Benutzeranleitung / Produktwartung 50g des Produzenten HP (Hewlett-Packard)
Zur Seite of 887
HP g gr aphing calc ulator user ’s guide H Ed it i on 1 HP part number F2 2 2 9AA-900 06.
Notice REG ISTER Y OUR PRODUCT A T: w ww .regis ter .hp .com THI S MANUAL AND ANY EX AMPLES CONT AINED H EREIN ARE PRO VIDED “ AS IS” AND A RE SUBJECT T O CHANGE WITHOUT NOTICE.
Pref ace Y ou have in y our hands a compact sy mbolic and numer ical comput er that will fac ilitate calc ulation and mathematical anal ysis o f pr oblems in a var iety of disc iplines, fr om elementary mathematic s to adv anced engineering and s c ience subjec ts.
F or sy mbolic oper ations the calc ulator includes a po we rful C omputer A lgebr aic S ystem (CA S) that lets you select diff er ent modes o f oper ation , e.g . , comple x numbers v s. r eal numbers , or ex act (sy mbolic) vs . appro ximat e (numer ical) mode .
Pa g e TO C - 1 T able of contents Chapter 1 - Getting started ,1-1 Basic Operations ,1-1 Batteries ,1-1 Turning the calculator on and off ,1-2 Adjusting the display contrast ,1-2 Contents of the calculator’s display ,1-2 Menus ,1-3 SOFT menus vs.
Pa g e TO C - 2 Chapter 2 - Introducing the calculator ,2-1 Calculator objects ,2-1 Editing expressions on the screen ,2-3 Creating arithmetic exp ressions ,2-3 Editing arithmetic expressions ,2-6 Cre.
Pa g e TO C - 3 Other flags of interest ,2-66 CHOOSE boxes vs. Soft M ENU ,2-67 Selected CHOOSE boxes ,2-69 Chapter 3 - Calculation with real numbers ,3-1 Checking calculators settings ,3-1 Checking c.
Pa g e TO C - 4 Physical constants in the calcula tor ,3-29 Special physical functions ,3-32 Function ZFACTOR ,3-32 Function F0 λ ,3-33 Function SIDENS ,3-33 Function TDELTA ,3-33 Function TINC ,3-34.
Pa g e TO C - 5 FACTOR ,5-5 LNCOLLECT ,5-5 LIN ,5-5 PARTFRAC ,5-5 SOLVE ,5-5 SUBST ,5-5 TEXPAND ,5-5 Other forms of substitution in algebraic expressions ,5-6 Operations with transcendental functions .
Pa g e TO C - 6 The PROOT function ,5-21 The PTAYL function ,5-21 The QUOT and REMAINDER functions ,5-21 The EPSX0 function and the CAS variable EPS ,5-22 The PEVAL function ,5-22 The TCHEBYCHEFF func.
Pa g e TO C - 7 Variable EQ ,6-26 The SOLVR sub-menu ,6-26 The DIFFE sub-menu ,6-29 The POLY sub-menu ,6-29 The SYS sub-menu ,6-30 The TVM sub-menu ,6-30 Chapter 7 - Solving multiple equations ,7-1 Ra.
Pa g e TO C - 8 List size ,8-10 Extracting and inserting elements in a list ,8-10 Element position in the list ,8-11 HEAD and TAIL functions ,8-11 The SEQ function ,8-11 The MAP function ,8-12 Definin.
Pa g e TO C - 9 Changing coordinate sy stem ,9-12 Application of vector operations ,9-15 Resultant of forces ,9-15 Angle between vectors ,9-15 Moment of a force ,9-16 Equation of a plane in space ,9-1.
Pa g e TO C - 1 0 Function VANDERMONDE ,10-13 Function HILBERT ,10-14 A program to build a matrix out of a number of lists ,10-14 Lists represent columns of the matrix ,10-15 Lists represent rows of t.
Pa g e TO C - 1 1 Function TRAN ,11-15 Additional matrix operations (The matri x OPER menu) ,11-15 Function AXL ,11-16 Function AXM ,11-16 Function LCXM ,11-16 Solution of linear systems ,11-17 Using .
Pa g e TO C - 1 2 Function QXA ,11-53 Function SYLVESTER ,11-54 Function GAUSS ,11-54 Linear Applications ,11-54 Function IMAGE ,11-55 Function ISOM ,11-55 Function KER ,11-56 Function MKISOM ,11-56 C.
Pa g e TO C - 1 3 Fast 3D plots ,12-34 Wireframe plots ,12-36 Ps-Contour plots ,12-38 Y-Slice plots ,12-39 Gridmap plots ,12-40 Pr-Surface plots ,12-41 The VPAR variable ,12-42 Interactive drawing ,12.
Pa g e TO C - 1 4 The SYMBOLIC menu and graphs ,12-49 The SYMB/GRAPH menu ,12-50 Function DRAW3DMATRIX ,12-52 Chapter 13 - Calculus Applications ,13-1 The CALC (Calculus) menu ,13-1 Limits and derivat.
Pa g e TO C - 1 5 Integration with units ,13-21 Infinite series ,13-22 Taylor and Maclaurin’s serie s ,13-23 Taylor polynomial and reminder ,13-23 Functions TAYLR, TAYLR0, and SERIES ,13-24 Chapter .
Pa g e TO C - 1 6 Checking solutions in the calc ulator ,16-2 Slope field visualization of solutions ,16-3 The CALC/DIFF menu ,16-3 Solution to linear and non-linear equations ,16-4 Function LDEC ,16-.
Pa g e TO C - 1 7 Numerical solution of first-order ODE ,16-57 Graphical solution of first-order ODE ,16-59 Numerical solution of second-order ODE ,16-61 Graphical solution for a second-order ODE ,16-.
Pa g e TO C - 1 8 Chapter 18 - Statistical Applications ,18-1 Pre-programmed statistical features ,18-1 Entering data ,18-1 Calculating single-variable statistics ,18-2 Obtaining frequency distributio.
Pa g e TO C - 1 9 Paired sample tests ,18-41 Inferences concerning one proportion ,18-41 Testing the difference betw een two proportions ,18-42 Hypothesis testing using pre-programmed features ,18-43 .
Pa g e TO C - 2 0 Custom menus (MENU and TMENU functions) ,20-2 Menu specification and CST variable ,20-4 Customizing the keyboard ,2 0-5 The PRG/MODES/KEYS sub-menu ,20-5 Recall current user-defined .
Pa g e TO C - 2 1 “De-tagging” a tagged quantity ,21-33 Examples of tagged output ,21-34 Using a message box ,21-37 Relational and logical operators ,21-43 Relational operators ,21-43 Logical oper.
Pa g e TO C - 22 Examples of program-generated plots ,22-17 Drawing commands for use in programming ,22-19 PICT ,22-20 PDIM ,22-20 LINE ,22-20 TLINE ,22-20 BOX ,22-21 ARC ,22-21 PIX?, PIXON, and PIXOF.
Pa g e T O C - 2 3 Chapter 24 - Calculator objects and flags ,24-1 Description of calculator objects ,24-1 Function TYPE ,24-2 Function VTYPE ,24-2 Calculator flags ,24-3 System flags ,24-3 Functions .
Pa g e TO C - 24 Storing objects on an SD card ,26-9 Recalling an object from an SD card ,26-10 Evaluating an object on an SD card ,26-10 Purging an object from the SD card ,26-11 Purging all objects .
Pa g e TO C - 2 5 Appendix F - The Applications (APPS) menu ,F-1 Appendix G - Useful shortcuts ,G-1 Appendix H - The CAS help facility ,H-1 Appendix I - Command catalog list ,I-1 Appendix J - MATHS me.
Pa g e 1 - 1 Chapter 1 Get ting started This c hapter pr ov ides basi c inf ormatio n about the operati on of your calc ulator . It is designed to familiar iz e y ou w ith the basic oper ations and settings befo re y ou perfor m a calc ulation.
Pa g e 1 - 2 b . Insert a new CR20 3 2 lithium battery . Make sur e its positi ve (+) si d e is fac ing up. c. Replace the plate and push it to the or iginal place .
Pa g e 1 - 3 At the top of the displa y you w ill hav e two line s of inf ormati on that descr i be the settings of the calc ulator . T he first line sho ws the c har acter s: R D XYZ HEX R = 'X' F or details on the meaning of thes e s ymbo ls see Chapter 2 .
Pa g e 1 - 4 E ach gr oup of 6 entr ies is called a Menu page . The c urr ent menu , know n as the T OOL menu (see be lo w) , has ei ght entri es arr anged in two page s. The ne xt page , containing the next tw o entries o f the menu is av ailable by pr essing the L (NeXT menu) k e y .
Pa g e 1 - 5 This CHOO SE box is labeled B ASE MENU and pr ov ides a list of n u mber ed fun cti ons, from 1 . H EX x to 6. B R. This dis play w ill constitute the f irst page of this CHOOSE bo x menu sho wing si x menu functi ons.
Pa g e 1 - 6 If y ou no w pres s ‚ã , instead of the CHOO SE box that y ou sa w earli er , the displa y will no w show si x soft men u labels as the f irst page o f the S T A CK menu: T o nav igate.
Pa g e 1 - 7 The T OOL menu Th e soft men u ke ys f or the men u cur ren tly dis pla yed , kno wn as the T OOL menu , are a ssoc iated with oper ations r elated to manipulation of v ariable s (see pag.
Pa g e 1 - 8 9 k ey the T IME choo se bo x is acti vated . This oper ation can also be r epre sented as ‚Ó . Th e TI ME cho os e b ox i s s hown in th e figu re b el ow: As indicated a bov e, the T IME menu pr ov ides f our differ ent options, number ed 1 thr ough 4.
Pa g e 1 - 9 Let’s c hange the m inute f ield to 2 5 , b y pressing: 25 !!@@OK#@ . T he seconds fi eld is now hi ghlighted . Suppose that y ou want to c hange the seconds fi eld to 4 5, use: 45 !!@@OK#@ The time f ormat f ield is no w highlighted .
Pa g e 1 - 1 0 Setting the date After setting the time f ormat option , the SET T IME AND D A TE input f orm w ill look as fo llo ws: T o set the date , f irst s et the date f ormat . The def ault for mat is M/D/Y (month/ day/y ear). T o modif y this f ormat, pr ess the do wn arr o w k ey .
P age 1-11 Introducing the calc ulator ’s k eyboar d The f igur e below sho ws a diagr am of the calculator ’s k ey board w ith the numbering of its r ow s and columns. T h e f i g u r e s h o w s 1 0 r o w s o f k e y s c o m b i n e d w i t h 3 , 5 , o r 6 c o l u m n s .
P age 1-12 shift ke y , k ey ( 9 ,1) , and the ALPHA k ey , ke y (7 ,1) , can be combined with some of the other k ey s to acti vate the alternati ve f unctions sho wn in the k ey board .
Pa g e 1 - 1 3 Pr ess the !!@@OK#@ soft men u k ey t o r etur n to normal display . Ex amples of selecting diffe ren t calc ulator modes ar e show n next . Oper ating Mode The calc ulator offer s two oper ating modes: the Algebr aic mode , and the Re vers e P olish Notatio n ( RPN ) mode .
Pa g e 1 - 1 4 T o enter this e xpres sion in the calc ulator w e will f irst us e the equati on wr iter , ‚O . P lease identify the f ollo wing k ey s in the k ey board , besides the numer ic k ey pad k e ys: !@.
Pa g e 1 - 1 5 Change the oper ating mode to RPN by f irst pr essing the H butto n. S elect th e RPN operating mode b y either using the k ey , or pr essing the @CHOOS soft m e n u k e y . P r e s s t h e !!@@OK#@ soft men u ke y to complete the oper ation .
Pa g e 1 - 1 6 3.` Ent er 3 in lev el 1 5.` Ent er 5 in lev el 1, 3 mov es to y 3.` Ent er 3 in lev el 1, 5 mov es to lev el 2 , 3 to lev el 3 3.* P lace 3 and multiply , 9 appears in le vel 1 Y 1/(3 × 3), last value in lev .
Pa g e 1 - 1 7 Notice ho w the expr ession is placed in stac k lev el 1 after pressing ` . Pr essing the EV AL k ey at this po int will e valuate the numer ical value of that e xpr essi on Note: In RP.
Pa g e 1 - 1 8 mor e about r eals, see Cha pter 2 . T o illu str ate this and other number f ormats try the fo llo w ing ex erc ises: Θ Standard f ormat : This mode is the mos t used mode as it sho ws number s in the most famili ar notation .
Pa g e 1 - 1 9 Notice that the Number F ormat mode is set t o Fix follo wed b y a z er o ( 0 ). This n umber indicate s the number of dec imals to be sho wn after t he dec imal point in the calc ulator’s displa y . Press the !!@ @OK#@ sof t menu ke y to r eturn to the calc ulator display .
Pa g e 1 - 2 0 Press the !!@@OK#@ soft menu ke y to complete the selection: Pr ess the !!@@OK#@ s oft menu k ey r eturn to the calc ulator displa y . The n umber now is sho wn as: Notice ho w the number is r ounded, not tr uncated . Thu s, the number 12 3 .
Pa g e 1 - 2 1 same fashi on that we c hanged the Fixe d number of dec imals in the exa mp l e a b ove ) . Pr ess the !!@@OK#@ soft menu ke y retur n to the calc ulator display . The n umber now is sho wn as: This r esult , 1.2 3E2 , is the calculator ’s versi on of po wer s-of-ten notatio n, i.
Pa g e 1 - 22 Pr ess the !!@@OK#@ s oft menu k ey r eturn to the calc ulator displa y . The number now is sho wn as: Becaus e this number has thr ee fi gur es in the intege r part, it is sho wn w ith four si gnificati ve f igur es and a zer o pow er of ten , while using the Engineer ing for mat.
Pa g e 1 - 23 Θ Pr ess the !!@@OK#@ s oft menu k ey r eturn to the calc ulator displa y . The number 12 3 .45 6 7 8 9012 , enter ed earlie r , no w is sho wn as: Angle Me asure T ri gonometric functi ons, for e xample , requir e arguments r epre senting plane angles .
Pa g e 1 - 24 ke y . If using the latter appr oach, u se up and do wn arr ow k ey s, — ˜ , to selec t the pref err ed mode , and pr ess the !!@@OK#@ soft m enu key to complete the ope rati on.
Pa g e 1 - 25 fr om the positi ve z ax is to the r adial distance ρ . The R ectangular and Spher ical coordinate s ys tems are r elated by the follo w ing quantities: T o change the coor dinate s ys tem in yo ur calculat or , follo w these step s: Θ Pr ess the H bu t ton .
Pa g e 1 - 26 _La st St ack : K eep s the contents o f the last stac k entry for use w ith the functi ons UNDO and ANS (s ee Chapter 2). The _Beep option can be us eful t o adv ise the user abou t err ors. Y ou ma y want to deselec t this option if using y our calc ulator in a cla ssr oom or library .
Pa g e 1 - 27 Selecting Display modes The calc ulator display can be c ustomi z ed to your pr efer ence by selec ting different disp lay mod es. T o see t he opt ional disp lay setti ngs use t he follow ing: Θ F irst , pr ess the H button to ac tivat e the CAL CULA T OR MODE S input f orm .
Pa g e 1 - 28 Pr essing the @CH OOS soft men u k ey w ill pr ov ide a list of a vailable s yst em fo nts, as sho wn belo w: The opti ons availa ble ar e three standar d Sys t e m Fo n t s (siz es 8, 7 , and 6 ) and a Br ow se .
Pa g e 1 - 2 9 displa y the DISPLA Y MOD E S input for m. Pr ess the do wn arr ow k ey , ˜ , twi ce, to get to the St ack line . This line sho ws tw o propertie s that can be modifi ed. When these pr operties ar e select ed (chec ked) the fo llo wi ng effec ts are acti vated: _Small Changes f ont si ze to small .
Pa g e 1 - 3 0 times , to get t o the EQW (E quation W riter ) line. T his line shows tw o propertie s that can be modifi ed. When these pr operties ar e select ed (chec k ed) the fo llow ing eff ects.
Pa g e 1 - 3 1 ri ght arr ow k ey ( ™ ) t o select the under line in fr ont of the options _Clock or _Analog . T oggle the @ @CHK@@ soft menu k e y until the desir ed setting is achie ved . If the _Clock opti on is selected , the time of the da y and date w ill be sho wn in the upper r ight corner of the display .
Pa g e 2 - 1 Chapter 2 Intr oducing the calc ulator In this chapter w e present a n umber of basic operati ons of the calculator including the u se of the E quation W riter and the manipulatio n of data obj e c ts in the calc ulator .
Pa g e 2 - 2 the CAS , it might be a good idea to sw itch dir ectl y into appr ox imate mode. Re fer t o Appendi x C for mor e det ails. Mi xing integers and reals together or mi staking an integer for a real is a common occ urre nce.
Pa g e 2 - 3 Binary integers , obje cts of t ype 10 , are used i n some computer science applications . Graphics objec ts , ob jects o f t ype 11, s tor e graphi cs produced b y the calculator . T agged objec ts , obj ects of ty pe 12 , ar e used in the ou tput of man y progr ams to identify r esults .
Pa g e 2 - 4 The r esulting e xpres sion is: 5.*(1.+1./7.5)/( √ 3.-2.^3). Press ` to get the e xpres sion in the display as f ollow s: Notice that , if your CA S is set to EXACT (s ee Appendix C) and y ou enter y our e xpr essi on using integer number s for in teger v alues, the r esult is a sy mbolic quantity , e .
Pa g e 2 - 5 T o e valuate the e xpr essi on w e can use the EV AL f u ncti on, as f ollo ws: μ„î` As in the pre vi ous e xample , you w ill be ask ed to appr ov e changing the CAS setti ng to Appro x . Once this is done , you w ill get the same r esult as befo r e.
Pa g e 2 - 6 This latter r esult is pur ely numer ical , so that the t w o re sults in the stack , although r epre senting the same e xpres sion, seem diff erent .
Pa g e 2 - 7 The editing c ursor is sho wn as a blinking le ft arr ow o ver the f irst c harac ter in the line to be edited. Since the editing in this case consists of r emov ing some char acter s and.
Pa g e 2 - 8 W e set the calc ulator operating mode t o Algebr aic, the CA S to Exact , and the displa y to T extbook . T o ent er this algebr aic e xpre ssion w e use the f ollo wing keyst ro kes : .
Pa g e 2 - 9 Θ Pr ess the r ight arr ow k ey , ™ , until the c ursor is to the ri ght of the x Θ Ty p e Q2 to enter the pow er 2 fo r the x Θ Pr ess the r ight arr ow k ey , ™ , until the c ursor is to the ri ght of the y Θ Pr ess the delet e ke y , ƒ , once to er ase the char acters y.
Pa g e 2 - 1 0 Θ Pr essi ng ` once more to re turn to normal display . T o see the entir e expr essi on in the scr een, w e can c hange the option _Small Stack Disp in the DISP LA Y MODE S input for m (see Chapter 1).
Pa g e 2 - 1 1 The si x soft menu k ey s for the E quation W rit er acti vate the fo llow ing functi ons: @EDIT : lets the u ser edit an entry in the line editor (see e xample s abo ve) @CURS : highli.
Pa g e 2 - 1 2 The r esult is the e xpr essi on The cur sor is sho wn a s a left-fac ing ke y . T he curs or indicat es the c urr ent edition location . T yp ing a char acter , f unctio n name, or oper ation w ill enter the corr esponding char acter or c h ar acters in the c ursor location .
Pa g e 2 - 1 3 Suppos e that no w y ou want t o add the fr action 1/3 to this entir e expr ession , i .e., y ou wan t to ent er the expr ession: F irst , w e need to highli ght the entir e firs t ter m by using either the r ight ar ro w ( ™ ) or the upper arr o w ( — ) k ey s, r epeatedly , until the entire e xpre ssion is highli ghted , i.
Pa g e 2 - 1 4 Show i ng the expr ession in smaller-si ze T o show the e xpres sion in a smaller -siz e fo nt (w hic h could be usef ul if the e xpre ssi on is long and con volut ed) , simply pr ess the @BIG soft men u k ey .
Pa g e 2 - 1 5 If y ou wan t a floating-point (n umerical) e valuation , use the NUM fun ction (i .e., …ï ). T he r esult is as follo ws: Use the function UNDO ( …¯ ) o n c e mo re to re c o.
Pa g e 2 - 1 6 A sy mbolic ev aluation once more . Suppose that , at this point , we w ant to ev aluate the left -hand side fr action onl y . Pre ss the upper ar r o w ke y ( — ) three times to s el.
Pa g e 2 - 1 7 Editing arithmetic e x pr essions W e will sho w some of the editing featur es in the Equati on W riter as an e xer cis e. W e start by e ntering the f ollow ing expr essi on used in th.
Pa g e 2 - 1 8 Pr ess the do wn ar ro w ke y ( ˜ ) to trigger the c lear editing c ursor . The scr een now looks lik e this: By using the left arr ow k ey ( š ) y ou ca n mo ve the c urso r in the gener al left dir ecti on, bu t stopping at eac h indiv idual component of the e xpres sion .
Pa g e 2 - 1 9 Ne xt, w e’ll con vert the 2 in front of the par enth eses in the denominator into a 2/3 by using: šƒƒ2/3 At this point the e xpr essi on looks as fo llow s: The f inal step is to r emov e the 1/3 in the ri ght -hand side o f the expr essi on.
Pa g e 2 - 2 0 Use the follo wing k ey str okes: 2 / R3 ™™ * ~‚n+ „¸ ~‚m ™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c ~„y ——— / ~‚t Q1/3 This r esults in the output: In this ex ample we us ed se ve ral lo we r- case English letter s, e .
Pa g e 2 - 2 1 Editing algebraic e xpressions The editing of algebr aic equations f ollow s the same rules as the editing of algebrai c equations. Name ly : Θ Use the ar r ow k ey s ( š™—˜ ) to highli ght expr essions Θ Use the do wn arr o w ke y ( ˜ ) , re peatedly , to tri gger the clear editing c ursor .
Pa g e 2 - 2 2 2. θ 3. Δ y 4. μ 5. 2 6. x 7. μ in the expone ntial func tion 8. λ 9. 3 i n t h e √ 3 te rm 10. the 2 in the 2/ √ 3 fr action At an y point we can c hange the clear editing cur sor into the insertio n cur sor by pr essing the delet e k ey ( ƒ ).
Pa g e 2- 23 Ev aluating a sub-expr ession Since w e alread y have the sub-e xpre ssion highli ghted , let’s pre ss the @EVAL so ft menu ke y to ev aluate this sub-expr ession . The re sult is: Some algebr aic expr essions cannot be simplif ied any more .
Pa g e 2- 24 3 in the fi rst ter m of the numerator . Then , pres s the ri ght a rr o w k ey , ™ , to nav igate through the e xp r ession . Simplifying an e xpression Pr ess the @BIG soft menu k ey to get the sc r een to look as in the pre vi ous f igur e (see abo ve).
Pa g e 2- 25 Press ‚¯ to reco ver the or iginal expr ession . Next , enter the follo wing keyst ro kes : ˜˜˜™™™™™™™———‚™ to sele ct the last two ter ms in the e xpre ssion , i.e ., pr ess the @F ACTO soft menu k ey , to get Press ‚¯ to reco ver the ori ginal expr ession .
Pa g e 2 - 2 6 Ne xt, s elect the command DERVX (the de ri vati ve w ith r espec t to the var iable X, the c urr ent CAS independent v ariable) b y using: ~d˜˜˜ .
Pa g e 2- 27 Detailed explanati on on the u se of the help fac ilit y fo r the CA S is pr esented in Chapter 1. T o retur n to the E quation W r iter , pre ss the @EXIT so ft menu k ey .
Pa g e 2 - 2 8 Ne xt, w e’ll cop y the fr actio n 2/ √ 3 from t he l ef tm ost fa ctor in the exp ression, and place it in the numerator o f the ar gument for the LN functi on.
Pa g e 2 - 2 9 W e can no w cop y this expr essio n and place it in the denominator o f the LN argume nt, as f ollow s: ‚¨™™ … (2 7 times ) … ™ ƒƒ … (9 times) … ƒ ‚¬ The li ne e.
Pa g e 2 - 3 0 T o see the corr esponding e xpres sion in the line editor , pr ess ‚— and the A soft menu k ey , to show : This e xpres sion sho ws the gener al for m of a summation typed dir ectly in the stack or line editor : Σ ( inde x = starting_value , ending_value , summation e xpres sion ) Press ` to r eturn to the E quation W r iter .
Pa g e 2 - 3 1 and the var iable of diff erentiati on. T o f i ll thes e input locatio ns, us e the follo wing keyst ro kes : ~„t™~‚a*~„tQ2 ™™+~‚b*~„t+~‚d The r esu lting scr een is .
Pa g e 2- 32 Definite integr als W e wi ll use the E quation W r iter to ente r the follo wing def inite integr al: . Pr ess ‚O to acti vate the E quation W r iter .
Pa g e 2- 33 Double integr als are als o possible . F or ex ample, whi ch ev aluates to 3 6. P artial e valuati on is possible , fo r ex ample: This integr al ev aluates to 3 6. Organizing data in the calculator Y ou can organi z e data in y our calculat or by st oring v aria bles in a direc tor y tr ee .
Pa g e 2 - 3 4 @CHDIR : Change to selected dir ectory @CANCL : Cancel a ction @@OK@@ : Appro ve a selec tion F or ex ample, to c hange directory to the CA SD IR, pr ess the do wn-arro w ke y , ˜ , and pr ess @ CHDIR . T his action c loses the Fi l e M a n a g e r w indow and r eturns us to nor mal calculator displa y .
Pa g e 2 - 3 5 T o mov e between the differ ent soft men u commands, y ou can use not only the NEXT ke y ( L ), but also the PREV k ey ( „« ). The u ser is in vited to try these f uncti ons on his or her o wn . The ir applicati ons ar e strai ghtforw ard .
Pa g e 2- 3 6 This time the CA SD IR is highlight ed in the scr een. T o see the contents of the dir ectory pr ess the @@ OK@@ sof t me nu ke y or ` , to get the f ollow ing scr een: The s cr een sho w s a table des cr ibing the var iables cont ained in the CA SDIR dir ectory .
Pa g e 2 - 37 Pr essing the L k ey show s one more var iable stor ed in this directory: • T o see the contents o f the var iable EPS , for e xam p le , use ‚ @EPS@ . T his sho ws the va lue of EP S to be .00 00000001 • T o see the value of a n umeri cal vari able , we need t o pres s only the soft menu k ey f or the v ari able .
Pa g e 2- 3 8 lock the alpha betic k ey board tempor aril y and enter a f ull name bef or e unloc king it again. T he follo w ing combination s of k ey str okes will lock the alphabetic k e yboar d: ~~ locks the alphabeti c ke yboar d in upper case .
Pa g e 2- 39 Creating subdir ectories Subdir ector ies can be cr eated by using the FILES en vir onment or by using the co mm a nd CR D IR. Th e t wo ap proa che s fo r cre at i ng su b- di rect orie s a re pr esent ed next .
Pa g e 2 - 4 0 The Object input f ield, the f irst input f ield in the fo rm , is highlight ed by def ault. This input f ield can hold the contents of a new v ariable that is be ing cr eated. Since w e hav e no contents f or the new sub-dir ectory at this point, we simpl y skip this input fi eld by pr essing the do wn-arr o w ke y , ˜ , once .
Pa g e 2 - 4 1 T o mo ve into the MAN S direct ory , pr ess the co rr esponding so ft menu k ey ( A in this case), and ` if in algebraic mode . T he direc tor y tr ee will be show n in the second line of the displa y as {HOME M NS} .
Pa g e 2- 42 Use the do wn ar ro w ke y ( ˜ ) to select the option 2. M E M O RY … , or j ust press 2 . Then, pr ess @@OK@@ . This w ill pr oduce the fo llow ing pull-dow n menu: Use the do wn arr ow k ey ( ˜ ) t o select the 5 . DIRE CT OR Y opti on, or j ust press 5 .
Pa g e 2- 4 3 Pr ess the @@ OK@ soft menu ke y to activ ate the command, to cr eate the sub- dir ectory: Mov ing among subdirectories T o mov e dow n the dir ector y tr ee, y ou need to press the s oft menu ke y corr esponding to the sub-dir ectory you w ant to mo ve to .
Pa g e 2 - 4 4 The ‘S2’ str ing in this f orm is the name o f the sub-direct ory that is being de leted . The s oft menu k ey s pro vi d e the f ollow ing options: @YES@ Pr oceed with deleting the.
Pa g e 2 - 4 5 Use the do wn ar ro w ke y ( ˜ ) to select the option 2. M E M O RY … T h e n , press @@OK@@ . This w ill produce the f ollo w ing pull-do wn menu: Use the do wn ar r o w ke y ( ˜ ) to select the 5 . DIRE CT OR Y option . The n, press @@OK@@ .
Pa g e 2 - 4 6 Press @@OK@@ , to get: Then , press ) @ @S3@@ to enter ‘S3’ as the ar gument to PGDIR. Press ` to delete the sub-dir ectory: Command PGDIR in RPN mode T o use the P GD IR in RPN mode y ou need to ha ve the name o f the direc tor y , between q uotes , alread y availa ble in the stac k befor e accessing the command.
Pa g e 2- 47 Using the PURGE command fr om the TOOL menu The T OOL men u is av ailable by pr essing the I ke y ( Algebr aic and RPN modes sho wn): The P URGE command is av ailable by pr essing the @PURGE s oft menu k e y .
Pa g e 2- 4 8 Using the FILES menu W e wi ll use the FILE S menu to enter the v ari able A. W e assume that w e are in the sub-dir ectory {HOME M NS IN TRO}. T o get to this sub-dir ectory , use the fo llo wing: „¡ and sel ect the INTR O sub-direc tor y as sho wn in this scr e en: Press @@OK@@ t o enter the dir ectory .
Pa g e 2- 49 T o enter var iable A (see table abov e ), we fir st enter its contents , namely , the number 12 .5, and then its name , A, as follo ws: 12.5 @@OK@@ ~a @@OK@@ . Resulting in the f ollow ing scr een: Press @@OK@@ once more to cr eate the vari able.
Pa g e 2- 5 0 Using the ST O command A simpler wa y to cr eate a var iable is by u sing the S T O command (i.e ., the K k ey). W e pr ov ide e xamples in both the A lgebrai c and RPN modes, b y cr.
Pa g e 2 - 5 1 z1: 3+5*„¥ K~„z1` (if needed , accept change t o Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ K~„p1` . The s cr een, at this po int, w ill look as follo ws: Y ou will see si x of the se ven v ariable s listed at the bottom of the sc reen: p1, z1, R, Q, A12 , α .
Pa g e 2 - 52 z1: ³3+5*„¥ ³~„z1 K (if needed, accept c hange to Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . The s cr een, at this po int, w ill look as follo ws: Y ou will s ee si x of the se ven v ar iables listed at the bottom o f the sc reen: p1, z1, R, Q, A12 , α .
Pag e 2- 53 Pr essing the s oft menu k ey cor r esponding t o p1 w ill pr ov ide an err or messa ge (try L @@@p1@@ ` ): Note: By pre ss i ng @@@p1@@ ` we ar e try ing to acti vate (run) the p1 progr am. Ho we ver , this progr am expects a nume ri cal input .
Pa g e 2 - 5 4 At this point , the scr een looks like this: T o see the contents o f A, use: L @@@A@@@ . To r u n p r o g r a m p1 w ith r = 5, use: L5 @@@p1@@@ . Notice that to run the progr am in RPN mode, y ou only need to enter the input (5) and pr ess the corr esponding soft menu k ey .
Pag e 2- 55 Notice that this time the contents o f pr ogr am p1 are liste d in t he scr een. T o see the r emaining var iables in this dir ectory , pr ess L : Listing the content s of all var iables in the screen Use the k ey str oke combinati on ‚˜ to list the contents of all var iables in the sc r een.
Pa g e 2- 56 follo wed b y the var iable ’s soft menu k ey . For e xample , in RPN, if we w ant to change the contents of v ariable z1 to ‘ a+b ⋅ i ’, u s e : ³~„a+~„b*„¥` This w ill place the algebrai c expr ession ‘ a+b ⋅ i ’ in le ve l 1: i n t h e st a ck.
