HP 39g+ Master Manual

Mastering the hp 39g+ A guide for teachers, students and other users of the hp 39g+, hp 39g & hp 40g Edition 1.1 HP part number F2224-90010 Printed Date: 2005/10/11

Introduction.................................................................................................12 How to use this Manual..............................................................................13 Early High School.........................................................................................................13 Pre-Calculus.................................................................................................................14 Calculus........................................................................................................................14 Where’s the ON button?.............................................................................15 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 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 An introduction to the MATH Menu...............................................................................47 Resetting the calculator..................................................................................................48 Soft reboot....................................................................................................................48 Soft reboot (Hardware).................................................................................................49 Hard reboot (with memory wipe) ..................................................................................49 Summary ..........................................................................................................................50

Choose the aplet ..........................................................................................................51 The SYMB view .............................................................................................................52 The XTθ button .............................................................................................................52 ing your function ...............................................................................................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 Bar ...............................................................................................................57 The MENU toggle.........................................................................................................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 x4 and Y-Zoom In/Out x4 ....................................................................62 Square..........................................................................................................................62 Auto Scale, Decimal, Integer and Trig .........................................................................62 The FCN 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 Tips & Tricks - Function.............................................................................69 Finding a suitable set of axes.......................................................................................69 Composite functions.....................................................................................................71 Using functions in the HOME view...............................................................................72 Differentiating ...............................................................................................................73 Circular functions..........................................................................................................74 Trig functions................................................................................................................76 Retaining calculated values..........................................................................................77 The NUM view revisited ..............................................................................................77 ZOOM...........................................................................................................................78 Integration: The definite integral using the ∫ function ...............................................80 Integration: The algebraic indefinite integral ................................................................81 The hp 40g Computer Algebra System........................................................................82 Integration: The definite integral using PLOT variables ...............................................83 Piecewise defined functions.........................................................................................85 ‘Nice’ scales..................................................................................................................86 Use of brackets in functions .........................................................................................87

Gradient at a point........................................................................................................90 Finding and accessing polynomial roots ......................................................................91 The VIEWS menu........................................................................................92 Plot-Detail.....................................................................................................................93 Plot-Table .....................................................................................................................94 Overlay Plot..................................................................................................................95 Auto Scale ....................................................................................................................96 Decimal, Integer & Trig.................................................................................................97 Downloaded 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 Tips & Tricks - Parametric equations .....................................................103 Fun and games ..............................................................................................................103 Example 1...................................................................................................................103 Example 2...................................................................................................................103 Vectors ...........................................................................................................................104 Example 1...................................................................................................................104 Example 2...................................................................................................................105 The Polar Aplet .........................................................................................106 Choose XRng, YRng & θRng .....................................................................................106 θStep and smoothness...............................................................................................106 Changing the default for θStep...................................................................................106 Circular circles............................................................................................................106 The Sequence Aplet .................................................................................107 Recursive or non-recursive ........................................................................................107 First, second & general terms ....................................................................................107 Convenient screen keys provided ..............................................................................108 Tips & Tricks - Sequences & Series........................................................110 Defining a generalized GP and the sum to n terms. ..................................................110 Solving sequence problems .......................................................................................110 Modeling loans ...........................................................................................................112 The Solve Aplet.........................................................................................113 Equations vs. expressions..........................................................................................113 Entering the equation .................................................................................................113 Solving for a missing value.........................................................................................114 The INFO report .........................................................................................................114 Multiple solutions and the initial guess.......................................................................115 Graphing in Solve.......................................................................................................116 Transferring approximate solutions............................................................................116 Referring to functions from other aplets.....................................................................117 A detailed explanation of PLOT in Solve....................................................................118 The meaning of messages..............................................................................120 Tips & Tricks - Solve ................................................................................121 Easy problems............................................................................................................121 Harder problems.........................................................................................................121

Uni vs. Bi-variate data ................................................................................................122 Clearing data ..............................................................................................................122 Sorting data ................................................................................................................123 The STATS key ..........................................................................................................123 Functions of columns .................................................................................................124 Registering columns as ‘in use’..................................................................................124 Working with frequency tables ...................................................................................125 Auto scale...................................................................................................................125 Plot Setup options ......................................................................................................126 Box and whisker graphs.............................................................................................126 The effect of HRng .....................................................................................................127 Grouped data & HWidth .............................................................................................127 Tips & Tricks - Univariate Data................................................................129 New columns as functions of old................................................................................129 Simulating Dice ..........................................................................................................129 Simulating Random Variables....................................................................................130 The Stats Aplet - Bivariate Data ..............................................................132 Uni vs. Bi-variate data ................................................................................................132 Clearing data ..............................................................................................................132 Entering data as ordered pairs...................................................................................133 Adjusting the symbols used to plot points..................................................................133 The cursor ..................................................................................................................133 Sorting paired columns...............................................................................................134 Specifying the fit model ..............................................................................................134 Multiple data sets .......................................................................................................134 Choosing from available fit models ............................................................................135 The User Defined model ............................................................................................135 Connected data ..........................................................................................................136 Two Variable Statistics...............................................................................................137 Showing the line of best fit .........................................................................................138 Predicting using PREDY..............................................................................................140 Predicting using the PLOT view..................................................................................140 RelErr as a measure of non-linear fit.......................................................................141 Tips & Tricks - Bivariate Data..................................................................143 New columns as functions of old................................................................................143 Using values from in calculations ..................................................................143 Obtaining coefficients from the fit model....................................................................145 Finding Fit Coefficients...............................................................................................145 Correct interpretation of the PREDX function ............................................................146 Assigning rank orders to sets of data.........................................................................147 Using Stats to find equations from point data ............................................................148 The Inference aplet...................................................................................150 Using the Chi2 test on a frequency table ...................................................................150 Hypothesis test: T-Test 1-µ .......................................................................................151 Confidence interval: T-Int 1-µ ....................................................................................153 Hypothesis test: T-Test µ1 -µ2.....................................................................................154 Hypothesis test: Z-Test 1-µ .......................................................................................156 Tips & Tricks - Inference..........................................................................158 Importing from a frequency table ...............................................................................158

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 Using Matrices on the hp 39g+................................................................170 The MATRIX Catalog ................................................................................................170 Matrix calculations in the HOME view..........................................................................171 Solving a system of equations....................................................................................172 Finding an inverse matrix ...........................................................................................174 The dot product ..........................................................................................................175 Using Lists on the hp 39g+......................................................................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 Using the Notepad Catalog......................................................................178 Aplet notes vs. independent notes .............................................................................178 Independent Notes and the Notepad Catalog..........................................................179 Transferring notes using IR........................................................................................180 Editing software..........................................................................................................180 Software .........................................................................................................................181 For the hp 38g, hp 39g & hp 40g................................................................................181 For the hp 39g+..........................................................................................................181 Creating a Note..............................................................................................................182 Locking ALPHA mode .................................................................................................182 The CHARS view .........................................................................................................183 Corrupting notes.........................................................................................................183 Using the aplet Sketchpad.......................................................................184 Adding text to a sketch...............................................................................................184 The DRAW menu ...........................................................................................................185 DOT+..........................................................................................................................185 LINE............................................................................................................................185 BOX............................................................................................................................185 CIRCLE ......................................................................................................................186 Cut and paste (sort of…)............................................................................................186 Capturing the PLOT screen .......................................................................................187

Creating a copy of a Standard aplet............................................................................190 Copying and adding to the Function aplet..................................................................190 Copying and adding to the Stats aplet .......................................................................191 Some examples of saved aplets ..................................................................................192 The Triangles aplet.....................................................................................................192 The Prob. Distributions aplet......................................................................................192 The Transformer aplet................................................................................................194 Copying from hp 39g+ to hp 39g+ via the infra-red link............................................197 The infra-red port........................................................................................................197 Time out......................................................................................................................199 Attached programs.....................................................................................................199 Downloading aplets from the Internet.........................................................................200 Finding aplets .............................................................................................................200 Organizing your collection..........................................................................................201 Software for the hp 38g, hp 39g & hp 40g .................................................................202 Software for the hp 39g+............................................................................................203 The HPGComm Connectivity Program ......................................................................204 Deleting downloaded aplets from the calculator ........................................................207 Saving notes, aplets and sketches via the Connectivity Kit.......................................208 Capturing screens using the Connectivity Kit.............................................................210 Editing Notes with the Aplet Development Kit............................................................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 in the SETVIEWS command...............................................................216 The ‘Start’ entry ..........................................................................................................217 Example aplet #1........................................................................................................217 Example aplet #2........................................................................................................222 Example aplet #3........................................................................................................224 Example aplet #4........................................................................................................226 Programming Commands........................................................................233 The Aplet commands ....................................................................................................233 CHECK & UNCHECK................................................................................................233 SELECT......................................................................................................................233 SETVIEWS.................................................................................................................234 The Branch commands.................................................................................................234 IF THEN [ELSE] END...............................................................................................234 CASE..........................................................................................................................234 IFFERR THEN [ELSE] END......................................................................................234 RUN............................................................................................................................235 STOP..........................................................................................................................235 The Drawing commands...............................................................................................235 ARC............................................................................................................................235 BOX............................................................................................................................235 ERASE .......................................................................................................................236 FREEZE .....................................................................................................................236 LINE............................................................................................................................236 PIXON and PIXOFF ..................................................................................................236 TLINE .........................................................................................................................236

The Loop commands ....................................................................................................237 FOR = TO [STEP] END.....................................237 DO UNTIL...................................................................................................................237 WHILE REPEAT.........................................................................................................237 BREAK .......................................................................................................................238 The Matrix commands...................................................................................................238 EDITMAT....................................................................................................................238 REDIM........................................................................................................................238 The Print commands.....................................................................................................239 PRDISPLAY ...............................................................................................................239 PRHISTORY...............................................................................................................239 PRVAR .......................................................................................................................239 The 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 of functions.......................................................................................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 % ................................................................................................................................250 %CHANGE.................................................................................................................251 %TOTAL.....................................................................................................................251 RAD DEG ................................................................................................................251 ROUND ......................................................................................................................252 SIGN...........................................................................................................................252 TRUNCATE................................................................................................................253 XPON .........................................................................................................................253 The ‘Stat-Two’ group of functions...............................................................................254 PREDY .......................................................................................................................254 PREDX .......................................................................................................................254 The ‘Symbolic’ group of functions ..............................................................................255 The = ‘function’...........................................................................................................255 ISOLATE ....................................................................................................................255 LINEAR?.....................................................................................................................256 QUAD .........................................................................................................................256 QUOTE.......................................................................................................................257 The | function..........................................................................................................258

The ‘Trigonometric’ & ‘Hyperbolic’ groups of functions...........................................259 COT, SEC etc.............................................................................................................259 EXP ............................................................................................................................260 ALOG..........................................................................................................................260 EXPM1 .......................................................................................................................260 LNP1...........................................................................................................................261 The ‘Calculus’ group of functions ...............................................................................261 The ‘Complex’ group of functions...............................................................................262 ABS ............................................................................................................................263 SIGN...........................................................................................................................263 ARG............................................................................................................................264 CONJ..........................................................................................................................264 IM & RE ......................................................................................................................264 The ‘List’ group of functions........................................................................................265 CONCAT ....................................................................................................................265 ∆LIST..........................................................................................................................266 MAKELIST..................................................................................................................266 πLIST..........................................................................................................................267 POS............................................................................................................................267 SIZE............................................................................................................................268 ΣLIST..........................................................................................................................268 REVERSE ..................................................................................................................268 SORT..........................................................................................................................268 The ‘Loop’ group of functions .....................................................................................269 ITERATE ....................................................................................................................269 RECURSE..................................................................................................................270 Σ (SUMMATION)........................................................................................................270 The ‘Matrix’ group of functions....................................................................................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

POLYCOEF................................................................................................................277 POLYEVAL.................................................................................................................277 POLYFORM ...............................................................................................................278 POLYROOT................................................................................................................279 The ‘Probability’ group of functions............................................................................280 COMB.........................................................................................................................280 The ! function..............................................................................................................280 PERM .........................................................................................................................281 RANDOM....................................................................................................................281 RANDSEED................................................................................................................281 UTPN..........................................................................................................................282 UTPC..........................................................................................................................283 UTPF ..........................................................................................................................283 UTPT ..........................................................................................................................283 Appendix A: Worked Examples...............................................................284 Finding the intercepts of a quadratic ..........................................................................284 Method 1 - Using the QUAD function in HOME.............................................................284 Method 2 - Using the Function aplet. .........................................................................284 Method 3 - Using the POLYROOT function..................................................................285 Finding complex solutions to a complex equation....................................................285 Method 1 - Using the QUAD function.........................................................................285 Method 2 - Using POLYROOT...................................................................................285 Finding critical points and graphing a polynomial....................................................286 Solving simultaneous equations. ................................................................................288 Method 1 - Graphing the lines....................................................................................288 Second method - using a matrix.................................................................................288 Third method - using the 3x3 Solver aplet ................................................................289 Expanding polynomials ................................................................................................290 Exponential growth .......................................................................................................291 Solution of matrix equations........................................................................................293 Inconsistent systems of equations .............................................................................294 Using the RREF function............................................................................................294 Finding complex roots..................................................................................................295 Analyzing vector motion and collisions......................................................................296 Circular Motion and the Dot Product...........................................................................297 Inference testing using the Chi2 test...........................................................................299 Appendix B: Teaching Calculus with an hp 39g+..................................301 Investigating n y x = for n an integer..........................................................................301 Domains and Composite Functions............................................................................302 Gradient at a Point.........................................................................................................303 Gradient Function..........................................................................................................304 The Chain Rule ..............................................................................................................305 Optimization...................................................................................................................305 Area Under Curves........................................................................................................306 Fields of Slopes and Curve Families...........................................................................306 Inequalities.....................................................................................................................307 Rectilinear Motion .........................................................................................................307 Limits ..............................................................................................................................307 Piecewise Defined Functions.......................................................................................308 Sequences and Series ..................................................................................................308 Transformations of Graphs ..........................................................................................309

