimal	80
ZSquare	80
ZStandard	80
ZTrig	80
ZInteger	80
ZoomStat	80
Using ZOOM MEMORY	81
ZOOM MEMORY Menu	81
ZPrevious	81
ZoomSto	81
ZoomRcl	81
Using ZOOM MEMORY from the Home Screen or a Program	81
Setting ZOOM FACTORS	82
ZOOM FACTORS	82
Checking XFact and YFact	82
Changing XFact and YFact	82
Leaving ZOOM FACTORS	82
Using CALC (Calculate) Operations	83
CALCULATE Menu	83
value	83
root	84
minimum, maximum	85
intersect	85
dy/dx	86
Integral f(x)dx	86
Chapter 4: Parametric Graphing	87
Getting Started: Path of a Ball	88
Defining and Displaying a Parametric Graph	89
Defining a Parametric Graph	89
Setting Parametric Graph Modes	89
Displaying Parametric Equations	89
Defining Parametric Equations	89
Selecting Parametric Equations	90
Setting WINDOW Variables	90
Setting the WINDOW FORMAT	91
Displaying a Graph	91
WINDOW Variables and Y-VARS Menus	91
Exploring a Parametric Graph	92
Free-Moving Cursor	92
TRACE	92
ZOOM	92
CALC	92
Chapter 5: Polar Graphing	93
Getting Started: Polar Rose	94
Defining and Displaying a Polar Graph	95
Defining a Polar Graph	95
Setting Polar Graph Modes	95
Displaying Polar Equations	95
Defining Polar Equations	95
Selecting Polar Equations	95
Setting WINDOW Variables	96
Setting the WINDOW FORMAT	96
Displaying a Graph	97
WINDOW Variables and Y-VARS Menus	97
Exploring a Polar Graph	98
Free-Moving Cursor	98
TRACE	98
ZOOM	98
CALC	98
Chapter 6: Sequence Graphing	99
Getting Started: Forest and Trees	100
Defining and Displaying a Sequence Graph	101
Defining a Sequence Graph	101
Setting Sequence Graph Modes	101
Displaying Sequence Functions	101
Defining Sequence Functions	101
Selecting Sequence Functions	102
Setting the WINDOW Variables	102
Setting WINDOW FORMAT	103
Displaying a Graph	103
Evaluating Un and Vn	103
Exploring a Sequence Graph	104
Free-Moving Cursor	104
TRACE	104
ZOOM	104
CALC	104
Chapter 7: Tables	105
Getting Started: Roots of a Function	106
Defining the Variables	107
TABLE SETUP Screen	107
TblMin and Delta Tbl	107
Indpnt: Auto or Ask	107
Depend: Auto or Ask	107
Setting Up a Table from the Home Screen or a Program	107
Defining the Dependent Variable	108
From the Y= Editor	108
From the Table Editor	108
Displaying the Table	109
The Table	109
Displaying More Independent Values	110
Displaying Other Dependent Variables	110
Clearing the Table	110
Chapter 8: DRAW Operations	111
Getting Started: Shading a Graph	112
DRAW DRAW Menu	113
DRAW DRAW Menu	113
Before Drawing on a Graph	113
Drawing on a Graph	113
Drawing Lines	114
Directly on a Graph	114
From the Home Screen or a Program	114
Drawing Horizontal and Vertical Lines	115
Directly on a Graph	115
From the Home Screen or a Program	115
Drawing Tangent Lines	116
Directly on a Graph	116
From the Home Screen or a Program	116
Drawing Functions and Inverses	117
Drawing a Function	117
Drawing an Inverse of a Function	117
Shading Areas on a Graph	118
Shading a Graph	118
Drawing Circles	119
Directly on a Graph	119
From the Home Screen or a Program	119
Placing Text on a Graph	120
Directly on a Graph	120
From the Home Screen or a Program	120
Split Screen	120
Using Pen to Draw on a Graph	121
Using Pen	121
Drawing Points	122
DRAW POINTS Menu	122
Directly on a Graph	122
Pt-Off(, Pt Change(	122
From the Home Screen or a Program	122
Drawing Pixels	123
TI.82 Pixels	123