Pa g e 2 - 5 7 Use th e up ar r o w ke y — to select the sub-dir ectory MANS and pres s @@OK@@ . If you no w press „§ , the scr een will sho w the contents of sub-directory MANS (notice that v ariable A is show n in this list, as e xp ect ed): Press $ @INTRO@ ` ( Algebrai c mode), or $ @IN TRO@ (RPN mode) to r eturn to the INTRO dir ectory .
Pa g e 2- 58 Ne xt, u se the delete k ey thr ee times, to r emo ve the las t three lines in the dis play : ƒ ƒ ƒ . At this poin t , the stac k is r eady to e xec ute the command ANS(1) z1 . Pr ess ` to e xec ute this command. Then , us e ‚ @@z1@ , to ver ify the contents of the v ariable .
Pa g e 2 - 59 Copy ing two or more v ariables using the stack in RPN mod e The f ollow ing is an ex erc ise to demonstr ate ho w to copy two or mor e var iables using the stac k when the calc ulator is in RPN mode.
Pa g e 2- 6 0 The s cr een no w show s the ne w order ing of the var iables: RPN mode In RPN mode, the list o f r e -orde red v ariables is listed in the st ack bef ore apply ing the command ORDER. Su ppose that w e start fr om the same situation as abov e, but in RPN mode , i.
Pa g e 2 - 6 1 Notice that v ariable A12 is no longer ther e. If y ou no w press 㤠, the sc r een w ill sho w the contents of sub-dir ectory MANS, inc luding vari able A12 : Deleting var iables V ari ables can be deleted using functi on PUR GE .
Pa g e 2 - 62 vari ab le p1 . Pr ess I @PURGE@ J @@p1@@ ` . The scr een will now sho w vari ab le p1 re m ove d : Y ou can use the P URGE command to era se mor e than one var iable b y plac ing their names in a lis t in the argument o f PUR GE.
Pa g e 2 - 6 3 the HIS T ke y: UNDO r esults fr om the ke ystr oke s equence ‚¯ , w hile CMD r esults fr om the k ey str oke se quence „® . T o illustr ate the us e of UNDO , try the follo w ing ex er c ise in algebr aic (AL G) mode: 5*4/3` . T he UNDO command ( ‚¯ ) will simply er ase the re sult.
Pa g e 2 - 6 4 As you can s ee, the number s 3, 2 , and 5, us ed in the fi rst calc ulation abov e, ar e listed in the se lecti on bo x, as w ell as the algebr a i c ‘SIN(5x2)’ , but not the SIN f u ncti on entered pr ev ious to the algebr aic.
Pa g e 2- 6 5 Ex ampl e of flag setting: general solutions v s. principal value F or ex ample, the def ault v a lue f or s yst em flag 01 is Gener al soluti ons . What this means is that, if an equati on has multiple soluti ons, all the s olutions w ill be r eturned b y the calculator , most lik ely in a list .
Pa g e 2- 6 6 ` (keep ing a second copy in the RPN stack) ³~ „t` Use the follo wing k ey strok e sequence to enter the QU AD command: ‚N~q (use the up and do wn arr ow k ey s, —˜ , to s elect command QU AD) , pr ess @@OK@@ .
Pa g e 2 - 67 CHOOSE bo x es vs. So f t MENU In some of the e xer cises pr esented in this chapter w e ha ve seen men u lists of commands displa yed in the sc reen .
Pa g e 2- 6 8 The s cr e en sh ow s flag 117 not s et ( CHOO SE box es ), as sho wn here: Pr ess the @ @CHK@@ soft menu k ey to set flag 117 t o soft MENU . The sc reen w ill r efl ect that c hange: Press @@OK@@ twice to r eturn to normal calculator displa y .
Pa g e 2- 69 Note: most o f the e xam p les in this us er guide assume that the c urre nt setting of flag 117 is its de fault s etting (that is, not set). If yo u hav e set the flag but w ant to str ictly f ollow the e xam ples in this guide , you should c lear the flag be for e contin uing.
Pa g e 2- 70 • The CMD S (CoMmanD S) menu , acti vat ed w ithin the Equati on W riter , i. e. , ‚O L @CMDS.
Pa g e 3 - 1 Chapter 3 Calculation with real numbers This c hapter demonstr ates the use of the calc ulator for oper ations and func tions r elated to r eal numbers . Oper ations along the se lines ar e usef ul for mos t common calc ulations in the ph ysi cal sc iences and engineer ing.
Pa g e 3 - 2 2 . Coordinate sy stem specifi cat ion (XYZ , R ∠ Z, R ∠∠ ). T he s y mb ol ∠ stands f or an angular coor dinate . XYZ: Cartesi an or rectangular (x ,y ,z) R ∠ Z: cylindr ical P olar co or dinates (r , θ ,z) R ∠∠ : Spher ical coordinat es ( ρ,θ,φ ) 3 .
Pa g e 3 - 3 Real n u mber calc ulations w ill be demonstr ated in both the Algebr aic ( AL G) and Re ver se P olish Notation (RPN) modes . Changing sign of a number , variable, or e xpression Use the ke y . In AL G mode , yo u can pr ess be for e entering the number , e.
Pa g e 3 - 4 Alter nativ ely , in RPN mode , y ou can separate the oper ands with a space ( # ) befo re pr essing the oper ator ke y . Example s: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / Using parentheses P arentheses can be used to gr oup operations , as well as to enc lose arguments of func tions .
Pa g e 3 - 5 Squares and squar e roots The s quar e function , SQ, is a vailable thr ough the ke ystr ok e combination: „º . When calc ulating in the stack in AL G mode , enter the func tion befo r e the argument , e.g ., „º2.3` In RPN mode, ent er the number f irst , then the functi on, e .
Pa g e 3 - 6 Using po wers o f 10 in entering data P owe rs of te n, i.e. , n u mb e rs of th e for m - 4 .5 ´ 10 -2 , etc., ar e entered b y using the V k ey . For e xample , in AL G mode: 4.5V2` Or , in RPN mode: 4.5V2` Natural logar ithms and exponential function Natur al logarithms (i .
Pa g e 3 - 7 the inv erse tr igonometri c functi ons re pr esent angles, the ans w er fr om these func tions w ill be give n in the select ed angular measur e (DEG , RAD, GRD). Some e xamples ar e show n next: In AL G mode: „¼0.25` „¾0.85` „À1.
Pa g e 3 - 8 combinati on „´ . With the default s et ting of CHOO SE box es fo r syst em flag 117 (see Chapter 2), the MTH menu is show n as the follo wing menu list: As the y are a gr eat number of mathematic f unctions a vailable in the calc ulator , the MTH menu is so rted by the ty pe of obj ect the fu nctio ns apply on .
Pa g e 3 - 9 Hy perbolic func tions and their inverses Selecting Option 4. HYP ERBOLIC.. , in the MTH menu , and pres sing @@OK@@ , pr oduces the h yperboli c function men u: The h yperbolic f unction.
Pa g e 3 - 1 0 The r esult is: The ope rati ons show n abov e assume that you ar e using the defa ult setting for s ys tem flag 117 ( CHOO SE box es ) .
Pa g e 3 - 1 1 F or ex ample, to calc ulate tanh( 2 . 5), in the AL G mode, w hen using SOF T m en us over CHOO S E bo xe s , f ollow this pr ocedure: „´ Select MTH menu ) @@HYP@ Selec t the HYPERBOLIC.. men u @@TANH@ Selec t the TA N H func tion 2.
Pa g e 3 - 1 2 Option 19 . MA TH.. retur ns the user to the MTH menu . The r emaining functi ons ar e gr ouped into si x differ ent grou ps descr ibed belo w .
Pa g e 3 - 1 3 The r esult is sho wn ne xt: In RPN mode , recall that ar gument y is located in the second le v el of the st ack , while ar gument x is located in the f irst le vel o f the stac k. T his means, y ou should enter x fir st, and then , y , j ust as in AL G mode.
Pa g e 3 - 1 4 P lease notice that MOD is not a function, but r ather an operator , i .e., in AL G mode , MOD should be us ed as y MOD x , and not as MOD(y,x) .
Pa g e 3 - 1 5 G AMMA: The Gamma func tion Γ ( α ) P SI: N -th deri vati ve of the digamma f unctio n P si: Digamma f unction , deri vati ve of the ln(Gamma) The Gamma f unction is def ined by . This f unction has applicati ons in applied mathemati cs f or sc ience and engineering , as well as in pr obabil ity and statisti cs.
Pa g e 3 - 1 6 Example s of these s pec ial func tions ar e show n her e using both the AL G and RPN modes. As an e xe r c ise , verify that G AMMA(2 .
Pa g e 3 - 1 7 Selecting an y of these en tri es will place the v alue select ed, w hether a sy mbol (e .g., e , i , π , MINR , or MAXR ) or a v alue ( 2 .7 1.., (0,1) , 3 . 14.., 1E-4 99 , 9. 9 9. . E 4 9 9 ) in the stac k. P lease notice that e is a vailable fr om the ke yboar d as exp ( 1) , i.
Pa g e 3 - 1 8 The u ser w ill recogni z e mos t of these units (s ome , e.g ., dy ne , are not u sed v ery often no wada ys) fr om his or her ph ysics c lasses: N = ne wtons, dyn = dyne s, gf = gr am.
Pa g e 3 - 1 9 A vailable units The f ollow ing is a list of the units av ailable in the UNI TS men u . T he unit s ymbo l is show n first f ollow ed by the unit name in parenth eses: LENG TH m (meter.
Pa g e 3 - 2 0 SPEED m/s (meter per se cond), cm/s (centimeter per second), ft/s (feet per second), kph (kilometer per hour ) , mph (mile per hour), knot (nautical mile s per hour), c (speed of li ght.
Pa g e 3 - 2 1 ANGLE (planar and solid angle mea sur ements) o (se xage simal degree), r (radi an) , gr ad (gr ade) , arcmin (minut e of ar c) , arc s (second of ar c) , sr (ster adian) LIGHT (Illumin.
Pa g e 3 - 22 Conv er ting to base units T o conv ert an y of these units to the def ault units in the SI s yst em, u se the functi on UB ASE . F or e xample , to find out what is the v alue of 1 pois.
Pa g e 3 - 23 ` Con vert the units In RPN mode , s y stem flag 117 s et to SO FT m e nu s : 1 Enter 1 (no under line) ‚Û Select the UNIT S menu „« @ ) VISC Select the VIS COS ITY option @@@P@@ S.
Pa g e 3 - 24 Notice that the under scor e is entered a utomati cally when the RPN mode is acti ve . The r esult is the fo llow ing scr een: As indicated earl ier , if sy stem flag 117 is s et to SO FT m e n us , then the UNIT S menu w ill show up as labels f or the soft menu k eys .
Pa g e 3 - 25 Yy o t t a + 2 4 dd e c i - 1 Z z etta + 21 c c enti - 2 E ex a +18 m milli -3 P peta +15 μ mic ro -6 T ter a +12 n nano - 9 Gg i g a + 9 p p i c o - 1 2 Mm e g a + 6 f f e m t o - 1 5 .
Pa g e 3 - 26 whi ch sho ws as 6 5_(m ⋅ yd). T o conv ert to units of the SI s ys tem , use f unctio n UB ASE: T o calculat e a div ision, s ay , 3 2 50 mi / 5 0 h, enter it a s (3 2 50_mi)/(5 0_h) .
Pa g e 3 - 27 Stac k calculations in the RPN mode , do not r equir e y ou to enc lose the diff er ent terms in par enth eses, e.g . , 12_m ` 1.5_y d ` * 3 2 50_mi ` 5 0_h ` / The se oper ations pr odu.
Pa g e 3 - 2 8 UF ACT(x ,y) : fac tors a un it y fr om unit object x UNIT(x ,y): combines value of x w ith units of y The UB ASE f unction w as discu ssed in detail in an earli er secti on in this cha pter . T o access an y of these f unctions f ollow the e xamples pr ov ided earlier f or UB ASE .
Pa g e 3 - 2 9 Ex amples of UNI T UNIT( 25,1_m) ` UNIT(11. 3,1_mph) ` Ph ysical constants in the calculator F ollow ing along the treatment of units , we dis cu ss the use of ph ysical const ants that are a vailable in the calc ulator’s memory .
Pa g e 3 - 3 0 The s oft menu k ey s corre sponding to this CONS T ANT S LIBR AR Y sc r een include the fo llo wing f unctions: SI whe n select ed, cons tants values ar e show n in SI units ENGL w hen.
Pa g e 3 - 3 1 T o see the v alues of the constants in the English (or Imper ial) s ys tem , pre ss the @ENGL optio n: If we de-select the UNIT S opti on (pre ss @UNITS ) only the values ar e shown (English units se lected in this case): T o cop y the value of Vm to the st ack , select the v ariable name , and pre ss ! , then, pr ess @QUIT@ .
Pa g e 3 - 32 Special phy sical func tions Menu 117 , tri ggered b y using MENU(117) in AL G mode, or 117 ` MENU in RPN mode , produce s the fol low ing menu (labels lis ted in the displa y by u sing .
Pa g e 3 - 3 3 ZF ACT O R(x T , y P ) , wher e x T is the reduced temper ature , i . e ., the ratio o f actual temper ature to p seudo -cri tical temper ature , and y P is the r educed pr essur e, i .e., the r atio of the actual pr essur e to the pseudo -cr itical pr essur e .
Pa g e 3 - 3 4 Function TINC F unction TI NC(T 0 , Δ T) calculat es T 0 +D T . The ope rati on of this functi on is similar to that of f uncti on TDEL T A in the sens e that it retur ns a re sult in the units of T 0 . Otherwise , it retur ns a simple addition of value s, e .
Pa g e 3 - 3 5 Pr ess the J k ey , and you w ill notice that ther e is a ne w vari able in y our so ft menu k ey ( @@@H@@ ). T o see the contents of this var iable pr ess ‚ @@@H@@ .
Pa g e 3 - 3 6 The cont ents of the v ari able K are: << α β ‘ α+β ’ >>. Functions defined b y more than one expr ession In this secti on we disc uss the tr eatment of f unctions that ar e def ined b y two or mor e expr essio ns.
Pa g e 3 - 37 Combined IFTE functions T o pr ogram a mor e complicated f u ncti on such as y ou can combine se ver al leve ls of the IFTE func tion, i .
Pa g e 4 - 1 Chapter 4 Calculations with complex numbers This c hapter sho ws e xam ples of calc ulations and applicati on of func tions to comple x numbers . Definitions A complex number z is a n umber w r itten as z = x + iy , where x and y ar e real numbers , and i is the imaginary unit defined b y i 2 = - 1.
Pa g e 4 - 2 Press @@OK@@ , t w ice , to r eturn to the stack . Entering comple x numbers Comple x numbers in the calc ulator can be enter ed in either of the tw o Car tesian representations, nam ely , x+iy , or (x,y) . The r esults in the calc ulator w ill be show n in the or der ed-p air for mat, i .
Pa g e 4 - 3 Notice that the last entry sho ws a complex n umber in the for m x+iy . This is so because the n u mber w as enter ed between single quot es, w hich r eprese nts an algebrai c expr essi on. T o ev aluate this number use the EV AL ke y( μ ).
Pa g e 4 - 4 On the other hand , if the coordinate s yste m is set t o cy lindrical coor dinates (use CYLIN), enter ing a complex number (x ,y) , wher e x and y are r eal numbers, w ill pr oduce a polar repr esentati on. F or e xample , in cy lindrical coor dinates, en ter the number (3 .
Pa g e 4 - 5 Changing sign of a complex number Changing the sign o f a complex n umber can be accomplished b y using the k e y , e.g ., -(5-3i) = -5 + 3i Entering the unit imaginary number T o enter the unit imaginary number type : „¥ Notice that the n umber i is enter ed as the order ed pair (0,1) if the CA S is set to APP RO X mode .
Pa g e 4 - 6 CMP LX menu through the MTH menu Assuming that s yst em flag 117 is se t to CHOOSE bo xes (see Chapter 2), the CMPLX sub-men u within the MTH men u is acc essed by using: „´9 @@OK@@ .
Pa g e 4 - 7 This f irst sc reen sho ws f unctions RE , IM, and C R. Notice that the la st f unction r eturns a list {3 . 5.} r epre senting the r eal and imaginar y components of the comple x number : The f ollow ing scr een show s functi ons R C, ABS , and ARG .
Pa g e 4 - 8 The r esulting menu inc lude some of the f unctions alr eady intr oduced in the pr ev ious s ecti on , namely , ARG , ABS , CONJ , IM, NE G, RE , and S IGN. It also include s func tion i whi ch serve s the s ame purpo se as the ke ys trok e combination „¥ , i .
Pa g e 4 - 9 Functions from the MTH menu The h yper bolic functi ons and their inv erses , as well as the Gamma, P SI, and P si functi ons (special f unctions) w er e introduced and appli ed to r eal numbers in Chapter 3 . Thes e functi ons can also be applied to comple x numbers by follo w ing the procedur es pre sented in Chapter 3 .
Pa g e 4 - 1 0 F unction DROI TE is found in the command catalog ( ‚N ). Using EV AL(ANS(1)) simplif ies the r esult to:.
Pa g e 5 - 1 Chapter 5 Algebraic and arithmetic operations An algebr aic obj ect , or simply , algebrai c, is an y number , vari able name or algebrai c expr essi on that can be oper ated upon , manipulated , and combined accor ding to the rules o f algebr a.
Pa g e 5 - 2 (e xponential , logarithmic , trigonometry , hy perboli c, etc.), as y ou w ould any r eal or comple x number . T o demonstr ate basic oper ations w ith algebr aic obj ects , let’s cr e.
Pa g e 5 - 3 ‚¹ @@A1@@ „¸ @@ A2@@ The s ame r esults ar e obtained in RPN mode if using the follo w ing ke ys tr ok es: @@A1@@ @@A2@@ +μ @@A1@ @ @@A2@@ -μ @@A1@@ @@A2@@ *μ @@A1@@ @@A2@ @ /μ .
Pa g e 5 - 4 W e notice that , at the bottom of the sc reen , the line See: E XP AND F A CT OR suggests links to other help fac ility entr ies , the f unctions E XP AND and F A CT OR. T o mov e direc tly to tho se entr ies, pr ess the soft men u ke y @SEE1! for E XP AND , and @SEE2! f or F A CT OR.
Pa g e 5 - 5 F A CT OR: LNCOLLE CT : LIN: P A R TFR A C: S OL VE: SUB S T : TEXP AND: Note : Re call that, to u se these , or any other f unctions in the RPN mode, y ou mus t enter the ar gument fi rst , and then the func tion .
Pa g e 5 - 6 Other forms of substitution in algebraic e xpressions F unctions SUB ST , show n ab o ve , is us ed to substitute a v ariable in an e x pr ession . A second f orm of sub stitution can be accomplished b y using the ‚¦ (assoc iated w ith the I k e y) .
Pa g e 5 - 7 A differ ent approac h to subs titution consists in def ining the substitution e xpre ssi ons in calc ulator v ari ables and placing the name o f the var iables in the ori ginal expr ession .
Pa g e 5 - 8 LNCOLLE CT , and TEXP AND are also contained in the AL G menu pr esented earli er . Functi ons LNP1 and EXP M wer e intr oduced in menu HYPERB OLIC, under the MTH menu (S ee Chapte r 2) .
Pa g e 5 - 9 Functions in the ARITHME TIC menu The ARI THMET IC menu contains a number o f sub-menu s fo r spec ific appli c ati ons in number theo ry (integers , poly nomials , et c.), as w ell as a n umber of f unctions that appl y to gener al arithme tic ope rati ons.
Pa g e 5 - 1 0 L GCD (Greatest C ommon Denominator): PROPFRA C (proper f rac tion) SIM P2: The f unctions assoc iated w ith the ARI THMETIC submenu s: INTE GER, POL YNOMIAL , MODUL O , and P ERMUT A T.
Pa g e 5 - 1 1 F ACT OR F actori z es an integer number or a poly nomial FCOEF Gener ates fr action gi ven r oots and multipli city FROO T S Returns r oots and multiplic ity giv en a fr action GCD Gr .
Pa g e 5 - 1 2 Applications of the ARI THMET IC m enu This sec tion is inte nded to pr esent some of the back ground necessary for applicati on of the ARITHMET IC menu f unctions. Def initions ar e pres ented next r egarding the su bjec ts of poly nomials , poly nomial fr actions and modular arithme tic .
Pa g e 5 - 1 3 multiply ing j times k in modulus n arithmeti c is, in essence , the integer r emainder of j ⋅ k / n in infinit e arithmeti c , if j ⋅ k>n . F or ex ample, in modulu s 12 arithme tic we ha ve 7 ⋅ 3 = 21 = 12 + 9 , (or , 7 ⋅ 3/12 = 21/12 = 1 + 9/12 , i .
Pa g e 5 - 1 4 Notice that , whene ver a r esult in the ri ght-hand side of the “ c ongr uence ” sy mbol pr oduces a r esult that is larger than the modulo (in this case , n = 6) , you can alw ay s subtr act a multiple of the modulo fr om that result and simplify it to a number smaller than the modulo.
Pa g e 5 - 1 5 [SPC ] entry , and the n pr ess the cor re sponding modular ar ithmetic f uncti on. F or e xam ple , using a modulus o f 12 , try the f ollo wing oper ations: ADDTMOD e xamples 6+5 ≡ .
Pa g e 5 - 1 6 oper ating on them. Y ou can also con vert an y number into a r ing number by using the func tion EXP ANDM OD . For e xample , EXP AN DMO D(1 2 5) ≡ 5 (mod 12) EXP AN DMOD (17 ) ≡ 5.
Pa g e 5 - 1 7 P ol ynomials P ol ynomials ar e algebraic e xpres sions consisting of one or mor e terms containing dec reasing po wer s of a giv en var iable . F or ex ample, ‘X^3+2*X^2 - 3*X+2’ is a third-o rder poly nomial in X, while ‘S IN(X)^2 - 2’ is a second-or d er poly nomial in SIN(X).
Pa g e 5 - 1 8 numbers (f unction ICHINREM). The input consists o f tw o vec tors [e xpressi on_1, modulo_1] and [e xpres sion_2 , modulo_2] . The o utput is a v ector cont aining [e xpre ssion_3, modulo_3] , wher e modulo_3 is related to the product (modulo_1) ⋅ (modulo_2) .
Pa g e 5 - 1 9 An alter nate def inition of the Hermite pol yn omials is wher e d n /dx n = n- th der i vati ve w i th r espect to x . This is the definiti on used in the calculat or . Example s: The Hermit e poly nomials of or ders 3 and 5 ar e giv en by: HERMITE( 3) = ‘8*X^3-12*X’ , And HERMITE(5) = ‘3 2*x^5-160*X^3+120*X’ .
Pa g e 5 - 2 0 F or ex ample, f or n = 2 , we w ill wr ite: Check this r esult w ith your calc ulator: LAGRANGE([[ x1,x2],[y1,y2] ]) = ‘((y1-y2)*X+(y2*x1-y1*x2))/(x1- x2)’ . Other e xam ples: L A GR ANGE([[1, 2 , 3][2 , 8, 15]]) = ‘( X^2+9*X-6)/2’ LAGRANGE([[0.
Pa g e 5 - 2 1 The P COEF function Gi ven an arr ay con taining the r oots of a poly nomial , the functi on PC OEF gener a tes an ar ra y containing the coeff ic ients of the cor r esponding pol ynomial . The coe ffi cients cor respond t o decr easing order o f the independent vari able.
Pa g e 5 - 22 The EP SX0 func tion and the CAS vari able EPS The va riab le ε (epsilon) is typi cally used in mathemati cal te xtbooks to repr esent a ve ry small number . The calc ulator’s CA S cr eates a v ariable EP S, w ith defa ult value 0. 000000000 1 = 10 -10 , when y ou use the EPSX0 f unction.
Pa g e 5 - 23 Frac ti on s F racti ons can be expanded and fact or ed by using func tions EXP AND and F ACT OR , fr om the AL G menu (‚×). F or e xam ple: EXP AND(‘(1+X)^3/((X-1)*(X+3))’) = ‘.
Pa g e 5 - 24 If y ou hav e the Complex mode ac ti ve , the re sult will be: ‘2*X+(1/2/(X+i)+1/2/(X- 2 )+5/(X-5)+1/2/X+1/2/(X -i))’ The FCOEF function The f unction FC OEF is used to obtain a r ational fr action, gi ven the roots and poles of the fr action .
Pa g e 5 - 25 mode selected , then the re sults wo uld be: [0 –2 . 1 –1. – ((1+i* √ 3)/2) –1. – ((1–i* √ 3)/2) –1. 3 1. 2 1.]. Step-b y-step operations w i th poly nomials and fracti.
Pa g e 5 - 26 The CONVER T M enu and algebraic operations The C ONVERT menu is acti vated b y using „Ú ke y (the 6 key ) . T hi s menu summar iz es all con ver sion menus in the calc ulator . The list o f these menu s is sho wn ne xt: The f unctions a vailable in eac h of the sub-menu s ar e show n next .
Pa g e 5 - 27 B ASE conv er t menu (Option 2) This men u is the same as the UNI T S menu obtained b y using ‚ã . The applicati ons of this menu ar e disc uss ed in detail in Chapter 19 . TRIGONOMETRIC convert menu (Option 3) This men u is the same as the TRIG men u obtained b y using ‚Ñ .
Pa g e 5 - 2 8 Fu n ct i o n NUM has the same eff ect as the ke ystr oke comb ination ‚ï (assoc iated w ith the ` key ) . Fu n ct ion N UM c onver t s a symb ol ic resu l t i nto its floating-poin t v alue . Func tion Q conv erts a floating-po int value into a fr action.
Pa g e 5 - 2 9 LIN LNCOLLE CT PO WEREXP AND SIMP LIFY.
Pa g e 6 - 1 Chapter 6 Solution to single equations In this chapte r we f eature those f unctions that the calc u lator pr ov ides for s olv ing single equations of the for m f(X) = 0. Assoc iated with the 7 k ey ther e are two men us of eq uation-sol v ing functi ons, the S ymbolic S OL V er ( „Î ), and the NUMer ical SoL V er ( ‚Ï ) .
Pa g e 6 - 2 Using the RPN mode, the soluti on is accomplished by enter ing the equation in the stac k, f ollo wed by the v ari able , befor e enter ing func tion I S OL. R ight bef ore the ex ecuti on of ISOL , the R PN st ack should look as in the fi gure to the left .
Pa g e 6 - 3 The sc reen shot sho wn abo ve dis plays tw o solutions . In the fir st one , β 4 -5 β =12 5, SOL VE pr oduces no so lutions { }. In the s econd one , β 4 - 5 β = 6, S OL VE pr oduces four s olutions , show n in the last output line .
Pa g e 6 - 4 In the fir st case S OL VEVX could not find a solu tion . In the second case , S OL VEVX found a single so lution , X = 2 . The foll owing screen s sh ow th e R PN sta ck for solvin g t h.
Pa g e 6 - 5 The S ymbolic So lv er functions pre sented abo ve pr oduce solutions to r ational equations (mainl y , poly nomial equations). If the equation to be so lv ed for has all numer ical coeffi ci ents, a numer ical solu tion is pos sible thr ough the use o f the Numer ical So lv er featur es of the calc ulator .
Pa g e 6 - 6 P ol ynomial Equations Using the Solv e p ol y… option in the calc ulator’s SO L V E en vir onment you can: (1) f ind the solutions to a pol ynomial equati on; (2) obtain the coeff ic ien ts of the pol yno mial ha ving a n umber of gi ven r oots; (3) obtain an algebr aic e xpressi on for the poly nomial as a functi on of X.
Pa g e 6 - 7 All the so lutions ar e complex n umbers: (0.43 2 ,-0. 38 9), (0.43 2 , 0.3 89), (-0.7 66 , 0.6 3 2) , (-0.7 66 , -0.6 3 2). Generating polynomial coe fficients giv en the polynomial's r oots Suppos e y ou want t o generate the pol ynomi al whose r oots are the nu mbers [1, 5, - 2 , 4].
Pa g e 6 - 8 Press ˜ to tri g ger the line editor to see all the coe ffi cients . Generating an algebraic expr ession for the poly nomial Y ou can use the calc ulator to gener ate an algebrai c expr ession f or a poly nomial giv en the coeffi c ients or the r oots of the pol yno mial .
Pa g e 6 - 9 T o e xpand the produ cts, y ou can use the EXP AND command. T he r esulting e xpr essi on is: ' X^4+-3*X^3+ -3*X^2+11*X-6' . A differ ent approac h to obtaining an expr essi on for the poly nomial is to gener ate the coeffi c ients firs t , then gener ate the algebrai c ex pre ssi on wi th the coeff ic ients highli ghted.
Pa g e 6 - 1 0 Ex ample 1 – Calc ulating pay ment on a loan If $2 milli on ar e borr ow ed at an annual inter est r ate of 6 .5% to be r epaid in 6 0 monthly pa yments , what should be the monthly pa yment? F or the debt to be totall y repaid in 6 0 months, the fu tur e value s of the loan should be z ero .
Pa g e 6 - 1 1 pay ments. Suppose that w e use 2 4 per iods in the first line of the amorti zati on scr e en, i .e., 24 @@OK@@ . Then , pr ess @@AMOR@@ . Y ou will get the f ollo w ing res u l t : This s cr een is interpr eted as indicating that after 2 4 months o f pay i ng bac k the debt , the borr ow er has paid up US $ 7 2 3,211.
Pa g e 6 - 1 2 ˜ Skip P MT , since w e will be s olv ing fo r it 0 @@OK@@ Enter FV = 0, the option End is hi ghlighted @@CHOOS ! — @@OK@@ Change pay ment option t o Begin — š @@SOLVE! H ighlight P MT and so lv e for it The s cr een now sho ws the v alue of P MT as –38 , 9 21.
Pa g e 6 - 1 3 ™ ‚í Enter a comma ³ ‚ @@PYR@ @ Enter name o f var iable P YR ™ ‚í Enter a comma ³ ‚ @@FV@@ . Enter name of var iable FV ` Exec ute P URGE command The follo w ing two s cr een shots sho w the P URGE co mmand for purging all the var iables in the dir ectory , and the r esult after e xec uting the command.
Pa g e 6 - 1 4 ³„¸~„x™-S„ì *~„x/3™‚Å 0™ K~e~q` Press J to see the ne wly c reated E Q var iable: Then , enter the SOL VE env ironment and select Sol ve equation… , by u s ing: ‚Ï @@OK@@ .
Pa g e 6 - 1 5 This , ho we ver , is not the only possible s olution f or this equati on. T o obtain a negativ e solutio n, f or e xampl e, ent er a negati ve number in the X: field be for e solv ing the equation. T ry 3 @@@OK@@ ˜ @SOLVE@ . The s olution is no w X: - 3.
Pa g e 6 - 1 6 The equati on is her e e xx is the unit strain in the x -directi on, σ xx , σ yy , and σ zz , are the normal str esses on the partic le in the dir ection s of the x -, y-, and z -axe.
Pa g e 6 - 1 7 With the ex: field hi ghlighted , pres s @SOLVE@ to solv e f or ex : The s oluti on can be seen fr om within the S OL VE E QUA TION in put for m by pr essing @EDIT whi l e t he ex: fi eld is highlighted . The r esulting value is 2.47 0 833333333 E- 3.
Pa g e 6 - 1 8 Spec ifi c energ y in an open channel is def ined as the energ y per unit wei ght measur ed with r espect to the c hannel bottom. L et E = spec ific ene rg y , y = chann el depth, V = f.
Pa g e 6 - 1 9 Θ Solv e for y . The r esult is 0.14 9 8 36 .., i.e ., y = 0.14 98 3 6 . Θ It is kno wn, how ev er , that ther e are ac tually two s oluti ons av ailable f or y in the spec ifi c energ y equation. T he soluti on we j ust found corr esponds to a numer ical soluti on with an initial v alue of 0 (the de faul t va lu e for y , i .
Pa g e 6 - 2 0 In the ne xt e xample w e will u se the D ARCY f unction f or finding fr icti on fac tors in pipelines . Thus , we def ine the functi on in the fo llow ing fr ame.