Introduction....................................................................................................................310 What is a CAS? ..........................................................................................................310 What is the difference between the hp 39g, hp 40g & hp 39g+?...............................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 The PUSH and POP commands................................................................................318 Pasting to an aplet......................................................................................................319 Evaluating algebraic expressions...............................................................................320 Using functions in the CAS..........................................................................................321 E.g. 1 Using LIMIT....................................................................................................321 E.g. 2 Factorizing expressions .................................................................................322 E.g. 3 Solving equations...........................................................................................323 E.g. 4 Solving simultaneous equations ....................................................................323 E.g. 5 Solving a simultaneous integration................................................................325 E.g. 6 Defining a user function .................................................................................327 On-line help....................................................................................................................328 Configuring the CAS .....................................................................................................329 Tips & Tricks - CAS .......................................................................................................330 REGISTER YOUR PRODUCT AT: www.register.hp.com THIS MANUAL AND ANY EXAMPLES CONTAINED HEREIN ARE PROVIDED "AS IS" AND ARE SUBJECT TO CHANGE WITHOUT NOTICE. HEWLETT-PACKARD COMPANY MAKES NO WARRANTY OF ANY KIND WITH REGARD TO THIS MANUAL, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY, NON-INFRINGEMENT AND FITNESS FOR A PARTICULAR PURPOSE. HEWLETT-PACKARD CO. SHALL NOT BE LIABLE FOR ANY ERRORS OR FOR INCIDENTAL OR CONSEQUENTIAL DAMAGES IN CONNECTION WITH THE FURNISHING, PERFORMANCE, OR USE OF THIS MANUAL OR THE EXAMPLES CONTAINED HEREIN. © Copyright 1993-1998, 2005 Hewlett-Packard Development Company, L.P. Reproduction, adaptation, or translation of this manual is prohibited without prior written permission of Hewlett-Packard Company, except as allowed under the copyright laws. Hewlett-Packard Company 4995 Murphy Canyon Rd, Suite 301 San Diego, CA 92123 Acknowledgements Hewlett-Packard would like to thank the author Colin Croft. Printing History Edition 1 February 2004 Edition 2 September 2005 Notice

also aimed at users of the hp 39g and hp 40g. These are very sophisticated calculators, having more capabilities than a mainframe computer of the ’70s, so don’t expect to come to grips with its abilities in one or two sessions. However, if you persevere you will gain efficiency and confidence. The majority of readers may only have used a Scientific calculator before so explanations are as complete as possible. However it is not the purpose of this book to teach mathematics so knowledge will be assumed. Those who are already familiar with another brand or type, may find that a quick skim is sufficient, perhaps with detailed reading of the “Tips and Tricks” sections. The impact these calculators are having on the topics taught and even more, the way they are taught is proving to be profound. The inventiveness and flexibility of teachers of mathematics is being stretched to the limit as we gradually change the face of teaching in the light of these machines. For those concerned with the impact of a graphical calculator on the ‘fundamentals’ of mathematics, it should be recalled that the same fears were held for scientific calculators when they were introduced earlier. History has shown that these fears were generally groundless. Students are learning topics in high school that their parents did not cover until university years. In particular, the scientific calculator proved to be a great boon to students of middle to lower ability in mathematics, relieving them of the burden of tedious calculations and allowing them to concentrate on the concepts. This is also the case with graphical calculators. This manual is not intended to replace the one supplied with the calculator. It fills in the holes, and also provides tips to make your work smother and more confident. Happy calculating. About the author Colin Croft is a teacher at St. Hilda’s Anglican School for Girls in Perth, Western Australia. Colin has worked extensively with Hewlett Packard on the graphic calculator family of which the hp 39g+ is a member, and was part of the team which created the hp 39g & hp 40g in 2000. He maintains an extensive website of material for the hp 39g/40g/39g+ series called The HP Home view, at http://www.hphomeview.com.

MAANNUUAALL It has been attempted to design this manual to cover the full use of the hp 39g+ calculator. This means explanations which will be useful to anyone from a student who is just beginning to use algebra seriously, to one who is coming to grips with advanced calculus, and also to a teacher who is already familiar with some other brand of graphic calculator. Readers may encounter one of two difficulties. Firstly, the information in here will be beyond the needs of some readers and secondly, the explanations may be too detailed for more advanced users. For students who don’t yet need the more advanced capabilities, suggestions on which parts of the manual to read are given below. For more advanced users, it is suggested that you read the sections on the Function, Sequence, Statistics, Inference, Finance and Solve aplets, and also read the ‘Tips and Tricks’ sections which follow many of the chapters. Early High School Typical topics covered include… Solving linear equations, graphing linear equations and possibly simple quadratics, examining number patterns, multiplying polynomials, factoring simple polynomials, calculations involving powers ( 2 3 , ,...... x x ), square roots, cube roots, order of operations, positives and negatives, scientific notation, indices, systems of equations and inequalities, parallel and perpendicular lines, dividing polynomials, solving quadratics, rational expressions and equations. Suggestions… Read about the Function aplet in full, ignoring any sections that seem to advanced. Learn about Intersection and Root in the menu and make sure you know how the menu, Autoscale and PLOT SETUP work. Learn to Build Your Own in the NUM view because it lets you find values for rules easily. Learn to use the HOME view for routine calculations, the MODES view, and how to use the calculator’s memories. Read about the Solve aplet and how to use it to solve equations. Read about the Statistics aplet and how to use it to find

functions ROUND, POLYFORM and POLYROOT. Make sure you know how to save and transfer aplets. Learn about the Sketch view and the Notes catalog for a bit of fun. Pre-Calculus Typical topics covered include… Solving complex linear and non-linear simultaneous equations, and trig, exponential & complex quadratic equations, factoring of any quadratic, use and re-arrangement of formulas, indices, trigonometry, some statistics, absolute value and greatest integer functions, matrices, logarithms and parametric equations. Suggestions… Cover all of the material mentioned for high school students. Read the suggestions on how to deal with graphs whose shape you don’t know in advance. Learn how to use the Parametric aplet. Your teacher might best advise on which portions of the Statistics aplet will be relevant to you. In the MATH menu, also learn about functions CEILING, ABS and FLOOR, POLYCOEF, and POLYEVAL. Read the section covering the Matrices catalog and the functions DET, RREF, INVERSE, and TRN. Calculus Topics covered here will vary according to which course students undertake but there are very few skills covered in this manual which will not be of use at some time. It is suggested that students skim the whole manual, and then re-read it at intervals as material is covered in their courses and they begin to see which parts of the manual are particularly relevant.

ONN BBUUTTTTO ONN?? Let’s begin by looking at the fundamentals - the layout of the keyboard and which are the important keys that are used frequently. The sketch below shows most of the important keys. These are the ones which control the operation of the calculator - others are used to do calculations once the important keys have set up the environment to do it in. Examples of the effects of each of these keys and many more are shown on the pages that follow. These six screen keys change their function in different contexts. The bar at the bottom of the screen labels them. Check this bar for special functions in any given context. The SYMB key nearly always takes you to a view in which you can enter equations The PLOT key displays the graph view for any given environment. HOME is where you will do most of your calculations. It is shared by all the aplets and oversees them all. The NUM key gives you a tabular view of your function, sequence or data. These are the cursor (or arrow) keys. They let you move within a window. The VIEWS key gives a different menu in each aplet. It can be very useful, and is always worth checking. The APLET key is central. This key allows you to choose which mathematical environment you wish to working. So where is the ON button?

you press each of the keys shown on the previous page. Exactly what you see depends on which aplet is active at the time. The aplet used below to illustrate this is the Function aplet, which is used to graph and analyze Cartesian functions. Notice how the meanings of the row of blank screen keys under the screen changes in different views. The SYMB key - in this case it is set to graph the function 3 2 7 6 y x x x = − − + . The PLOT key - used to graph the function. The NUM key showing a tabular view of the function. The APLET key is used to choose which aplet is active. There are 10 aplets provided with the calculator and more can be downloaded from the internet. The MATH key gives access to more than a hundred extra functions, grouped by category. The view shown right is currently showing the Probability section.

OTTAATTIIO ONN CCO ONNVVEENNTTIIO ONNSS There are a number of types of keys/buttons that are used on the hp 39g+. SSoom mee eesssseennttiiaall kkeeyyss The basic keys are those that you see on any calculator including scientific ones, such as the numeric operators and the trig keys. Most of these keys have two or more functions. Take for example the COS key shown left. If you just press the key, you get the COS function. However above left of the key and below right you will see two additional meanings assigned via the SHIFT and ALPHA keys. All references to keys, whether they need the SHIFT key or not, are written in this typeface: KEY. The SHIFT key The SHIFT key gives you the second function for each key. In the case of the COS key, the second function is ACOS, sometimes referred to as arc-cos or cos-1 or inverse cos. Most keys have these second functions that are obtained via the SHIFT key. When I want you to use one of these keys that needs to have the SHIFT key pressed first I usually won’t say so. It seems to me that you’re intelligent enough to work out for yourself when the SHIFT key needs to be pressed.

The next modifier key is the ALPHA key. This is used to type alphabetic characters, and these appear in orange just below most keys. The Screen keys A special type of key unique to the hp 39g+ and family is the row of blank keys directly under the screen. These keys change their function depending on what you are doing at the time. The easiest way to see this is to press the APLET key. As you can see right, the functions are listed at the bottom of the screen. All you have to is to press the key under the screen definition you want to use. All references to keys of this type are shown as images of the label. For example, if I want you to press the key under the SORT label it would be written as . Do it now and you’ll see the screen shown on the right. Notice that the keys have now changed function. Press the one under . Pop-up menus & short-cuts Sometimes pressing a key pops up a menu on the screen as you just saw. You use the up/down arrow keys to move the highlight through the menu and make choices by pressing the ENTER key. Choices that are listed in a menu will usually be written using italics. As an example, I might say to press and choose Chronologically. The manual you are given with your calculator uses a different convention. As mentioned before, the third way a key can be used is to get letters of the alphabet. This is not so that you can write letters to your friends (although you can do that with the Notepad) but so that you can use variables like X and Y or A and B. The key above the SHIFT key labeled ALPHA is used to type in letters of the alphabet. Lower case letters are obtained by pressing the SHIFT key before the ALPHA key. If you want to type in more than just a single letter, hold down the ALPHA key. Unfortunately, this doesn't work for lowercase.

out of the menu from the previous screen. Press the HOME key to see the screen on the right. Yours may not be blank like mine but that doesn’t matter. Press 12 and then press the screen key labeled . Now press the ALPHA key and then the alphabetic D key (on the XTθ key). Finally, press the ENTER key. Your screen should look like mine on the right. You have now stored the value 12 into memory D. Each alpha key can be used as a memory. You can also use these memories in calculations. Type in the following (not forgetting the ALPHA key before the D)…. (3+D)/5 ENTER The calculator will use the value of 12 stored earlier in D to evaluate the expression (see right). In case you haven’t worked it out for yourself, the / symbol comes from the divide key ( ) and the * symbol from the multiply key. More information on memories and detailed information on the HOME view in general is given on pages 45. The calculator also comes with a large number of mathematical functions that are very useful. They can all be obtained via menus through the MATH key. Try pressing the MATH key now and you should find your screen looks like the screen shot on the right. The MATH menu is covered in detail on pages 243 but we will have a brief look now. The left side of the menu lists the categories of functions. As you use the up/down arrows to scroll through the topics, you’ll see the actual list on the right change. Move down through the menu until you reach Prob. (short for Probability) and then one step more and you’ll find yourself back at Real. Now press the right arrow key and your highlight will move into the right hand menu (see above). Move the highlight down through this menu until you reach Round. Press ENTER.

