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HP 39g hp 39g+ (39g & 40g)_mastering the hp 39g+_English_E_F2224-90010.pdf - Page 89

that it rounds off to a smaller number than 1.03, which is

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The problem lies in the fact that the slow convergence will mean that people will often try to graph this function for very large values of x. The first graph on the right shows the graph of this function for the domain of 0 to 100. The second graph shows how instability develops in the domain 0 to 1E11 (1×1011 ). This apparent instability is caused by the internal rounding of the calculator. It works to 16 bits accuracy, which means that it can store 12 significant digits (for reasons only of interest to programmers). This means that when you invert a really large number and add it to one, you lose a lot of accuracy. For example: if X = 2⋅85×1010 then 1/X is 2⋅5087719298×10-11. When you add 1 to this, the calculator is forced to discard all but the last decimal place. Thus 1 + 1/X = 1.00000000003 (rounded off from 1.00000000002508...) There are naturally a whole range of numbers which will all round off to the same value of 1.00000000003, so that (for that range of numbers) the expression (1+1/X)^X is equivalent mathematically (on the HP) to (1.00000000003)^X. This produces a short section of an exponential graph, which only looks linear because you don't see enough of it. Eventually the calculator reaches a value on the x axis which is large enough that it rounds off to a smaller number than 1.00000000003, which is 1.00000000002. This produces the sudden drop in the graph as the plot changes from a section of a 1.00000000003^X graph to a section of a 1.00000000002^X graph (which has a shallower gradient). This section is maintained until the next drop, and so on. Finally, at the value x = 2×1011 the inverted value is so small that 1+1/X becomes exactly 1 and the graph becomes horizontal. Of course this is completely the wrong value! Although this explanation may be beyond the level of many students it is quite important that they have some understanding of these ideas if they use the calculator to evaluate limits. The solution to all problems of this type is to simply be aware of their existence and to allow for them rather than simply accepting the results shown in the NUM view. 89