Pa g e 6 - 2 1 Ex ample 3 – Flow in a pipe Y ou may want t o cr eate a separate sub-dir ectory (PIPE S) to try this ex ample . The main eq uation go vernin g flo w in a pipe is, of cour se, the Dar cy- W eisbac h equation .
Pa g e 6 - 2 2 The comb ined equation has pr imitiv e v a r iables: h f , Q, L, g, D, ε , and Nu . Laun ch t he nume rical solver ( ‚Ï @@OK@ @ ) to see the primiti ve v ari ables listed in the S OL VE EQU A T ION input f orm: Suppo se that w e use the v alues hf = 2 m, ε = 0.
Pa g e 6 - 23 Ex ample 4 – Universal gr av itation Ne wton ’s law of uni versal gr av itation indi cates that the magnitude of the attrac ti ve fo r ce betw een tw o bodies of mass es m 1 and m 2 .
Pa g e 6 - 24 Sol ve for F , and pres s to r eturn to nor ma l calc ulator display . The solu tion is F : 6. 6 7 2 5 9E-15_N , or F = 6.6 7 2 5 9 × 10 -15 N.
Pa g e 6 - 2 5 T y pe an equati on, sa y X^2 - 125 = 0, dir ectly on the s tack , and pres s @@@OK@@@ . At this point the equati on is r eady for so lution . Alter nati vel y , y ou can activ ate the equation w riter after pr essing @E DIT to enter y our equation.
Pa g e 6 - 2 6 The S OL VE soft menu The SOL VE sof t menu allows acc ess to som e of th e num erical solver funct ions thr ough the soft men u ke ys . T o access this menu us e in RPN mode: 7 4 MENU, or in AL G mode: MENU(7 4) . Alternati vely , y ou can use ‚ (hold) 7 to acti vate the S OL VE soft men u .
Pa g e 6 - 27 Example 1 - Sol ving the equati on t 2 -5t = - 4 F or ex ample, if y ou stor e the equation ‘t^2 -5*t=- 4’ into E Q, and pr ess @) SOLVR , it w ill acti vate the f ollo wing menu: This r esult indicates that y ou can solv e for a value o f t for the equati on listed at the top of the display .
Pa g e 6 - 2 8 Y ou can also solv e mor e than one equation by sol ving one equation at a time , and repeating the pr ocess until a soluti on is found .
Pa g e 6 - 2 9 Using units with the SOL VR sub-menu The se are s ome rules o n the use o f units w ith the SO L VR sub-menu: Θ Enter ing a guess w ith units for a gi ven v ari able , will intr oduce the use of those units in the s olution .
Pa g e 6 - 3 0 This f unction pr oduces the coeff ic ients [a n , a n- 1 , … , a 2 , a 1 , a 0 ] of a poly nomial a n x n + a n-1 x n- 1 + … + a 2 x 2 + a 1 x + a 0 , give n a ve c to r of it s ro ot s [r 1 , r 2 , …, r n ].
Pa g e 6 - 3 1 Press J to e x it the S OL VR env ir onment . F i nd y our wa y bac k to the TVM sub- menu w ithin the S OL VE sub-me nu to try the other functio ns available . Function TVM ROO T This function requires as argument t he na me of one of the var iables in t he T VM pr oblem.
Pa g e 7- 1 Chapter 7 Solv ing multiple equations Many pr oblems of sc ience and engineer ing req uir e the simultaneous so lutions of mor e than one equation . The calculator pr ov ides se ve ral pr ocedure s for solv ing multiple equations as pr esented belo w .
Pa g e 7- 2 Use co mmand S OL VE at this poin t (fr om the S. SL V men u: „Î ) After a bout 40 seconds , may be more , you get as r esult a list: { ‘t = (x- x0)/(COS( θ 0)*v0)’ ‘ y 0 = (2*C .
Pa g e 7- 3 the conten ts of T1 and T2 to the stac k and adding and subtr acting them. Her e is how t o do it with the equati on writ er : Enter and st ore ter m T1: Enter and stor e term T2 : Notice that w e are using the RPN mode in this ex ample, ho we ver , the pr ocedur e in the AL G mode should be v ery similar .
Pa g e 7- 4 Notice that the r esult includes a v ector [ ] contained w ithin a list { }. T o remo ve the list s ymbol, u se μ . F inally , to decompose the vec tor , use functi on OB J .
Pa g e 7- 5 Ex ampl e 1 - Ex ampl e fr om the help facilit y As w ith all functi on entries in the help f acility , ther e is an ex ample at tac hed to the MSL V entry as show n abov e. Notice that f unction M SL V r equire s thr ee argume nts: 1. A vec tor containing the equati ons, i .
Pa g e 7- 6 disc harge (m 3 /s o r ft 3 /s) , A is the cr oss-sec tional ar ea (m 2 or ft 2 ), C u is a coeff ic ient that depends on the s yst em of units (C u = 1. 0 for the S I, C u = 1.4 86 fo r the English sy stem of units), n is the Manning’s coe ffi cie nt , a measure o f the channel surface r oughness (e .
Pa g e 7- 7 μ @@@EQ1@@ μ @@@EQ2@ @ . T he equations ar e listed in the s tack as follo ws (small font opti on selected): W e can see that these equati ons are indeed gi ven in ter ms of the pr imitiv e var iables b, m , y , g , S o , n, Cu , Q, and H o .
Pa g e 7- 8 Ne xt, w e’ll ente r var iable EQS: LL @ @EQS@ , follo wed b y vector [y ,Q]: ‚í„Ô~„y‚í~q™ and b y th e init ial guesses ‚í„Ô5‚í 10 . Bef ore pr essing ` , the sc reen w ill look like this: Press ` to sol ve the s yst em of equations .
Pa g e 7- 9 The r esult is a list of thr ee v ectors. The f irst v ector in the list will be the equati ons sol ved . The second v e ctor is the list of unkno wns . The thir d vecto r repr esents the soluti on. T o be able to see the se v ector s, pr ess the do wn-arr ow k ey ˜ to acti vate the line editor .
Pa g e 7- 1 0 The co sine la w indicate s that: a 2 = b 2 + c 2 – 2 ⋅ b ⋅ c ⋅ cos α , b 2 = a 2 + c 2 – 2 ⋅ a ⋅ c ⋅ cos β , c 2 = a 2 + b 2 – 2 ⋅ a ⋅ b ⋅ cos γ . In orde r to solv e any tr iangle , yo u need to know at leas t thr ee of the fol lo w ing si x v ari ables: a, b, c, α, β, γ .
Pa g e 7- 1 1 ‘SIN( α )/a = S IN( β )/ b ’ ‘SIN( α )/a = S IN( γ )/c’ ‘SIN( β )/b = S IN( γ )/c’ ‘ c^2 = a^2+b^2 - 2*a*b*COS( γ )’ ‘b^2 = a^2+c^2 - 2*a*c*CO S( β )’ ‘ a^2.
Pa g e 7- 1 2 Press J , if needed , to get y our var iables me nu . Y our men u should sho w the vari ab le s @LVARI! !@ TITLE @@EQ@@ . Preparing to run t he ME S The ne xt step is to acti vate the ME S and try one sample solution .
Pa g e 7- 1 3 Let ’s tr y a simple s oluti on of Case I, using a = 5, b = 3, c = 5 . Us e the follo w ing entr ies: 5 [ a ] a:5 is listed in the top left corner of the displa y . 3 [ b ] b:3 is listed in the top left corner of the displa y . 5 [ c ] c:5 is listed in the top left corner of the display .
Pa g e 7- 1 4 Pr essi ng „ @@ALL@@ will so lv e for all the v ari ables, tempor aril y show ing the intermediate re sults. Press ‚ @@ALL@@ to see t he solu tions: When done , pres s $ to retur n to the MES en vir onment. Pr ess J to ex i t the ME S env ir onment and r eturn to the normal calc ulator display .
Pa g e 7- 1 5 Progr amming t he MES triangle solution using User RP L T o fac ilitate acti vating the ME S for f utur e so lutions , we w ill cr eate a pr ogr am that w ill load the MES w ith a single ke ystr ok e .
Pa g e 7- 1 6 Use a = 3, b = 4 , c = 6. T he solution pr ocedure us ed her e consists of sol ving fo r all var iables at once , and then recalling the soluti ons to the stack: J @TRISO T o clear up data and r e -start ME S 3 [ a ] 4 [ b ] 6 [ c ] T o ent er data L T o mov e to the next v ariable s menu.
Pa g e 7- 1 7 Adding an I NFO but ton to your directory An inf ormati on button can be us eful f or your dir ectory to help y ou remember t he oper ation o f the functi ons in the direc tory . In this dir ectory , all we need to r emember is to pr ess @ TRISO to get a tr iangle solution s tarted.
Pa g e 7- 1 8 An e xplanation of the v ari ables follo ws : SOL V EP = a progr am that tri g gers the m u ltiple equati on sol ver f or the partic ular set of equations s tor ed in var iable PEQ ; NAME = a var iable st oring the name o f the multiple equati on solv er , namel y , "ve l.
Pa g e 7- 1 9 Notice that after y ou enter a partic ular value , the calc ulator displa ys the var iable and its value in the upper left co rner of the dis play . W e hav e now enter ed the kno wn v aria bles . T o calc ulate the unkno wns w e can proceed in tw o ways: a).
Pa g e 7- 2 0.
Pa g e 8 - 1 Chapter 8 Operations w ith lists L ists ar e a type of calc ulator’s ob ject that can be u seful f or data pr ocessing and in pr ogramming .
Pa g e 8 - 2 The f igur e belo w show s the RPN stack be fo r e pre ssing the K key : Composing and decomposing lists Compo sing and decomposing lis ts mak es sense in RPN mode onl y . Under suc h oper ating mode , decomposing a list is achi ev ed by u sing functi on OBJ .
Pa g e 8 - 3 In RPN mode, the f ollow ing scr een show s the three lists and the ir names read y to be stor ed. T o stor e the lists in this case y ou need to pres s K three time s. Changing sign The si gn -change k ey ( ) , w hen applied t o a list of number s, will c hange the sign o f all elements in the list .
Pa g e 8 - 4 Subtr action , multiplication, and di vision o f lists of numbers o f the same length pr oduce a list of the same length w ith term-by-ter m oper ations.
Pa g e 8 - 5 ABS EXP and LN L OG and ANTIL OG S Q and squar e root SIN, ASIN COS, ACOS T AN, A T AN INVER SE (1/x) Real number functions from the MTH menu F unctions of inter est fr om the MTH menu in.
Pa g e 8 - 6 T ANH , A T ANH SIGN , MANT , XPON IP , FP FL OOR, CEIL D R, R D Ex ampl es of functions that use two arguments The s cr een shots below sho w applications o f the functi on % to lis t arguments . F unction % r equires two ar g uments.
Pa g e 8 - 7 %({10,20, 30},{1,2 ,3} ) = {%(10,1),%(20,2),%(30, 3)} This de sc ripti on of func tion % for lis t ar guments sh o ws the gener al pattern of ev aluation of an y functi on w ith two ar guments when one or both ar guments are lists .
Pa g e 8 - 8 The f ollow ing ex ample sho ws appli cations o f the functi ons RE(Real part) , IM(imaginary par t), AB S(magnitude), and ARG(argument) of comple x numbers .
Pa g e 8 - 9 This me nu cont ains the fo llo w ing func tions: Δ LIS T : Calculate inc rement among consec utiv e elements in list Σ LIS T : Calc ulate summation o f elemen ts in the list Π LIS T :.
Pa g e 8 - 1 0 M anipulating elements of a list The P RG (pr ogramming) men u includes a LI ST su b-menu w ith a number o f func tions to mani pulate ele ments of a li st .
Pa g e 8 - 1 1 F unctions GET I and P UTI , als o av ailable in sub-menu PR G/ ELEMENT S/, can also be used to extr act and place elements in a list . Thes e two f unctions , ho we ver , ar e usef ul mainly in pr ogramming .
Pa g e 8 - 1 2 SEQ is u seful to pr oduce a list of v alues gi ven a partic ular expr essi on and is desc r ibed in more de tail her e . The SE Q functi on tak es as arguments an e xpressi on in terms.
Pa g e 8 - 1 3 In both cases , you can either ty pe out the M AP command (as in the e xamples abo ve) or s elect the command from the CA T menu . The f ollow ing call to func tion MAP us es a pr ogram.
Pa g e 8 - 1 4 to r eplace the plus sign (+) w ith ADD: Ne xt, w e stor e the edited expr ession in to v ari able @@@G@@@ : Ev alua ting G(L1,L2) no w produces the f ollow ing result: As an alternati ve , yo u can define the f unction w ith ADD rather than the plus sign (+), fr om the start, i .
Pa g e 8 - 1 5 Applications of lists This sec tion show s a couple of applications o f lists to the calc ulation of statisti cs of a samp le. B y a sample we un derstand a list of valu es, say , { s 1 , s 2 , …, s n }.
Pa g e 8 - 1 6 3 . Di vide the r esult abov e by n = 10: 4. Apply the INV() f unction to the late st re sult: Thu s, the harmonic mean of lis t S is s h = 1.63 48… Geometric mean of a list The geometr ic mean of a sample is def ined as T o find the geometr ic mean of the list stor ed in S, we can u se the follo wing pr ocedur e: 1.
Pa g e 8 - 1 7 Thu s, the geometri c mean of list S is s g = 1.00 3 2 0 3… W eighted aver age Suppos e that the data in list S , defined a bo ve , namely : S = {1,5,3,1,2 ,1,3,4,2,1} is affec ted b .
Pa g e 8 - 1 8 3. Us e fu n c t io n Σ LIS T , once mor e, to calc ulate the denominator o f s w : 4. Use the e xpre ssion AN S(2)/ANS(1) to cal culat e the w eigh ted av er age: Thu s, the wei ghted av er age of list S w i th w eights in list W is s w = 2 .
Pa g e 8 - 1 9 The c lass mark dat a can be stor ed in var iable S , whi le the fr equency coun t can be stored in v ariable W , as f ollow s: Giv en the list of class marks S = {s 1 , s 2 , …, s n .
Pa g e 8 - 2 0 T o calc ulate this last r esult , we can us e the fo llow ing: The s tandar d dev iation o f the gr ouped data is the squar e r oot of the var iance: N s s w w s s w V n k k k n k k n .
Pa g e 9 - 1 Chapter 9 V ectors This Cha pter pr ov ides e xamples o f enter ing and oper ating with v ectors , both mathematical ve ctors o f many e lements, as w ell as ph ysi cal vectors of 2 and 3 components . Definitions F rom a mathematical po int of v ie w , a vec tor is an arr ay of 2 or mor e elements arr anged into a r ow or a column .
Pa g e 9 - 2 wher e θ is the angle between the two v ectors . The cr oss pr oduct pr oduces a vec tor A × B whos e magnitude is | A × B | = | A || B |sin( θ ), and its dir ection is gi ven b y the so -called right-hand rule (consult a te xtbook on Math, Ph ysi cs, or Mechani cs to see this oper ation illustr ated gra phically).
Pa g e 9 - 3 Stor ing vectors into v ariables V ectors can be stor ed into var iables . The sc reen shots belo w show the vec tors u 2 = [1, 2] , u 3 = [-3, 2, -2] , v 2 = [3,- 1] , v 3 = [1, -5, 2] stored into var iabl es @ @@u2@@ , @@@u3@@ , @@@v2@@ , and @@@v3@@ , r especti vel y .
Pa g e 9 - 4 The ← WID k ey is used to decr ease the wi dth of the columns in the spr eadsheet . Pr ess this k ey a couple of time s to see the column w idth decr ease in y our Matri x W riter . The @ W I D → k e y is used to incr ease the w idth of the columns in the spr eadsheet .
Pa g e 9 - 5 The @+ROW@ k ey w ill add a ro w full of z er os at the location o f the selected cell of the s pr eadsheet . The @-ROW ke y will dele te the ro w corr esponding to the selec ted cell of the spr eadsheet. The @+COL@ k ey w ill add a column full of z er os at the location of the select ed cell of the spr eadsheet .
Pa g e 9 - 6 Building a vector with ARR Y The fun ct ion → ARR Y , av ailable in the function catalog ( ‚N‚é , u se —˜ to locate the f unction), can also be used to build a ve ctor or arr ay in the f ollo wing wa y . In AL G mode , enter ARR Y( vector elements, nu mber of elements ), e.
Pa g e 9 - 7 In RPN mode, the f unction [ → ARR Y] tak es the obje cts fr om stac k lev els n+1, n, n-1 , …, dow n to stack le vels 3 and 2 , and conv erts them into a vec tor of n elements . The ob ject or iginally at s tack le vel n+1 becomes the f irst element , the objec t ori gina ll y at lev el n becomes the second element, and so on .
Pa g e 9 - 8 Highli ghting the entire e xpr essio n and using the @ EVAL@ so ft menu k e y , w e get the res u l t : -15 . T o r eplace an element in an arr ay use f unction P UT (y ou can find it in the func tion catalog ‚N , or in the P RG/LI S T/ELEMENTS su b-menu – the later wa s intr oduced in Chapter 8).
Pa g e 9 - 9 Simple operations w it h vectors T o illustr ate oper atio ns wi th vec tors w e will u se the ve ctor s A, u2 , u3, v2 , and v3, stor ed in an earli er ex er cise .
Pa g e 9 - 1 0 Absolute value function The ab solute v alue functi on (ABS), when appli ed to a vec tor , produce s the magnitude of the vec tor . F or a vec tor A = [ A 1 ,A 2 ,…,A n ], the magnitude is def ined as . In the AL G mode, ent er the functi on name follo wed b y the vector ar gument .
Pa g e 9 - 1 1 Dot pr oduc t F unction DO T is used to calc ulate the dot produc t of two vect ors o f the same length. So me ex amples of applicati on of functi on DO T , using the v ectors A, u2 , u3, v2 , and v3, stor ed earlier , ar e sho wn ne xt in AL G mode.
Pa g e 9 - 1 2 In the RPN mode , application o f func tion V w ill list the components of a vec tor in the stac k, e .g., V (A ) will pr oduce the fo llo w ing output in the RPN stack (vector A is listed i n stack lev el 6:) .
Pa g e 9 - 1 3 When the r ectangular , or Cartesian , coordinat e s ys tem is selec ted , the top line of the displa y will sho w an XY Z fi eld, and an y 2 -D or 3-D vector e nter ed in the calculator is r eproduced as the (x ,y ,z) components of the vec tor .
Pa g e 9 - 1 4 The f igur e belo w show s the tr ansfor mation of the v e ct or fr om spheri cal to Cartesi an coor dinates , with x = ρ sin( φ ) co s( θ ), y = ρ sin ( φ ) cos ( θ ), z = ρ cos( φ ). F or this case , x = 3.20 4, y = 1.4 9 4 , and z = 3.
Pa g e 9 - 1 5 equi valent (r , θ ,z) w ith r = ρ sin φ , θ = θ , z = ρ cos φ . F or ex ample , the follo w ing fi gure sho ws the v ector enter ed in spheri cal coordinat es, and tr ansformed to polar coor dinates . F or this case, ρ = 5, θ = 2 5 o , and φ = 4 5 o , w hile the transf ormation sho ws that r = 3.
Pa g e 9 - 1 6 Suppose that y ou want t o find the angle between v ectors A = 3 i -5 j +6 k , B = 2 i + j -3 k , y ou could try the f ollow ing operati on (angular measur e set to degree s) in AL G mode: 1 - Enter vect ors [3,-5, 6], press ` , [2 ,1,-3], pre ss ` .
Pa g e 9 - 1 7 Thus, M = (10 i +2 6 j +2 5 k ) m ⋅ N. W e kno w that the magnitude of M is such that | M | = | r || F |sin( θ ), where θ is the angle betw een r and F .
Pa g e 9 - 1 8 Ne xt, w e calculate v e ct or P 0 P = r as ANS(1) – ANS( 2), i.e ., F inally , we tak e the dot produc t of ANS(1) and AN S( 4) and make it equal to z ero to complete the operatio n .
Pa g e 9 - 1 9 In this secti on w e will sho wing y ou wa ys to transf orm: a column vec tor into a r o w vect or , a ro w vecto r into a column v ector , a list into a vec tor , and a vec tor (or matr ix) into a list . W e fir st demonstr ate these tr ansfor mations using the RPN mode.
Pa g e 9 - 2 0 If w e no w apply f uncti on OB J once more , the list in st ack le vel 1:, {3 .}, w ill be decomposed as f ollows: Function LIS T This f uncti on is used to c reate a list gi ven the elements o f the list and the list length or si ze .
Pa g e 9 - 2 1 3 - Use f u ncti on ARR Y to build the column vec tor The se thr ee steps can be put toge ther into a U serRP L progr am, e nter ed as follo ws (in RPN mode , still): ‚å„° @) .
Pa g e 9 - 22 2 - Use f u ncti on OBJ to deco mpose the list i n stack level 1: 3 - Pr ess the delet e k ey ƒ (als o kno wn as f unction DROP) t o eliminate the number in stac k lev el 1: 4 - Use.
Pa g e 9 - 23 Thi s va riab le, @@CXR@@ , can no w be used to dir ectl y transf orm a column v ector to a r ow v ector . In RPN mode, enter the column v ector , a nd then pr ess @@CXR@ @ .
Pa g e 9 - 24 A ne w var iabl e , @@LX V@@ , w ill be av ailable in the soft menu labels after pr essing J : Press ‚ @@LXV@@ to see the pr ogram co ntained in the var iable LXV : << OBJ 1 LIST RRY >> Thi s vari ab le, @@LXV@@ , can no w be used to dir ectly tr ansfor m a list into a vec tor .
Pa g e 1 0 - 1 Chapter 10 ! Creating and manipulating matr ices This c hapter sho ws a number of e xamples aimed at cr eating matri ces in the calc ulator and demonstrating manipulati on of matri x elements. Definitions A matri x is simpl y a rec tangular arr ay of ob ject s (e.
Pa g e 1 0 - 2 Entering matr ices in the stac k In this secti on w e pre sent tw o differ ent methods to enter matr ices in the calc ula tor s tack: (1) using the Matr ix W r iter , and (2) ty ping the matri x direc tly in to th e s ta ck.
Pa g e 1 0 - 3 If y ou hav e selected the te xtbook display opti on (using H @) DISP! and c hecking off Textbook ), the matr ix w ill look lik e the one sho wn abo ve . Otherwise , the displa y w ill sho w: The dis play in RPN mode w ill look very similar to these .
Pa g e 1 0 - 4 or in the MA TRICE S/CREA TE menu a vaila ble thr ough „Ø : The MTH/MA TRIX/MAKE sub menu (let’s call it the MAKE menu) contains the fo llo w ing func tio ns: while the MA TRICE S/.
Pa g e 1 0 - 5 As yo u can see f rom e xploring these men us (MAKE and CREA TE) , they both hav e the same functi ons GET , GE TI , PUT , P U T I, S UB, REPL , RDM, R ANM, HILBERT , V ANDERMOND E , IDN, CON, → DIA G , and DIA G → . The CREA TE menu inc ludes the C OL UMN and RO W sub-menus , that are also av ailable under the MTH/MA TRIX menu .
Pa g e 1 0 - 6 Functions GET and P UT F unctions GET , GET I, PUT , and PUT I, oper ate w ith matri ces in a similar manner as w ith lists or vec tors , i.
Pa g e 1 0 - 7 Notice that the s cr een is prepar ed for a su bseq uent appli cation o f GET I or GET , by inc reasing the column index o f the original r efer ence by 1, (i .e., fr om {2 ,2} to {2 , 3}) , whil e sho wing the ex trac ted value , namely A(2 ,2) = 1.
Pa g e 1 0 - 8 If the ar gument is a real matr ix , TRN simply pr oduces the tr anspose of the r eal matri x. T ry , f or ex ample , TRN(A ) , and compare it w ith TRAN(A). In RPN mode, the tr ansconjugat e of matri x A is calculated by using @@@A@@@ TRN .
Pa g e 1 0 - 9 In RPN mode this is accomplished by u sing {4,3} ` 1.5 ` CON . Function IDN F unction IDN (IDeNtit y matri x) cr eates an identity matri x giv en its si ze . Recall that an identity matr i x has to be a squar e matri x, ther efor e, onl y one value is r equir ed to des cr ibe it completely .
Pa g e 1 0 - 1 0 vec tor ’s dimension , in the latter the number of r ow s and columns of the matri x. The f ollow ing ex amples illus tr ate the use o f functi on RDM: Re-dim ensioning a vector int.
Pa g e 1 0 - 1 1 If using RPN mode , we as sume that the matr ix is in the st ack and u se {6} ` RDM . Function RANM F unction RANM (R ANdom Matr ix) w ill gener ate a matri x with r andom integer elements gi ven a list w ith the number of r ow s and columns (i .
Pa g e 1 0 - 1 2 In RPN mode , assuming that the ori ginal 2 × 3 matr ix is alr eady in the stack , use {1,2} ` {2,3} ` SUB . Function REP L F unction REPL r eplaces or inserts a sub-matr ix int o a larger one .
Pa g e 1 0 - 1 3 In RPN mode, w ith the 3 × 3 matri x in the stack , we simpl y have to acti vate fun ctio n DI G to obtain the same r esult as abov e.
Pa g e 1 0 - 1 4 F or ex ample, the f ollo wing command in AL G mode f or the list {1,2 , 3, 4}: In RPN mode, enter {1, 2,3,4} ` V NDERMON DE . Function HILBERT F unction HILBERT c reates the Hilbert matr i x corr esponding to a dimension n .
Pa g e 1 0 - 1 5 enter ed in the display as y ou perform tho se ke ystr ok es . F irst , we pres ent the steps ne cessar y to produce program CRMC. Lists r epresent columns of the matri x The p rogra m @CRMC allo ws y ou to put together a p × n matri x (i .
Pa g e 1 0 - 1 6 ~„n # n „´ @) MATRX! @ ) COL! @COL! COL ` Pr ogram is dis play ed in lev el 1 To s a v e t h e p r o g r a m : ! ³~~crmc~ K T o see the contents o f the progr am use J ‚ @CRMC .
Pa g e 1 0 - 1 7 Lists r epresent ro ws of the matrix The pr ev ious pr ogram can be easil y modified to c reate a matr ix w hen the input lists w ill become the r ow s of the r esulting matri x. The onl y change to be perfor med is to change C OL → for ROW → in the progr am listing.
Pa g e 1 0 - 1 8 Both appr oaches w ill show the same f unctions: When s ystem f lag 117 is set to S OFT menus , the COL menu is acces sible thr ough „´ !) MATRX ) !)@@COL@ , or thr ough „Ø !) @CREAT@ ! ) @@ COL@ . Both appr oaches w ill sho w the same set of f unctions: The operation of these functions is presented be lo w .
Pa g e 1 0 - 1 9 In this re sult, the f irst column occ upies the highe st stac k lev el after decompositi on, and st ack le vel 1 is occ upied b y the number of co lumns of the ori ginal matri x. T he matri x does not survi ve decompositi on, i .e., it is no longer av ailable in the stack .
Pa g e 1 0 - 2 0 In RPN mode, ent er the matr i x fir st , then the v ector , and the column number , bef or e apply ing func tion COL+. T he fi gure belo w show s the RPN stack be fo re and after apply ing functi on COL+.
Pa g e 1 0 - 2 1 In RPN mode, f unction CS WP lets you s wap the columns of a matr ix listed in stac k lev el 3, who se indices ar e listed in stac k lev els 1 and 2 .
Pa g e 1 0 - 2 2 When s yst em flag 117 is set to S OFT menus , the RO W menu is acces sible thr ough „´ !) MATRX ! )@@ROW@ , or thr ough „Ø !) @CREAT@ ! ) @@ROW@ . Both appr oaches w ill sho w the same set of f unctions: The operation of these functions is presented be lo w .
Pa g e 1 0 - 23 matri x does not survi ve decompo sition , i.e ., it is no longer av ailable in the stack. Function RO W → Fu n ct i o n R OW → has the opposite e ffect o f the functio n → RO W , i .
Pa g e 1 0 - 24 Function RO W- F unction RO W - tak es as argument a matr ix and an in teger number r epre senting the position o f a r ow in the matri x. T he functi on returns the or iginal matr ix , minus a r o w , as w ell as the e xtrac ted ro w sho wn as a v ector .
Pa g e 1 0 - 2 5 As y ou can see , the ro ws that or iginally occ upi ed positions 2 and 3 ha ve been s wapped . Function RCI F unction R CI stands f or multipl y ing R ow I by a C ons tant v alue and r eplace the r esulting r ow at the same location .
Pa g e 1 0 - 26 In RPN mode, ent er the matr ix f irst , follo wed by the const ant value , then by the r o w to be multiplied b y the constant value , and finall y enter the ro w that will be r eplaced.
P age 11-1 Chapter 11 M atr ix Operations and L in ear Algebr a In Chapter 10 w e introduced the concept of a matr ix and pr esent ed a number of func tions f or enter ing, c r eating, o r manipulating matri ces. In this Chapt er w e pr esent e xamples o f matr ix oper ations and applicatio ns to pr oblems of linear algebra .
P age 11-2 Addition and subtr action Consi der a pair of matr ices A = [a ij ] m × n and B = [b ij ] m × n . A ddition and subtr action of thes e t w o matri ces is only pos sible if the y have the s ame number of r ow s and columns. The r esulting matr i x, C = A ± B = [c ij ] m × n has elem ents c ij = a ij ± b ij .
P age 11-3 By comb ining add ition and subtr action w ith multiplicatio n by a scalar w e can fo rm linear combinati ons of matr ices o f the same dimensions , e.g ., In a linear combinati on of matr ices, w e can multiply a matr i x by an imaginary number to obtain a matr ix o f complex n umbers, e .
P age 11-4 Matrix multiplication Matri x multiplicati on is defined b y C m × n = A m × p ⋅ B p × n , w h er e A = [a ij ] m × p , B = [b ij ] p × n , and C = [c ij ] m × n . Notice that matr ix multipli cation is onl y possible if the number of columns in the f irst oper and is equal to the number o f r o ws of the second oper and.
P age 11-5 (another r ow vect or). Fo r the calculator to identify a ro w vector , you mus t use double br acke ts to enter it: T erm -b y-term multiplication T erm-b y-term multiplication o f two matri ces of the same dimensions is possible thr ough the us e of func tion HAD AMARD .
P age 11-6 In algebr aic mode , the k eys trok es are: [enter or s elect the matri x] Q [enter the po wer] ` . In RPN mode , the ke ys tr ok es ar e: [enter or select the matr ix] † [enter the po we r] Q` . Matri ces can be r aised to negativ e po we rs .
P age 11-7 T o ver ify the pr operties of the in verse matr ix , consider the follo wing multiplications: Characteri zing a matrix (T h e matr ix NORM menu) The matr ix NORM (NORMALI ZE) menu is acces.
P age 11-8 Function ABS F unction ABS calc ulates what is kno wn as the F robeniu s norm of a matr ix . For a matri x A = [a ij ] m × n , the F robeniu s norm of the matr ix is def ined as If the matri x under consider ation in a ro w vec tor or a column vector , then the F robeniu s norm , || A || F , is simply the v ector ’s magnitude .
P age 11-9 Functions RNRM and CNRM F unction RNRM r eturns the Ro w NoRM of a matr i x , while f unction CNRM r eturns the C olumn NoRM of a matri x. Ex amples, Singular value decomposition T o underst and the oper ation of F uncti on SNRM, w e need to introduce the concept of matri x decompositi on.
P age 11-10 Function SRAD F unction SRAD determine s the Spectr al R ADius o f a matri x, def ined as the large st of the ab solute v alues of its eigen values .
P age 11-11 T ry the follo wing e xer cis e fo r matri x condition nu mber on matr i x A3 3. T he condition number is C O ND( A3 3 ) , r o w norm, and column norm for A3 3 are sho wn to the left .
P age 11-12 F or ex ample, try finding the r ank for the matr ix: Y ou will f i nd that the r ank is 2 . That is because the second r ow [2 , 4, 6] is equal to the f irst r ow [1,2 , 3] multiplied b y 2 , thu s, ro w two is linear ly dependent o f r o w 1 and the max imum number of linearl y independent r o ws is 2 .
P age 11-13 The determinant of a matri x The de ter minant of a 2x2 and o r a 3x3 matri x ar e r e pr esented b y the same arr angement of elements o f the matr ices, but enc losed betw een ve rtical lines , i.