ROUND( entered in the display as shown right. You can also achieve the same effect by using ALPHA to type in the word letter by letter. Some people prefer to do it that way. Now type in: 4+D/18,3) and press ENTER As you can see, the effect was to round off the answer of 4.666666.. to 3 decimal places. There are shortcuts for obtaining things from the MATH menu that are covered later (see page 47).

effectively mean that it is not just one calculator but nine (or more), changing capabilities according to which aplet is chosen. The best way to think of these aplets is as “environments” or “rooms” within which you can work. Although these environments may seem dissimilar at first, they all have things in common, such as that the PLOT key produces graphs, that the SYMB key puts you into a screen used to enter equations and rules, and that the NUM key displays the information in tabular form. There are ten standard aplets available via the APLET key. More can be created by you or obtained via the Internet (see pages 212 & 200) The Function aplet (see page 51) Provides f(x) style graphs, calculus functions etc. It will not only graph but find intercepts, intersections, areas and turning points. The Inference aplet (see page 150) Allows the investigation of inferential statistics via hypothesis testing and confidence intervals. This was not available on the hp 38g, the original calculator upon which the hp 39g+ was based. The Parametric aplet (see page 100) Handles x(t), y(t) style graphs. Can also be used to help with vector motion. The Finance aplet (see page 160) Performs calculations involving time/value of money. The Polar aplet (see page 106) Handles r(θ) style graphs. Quite apart from their mathematical use, they produce some really lovely patterns!

This is a teaching aplet, allowing the student to investigate the properties of quadratic graphs. The Sequence aplet (see page 107) Handles sequences such as 1 1 2 3; 2 n n T T T − = + = or 1 2n nT − = . Allows you to explore recursive and non-recursive sequences. The Solve aplet (see page 113) Solves equations for you. Given an equation such as ( ) 2 A r r h π = + it will solve for any variable if you tell it the values of the others. The Statistics aplet (see page 122 & 132) Handles descriptive statistics really well. Data entry is easy, as is editing. It analyzes univariate and bivariate data, drawing scatter graphs, histograms and box & whisker graphs. The Trig Explorer aplet (see page 167) This is a teaching aplet, allowing the student to investigate the properties of sine and cosine graphs. The Function aplet is probably the easiest to understand and also the one you will use most often, so we will have a very quick look at this aplet. Some typical aplet views The APLET key is used to list all the aplets and start, reset or save them. The SYMB view is used to enter equations…. It can store up to ten functions. The NUM view shows the function in table form…

graph… The key gives access to a number of other useful tools allowing further analysis of the function. Although these views are superficially different in other aplets, the basic idea is usually similar. Having said that aplets are best thought of as “working environments”, it is equally true that aplets are essentially programs, with the standard ones simply being built into the calculator. This is a programmable calculator, having its own programming language and able to perform quite sophisticated tasks. Unless you particularly want to learn about the programming language, there is no reason why you should worry about it. The standard aplets will cover all of your normal requirements in mathematics. However one of the great strengths of the hp 39g+ is its ability to “download” additional aplets from other calculators, from a PC or Mac and from the Internet. See page 200. A cable and software were provided with your hp 39g+ which you can use to connect your PC or Mac to your calculator and then download aplets from the computer to the calculator or to save your work to the computer. If you have an hp 38g, hp 39g or hp 40g then you need to buy the cable separately. If you are using an hp 39g+ then the cable connects to the USB port on your computer. If you are using an hp 39g or an hp 40g then you will need to purchase the cable and download the software from Hewlett-Packard’s website. More information on this can be found on pages 200 - 226. Calculator Tip Search on the web using the key word “hp 39g” and you will find a variety of sites which contain information and aplets.

takes only seconds using the built in infra-red link at the top of each calculator. This is exactly like the remote control of a VCR, and allows two calculators to talk to each other. In the interests of security in examinations the distance over which they can communicate is limited to about 8 - 10cm (about 3 - 4 inches). See page 197 for details on this process. Aplets are available to do many mathematical tasks such as statistical simulations, time series analysis and many tasks called for in calculus, physics and chemistry. There are a number of web sites which offer aplets. The Hewlett-Packard site is found at… http://www.hp.com/calculators/ (follow the links to graphical calculators and then to the library of aplets) In addition to this you should check the site called The HP HOME view which can be found at… http://www.hphomeview.com (contains not only aplets and games, but also a huge amount of detailed information on the calculator.) Calculator Tip The aplets for an hp 39g, hp 40g and hp 39g+ are interchangeable but not those of an hp 38g. If you load an aplet from an hp 38g onto an older model then the download will appear to be successful but the calculator will “crash” when the aplet is run. The entire topic of aplets is discussed in more detail in the chapter entitled “Using, copying & creating aplets” on page 189.

thought of as a scratch pad for all the others. This is accessed via the HOME key (just below the APLET key) and is the view in which you will do your routine calculations such as working out 5% of $85, or finding √35. The HOME view is the view that you will most often use, so we will explore that view first. W Whhaatt iiss tthhee HHO OM MEE vviieew w?? This is the HOME base for the calculator. All other aplets can be accessed from it and affect it to varying degrees, and all mathematical functions are available in this view. Learn to use this view as efficiently as possible, since a great deal of work will be done here. We will explore the HOME view in the following order: 1. Exploring the Keyboard 2. Angle and numeric settings 3. Memory management 4. Fractions on the hp 39g+ 5. The HOME History 6. Storing and retrieving memories 7. Referring to other aplets from the HOME view 8. An introduction to the MATH menu 9. Resetting the calculator

It is worth familiarizing yourself with the mathematical functions available on the keyboard. If we examine them row by row, you will see that they tend to fall into two categories - those which are specific to the use of aplets, and those which are commonly used in mathematical calculations. The screen keys The first row of blank keys are context defined. The reason they have no label is that their meaning is redefined in different situations - they are the ‘screen keys’. The current meaning of each key is listed in the row of boxes at the bottom of the screen. A common abbreviation used for these keys is SK1 or SK2 etc (for “screen key 1” ). In the PLOT view shown above, some of the screen keys are labeled, such as the key. When you press this the row of screen keys labels appear or disappear. To see another view where all the keys are in use, change to the APLET view. Calculator Tip Develop the habit of checking the screen to see if any of those keys have been given meanings. In many views, the screen keys have been set up with useful shortcuts and functions. Aplet related keys The next two rows of keys and part of the third are mainly aplet related, so we’ll deal with them as a group. The arrow keys The arrow keys on the right are used in most views, usually to move the cursor (a small cross) or the highlight around on the screen.

various different aplets available. Everything in the calculator revolves around aplets, which you can think of either as miniature programs or as environments within which you can work. The hp 39g+ comes with ten standard aplets - Finance, Function, Inference, Parametric, Polar, Quadratic Explorer, Sequence, Solve, Statistics and Trig Explorer. Which one you want to work with is chosen via the APLET key. Calculator Tip The name of the active aplet is shown at the top of the screen, as above. It is important to bear this in mind because the angle and numeric settings are tied to the active aplet. Changing aplets may therefore cause these settings to change in HOME too. See page 35 for more details on this. In addition to the standard ten, covered in great detail in the chapters following, many more aplets are available from the Internet written by other programmers. Once these are downloaded into your calculator they can also be accessed via the APLET key. For more detail on this type of aplet, see the brief summary later in this section, and the chapter entitled “Programming the hp 39g+” on page 212. The SYMB, PLOT and NUM keys The SYMB, PLOT and NUM keys are used within aplets to move from view to view. In most aplets the PLOT view shows the graph, the SYMB view shows the equations and the NUM view shows the equations in tabular (numeric) format. The VIEWS key pops up a menu from which you can choose various options. Part of the VIEWS menu for the Function aplet is shown right. See page 92 for more information.

Intro to the VIEWS menu Firstly, within the standard aplets (Function, Sequence, Solve etc.) it provides a list of special views available to enhance the PLOT view. For example the standard PLOT screen provides a graph, but the VIEWS menu lets you use a split screen such as shown right. Information on the VIEWS menu is given in the chapter dealing with the Function aplet. In addition to this, the VIEWS key also has a critical role when using aplets which have been downloaded from the Internet. When a programmed aplet is created for the hp 39g+, a menu is provided by the programmer to let you control and use it. This menu is tied to the VIEWS key, replacing the menu normally found on the key. For example, the snapshot shown right is of a VIEWS menu taken from an aplet designed to analyze and graph Time Series data. The next important key is the HOME key. It allows you to change into the HOME view from wherever you are. Above it is the MODES key, accessed by pressing SHIFT first. Far more detailed information on these two views follows later. The VARS key The VARS key is used (mainly by programmers) to access all the different variables stored by the calculator. Shown right are two views of the VARS screen, the first from the HOME list showing the graphic variables (memories) G1, G2…. and the next from the APLET list showing some of the variables in the set controlling PLOT. The VARS key is not generally used much, and you may not have followed this explanation. Don’t worry. The VARS key is mostly used by programmers who want to produce aplets of their own and is discussed in more detail in the chapter entitled “Programming on the hp 39g+” on page 212. Uses are also detailed in the Function aplet’s “Tips and Tricks” section on page 69.

library of mathematical functions. The more common functions have keys of their own, but there is a limit to the number of keys that one can put on a calculator before it takes too long to find the key required. The MATH menu lists all those functions that would not fit onto the keyboard plus some which also appear on the keyboard. Shown in the screen snapshot above is a small selection of the total list. For a listing of almost all the functions, with examples of their use, see the chapter entitled “The MATH Menu” on page 243. Most of the keys have another function in light blue above the key. The hp 39g+ gets twice the action from each key by having this second function. The second function is accessed via the SHIFT key on the left side of the calculator. Although this book will sometimes tell you explicitly to press this key, in most cases it will be assumed that you are intelligent enough to work out for yourself when it is necessary to press it. The ALPHA key gives alphabetical characters, shown below right of most keys. Pressing SHIFT ALPHA gives lower case. If you press and hold down the ALPHA key you can ‘lock’ alpha mode but this doesn’t work for lower case. Many people use this to type in functions by hand rather than using the MATH menu. Calculator Tip You can use the ALPHA button to type in functions by hand instead of going through the MATH menu. This is often faster.

The SETUP views, above PLOT, SYMB and NUM, are used to customize their respective views. For example, the PLOT SETUP screen controls things like axes, labels etc. Their use changes in different aplets, so for more information see the explanations in the chapters dealing with the various aplets, particularly with the Function aplet on page 55. In particular, the SYMB SETUP key is only used in one place, which is to choose the data model for bivariate statistics in the Statistics aplet. It is not available in the other aplets and trying to access it will result only in a quick flash of an exclamation mark on the screen to say “You’ve done something wrong!”. Information on the use of the SKETCH and NOTE keys (above APLET and VEIWS) can be found in the chapters “Using the Sketchpad” and “Using the Notepad Catalog” on pages 184 & 178. Briefly, any aplet (except for the Quad & Trig Explorer aplets) has a note and a sketch associated with it, which are usually blank unless you have added to them. The main use for them comes with aplets downloaded from the Internet. Instructions for using the aplet are sometimes included with the aplet in note form, and sometimes as an accompanying sketch. The MODES view The MODES view (see right) controls the numeric format used in displaying numbers and angles in aplets. At the bottom of the screen you will see that one of the screen keys has been given the function . Pressing this key pops up a menu of choices from which you can select the option which suits you. The default angle setting is radians. Calculator Tip If you don’t want to use the menu then, rather than pressing , highlight the field and then press the ‘+’ key repeatedly. This will cycle through the choices without popping up a menu. This can be much faster if the menu has only a few choices.

on the right. Standard is probably the best choice in most cases, although it can be a little annoying to constantly have 12 significant figures displayed. In Standard mode, very large and very small numbers are displayed in scientific notation. The Fixed, Scientific and Engineering formats all require you to specify how many decimal places to display. The screenshot right shows Fixed 4, which rounds everything off to 4 decimal places. Of course, you can change the 4 to any other number you want. A setting of Scientific notation ensures that any results are displayed in scientific notation. Of course, the calculator’s idea of scientific notation may not be the same as yours. Since the calculator has no way of displaying powers as superscripts, a result of 4 3 203 10 ⋅ × has to be displayed as 3.203E4. The alternative of Engineering notation is very similar to Scientific, except that powers are always displayed as multiples of 3. This is done to allow easy conversion in the Metric system, which also works in multiples of 1000. The screens right show the same two numbers displayed as in turn as; Fixed 4, Scientific 4 and Engineering 4. Calculator Tip If you have Labels turned on when you in (or out) on a graph then you may end up with axes whose numeric labels are horrible decimals (see below right).

deceptive to use and is discussed in more detail on page 40. The next alternative in the MODES view of ‘Decimal Mark’ controls the character which is used as a decimal point. In some countries a comma is used instead of a decimal point. If you opt to use a comma rather than a full stop then any places where a comma would normally be used (such as in lists) will swap to using a full stop. Any functions which might normally have terms separated by a comma will use a full stop instead. For example, ROUND(3.456,2) will become ROUND(3,456.2). Moving back to our tour of the keyboard, the next key is the ENTER key. This is used as an all purpose “I’ve finished - do your thing!” signal to the calculator. In situations where you would normally press the ‘=’ key on most calculators, press the ENTER key instead. The ANS key Above the ENTER key is the ANS key. This can be used to retrieve the final value of the last calculation done. An example of this is shown right. If you are not confident about using brackets, then the ANS key can be quite useful. For example, you could calculate the value of 2 3 3 7 2 2 5 −× + by using brackets… …. or you could use the ANS key.

function. This is discussed on page 43. The negative key Another important key is the (-) key (shown right). If you want to calculate the value of (say) 2 ( 9) −−− then you must use the (-) key before the 2 and the 9 rather than the subtract key. If you press the subtract key before the 2 instead of the (-) key, then the calculator will enter instead: Ans - 2, meaning, subtract 2 from the previous answer. Similarly, entering ‘subtract, subtract 9’ instead of ‘subtract (-) 9’ will give an error message of “Invalid syntax”, meaning it does not make mathematical sense to have two subtract signs instead of a subtract of a negative. The reason for the Ans - 2 is that a subtract cannot start a ‘sentence’ in mathematics, while a negative sign can. Since the subtract can’t come first, the calculator decides that you must have intended to subtract from the previous answer. Hence the sudden appearance of an Ans. This occurs at other times too. One of the most common The CHARS key The next important key is the CHARS key (above VARS). This key is used to access all of the many characters that are used occasionally but not often enough to bother putting on the keyboard. Pressing SHIFT CHARS will pop up the screen above right. One of the screen keys is set to be a ‘Page Down’ key ( ), and will give access to two more pages of characters. These special characters are obtained by pressing the screen key labeled . You can press as many times as you need to in order to obtain multiple characters. When you have as many as you need, press the key.