Section	Page
Title Page	1
Introduction	12
Some Keyboard Examples	16
Keys & Notation Conventions	17
Some essential keys	17
The SHIFT key	17
The ALPHA key	18
The Screen keys	18
Pop-up menus & short-cuts	18
Everything revolves around Aplets!	21
What are aplets?	21
A quick list of aplets	21
The Function aplet (see page 51)	21
The Inference aplet (see page 150)	21
The Parametric aplet (see page 100)	21
The Finance aplet (see page 160)	21
The Polar aplet (see page 106)	21
The Quadratic Explorer aplet (see page 164)	22
The Sequence aplet (see page 107)	22
The Solve aplet (see page 113)	22
The Statistics aplet (see page 122 & 132)	22
The Trig Explorer aplet (see page 167)	22
Some typical aplet views	22
The HOME view	25
What is the HOME view?	25
Exploring the keyboard	26
The screen keys	26
Aplet related keys	26
The arrow keys	26
The SYMB, PLOT and NUM keys	27
Intro to the VIEWS menu	28
The VARS key	28
The SETUP views	30
The MODES view	30
Numeric formats	31
The ANS key	32
The negative key	33
The CHARS key	33
The DEL and CLEAR keys	34
Angle and Numeric settings	35
Memory Management	37
The MEMORY MANAGER view	37
Downloaded aplets & memory	38
The GRAPHICS MANAGER	39
The LIBRARY MANAGER	39
Fractions on the hp 39g+	40
Pitfalls to watch for	42
The HOME History	43
COPYing calculations	43
Clearing the History	44
SHOWing results	44
Storing and Retrieving Memories	45
Referring to other aplets from the HOME view.	46
Introducing the MATH Menu	47
Resetting the calculator	48
Soft reboot	48
Soft reboot (Hardware)	49
Hard reboot (with memory wipe)	49
Summary	50
The Function Aplet	51
Introduction	51
Choose the aplet	51
The SYMB view	52
The XT-theta button	52
The NUM view	53
The PLOT view	53
Auto Scale	54
The PLOT SETUP view	55
Detail vs. Faster	55
Simultaneous	56
Connect	56
Axes	56
Labels	56
Grid	56
The default axis settings	57
The MENU bar	57
The MENU bar functions	58
Trace	58
Defn	58
Goto	59
The Zoom Sub-menu	60
Center	60
In/Out	60
Box…	61
X-Zoom In/Out and Y-Zoom In/Out	62
Square	62
Auto Scale, Decimal, Integer and Trig	62
The Function Tools menu	63
Root	63
Intersection	64
Slope	64
Signed area…	65
Definite integrals	65
Tracing the integral in PLOT	66
Areas between and under curves	67
Extremum	68
The VIEWS menu	92
	93
Plot-Detail	93
Plot-Table	94
Nice table values	95
Overlay Plot	95
Auto Scale in detail	96
Decimal, Integer & Trig	97
Aplets from the Internet	99
Curve Areas	99
Linear Programming	99
The Parametric Aplet	100
Choose XRng, YRng & TRng	100
The effect of TRng	101
TStep controls smoothness	101
The Polar Aplet	106
	106
Choose XRng, YRng & thetaRng	106
thetaStep and smoothness	106
Changing the default thetaStep	106
Circular circles	106
The Sequence Aplet	107
Recursive or non-recursive	107
First, second & general terms	107
Convenient screen keys provided	108
The Solve Aplet	113
Equations vs. expressions	113
Entering the equation	113
Solving for a missing value	114
INFO	114
Mult solns & the initial guess	115
Graphing in Solve	116
Transferring approx solutions	116
Referring to other aplets	117
A detailed explanation of PLOT in Solve	118
INFO messages in detail	120
The Stats Aplet - Univariate Data	122
Uni vs. Bi-variate data	122
Clearing data	122
Sorting data	123
The STATS key	123
Functions of columns	124
Registering columns	124
Working with freq. tables	125
Auto scale	125
Plot Setup options	126
Box and whisker graphs	126
The effect of HRng	127
Grouped data & HWidth	127
The Stats Aplet - Bivariate Data	132
Uni vs. Bi-variate data	132
Clearing data	132
Entering data	133
Plot symbols used	133
Tracing with the cursor	133
Sorting paired columns	134
Specifying the fit model	134
Multiple data sets	134
Fit models available	135
The User Defined model	135
Connected data	136
Two Variable Statistics	137
Showing the line of best fit	138
A caveat…	139
Predicting using PREDY	140
Predicting in the PLOT view	140
RelErr as a measure of non-linear fit	141
The Inference Aplet	150
Chi-Squared on a freq table	150
Hypoth test: T-Test 1-mu	151
Conf interval: T-Int 1-mu	153
Hypoth test: T-Test mu1-mu2	154
Hypoth test: Z-Test 1-mu	156
The Finance Aplet	160
Parameters	160
Straightforward compound interest	161
Annuity	162
Loan calculations	162
Amortization	163
The Quad Explorer Teaching Aplet	164
Objectives	164
Choosing the level	164
GRAPH mode	164
SYMB mode	165
Self test mode	166
The Trig Explorer Teaching Aplet	167
Objectives	167
SIN vs. COS	167
SYMB vs. GRPH mode	167
Matrices	170
The MATRIX Catalog	170
Matrix calcs in HOME	171
System of equations	172
Finding an inverse matrix	174
The dot product	175
Lists	176
The list variables	176
Operations on lists	176
Statistical columns as lists	176
List functions	177
Editing a list	177
Operations on elements	177