P age 11-14 Function TRACE F unction TRA CE calculates the tr ace of squar e matri x, def ined as the sum of the elements in its main diagonal , or . Example s: F or squar e matrice s of hi gher or der determinants can be calc ulated by using smaller or der determinant called cof actors .
P age 11-15 Function TRAN F unction TRAN re turns the tr anspose o f a r eal or the conj ugate transpo se of a comple x matri x. TRAN is equi valent t o TRN.
P age 11-16 MAD and RSD ar e related t o the soluti on of s yste ms of linear equati ons and wil l be pr esent ed in a subsequen t sec tion in this Cha pter .
P age 11-17 The im plementation of functi on L CXM f or this case r equires y ou to enter : 2`3`‚ @@P1@@ LCXM ` The f ollow ing fi gure sho ws the RPN st a c k befo r e and after apply ing func tion LC X M : In AL G mode , this ex ample can be obtained b y using: The pr ogram P1 mu st still ha ve been c reated and stor ed in RPN mode.
P age 11-18 , , Using the numerical solv er for linear s ystems Ther e are man y way s to solv e a sy stem of linear equations w ith the calculator . One possib ility is through the numer ical sol v er ‚Ï . Fr om the numer ical sol ver s cr een, sho wn belo w (left) , select the opti on 4.
P age 11-19 This s yst em has the same number of equations as of unknow ns, and will be r efer red to as a squar e sy stem. In gener al, there sho uld be a unique soluti on to the s ystem . The soluti on will be the po int of intersec tion o f the thr ee planes in the coor dinate sy stem (x 1 , x 2 , x 3 ) r epre sented b y the three eq uations.
P age 11-20 T o chec k that the solution is cor r ect , ent er the matri x A and multiply times this soluti on vector (e xample in algebr aic mode) : Under-deter mined sy stem The s ys tem of linear e.
P age 11-21 T o see the details of the so lution v ector , if needed , press the @E DIT! button . T his w ill acti vate the Matr ix W r iter . Within this en vir onment, u se the r ight- and left- arr ow k e ys t o mov e about the vec tor: Thu s, the solution is x = [15 .
P age 11-22 Let’s stor e the latest result in a v ari able X, and the matr i x into var iable A, as fo llow s: Press K~x` to stor e the solution vect or into var iable X Press ƒ ƒ ƒ to clear thr .
P age 11-2 3 can be wr itten as the matri x equation A ⋅ x = b , if This s ystem has mo r e equations than unkno wns (an ov er-determined s yst em) .
P age 11-2 4 Press ` to retur n to the numer ical solv er env ironment . T o check that the soluti on is correc t, try the follo wing: • Press —— , to highlight the A: field . • Press L @CALC@ ` , to cop y matri x A onto the stack . • Press @@@OK@@@ to r eturn to the n umeri cal sol ver e nv ironment .
P age 11-2 5 • If A is a squar e matri x and A is non-singular (i .e., it’s in ver se matr ix e xis t , or its determinant is non- z ero), LSQ r eturns the ex act soluti on to the linear s y stem .
P age 11-2 6 Under-deter mined sy stem Consider the s yst em 2x 1 + 3x 2 –5x 3 = -10, x 1 – 3x 2 + 8x 3 = 85, wi th The s oluti on using LS Q is sho wn ne xt: Over-determin ed s ystem Consider the s yst em x 1 + 3x 2 = 15, 2x 1 – 5x 2 = 5, -x 1 + x 2 = 2 2 , wi th The s oluti on using LS Q is sho wn ne xt: .
P age 11-2 7 Compar e these thr ee soluti ons w ith the ones calculated w ith the numer ical solver . Solution with the in verse matri x The s olution t o the sy stem A ⋅ x = b , w here A is a squar e matri x is x = A -1 ⋅ b . This r esults fr om multiply ing the firs t equation b y A -1 , i.
Pa g e 1 1 - 2 8 The pr ocedure f or the case of “ di viding ” b by A is illustr ated belo w for the cas e 2x 1 + 3x 2 –5x 3 = 13, x 1 – 3x 2 + 8x 3 = -13, 2x 1 – 2x 2 + 4x 3 = -6 , The pr ocedure is show n in the follo wing s cr een shots: The s ame soluti on as found abo ve w ith the inv erse matr i x.
P age 11-29 [[14,9,-2] ,[2,-5,2],[5, 19,12]] ` [[1,2,3],[3, -2,1],[4,2,- 1]] `/ The r esult of this oper ation is: Gaussian and Gauss-Jordan elimination Gaussi an elimination is a pr ocedure b y whi c.
P age 11-30 T o start the pr ocess of f orwar d elimination , we di vi de the firs t equation (E1) b y 2 , and st or e it in E1, and sho w the three eq uatio ns again to pr oduce: Next , we r eplac e the second equati on E2 by (equation 2 – 3 × equation 1, i .
P age 11-31 an expr ession = 0. T hus, the las t set of equati ons is interpr eted to be the follo w ing equiv alent set of equatio n s: X +2Y+3Z = 7 , Y+ Z = 3, - 7Z = -14. The pr ocess of backw ard subs titution in Gaussi an elimination consis ts in finding the value s of the unknow ns, starting fr om the last equation and w orking upw a r d s.
P age 11-3 2 T o obtain a solution to the s yst em matr ix equati on using Gaussian eliminati on, we f i rs t c re a t e w h a t i s k n ow n a s t h e augmented matri x corr esponding to A , i .e., The matr ix A aug is the same as the or iginal matr ix A w ith a new r ow , corr esponding to the elements o f the vec tor b , added (i.
P age 11-3 3 Multiply r ow 2 by –1/8: 8Y2 @RCI! Multiply r ow 2 by 6 add it to r ow 3, r eplacing it: 6#2#3 @RCIJ! If y ou we r e perfor ming these oper ations by hand , you w ould wr ite the fo llo.
P age 11-34 Multiply r ow 3 by –1/7 : 7Y 3 @RCI! Multiply r ow 3 b y –1, add it to ro w 2 , r eplac ing it: 1 # 3 #2 @RCIJ! Multiply r ow 3 by –3, add it to r ow 1, r eplacing it: 3#3#1 @RCIJ! M.
Pa g e 1 1 - 3 5 While perfo rming pi voting in a matr ix elimination pr ocedure , yo u can impr ov e the numer ical solutio n e ven more b y selecting as the pi vot the ele ment wi th the large st absolute v alue in the column and r ow o f inter est .
Pa g e 1 1 - 3 6 No w we ar e read y to start the Gauss-Jor dan elimination w ith full pi vo ting. W e will need to k eep track of the per mutation matri x by hand, s o take y our notebook and w rite the P ma t rix s how n a b ove. F irst, w e check the piv ot a 11 .
P age 11-3 7 Hav ing f illed up w ith zer os the elements o f column 1 belo w the pi vot , now w e pr oceed to chec k the piv ot at position (2 ,2) . W e find that the number 3 in position ( 2 ,3) w i.
P age 11-38 2 Y #3#1 @RCIJ F inally , w e eliminate the –1/16 fr om position (1,2) b y using: 16 Y # 2#1 @RCIJ W e now ha ve an identity matri x in the por tion o f the augmented matr ix corr espond.
P age 11-3 9 Then , for this partic ular ex ample , in RPN mode , use: [2,-1,41] ` [[1,2,3],[2,0,3],[8 ,16,-1]] `/ The calc ulator sho ws an a ugmented matr ix consis ting of the coeff ic ients matr i.
P age 11-40 T o see the int ermedi ate steps in calc ulating and inv erse , jus t ente r the matri x A fr om abov e, and pr ess Y , while k eeping the step-b y-step option acti ve in the calc ulator’s CA S.
P age 11-41 The r esult ( A -1 ) n × n = C n × n / det ( A n × n ) , is a gener al result that appli es to any non -singular matr i x A . A general f orm for the ele ments of C can be written based on the Gaus s-Jor dan algorithm .
P age 11-4 2 LINSOLVE([X- 2*Y+Z=-8,2*X+ Y-2*Z=6,5*X-2 *Y+Z=-12], [X,Y,Z]) to pr oduce the solution: [X=-1, Y=2,Z = -3]. F unction LINS OL VE w orks w ith s ymboli c expr essions . F unctions REF , rre f, and RREF , w ork wi th the augment ed matri x in a Gaussi an elimination a ppr oach .
P age 11-43 The di agonal matr ix that r esults f r om a Gaus s -Jor dan elimination is called a r o w-reduced ec helon for m. F unction RREF ( R ow-R educed Ec helon F orm) The r esult of this f unction call is to pr oduce the r o w-r educed echelon f orm so that the matri x of coeff ici ents is r educed to an identity matri x.
P age 11-44 The r esult is the augmented matr i x corr esponding to the sy stem of equations: X+Y = 0 X- Y =2 Residual err ors in linear sy stem solutions (Function RSD) F unction R SD calculate s the Re SiDuals or error s in the so lution of the matri x equation A ⋅ x = b , repr esenting a s ystem o f n linear equations in n unkno wns.
P age 11-45 Eigenv alues and eig env ec tors Gi ven a sq uare matr ix A , w e can w rite the ei gen value equation A ⋅ x = λ⋅ x , wher e the values of λ that satisfy the equation ar e know n as the eigen values of matri x A .
Pa g e 1 1 - 4 6 Using the var iable λ to r epres ent e igen values , this char acter istic pol ynomial is t o be interpr eted as λ 3 -2 λ 2 -2 2 λ +21=0. Function EG VL F unction E GVL ( E iGenV aL ues) pr oduces the ei gen value s of a sq uar e matri x.
P age 11-4 7 of a matri x, w hile the cor r esponding ei genv alues are the compone nts of a vec tor . F or ex ample, in AL G mode , the e igen vector s and ei genv alues of the matr i x listed belo w are f ound by a pply ing func tion E G V: The r esult sho ws the e igen values as the columns of the matr ix in the r esult list .
P age 11-48 • A list w ith the eigen vect ors cor res ponding to eac h eigen value of matr ix A (stac k level 2) • A v ector w ith the eige nv ector s of matr i x A (stac k le ve l 4) F or ex ampl.
P age 11-4 9 Notice that the equati on ( x ⋅ I - A ) ⋅ p( x )=m( x ) ⋅ I is similar , in f orm , to the eige nvalue equati on A ⋅ x = λ⋅ x . As an e xample , in RPN mode , tr y: [[4,1,-2] [1, 2,-1][-2,-1,0 ]] M D The r esult is: 4: -8. 3: [[ 0.
P age 11-50 Function L U F unction L U tak es as input a s quar e matri x A , and r eturns a low er - tri angular matri x L , an upper tr iangular matri x U , and a perm utation matri x P , in stack lev els 3, 2 , and 1, respec ti ve ly . The r esults L , U , and P , satisfy the equation P ⋅ A = L ⋅ U .
P age 11-51 decomposition , while the v ector s r epresents the main di agonal of the matr ix S used earli er . F or ex ample, in RPN mode: [[5,4 ,-1],[2,-3,5 ],[7,2,8]] S VD 3: [[-0.2 7 0.81 –0. 5 3][-0. 3 7 –0. 5 9 –0.7 2][-0.8 9 3 . 09E -3 0.
Pa g e 1 1 - 52 Function QR In RPN, f unction QR pr oduces the Q R fact oriz at ion of a matrix A n × m r eturning a Q n × n orthogonal matri x, a R n × m upper tr apez oi dal matri x, and a P m × m permut ation matri x, in s tack le ve ls 3, 2 , and 1.
Pa g e 1 1 - 5 3 This menu inc ludes functi ons AXQ, CHOLE SKY , G A US S, QXA, and S YL VE STER . Function AX Q In RPN mode , function AXQ pr oduces the quadr atic f orm cor responding t o a matri x A n × n in stac k leve l 2 using the n var iables in a v ector placed in st ack lev el 1.
P age 11-54 suc h that x = P ⋅ y , by using Q = x ⋅ A ⋅ x T = ( P ⋅ y ) ⋅ A ⋅ ( P ⋅ y ) T = y ⋅ ( P T ⋅ A ⋅ P ) ⋅ y T = y ⋅ D ⋅ y T .
Pa g e 1 1 - 5 5 Infor mation on the func tions list ed in this menu is pr esented belo w by using the calc ulator’s o w n help fac ility . The f igure s show the help f acility entry and the attached e xamples .
P age 11-5 6 Function KER Function MKISOM.
Pa g e 1 2 - 1 Chapter 12 Gr aphi c s In this chapt er we intr oduce some of the gr aphics capab ilities o f the calc ulator . W e wi ll pre sent graphi cs of functi ons in Cartesian coor dinates and polar coor dinates , parametr ic plots , gr aphics of co nics , ba r plots, s cat ter plots, and a var iety of thr ee -dimensi onal gr aphs.
Pa g e 1 2 - 2 The se gr aph options ar e desc ri bed bri efl y next . Fu n ct i o n : f or equations of the f orm y = f(x) in plane Cartesi an coordinates P olar : for equations o f the f ro m r = f(.
Pa g e 1 2 - 3 Θ Enter the P L O T env ir onment b y pres sing „ñ (pre ss them simultaneou sly if in RPN mode). Pr ess @ADD to get y ou into the equati on wr iter .
Pa g e 1 2 - 4 Θ Enter the P L OT WINDO W env ironme nt by ent er ing „ò (pr ess them simultaneously if in RPN mode). Use a r ange of –4 to 4 for H- VIEW , t hen press @AUT O to generate the V - VIEW automatically .
Pa g e 1 2 - 5 Some useful PL O T operations fo r FUNCTION plots In orde r to disc uss these P L O T options , w e'll modif y the func tion to f or ce it to hav e some real r oots (Since the curr ent curve is totall y contained abov e the x axis , it has no real r oots.
Pa g e 1 2 - 6 ROO T : 1.66 3 5... T he calculator indicated , befor e show ing the root , that it wa s found thr ough SIGN REVER SAL . Press L to r ecover the menu . Θ Pr es sing @ISECT w ill gi ve y ou the intersecti on of the c urve w i th the x -axis, whi ch is esse ntiall y the roo t .
Pa g e 1 2 - 7 Θ Enter the PL O T env i r onment by pres sing, simultaneousl y if in RPN mode, „ñ . Notice that the highlighted f ield in the PL O T en v ir onment now contains the deri vati ve of Y1(X) . Pr ess L @@@OK@@@ to r eturn to return to nor mal calculator dis play .
Pa g e 1 2 - 8 T o r eturn t o nor mal calculato r func tion , pres s @) PICT @CANCL . Graphics of tr anscendental func tions In this secti on we us e some of the gr aphics f eatures of the calc ula tor t o sho w the typi cal beha vior of the natur al log, e x ponenti al, tr igonometr ic and h yperboli c functi ons.
Pa g e 1 2 - 9 10 by us i ng 1 @@@OK@@ 10 @@@OK@@@ . Ne xt , pr ess the soft k ey labeled @AUTO to let the calc ulator determine the cor r esponding v er ti cal range . After a co uple of seconds this r ange w ill be shown in the P L O T WINDO W-FUNCT ION w indo w .
Pa g e 1 2 - 1 0 Graph of the e x ponential function F irst , loa d the f unction e xp(X) , by pr essing , simultaneousl y if in RPN mode , the left-shif t k ey „ and the ñ ( V ) k ey to acce ss the PL O T-FUNCTI ON windo w . Pr ess @@DEL@@ to remo ve the f unction LN( X) , if y ou didn’t dele te Y1 as suggested in the pr ev ious no te .
Pa g e 1 2 - 1 1 The P P AR v ariable Press J to reco ver y our vari ables menu , if needed. In your v ariables me nu y ou should ha ve a v ar iable labe led PP AR .
Pa g e 1 2 - 1 2 As indicated earl ier , the ln(x) and exp(x) f unctions ar e inv erse of eac h other , i .e., ln(e xp(x)) = x, and e xp(ln(x)) = x. This can be v erif ied in the calculato r b y typing and e valuating the f ollow ing expr essi ons in the Eq uation W r iter: LN(EXP(X)) and EXP(LN(X)).
Pa g e 1 2 - 1 3 Summary of FUNCTION plot oper ation In this secti on w e pre sent inf ormati on regar d ing the PL O T SETUP , PL O T - FUNCTION , and P L O T WINDOW sc reens accessible thr ough the left-shif t k ey combined w ith the soft-menu k ey s A thr ough D .
Pa g e 1 2 - 1 4 Θ Use @CANCL to cancel an y changes to the P L O T SE TUP windo w and retur n to nor mal calc ulator displa y . Θ Press @@@OK@@@ to save c h anges to the op tions in the PL O T S E TUP window and r eturn t o normal calc ulator display .
Pa g e 1 2 - 1 5 Θ Enter lo w er and upper limits fo r hor iz ont al v ie w (H-V ie w) , and pr ess @AUTO , whi le the cur sor is in one of the V - Vie w fi elds, to gener ate the verti cal vie w (V- Vie w) range automatically .
Pa g e 1 2 - 1 6 „ó , simultaneou sly if in RPN mode: Plots the gr aph based on the setting s stor ed in var iable P P AR and the cur rent func tions def ined in the PL O T – FUNCTION s cr een.
Pa g e 1 2 - 1 7 Generating a table of v alues for a function The co mbinati ons „õ ( E ) and „ö ( F ), pressed simultaneousl y if in RPN mode , let’s the us er pr oduce a table of values o f functi ons.
Pa g e 1 2 - 1 8 the corr esponding values o f f(x) , listed as Y1 b y default . Y ou can us e the up and do wn ar ro w k ey s to mo ve abou t in the table . Y ou w ill notice that w e did not hav e to indicate an ending value f or the independent var iable x.
Pa g e 1 2 - 1 9 W e wi ll try to plot the f unction f( θ ) = 2(1-sin( θ )), as follo ws: Θ F irst , mak e sure that y our calculator ’s angle measure is s et to r adians. Θ Press „ô , simultaneousl y if in RPN mode, to access to the PL O T SETUP w indo w .
Pa g e 1 2 - 2 0 Θ Press L @CAN CL to retu rn to th e PL O T W I ND OW scre en. Press L @@@OK@@@ to r eturn t o normal calc ulator display . In this ex erc ise w e ent er ed the equation to be plotted dir e ctl y in the PL O T SETUP w indo w . W e can also enter equations f or plotting using the PL O T w indow , i.
Pa g e 1 2 - 2 1 The calc ulator has the ability of plotting one or more coni c curv es b y selecting Con ic as the functi on TYPE in the PL O T en vir onment .
Pa g e 1 2 - 2 2 Θ T o see labels: @EDIT L @) LABEL @MENU Θ T o r ecov er the menu: LL @) PICT Θ T o estimate the coordinate s of the po int of inter section , pres s the @ ( X, Y ) @ menu k ey and mo ve the c ursor as c lose as pos sible to those points using the arr ow k ey s.
Pa g e 1 2- 23 whi ch inv olve constant values x 0 , y 0 , v 0 , and θ 0 , we need to stor e the values of those par ameters in v ari ables . T o de ve lop this ex ample, c reate a sub-dir ectory cal.
Pa g e 1 2 - 24 Θ Press @AUTO . This w i ll gener ate automatic values of the H-V iew and V - Vi ew r anges based on the v alues of the independent v ariable t and the def initions o f X(t) and Y(t) us ed. The r esult w ill be: Θ Press @ERASE @DRAW to dr aw the par ametri c plot .
Pa g e 1 2 - 2 5 parameter s. The other v ariables contain the v alues of constants us ed in the def initions of X(t) and Y(t). Y ou can store diff erent v alues in the var iables and pr oduce ne w parametr ic plots of the pr ojectile equati ons used in this ex ample.
Pa g e 1 2 - 26 P lot ting the solution to simpl e differ ential equations The plot o f a simple differ ential equati on can be obtained by selecting Diff Eq in the TYPE fi eld of the PL O T SETUP en .
Pa g e 1 2- 27 Θ Press L to reco ver the menu . P r ess L @) PICT to reco ver the original graphi cs menu. Θ When w e observed the gr aph being plotted, y ou'll notice that the gr aph is not v er y smooth . That is becau se the plotter is u sing a time step th at is too lar ge .
Pa g e 1 2 - 28 T ruth plots T ruth plots ar e used to pr oduce two -dimensi onal plots of r egions that satisfy a certain mathematical co ndition that can be eithe r true or fals e.
Pa g e 1 2 - 2 9 Θ Press „ô , sim ultaneously if in RPN mode , to access t o the PL O T SETUP wi nd ow . Θ Press ˜ and ty pe ‘(X^2/3 6+Y^2/9 < 1) ⋅ (X^2/16+Y^2/9 > 1)’ @@@OK@@@ to def ine the conditions to be plotted . Θ Press @ERASE @DRAW to dr aw the truth plot .
Pa g e 1 2 - 3 0 [4. 5,5.6 ,4.4],[4.9 , 3.8 ,5 .5],[5 .2 ,2 .2 ,6 .6]] ` to stor e it in Σ D A T , use the f unction S T O Σ (a vailable in the functi on c atalog , ‚N ) . Pr ess V AR to reco ve r your v ariable s menu . A soft menu ke y labeled Σ D A T should be a vailable in the s tac k.
Pa g e 1 2 - 3 1 accommodate the max imum value in column 1 of Σ D A T . Bar plots ar e use ful when plotting categori cal (i .e., non -numer ical) data. Suppos e that y ou want t o plot the data in column 2 of the Σ DA T m a t rix : Θ Press „ô , sim ultaneously if in RPN mode , to access t o the PL O T SETUP wi nd ow .
Pa g e 1 2- 32 Θ Press @ERASE @DR AW to dr aw the bar plot . Pre ss @EDIT L @LABEL @ MENU to see the plot unenc umber ed by the menu and w i th ide ntifying la bels (the c ursor w i ll be in the middle of the plot , how ev er ): Θ Press LL @) PICT to lea ve the EDI T en vir onment.
Pa g e 1 2- 3 3 Slope fields Slope fi elds ar e us ed to vi suali z e the solutions to a diffe r ential equation of the fo rm y’ = f(x ,y) . Basi cally , w hat is pres ented in the plot ar e segmen .
Pa g e 1 2 - 3 4 of y(x ,y) = constant , for the soluti on of y’ = f(x,y). Thus , slope fi elds ar e usef ul tools f or v isualizing parti cul arl y diffic ult equations to sol ve .
Pa g e 1 2 - 35 Θ Press @ERASE @DRAW to dr aw the thr ee -dimensional surface . The r esult is a w i r efr ame pi ctur e of the surface w ith the r efer ence coor dinate s y stem sho wn at the lo wer le ft corner of the sc reen . B y using the arr o w k ey s ( š™— ˜ ) you can c hange the orient ation of the surface .
Pa g e 1 2 - 36 Θ Press „ô , simult aneousl y if in RPN mode, t o access the P L O T SETUP wi nd ow . Θ Press ˜ and type ‘S IN(X^2+Y^2)’ @@@OK@@@ . Θ Press @ERASE @DRAW to dra w the plot . Θ When done , pres s @EXIT . Θ Press @CANCL to r eturn to PL O T W INDO W .
Pa g e 1 2 - 37 Θ Press @EDIT L @ LABEL @MENU to s ee the gr aph w ith labels and range s. This partic ular v ersio n of the gr aph is limited to the lo we r part of the display . W e can change the v ie wpoint to see a differ ent versi on of the graph .
Pa g e 1 2 - 3 8 T ry also a Wir efr ame plot for the surface z = f(x ,y) = x 2 +y 2 Θ Press „ô , simult aneousl y if in RPN mode, t o access the P L O T SETUP wi nd ow . Θ Press ˜ and type ‘X^2+Y^2’ @@@OK@@@ . Θ Press @ERASE @DRAW t o dr aw the slope f ield plot .
Pa g e 1 2 - 3 9 Θ Press @EDIT ! L @LABEL @MENU t o see the gr aph w ith labels and r anges. Θ Press LL @) PICT@CANCL to r eturn to the PL O T WINDOW en vir onment . Θ Press $ , or L @@@OK@@@ , to return to nor mal calculator displa y . T ry also a P s-Contour plot f or the sur face z = f(x ,y) = sin x cos y .
Pa g e 1 2 - 4 0 Θ Make sur e that ‘X’ is selected as the Indep: and ‘ Y’ as the Depnd: variab le s. Θ Press L @@@OK@@@ to re turn to nor mal calculator dis play . Θ Press „ò , simultaneou sly if in RPN mode , to access the P L O T WINDO W scr e en .
Pa g e 1 2 - 4 1 Θ Press „ô , sim ultaneou sly i f in RPN mode , to access t o the P L O T SETUP w indow . Θ Cha nge TYPE to Gr idmap. Θ Press ˜ and type ‘SIN(X+i*Y)’ @@@OK@@@ . Θ Make sur e that ‘X’ is selected as the Indep: and ‘ Y’ as the Depnd: variab le s.
Pa g e 1 2 - 42 F or ex ample, to pr oduce a Pr- Sur fa ce plot for the surf ace x = x(X,Y) = X sin Y , y = y(X,Y) = x co s Y , z=z(X,Y)=X, u se the f ollo w ing: Θ Press „ô , sim ultaneou sly i f in RPN mode , to access t o the P L O T SETUP w indow .
Pa g e 1 2 - 4 3 Interactiv e draw ing Whene ve r we pr oduce a two-dimensional gr aph, w e find in the gr aphics s cr een a soft menu k e y label ed @) EDIT .
Pa g e 1 2 - 4 4 Ne xt, w e illustr ate the use o f the differ ent dra w ing functi ons on the resulting gr aphic s sc reen . The y requir e use of the c ursor and the arr ow k ey s ( š™— ˜ ) to mo ve the cu rsor about the gr aphic s scr een.
Pa g e 1 2 - 4 5 should hav e a str aight angle tr aced by a hor iz on tal and a ve rtical segmen ts. The c ursor is still acti ve . T o deacti vate it , without mo ving it at all , pre ss @LINE . The c ursor r eturns to its n ormal shape (a c ro ss) and the LINE f unction is no longer acti ve .
Pa g e 1 2 - 4 6 DEL This command is us ed to remo ve parts of the gr aph betw een two MARK positions . Mov e the cur sor to a point in the gr aph, and pre ss @MARK . Mov e the cu rsor to a diff er ent point , press @MARK again. T hen, pr ess @@DEL@ .
Pa g e 1 2- 47 X,Y This command copi es the coordinates o f the cur r ent cur sor position, in us er coor dinates , in the stac k . Zooming in and out in the gr aphics display Whene ve r y ou prod.
Pa g e 1 2 - 4 8 Y ou can alwa ys r eturn to the v ery last z oom windo w by using @ZLAST . BO XZ Z ooming in and out of a gi ven gr aph can be perfor med by u s ing the so ft-menu ke y BO XZ . With BO XZ you selec t the rect angular sector (the “bo x ”) that y ou want to z oom in into.
Pa g e 1 2 - 4 9 cu rsor at the center of the scr een, the w indow gets z oomed so that the x -ax is extends f rom –64. 5 to 6 5 . 5 . ZSQR Z ooms the graph s o that the plotting scale is maintained at 1:1 b y adjus ting the x scale , keeping the y s c ale f ix ed, if the w indow is w ider than taller .
Pa g e 1 2- 5 0 S OL VER .. „Î (the 7 key ) C h. 6 TRIGONO METRIC. . ‚Ñ (the 8 key ) Ch. 5 EXP&LN .. „Ð (the 8 key ) Ch. 5 The S YMB/GRAPH menu The GRAP H sub-men u w ithin the S YMB menu.
Pa g e 1 2 - 5 1 T AB V AL(X^2 -1,{1, 3}) produ ces a list of {min max} v alues of the f u ncti on in the interval {1, 3}, w hile SIGNT AB(X^2 -1) show s the sign of the fu ncti on in the interval ( - ∞ ,+) , w ith f(x) > 0 in (- ∞ ,-1), f(x) <0, in (-1,1) , and f(x) > 0 in (1,+ ∞ ).
Pa g e 1 2 - 52 of F . The question ma rks indicates u ncertai nty or non -definition. Fo r example , for X<0, LN(X) is not def ined, thu s the X lines sho ws a que stion mark in that interval . Ri ght at z er o (0+0) F is inf inite, f or X = e, F = 1/e.
P age 13-1 Chapter 13 Calculus Applications In this Chapter w e discu ss applicati ons of the calculator ’s functions to oper ations r elated to Calc ulus, e .
P age 13-2 Function lim The calc ulator pro vi des functi on lim t o ca l cu l a t e l im i t s o f fu n ct i on s . Th i s fu n c ti o n uses a s input an expr ession r epr esenting a fu nction and the v alue wher e the limit is to be calculated . Functi on lim is av ailable thro ugh the command catalog ( ‚N~„l ) or thr ough option 2 .
P age 13-3 T o calculat e one -sided limits, add +0 or -0 to the v alue to the vari able. A “+0” means limit fr om the ri ght , w hile a “-0” means limit fr om the left .
P age 13-4 in AL G mode . R ecall that in RPN mode the arguments mu st be en ter ed befor e the functi on is applied. The DERI V&INTEG menu The f unctions a vailable in this sub-me nu ar e listed belo w: Out of thes e functi ons D ERIV and DER VX ar e used for der iv ati ve s.
P age 13-5 be differ entiated . Thus , to calculate the deri vati ve d(sin(r ) ,r), use , in AL G mod e: ‚¿~„r„ÜS~„r` In RPN mode , this expr ession mu st be enclos ed in quot es befo re ente ring it in to th e sta ck.
P age 13-6 T o ev aluate the deri vati ve in the E quation W r iter , pres s the up-arr o w k e y — , fo ur times, to s elect the entir e expr essi on, then , pr ess @EVAL . The der ivati ve w ill be ev aluated in the E quation W riter as: The chain rule The c hain rule for der ivati ves appli es to deri vati ves of composit e functi ons.
P age 13-7 Deri vati ves of equations Y ou can us e the calculat or to calculat e deri vati ves of eq uations , i .e., expr essi ons in whi ch deri vati ves w ill e xist in both side s of the equal sign .
P age 13-8 Analyzing gr aphic s of functions In Chapter 11 w e pres ented some functi ons that ar e a vailable in the gr ap hic s sc r een f or anal yzing gr aphics of f unctions of the for m y = f(x) . The se functi ons include (X,Y ) and TRACE f or determining point s on the gra ph, a s wel l as functi ons in the Z OOM and FCN menu .
P age 13-9 Θ Press L @PICT @CANCL $ to re turn to nor mal calculator dis play . Notice that the slope and tangent line that y ou reques ted ar e listed in the stac k. Function DOMAIN F unction DOMAIN , av ailable through the command catalog ( ‚N ), pr o vi des the domain of def inition of a func tion as a list of numbers and spec ificati ons.
P age 13-10 This r esult indicat es that the r ange of the functi on corr esponding to the domain D = { -1,5 } is R = . Function SIGNT AB F unction SIGNT AB, av ailable through the command cat alog ( ‚N ), pro vides informa tion on th e sign of a function th r ou gh it s domai n .
P age 13-11 Θ Le vel 3: the f uncti on f(VX) Θ T wo lists , the fir st one indicate s the vari ation of the f unction (i .e., whe r e it incr eases or dec reas es) in ter m s of the independent v ari able VX, the second one indicate s the var iation of the f unction in ter ms of the dependent v ariable .
P age 13-12 The interpr etation of th e var iation table show n abo ve is as f ollow s: the functi on F(X) incr eases f or X in the interval (- ∞ , -1), reac hing a max imum equal to 3 6 at X = -1. Then, F(X) dec reas es until X = 11/3, reac hing a minimum of –400/2 7 .
P age 13-13 W e find tw o cr itical po ints, one at x = 11/3 and one at x = -1. T o ev aluate the second der iv ativ e at each point use: The last sc reen sho ws that f ”(11/3) = 14, thus , x = 11/3 is a r elativ e minimum.
P age 13-14 Anti-deri vativ es and integrals An anti-der iv ative o f a func tion f(x) is a func tion F(x) such that f(x) = dF/dx. F or e xam ple , since d(x 3 ) /dx = 3x 2 , an anti-der iv ativ e of f(x) = 3x 2 is F(x) = x 3 + C, wher e C is a constant.