The next important key is the DEL key at the top right of the keyboard. This serves as a backspace key when typing in formulas or calculations, erasing the last character typed. If you have used the left/right arrow keys to move around within a line of typing, then the DEL key will delete the character above the cursor. The CLEAR key above DEL can be thought of as a kind of ‘super delete’ key. For example, if pressing DEL would erase one function only in the SYMB view then CLEAR will erase the whole set. Calculator Tip Another use for the DEL key is to restore factory settings. For example, if you move back into the MODES screen and change to Degree mode, then pressing the DEL key will restore the default of Radians. Pressing CLEAR in the MODES view would restore factory settings to all the entries. This is particularly useful in the PLOT SETUP view. The remaining keys of LIST, MATRIX, MEMORY , NOTEPAD and PROGRAM have special chapters of their own.

meerriicc sseettttiinnggss It is critical to your efficient use of the hp 39g+ that you understand how the angle and numeric settings work. For those upgrading from the hp 38g this is particularly important, since the behavior is significantly different. On the hp 39g+, when you set the angle measure or the numeric format in the MODES view, it applies both to the aplet and to the HOME view. However, this setting applies only to the currently active aplet (the one highlighted in the APLET view). This means that if you change active aplet then these settings may change also, not only for the aplet but in the HOME view too. For example, suppose you have been performing trig calculations in the HOME view with the Function aplet being currently active, and have set the angle measure to DEG. If you were to now change to the Solve aplet in order to solve an equation then the settings would revert to those of the Solve aplet; probably be radians unless you had also changed those as well. Radian measure is the default for all aplets and thus also for HOME unless you change it. Performing the same calculation in HOME would now give a different result (see right). Although this may seem to be a strange way of doing things, there is actually a very good reason for it. The hp 38g did not behave this way and it caused a number of users to have difficulties as you'll see on the next page.

the Function aplet as shown. The default setting for the Function aplet is radians, so if we set the axes to extend from -π to π , the graph would look as shown right. In the PLOT view shown, the first positive root has been found (see page 63) as x=1.0471… On the hp 39g+, if we now change to the HOME view and perform the calculation shown right, we expect that the answer should be zero, as indeed it is. However, this is only the case because the angle measures of HOME and the Function aplet agree. The problem was that on the hp 38g the default setting for the Function aplet was radians, while HOME had a default setting of degrees and its setting was independent of those of the aplet. This meant that a calculation such as the one above would give incorrect results, and caused considerable confusion to some students. It even resulted in users returning their hp 38g to dealers as ‘faulty’! The only drawback of this method is that you might change aplets and forget that it may also change settings. For this reason, the name of the active aplet is shown at the top of the HOME view as a reminder. On the hp 39g+ you can see that if we turn Labels on and then PLOT, the numeric mode also affects the axis labels.

displayed using the SHOW command. Calculator Tip Under the system used on the HP39+, if you want to work in degrees then you will need to choose that setting in the MODES view and possibly set it again if you change to another aplet. Some people choose to go through and change the setting on all the aplets at once so that they don’t have to remember that it might change. However, if you the aplet the default setting will return. M Meem moorryy M Maannaaggeem meenntt One of the major complaints about the hp 38g was its memory - mainly the lack of it, but also the inability to control or manage it. This problem has been addressed on the hp 39g+ in two ways. Firstly, the hp 39g+ has over ten times the useable memory of the hp 38g. At 232 Kb (vs. only 23 Kb), there are very few users who will come close to filling the hp 39g+. Depending on size, there is enough room for at least fifty aplets, or for over 10,000 data points. The MEMORY MANAGER view In addition to extra memory, the hp 39g+ supplies a better way to control it through the MEMORY MANAGER view. If you press the MEMORY key you will see the view shown right. Scrolling through it will show you exactly how the available memory is currently being used. The remaining memory, in Kb, is shown at the top right of the screen. This view gives an overview of the memory. For detailed management the key is provided. Pressing on any entry will take you a relevent screen in which you can delete entries no longer needed. For example, with the highlight on Aplets, pressing will take you to the APLET view (right), where you can choose to delete or reset any aplets no longer required.

has a number of extra aplets. Two of them, Statistics2 and Statistics3 are simply copies of the normal Statistics aplet containing data that I did not want to lose. The top two aplets Curve Area and Coin Tossing are teaching aplets that I have downloaded from the internet. Downloaded aplets & memory If you use teaching aplets that you download from the internet via the Connectivity Kit, or which are supplied to you by your teacher via the infra-red link on your hp 39g+, then you need to bear in mind that most of them have ‘helper’ programs that aid them in performing their tasks. In the screens shown right you can see some of the programs which are attached to these two teaching aplets. The convention which most hp 39g+ programmers follow is to name these ‘helper’ programs in a way that associates them clearly with the parent aplet. The memory associated with these programs is not included in that shown for the aplet in the APLET view but will not usually be a large amount. Calculator Tip The reason for this naming convention for ‘helper’ programs is that when you delete the parent aplet in the APLET view the ‘helper’ programs are not automatically deleted with it. You must change to the Program Catalog view and delete them manually after you have finished with the teaching aplet and deleted it in the APLET view. If you don’t do this then they will continue to take up memory on the calculator. Even on the hp 39g+ this is not infinite and too many left over programs will eventually cause problems. As mentioned on the earlier, pressing in the MEMORY MANAGER screen takes you to a relevent view showing greater detail. For example, the Matrices entry right shows 0.1 Kb in use. Pressing will take you to the MATRIX CATALOG view which shows exactly where the memory is being used and allows you to delete any or all of the matrices, or you can enter the view in the normal way by pressing SHIFT 4.

CLEAR will clear the History. The GRAPHICS MANAGER There are two views, shown right, for which the only access is via the MEMORY MANAGER screen. The first of these, the GRAPHICS MANAGER, shows some memory in use on my calculator due to the screen captures I am performing to show you these views. Yours will probably be empty. The LIBRARY MANAGER The second of these is the LIBRARY MANAGER, and this will almost certainly be empty unless you have games loaded. Generally, the only aplets which use libraries are those such as games which are written by expert programmers in machine code in order to make them run as fast as possible. These games, available on the internet, are listed in the APLET view along with the normal aplets and when you delete them the associated library is automatically deleted with them, unlike the case of the ‘helper’ programs. Calculator Tip Because of the amount of memory available on the hp 39g+, the Memory View is not one that you will normally need to worry about. It is probably of more interest to programmers.

Earlier we examined the use of the MODES view, and the meaning of Number Format. We discussed the use of the settings Fixed, Scientific and Engineering, but left the setting of Fraction for later. The Fraction setting can be a little deceptive. Most calculators have a fraction key, often labeled b a c , that allows you to input, for example, 2 1 3 as 1 2 3 ¬ ¬ or something similar. What these calculators usually won’t do is allow you to mix fractions and decimals. On most calculators 2 1 3 7 3 + ⋅ will give a decimal result - the calculator will not attempt to convert the 3 7 ⋅ into a fraction. The reason for this is that while some decimals like 0.25 are easy to convert to a fraction, others, such as recurring ones, are not so easy. Most calculators opt for the easy option of switching to a decimal answer in any mixture of fractions and decimals. The makers of the hp 39g+ took a very different approach. Once you select Fraction mode, all numbers become fractions - including any decimals. The first point to remember is that there is no provision for inputting mixed fractions such as 2 1 3 . Fractions are entered using the divide key and, while the calculator is quite happy with improper fractions such as 5/3, it correctly interprets 1/2/3 as one half divided by 3 and gives a result of 1/6 . The solution to this is simply to enter mixed fractions as (1+2/3). Calculator Tip You need to be careful with brackets or “order of operations” problems may occur, such as 2 1 3 * 15 being interpreted as 1+(2/3*1/5) rather than as it should be: (1+2/3)*1/5. When in doubt, use brackets for mixed fractions.

1. 1 4 17 3 5 15 + = 2. 1 1 1 3 4 1 3 2 6 − = − The second point to remember involves the method the hp 39g+ uses when converting decimals to fractions, which is basically to generate (internally and unseen by you) a series of continued fractions which are approximations to the decimal entered. The final fractional approximation chosen for display is the first one found which is ‘sufficiently close’ to the decimal. The trap lies in what constitutes ‘sufficiently close’, and this is determined by the ‘4’ in Fraction 4. Roughly, the calculator will use the first fraction it finds in its process of approximation which matches the decimal to that number of significant digits. For example, a setting in the MODES view of… Fraction 1 changes 0.234 to 3 13 which is actually 0.2307692… (matching to at least 1 sig. fig.) Fraction 2 changes 0.234 to 7 30 which is actually 0.2333333… (matching to at least 2 sig. fig.) Fraction 3 changes 0.234 to 11 47 which is actually 0.2340425… (matching to at least 3 sig. fig.) Fraction 4 changes 0.234 to 117 234 500 1000 or       which is exactly 0.234 Essentially, the value of ‘n’ in ‘Fraction n’ affects the degree of precision used in converting the decimal to a fraction. As was said earlier, the calculator will use the first fraction it finds in its process of approximation which matches the decimal to that number of significant digits.

powerful than most calculators but can require that you understand what is happening. It should also be clear why a special fraction button was not provided: the ‘fractions’ are never actually stored or manipulated as fractions at all! Pitfalls to watch for As you can see in the screenshot right, a setting of Fraction 4 produces a strange (but actually correct) result for 0.666, while adding one more 6 (to take the decimal beyond 4 d.p.) will give the desired result of 2/3. In other words, so long as you understand the approach taken by the hp 39g+ it is capable of producing results which are closer to what was probably intended by the user in entering 0.66666. If you want to use this facility to convert decimals to fractions, here are some tips… ! if you are converting a recurring decimal to a fraction, then make sure you include at least one more digit in the decimal than the setting of Fraction in MODES. As you can see in the second screenshot, failing to include enough decimal places does not produce the desired result. ! if you are converting an exact decimal to a fraction, then set a Fraction n value of at least one more than the number of decimal places in the value entered. Both examples right were done at Fraction 6. Not understanding the significance of the setting of Fraction can produce some unfortunate effects. For example, at Fraction 2, the value of 123.456 becomes 123, with the 0.456 dropped entirely. Similarly, one of the earlier examples at the top of the previous page was 1/3 + 4/5 = 17/15.

perform this, you will find to your amazement that 1/3 + 4/5 = 8/7 , whereas using Fraction 6 gives the correct answer. The reason for this ‘error’ is that the 1/3 and 4/5 were converted to decimals and added to give 1.133333…. This was converted back to a fraction to give 8/7 (1.1428..) matching sufficiently closely in Fraction 2 to be accepted. TThhee HHOOMMEE HHiissttoorryy The HOME page maintains a record of all your calculations called the History. You can re-use any of the calculations or their results in subsequent calculations. Try this for yourself now. Type in at least four calculations, pressing the ENTER key after each one to tell the calculator to perform the calculation. You will now be looking at a screen similar to the one on the right (except probably with different calculations). If you now press the up arrow key, a highlighted bar moves up the screen. When you reach the top of the screen the previous calculations will scroll into view. Pressing SHIFT up-arrow takes the highlight to the top in one movement. COPYing calculations You may have noticed that as soon as the highlight appeared so did two labels at the bottom of the screen. If you now press the screen key under you will find that the highlighted calculation will be copied on the edit line. This is shown in the screen shot on the right.