The Notepad Catalog	178
Aplet notes vs. independent notes	178
Independent Notes	179
Transferring notes using IR	180
Editing software	180
Software	181
For the 38g, 39g & 40g	181
For the hp 39g+	181
Creating a Note	182
Locking ALPHA mode	182
The CHARS view	183
Corrupting notes	183
The aplet Sketchpad	184
Adding text to a sketch	184
The DRAW menu	185
DOT+	185
LINE	185
BOX	185
CIRCLE	186
Cut & paste (sort of…)	186
Storing to a GROB	186
Using VAR	186
Animations	187
Capturing the PLOT screen	187
Using, Copying & Creating aplets	189
Creating a copy of an aplet	190
Examples of saved aplets	192
The Triangles aplet	192
The Prob. Dist. aplet	192
The Transformer aplet	194
Copying via IR	197
The infra-red port	197
The 40g (Wire) option	198
The ‘Disk drive…’ option	198
Time out	199
Attached programs	199
Downloading from the Internet	200
Finding aplets	200
Organizing aplets	201
Software (38g, 39g & 40g)	202
Software (39g+)	203
HPGComm Connectivity	204
Deleting downloaded aplets	207
Saving via HPGComm	208
Capturing screens	210
Editing with the ADK	211
Programming the hp 39g+	212
The design process	212
An overview	212
Choosing the parent aplet	212
Naming conventions	213
Planning the VIEWS menu	214
The SETVIEWS command	214
Special entries	216
The ‘Start’ entry	217
Example aplet #1	217
Example aplet #2	222
Example aplet #3	224
Example aplet #4	226
Programming Commands	233
Aplet commands	233
CHECK, UNCHECK	233
SELECT	233
SETVIEWS	234
Branch commands	234
IF THEN [ELSE] END	234
CASE END	234
IFFERR THEN [ELSE] END	234
RUN	235
STOP	235
Drawing commands	235
ARC	235
BOX	235
ERASE	236
FREEZE	236
LINE	236
PIXON & PIXOFF	236
TLINE	236
Graphics commands	237
Loop commands	237
FOR TO [STEP] END	237
DO UNTIL END	237
WHILE REPEAT END	237
BREAK	238
Matrix commands	238
EDITMAT	238
REDIM	238
Print commands	239
PRDISPLAY	239
PRHISTORY	239
PRVAR	239
Prompt commands	240
BEEP	240
CHOOSE	240
DISP	241
DISPXY	241
DISPTIME	241
GETKEY	241
INPUT	242
MSGBOX	242
PROMPT	242
WAIT	242
The MATH menu functions	243
The ‘Real’ group	246
CEILING	246
DEG->RAD	246
FLOOR	246
FNROOT	247
FRAC	247
HMS->	248
->HMS	248
INT	249
MANT	249
MAX	249
MIN	250
MOD	250
The % function	250
%CHANGE	251
%TOTAL	251
RAD->DEG	251
ROUND	252
SIGN	252
TRUNCATE	253
XPON	253
The ‘Stat-Two’ group	254
PREDY	254
PREDX	254
The ‘Symbolic’ group	255
The = ‘function’	255
ISOLATE	255
LINEAR?	256
QUAD	256
QUOTE	257
The \| function	258
The ‘Tests’ group	259
The ‘Trig’ & ‘Hyperbolic’ groups	259
COT, SEC etc	259
EXP	260
ALOG	260
EXPM1	260
LNP1	261
The ‘Calculus’ group	261
The ‘Complex’ group	262
ABS	263
ARG	264
CONJ	264
IM & RE	264
The ‘Constants’ group	265
The ‘List’ group	265
CONCAT	265
1st order differences	266
MAKELIST	266
Product of LIST	267
POS	267
SIZE	268
Sum of LIST	268
REVERSE	268
SORT	268
The ‘Loop’ group	269
ITERATE	269
RECURSE	270
SUMMATION	270
The ‘Matrix’ group	271
COLNORM	271
COND	271
CROSS	271
DET	272
DOT	272
EIGENVAL	272
EIGENVV	272
IDENTMAT	272
INVERSE	273
LQ	273
LSQ	274
LU	274
MAKEMAT	274
QR	274
RANK	274
ROWNORM	274
RREF	274
SCHUR	275
SIZE	275
SPECNORM	275
SPECRAD	275
SVD	275
SVL	275
TRACE	276
TRN	276
The ‘Polynomial’ group	277
POLYCOEF	277
POLYEVAL	277
POLYFORM	278
POLYROOT	279
The ‘Probability’ group	280
COMB	280
The ! function	280
PERM	281
RANDOM	281
RANDSEED	281
UTPN	282
UTPC	283
UTPF	283
UTPT	283
App. A: Worked Examples	284
Finding intercepts of a quad	284
Using the QUAD.	284
Using Function	284
Using POLYROOT	285
Finding complex solutions	285
Using QUAD	285
Using POLYROOT	285
Finding critical points	286
Simultaneous equations.	288
By graphing	288
Using a matrix	288
Using the 3x3 Solver aplet	289
Inconsistent systems	294
Using RREF	294
Finding complex roots	295
Vector motion & collisions	296
Circular Motion and the Dot Product	297
Inference testing using Chi-Squared	299
App. B: Teaching Calculus with an hp 39g+	301
Investigating x^n	301
Domains & Composite Functions	302
Gradient at a Point	303
The Gradient Function	304
The Chain Rule	305
Optimization	305
Area Under Curves	306
Fields of Slopes	306
Inequalities	307
Rectilinear Motion	307
Limits	307
Piecewise Defined Functions	308
Sequences and Series	308
Transformations of Graphs	309
App. C: CAS & the hp 40g	310
Introduction	310
What is a CAS?	310
The hp 39g vs. the hp 40	312
Using the CAS	313
Entering and editing an expression	313
In-line editing mode	316
Erasing, copying, cutting and pasting	316
Cursor mode	317
The CAS HOME History	317
PUSH & POP	318
Pasting to an aplet	319
Evaluating algebraic expressions	320
Worked examples	321
Using LIMIT	321
Factorizing expressions	322
Solving equations	323
Simultaneous equations	323
Simultaneous integration	325
Defining a user function	327
On-line help	328
Configuring the CAS	329