P age 13-15 abo ve . Their r esult is the so -called discr ete der iv ativ e, i .e., one de fined f or integer number s only . Definite integr als In a def inite integr al of a f unction , the resulting an ti-der i vati ve is ev aluated at the upper and lo wer limit of an interv al (a,b) and the e valuated v a lues subtr acted .
P age 13-16 This is the gener al format f or the definite integr al when typed dir ectly into the stack , i .e., ∫ (low er limit , upper limit, int egr and, v ari able of in tegr ation) Pr essi ng `.
P age 13-17 The f ollow ing ex ample sho ws the ev aluation of a def inite integral in the E quation W r iter , step-by-s tep: ʳʳʳʳʳ Notice that the step-b y-step pr ocess pr ov ides infor mation on the inter mediate steps f ollow ed by the CAS to solv e this integral .
P age 13-18 T echniques of integration Se ver al techni ques of integr ation can be implemented in the calc ulators, as sho wn in the f ollo wing e xamples .
P age 13-19 Integration b y par ts and differentials A differ ential of a f unction y = f(x), is defined as d y = f’(x) dx , wher e f’(x) is the deri vati ve of f(x).
P age 13-20 Integration b y par tial fr actions F unction P AR TFRAC, presented in Chapte r 5, pr ovi d es the decomposition of a fr action into partial f rac tions. This t echni que is use ful to r educe a complicated fr action into a sum of simple fr actio ns that can then be integrated t erm b y te rm .
P age 13-21 Using the calc ulator , we pr oceed as fo llo ws: Alternati vel y , you can e valuate the integra l to inf inity from the start , e.g . , Integration w it h units An integr al can be calcu.
P age 13-22 Some no tes in the us e of units in the limits of integr ations: 1 – The units of the lo wer limit o f integrati on will be the ones used in the f inal r esult , as illustr ated in the two e xamples belo w: 2 - Upper limit units mus t be consiste nt w ith lo we r limit units .
Pa g e 1 3 - 23 T ay lor and M aclaurin’s series A func tion f(x) can be e xpanded into an infinit e ser ies ar ound a point x=x 0 by using a T a ylor ’s ser ies, namel y , , wher e f (n) (x) represen ts the n- th der iv ativ e of f(x) with r esp ect to x , f (0) (x) = f(x).
P age 13-2 4 wher e ξ is a number near x = x 0 . Since ξ is typ icall y unknow n, inst ead of an estimate o f the residual , we pr ov ide an estimate of the or der of the re sidual in ref e ren c e t o h , i. e., we s ay t h a t R k (x) has an err or of order h n+1 , or R ≈ O(h k+1 ).
P age 13-2 5 incr ement h. T he list r eturned as the fir st output objec t includes the f ollow ing items: 1 - Bi-dir ectional limit o f the funct ion at po int of e xpansion , i .
Pa g e 1 4 - 1 Chapter 14 Multi-var iate Calculus Applications Multi-vari ate calculus re fers to functi ons of two or mor e vari ables. In this Chapter w e discu ss the basi c concepts of multi-v ari ate calculu s including partial deri vati ves and multiple int egrals .
Pa g e 1 4 - 2 . Similarl y , . W e wi ll use the multi-var iate functi ons def ined earlier to calc ulate partial deri vati ves using the se def initions.
Pa g e 1 4 - 3 ther ef or e , w ith DERVX y ou can only calculate der iv ativ es with r espect to X. Some e xamples of f irst-order partial der iv ative s are sho wn ne xt: ʳʳʳʳʳ Hi gh er- ord e .
Pa g e 1 4 - 4 Thir d-, fourth-, and higher or der der i vati ves ar e def ined in a similar manner . T o calc ulate higher o rde r der iv ativ es in the calculator , simply r epeat the deri vati ve func tion as man y times as needed.
Pa g e 1 4 - 5 A diffe r ent ver sion of the c hain rule appli es to the case in whi ch z = f(x ,y) , x = x(u ,v) , y = y(u ,v), so that z = f[x(u, v) , y(u ,v)].
Pa g e 1 4 - 6 W e find c riti cal points at (X,Y) = (1, 0) , and (X,Y) = (-1, 0) . T o calculate the disc riminant , we pr oceed to calculate the second der iv ativ es, fXX(X,Y) = ∂ 2 f/ ∂ X 2 , fXY(X,Y) = ∂ 2 f/ ∂ X/ ∂ Y , and fYY(X,Y) = ∂ 2 f/ ∂ Y 2 .
Pa g e 1 4 - 7 Applicati ons of function HE S S are easi er to visuali z e in the RPN mode . Consi der as an ex ample the function φ (X,Y ,Z) = X 2 + XY + XZ , we ’ll apply fun ctio n H E SS to fu nct ion φ i n t h e f ol l owi n g e xa m p l e. T h e s cr e e n s h o t s s h ow t h e RPN stac k befo re and after appl y ing functi on HES S .
Pa g e 1 4 - 8 The r esulting matri x has elements a 11 = ∂ 2 φ / ∂ X 2 = 6 ., a 22 = ∂ 2 φ / ∂ X 2 = - 2 ., and a 12 = a 21 = ∂ 2 φ / ∂ X ∂ Y = 0. The discr iminant , f or this criti cal point s2(1, 0) is Δ = ( ∂ 2 f/ ∂ x 2 ) ⋅ ( ∂ 2 f/ ∂ y 2 )- [ ∂ 2 f/ ∂ x ∂ y] 2 = (6 .
Pa g e 1 4 - 9 Jacobian of coordinate tr ansformation Consi der the coordinate tr ansfor mation x = x(u ,v) , y = y(u ,v) . The Jacobi an of this tr ansfor mation is def ined as . When calc ulating an integr al using suc h transf ormation , the e xp r ession to us e is , whe re R’ is the r egion R expr essed in (u ,v) coordinates .
Pa g e 1 4 - 1 0 wher e the region R’ in polar coor dinates is R ’ = { α < θ < β , f ( θ ) < r < g( θ )}. Double integr als in polar coordinat es can be enter ed in the calculator , making sur e that the Jacobian |J| = r is inc lud ed in the integrand.
P age 15-1 Chapter 15 V ector Analy sis Applications In this Chapter w e pres ent a number of functio ns fr om the CAL C menu that apply t o the analy sis of scalar and vec tor fields .
P age 15-2 At an y partic ular point , the maximum r ate of change of the functi on occurs in the dir ection o f the gradien t , i .e ., along a unit vec tor u = ∇φ /| ∇φ |.
P age 15-3 as the matri x H = [h ij ] = [ ∂φ / ∂ x i ∂ x j ], the gradie nt of the f unction w ith r espec t to the n-vari ables, gr ad f = [ ∂φ / ∂ x 1 , ∂φ / ∂ x 2 , … ∂φ / ∂ x n ], and the list of vari ab le s [ ‘ x 1 ’ ‘ x 2 ’…’x n ’].
P age 15-4 not hav e a potential functi on asso c iated with it , sinc e , ∂ f/ ∂ z ≠∂ h/ ∂ x. Th e calcula tor response in th is case is shown below : Div ergence The di ver gence of a vect.
P age 15-5 Cur l The c url of a v ector field F (x ,y ,z) = f(x ,y ,z) i +g(x ,y ,z) j +h(x,y ,z) k , is def ined by a “ c r oss-pr oduct” of the del oper ator with the v ector fi eld, i .e. , The c url of v ector fi eld can be calc ulated with f unction C U RL .
P age 15-6 As an ex ample, in an ear lier ex ample w e attempted to f ind a potenti al func tion for the vect or fie ld F (x,y ,z) = (x+y) i + (x-y+z) j + xz k , and got an err or message back f r om functi on PO TENT IAL. T o ver if y that this is a r otational f ield (i .
P age 15-7 pr oduces the v ector potential f unction Φ 2 = [0, ZYX- 2YX, Y -( 2ZX-X)], w hic h is differ ent fr om Φ 1 . The last command in the sc r een shot show s that indeed F = ∇× Φ 2 . Thu s, a vector potenti al function is not uniquel y d eter mined.
Pa g e 1 6 - 1 Chapter 16 Differential Equations In this Chapter w e pres ent e xample s of sol ving or dinary differ ential equati ons (ODE) using calc ulator functi ons. A differ ential equatio n is an equati on inv olv ing deri vativ es of the independent var iable .
Pa g e 1 6 - 2 ( H @) DISP ) is not selec ted. Pr ess ˜ to see the equati on in the Equati on Wr i t e r. An alter native no tation fo r deri vati ves typed dir ectly in the s tack is to use ‘ d1.
Pa g e 1 6 - 3 EV AL(ANS(1)) ` In RPN mode: ‘ ∂ t( ∂ t(u(t)))+ ω 0^2*u(t) = 0’ ` ‘ u(t)=A*SIN ( ω 0*t)’ ` SUBST EVAL The r esult is ‘0=0’ . F or this ex ample, yo u could also use: ‘ ∂ t( ∂ t(u(t))))+ ω 0^2*u(t) = 0’ to enter the differ ential equation .
Pa g e 1 6 - 4 The se func tions ar e brief ly desc ribed next . The y will be desc ribed in mor e detail in later parts of this Chapte r . DE SOL VE: Differe ntial E quation S OL VEr , pro vi des a s.
Pa g e 1 6 - 5 Both of these inputs mu st be giv en in terms of the def ault independent v ari able for the calc ulator’s CAS (ty pi cally ‘X’). The output fr om the functi on is the gener al solution o f the ODE . The functi on LDEC is av ailable thr ough in the CAL C/DIFF menu .
Pa g e 1 6 - 6 The s olution , show n partially her e in the Equati on W riter , is: Replac ing the combination o f constants accompan ying the e xponential ter ms with sim pler values , the e xpres sion can be simplifi ed to y = K 1 ⋅ e –3x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + ( 4 50 ⋅ x 2 +3 30 ⋅ x+2 41)/13 500.
Pa g e 1 6 - 7 2x 1 ’(t) + x 2 ’(t) = 0. In algebr aic fo rm , this is wr it te n as: A ⋅ x ’(t) = 0, wher e . T he s ys tem can be so lv ed by using fu nctio n LDEC w ith argumen ts [0, 0] an.
Pa g e 1 6 - 8 Example 2 -- Sol ve the second-o rde r ODE: d 2 y/dx 2 + x (dy/dx) = e xp(x) . In the calculator u se: ‘ d1d1y(x)+x*d 1y(x) = EXP(x) ’ ` ‘ y(x) ’ ` DESOLVE The r esult is an e x.
Pa g e 1 6 - 9 P erf orming the int egr ation b y hand, we can onl y get it as far as: because the in tegr al of exp(x)/x is no t av ailable in c losed f orm. Example 3 – Sol v ing an equatio n w ith initial conditi ons. Sol ve d 2 y/dt 2 + 5y = 2 cos(t/2) , w ith initial conditions y(0) = 1.
Pa g e 1 6 - 1 0 Press J @ODETY to get the str ing “ Linear w/ cst coeff ” for the ODE type in this case . Laplace T ransf orms The L aplace transf orm o f a functi on f(t) pr oduces a functi on F.
Pa g e 1 6 - 1 1 Laplace tr ansform and in verses in the calc ulator The calc ulator pr o vi des the f uncti ons LAP and IL AP to calc ulate the L aplace transf orm and the in verse L aplace transf orm, r especti vel y , of a f unction f(VX), wher e VX is the CAS def ault independent var iable, w hich y ou should set to ‘X’ .
Pa g e 1 6 - 1 2 Example 3 – Deter mine the in ver se Laplace tr ansfor m of F(s) = sin(s). Use: ‘SIN(X)’ ` ILAP . The calc ulator tak es a fe w seconds to r eturn the re sult: ‘ILAP(SIN(X))’ , meaning that there is no cl osed-f orm expr essi on f(t), such that f(t) = L -1 {sin(s)}.
Pa g e 1 6 - 1 3 Θ Differ entiati on theorem f or the n -th deri vati ve . Let f (k) o = d k f/dx k | t = 0 , and f o = f(0), then L{d n f/dt n } = s n ⋅ F(s) – s n-1 ⋅ f o − …– s ⋅ f (n- 2) o – f (n-1) o . Θ L inearity theor em . L{af(t)+bg(t)} = a ⋅ L{f(t)} + b ⋅ L{g(t)}.
Pa g e 1 6 - 1 4 Θ Shift theore m for a shift to the r ight . Let F(s) = L{f(t)}, then L{f(t-a)}=e –as ⋅ L{f(t)} = e –as ⋅ F(s). Θ Shift theore m f or a shift to the left . Let F(s) = L{f(t)}, and a >0, then Θ Similar ity theor em . Let F(s) = L{f(t)}, and a>0, then L{f(a ⋅ t)} = (1/a) ⋅ F(s/a).
Pa g e 1 6 - 1 5 Dirac’s delta function and Heaviside’s step function In the analy sis of contr ol s ys tems it is cu stomary to utili ze a ty pe of functi ons that r epre sent certain ph ysi cal .
Pa g e 1 6 - 1 6 Y ou can pro ve that L{H(t)} = 1/s, from whi ch i t fol lows th at L {U o ⋅ H(t)} = U o /s, wher e U o is a constant. Also , L -1 {1/s}=H(t), and L -1 { U o /s}= U o ⋅ H(t).
Pa g e 1 6 - 1 7 Applications of Laplace tr ansform in the solution of linear ODEs At the beginning of the se ction on L aplace transf orms w e indicated that y ou could us e these tr ansfor ms to conv er t a linear ODE in the time domain into an algebrai c equation in the image domain .
Pa g e 1 6 - 1 8 The r esult is ‘H=((X+1)*h0+a)/(X^2+(k +1)*X+k)’ . T o fi nd the soluti on to the ODE , h(t) , w e need to use the inv erse L aplace transf orm, as f ollow s : OB J ƒ ƒ Isolate s ri ght-hand side of last e xpres sion ILAP μ Obtains the inv erse L aplace tr ansfor m The r esult is .
Pa g e 1 6 - 1 9 With Y(s) = L{y(t)}, and L{d 2 y/dt 2 } = s 2 ⋅ Y(s) - s ⋅ y o – y 1 , w here y o = h(0) and y 1 = h ’(0), the transf ormed eq uation is s 2 ⋅ Y(s) – s ⋅ y o – y 1 + 2 ⋅ Y(s) = 3/(s 2 +9).
Pa g e 1 6 - 2 0 Example 3 – Consider the equati on d 2 y/dt 2 +y = δ (t-3) , wher e δ (t) is Dir ac’s delta functi on. Using Laplace tr ansforms , we can wr ite: L{d 2 y/dt 2 +y} = L{ δ (t-3)}, L{d 2 y/dt 2 } + L{y(t)} = L{ δ (t-3)}. Wit h ‘ Delta(X-3) ’ ` LA P , the calc ulator produc es EXP(-3*X), i.
Pa g e 1 6 - 2 1 Check w hat the solution t o the ODE would be if y ou use the f unction LDEC: ‘Delta(X- 3)’ ` ‘X^2+1’ ` LDEC μ Notes : [1]. An alter nativ e wa y to obtain the in ver se Laplace tr ansform of the e xpr essi on ‘(X*y0+(y1+EXP(-(3*X))))/(X^2+1)’ is b y separating the e xpr essi on into partial f r actions , i.
Pa g e 1 6 - 2 2 The r esult is: ‘SI N(X-3)*Heav iside(X-3) + cC1*SIN(X) + cC0*CO S(X)’ . P lease notice that the var iable X in this expr essi on actuall y re presen ts the var iable t in the ori ginal ODE .
Pa g e 1 6 - 23 Use of the func tion H(X) w ith LD E C, L AP , or IL AP , is not allo wed in the calc ulator . Y ou have to u se the main re sults pr ov ided earlier w hen dealing with the Heav iside step f unction , i .
Pa g e 1 6 - 24 wher e H(t) is Heavisi de’s step f u ncti on. Using L aplace transfo rms, w e can writ e: L{ d 2 y/dt 2 +y} = L{H(t-3 )}, L{d 2 y/dt 2 } + L{y(t)} = L{H(t-3)}.
Pa g e 1 6 - 2 5 Example 4 – P lot the solution to Ex ample 3 using the same v alues of y o and y 1 used in the plot of Example 1, abo ve . W e no w plot the functi on y(t) = 0.
Pa g e 1 6 - 26 f(t) = U o ⋅ [1-(t-a)/(b-1)] ⋅ [H(t-a) -H(t -b)]. Example s of the plots generated b y these functi ons, fo r Uo = 1, a = 2 , b = 3, c = 4, hor iz ontal r ange = (0,5) , and verti cal range = (-1, 1.
Pa g e 1 6 - 27 The f ollow ing ex erc ises ar e in AL G mode, w ith CAS mode s et to Ex act . (When y ou pr oduce a gr aph, the CA S mode wi ll be re set to A ppr o x. Mak e sure to s et it back to Ex act after pr oduc ing the gr aph .) Suppose , for e xample , that the functi on f(t) = t 2 +t is periodi c with per iod T = 2 .
Pa g e 1 6 - 28 Function FOURIER An alter nati ve w ay to def ine a F ouri er ser ies is by using comple x numbers as fo llow s: whe re F unction FOURIER pr ov ides the coeff ic ient c n of the comple x -for m of the F our ier ser ies giv en the functi on f(t) and the value of n .
Pa g e 1 6 - 2 9 Next , we mo ve to the CASDIR sub-dir ectory under HOME to change the value of var iable PERIOD, e .g., „ (hold) §`J @) CASDI `2 K @PERIOD ` Retur n to the sub-dir ectory wher e yo.
Pa g e 1 6 - 3 0 The f itting is somew hat acceptable for 0<t<2 , although not as good as in the pr ev ious e xample . A general expr ession for c n The f unction FO URIER can pro vi de a gener al expr ession for the coe ffi cien t c n of the comple x F our ier ser ies e xpansion.
Pa g e 1 6 - 3 1 The r esult is c n = (i ⋅ n ⋅π +2)/(n 2 ⋅π 2 ). P utting t ogether the comple x Fou rier series Hav ing determined the gener al expr ession f or c n , w e can put together a f.
Pa g e 1 6 - 32 Or , in the calc ulator entry line as: DEFINE(‘F(X,k,c0) = c0+ Σ (n=1,k ,c(n)*EXP(2*i* π *n*X/T)+ c(-n)*EXP(-( 2*i* π *n*X/T))’), wher e T is the period , T = 2 .
Pa g e 1 6 - 3 3 Accept change t o Approx mode if reque sted . The re sult is the value –0.40 46 7…. The ac tual value o f the functi on g(0.5) is g(0. 5) = -0.2 5. T he fo llow ing calc ulations sho w ho w well the F our ier se ri es appr o ximat es this value as the number of componen ts in the ser ies, gi ven b y k , inc reas es: F (0.
Pa g e 1 6 - 3 4 peri odic ity in the graph of the ser ies. T his periodi city is eas y to v isuali ze b y expanding the hori z ontal range of the plot to (-0.5, 4) : Four ier series for a triangular w ave Consider the f unction whi ch we assume to be per iodic w ith per iod T = 2 .
Pa g e 1 6 - 35 The calc ulator r eturns an int egr al that cannot be evaluat ed numer icall y because it depends on the parame ter n . The coeff ic ient can still be calc ulated by typing its def inition in the calc ulator , i.e ., wher e T = 2 is the perio d.
Pa g e 1 6 - 36 Press `` to copy this r esult to the scr een. T hen , re acti vat e the Eq uation W rit er to calc ulate the second integral de fining the coeff ic ient c n , namel y , Once again, r eplacing e in π = (-1) n , and u sing e 2in π = 1, we get: Press `` to cop y this second re sult to the sc reen .
Pa g e 1 6 - 37 This r esult is used to def ine the functi on c(n) as follo ws: DEFINE(‘ c(n) = - (((-1)^n-1)/(n^2* π ^2*(-1)^n)’) i. e. , Next , we def ine function F(X,k,c0) to calculate the F .
Pa g e 1 6 - 3 8 F rom the plot it is very diffi cult to distinguish the or iginal f unction f rom the F ourier ser ies appr ox imation . Using k = 2 , or 5 terms in the seri es, show s not so good a .
Pa g e 1 6 - 3 9 In thi s case, the peri od T , is 4. Make sure to c hang e the valu e of var iabl e @@@T@@@ to 4 (use: 4K @@@T@@ ` ) . F unction g(X) can be def ined in the calculator by u s in g DEFINE(‘ g(X) = IFTE((X>1) AND (X<3),1, 0)’) The function plot ted as follo ws (hori zontal r ange : 0 to 4, vertical range: 0 to 1.
Pa g e 1 6 - 4 0 Th e s i mp l i fica t io n o f th e rig ht -h a nd s id e of c (n ) , a bove, i s ea si er d on e on p ap e r (i .e., b y hand). Then, r etype the e x pr ession f or c(n) as sho wn in the f igure to the left abo ve , to define fu ncti on c(n).
Pa g e 1 6 - 4 1 W e can use this r esult as the firs t input to the f unction LDE C when used to obt ain a soluti on to the s yste m d 2 y/dX 2 + 0.2 5y = SW(X), wher e S W(X) stands for Squar e W av e function o f X. The second inpu t item w ill be the char acter istic equation cor responding t o the homogeneous ODE sho wn abo ve , i.
Pa g e 1 6 - 42 The s olution is sho wn belo w: Four ier T ransfor ms Befor e presen ting the concept of F our ier tr ansforms , we ’ll discus s the general def i nitio n of an integr al transf orm.
Pa g e 1 6 - 4 3 The am plitudes A n w ill be ref err ed to as the spectr um of the functi on and will be a measure of the magnitude of the component of f(x) with f requency f n = n/T . The basi c or fundamental fr equency in the F ouri er ser ies is f 0 = 1/T , th us, all other fr equenc ies ar e multiples o f this basic fr equenc y , i .
Pa g e 1 6 - 4 4 and The continuous spectrum is giv en by The fun ct ion s C ( ω ), S ( ω ), and A( ω ) ar e continuous functions of a var iable ω , whi ch becomes the tr ansfor m vari able for the F ourier tr ansforms de fined below .
Pa g e 1 6 - 4 5 Define this expr essio n as a f unction by u s ing func tion DEFINE ( „à ) . T hen, plot the continuou s spectr um, in the r ange 0 < ω < 10, as: Definition of F ourier transfor ms Differ ent types of F our ier tr ansforms can be def ined.
Pa g e 1 6 - 4 6 The continuous spectrum, F( ω ) , is calculated w ith the integral: This r esult can be r ationali z ed b y multiply ing numer a to r and denominator b y the conjugate o f the denominator , namely , 1-i ω . The r esult is now: which is a co mpl ex fun ct ion.
Pa g e 1 6 - 47 Properties of the F ourier transf orm L inearity: If a and b ar e constants , and f and g functi ons, then F{a ⋅ f + b ⋅ g} = a F{f }+ b F{g}.
Pa g e 1 6 - 4 8 the number o f oper ations using the FF T is reduced b y a factor of 10000/66 4 ≈ 15 . The FFT operates on t he sequenc e {x j } by partitioning it into a n umber of shorte r sequence s. The DFT ’s of the shorter seq uences ar e calculated and later combined t ogether in a highl y effi c ient manner .
Pa g e 1 6 - 4 9 The f igur e belo w is a box plot o f the data pr oduced. T o obtain the gra ph, f irst copy the arr ay ju st cr eated, then tr ansform it into a column vector b y using: OB J 1 + ARR Y (F unctions OB J and ARR Y are av ailable in the command catalog , ‚N ).
Pa g e 1 6 - 5 0 Example 2 – T o produce the si gnal giv en the spectrum , we modify the progr am GD A T A to inc lude an abso lute v alue, so that it r eads: << m a b << ‘2^m’ E.
Pa g e 1 6 - 5 1 Except f or a large peak at t = 0, the signal is mo stl y noise . A smaller vertical scale (-0. 5 to 0.5) sho ws the si gnal as follo ws: Solution to specific second-order diff erenti.
Pa g e 1 6 - 52 wher e M = n/2 or (n -1)/2 , whi chev er is an integer . Legendr e’s pol ynomials ar e pre -pr ogrammed in the calculat or and can be r ecalled by us ing the func tion LE GEND RE gi ven the or der of the poly nomial , n.
Pa g e 1 6 - 53 wher e ν is not an integer , and the function Gamma Γ ( α ) is def ined in Chapter 3. If ν = n, an int eger , the Bes sel func tio ns of the fi rst kind f or n = integer ar e def i.
Pa g e 1 6 - 5 4 Y ν (x) = [J ν (x) co s νπ – J −ν (x)]/sin νπ , for n on-integer ν , and fo r n integer , w ith n > 0, by wher e γ is the Euler constant , def ined by and h m repr ese.
Pa g e 1 6 - 55 The modif ied Bessel f unctions o f the second kind, K ν (x) = ( π /2) ⋅ [I - ν (x) − I ν (x)]/sin νπ , ar e also solu tions of this ODE .
Pa g e 1 6 - 5 6 Laguerr e’s equation Laguer re ’s equation is the second-o rde r , linear ODE of the for m x ⋅ (d 2 y/dx 2 ) +(1 − x) ⋅ (d y/dx) + n ⋅ y = 0. Laguer re poly nomials, de fined as , ar e soluti ons to L aguerr e’s equati on.
Pa g e 1 6 - 57 L 2 (x) = 1- 2x+ 0. 5x 2 L 3 (x) = 1-3x+1.5x 2 -0 . 16666 … x 3 . W eber ’s equation and Hermite poly nomials W eber’s equati on is defined as d 2 y/dx 2 +(n+1/2 - x 2 /4)y = 0, .
Pa g e 1 6 - 5 8 F irst , cr eate the e xpressi on defi ning the der iv ativ e and stor e it into var iable E Q. The f igur e to the left sho ws the AL G mode command, w hile the ri ght -hand side fi gure sho ws the RPN s tack be for e pre ssing K .
Pa g e 1 6 - 59 @@OK@@ @INIT+ — .7 5 @@OK@@ ™™ @SOLVE (wait ) @EDIT (Changes initial v alue of t to 0.5, and f inal value of t to 0.7 5, sol ve f or v(0.7 5 ) = 2 . 066…) @@OK@@ @INIT+ — 1 @@OK@@ ™ ™ @SOLVE (wa it ) @EDIT (Changes initial v alue of t to 0.
Pa g e 1 6 - 6 0 Θ „ô (simultaneousl y , if in RPN mode) to enter PL O T en vir onment Θ Highl ight t he fi eld in fr ont o f TYPE , using the —˜ ke ys .
Pa g e 1 6 - 6 1 LL @) PICT T o re c ove r m e nu a n d ret u rn to P IC T en viro n m en t. @ ( X,Y ) @ T o deter mine coordinates o f any po int on the gr aph . Use the š™ ke ys to mov e the cursor ar ound t he plot area. At the bottom of the sc r een yo u w ill see the coor dinates of the c ursor as (X,Y ) , i .
Pa g e 1 6 - 62 time t = 2 , the input for m for the diff erenti al equati on sol ver sho uld look as fo llow s (notice that the Init: v alue for the Soln: is a v ector [0, 6]): Press @SOLVE (wai t) @EDIT to so lv e f or w(t=2). The solution r eads [.
Pa g e 1 6 - 63 (Changes initial v alue of t to 0.7 5, and final v alue of t to 1, solv e again for w(1) = [-0.4 6 9 -0.6 0 7]) Repeat f or t = 1.2 5, 1.5 0, 1.7 5, 2 . 00. Pre ss @@OK@@ after v ie wing the last r esult in @EDIT . T o return to nor mal c alc u lator displa y , pr ess $ or L @@OK@@ .
Pa g e 1 6 - 6 4 Notice that the opti on V- V ar : is set to 1, indicating that the fi rst element in the vec tor so lution , namely , x’ , is to be plotted against the independent var iable t . Accept c hanges to PL O T SETUP b y pres sing L @@OK@@ .
Pa g e 1 6 - 65 Press LL @PICT @CANCL $ to r eturn t o nor mal calc ulator displa y . Numerical solution for stiff first-order ODE Consi der the ODE: dy/dt = -100y+100t+ 101, sub ject t o the initial conditi on y(0) = 1.
Pa g e 1 6 - 6 6 Here w e are try ing to obtain the value of y( 2) giv en y(0) = 1. With the Soln: Final fi eld highlighted , pres s @SOLVE . Y ou can chec k that a so lution t ak es about 6 s ec on ds, whi le in t he previous fir st - orde r exa mp le th e s olu tio n wa s a lm ost instantaneous .
Pa g e 1 6 - 67 Note: T he option Stiff is also av ailable for gr aphical soluti ons of differ ential equations . Numerical solution to ODEs with the S OL VE/DIFF menu The S OL VE soft menu is acti vated b y using 7 4 MENU in RPN mode. This men u is pre sented in detail in Cha pter 6 .
Pa g e 1 6 - 6 8 The value o f the solu tion , y fi n al , will be av ailable in var iable @@@y@@@ . This f unctio n is appr opriate f or progr amming since it leav es the differ ential equation spec ificati ons and the tolerance in the stac k read y for a ne w solution .
Pa g e 1 6 - 69 contain only the v alue of ε , and the s tep Δ x w ill be taken as a small def ault value . After running func tion @@RKF@@ , the stack w ill show the lines: 2 : {‘ x’ , ‘ y’.
Pa g e 1 6 - 70 The se r esults indicate that ( Δ x) ne xt = 0.3 404 9… Function RRKS TEP This f unction u ses an input list similar to that o f functi on RRK, as w ell as the toler ance for the so.
Pa g e 1 6 - 7 1 The se r esults indicate that ( Δ x) ne xt = 0.00 5 5 8… and that the RKF method (CURRENT = 1) should be used. Function RKFERR This f unction r eturns the abs olute er r or estimate f or a gi ven s tep w hen sol v ing a pr oblem as that desc ribed f or func tion RKF .
Pa g e 1 6 - 72 The f ollow ing scr een shots sho w the RPN stack be for e and after applicati on of functi on RSBERR: The se r esults indicate that Δ y = 4.1514… and err or = 2 .7 6 2 ..., for Dx = 0.1. Chec k that , if Dx is reduced t o 0. 01, Δ y = -0.
Pa g e 1 7- 1 Chapter 17 Pr obability Applications In this Chapter we pr ov ide ex amples of appli cations of calc ulator’s func tions to pr obabil ity distribu tions . The MTH/P ROB ABILITY .. sub-m enu - part 1 The MTH/P ROB ABI LI TY .. sub-menu is acces s ible thr ough the ke ystr ok e sequence „´ .
Pa g e 1 7- 2 T o simplify notation , use P(n ,r) fo r p er mutations , and C(n,r ) for combinati ons. W e can calculate comb inations , perm utations , and factor ials with f unctions COMB , PERM, and ! fr om the MTH/PROB ABILITY .
Pa g e 1 7- 3 Random n u mber gene rat ors , in gener al, oper ate b y taking a v alue, called the “ seed” of the gener ator , and perfor ming some mathematical algor ithm on that “ seed” that gener ates a ne w (pseudo)r andom number .
Pa g e 1 7- 4 fun ctio n (pmf) is r e pr esente d by f (x) = P[X=x], i .e., the pr obability that the ran d om vari ab le X ta kes th e va l ue x. The mas s distr ibution func tion mu st satisfy the c.
Pa g e 1 7- 5 P oisson distribution The probabilit y mass function of the P oisson di stribution is g i ven by . In this expr ession , if the random var iable X r epresen ts the number of occur rences of an ev ent or observati on per unit time, length , area , vo lume , etc.
Pa g e 1 7- 6 Continuous probability distr ibutions The pr obability distributi on for a continuou s random v ari able , X, is char acter i zed b y a function f(x) kno wn as the pr obability density functi on (pdf) .
Pa g e 1 7- 7 , while its cdf is gi ven b y F(x) = 1 - exp(- x/ β ) , for x>0, β >0. The beta distr ibution The pdf f or the gamma distributi on is giv en by As in the case of the gamma distr ibution , the corres pond ing cdf f or the beta distr ibution is also gi ven b y an integr al wi th no clo sed-f orm soluti on.