calculation by removing some of the characters and/or adding to it. For example, in the screenshot right, the calculation of 3*2*COS(35) has been edited to 3*COS(35). Clearing the History Pressing enter will now cause this new calculation to be performed. Calculator Tip ! Pressing ON during editing will erase the whole line. ! Pressing SHIFT CLEAR erases the whole history. This is worth doing regularly, since the history uses memory that may be needed for other things, even with the immense amount of user memory the hp 39g+ has. ! You can calculations and results from any number of different lines in building your new expression. SHOWing results Next to the key you will see another screen key labeled . This key will display an expression the way you would write it on the page rather than in the somewhat difficult to read style that is forced on the calculator when it must show the whole expression on one line. This works anywhere the label appears, not just in HOME. Some examples…

Meem moorriieess Each of the alphabetic characters shown in orange below the keys can function as a memory. Some examples of this are shown in the third and fourth examples above where the values of 1, -3 and -4 are stored into A, B and C and the value of 3 is stored into X. All of this ‘storing’ of values is done with the key, which is one of the screen keys listed at the bottom of the HOME view. There are ways of obtaining even more memories than these 26 alphabetic ones, such as storing values into a list (see page 176), but 26 is enough for most people. Once stored into memory, a value can be used in a calculation by typing the letter into the place where you would normally use the value. Typing a letter and pressing ENTER will display memory’s contents. There is an advantage to storing results in memories, particularly if they are long decimals, or if you’re going to be re-using the result a number of times. As an example, we will perform the calculation of 4 ( 3) 2 5 2 3 5 × − + ⋅− . We will do this in two stages, calculating the top and bottom of the fraction and storing the results in memories. Firstly the top of the fraction, storing the result in memory A, then the bottom, storing in B. And then finally the result…

m tthhee HHOOMMEE vviieew w.. Once functions or sequences have been defined in other aplets, they can be referenced in the HOME view. e.g. 1 Suppose we use the Function aplet to define F1(X)=X²-2 and F2(X)=e^X as shown right. These functions now become accessible not only from within the HOME view but also within any other aplet also. This is shown by the screen shots below. The results shown will (of course) depend on your settings in the MODES view. The reason for the QUOTE(X-2) rather than just X-2, is that using X-2 would tell the calculator to use the value currently stored in memory X, while QUOTE(X-2) tells it to use the symbol. The QUOTE function is available through the MATH menu under Symbolic (see page 257). This type of work is actually far more easily done in the Function aplet, where QUOTE is not needed and the key does a better job. See page 71. e.g. 2 Suppose we use the Sequence aplet to define a sequence with 1 2 3, 1 T T = = and 1 2 2 n n n T T T − − = + . In the HOME view, the sequence values can be referred to as easily as the function values in the previous example.

MAATTHH M Meennuu The MATH menu holds all the functions that are not used often enough to be worth a key of their own. There is a very large supply of functions available, many of them extremely powerful, listed in their own chapter later in the book. When you press the MATH key you will see the pop up screen shown right. The left hand menu is a list of topics. Scroll through the topics until you find the one you want, then use the right arrow key to move into the list of functions for that topic. For example: The function ROUND will round off to a specified number of decimal places. E.g. round off 145.25667 to 3 decimal places. ! Press the MATH key & right arrow (→) into the Real group of functions. ! Press the ‘R’ key (the 9 key) to move to the first of the functions beginning with ‘R’, then move down one more function to ROUND. Press ENTER. ! Now finish off the command and press ENTER again to see the result. See page 243 for a far more detailed look at the Math Menu.

It is probably inevitable as the line between calculators and computers becomes blurred that calculators will inherit one of the more frustrating characteristics of computers: they crash! If you find that the calculator is beginning to behave strangely, or is locking up then there are a number of ways to deal with this. Calculator Tip If you are a user of external aplets then you may find that one will stop working with the message “Invalid syntax. Edit program?”. There is almost certainly nothing wrong. Press , try a soft reboot as below, then run the aplet again. Soft reboot Pressing ON+SK3 will perform a soft reboot. It is perfectly safe and will not cause any memory loss except that your HOME History will be cleared. Hold down the ON key and, while still holding it down, press the 3rd screen key from the left. The calculator will very briefly display a boot screen and then redisplay the HOME view, with the Function aplet as the active one.

completely that the keyboard will not respond a method of reset is provided which is independent of the keyboard. This should never happen but it is important to know how to deal with it in case it happened during a test or an exam. Soft reboot (Hardware) On the back of the calculator is a small hole. Poke a paper clip or a pin into this hole and press gently on the switch inside. To the calculator this is exactly equivalent to a soft reboot as outlined on the previous page. If you do this when the calculator is locked up then the chances are that you will find that the procedure will not only unlock the calculator but retain your data intact. Hard reboot (with memory wipe) To completely wipe the calculator’s memory press ON+SK1+SK6. Hold down the ON key and, while still holding it down, press the first and then the last screen keys. Release them in the opposite order. Don't release SK1 and SK6 together - release SK6, then SK1, then ON. This method will always cause complete loss of data. If you find that the screen fills with garbage, or if the calculator’s in-built diagnostic routine starts to run, then it is just that you have not released them in the right order. Simply try again. Calculator Tip As a last resort, take the batteries out, including the disk shaped back-up battery and wait a few hours. Reinsert the batteries. Make SURE they go in correctly as you can very seriously damage the calculator if you insert batteries the wrong way around.

of previous calculations. When the highlight is visible, the key can be used to retrieve any earlier results for editing using the left/right arrow keys and the DEL key. 2. Care must be taken to ensure the your idea of order of operations agrees with the calculator’s. For example, (-5) 2 must be entered as (-5) 2 rather than as -52, and 5 4 + must be entered as √(5+4) rather than √5 + 4. 3. The ANS key can be used to retrieve the results of the calculation immediately preceding the one you’re working on. E.g. √(5+Ans) 4. The DEL key can be used to erase single lines in the history. The SHIFT CLEAR key will delete the entire history. Regular clearing will ensure that memory is not gradually eroded. 5. The key displays a calculation as you would see it written. 6. The MODES view can be used to set the format in which numbers are displayed on the HOME page, and to choose the angle measure which is to be used. 7. Make sure you understand Fraction mode before using it. Remember that the angle and numeric mode settings may change if you change aplets in the APLET view. 8. Numbers are stored in memory using the key labeled . The stored values can then be used by simply putting the letter in the expression in place of the number. 9. You can reboot the calculator if it locks up. Make sure you know how to do this in case it happens during a test or an examination. For more information on the complete set of mathematical functions available in the HOME view (and anywhere else) see the chapter “The MATH menu” on page 243.

most of all. It allows you to: ! graph equations ! find intercepts ! find turning points (maxima/minima) ! find areas under curves ! find areas between curves ! find gradients ! find derivatives algebraically ! find simple integrals algebraically ! evaluate functions at particular values ! graph and evaluate algebraically expressions such as f(g(x)) or f(x+2) Choose the aplet The first step for any aplet is to choose it in the APLET LIBRARY. Press the APLET key and you will see something similar to the screen on the right. Use the arrow keys to move the highlight up or down until the Function aplet is selected. Now, looking at the list of programmable functions at the bottom of the screen, you should see labels of and . Press the key under first. You’ll see the message shown right - press the key. The reason for doing this is to clear out any functions that you may have put in there while playing around, and so to make sure that what you see will be the same as the screen snapshots.

Now press the key. When you do, your screen should change so that it appears like the one on the right. This is the SYMB view. Notice the screen title so that you will know where you are (if you didn’t already). Calculator Tip Pressing ENTER here would have had the same effect. Whenever there is an obvious choice pressing ENTER will usually produce the desired effect. The XTθ button Whenever you enter an aplet, one of the keys which usually changes its function is the key labeled XTθ. As its label suggests, it supplies an X, or a T, or a θ, depending on which aplet you are in. Let's use that key to produce a graph of the quadratic we dealt with in the earlier section on the HOME view. Using the up/down arrows, move the cursor (if necessary) to the line F1(X) = . Type in: 3 XTθ X² - 5 XTθ - 4 ENTER This will produce the screen shown on the right. Notice the check/tick mark next to the function F1(X). This signifies that this function is to be graphed, so that if you had five functions entered but only wanted numbers 1, 3 and 4 graphed, you could simply ensure that only F1(X), F3(X) and F4(X) were checked. ing your function The key that turns the check on or off is a screen key - the one labeled with the . It is a toggle key - when the check is off it turns it on and when on it turns it off. In some countries a ‘check’ mark is called a ‘tick’ mark but the idea is the same.

highlight has to be on the function before the check can be changed. Make sure it’s checked when you have finished. The NUM view If you now press the NUM key, you will see the screen on the right. It shows the calculated function values for F1(X), starting at zero and increasing in steps of 0.1 Make sure the highlight is in the X column, and then press 4 and ENTER. You will find that the numbers will now start at 4 instead of zero. It is also possible to change the step size via the key, which can be convenient at times, particularly for trig functions. The simplest way to set the start value and increment is in the NUM SETUP view. This will be covered in detail later (see page 77). The PLOT view Now try pressing the PLOT key. The graph you’ll see will not be a terribly useful one (see right) because the axes will not be set up correctly. We’ll look next at how to do this. One of the easiest ways to set up the axes properly for a function whose shape is not known in advance is to let the calculator suggest a suitable scale. There are a number of ways of doing this. See page 86 for a discussion of ways of finding ‘nice’ axes. One method of finding a viable scale is to use the Auto Scale option in the VIEWS menu (below). More information on the VIEWS menu can be found on page 92. Another method is to use the NUM view to investigate the range of values that will be required for the function.

scroll down to Auto Scale and press ENTER. The calculator will adjust the y axis in an attempt to fit as much of the graph on to the screen as possible. Some points to bear in mind; ! the y axis is scaled only on the first function which has a . ! the y axis is scaled for the x axis you have chosen in PLOT SETUP. If you’ve not chosen wisely then your result will not be good. ! it doesn’t choose ‘nice’ scales such as we would choose (going up in 0.2’s or 5’s or 10’s etc.) so we generally need to tidy up its choice a little. If you look at the y axis of the graph you’ve just produced, you’ll see that the axis tick marks are so close together that it looks like a solid line! To tidy this up you must change to the PLOT SETUP view. If you look above the PLOT, SYMB and NUM keys you will see the word ‘SETUP’. The SETUP view for each of these keys is obtained via the SHIFT key. See “Tips & Tricks - Function” on page 69 for more information on how to find good choices for axes. The Auto Scale function is covered in much greater detail on page 96.

w If you press SHIFT then PLOT you will see something like the view on the right. The highlight should be on the first value of ‘XRng:’. Enter the value -4. Calculator Tip Don’t use the subtract key to input a negative. You MUST use the negative key labeled (-) which is in the same row as the ENTER key. Type in 4 for the other ‘XRng:’ value, then -20 and 20 for the ‘YRng:’ values. When you’ve done this use the arrow keys move to ‘Ytick:’ and change it to 5. Detail vs. Faster At the bottom of the screen you will see Res: (short for ‘Resolution’). If you highlight it and press the key you will see that you have a choice of Faster or More Detail. More Detail should be selected. If you choose Faster then every second dot is plotted instead of every dot. This is quicker but may make some graphs appear less smooth, particularly graphs with steep gradients. There are two pages to this view (see the key at the bottom of the screen). The first page is used to set axes, the second to control certain features of them. Press the key. You will now be looking at the screen shown above right. Using the arrow keys to move the highlight, make sure that your checks/ticks match the ones in my snapshot. Now press PLOT again. Perfect!

page of PLOT SETUP. Although they are not used often they can be quite useful and I recommend highly that you use Simult: Simultaneous The first option Simult: controls whether each graph is drawn separately (one after the other) or whether they are all drawn at the same time, sweeping from left to right on the screen. My preference is to turn this off. I find that if there are more than two functions defined then drawing them all at the same time can be confusing. Turning off Simult: means that they are drawn one after the other, in the order that they are defined. This is obviously a bit slower. Connect The second option Connect: controls whether the separate dots that make up a graph are connected with lines or left as dots. vs… Axes The third option Axes: controls whether axes are drawn. The fourth Inv.Cross: controls the appearance of the cursor that is moved by the arrow keys. It is best if you try this one yourself to see the effect. Labels The fifth option Labels: controls whether labels (X, Y and numbers) are put on the axes. The only time this causes problems is if the scale is an odd one, causing the labels to have too many decimal places. Grid The sixth and last check Grid: causes a grid of dots to be drawn on the screen (see right). The density of the grid is controlled by the values of Xtick and Ytick. This can be quite useful.