Turning Pixels On and Off	123
pxl-Test(	123
Split Screen	123
Storing and Recalling Graph Pictures	124
DRAW STO Menu	124
Storing a Graph Picture	124
Recalling a Graph Picture	124
Deleting a Graph Picture	124
Storing and Recalling Graph Databases	125
Graph Databases	125
Storing a Graph Database	125
Recalling a Graph Database	125
Deleting a Graph Database	125
Clearing a Drawing	126
When a Graph is Displayed	126
From the Home Screen or a Program	126
Chapter 9: Split Screen	127
Getting Started: Polynomial Coefficients	128
Using Split Screen	129
Setting Screen MODE	129
Split Screen: Top	129
Split Screen: Bottom	129
Exceptions	129
TI-82 Pixels	130
DRAW Pixels Instructions	130
DRAW Text Instruction	130
PRGM Output Instruction	130
Setting Screen MODE from the Home Screen or a Program	130
Chapter 10: Matrices	131
Getting Started: Systems of Linear Equations	132
Defining a Matrix	134
Selecting a Matrix	134
Accepting or Changing Matrix Dimensions	134
Viewing Matrix Elements	135
Displaying Matrix Elements	135
Leaving the MATRX Edit Screen	135
Deleting a Matrix	135
Editing Matrix Elements	136
Viewing a Matrix	136
Viewing Context Keys	136
Editing a Matrix Element	137
Editing Context Keys	137
About Matrices	138
Using a Matrix in an Expression	138
Entering a Matrix in an Expression	138
Displaying a Matrix	139
Copying One Matrix to Another	139
Accessing a Matrix Element	139
Matrix Math Functions	140
+ (Add), – (Subtract	140
* (Multiply)	140
- (Negate)	140
abs	140
round(	141
-1 (Inverse)	141
Powers	141
Relational Operations	141
iPart, fPart, int	141
MATRX MATH Operations	142
MATRX MATH Menu	142
det	142
T (Transpose)	142
Accessing Matrix Dimensions with dim	142
Creating a Matrix with dim	143
Redimensioning a Matrix with dim	143
Fill(	143
identity	143
randM(	143
augment(	144
Row Operations	144
rowSwap(	144
row+(	144
*row(	144
*row+(	144
Chapter 11: Lists	145
Getting Started: Generating a Sequence	146
About Lists	147
Using a List in an Expression	147
Entering a List in an Expression	147
Saving a List in Memory	147
Displaying a List	148
Copying One List to Another	148
Accessing a List Element	148
Lists in Graphing	148
Notes about Using Math Functions with Lists	149
LIST OPS Operations	150
LIST OPS Menu	150
SortA(, SortD(	150
Accessing List Dimensions with dim	151
Creating a List with dim	151
Redimensioning a List with dim	151
Fill(	152
seq(	152
LIST MATH Operations	153
LIST MATH Menu	153
min(, max(	153
mean(, median(	153
sum	154
prod	154
Sums and Products of Numeric Sequences	154
Chapter 12: Statistics	155
Getting Started: Building Height and City Size	156
Setting Up a Statistical Analysis	163
Steps	163
Displaying the STAT List Editor	163
Leaving the STAT List Editor	163
Deleting a List	163
Viewing List Elements	164
Viewing Context Keys	164
Editing List Elements	165
Editing Context Keys	165
Editing a List Element	165
STAT EDIT Menu	166
STAT EDIT Menu	166
SortA(, SortD(	166
ClrList	166
Statistical Analysis	167
STAT CALC Menu	167
SET UP CALCS Screen	167
Changing Settings	167
Statistical Variables	168
Q1 and Q3	168
Types of Statistical Analysis	169
1-Var Stats	169
2-Var Stats	169
Med.Med	169
LinReg (ax+b)	169
QuadReg	169
CubicReg	169
QuartReg	169
LinReg (a+bx)	170
LnReg	170
ExpReg	170
PwrReg	170
Statistical Analysis in a Program	171
Entering Stat Data	171