Pa g e 1 7- 8 Exponential pdf: 'epdf(x) = EXP(-x/ β )/ β ' Exponential cdf: 'ecdf(x) = 1 - EXP(-x/ β )' W eibull pdf: 'Wpdf(x) = α * β *x^( β -1)*EXP(- α *x^ β )' W eibull cdf: 'Wcdf(x) = 1 - EXP(- α *x^ β )' Use f uncti on DEFINE to define all the se func tions .
Pa g e 1 7- 9 Continuous distributions f o r statistical infer ence In this secti on we dis cu ss f our continu ous pr obability distr ibutions that ar e commonl y used f or pr oblems relat ed to statis tical infer ence .
Pa g e 1 7- 1 0 wher e μ is the mean , and σ 2 is the v ariance of the distr ibution . T o calculate the val ue of f( μ , σ 2 ,x) for the normal distr ibution , use f unction NDI S T with the follo w ing arguments: the mean , μ , the v ariance , σ 2 , and, the v alue x , i.
Pa g e 1 7- 1 1 wher e Γ ( α ) = ( α -1)! is the GAMMA f unction de fined in Chapt er 3 . The calc ulator pro vi des for v alues of the upper - tail (cumulati ve) distr ibution functi on for the t-distr ibution , functi on UTPT , giv en the paramet er ν and the v alue of t , i .
Pa g e 1 7- 1 2 The calc ulator pro vi des for v alues of the upper - tail (cumulati ve) distr ibution fun ctio n for th e χ 2 -distribu tion u sing [UTPC] gi ve n the value of x and the paramet er ν .
Pa g e 1 7- 1 3 The calc ulator pro vi des for v alues of the upper - tail (cumulati ve) distr ibution functi on for the F distr ibution, f unction UTPF , gi ven the paramete rs ν N and ν D, and t he value of F . The definition of this function is, t heref ore , F or ex ample, to calc ulate UTPF(10,5, 2 .
Pa g e 1 7- 1 4 Exponential: W eibull: F or the Gamma and Beta distr ibutions the e xpressi ons to sol ve w ill be mor e compli cated due to the pr esence of in tegr als, i . e ., • Gamma, • Beta, A numer ical soluti on w ith the numerical s olv er will n ot be feasible beca use of the integr al sign in vo lv ed in the expr ession .
Pa g e 1 7- 1 5 Ther e are two r oots of this functi on found b y using function @ROOT w ithin the plot env iro nment . Becaus e of the integr al in the equation , the r oot is appr o ximated and w ill not be sho wn in the plot sc reen . Y ou will onl y get the message Constant? Sho wn in the sc reen.
Pa g e 1 7- 1 6 Notice that the second par ameter in the UTPN functi on is σ 2, n o t σ 2 , r epre senting the var iance of the distr ibution . Also , the s ymbol ν (the lo wer -case Gr eek letter no) is not a vailable in the calc ulator . Y ou can u se , for e xample , γ (gamma) instead o f ν .
Pa g e 1 7- 1 7 Thu s, at this point, y ou will hav e the f our equations av ailable for solution . Y ou needs ju st load one of the equations into the E Q f ield in the numer ical sol ver and proceed w ith solv ing fo r one of the var iables .
Pa g e 1 7- 1 8 With thes e four equati ons, w henev er you launch the n u mer ical solv er you hav e the fo llo w ing cho ices: Example s of soluti on of equations E QNA, E QT A, E QCA, and EQ F A ar.
P age 18-1 Chapter 18 Statistical Applications In this Chapter w e introdu ce statisti cal applicati ons of the calc ulator including statisti cs of a sample , fr equency dis tributi on of data, simple r egre ssi on, conf idence int ervals , and hy pothesis te sting .
P age 18-2 Stor e the progr am in a var iable called LX C. After s tor ing this pr ogram in RPN mode yo u can also use it in AL G mode . T o stor e a column vec tor into v ariable Σ DA T use function S T O Σ , av ailable thr ough the catalog ( ‚N ) , e .
P age 18-3 Example 1 -- F or the data stor ed in the pr ev ious ex ample, the single -v ari able statis tics re sults ar e the f ollo wing: M e a n : 2. 1 3333333333 , S t d D e v: 0 . 96 42 0 79 49 4 0 6 , Va r i a n c e : 0 . 9 2969696969 7 T otal: 2 5 .
P age 18-4 Example s of calculati on of these measur es, using lis ts, ar e available in C hapter 8. The medi an is the v alue that splits the data set in the mi ddle w hen the elements ar e placed in incr easing order . If you ha ve an odd number , n, of or der ed elements, the medi an of this sample is the v alue located in position (n+1)/2 .
P age 18-5 The ran ge of the sample is the differ ence between the max imum and minimum value s of the sample . Since the calculat or , through the pr e -progr ammed statisti cal functi ons pro vides the max imum and minimum v alues of the sample , y ou can easily calculate the r ange.
P age 18-6 Definitions T o unders tand the meaning of thes e paramet ers w e pre sent the f ollow ing def initions : Gi ven a set o f n data values: {x 1 , x 2 , … , x n } listed in no partic ular o.
P age 18-7 Θ Gener ate the list of 200 number b y using RDLIS T(200) in AL G mode , or 200 ` @RD LIST@ in RPN mode . Θ Use pr ogram LX C (see abo ve) to conv ert the list thus gener ated into a column vec tor . Θ Stor e the colu mn v ector into Σ DA T, b y us i n g f u n c t io n STO Σ .
P age 18-8 to calculate f or uniform-si ze classes (or b ins) , and the class mark is j ust the av erage of the c lass boundari es for eac h class . F inally , the c u mulati ve fr equency is obtained.
P age 18-9 « DUP S IZE 1 GET f req k « {k 1} 0 CON cfr eq « ‘fr eq(1,1)’ EV AL ‘ cfr eq(1,1)’ S T O 2 k FOR j ‘ cf r eq(j-1,1) +fr eq(j,1)’ EV AL ‘ cfr eq (j,1)’ ST O NEXT cfr eq » » » Sa ve it unde r the name CFREQ.
P age 18-10 Θ Press @CANCEL to r eturn t o the pre vi ous sc reen . Change the V-v iew and Bar Wi dth once mor e, n o w to r ead V - Vi ew: 0 3 0, Bar Width: 10. T he new histogr am, based on the same dat a set , now looks lik e this: A plot of fr equency count , f i , v s.
P age 18-11 Θ F irst, ente r the two r ow s of data into column in the var iable Σ DA T by u s i n g the matri x wr iter , and function S T O Σ . Θ T o access the pr ogr am 3. Fit data.. , use the f ollow ing k ey str okes: ‚Ù˜˜ @@@OK@@@ The input f orm w ill show the c urr ent Σ D A T , already loaded.
P age 18-12 Wher e s x , s y ar e the standar d dev iations of x and y , res pectiv ely , i.e . The va lu es s xy and r xy ar e the "Cov ariance" and "Cor relati on," r especti vel y , obtained by u sing the "F it data" featur e of the calc ulator .
P age 18-13 The ge neral f orm of the r egressi on equation is η = A + B ξ . Best data fitting The calc ulator can determine w hich one of its linear or lineari z ed relati onship offer s the best fitting f or a set of (x ,y) data points. W e w ill illustrate the u se of this featur e wit h an e xample .
P age 18-14 X-Col, Y -Col: these opti ons apply only w hen you ha ve mor e than two columns in the matr ix Σ D A T . By def ault, the x co lumn is column 1, and the y column is co lumn 2 .
P age 18-15 B. I f n ⋅ p is an integer , say k , calculat e the mean of the k - th and (k -1) th or der ed observati ons. This algor ithm can be implemented in the fo llo w ing pr ogr am typed in RP.
P age 18-16 The D A T A sub-menu The D A T A sub-menu contains fu nctions u sed to manipulate the s tatisti cs matr ix Σ DA TA: The oper ation of thes e func tions is as f ollo ws: Σ + : add ro w in lev el 1 to bottom of Σ DA T A m a trix. Σ - : r emov es last r ow in Σ D A T A matri x and places it in le vel of 1 of the st ack .
P age 18-17 Σ P A R: sho ws statisti ca l par ameters. RE SET : r eset parameter s to default v alues INFO: sho ws s tatist ical par ameter s The MODL sub-menu within Σ PA R This sub-me nu cont ains func tio ns that let yo u change the data-fitting model t o LINFIT , L OGFIT , EXPFIT , PWRFIT o r BE STFI T by pr essing the appr opri ate button .
P age 18-18 The f unctions inc luded ar e: B A RP L: pr oduces a bar plot with dat a in Xcol column of the Σ D ATA m a t r i x . HIS TP: produce s histogr am of the data in Xcol column in the Σ DA T.
P age 18-19 Σ X^2 : pr o vi des the sum of squar es of values in Xcol co lumn. Σ Y^2 : pro vi des the sum of squar es of values in Ycol column . Σ X*Y : pr ov ides the sum of x ⋅ y , i.e ., the pr od uc ts of data in columns Xcol and Ycol. N Σ : pr o vi des the number of col umns in the Σ DA T A m a trix.
P age 18-20 @) STAT @ ) £PAR @ RESET resets s tatistical paramet ers L @ ) STAT @PLOT @ SCATR pr oduces scatt erpl ot @STATL dr aws data f it as a strai g ht line @CANCL r eturns to main displa y Θ Deter mine the fitting equati on and some of its statis tics: @) STAT @ ) FIT@ @ £LINE pr oduces '1.
P age 18-21 Θ F it data using columns 1 (x) and 3 (y) using a logarithmi c fitting: L @ ) STAT @ ) £PAR 3 @YCOL selec t Ycol = 3, and @) MODL @LOGFI select Mod el = Logfit L @ ) STAT @PLOT @SCATR produce s catter gram o f y vs . x @STATL sho w line for log f itting Obv iousl y , the log-f it is not a good choi ce.
P age 18-22 L @ ) STAT @PLOT @SCATR produce s catter gram o f y vs . x @STATL sho w line for log f itting Θ T o r eturn to S T A T menu use: L @) STAT Θ T o get y our var iable menu back use: J . Confidence inter vals Statis tical infer ence is the proces s of making conclusi ons about a population based on info rmation f rom sample dat a.
P age 18-2 3 Θ P o int estimati on: when a single value of the par ameter θ is pro vided . Θ Conf idence interval: a n umerical interv al that contains the par ameter θ at a giv en leve l of pr obability . Θ Es timator : rule or method of estimati on of the par ameter θ .
P age 18-2 4 Θ The par ameter α is kno wn as the si gnificance le vel . T ypi cal values of α ar e 0.01, 0. 05, 0.1, corr esponding to conf idence lev els of 0.
P age 18-2 5 Small samples and large samples The beha vi or of the Student’s t distr ibution is such that f or n>30, the distr ibution is indistinguishable fr om the standar d normal distributi on.
P age 18-2 6 Es timators for the mean and s tandar d dev iation o f the diff er ence and sum of the statisti cs S 1 and S 2 ar e giv en by : In t hese expressions, ⎯ X 1 and ⎯ X 2 ar e the values .
P age 18-2 7 In this case , the centered conf idence intervals fo r the sum and difference o f the mean value s of the populations , i .e., μ 1 ±μ 2 , are gi ven b y: wher e ν = n 1 +n 2 - 2 is the number of degr ees of fr eedom in the Student’s t distr ibution .
P age 18- 28 These options are to be interpr eted as follow s : 1. Z -INT : 1 μ .: Single sample confi dence in terval for the populati on mean, μ , w ith know n population var iance , or for lar ge samples with unkno wn populatio n var iance . 2. Z - I N T : μ1−μ2 .
P age 18-29 Press @HELP to obtain a scr een explaining the meaning of the conf idence interval in terms o f random number s generated b y a calculator . T o scr oll d o wn the r esulting sc r een use the do wn-arr ow k ey ˜ . Pr ess @@@OK@@@ when done w ith the help sc ree n.
P age 18-30 Example 2 -- Data f r om two s amples (sample s 1 and 2) indicate that ⎯ x 1 = 5 7 .8 and ⎯ x 2 = 60. 0. The sample si z es ar e n 1 = 45 and n 2 = 7 5 . If it is kno wn that the populations ’ standar d dev iations ar e σ 1 = 3.2 , and σ 2 = 4.
P age 18-31 When done , pres s @@@OK@@@ . The r esults, as t ext and gr aph, are sho wn be lo w: Example 4 -- Determine a 90% conf idence interval for the diff er ence between two pr oportions if sample 1 sho ws 20 succe sses out of 120 tr ials, and sample 2 shows 15 s uccesses out of 1 00 trial s.
P age 18-3 2 Example 5 – Determine a 9 5% confi dence interval f or the mean of the population if a sample of 5 0 elements has a mean of 15 .5 and a standar d dev iatio n of 5 . The population ’s standar d dev iation is unkno wn . Press ‚Ù— @@@OK@@@ to access the confidence inte rval featur e in the calc ulator .
P age 18-3 3 The se re sults assume that the v alues s 1 and s 2 ar e the population s tandard dev iations . If these v alues actually r eprese nt the samples ’ standar d d e viati ons, y ou should enter the same v alues as befor e, but w ith the option _pooled selected .
P age 18-34 The conf idence interv al for the populati on var iance σ 2 is t herefor e, [(n -1) ⋅ S 2 / χ 2 n-1 , α /2 ; (n-1) ⋅ S 2 / χ 2 n-1,1- α /2 ].
P age 18-35 Hy pot hesis testing A h ypo thesis is a declar ation made about a population (f or instance , w ith r espect t o its mean) . Acceptance o f the h ypothesis is bas ed on a statis tical te st on a sample tak en fr om the population . The consequent acti on and decision- making ar e called h ypo thesis testing .
Pa g e 1 8 - 3 6 Err ors in h ypothesis testing In hy pothesis testing w e use the ter ms err ors of T ype I and T ype II to def ine the cases in w hich a true h ypothesis is r ejec ted or a false h ypothe sis is accepted (not rejected) , respect i vely .
P age 18-3 7 The va lu e of β , i .e ., the pr obability of making an err or of T ype II, depends on α , the sample si z e n, and on the true v alue of the paramete r tested . Thus , the val ue of β is determined after the h ypothesis te sting is perfor med.
P age 18-38 The c riter ia to us e for h ypothesis t esting is: Θ Rej ec t H o if P - v alue < α Θ Do not r ej ect H o if P -value > α . The P -value fo r a two-sided test can be calculat ed.
P age 18-3 9 Next , we us e the P -value assoc iated with eithe r z ο or t ο , and compare it to α to dec ide whether or no t to r ej ect the nul l hy pothesis. T he P -value f or a two-sided test is def ined as either P -value = P(z > |z o |), or , P -value = P(t > |t o |) .
P age 18-40 val ue s ⎯ x 1 and ⎯ x 2 , and standar d dev iations s 1 and s 2 . If the populations standar d dev iati ons cor re sponding to the samples, σ 1 and σ 2 , ar e kno wn, or if n 1 >.
P age 18-41 The c riter ia to us e for h ypothesis t esting is: Θ Rej ec t H o if P - v alue < α Θ Do not r ej ect H o if P -value > α . P aired sample tests When w e deal with two s a mple .
P age 18-4 2 wher e Φ (z) is the c umulativ e distributi on func tion (CDF ) of the standar d normal distr ibution (see Cha pter 17). Re ject the null hy pothesis, H 0 , if z 0 >z α /2 , or if z 0 < - z α /2 .
P age 18-43 T w o - tail ed test If using a two- tailed test w e will f ind the v alu e of z α /2 , f r om Pr[Z> z α /2 ] = 1- Φ (z α /2 ) = α /2 , or Φ (z α /2 ) = 1- α /2 , wher e Φ (z) is the c umulativ e distributi on func tion (CDF ) of the standar d normal distr ibution .
P age 18-44 1. Z - T es t: 1 μ .: Single s ample h ypothesis tes ting for the populati on mean, μ , w ith kno wn populati on var iance , or for lar ge samples w ith unknow n populatio n var iance .
P age 18-45 Then , we r ejec t H 0 : μ = 150 , against H 1 : μ ≠ 150 . The test z v alue is z 0 = 5. 656854. T he P- va l u e i s 1. 54 × 10 -8 . Th e cri ti ca l va lu es of ± z α /2 = ± 1.9 5 99 64, cor r esponding t o cr itical ⎯ x r ange of {14 7 .
P age 18-46 W e re ject the null h ypothe sis, H 0 : μ 0 = 15 0, against the alternati ve h ypothesis , H 1 : μ > 15 0. The te st t value is t 0 = 5.6 5 68 5 4, w ith a P - v a lue = 0. 0000003 9 35 2 5. T he c ri tical v alue of t is t α = 1.6 7 65 51, corr esponding to a crit ica l ⎯ x = 15 2 .
P age 18-4 7 Th us, w e accept (mor e accurat el y , we do not r ejec t) the h ypothe sis: H 0 : μ 1 −μ 2 = 0 , or H 0 : μ 1 =μ 2 , against the alternati ve h ypothesis H 1 : μ 1 −μ 2 < 0 , or H 1 : μ 1 =μ 2 . The test t value is t 0 = -1.
P age 18-48 The t est c r iter ia are the s ame as in h ypothesis te sting of means, name ly , Θ Rej ec t H o if P - v alue < α Θ Do not r ej ect H o if P -value > α . P lease notice that this pr ocedure is v alid only if the populati on fr om whic h the sample wa s tak en is a Nor mal population .
P age 18-4 9 The f ollow ing table sho ws ho w to select the n umer ator and denominator f or F o depending on the alternati ve h ypothe sis cho sen: _______________ ____________________ _____________.
P age 18-50 Ther efor e, the F test statistics is F o = s M 2 /s m 2 =0.3 6/ 0.2 5=1. 44 The P -v alue is P -value = P(F>F o ) = P(F>1.44) = UTPF( ν N , ν D ,F o ) = UTPF(20, 3 0,1.44) = 0.17 88… Since 0.17 88… > 0. 05, i .e., P -value > α , theref ore , we cannot r ej ect the null h ypothesis that H o : σ 1 2 = σ 2 2 .
P age 18-51 W e get the, s o -called, nor mal equations: This is a s ys tem o f linear equati ons w ith a and b as the unkno wns , whi ch can be sol ved u sing the linear equation featur es of the calculator . Ther e is, ho we ver , no need to bother w ith these calculati ons because you can use the 3.
Pa g e 1 8 - 52 F rom w hic h it fo llow s that the standar d dev iations o f x and y , and the cov ariance of x ,y are giv en, r especti ve ly , by , , and Also , the sample corr elation coeff ici en.
Pa g e 1 8 - 5 3 Θ Conf idence limits for r egres sion coeff ic ients: F or the slope ( Β ): b − (t n- 2 , α /2 ) ⋅ s e / √ S xx < Β < b + (t n- 2 , α /2 ) ⋅ s e / √ S xx , F or t.
P age 18-54 a+b ⋅ x+(t n- 2 , α /2 ) ⋅ s e ⋅ [1+(1/n)+(x 0 - ⎯ x) 2 /S xx ] 1/2 . Procedur e for inference statistics f or linear regression using the calculator 1) Enter (x,y) as columns of data in the statis tical matr ix Σ D AT.
Pa g e 1 8 - 5 5 1: Covariance: 2.025 The se r esults are int erpr eted as a = -0.8 6 , b = 3 .2 4 , r xy = 0.9 89 7 20 22 9 7 4 9 , and s xy = 2 . 02 5 . The corr elation coeff ic ient is clo se enough to 1.0 t o conf irm the linear tr end obse rved in the gr aph .
P age 18-5 6 Example 2 -- Su ppose that the y-data used in Ex ample 1 repr esent the elongation (in h undr edths of an inc h) of a me tal w ire w hen sub jec ted to a f orce x (in tens of pounds) . T he phy sical phenomenon is suc h that we e xpect the inter cept, A, to be z er o.
P age 18-5 7 Multiple lin ear fitting Consi der a data set of the for m Suppos e that w e searc h for a data f itting of the fo rm y = b 0 + b 1 ⋅ x 1 + b 2 ⋅ x 2 + b 3 ⋅ x 3 + … + b n ⋅ x n .
P age 18-5 8 With the calc ulator , in RPN mode , yo u can pr oceed as fo llo ws: F irst , w ithin your HO ME direc tory , c r eate a sub-dir ectory to be called MPFIT (Multiple linear and P o ly nomial data FI Tting) , and enter the MPFI T sub- dir ectory .
P age 18-5 9 Compar e these f itted values w ith the ori ginal data as sho wn in the ta ble belo w: P ol ynomial fitting Consider the x -y data set {(x 1 ,y 1 ), ( x 2 ,y 2 ), …, (x n ,y n )}. Suppose that w e w ant to fit a po ly nomial or order p to this data s et .
P age 18-60 If p > n-1 , then add columns n+1, …, p-1, p+1 , to V n to f orm matr ix X . In step 3 f r om this lis t , w e hav e to be aw are that column i ( i = n+1, n+2 , …, p+1 ) is the vec tor [x 1 i x 2 i … x n i ]. If we w er e to us e a list of data value s for x rathe r than a vec tor , i.
P age 18-61 « Open pr ogram x y p Enter lis ts x and y , and p (lev els 3,2 ,1) « Open subpr ogram 1 x SI ZE n Determine si z e of x list « Open subpr ogram 2 x V ANDERMONDE Place x in stac.
P age 18-6 2 Becaus e w e w ill be using the same x -y data for f itting poly nomials of diff er ent or ders , it is adv isable to sav e the lists of data v alues x and y into var iables xx and yy , r especti vel y . This w ay , we w ill not hav e to type the m all ov er again in each a pplicati on of the pr ogram P OL Y .
P age 18-63 Θ The cor relation coe ffi cient , r . T h is value is constr ained to the r ange –1 < r < 1. Th e cl ose r r is to +1 or –1, the better the data fitting . Θ The sum o f squar ed erro rs, S SE . This is the quantity that is to be minimi zed b y least-squar e approac h.
P age 18-64 x V ANDERMONDE P lace x in stac k, obtain V n IF ‘ p<n -1’ THEN This IF is step 3 in algo rithm n P lace n in stac k p 2 + Calc ulate p+1 FOR j Start loop, j = n-1 to p+1, step = -1.
P age 18-65 “SSE” T A G T ag r esult as S SE » Close sub-pr ogram 4 » Clos e sub-progr am 3 » Clo se sub-pr ogram 2 » Clo se sub-pr ogram 1 » Clos e main progr am Sa ve this pr ogram unde r the name PO L YR, to em phasi z e calc ulation of the correlation c oeffic ient r .
P age 19-1 Chapter 19 Numbers in Different Bases In this Chapter w e pre sent e xamples o f calculati ons of number in base s other than the dec ima l basis .
P age 19-2 With s yst em flag 117 set to S OFT menus , the B ASE men u show s the follo wing: With this f ormat , it is ev ident that the L OGIC, BIT , and B Y TE entr ies w ithin the B ASE menu a r e themselves sub-menus. These me nus are discussed later in this Chapter .
P age 19-3 As the deci mal (DEC) s ystem has 10 di gits (0,1,2 , 3,4 ,5, 6, 7 , 8, 9) , the hex adecimal (HEX) s yst em has 16 digits (0, 1,2 , 3, 4,5, 6, 7 , 8, 9 ,A,B ,C,D ,E ,F), the octal (OCT) sy stem has 8 digits (0,1,2 , 3,4 ,5,6 , 7) , and the binary (BIN) sy stem has only 2 digits (0,1).
P age 19-4 The onl y effect o f selecting the DEC imal sy stem is that dec imal number s, w hen started w ith the sy mbol #, are w ritten with the suff ix d . W ordsi ze The w ordsi z e is the number of bits in a binary obj ect . By defa ult , the wor dsiz e is 64 bites .
P age 19-5 The L OGIC m enu The L OGIC menu , av ailable through the B ASE ( ‚ã ) pr ov ides the f ollo wing fun ctio ns : The f unctions AND , OR, X OR (ex clusi ve OR), and NO T ar e logical f uncti ons.
P age 19-6 AND (BIN) OR (BIN) XO R (BIN) NO T (HEX) The BI T menu The BI T menu , available thr ough the BA SE ( ‚ã ) pro vide s the follo wing fun ctio ns : F unctions RL, SL , ASR , SR, RR , contained in the BI T menu , are u sed to manipulate bits in a b inar y integer .
P age 19-7 The B Y TE menu The B YTE menu , a vaila ble thr ough the B A SE ( ‚ã ) pr ov ides the f ollo wing fun ctio ns : F unctions RLB, SLB , SRB, RRB, co ntained in the BIT menu , ar e used to manipulate bits in a b inar y integer . The de fi nition of these f unctions ar e show n belo w: RLB: Rotate L ef t one byte , e.
Pa g e 2 0 - 1 Chapter 20 Customi zing menus and k ey board Thr ough the use of the man y calculator menu s yo u hav e become familiar w ith the oper ation of men us f or a var iety of a pplicatio ns.
Pa g e 2 0 - 2 Menu numbers (RCLMENU and MENU func tions) E ach pr e -defined men u has a number attached to it . F or e xample , suppose that y ou acti vate the MTH menu ( „´ ). Then , using the functi on catalog ( ‚N ) find f u nc tion R CLMENU and acti vate it .
Pa g e 2 0 - 3 T o acti vate an y of those f unctions y ou simply need to enter the function argume nt (a number ) , and then pr ess the corr esponding soft menu k ey .
Pa g e 2 0 - 4 Y ou can tr y using this list w ith TMENU or MENU in RPN mode to ver if y that y ou get the same menu as obt ained earli er in AL G mode.
Pa g e 2 0 - 5 Customizing the k e yboard E ach k ey in the k ey board can be iden tifi ed by two n umbers r e pr esenting their r o w and column. F or ex ample , the V AR k ey ( J ) is located in r ow 3 of column 1, and w ill be r eferr ed to as k ey 31.
Pa g e 2 0 - 6 The f unctions av ailable are: AS N: Assigns an objec t to a ke y specif ied by XY .Z S T OK E Y S: Stores user -defined key list RCL KEYS: Ret ur ns cu rrent us er-d efin ed ke y l ist.
Pa g e 2 0 - 7 Operating user-defined ke ys T o operate this us er -defined k ey , enter „Ì befo re pre ssing the C key . Notice that after pr essing „Ì the sc reen sho ws the spec ificati on 1USR in the second displa y line.
Pa g e 2 0 - 8 T o un -assign all user-def ined k eys use: AL G mode: DELKEYS (0) RPN mode: 0 DELKEYS Chec k that the user -k e y def initions w ere r emov ed by using f u ncti on RCLKEY S.
P age 21-1 Chapter 21 Pr ogramming in User RP L language Use r RPL language is the pr ogramming language mo st commonl y used to pr ogram the calc ulator . The progr am components can be put together in the line editor by inc luding them bet w een progr a m containers « » in the appr opriat e orde r .
P age 21-2 „´ @LIST @ADD@ ADD Calc ulate (1+x 2 ), / / then di v ide ['] ~„x™ 'x' „° @) @MEM@@ @ ) @DIR@@ @ PURGE PURGE Purg e variab l e x ` Pr ogram in le vel 1 ____________.
P age 21-3 use a local v ari able within the pr ogram that is only de fined f or that progr am and w ill not be availa ble fo r use after pr ogram e xec ution.
P age 21-4 Global V ariable Scope An y vari able that you def ine in the HOME direc tory or any o ther dir ectory or sub-dir ectory will be consider ed a global var iable fr om the point of v iew o f pr ogram de velopment . How ev er , the scope of such v ariable , i .
P age 21-5 Local V ariable Scope Local v ariable s are ac tiv e only w ithin a progr am or sub-pr ogr am. The ref ore , their s cope is limited to t he pr ogram or sub-pr ogram w her e the y’r e defined .
P age 21-6 S T ART : S T ART -NEXT- STEP cons truc t for br anching FOR: FOR-NE XT- STEP constr uct for loops DO: DO-UNTIL -END constr uct f or loops WHILE: WHILE -REPEA T-END c onstru ct for loop s T.
P age 21-7 Functions listed b y sub-menu The f ollow ing is a listing of the func tions w ithin the PRG sub-me nus list ed by sub- menu . ST A CK MEM/DIR BRCH/IF BRCH/WHILE TYP E DUP PUR GE IF WHILE O.
P age 21-8 LIST/ELEM GROB CHARS MODES/FLAG MODES/MISC GET GR OB SUB SF BEEP GET I BLANK REPL CF CLK PUT G O R POS FS? S Y M PUTI G XOR SIZ E F C? S T K SI ZE SUB NUM F S?C ARG PO S REPL CHR F S?C .
P age 21-9 Shortc uts in the PR G menu Many o f the functi ons listed abo ve f or the PRG menu ar e readil y av ailable thr ough other means: Θ Compar ison operators ( ≠ , ≤ , <, ≥ , >) ar e av ailable in the k eyboar d.
P age 21-10 „ @ ) @IF@@ „ @CASE@ „ @ ) @IF@@ „ @CASE@ „ @ ) START „ @) @FOR @ „ @ ) START „ @) @FOR @ „ @ ) @@DO@@ „ @WHIL E Notice that the inse rt prompt ( ) is a vailable after the k ey w or d for eac h constr uct so y ou can start typing at the r ight location.
P age 21-11 @) STACK DUP „° @) STACK @@ DUP@@ SW A P „° @) STACK @SWAP@ DRO P „° @) STACK @DROP@ @) @MEM@@ @ ) @DIR@@ PUR GE „° @) @ME M@@ @ ) @DIR@ @ @PURGE ORDER „° @) @MEM@@ @ ) @DIR.
P age 21-12 @) @BRCH@ @ ) WHILE@ WHILE „° @) @BRCH@ @ ) WHILE@ @WHILE REPE A T „° ) @BRCH@ @ ) WHILE@ @REPEA END „° ) @BRCH@ @ ) WHILE@ @@ END@ @ ) TEST@ == „° @ ) TEST@ @@ @ ≠ @@@ AND .
P age 21-13 @) LIST@ @ ) PRO C@ REVLI S T „° @) LIST@ @ ) PROC@ @REVLI@ SO RT „° @) LIST@ @ ) PROC@ L @SORT@ SEQ „° @) LI ST@ @ ) PROC@ L @@SEQ@@ @) MODES @ ) ANGLE@ DE G „°L @) MODES @ ) .
P age 21-14 fun ction s from the MT H m enu. Specific ally , you ca n use fun ctio ns for li st oper ations such as S ORT , Σ LI S T , etc. , av ailable thr ough the MTH/LIS T menu .
P age 21-15 Ex amples of sequential progr amming In gener al, a pr ogram is an y sequence o f calculato r instructi ons enclosed between the pr ogram container s and ». Subprogr ams can be inc luded as part of a progr am. The e xamples pr esented pr ev iousl y in this guide (e.
P age 21-16 wher e C u is a constant that depends on the sy stem of units used [C u = 1.0 f or units of the International S ys tem (S.I .) , and C u = 1.
P age 21-17 Y ou can also separate the in p ut data w ith spaces in a single stac k line rathe r than using ` . Progr ams that simulate a sequence of stac k operations In this case , the terms to be inv olv ed in the sequence of oper ations are assumed to be pr esent in the stac k.
P age 21-18 As yo u can see , y is used fir st, then w e use b , g, and Q, in that or der . Ther efor e, f or the purpose of this calc ulation we need to enter the v ariables in the inv erse or der , i .e., (do not ty pe the fo llo w ing): Q ` g ` b ` y ` F or the spec ifi c values under consider ation w e use: 23 ` 32.
P age 21-19 Sav e the progr am into a v ari able called hv: ³~„h~„v K A ne w var iable @@@hv @@@ should be a vailable in y our soft k ey men u . (Pre ss J to see y our var iable list .) The pr ogram le f t in the s tack can be e valuated b y using functi on EV AL.
P age 21-20 it is alw ay s pos s ible to r ecall the pr ogr am def inition into the s tack ( ‚ @@@q@@@ ) to see the or der in whic h the v ariabl es mus t be enter ed, namel y , → Cu n y0 S0 .
P age 21-21 whi ch indicates the positi on of the differ ent stack in put lev els in the form ula. B y compar ing this r esult with the or iginal f ormula that w e progr ammed, i .e., w e find that w e must enter y in st ack lev el 1 (S1) , b in stac k lev el 2 (S2), g in stac k leve l 3 (S3), and Q in stack le ve l 4 (S4) .