The default scale is displayed in the PLOT SETUP view shown right. It may seem a strange choice for axes but it reflects the physical properties of the LCD screen, which is 131 pixels wide by 63 pixels tall. A ‘pixel’ is a ‘picture element’ and means a dot on the screen. The default scale means that each dot represents a ‘jump’ in the scale of 0.1 when tracing graphs. The y value is determined by the graph, of course, and has nothing to do with your choice of scale. Once the scale changes, the cursor jumps from dot to dot are often not a useful size. See page 69 for information on choosing ‘nice’ scales. TThhee BBaarr The MENU toggle If you look at the screen key list at the bottom of the screen you will see only a single entry, labeled . Press the key under it and your screen will change to look like the one above right. Press it again and the screen will clear completely. Once more and you are back to the original appearance. Try pressing it a few times to get the feel for its behavior. This is what is known as a ‘toggle’ switch. The key is a triple toggle, cycling through each of the display modes shown right. The first default mode is (X,Y) mode, which displays the coordinates of the current cursor position. In any of these modes the up/down arrows move the cursor from function to function, while the left/right arrows move along the currently selected function. Calculator Tip Pressing SHIFT right arrow or SHIFT left arrow will jump the cursor directly to the right or left side of the screen.

In the examples and explanations which follow, the functions and settings used are: Trace is quite a useful tool. The dot next to the word means that it is currently switched on. If yours isn’t then press the key underneath to turn it on. Leave it on for now. Press the left arrow 5 or 6 times to see a similar display to that shown right. Pressing up or down arrow moves from function to function. The order used is not related graph but solely to the order that they are defined in the SYMB view. If it is turned off then the cursor is free to move anywhere on the screen. Try this and then turn it on again. Defn Press the key labeled (short for Definition). You will find that the equation is now listed at the bottom of the screen. The up/down arrows will move the cursor from F1(X) to F2(X), with the definition changing as it does so. If Trace is switched off then Defn will not work correctly.

having to trace along the graph. It is very powerful and useful. Suppose we begin with the cursor at x=0 on F1(X) as shown right. Press and then to see the input form shown right. Type the value 3 and press ENTER. The cursor will jump straight to the value x = 3, displaying the (X,Y) coordinates at the bottom of the screen. A very nice feature of the key is that it will jump to values which are not on the current screen, or which would be inaccessible for the current scale. For example, we can jump to the value x = 100 and see the (X,Y) coordinate displayed, with the cursor positioned at the far right side of the screen. If the left arrow is now pressed, tracing will resume from the right edge of the screen, in this case at one pixel point back from x = 4. Calculator Tip The key will also accept calculated values. You could, for example, jump to a value such as e2+2. If you had recently found an intersection, then jumping to a value of Isect would return the cursor to that point. See page 64 for information on Isect.

The next menu key we’ll examine is . Pressing the key under pops up a new menu, shown right. The menu is longer than will show in one screen, so two screen shots have been included to show most of the menu. The list which follows covers the purpose of each of the ten options shown right. The four extra options which follow these are covered as part of the detailed examination of the VIEWS menu on page 92. Center This redraws the graph with proportionally the same scale for each axis but re-centered around the current position of the cursor. If you already have a ‘nice’ scale, this will preserve it, while perhaps showing a more interesting section of graph. In/Out These two options zoom in or out by adjusting the scales by the factor shown. The default factor is 4x4 but this can be adjusted through the Set Zoom Factor option later in the menu. The most useful settings are either 5x5 or 2x2 as these are more likely to give and preserve ‘nice’ scales.

commands. When you choose this option a message will appear at the bottom of the screen asking you to Select first corner. If you use the arrow keys to move the cursor to one corner of a rectangle containing the part of the graph you want to zoom into and then press ENTER, the message will change to Select second corner. As you move the cursor this time to a position at the diagonally opposite corner of a box, the box will appear on the screen. Pressing expands the box to fill the screen. You’ll notice that the scale has been disrupted so that the labels are no longer very helpful. PLOT SETUP would give better end points for the axes or let you switch off the labels option. Rather than doing that however, scroll down the menu to a new option of UnZoom. This option puts the screen back to the way it was before you did the . Calculator Tip Zoom is very handy for allowing you to examine small sections of a graph in detail, but remember Un-Zoom! The cursor positioned at one corner of a …and at the other corner.

These two options allow you to zoom in (or out) by a factor of 4 on either axis. The factors can be set using the Set Factors… option, which gives you access to the view shown above right. You will also see a CHK mark next to an option called Recenter. If this is CHKed then the graph will be redrawn after zooming in or out with the current position of the cursor as its center. Changing the x factor is reflected in the menu as you can see in the second screen snapshot. Square This option changes the vertical scale to proportionally match the horizontal scale so that circles will appear circular rather than elliptical. Auto Scale, Decimal, Integer and Trig These four options are duplicates of those found on the VIEWS menu, and are provided in both places simply for convenience. Information on them can be found in the section on the VIEWS menu on pages 92.

meennuu Before continuing, set the axes back to the way we set them at the start of the section on the Menu bar. Looking at the menu functions again, you will see that the only one we have not yet examined is the one labeled . This key pops up the Function Tools FCN menu. Before you use this key, make sure is switched on and move the cursor so that it is in roughly the position shown right. Root Press the key. As you can see on the right, this key gives you access to a number of useful tools. If you leave the highlight on Root (as shown) and press ENTER (or ) then the cursor will jump to the nearest root or x intercept for the function it is on, starting its search at the current position. Try it now. Notice the message at the bottom of the screen giving the value of the root that was found. To find the other root, you need to move the cursor so that it is closer to the other root than to the present one. In general that means moving it past the turning point. Calculator Tip If you are working with a function which has asymptotes then make sure the cursor is positioned on the same side of the asymptote as the root. The internal algorithm used does not seem to cope well with crossing asymptotes. Move to about this position.

menu is Intersection. If you choose this option, then you will be presented with a choice similar to the one in the screen shown right. Exactly what is in the menu depends on how many functions you have showing. In the case shown here we only have two, so the choice is of finding the intersection of F1(X), which is the one the cursor is on, with either the X axis, or the other function F2(X). The results of choosing F2(X) are shown right. Calculator Tip When you find an intersection or a root the value of the x coordinate is stored in the memory X. If you immediately change to the HOME view and type X and hit ENTER then you can retrieve and use this value. See “Tips and Tricks - Function” on page 83 for more detail and examples. Slope This gives the numerical value of the derivative at the point of the cursor for the current function. Calculator Tip If the value at which you wish to find the slope is not accessible for the current scale then you will need to use the key to jump to the desired point before choosing Slope. The value is remembered and used in the Slope calculation, despite not being on the current scale. In the same way, choosing Slope immediately after finding a root or intersection will give the slope at the remembered value rather than the nearest dot.

menu is the Signed Area… tool. Before we begin to use it, make sure that is switched on, and that the cursor is on F1(X) - the quadratic. The Signed Area… tool is similar to the Box Zoom in that it requires you to indicate the start and end points of the area to be calculated. Definite integrals Suppose we want to find the definite integral: 3 2 2 5 4 x x dx − − − ∫ Choose and then Signed Area… You will see the message shown right, asking you to choose a starting point. Press the key, enter the value -2 and press or ENTER. The starting value will then be -2 so press again (or ENTER) to accept it. Another menu will appear, asking you to indicate what area you wish to calculate. In this case there are only two choices - between F1(X) and the X axis, or between F1(X) and F2(X). If we had defined more functions in the SYMB view then this menu would be longer. In this case we want the area between F1(X) and the x axis, so position the highlight as shown and press ENTER. The graphs will then reappear, with a message requesting that you choose an end point. In the screenshot shown right I have pressed and entered the value 3 to move to that point directly.

end point, the hp 39g+ will calculate the signed area and display the result at the bottom of the screen. Calculator Tip It should be clearly understand that although the label at the bottom of the screen is Area it is a little misleading. What has actually been calculated is the definite integral (right), with ‘areas’ below the x axis included as negatives. This is why the label on the original menu reads “Signed area” instead of just “Area”. 2ndpoint 1st point 1( ) F X dx ∫ Tracing the integral in PLOT Rather than using the key, an alternative method is to use the tracing facility. The advantage of this is that the ‘area’ is shown visually as you go by shading, as can be seen right. The disadvantage of this is that you can only trace to values which are permitted in the scale you are using. As soon as you use the shading stops. In this case, due to the scale we chose, if you try to trace to the values x = -3 or x = 2 you will find that they are not accessible so we will need to change scale. Change to the PLOT SETUP view and change the x axis (only) back to -6.5 to 6.5, then PLOT again.

Area… as before. Use the left/right arrow keys or the key to move the cursor to x = -2. Press ENTER to accept the starting point. This time, choose the boundary as F2(X) instead of the x axis so that we will be finding the ‘area’ between curves instead of the area under one. Again, the result will be a signed area (definite integral) not a true area. See page 83 for a simple method of finding true areas. We now need to choose the end point. This time do it by tracing with right arrow to move the cursor. As you do the area will be shaded by the calculator. The current position is shown at the bottom of the screen. When you reach the end point you are looking for, press the key and the area will be calculated as before. This is shown right. To remove the shading, press PLOT again. Calculator Tip Note that common sense tells us that the answer is almost certainly -3.75 rather than -3.75000000002. The small error is simply due to accumulated rounding error in the internal methods used by the calculator. For example, an answer of 0.4999999999 should be read as 0.5. This is quite common and students should be aware of the need for common sense interpretation. Areas between and under curves If we are wanting to find true areas rather than the ‘signed areas’ given by a simple definite integral then we must take into account any roots of the function. This process is shown in detail on page 83, as there are certain tricks which can be used to make the process far simpler.

menu is the Extremum tool. This is used to find relative maxima and minima for the graphs. Ensure that is switched on and that the cursor is positioned on the cubic F2(X) in the vicinity of the left hand maximum (turning point) as shown right. Press and choose Extremum from the menu. You should find that the cursor will jump to the position of the maximum. Calculator Tip If your graph has asymptotes then make sure that the cursor is positioned on the side of the asymptote containing the extremum before initiating the process. The internal algorithm used does not cope well with intervening asymptotes.

ONN Finding a suitable set of axes This is probably the most frustrating aspect of graphical calculators for many users and there is unfortunately no simple answer. Part of the answer is to know your function. If you know, for example, that your function is hyperbolic then that immediately gives information about what to expect. If you don’t have knowledge then here are a few tips: 1. Try just plotting the function on the default axes. You may find that enough of the function is showing to give you a rough idea of how to adjust them to display it better. Remember that ZOOM can work on either axis or on both. See Tip #4 on the next page. 2. The NUM view can be very helpful. Try changing to NUM SETUP and setting the value of NumStep to 1, or even 5 or 10. Now scroll through the NUM view and look at what is happening to the F(X) values. Look for two things. Firstly, where is the function most active? For what domain on the x axis is it changing fastest? This is likely to be the domain you are most interested in. Secondly, what is the range? What sort of values will you need to display on the y axis? Now change to the PLOT SETUP view and set what you think may be appropriate axes. From those you can PLOT and then zoom in or out. 3. If the graph is part of a test or an examination then the wording of the question will often give a clue as to what x axis domain you should work with. 4. I most often use Auto Scale to get a first approximation to a good set of axes. To do this you must choose your x axis domain first so try Tip #2 above and use your knowledge of what the function might look like.

perhaps importantly, always gives ‘nice’ scales is to use ZOOM. ! Enter your graphs into the SYMB view. Remember that Auto Scale only works on the first ticked graph. ! Press VIEWS and choose Decimal, or press SHIFT CLEAR in the PLOT SETUP view. This will give you the default axes, probably not showing the graph very well. ! Place the cursor so that it is in the center of the area you are most interested in. ! Use the menu to adjust the view. You may choose first to change the zoom factors to something other than 4x4, and to ensure that Recenter: is ed. The advantage of doing it this way is that if you zoom in or out by a factor of 2 or 4 or 5, the cursor jumps will stay at (relatively) nice values allowing you to trace more easily. In this case, the cursor now moves in jumps of 0.05, which is ideal for most purposes. On the other hand you may not be interested in tracing along the graph.

functions such as ( ) 2 f x + or ( ) ( ) f g x in its SYMB view. The and keys are particularly helpful with this. For example, if we define 2 1( ) 1 F x x = − and 2( ) F x x = , then we can use these in our defining of F3, F4. See the screen shot on the left below. If the highlight is now positioned on each of these in turn, and the key pressed then the substitution is performed. The result is shown in the upper right hand snapshot and the F4(X) function is shown right after pressing SHOW. Notice that the calculator is smart enough to realize in F3(X) that ( ) 2 1 x − is the same as 1 x −, although not, unfortunately, smart enough to keep track of the implications for the domain that F3(X) should be defined only for non- negative x. There is a limit to this however. If you define 2 1( ) 1 F x x x = − − and then 2( ) 1( 1) F x F x = + , then the routine will not simplify ( ) ( ) 2 1 1 1 x x + − + − to 2 1 x x + −.