Match case Limit results 1 per page

The problem lies in the fact that the slow

convergence will mean that people will often

try to graph this function for very large values

of x.

The first graph on the right shows the

graph of this function for the domain of

0 to

100. The second graph shows how instability

develops in the domain 0 to 1E11 (

1 10

This apparent instability is caused by the

internal rounding of the calculator. It works to

16 bits accuracy, which means that it can store 12 significant digits (for

reasons only of interest to

programmers). This means that when you invert a

really large number and add it to one, you lose a lot of accuracy.

For example: if X =

2 85 10

⋅

then 1/X is

-11

2 5087719298 10

⋅

When you

add 1 to this, the calculator is forced to discard all but the last decimal place.

Thus 1 + 1/X = 1.00000000003 (rounded off from 1.00000000002508...)

There are naturally a whole range of numbers

which will all round off to the same value of

1.00000000003, so that (for that range of

numbers) the expression (1+1/X)^X is

equivalent mathematically (on the HP) to

(1.00000000003)^X. This produces a short section of an exponential graph,

which only looks linear because you don’t see enough of it.

Eventually the calculator reaches a value on the x axis which is large enough

that it rounds off to a smaller number than 1.00000000003, which is

1.00000000002. This produces the sudden drop in the graph as the plot

changes from a section of a 1.00000000003^X graph to a section of a

1.00000000002^X graph (which has a shallower gradient).

This section is maintained until the next drop, and so on.

Finally, at the value

x =

2 10

the inverted value is so small that 1+1/X becomes exactly 1 and

the graph becomes horizontal.

Of course this is completely the wrong value!

Although this explanation may be beyond the level of many students it is

quite important that they have some understanding of these ideas if they use

the calculator to evaluate limits.

The solution to all problems of this type is to

simply be aware of their existence and to allow for them rather than simply

accepting the results shown in the

NUM

view.