P age 21-22 The r esult is a stac k prom pting the user for the value o f a and placing the c ursor ri ght in fr ont of the pr ompt :a: Enter a value fo r a, sa y 35, then pr ess ` .
P age 21-2 3 @SST ↓ @ R esult: empty stac k, e xec uting → a @SST ↓ @ R esult: empty stack , entering subpr ogram « @SST ↓ @ R esult: ‘2*a^2+3’ @SST ↓ @ R esult: ‘2*a^2+3’ , leav in.
P age 21-2 4 Fi xing the program The onl y possible explanati on fo r the failur e of the pr ogram to pr oduce a numer ical re sult seems to be the lac k of the command NUM after the algebrai c expr ession ‘2*a^2+3’ . Let ’s edit the progr am by adding the missing EV AL f u ncti on.
P age 21-2 5 Input string pr ogram for two input v alues The in put str ing pr ogram for tw o input values, sa y a and b, looks as f ollo ws: « “ Enter a and b: “ { “ :a: :b: “ {2 0} V } INPUT OBJ → » This pr ogram can be easil y cr eated by modify ing the contents of INP T a.
P age 21-2 6 ` . The r esult is 4 98 8 7 . 06_J /m^3 . The units of J/m^3 are equi valent to P ascals (P a) , the pref err ed pressur e unit in the S.I .
P age 21-2 7 Enter v alues of V = 0. 01_m^3, T = 300_K , and n = 0.8_mol. Be fo r e pre ssing ` , the stac k will look lik e this: Press ` to get the result 19 9 54 8.2 4_J/m^3, or 199 54 8.2 4_P a = 199 . 5 5 kP a . Input through input f orms F unction INFORM ( „°L @) @@IN@ @ @INFOR@ .
Pa g e 2 1 - 2 8 The lis ts in items 4 and 5 can be em pty lists. A lso , if no value is t o be selected f or these options y ou can use the NO V AL command ( „°L @) @@IN@@ @NOVAL@ ).
P age 21-29 3 . F ield f ormat info rmation: { } (an empty list , thu s, def ault values us ed) 4. List of reset values: { 120 1 .0001} 5 . L ist of initial values: { 110 1.5 .00001} Save th e prog ram in to vari ab le IN FP 1 . P ress @INFP1 t o run the pr ogram .
P age 21-30 Th us , we demons tr ated the us e of f unction INF ORM. T o see how t o use thes e input v alues in a calculati on modif y the pr ogram as fo llo ws: « “ CHEZY’S EQN” { { “C:” “Che zy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { } { 120 1 .
P age 21-31 « “ CHEZY’S EQN” { { “C :” “Chezy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { 2 1 } { 120 1 .
P age 21-3 2 Acti vati on of the CHOO SE function w ill r eturn e ither a ze r o , if a @CANCEL ac tion is used , or , if a ch oice is made , the cho ice selected (e .
P age 21-3 3 commands “Operation cancelled” MSGBOX w ill sho w a message bo x indicating that the oper ation wa s cancelled. Identif y ing output in progr ams The simple st wa y to identify numer ical progr am output is to “tag” the pr ogram r esults .
P age 21-34 Ex ampl es of tagged output Example 1 – tagging output fr om function FUNC a Let ’s modify the function FUNCa , defined ear lier , to pr oduce a tagged output .
Pa g e 2 1 - 3 5 « “ Enter a: “ { “ :a: “ {2 0} V } INPUT OBJ →→ a « ‘ 2*a^2+3 ‘ EVAL ” F ” → TAG a SWAP »» (Recall that the f uncti on S W AP is availa ble by u sing „° @) STACK @SWAP@ ). Stor e the progr am back into FUNCa b y using „ @FUNCa .
Pa g e 2 1 - 3 6 Example 3 – tagging input and output f rom func tion p(V ,T) In this ex ample we modify the pr ogram @@@p@@@ so that the o utput tagged inpu t value s and tagged r esult .
P age 21-3 7 Stor e the progr am back into var iable p by using „ @@@p@@@ . Ne xt, r u n the pr ogram b y pres sing @@@p@@@ . Ent er v alues of V = 0.
P age 21-38 The r esult is the f ollo wing message bo x: Press @@@OK@@@ to cancel the mes sage bo x. Y ou could use a mes sage bo x for o utput fr om a progr am by us ing a tagged output , conv erted to a str ing, as the output st ring f or MS GBOX .
P age 21-3 9 Press @@@OK@@@ to can cel message box output. T he stack will now look lik e this: Including input and output in a m essage bo x W e could modify the p r ogram so that not onl y the output , but also the input , is included in a mes sage bo x.
P age 21-40 Y ou will noti ce that after typing the k ey strok e sequence ‚ë a ne w line is gener ated in the stack . The las t modificati on that needs to be included is to type in the plus si gn three times after the call to the f unction at the v ery end of the sub-pr ogram .
P age 21-41 Incorporating units w ithin a program As yo u have bee n able to obse rve fr om all the ex amples fo r the diffe r ent vers ion s of prog ram @@@p@@@ pr esented in this cha pter , attaching units to inpu t value s may be a tedi ous proce ss.
P age 21-4 2 2. ‘ 1_m^3 ’ : T he S.I. units cor respo nding to V are then placed in stac k lev el 1, the tagged input f or V is mo ved to stack lev el 2 . 3 . * : By multiply ing the contents of s tack le vels 1 and 2 , we gen er a te a nu mber wi th units (e .
P age 21-43 Press @@@OK@@@ to cancel mes sage box ou tput . Messag e bo x output without units Let ’s modify the progr am @@@p@@@ once mor e to eliminate the u se of units thr oughout it . T he unit-less progr am will look lik e this: « “ Enter V,T,n [S.
P age 21-44 oper ators ar e used to mak e a statement r egarding the r elativ e position of tw o or mor e real number s. Depending on the actual numbers used , such a st atement can be true (r epres ented b y the numer ical value o f 1. in the calc ulator), or false (r epr esented b y the numeri cal value of 0.
P age 21-45 Logical oper ators Logi cal oper ators ar e logical partic les that are u sed to jo in or modify simple logical stat ements. T he logical operat ors a vailable in the calc ulator can be easily acc essed thr ough the ke ystr ok e sequence: „° @ ) TEST@ L .
Pa g e 2 1 - 4 6 The calc ulator include s also the logi cal oper ator S AME . This is a non-standar d logical oper ator used t o determi ne if two ob jects ar e identical . If they ar e identi cal, a v alue of 1 (true) is r eturned, if not , a value of 0 (f alse) is r eturned.
P age 21-4 7 Branching with IF In this secti on w e pre sents ex amples using the constr ucts IF…THEN…END and IF…THEN…ELSE…END . The IF…THEN…END construct The IF…THEN…END is the simple st of the IF pr ogram constr ucts . T he general for mat of this construc t is: IF logical_statement THEN program_statements END .
P age 21-48 With the c ursor in fr ont of the IF stat ement pr ompting the user f or the logical statement that w ill acti vate the IF constr uct w h en the pr ogram is e xecu ted.
P age 21-4 9 Example : T ype in the follo wing pr ogram: « → x « IF ‘ x<3 ’ THEN ‘ x^2 ‘ ELSE ‘ 1-x ’ END EVAL ” Done ” MSGBOX » » and sa ve it under the name ‘f2 ’ . Pres s J and ver ify that vari able @@@f2@@@ is indeed av ailable in your v aria ble menu .
P age 21-50 IF x<3 THEN x 2 ELSE 1-x END While this simple constr uct w orks f ine when y our functi on has only tw o branc hes, y ou may need to nes t IF…THEN…ELSE…END constru cts to deal with func tion w ith three or mor e branc hes .
P age 21-51 A comple x IF construct lik e this is called a set of neste d IF … THEN … ELSE … END constr ucts . A possible w ay to e valuate f3(x), based on the nested IF constr uct show n abov e.
Pa g e 2 1 - 52 pr ogram_s tatements , and pa sses pr ogr am flo w to the statement follo wing the END statement . The CA SE , THEN, and END stat ements ar e available f or selecti ve typ ing by using „° @) @BRCH@ @ ) CASE@ . If y ou are in the BR CH menu, i .
Pa g e 2 1 - 5 3 5. 6 @@f3c@ Resu l t : -0.6312 66… (i .e., sin(x), with x in r adians) 12 @@f3c@ Res ul t : 16 2 7 54.7 91419 (i.e ., e xp(x)) 23 @@f3c@ Res ul t - 2 . (i .e., - 2) As yo u can see, f3c pr oduces ex actly the same r esults as f3. The onl y diffe rence in the pr ogr ams is the branc hing constructs u sed .
P age 21-54 Commands in volv ed in the ST ART constr uct are a vailable thr ough: „° @) @BRCH@ @ ) START @START Within the BRCH men u ( „° @) @BRCH@ ) the follo wing k ey str ok es are a vaila b.
Pa g e 2 1 - 5 5 1. This pr ogram needs an in teger numbe r as inpu t . Th us , bef or e e xec ution , that number (n) is in stac k lev el 1. T he progr am is then e xec uted .
P age 21-5 6 „°LL @) @RUN@ @ @DBG@ Start the debugger . SL1 = 2 . @SST ↓ @ SL1 = 0., SL2 = 2 . @SST ↓ @ SL1 = 0., SL2 = 0. , SL3 = 2 . (DUP) @SST ↓ @ Empty stac k (-> n S k) @SST ↓ @ Empty stac k ( « - start subpr ogram) @SST ↓ @ SL1 = 0.
P age 21-5 7 @SST ↓ @ SL1 = 1. (S + k 2 ) [S tore s value of SL2 = 2 , into SL1 = ‘k ’] @SST ↓ @ SL1 = ‘S’ , SL2 = 1. (S + k 2 ) @SST ↓ @ Empty stac k [Stor es value of SL2 = 1, into SL1 = ‘S’] @SST ↓ @ Empty stac k (NEXT – end of loop) --- loop e xec ution nu mber 3 f or k = 2 @SST ↓ @ SL1 = 2 .
P age 21-5 8 3 @@@S1@@ Res u lt : S:14 4 @@@S1@@ Res ul t: S:30 5 @@@S1@@ Res u lt : S:55 8 @@@S1@@ Res ul t: S:204 10 @@@S1@@ Res ul t: S:385 20 @@@S1@@ Resu l t: S:2870 30 @@@S1 @@ Resu lt : S:9455 .
P age 21-5 9 J 1 # 1.5 # 0. 5 ` Enter p ar ameters 1 1. 5 0.5 [ ‘ ] @GLIST ` Ent er the pr ogram name in lev el 1 „°LL @) @RUN@ @ @DBG@ St art the debugger . Use @SST ↓ @ to s tep into the pr ogram and see the detailed oper ation of eac h command .
P age 21-60 T o av oid an infinit e loop , make sur e that start_value < end_value . Ex am ple – calculate the summati on S using a FOR…NEXT construct The f ollow ing progr am calculat es the s.
P age 21-61 Example – gene rat e a list of numbers u sing a FOR…S TEP construc t T ype in the pr ogram: « → xs xe dx « xe xs – dx / ABS 1. + → n « xs xe FOR x x dx STEP n → LIST » » » and stor e it in var iable @GLI S2 . Θ Chec k out that the pr ogram call 0.
P age 21-6 2 The f ollow ing progr am calculat es the summation Using a DO…UNTIL…END loop: « 0. → n S « DO n SQ S + ‘ S ‘ STO n 1 – ‘ n ‘ STO UNTIL ‘ n<0 ‘ END S “ S ” → TAG » » Stor e this progr am in a var iable @@S3@@ .
Pa g e 2 1 - 6 3 The WHILE const ruct The ge ner al stru ctur e of this command is: WHILE logical_statement REPEAT program_statements END The WHILE st atement w ill r epeat the program_statements whi l e logical_statement is true (n on z er o ). If not , pr ogram contr ol is passed to the stateme nt right afte r END .
P age 21-64 and stor e it in var iable @GLI S4 . Θ Chec k out that the pr ogram call 0. 5 ` 2. 5 ` 0. 5 ` @GLI S4 pr oduces the list {0. 5 1. 1.5 2 . 2 .5}. Θ T o see step-b y-step operation u se the pr ogram DBUG for a short list, f or e xample: J 1 # 1.
P age 21-65 If y ou enter “ TR Y A GAIN” ` @DOERR , p r od uces t he follow in g message: TR Y AGA I N F inally , 0` @DOERR , produces t he messag e: Interrupted ERRN This f unction r eturns a number r epres enting the most r ecent err or . For e xample , if y ou try 0Y$ @ERRN , y ou get the number #30 5h.
P age 21-66 The se ar e the components of the IFERR … THEN … END construc t or o f the IFERR … THEN … ELSE … END construc t. Both logical cons truc ts ar e used f or tra pping er ror s dur ing pr ogram e xec ution .
P age 21-6 7 User RP L progr amming in alg ebr aic mode While all the pr ograms pr esent ed earlier ar e pr oduced and run in RPN mode, y ou can alw ay s type a pr ogram in U ser RP L when in algebr aic mode by us ing functi on RPL>. T his functi on is availa ble thr ough the command catalog .
P age 21-68 Wher eas, using RP L, ther e is no proble m when loading this pr ogram in algebrai c mode:.
Pa g e 2 2- 1 Chapter 2 2 Pr ograms f or graphics manipulation This c hapter includes a n u mber o f ex amples show ing how to u se the calc ulator’s func tions f or manipulating graphi cs inte rac tiv ely or thr ough the us e of pr ograms . As in Chapt er 21 w e recommend u sing RPN mode and setting s yst em flag 117 to S OFT menu labels.
Pa g e 2 2- 2 T o user -def ine a k ey y ou need to add to this list a command or pr ogram fo llow ed by a r efer ence to the k ey (see det ails in Chapter 20).
Pa g e 2 2- 3 LABEL (10) The f unction LABEL is used to label the axe s in a plot including the v ari able names and minimum and max imum values of the axes .
Pa g e 2 2- 4 EQ (3 ) The v ariable name E Q is res erved b y the calc ulator to stor e the c urren t equatio n in plots or soluti on to equations (see c hapter …). The soft menu k ey la beled E Q in this menu can be used a s it wo uld be if you ha ve y our var iable menu av ailable, e .
Pa g e 2 2- 5 The f ollow ing diagr am illustr ates the functi ons available in the P P AR menu . The letters attac hed to each f unction in the di agram ar e used for r efe r ence purpo ses in the desc ription o f the functi ons show n below .
Pa g e 2 2- 6 INDEP (a) The command INDEP spec ifie s the independent var iable and its plotting r a nge . The se spec ificati ons are st or ed as the third par ameter in the var iable P P AR. T he def ault value is 'X'. T he values that can be as signed to the independen t var iable spec ificati on are: Θ A var iable name , e .
Pa g e 2 2- 7 CENTR (g) The command CENTR tak es as ar gument an order ed pair (x,y) or a v alue x, and adjus ts the f irst tw o elements in the var iable P P AR, i .e ., (x min , y min ) and (x max , y max ), so that the center of the plot is (x,y) or (x , 0) , res pecti vel y .
Pa g e 2 2- 8 A list of tw o binary integers {#n #m}: sets the ti ck annotations in the x - and y- axes t o #n and #m pix els, r espectiv ely . AXE S (k) The in put value f or the axes command consis ts of either an order ed pair (x,y) or a list {(x ,y) atick "x -axis la bel" "y-axis la bel"}.
Pa g e 2 2- 9 The PTYP E menu within 3D (IV) The P TYPE menu under 3D contains the f ollow ing functi ons: The se fu nctions cor res pond to the gr aphics options Slope field , Wir efr ame, Y - Slice , Ps-C ontour , Gridmap and Pr -Sur face pr esented ear lier in this c hapter .
Pag e 22- 1 0 XV OL (N) , YV OL (O) , and ZVOL (P) The se func tions tak e as input a minimum and maxi mum value and ar e used to spec ify the extent of the par allelepiped wher e the graph w ill be generated (the vi ew ing parallelepiped). Thes e v alues ar e stor ed in the var iable VP AR .
Pag e 22- 1 1 The ST A T menu within PL OT The S T A T menu pro vi des access to plots r elated to statis tical anal ysis. W ithin this menu w e find the fo llow ing menus: The di agr am belo w show s the branc hing of the S T A T menu wi thin PL O T .
Pag e 22- 1 2 The PTYP E m enu within ST A T (I) The P TYPE menu pr ov ides the follo w ing func tions: Thes e ke ys corr espond to the plot t y pes Bar (A ) , Histogr am (B) , and Scatter(C ) , pr esented ear lier . Pres sing one of these so ft menu ke ys , while typ ing a pr ogram , will pl ace the corr esponding f uncti on call in the pr ogram .
Pag e 22- 1 3 XC OL (H) The command X COL is used to in dicate w hich o f the columns of Σ DA T , if mor e than one , w ill be the x - column or independent var iable column. YC O L ( I ) The command Y COL is used to indicat e whi ch o f the columns of Σ DA T, i f m o re than one , w ill be the y- column or dependent v ari able column.
Pag e 22- 1 4 Θ SIMU: w hen selec ted , and if more than one gr aph is to be plotted in the same set o f axe s, plots all the gr aphs simultaneousl y .
Pag e 22- 1 5 Thr ee -dimensional graphics The thr ee -dimensional gr aphics a vaila ble , namely , options Slopefi eld, Wir efr ame , Y -Slice , P s -Co ntour , G r idmap and Pr -Surface , use the VP.
Pag e 22- 1 6 @) PPAR Sho w plot paramet ers ~„r` @INDEP Def i ne ‘ r’ as the indep . var iable ~„s` @DEPND Def ine ‘ s ’ as the dependent va ri able 1 # 10 @XRNG De fine (- 1, 10) as the x -r ange 1 # 5 @YRNG L Def ine (-1, 5) as t he y-r ange { (0, 0) {.
Pag e 22- 1 7 @) PPAR Sho w plot paramet ers { θ 0 6. 2 9} ` @INDEP De fine ‘ θ ’ as the indep. V ari ab le ~y` @DEPND Def ine ‘Y’ as the dependent v ariab le 3 # 3 @XRNG Def ine (-3,3) as the x -range 0. 5 # 2.5 @YRNG L Def ine (-0.5,2 .5) as the y-r ange { (0, 0) {.
Pag e 22- 1 8 « St art pr ogram {PPAR EQ} PURGE P urge c urr ent P P AR and E Q ‘ √ r’ STEQ St ore ‘ √ r’ into E Q ‘r’ INDEP S et independent v ari able to ‘ r’ ‘s’ DEPND S et dependent v ariable t o ‘ s ’ FUNCTION Select FUNCT ION as the plot type { (0.
Pag e 22- 1 9 Example 3 – A polar plot . Enter the follo wing pr ogram: «S t a r t p r o g r a m RAD {PPAR EQ} PURGE Change to r adians , purge v ars.
Pag e 22- 2 0 PICT This so ft ke y re fer s to a v ari able called PICT that stor es the c urr ent contents of the gr aphic s w indow . This v ariable name , ho w ev er , cannot be placed w ithin quotes, and ca n only store graph ics obje cts. In tha t sen se, PICT i s li k e n o oth er calc ulator va ri ables.
Pag e 22- 2 1 BO X This command t ake s as input two or dered pair s (x 1 ,y 1 ) (x 2 , y 2 ) , or two pair s of pi xel coor dinates {#n 1 #m 1 } {#n 2 #m 2 }. It dr aws the bo x who se diagonals ar e r epre sented b y the t w o pairs of coor dinates in the input.
Pag e 22- 22 Θ PI X? Checks if pi xel at locati on (x,y) or {#n , #m} is on. Θ PI XOFF turns o ff pi xel at location (x ,y) or {#n, #m}. Θ PI XON turns on p ix el at location (x ,y) or {#n, #m}.
Pa g e 22- 23 (5 0., 50.) 12 . –180. 180. ARC Dr aw a c irc le center (5 0,50), r= 12 . 1 8 FOR j Dr aw 8 lines w ithin the ci rc le (50., 5 0.) D UP L ines ar e centered as (5 0,5 0) ‘12*COS( 45*.
Pa g e 22 - 24 It is suggested that y ou cr eate a separate sub-dir ectory to stor e the progr ams. Y ou could call the sub-direc tory RIVER , since we ar e dealing with ir regular open channel c r oss-s ectio ns, typ ical of ri ver s. T o see the pr ogram XSE CT in action , use the f ollow ing data sets .
Pa g e 22 - 2 5 P ixel coor dinates The f igur e belo w show s the graphi c coordinat es fo r the t yp ical (minimum) scr e en of 131 × 64 pix els. P ix els coordinates ar e measured fr om the top left corner of the screen {# 0 h # 0h}, w hich corresponds to user-defined c oordinates Data set 1 Da ta set 2 xy x y 0.
Pag e 22- 26 (x min , y max ) . The max imum coor dinates in terms of p i xels cor respond to the lo wer r ight corner of the sc reen {# 8 2h #3Fh}, whic h in user-coor dinates is the point (x max , y min ) . The coordinates of the two other cor ners both in pi xel as well as in user-defined coordinates ar e show n in t he fi gure .
Pa g e 22- 27 Animating a collection of graphics The calc ulator pr ov ides the func tion ANIMA TE t o animate a n umber of gr aphic s that hav e been placed in the stac k. Y ou can generat e a gra ph in the gr aphic s sc r een by u sing the commands in the PL O T and PICT menu s.
Pag e 22- 28 ANIMA TE is available by u sing „°L @) GROB L @ANIMA ) . The animati on will be r e -started. Pr ess $ to stop the animati on once more . Notice that the number 11 w ill still be list ed in stac k leve l 1. Pr ess ƒ to dr op it fr om the stack.
Pa g e 2 2- 2 9 Example 2 - Animating the plotting of differ ent po wer f unctions Suppose that y ou want t o animate the plotting of the functi ons f(x) = x n , n = 0, 1, 2 , 3, 4, in the same set of ax es.
Pag e 22- 3 0 pr oduced in the calculator ’s scr een. T here for e, w hen an image is converted into a GR OB, it becomes a s equence of binary digits ( b inary dig its = bits ), i . e . , 0’s and 1’s . T o illus trate GR OBs and con ve rsi on of image s to GR OBS consider the fo llo w ing ex er c ise .
Pag e 22- 3 1 1` „°L @) GROB @ GROB . Y ou w ill now ha ve in le vel 1 the GR OB desc r ibed as: As a gra phic obj ect this equation can no w be placed in the graphi cs display . T o re cov er the graphic s display pr ess š . Then , mov e the c u rs or to an empty sector in the gr aph, and pr ess @) EDIT LL @REPL .
Pa g e 2 2- 32 BLANK The f unction BLANK, w ith arguments #n and #m , cr eates a blank gra phics objec t of w idth and height spec ifie d by the v alues #n and #m, r especti vely .
Pa g e 2 2- 3 3 An ex ample of a progr am using GROB The f ollow ing progr am produ ces the gr aph of the sine f unctio n inc luding a fr ame – dra wn w ith the func tion B OX – and a GROB to label the gr aph.
Pa g e 2 2- 3 4 sho ws the state of s tres ses w hen the element is r otated b y an angle φ . In this case, the normal st r esses are σ ’ xx and σ ’ yy , while the shear str esses are τ ’ xy and τ ’ yx .
Pa g e 22- 35 The stress condit ion for which the she ar stress, τ ’ xy , is ze ro , indicated by segment D’E’ , produces the so-called pr inc i pal str esses , σ P xx (at point D’) and σ P yy (at point E’).
Pag e 22- 3 6 separ ate vari ables in the calculator . These sub-pr ograms ar e then linked b y a main pr ogram , that we w ill call MOHRCIRCL . W e will fir st cr eate a sub- dir ectory called MOHRC w ithin the HOME dir ectory , and mo ve into that dir ectory to type the pr ograms .
Pa g e 2 2- 37 At this point the pr ogram MOHR CI RC L starts calling the sub-progr ams to pr oduce the fi gure . Be patient. T he resulting Mohr’s c ir cle will loo k as in the pic ture to the left .
Pa g e 22 - 3 8 infor mation tell us is that some where betw een φ = 5 8 o and φ = 5 9 o , the shear stress, τ ’ xy , becomes z er o . T o f ind the actual v alue of φ n, press $ . T h en type the list cor re sponding to the value s { σ x σ y τ xy}, f or this case, it w ill be { 25 75 50 } [ENTER] Then , press @ CC&r .
Pa g e 22 - 3 9 necess ary to plot the c irc le. It is suggest that w e r e -order the v ari ables in the sub-dir ectory , s o that the progr ams @MOHRC and @PRNST ar e the two f irst v ari ables in the soft-menu k ey labels.
Pag e 22- 4 0 T o find the v alues of the str esse s corr esponding to a ro tation of 3 5 o in the angle of th e stressed pa rticle, we use: $š Clear scr e en, show PICT in g r aphics scr e en @TRACE @ ( x,y ) @ . T o move c ursor o ver the c irc le sho w ing φ and (x ,y) Ne xt, pr ess ™ until y ou read φ = 3 5.
Pag e 22- 4 1 Since pr ogr a m IND A T is us ed also f or pr ogr am @PRNST (P RiNc ipal ST r esses) , running that partic ular progr a m w ill now us e an input fo rm , f or e xample , The r esult , a.
Pa g e 2 3 - 1 Chapter 2 3 Character strings Char acter strings ar e calculator ob jects enc losed betw een double quotes . The y ar e tr eated as te xt by the calculat or . For e xample , the string “S INE FUNCTION” , can be transf ormed int o a GROB (Gr aphics Ob ject), to label a gr aph, o r can be used as output in a pr ogr am.
Pa g e 2 3 - 2 String concatenation Str ings can be concatenated (j oined together ) by using the plus sign +, f or exa mp le : Concatenating s tring s is a prac tical w ay to cr eate output in pr ogr ams.
Pa g e 2 3 - 3 The ope rati on of NUM, CHR, OB J , and S TR was pr esent ed earlie r in this Chapter . W e ha ve als o seen the func tions SUB and REP L in relati on to gra phics earli er in this chapter .
Pa g e 2 3 - 4 scr een the ke ystr oke sequence to get suc h char acter ( . f or this case) and the numer ical code corr esponding to the char acter (10 in this cas e) .
Pa g e 24 - 1 Chapter 2 4 Calculator objec ts and flags Numbers , lists, v ectors, matr ices, algebr aics, etc ., are calc ulator objects . The y ar e classif ied accor ding to its nature into 30 diff erent ty pes, w hic h are desc r ibed belo w . Fl ags ar e var iables that can be u sed to con tr ol the calculat or properties.
Pa g e 24 - 2 Number T ype Ex ample _______________ ____________________ _____________________ ____________ 21 Extended Real Number Long Real 2 2 Extended Comple x Number Long Complex 2 3 Link ed Arr .
Pa g e 24 - 3 Calculator flags A flag is a v ariable that can e ither be set or unse t . The statu s of a flag affec ts the behav ior of the calc ulator , if the flag is a s ys tem flag , or of a pr ogr am, if it is a user f lag. T hey ar e descr ibed in mor e detail next .
Pa g e 24 - 4 The f unctions contained w ithin the FL A G menu are the f ollow ing: The oper ation of thes e func tions is as f ollo ws: SF Set a flag CF Clear a f lag F S? Retur ns 1 if flag is set, .
Pa g e 25 - 1 Chapter 25 Date and T ime Functions In this Chapter w e demonstr ate some of the func tions and calc ulations using times and dates . The T I ME menu The T IME menu , av ailable thro ugh the ke ystr ok e sequence ‚Ó (the 9 k ey) pr o vi des the follo wing f uncti ons, w hich ar e desc ribed ne xt: Setting an alarm Option 2 .
Pa g e 25 - 2 Bro wsing alarms Option 1. Br o ws e alarms... in the TIME me nu lets yo u r ev iew your c urr ent alarms . F or ex ample, after ente ring the alarm u sed in the e xample abo ve , this o.
Pa g e 25 - 3 The appli cation of these f u ncti ons is demonstrated belo w . D A TE: P laces c urr ent date in the stack D A TE: Set sy stem date to specif ied value TIME: Place s cu rr ent time in 2 4 -hr HH.MMS S for mat TIME: Set sy stem time to spec ified v alue in 2 4 -hr HH.
Pa g e 25 - 4 Calculating with tim es The fun ct ion s HMS , HMS , HMS+, and HM S - are us ed to manipulate value s in the HH.MM SS f ormat . This is the same f ormat us ed to calc ulate with angle measur es in degree s, minu tes , and seconds.
Pa g e 2 6 - 1 Chapter 2 6 Managing memor y In Chapter 2 w e intr oduced the basic concepts of , and oper ations f or , cr eating and managing var iables and dir ector ies . In this Chapter w e disc uss the management of the calc ulator’s memory , inc luding the par tition o f memory and techni ques for backing u p data.
Pa g e 2 6 - 2 P ort 1 (ERAM ) can contain up to 12 8 KB of data. P or t 1, together w ith P ort 0 and the HOME direc tory , constitut e the calculator ’s R AM (R andom Access Memory) segment of calc ulator’s memory . T he RAM memor y segment r equires contin uous elec tri c pow er supply f r om the calculat or batter ies to operate .
Pa g e 2 6 - 3 Chec king objec ts in memory T o see the obj ects stor ed in memory you can u se the FILE S functi on ( „¡ ). Th e scre e n be l ow sh ows th e H OM E d i rec to r y wi th five d ire c to ri es, n a m ely , TRIANG , MA TRX , MPFIT , GRP HS, and CA SD IR.
Pa g e 2 6 - 4 Bac k up objec ts Back up obj ects ar e used to copy dat a fr om your home dir ectory into a memory port. T he purpose o f bac king up objects in me mory port is to pr eserve the contents of the ob jects f or futur e usage .
Pa g e 2 6 - 5 Bac king up and restor ing HOME Y ou can back up the contents of the c urr ent HOME dir ectory in a single back up obje ct . This ob jec t w ill contain all v ari ables , k ey as signments , and alarms c urr ently def ined in the HOME direc tory .
Pa g e 2 6 - 6 Stor ing, deleting, and rest oring bac k up objec ts T o cr eate a back up obj ect us e one of the f ollow ing appr oache s: Θ Use the F ile Manager ( „¡ ) t o c o p y t h e o b j e c t t o p o r t . U s i n g t h i s appr oach, the bac kup obj ect will ha ve the same name as the o ri ginal object .
Pa g e 2 6 - 7 Using data in backup objects Although y ou cannot directl y modif y the contents of back up objec ts, y ou can use thos e contents in calculat or oper ations. F or ex ample, y ou can run pr ograms stor ed as back up objec ts or use dat a fr om back up objects t o run pr ograms .
Pa g e 2 6 - 8 T o re move an SD car d, turn o f f the HP 5 0g, pr ess gentl y on the expo sed edge of the car d and push in . The car d should spring out of t he slot a small distance , allo w ing it now to be easil y r emov ed fr om the calculator .
Pa g e 2 6 - 9 Accessing objects on an SD card Acces sing an obj ect fr om the SD car d is similar to w hen an object is located in ports 0, 1, or 2 . Ho we ver , P ort 3 wi ll not appear in the menu when us ing the LIB func tion ( ‚á ). T he SD file s can only be managed using the F iler , or F ile M anager ( „¡ ).
Pa g e 2 6 - 1 0 Note that if the name of the object y ou intend to stor e on an SD card is longer than ei ght char acters , it will appear in 8. 3 DOS f ormat in port 3 in the Filer once it is stor ed on the card .
Pa g e 2 6 - 1 1 Note that in the case of obj ects with long f iles names , you can s pecify the full name of the ob ject , or its truncate d 8. 3 name , when ev aluating an objec t on an SD car d.
Pa g e 2 6 - 1 2 This w ill stor e the objec t pr ev iou sly on the stac k onto the SD card into the dir ectory named PR OGS into an objec t named PR OG1. Note: If PR OGS does not ex ist, the dir ectory will be au tomaticall y cr eated. Y ou can specify an y number of nes ted subdirect ori es.
Pa g e 2 6 - 1 3 Libr ar y numbers If y ou use the LIB men u ( ‚á ) and pr ess the soft menu k ey cor r esponding to port 0, 1 or 2 , you w ill see library numbers lis ted in the soft menu k ey labe ls. E ach libr ar y has a thr ee or four -digit number ass oc iated with it .