simplify the expression. If you now the result and enclose it with the POLYFORM function as shown right (note the final ‘,X’), then highlight it and press , the hp 39g+ will expand the brackets and gather terms. Calculator Tip These functions can all be graphed but the speed of graphing is slowed if you don’t press first. The 39G+ is fast enough that the result can be lived with but the 39G and 40G are usually too slow when graphing un-evaluated composite functions. Evaluating the function may also hide the domain. For example, if 2 1( ) F X X = and 2( ) F X X = then 3( ) 1( 2( )) F X F F X = will show the correct domain of 0 x ≥ in the NUM view. Pressing will destroy this. Using functions in the HOME view Once functions have been defined in the SYMB view of the Function aplet, they can be reused in the HOME view (and indeed in any other aplet!). For example suppose you needed to find some exact points (x = 0, 1, 2 and 3) for a graph of ( ) 2 ( ) 2 x f x x = − when doing a hand sketch of it. Type its definition into the Function aplet SYMB view, switch to the HOME view and then simply type F1(0) and press ENTER. The function will be evaluated for x = 0. Similar results can be obtained for F1(1), F1(2) and F1(3). Notice the error message for x = 2 caused by a divide by zero.

which are best done in the SYMB view of the Function aplet. The syntax of the differentiation function is: ( ) X function ∂ , where function is defined in terms of X. The function can either be already defined in the SYMB view of the Function aplet, or entered into the brackets as above. The ∂ symbol most easily obtained by pressing the key labeled d/dx. One point to remember is that if you use this function in the HOME view you may not receive the result you expect. If you try this yourself your result will probably not be the same as that shown. The reason for this is that the result you see is the derivative of 2x x − evaluated at whatever value of x happens to be currently in memory. This can be seen more clearly if we store a specific value into the memory X beforehand. In the example shown right, the answer of 3 is the value of the derivative 2 1 x − at the value of x = 2. But what of algebraic differentiation? It is possible but not very convenient to do this in the HOME view using a “formal variable” of S1. (..or S2 to S5) The drawback of this is simply the awkwardness of having to work with S1’s rather than with X’s.

the result is algebraic rather than numeric. The best method is to define your function as F1 and its derivative as F2 (see below)… …but you can also perform the whole process in one line. As you can see the calculator’s algebraic abilities do not extend to differentiating ( ) 2x f x = as ( ) ( ) ln 2 .2x f x ′ = , but at least it is numerically correct. Calculator Tip Doing your differentiation in the Function aplet is much easier and offers the additional advantage of being able to graph the two functions. Circular functions There are two issues that influence the graphing of circular functions, both related to the scale chosen. The first one, illustrated on the right, causes circles to be ellipses if you don’t choose scales for the x and y axes which are ‘square’ relative to each other. The two graphs above are both of the same function 2 2 9 x y + = . press press then .

scales which are multiples of the default scales. For example by using 13 13 x − ≤ ≤ and 6.2 6.4 y − ≤ ≤ (scale factor 2). You can also use the ‘Square’ option on the ZOOM menu. This adjusts the y axis so that it is ‘square’ relative to the x axis chosen. The second issue is caused by the domain of the circle being undefined for some values. The screen on your calculator is made up of small dots called pixels and is 131 pixels wide and 64 pixels high. As was mentioned earlier, this means that each pixel is 0.1 apart on the x and y axes. This can affect your graphs and it becomes particularly obvious with circles because the graph does not exist for the part of the x axis outside the circle. The two screen shots right are an example of two images of the same graph 2 2 9 x y + = using two slightly different scales. You can see that the second example has missing pieces. Let's look at the circle 2 2 9 x y + = as an example. This circle only exists from -3 to 3 on the x axis and is undefined outside this domain. In order to graph it on the hp 39g+ you have to rearrange it into two equations of F1(X)= (9-X2) for the top half & F2(X)= - (9-X2) for the bottom half. If you enter these equations and then graph them with the default axes then you get a perfect circle. However, if you change the x axis from the default setting to -6 to 6 and then re-PLOT then part of the circle disappears. This is shown in the snapshot at the start of the question. The reason for this is that when the calculator draws the graph it does so by ‘joining the dots’. For the default scale of -6.5 to 6.5 this is not a problem since the edges of the circle at -3 and 3 fall on a pixel. This means that the last segment of the graph plotted extends is from 2.9 to 3 and the circle reaches right down to the x axis.

If you try to trace the circle you'll see that the pixels fall on 0, 0.0923077, 0.1846154..... In particular, near x=3 the pixel values are 2.953846 and 3.046154. This means that the calculator can't draw anything past 2.953846 because the next value doesn't exist, being outside the circle. This is what causes the gap in the circle. There's nothing to join to past that last point. The solution is to use scales which allow the end points of your circle to fall on a pixel point. If your circle has an integer radius then this is easily done by starting with the default axes and then ZOOM in or out to show the circle. This will tend to give 'nice' pixel points. If your circle is not centered on the origin then just check/tick the box in the Set Zoom Factors box to Recenter. That will allow you to turn off TRACE, move the cursor closer to the point where you'd like the centre of screen to be and then ZOOM. Trig functions If graphing a trig function using radian measure then you should choose a scale which will cause pixel values (screen dots) to fall at convenient points. A good scale to use is provided on the VIEWS menu (see right). This scale sets each dot to be 24 π , ensuring that multiples and most factors of π will fall on dots and thus helping if you want to trace values on the graph. Ensure the angle measure is set to Radians in the MODES view if you intend to use this scale. Calculator Tip Make sure you set angle measure after you have ed the aplet, not before, so as to ensure you are setting the correct aplet.

When you find an extremum or an intersection, the point is remembered until you move again even if it is not actually on a value that would normally be accessible for the scale you have chosen. For example, if you find an intersection and then return to the menu and choose Slope, the slope calculated will be for the intersection just found rather than for the nearest pixel point. If you have recently found a root then pressing the key and entering the value Root will return the cursor to it. There are two ways to access these values. The first and simplest is via the value stored in memory X. If you move from PLOT to the HOME view and type X (using ALPHA) then the value it will contain will be the last position of the cursor. If you just found a root or an intersection then this will be the value displayed. The second way is via the reserved words of Root, Extremum, Area, Slope and Isect. Typing any of these reserved words in any situation will retrieve their last calculated values. If a value has not been found yet then they will return zero. This trick is particularly useful when calculating areas under or between curves. See page 83. The NUM view revisited We saw earlier that the NUM view gives a tabular view of the function. It is possible to manipulate this view through the NUM SETUP view.

NumStart & NumStep For example, values of 10 and 2 give: Automatic vs. Build Your Own Looking at the NUM SETUP view you will see an entry called NumType: with the default value of Automatic. The only alternative to Automatic is the setting of Build Your Own. Under this setting the NUM view will be empty, waiting for you to enter your own values for X. Typing in the values of (for example): 3 (ENTER) -2 (ENTER) 5 (ENTER) … will give… In this situation the function values are being calculated as you input the X values. This can be quite useful if you are wanting to evaluate the behavior of a function at selected points. ZOOM If you now use the NUM SETUP view to switch to Automatic you will find that there is a key at the bottom left of the NUM view. This can be quite useful as a fast way to reset the scale.

the right. The first option of In causes the step size to decrease from 0.1 to 0.025. This is a factor of 4 and is changeable via the NUM SETUP view. I find a zoom factor setting of 2 or 5 to be more useful. The second option of Out causes the opposite effect, changing the step size upwards by whatever the Zoom Factor is set to. The Decimal option restores the default settings. It changes from whatever is showing back to the step size of 0.1 The Integer option on the other hand, changes the scale so the step size is 1. The Trig option changes the scale so that the step size is 24 π exactly. This will obviously be useful when dealing with trigonometric functions.

The situation for integration is very similar to that of differentiation. The difference is that both the HOME view and the Function aplet require the use of a “formal variable” S1. As with differentiation, the results are better in the Function aplet. The ∫symbol is obtained via the keyboard. The syntax of the integration function is: ( , , , ) a b function X ∫ where: a and b are the limits of integration and function is defined in terms of X. Let’s look first at the definite integral… The screen left shows 2 2 1 1 x dx + ∫ = 3.3333… followed by ln2 0 xe dx ∫ = 1. It may help you to remember the syntax of the differentiation and integration functions if you realize that they are filled in with values in exactly the same way that they are spoken. E.g. 2 2 1 1 x dx + ∫ is read as: “the integral from 1 to 2 of 2 1 x + dx ” & entered: ∫( 1, 2, 2 1 X + , X ) A similar path was taken with the differentiation function, so that: 2 d X dx    , which is read as “the derivative with respect to X of X2” and entered ( ) 2 X X ∂ : ∂ X ( ) 2 X

Algebraic integration is also possible (for simple functions), in the following fashions: i. If done in the SYMB view of the Function aplet, then the integration must be done using the symbolic variable S1 (or S2, S3, S4 or S5). If done in this manner then the results are very good, except that there is no constant of integration ‘c’. The screenshot right shows the results of defining 2 1( ) 1 F X X = − and then ( ) 2 2( ) 0, 1, 1, F X S X X = − ∫ , together with the results of the same thing after pressing the key (the result is placed in F3 only for convenience of viewing). All that is now necessary is to read ‘-S1+S1^3/3’ as 3 3 x x −+ , or as it should be read as 3 3 x x c − + . ii. If done in the HOME view, then S1 must again be used as the variable of integration. i.e. 2 1 x dx − ∫ is entered as ∫( 1, S1, X2 - 1, X ). This is shown right, together with the results of highlighting the answer and pressing . The calculator assumes that X itself may be a function of some other variable and integrates accordingly as a ‘partial integration’ which, while mathematically correct, is not what most of us want. The way to simplify this answer to a better form is to highlight it, it, and press ENTER again, giving the result shown right. This is the result of the calculator performing the substitutions implied in the previous expression.

and so it is worth examining the process in more detail… 1 1 3 2 0 0 3 3 3 1 3 1 0 1 0 3 3 1 1 3 x S S x x x dx x S S S S = =   − = −         = − − −         = − ∫ The potential problem lies with the second line, where the substitution of zero results in the second bracket disappearing. This will not always happen. For example… ( ) ( ) ( ) ( ) ( ) 1 5 1 4 0 0 5 5 5 2 2 5 1 2 0 2 5 5 1 2 6 4 5 x S S x x x dx S S = =   −   − =         − −     = −         − = −⋅ ∫ The final constant of 6.4 comes from substituting zero into the expression and should not be there if we were doing this with the aim of finding an indefinite integral. On the other hand, we all know that the answer should have a constant of integration, so perhaps this extra constant will help you to remember the ‘+c’! In addition to this, there are strict limits to what the hp 39g+ can integrate. For example, if you try to evaluate 2 sin .cos x x dx ∫ using the calculator, it will not be able to do it. Essentially, beyond polynomials forget it. The hp 40g Computer Algebra System Owners of the hp 40g will be aware that it has a CAS which allows it to perform the most complex algebra, including complex integrations, with ease. More information on the CAS can be found in the supplementary appendix on the hp 40g. See the Table of Contents.

As was discussed earlier, when you find roots, intersections, extrema or signed areas in the PLOT view, the results are stored into variables for later use. For example, if we use Root to find the x intercept of 2 ( ) 2 f x x = − then the result is stored into a variable called Root, which can be accessed anywhere else. Similar variables called Isect, Area, and Extremum are stored for the other tools. You can access these names by typing them in using the ALPHA key, or by using the VARS key. Press VARS and you will see a list of the HOME variables. If you press SK2, labeled (not the APLET button on the keyboard), then the display changes to show the variables specific to whatever aplet you are currently using. Those shown right are for the Function aplet and the group of Plot FCN variables is shown. If you look at the screen shot you will see that the tag is currently selected (showing as ) and this means that when you press the name of the variable will be pasted into what ever view you were using when you pressed the VARS key. Pressing the screen key will cause the value of the variable to be pasted instead. There is no effective difference in most situations. This can be very useful in finding areas, as you will see in the example following.

2 ( ) 2 f x x = − and ( ) 0.5 1 g x x = −from x = -2 to the first positive intersection of the two graphs. From the shaded screenshot shown above right it can be seen that to find the area we need to split it into two sections, with the boundaries being -2 and the two intersections. The shortcut here is to use Intersection to find the first intersection, storing the results into memory variable A. We then do the same thing for the second intersection, storing the result into B. We can now calculate the area in the HOME view, using 1 2 f f − for the first and 2 1 f f − for the second. Use to duplicate the first integral and edit it to adjust the functions and limits. Finally, the two solutions and add them to give the final answer.