Pa g e 2 6 - 1 4 w ill indicate w hen this battery needs r eplacement. T he diagram belo w sho ws the location of the bac kup battery in the top compartment at the back of the calc ulator .
Pa g e 27- 1 Chapter 2 7 T he Equation Libr ar y The E quation L ibrary is a collection o f equations and commands that enable y ou to sol ve simple s c ience and engin eer ing pr oblems. T he library consists o f mor e than 300 equations gr ouped into 15 techni cal subj ects containing mor e than 100 pr oblem titles .
Pa g e 27- 2 7 . For eac h know n vari able, type its v alue and press the cor respo nding menu k ey . If a vari able is not sho wn , press L to display furth er v ariables. 8. Optional: suppl y a guess fo r an unknow n var iable . This can speed up the soluti on pr ocess or help to f o c us on one of se ver al solutions .
Pa g e 27- 3 Using the menu ke ys The ac tions of the unshifted and shifted var iable menu k ey s for both s olv ers ar e identi cal. Noti ce that the Multiple E quation S olver u ses tw o for m s of men u labels: black and whit e . The L k ey dis plays additi onal menu labels, if r equir ed.
Pa g e 27- 4 Bro wsing in the Equation L ibrary When y ou select a sub ject and title in the E quation L ibrary , you spec ify a set of one or mor e equati ons. Y ou can get the f ollow ing infor mation about the equation s et from the E quatio n Libr ary catalogs: The equations thems elves and the number of equati ons.
Pa g e 27- 5 Vie wing var iables and selec ting units After y ou select a subj ect and title , y ou can vi e w the catalog of names , desc r iptions , and units for the v ari ables in the equation s et b y pre ssing #VARS# . The t able belo w summari ze s the oper ations av ailable to y ou in the V ar iable catalogs .
Pa g e 27- 6 Pr ess to stor e the pictur e in PI C T , the gra phics memory . T hen y ou can use © PIC T ( or © PICTURE) t o vi ew the pic ture again after y ou hav e quit the Equati on Libr ar y . Pr ess a menu k ey or to v iew other equation inf ormati on.
Pa g e 27- 7 The men u labels for the v ariable k ey s are w hite at fir st, but c hange during the soluti on proces s as des cr ibed below . Becaus e a soluti on inv olv es man y equations and man y .
Pa g e 27- 8 Meani ngs of Menu Labe ls Defining a set of equations When y ou design a s et of eq uations , you sh ould do it w ith an understanding o f ho w the Multiple -Equati on Solv er uses the equati ons to sol ve pr oblems.
Pa g e 27- 9 F or ex ample, the f ollo wing thr ee equations def ine initial v elocity and acceler a ti on based on tw o observed dis tances and times. T he firs t two equations alone ar e mathematicall y suffi c ient f or solv ing the problem , but each equation con tains tw o unkno wn v aria bles.
Pa g e 27- 1 0 6. P ress !MSOLV! to launc h the sol ver w ith the new se t of equati ons. T o chang e the title and menu for a set of equations 1. Mak e sur e that the set o f equati ons is the c urr ent set (as the y are u sed when the Multiple -E quation Sol ver is launc hed) .
Pa g e 27- 1 1 Constant? T he initial value of a v ariable ma y be leading the r oot - finder in the w rong dir ection . Supply a guess in the oppo site dir e cti on fr om a cr itical v alue.
Pa g e 27- 1 2 Not r elated . A var iable ma y not be inv olv ed in the solution (no mark in the label), so it is not compatible w ith the var iables that w ere in volv ed. W r ong dir ection . The initial v alue of a v ariable ma y be leading the r oot - finder in the w rong dir ection .
Pa g e A - 1 Appendix A Using input forms This e xample o f setting time and date illu str ates the use o f input f orms in the calc ulator . S ome general r ules: Θ Use the arr ow k ey s ( š™˜— ) to mov e from one field to the ne xt in the input f orm.
Pa g e A - 2 In this particular ca se w e can giv e v alues to all but one of the var iables, sa y , n = 10, I%YR = 8. 5, PV = 10000, FV = 1000, and s ol ve f or var iable P MT (the meaning of thes e var iables w ill be pre sented later ) . T r y the f ollow ing: 10 @@OK@@ Enter n = 10 8.
Pa g e A - 3 !CALC Pr ess to access the stac k for calc ulations !TYPES Press to determine the type of object in highlighte d fi eld !CANCL Cancel o peration @@OK@@ Accep t en tr y If y ou pre ss !RES.
Pa g e A - 4 (In RPN mode , we w ould hav e used 113 6.2 2 ` 2 `/ ). Press @@ OK@@ to ent er this ne w value . The input f orm w ill now look lik e this: Press !TYPES to s ee the type of dat a in the P MT fi eld (the highligh ted fi eld) .
Pa g e B - 1 Appendix B T he calc ulator ’s ke y board The f igur e belo w show s a diagram o f the calc ulator ’s ke yboar d w ith the number ing of its ro ws and columns .
Pa g e B - 2 fi ve f uncti ons. T he main ke y f uncti ons ar e sho wn in the fi gure belo w . T o oper ate this main k ey func tions simpl y press the cor responding k ey . W e will r efer to the k ey s b y the r ow and column w here the y are located in the sk etc h abo ve , thus , ke y (10,1) is the ON key .
Pa g e B - 3 Main ke y functions Key s A through F ke ys ar e assoc iated w ith the soft menu opti ons that appear at the bottom of the calculat or’s displa y . Th us, these k e ys w ill acti vate a var iety of func tions that change acco rding t o the acti ve menu .
P age B-4 The left-s hi ft k ey „ and the ri ght -shift ke y … are combined w ith other ke ys to ac ti vate menu s, enter char acters , or calc ulate functi ons as descr ibed else wher e. The numerical k ey s ( 0 to 9 ) are used to ent er the digits of the decimal number sy stem.
P age B-5 the other three f unctions is ass oci ated with the left-shif t „ ( MTH ) , right-shift … ( CA T ) , and ~ ( P ) k ey s. Diagr am s sho w ing the function or c haracter r esulting fr om .
Pa g e B - 6 The CM D func tion sho ws the most r ecent commands, the PRG fun ctio n acti vates the pr ogramming men us, the MTR W functi on acti vates the Matri x Wr i t e r, Left-shift „ func tions of the calc ulator ’s ke yboard The CM D func tion sho ws the most r ecent commands.
Pa g e B - 7 The e x ke y calcul ates the e xponential func tion of x . The x 2 ke y calculat es the squar e of x (this is ref erred to as the SQ fun ctio n) . The A SIN , A COS , and A T AN func tions calc ulate the ar csine, ar ccosine , and ar ctangent f unctions, r especti vel y .
Pa g e B - 8 Righ t-s hif t … func tions of the calculator ’s ke yboard Right-shift functions The sk etch abo ve sho ws the functi ons, char acters, or menus ass o c iated w i th the differ ent calculator k ey s when the r ight-shift ke y … is activ ated.
Pa g e B - 9 The CA T function is u sed to acti vate the command catalog. The CLEAR functi on clear s the sc reen . The LN functi on calc ulates the natur al logarithm . The func tion calc ulates the x – th root o f y . The Σ function is us ed to enter summati ons (or the upper case Gr eek letter sigma).
Pa g e B - 1 0 is used mainl y to enter the upper -case letter s of the English alphabet ( A thr ough Z ) . T he numbers , mathematical s ymbols ( - , + ), decimal po int ( . ) , and the space ( SPC ) ar e the same as the main functions o f these k ey s.
Pa g e B - 1 1 Notice that the ~„ combinatio n is used ma inly to enter the lo wer -c ase letters of the English alphabet ( A thr ough Z ). The number s, mathematical sym bo l s ( - , +, × ), dec imal point ( . ) , and t he space ( SP C ) are the same a s the main functi ons of these ke ys .
Pa g e B - 1 2 Alpha-right-shift c har ac ters The f ollow ing sketc h show s the c har acter s assoc iated w ith the differ ent calc ulator k ey s when the ALP HA ~ is combined with the r ight -shift k ey … .
Pa g e B - 1 3 ~… combination inc lude Greek letters ( α, β, Δ, δ, ε, ρ, μ, λ, σ, θ, τ , ω , and Π ), other c har acter s gener ated by the ~… co mbinati on ar e |, ‘ , ^, =, <, >, /, “ , , __, ~, !, ?, <<>>, and @.
Pa g e C - 1 Appendix C CAS settings CA S stands f or C omputer A lgeb r a ic S yste m. T his is the mathemati cal core o f the calc ulator wher e the sy mbolic mathematical oper ations and func tions ar e pr ogrammed . The CA S offe rs a number of settings can be adj usted accor ding to the type of oper ation of inter est .
Pa g e C - 2 Θ T o r ecov er the ori ginal menu in the CAL CULA T OR MODES inpu t bo x, pr ess the L k e y . Of inter est at this point is the changing o f the CAS se ttings.
Pa g e C - 3 A var iable called VX ex ists in the calc ulator’s {HO ME CASDIR} directory that take s, by def ault , the value of ‘X’ . This is the name of the pr efe rr ed independent v ari able fo r algebr aic and calculu s applicati ons. F or that reason , most e xamples in this C hapter us e X as the unknow n var iable .
Pa g e C - 4 The s ame e xample , corr esponding to the RPN oper ating mode , is show n next: Appr o ximate v s. Ex act CAS mode When t he _ Appro x is selected, s ymbolic oper ations (e.g ., def inite integrals, squar e roots , etc .) , will be calc ulated numeri cally .
Pa g e C - 5 The k ey str ok es necessary for ent er ing these v alues in Algebrai c mode are the follo wing: …¹2` R5` The s ame calculati ons can be pr oduced in RPN mod e .
Pa g e C - 6 It is r ecommended that y ou se lect EXA CT mode as default CA S mode, and change t o APP ROX mode if r equested b y the calcul ator in the perfor mance of an oper ation . F or additional infor mation on real and integer n umbers , as we ll as other ca lcul ato r’s obje cts, refer to Cha pte r 2 .
Pa g e C - 7 If y ou pre ss the OK soft menu ke y () , then the _Comple x op ti on is for ced, and the r esult is the f ollo wing: The k ey str okes us ed abov e ar e the fo llo w ing: R„Ü5„Q2+ 8„Q2` When ask ed to change to C OMP LEX mode , use: F .
Pa g e C - 8 F or ex ample, ha v ing selec ted the St ep/step optio n, the f ollow ing scr eens show the step-b y-step di visi on of two pol ynomials , namely , (X 3 -5X 2 +3X- 2)/(X- 2). This is accomplished by u sing functi on DIV2 a s sho wn belo w .
Pa g e C - 9 . Increasing-po wer CA S mode When t he _Incr po w CA S option is se lected , polynomi als will be list ed so that the ter ms will ha ve incr easing po wer s of the independent var iable .
Pa g e C - 1 0 Rigor ous CAS set ting When t he _Rigor ous CAS option is se lected , the algebrai c expr essi on |X|, i.e ., the absolute v alue, is not simplif ied to X . If the _R igor ous CA S option is not select ed, the algebr aic e xpressi on |X| is simplified t o X .
Pa g e C - 1 1 Notice that , in this instance, s oft menu k ey s E and F are the onl y one wi th ass oci ated commands , namely : !!CANCL E CANCeL the help fac ility !!@@OK#@ F OK to acti vate help fa.
Pa g e C - 1 2 Notice that ther e are si x commands assoc iated w ith the soft menu k ey s in this case (y ou can chec k that there ar e only si x command s because pr essing the L pr oduces no additional menu items).
Pa g e C - 1 3 T o nav igate quic kly to a partic ular command in the help f ac ility list w ithout hav ing to use the arr o w k e ys all the time , we can us e a shortcu t consisting of typing the f irst letter in the command’s name .
Pa g e C - 1 4 In no ev ent unless r equired b y appli cable law w ill any cop yr ight holder be liable to y ou for damage s, inc luding any gene ral , speci al, inc idental or conseq uential damage s.
Pa g e D - 1 Appendix D Additional character set While y ou can use an y of the upper -case and low er -case English letter fr om the ke yboar d, ther e are 2 5 5 char acters usable in the calc ulator . Including spec ial cha ract ers li ke θ , λ , etc .
Pa g e D - 2 functi ons ass oc iated w ith the soft menu k ey s, f4, f5, and f6. T h ese f unctions ar e: @MODIF : O pens a gra phics scr een wher e the user can modify highlighted char acter . Use this option car efull y , since it will alt er the modifi ed charac ter up to the ne xt r eset o f the calc ulator .
Pa g e D - 3 Gr ee k lett ers α (alpha) ~‚a β (beta) ~‚b δ (delta) ~‚d ε (epsilon) ~‚e θ (theta) ~‚t λ (lambda) ~‚n μ (mu) ~‚m ρ (rh o) ~‚f σ (sigma) ~‚s τ (tau) ~‚u ω (.
Pa g e E - 1 Appendix E The Selec tion T ree in th e Equation W riter The e xpre ssion tr ee is a diagr am sho wing h o w the E quation W r iter interpr ets an ex pre ss io n. The fo rm of th e exp re ss io n t re e i s de t erm i ne d by a n u mb er o f r ul es kno wn as the hie rar ch y of oper ation .
Pa g e E - 2 Step A1 Ste p A2 Step A3 Ste p A4 Step A5 Ste p A6 W e notice the appli cation of the hier arc hy-of-oper ation rules in this selecti on. F irst the y (Step A1) . Then , y- 3 (S tep A2 , par enth eses ). Then , (y-3)x (Step A3, multiplicati on) .
Pa g e E - 3 Step B1 Step B2 Step B3 Step B4 = Step A5 Step B5 = S tep A6 W e can also follo w the ev aluation o f the expr essi on starting fr om the 4 in the argume nt of the SIN func tion in the denominator . Pr ess the dow n arr ow k e y ˜ , continuously , until the clear , editing cur sor is tr igger ed around the y , once mor e .
Pa g e E - 4 Step C3 Step C 4 Ste p C5 = Step B5 = S tep A6 The expr ession tree for the expression presented above is show n next: The s teps in the e valuation of the thr ee terms ( A1 through A6 , B1 thro u gh B5, and C1 thr ough C5) ar e sho wn ne xt to the c irc le containing number s, var iables , or oper ators .
Pa g e F - 1 Appendix F T he Applications (APP S) menu The A pplicati ons (AP PS) men u is av ailable through the G key ( fi rst key i n second r o w fr om the ke yboard’s top). The G ke y sho ws the f ollo w ing applications: The diff erent appli cations ar e desc ribed ne xt.
Pa g e F - 2 I/O functions.. Selecting opti on 2 . I/O f uncti ons .. in the APP S menu w ill produce the f ollow ing menu list o f input/output func tions The se appli cations ar e descr ibed next: S.
Pa g e F - 3 The C onstants Libr ar y is disc ussed in detail in C hapter 3 . Numeric solv er .. Selecting opti on 3. C onstants lib .. in the APP S menu pr oduces the numer ical solver menu : This oper ation is equi valent to the k ey strok e sequence ‚Ï .
Pa g e F - 4 Equation wr iter .. Selecting opti on 6.E quation w riter .. in the AP P S menu opens the equati on writ er: This oper ation is eq ui val ent to the k ey str oke s equence ‚O . The eq uation wr iter is intr oduced in detail in Chapte r 2 .
Pa g e F - 5 M atr ix W riter .. Selecting opti on 8.Matri x W riter .. in the APP S menu launc hes the matri x wr iter: This oper ation is eq ui val ent to the k ey str oke s equence „² .T he Matri x W rit er is pre sent ed in detail in Chapter 10.
Pa g e F - 6 This oper ation is eq ui val ent to the k ey str oke s equence „´ . T he MTH menu is intr oduced in Chapte r 3 (real n umb er s) . Other f uncti ons fr om the MTH menu ar e pr esented .
Pa g e F - 7 Note that flag –117 should be set if y ou are go ing to use the E quatio n Libr ary . Note too that the E quation L ibrary will onl y appear on the APP S menu if the two E quation L ibrary files ar e stor ed on the calculator . The E quation L ibrary is explained in det ail in chapter 2 7 .
P age G-1 Appendix G Useful shortc uts Pr esented her ein ar e a number of k eyboar d shortcuts commonl y used in the calc ulator : Θ Adjust di splay co ntrast: $ (hold) + , or $ (hold) - Θ T oggle between RPN and AL G modes: H @@@OK@@ or H` . Θ Set/c lear sy stem flag 9 5 (AL G v s.
P age G-2 Θ Set/clear s yst em flag 117 (CHOO S E bo xes vs . SOFT menu s): H @) FLAGS —„ — ˜ @@CHK@ Θ In AL G mode , SF(-117) selects S OFT menus CF(-117) se lects CHOO SE BOXE S .
P age G-3 Θ S ystem-lev el op er ation (H old $ , r elease it after enter ing second or thir d k e y): o $ (hold) AF : “Cold” r estart - all memory er ased o $ (hold) B : C ancels ke ys tr ok e o.
P age H-1 Appendix H T he CAS help facilit y The CA S help fac ility is available thr ough the ke ystr oke seq uence I L @HELP ` . The fo llo w ing scr een shots sho w the fir st menu page i n the listing of th e CAS help fac ili ty . The commands ar e listed in alphabeti cal or der .
P age H-2 Θ Y ou can t ype two or more letters of the com mand of interest, b y locking the alphabeti c ke yboar d. T his will t ake y ou to the command of int eres t , or to its neighbor hood. Afterwar d s, y ou need to unlock the alpha k ey board , and use the ve r tical ar r o w ke ys —˜ to locate the command , if needed.
Pa g e I - 1 Appendix I Command catalog list This is a list o f all commands in the command catalog ( ‚N ) . Those commands that belong to the CA S (C omputer A lgebrai c Sy stem) ar e listed also in Appendi x H.
Pa g e J - 1 Appendix J T he MA THS menu The MA THS menu , accessible thr ough the command MA THS (a vailable in the catalog N ), contains the follo w ing sub-menu s: The CMP LX sub-menu The CMP L X sub-men u contains functi ons per tinent to oper ations with comple x numbers: The se fu nctions ar e descr ibed in Chapter 4.
Pa g e J - 2 The HYP ERBOLI C sub-menu The HYP ERBOLIC sub-menu co ntains the h yperboli c functi ons and their in ver ses . The se func tions ar e descr ibed in Chapter 3 . The INTEGER sub-menu The INTE GER sub-menu pr ov ides functi ons f or manipulating integer numbers and some poly nomials.
Pa g e J - 3 The P OL YNOMIAL sub-menu The P OL YNO MIAL sub-menu inc ludes func tions for gener ating and manipulating poly nomials . The se func tions ar e pres ented in Chapt er 5: The TE ST S sub-m enu The TE S TS su b-menu inc ludes r elational oper ators (e .
Pa g e K- 1 Appendix K Th e M A I N m en u The MAIN men u is av ailable in the command catalog . This men u include the fo llow ing sub-menu s: The CA SCF G command This is the f irst entry in the MAIN menu . This command conf igure s the CAS . F or CAS conf igur ation inf ormatio n see Appendi x C.
Pa g e K- 2 The DIFF sub-menu The DIFF sub-menu contains the fo llo w ing funct ions: The se func tions ar e also av ailable thr ough the CAL C/DIFF sub-menu (start with „Ö ).
Pa g e K- 3 The se fu nctions ar e also av ailable in the TRIG menu ( ‚Ñ ) . Description of these f unctions is incl uded in Chapter 5 . The S OL VER sub-menu The S OL VER men u includes the follo w ing functi ons: The se fu nctions ar e av ailable in the CAL C/S OL VE menu (st art with „Ö ).
Pa g e K- 4 The su b-menus INTE GER, MODULAR, and POL YNOMIAL ar e pr esented in detail in Appendi x J. The E XP &LN sub-menu The EXP &L N menu contains the f ollow ing functi ons: This men u is also acces sible thr ough the k eyboar d by using „Ð .
Pa g e K- 5 The se f unctio ns are av ailable through the C ONVERT/REWR ITE me nu (start w ith „Ú ) . T h e f unctio ns ar e pres ented in Chapt er 5, e xcept f or func tions XNUM and XQ, w hich ar.
Pa g e L- 1 Appendix L L ine editor commands When y ou trigger the line editor b y using „˜ in the RPN stack or in AL G mode , the follo w ing soft menu f u ncti ons ar e pro vided (pr ess L to see the r emaining func tions): The f unctions ar e brief ly desc ribed as fo llo ws: SKIP: Skips char acter s to beginning of w ord .
Pa g e L- 2 The it ems sho w in this scr een are self-e xplanatory . F or ex ample , X and Y positions mean the positi on on a line (X) and the line number (Y). Stk Size means the number of obj ects in the AL G mode history or in the RPN stac k. Mem(KB) means the amount of fr ee memory .
Pa g e L- 3 The SE ARCH sub-menu The f unctions of the SE ARCH sub-me nu ar e: Fi n d : Use this functi on to find a str ing in the command line. The input f orm pr o vi ded w ith this command is show n next: Rep la c e : Use this command t o fi nd and replace a str ing.
Pa g e L- 4 The GO T O sub-menu The f unctions in the GO T O sub-men u are the f ollow ing: Goto L ine: to mo ve to a spec ifie d line. T he input fo rm pr ov ided w ith this command is: Goto P osition : mo ve to a spec ified positi on in the command line.
Pa g e L- 5.
Pa g e M - 1 Appendix M T able of Built-In Equations The E quation L ibrary consists o f 15 subj ects corr esponding to the s ections in the table belo w) and more than 100 titles. T he numbers in par e ntheses below indicate the n u mber of eq uations in the set and the number of v ariables in the set .
Pa g e M - 2 3: Fluids (29 , 2 9 ) 1: Pr essur e at D epth (1, 4) 3: F low w ith Lo sses (10, 17) 2 : Bernoulli E quation (10, 15) 4: Flo w in F ull P ipes (8 , 19) 4 : Forces and Energy ( 3 1 , 36) 1.
Pa g e M - 3 9: Op tics ( 1 1 , 1 4) 1: La w of Ref racti on (1, 4) 4: Spher ical Ref lection (3, 5 ) 2 : Criti cal Angle (1, 3) 5: Spheri cal Refr action (1, 5) 3: Bre wst er’s L aw (2 , 4) 6: Thin.
Pa g e N - 1 Appendix N Inde x A ABCUV 5-10 ABS 3-4, 4-6, 11-8 ACK 25-4 ACKALL 25-4 ACOS 3-6 ADD 8-9, 12-20 Additional character set D-1 ADDTMOD 5-11 Alarm functions 25-4 Alarms 25-2 ALG menu 5-3 Alge.
Pa g e N - 2 Bar plots 12-29 BASE menu 19-1 Base units 3-22 Beep 1-25 BEG 6-31 BEGIN 2-27 Bessel’s equation 16-52 Bessel’s functions 16-53 Best data fitting 18-13, 18-62 Best polynomial fitting 18.
Pa g e N - 3 Clock display 1-30 CMD 2-62 CMDS 2-25 CMPLX menus 4-5 CNCT 22-13 CNTR 12-48 Coefficient of variation 18-5 COL+ 10-19 COL 10-19 "Cold" calculator restart G-3 COLLECT 5-4 Colu.
Pa g e N - 4 Dates calculations 25-4 DBUG 21-35 DDAYS 25-3 Debugging programs 21-22 DEC 19-2 Decimal comma 1-22 Decimal numbers 19-4 decimal point 1-22 Decomposing a vector 9-11 Decomposing lists 8-2 .
Pa g e N - 5 DISTRIB 5-28 DIV 15-4 DIV2 5-10 DIV2MOD 5-11, 5-14 Divergence 15-4 DIVIS 5-9 DIVMOD 5-11, 5-14 DO construct 21-61 DOERR 21-64 DOLIST 8-11 DOMAIN 13-9 DOSUBS 8-11 DOT 9-11 Dot product 9-11.
Pa g e N - 6 ERRN 21-65 Error trapping in programming 21-64 Errors in hypothesis testing 18-36 Errors in programming 21-64 EULER 5-10 Euler constant 16-54 Euler equation 16-51 Euler formula 4-1 EVAL 2.
Pa g e N - 7 Function, table of values 12-17, 12-25 Functions, multi-variate 14-1 Fundamental theorem of algebra 6-7 G GAMMA 3-15 Gamma distribution 17-6 GAUSS 11-54 Gaussian elimination 11-14, 11-29 .
Pa g e N - 8 HELP 2-26 HERMITE 5-11, 5-18 HESS 15-2 Hessian matrix 15-2 HEX 3-2, 19-2 Hexadecimal numbers 19-7 Higher-order derivatives 13-13 Higher-order partial derivatives 14-3 HILBERT 10-14 Histog.
Pa g e N - 9 Integrals step-by-step 13-16 Integration by partial fractions 13-20 Integration by parts 13-19 Integration change of variable 13-19 Integration substitution 13-18 Integration techniques 1.
Pa g e N - 1 0 Left-shift functions B-5 LEGENDRE 5-11, 5-20 Legendre’s equation 16-51 Length units 3-19 LGCD 5-10 lim 13-2 Limits 13-1 LIN 5-5 LINE 12-44 Line editor commands L-1 Line editor propert.
Pa g e N - 1 1 Mass units 3-20 Math menu.. F-5 MATHS menu G-3, J-1 MATHS/CMPLX menu J -1 MATHS/CONSTANTS menu J-1 MATHS/HYPERBOLIC menu J-2 MATHS/INTEGER menu J-2 MATHS/MODULAR menu J-2 MATHS/POLYNOMI.
Pa g e N - 1 2 Multiple integrals 14-8 Multiple linear fitting 18-57 Multiple-Equation Solver 27-6 Multi-variate calculus 14-1 MULTMOD 5-11 N NDIST 17-10 NEG 4-6 Nested IF.
Pa g e N - 1 3 Partial fractions integration 13-20 Partial pivoting 11-34 PASTE 2-27 PCAR 11-45 PCOEF 5-11, 5-21 PDIM 22-20 Percentiles 18-14 PERIOD 2-37, 16-34 PERM 17-2 Permutation matrix 11-50, 11-.
Pa g e N - 1 4 17-6 Probability distributions discrete 17-4 Probability distributions for statistical inference 17-9 Probability mass function 17-4 Program branching 21-46 Program loops 21-53 Program-.
Pa g e N - 1 5 RCLMENU 20-1 RCWS 19-4 RDM 10-9 RDZ 17-3 RE 4-6 Real CAS mode C-6 Real numbers C-6 Real numbers vs. Integer numbers C-5 Real objects 2-1 Real part 4-1 RECT 4-3 REF.
Pa g e N - 1 6 SEARCH menu L-2 Selection tree in Equation Writer E-1 SEND 2-34 SEQ 8-11 Sequential programming 21-15 Series Fourier 16-26 Series Maclaurin 13-23 Series Taylor 13-23 Setting time and da.
Pa g e N - 1 7 Stiff differential equations 16-67 Stiff ODE 16-66 Stiff ODEs numerical solution 16-67 STOALARM 25-4 STOKEYS 20-6 STREAM 8-11 String 23-1 String concatenation 23-2 Student t distributio.
Pa g e N - 1 8 TINC 3-34 TITLE 7-14 TLINE 12-45, 22-20 TMENU 20-1 TOOL menu CASCMD 1-7 CLEAR 1-7 EDIT 1-7 HELP 1-7 PURGE 1-7 RCL 1-7 VIEW 1-7 TOOL menu 1-7 Total differential 14-5 TPAR 12-17 TRACE 11-.
Pa g e N - 1 9 Vector elements 9-7 Vector fields 15-1 Vector fields curl 15-5 Vector fields divergence 15-4 VECTOR menu 9-10 Vector potential 15-6 Vectors 9-1 Verbose CAS mode C-7 Verbose vs.
Pa g e N - 2 0 ! 17-2 % 3-12 %CH 3-12 %T 3-12 ARRY 9-6, 9-20 BEG L-1 COL 10-18 DATE 25-3 DIAG 10-12 END L-1 GROB 22-31 HMS 25-3 LCD 22-32 LIST 9-20 ROW 10-2.
Pa g e LW- 1 Limited W arranty HP 50g gr aphing calculator ; W arr anty period: 12 months 1. HP war rants to y ou, the end-us er cu stomer , that HP har dw a r e, access ori es and supplies w ill be fr ee fr om defects in mat er ials and w orkmanship after the dat e of pur chase , for the per iod s pecif ied abo ve .
Pa g e LW- 2 W ARRANTY S T A TEMENT ARE Y OUR SOLE AND E X CL USI VE REMED IE S. EX CEPT A S INDICA TED ABO VE , IN NO EVENT WILL HP OR I T S S UPPLIER S BE LIABLE FOR L O S S OF D A T A OR FOR DIRE C.
Pa g e LW- 3 Swi t ze r la n d +41-1-43 9 5 35 8 (Ger man) + 4 1 -2 2- 8 27878 0 ( F r e n c h ) +3 9-0 2 - 7 5 419 7 8 2 (Itali an) T urk ey +4 20 -5 - 414 2 25 2 3 UK +44 - 20 7 - 4 5 80161 Cz ech R.
Pa g e LW- 4 Regulatory infor mation Fe deral Communications Commission Notic e This eq uipment has bee n test ed and found t o compl y with the limits f or a Class B digital de vice , pursuant t o P art 15 of the FCC R ules.
Pa g e LW- 5 This de vi ce complies with P ar t 15 of the FCC Rules . Operati on is s ub ject to the follo wing two conditi ons: (1) this dev ice may not cause harmf ul interfer ence, and (2) this dev ice must accept an y interf er ence rece iv ed, incl uding interfer ence that may cau se undesir ed operation .
Pa g e LW- 6 This compliance is indi c ated b y the fo llow ing conformity marking placed on the pr oduc t: Japanese No tice 䈖 䈱ⵝ⟎䈲䇮 ᖱႎಣℂⵝ⟎╬㔚ᵄ㓚ኂ⥄ਥⷙද⼏ળ .
Ein wichtiger Punkt beim Kauf des Geräts HP (Hewlett-Packard) 50g (oder sogar vor seinem Kauf) ist das durchlesen seiner Bedienungsanleitung. Dies sollten wir wegen ein paar einfacher Gründe machen:
Wenn Sie HP (Hewlett-Packard) 50g noch nicht gekauft haben, ist jetzt ein guter Moment, um sich mit den grundliegenden Daten des Produkts bekannt zu machen. Schauen Sie zuerst die ersten Seiten der Anleitung durch, die Sie oben finden. Dort finden Sie die wichtigsten technischen Daten für HP (Hewlett-Packard) 50g - auf diese Weise prüfen Sie, ob das Gerät Ihren Wünschen entspricht. Wenn Sie tiefer in die Benutzeranleitung von HP (Hewlett-Packard) 50g reinschauen, lernen Sie alle zugänglichen Produktfunktionen kennen, sowie erhalten Informationen über die Nutzung. Die Informationen, die Sie über HP (Hewlett-Packard) 50g erhalten, werden Ihnen bestimmt bei der Kaufentscheidung helfen.
Wenn Sie aber schon HP (Hewlett-Packard) 50g besitzen, und noch keine Gelegenheit dazu hatten, die Bedienungsanleitung zu lesen, sollten Sie es aufgrund der oben beschriebenen Gründe machen. Sie erfahren dann, ob Sie die zugänglichen Funktionen richtig genutzt haben, aber auch, ob Sie keine Fehler begangen haben, die den Nutzungszeitraum von HP (Hewlett-Packard) 50g verkürzen könnten.
Jedoch ist die eine der wichtigsten Rollen, die eine Bedienungsanleitung für den Nutzer spielt, die Hilfe bei der Lösung von Problemen mit HP (Hewlett-Packard) 50g. Sie finden dort fast immer Troubleshooting, also die am häufigsten auftauchenden Störungen und Mängel bei HP (Hewlett-Packard) 50g gemeinsam mit Hinweisen bezüglich der Arten ihrer Lösung. Sogar wenn es Ihnen nicht gelingen sollte das Problem alleine zu bewältigen, die Anleitung zeigt Ihnen die weitere Vorgehensweise – den Kontakt zur Kundenberatung oder dem naheliegenden Service.