It is possible to graph piecewise defined functions using the Function aplet, although it involves literally splitting the function into pieces. For example: 2 3 ; 2 ( ) 2 ; 2 1 3 ; 1 x x f x x x x x + < −   = − − ≤ ≤  − ≥  To graph this we need to enter it into the SYMB view as three separate functions: F1(X)=(X+3)/(X < -2) F2(X)=(X2-2)/(X ≥ -2 AND X ≤ 1) F3(X)=(3-X)/(X ≥ 1) Note: AND can be found on the (-) key. The reason why this works is that the (X < -2) and the (X ≥ -2 AND X ≤ 1) expressions are evaluated as being either true (which for computers has a value of 1) or false (which has a value of 0). By dividing by the domain expression we are effectively dividing by 1 inside the range (no effect) or dividing by zero outside the domain, making the function undefined. Since undefined values are not graphed, this produces the desired effect.

for the seemingly strange default scale of -6.5 to 6.5 is to ensure that each dot on the screen is exactly 0.1 apart. There are other scales, basically multiples of these numbers, that also give nice values if you want to along the graph. For example, halving each of -6.5 and 6.5 will place the dots 0.05 apart. To zoom out instead of in simply double the values, producing dots that are 0.2 apart. Similarly, if you want to center the graph around a particular value then just add that value to the range values. The example right is centered around x = 1 by adding 1 to -3.25 and 3.25. A time when ‘nice’ scales are more important is when you use the Plot-Table option in the VIEWS menu. If you use the default axes you will find that the dots, and hence the table values are no longer ‘nice’ because of the 3 dots consumed by the line down the middle of the screen.

x axis scale to -6.4 to 6.4, which gives table values of 0.2. Using -3.2 to 3.2 is even better since it makes the graph ‘square’ again, with both axes proportional. Another good choice of scale for the Plot-Table view is -8 to 8, giving table values of 0.25. Basically any power of 2 is a good choice. Again, adding or subtracting a constant from each end of the axes will produce a graph where the y axis is not centred. Use of brackets in functions One problem commonly encountered by new users is misinterpretation of brackets. The hp 39g+ will correctly interpret F1(X) = X2(X+1) as X2*(X+1) but will not understand F(X)=X(X+1). When used in either Function or Solve, it will result in the error message of “Invalid User Function”. Similarly if you want to use the sum to n terms formula for a GP in the Solve aplet and enter it as S=A(1-R^N)/(1-R) then you will see a similar message until you change it to read S=A*(1-R^)/(1-R). The reason for this apparent ‘error’ is that all of the built-in functions such as SIN(....) and COS(....) and ROUND(....) work with brackets. When the calculator encounters X(X+1) it interprets this as asking it to evaluate a function called X(....) at the value X+1. Since there is no such function it returns the error message that you are trying to use a function that is unknown. The solution is simple: just remember to put the * sign in when you use letters immediately before a bracket.

In evaluating limits to infinity using substitution, problems can be encountered if values are used which are too large. For example: lim x x x e e →∞ + 2 6 It is possible to gain a good idea of the value of this limit by entering the function F1(X)=e^X/(2*e^X+6) into the Function aplet, changing to the NUM view and then trying increasingly large values. As you can see (right) the limit appears to be 0.5, which is correct. However, if you continue to use larger values then the limit appears to change to 1 (see right). The reason for this is that the value of ex passes the limit of the capacity of the calculator (10500), and so the top and bottom of the fraction become equal (both at a value of 10500 ) instead of the true situation of the bottom being roughly twice the top. This is most likely to happen with limits involving power functions as they will overflow for smaller values of x. A related effect happens when investigating the behavior of the commonly used calculus limit of 1 lim 1 n n n →∞ +       . Suppose that you wish to use the Function aplet to evaluate what happens to F1(X)=(1+1/X)^X by changing to the NUM view and choosing the ‘Build Your Own’ facility (see right). The convergence towards e can also be seen graphically in the PLOT view but is very slow to reach high accuracy.

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 ( 11 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 = 10 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 = 11 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.

This can be introduced via the Function aplet. In the Function aplet, enter the function being studied into F1(X). To examine the gradient at x=3, store 3 into memory A in the HOME view as shown right. Return to the SYMB view, un-CHK F1(X) and enter the expression F2(X)=(F1(A+X)-F1(A))/X in F2(X). This is the basic differentiation formula with X taking the role of h and A being the point of evaluation. Change to the NUM SETUP view and change the NumType to “Build Your Own”. By entering successively smaller values for X you can now investigate the limit as h tends towards zero. To investigate the gradient at a different point simply change back to the HOME view, enter a new value into A and then return to the NUM view. The disadvantage of the previous method is that it is not very visual. An alternative is to use an aplet downloaded from the web. ( ) ( ) ( ) 0 ' lim h f a h f a f a h → + − =

The POLYROOT function can be used to find roots very quickly, but the results are often difficult to see in the HOME view, particularly if they include complex roots. This can be dealt with easily by storing the results to a matrix. For example, suppose we want to find the roots of 3 2 ( ) 3 3 f x x x = − + . We will use the POLYROOT function and store the results into M1. The advantage of this is that you can now view the roots by changing to the Matrix Catalog. and pressing . See page 170 for more detailed information on matrices. In addition to this, you can access the roots in the HOME view as shown. Calculator Tip This trick is particularly helpful if you are working with complex roots. Not only does it make it easier to re-use them it makes it easier to tell at a glance which are real and which complex.

views), there is another key which we have so far only used very fleetingly - the VIEWS key. It may seem odd to devote an entire chapter to what might appear to be an inconsequential key. In fact, however, this button is very useful to the effective use of the calculator, and crucial if you intend to use aplets downloaded from the internet. The contents of the VIEWS menu changes according to which aplet you are currently using. The Function aplet contents are covered here but the others differ in only small ways. However, aplets downloaded from the Internet will usually have a radically different VIEWS menus created by the person who wrote the program for the aplet. See page 214 for more information on this process if you intend to program the calculator. The VIEWS key pops up the menu on the right. We have already seen the use of the Auto Scale option (see page 96) and this is probably the one most used. The other options are also very useful at times. Shown on the right is the graph of 2 1( ) 1 F X X = −. If we press the VIEWS key, the menu above will pop up.

screen into two halves and re-plots the graph in each half. The right hand side can now be used to without affecting the left screen. For example a Box zoom shows the result on the right allowing easy comparison of ‘before’ and ‘after’ views. The left hand graph is always the active one, with results of actions shown on the right. We can now use the left graph again to zoom in on another section of interest, or alternatively, press the key under the label. This switches the right hand graph onto the left screen. Using or the FCN menu you can then find or examine points of interest. Alternatively you can zoom in again using another Box zoom. Any of the normal tools such as Signed Area… or Extremum can be used in this split screen. All the normal function tools are available except for .

Table. This option plots the graph on the right, with the Numeric view on the right half screen. Using the left/right arrow keys moves the cursor in both the graph and the numeric windows. See page 95 for information on how to keep nice scales in the table view. When more than one function defined in the SYMB view, pressing the up or down arrows changes the table focus from one to another. In this case, with only one, it centers the table. Let us switch now to a graph of the two functions 2 1( ) 1 F X X = − & 3 2 2( ) 2 5 4 F X X X X = + − − . This shown on the right, using an XRng of -8 to 8. Changing to Plot-Table gives the result shown left. As you can see, the scale has been preserved unchanged, although without labels. The table on the right also uses a sensible scale of 0.25 because of the choice of an x axis scale of -8 to 8. Choice of scale in the Plot-Table view is discussed in detail in the “Tips and Tricks – Nice Scales” immediately following the chapter on the Function aplet. Looking at the table heading you will see that it currently shows the function F1(X). The left/right arrow keys move within that function, with the cursor keeping track. Pressing the up/down arrows now not only centers the table highlight but, more importantly switches from F1(X) to F2(X). The X column will not change.

What makes this view even more useful is that the table keeps its ‘nice’ scale even while the usual ‘FCN’ tools are being used. As you can see in the screenshot left, the table is automatically repositioned to show the closest pixel value to that of the extremum found. The Signed Area… tool is also available in this view and when the cursor is moved the values in the table follow it. Unfortunately, the highlighted value in the table doesn’t change as the cursor moves to create the shaded area. For this reason the best strategy is to use the key to jump to the end point. This means that the area will not be shaded but this should not be a problem. Overlay Plot Another possibility from the VIEWS menu is Overlay Plot. This option can be used to add another graph over the top of an existing one, without the screen being blanked first as it usually is. As an example, if you have already graphed functions F1(X) through to F6(X) and then add another one in the SYMB view, then you don’t really want to have to wait while all the earlier ones are redrawn. If you un- the earlier graphs and then use Overlay Plot for the new one then it will be drawn over the top of the existing ones. Of course the results will not be good if the scales don’t match!. You can even use this to combine different styles of graphs. For example, the screen shot right was produced by drawing a circle in the Parametric aplet and then superimposing the equations y = x and y = -x in the Function aplet. This could be used to show, for example, enclosing curves for conic sections. You can also use this technique to overlay functions on top of statistical graphs.

graph if you are not sure in advance of the scale. After using Auto Scale you can then use the PLOT SETUP view to adjust the results. It is important to understand two points about how Auto Scale works. 1. It works by using the X-axis range that is currently chosen in PLOT SETUP to adjust the Y-axis range to include as much of the graph as possible. It will not adjust the x axis. 2. Auto Scale is done only for the first graph with a . If there are other graphs and they don’t fit the scale then they will not benefit. As you can see in the example shown right, the quadratic shows well but the second graph (a cubic) shows only an ascending section. Zooming out would be an option at this stage, as would un- ing the quadratic in the hopes that Auto Scaling the cubic might give better results (unlikely). 3. The resulting y scale is often not a very ‘nice’ one. You will usually have to adjust it in the PLOT SETUP view to make it look good. The excessively thick axis in the first screen shot of the sequence of three on the right is a common result of Auto Scale and is caused by the value of Ytick being too small. The third graph has a Ytick value of 10 instead of 1.

The next option of Decimal resets the scales so that each pixel (dot on the screen) is exactly 0.1. The result is an X scale of 6 5 6 5 x −⋅≤ ≤ ⋅ and a Y scale of 3 1 3 2 y −⋅≤ ≤⋅. This may not give the best view of the function. Personally I don’t often use it, as it is generally easier to go to the PLOT SETUP view and press SHIFT CLEAR. The Integer option is similar to decimal, except that it sets the axes so that each pixel is 1 rather than 0.1 thus giving an X scale of 65 65 X − ≤ ≤ . The usual result of this is rather horrible. The final option of Trig is designed for graphing trig functions. It sets the scale so each pixel is 24 π . This means that if you were graphing 1( ) 2sin( ) F X X = then 24 presses of the left or right arrows would move you through exactly π and the value would be exactly 2 instead of a horrible decimal. If you zoom in or out from this, the jumps will still stay relatively nice, particularly since 24 has so many factors. For example, with a zoom factor of 2, zooming out once would mean each pixel was now 48 π , while zooming in would give a pixel jump of 12 π . As you can see, the cursor is nicely at a value of 2 π , giving a y value of exactly 1 instead of only very close.

− to 2π . If you are primarily interested in the first 2π of the graph then simply change Xmin to zero. Alternatively you can move the cursor to π (the middle) and then zoom in. The example below uses zoom factors of 2x2 with Recenter: ed. Calculator Tip In the graphs above the cursor is at x = π. The coordinates at the bottom of the screen should show F1(X)=0 but doesn’t due to the fact that the value of π stored internally is not (and of course cannot) be exact. The rounding of π in the 13th decimal place means that the resulting trig values will be ‘wrong’ in the 11th to 15th decimal place depending on the function used.

m tthhee IInntteerrnneett The most powerful feature of the hp 39g+ is that you can download aplets and programs from the internet to help you to learn and to do mathematics. Two quick examples of aplets that are available are shown here. More are listed in the supplementary appendix on “Teaching Calculus using the hp 39g+”. Notice that in each case the aplet is controlled by a menu. This menu is created by the programmer and ‘attached’ to the VIEWS button so that it displays in place of the normal menu. Curve Areas This aplet allows the user to find approximations to the area under a curve by finding either the lower rectangular area, the upper rectangular area, or the trapezoidal area. The user can choose the end points of the interval, the type of calculation and the number of rectangles to be used. The rectangles are drawn on the screen. A worksheet introduces the idea of integration to find areas. Linear Programming This aplet visually solves linear programming problems, finding the vertices of the feasible region and the max/min of an objective function. The final stage of finding the vertices is a bit slow on an hp 39g but more acceptable on an hp 39g+.

third independent variable t. It is generally very similar to the Function aplet and so we will look mainly at the way it differs. An example of a graph from this aplet is: ( ) ( ) ( ) 5cos 0 2 ( ) 3sin 3 x t t t y t t π  =  ≤≤  =  which gives: Although it you can graph equations of this type, only some of the usual PLOT tools are present. As you can see in the screen shot above, the key is not shown, meaning that none of its tools are available. Thinking about the nature of these equations will tell you why. As usual the first step is to choose it in the Aplet Library. Press the APLET key, highlight Parametric and press . If you wish to ensure that you see the same thing as the examples following then press the button before pressing . As with the Function aplet, this aplet begins in the SYMB view by allowing you to enter functions, but the functions are paired. Each function consists of a function in T for X and another for Y. Choose XRng, YRng & TRng Looking at the PLOT SETUP view, you will see that we now have to enter a range for T as well as the usual ranges for X and Y. It is crucial to understand the different effect of the T range to that of the X and Y. Calculator Tip The default setting for TStep is 0.1. In my experience this is too large and can result in graphs that are not sufficiently smooth. It is worth developing the habit of changing it to 0.05.