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HP g gr aphing calc ulator user ’s guide H Ed it i on 1 HP part number F2 2 2 9AA-900 06.
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Pref ace Y o u hav e in your hands a compact s ymbolic and numer ical computer that w ill 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 matrix OPER menu) ,11-15 Function AXL ,11-16 Function AXM ,11-16 Function LCXM ,11-16 Solution of linear systems ,11-17 Using t.
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 ey .
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 OO L menu , are a ssoc iated with oper ations r elated to manipulation of v ariable s (see pa.
Pa g e 1 - 8 9 ke y the TIME 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 mn 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#@ s oft menu k ey to r eturn t o normal dis play . Examples o f se lecting 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 bu tton. S elect the RPN oper ating mode by either u sing 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 k ey to co mplete 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 a ppears in le ve l 1 Y 1/(3 × 3), last v alue in le v .
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 . Pr ess the !!@@OK#@ soft menu k ey 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 . T he number 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 number 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 . T he 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 . T he number 12 3 .45 6 7 8 9 012 , enter ed earlier , now 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 , f ollo w these st eps: Θ Pr ess the H bu tton.
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 may 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 tiv ate the CAL CULA T OR MODE S input fo rm .
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 MODE S input fo rm . Press the do wn ar r ow k ey , ˜ , tw ice , to get to the St ack line . This line sho ws tw o properties that can be modified . 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 r iter ) line. This line sho ws tw o pr operties that can be modifi ed. When these pr operties ar e select ed (chec k ed) the fo llow ing eff ect.
Pa g e 1 - 3 1 ri ght arr ow k ey ( ™ ) to s elect the underline in f r ont of the options _Cloc k or _Analog . T oggle the @ @CHK@@ s oft menu k ey until the de sir ed setting is ac hie 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 r iter and the manipulation of data ob jects 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 cursor is t o the right o f 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 cursor is t o the right o f 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 retur n 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 M ODE S input f orm (see Chapte r 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 unction name , or operation 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 ow ( — ) 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 ( — ) thr ee times to s e.
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 u r sor . T he scr een now looks lik e this: By using the left arr ow k ey ( š ) y ou can mov e the cur sor 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 ( ˜ ) , repeat edly , t o trigger the cl ear 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 ter m 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 soft menu k ey to e valuate 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 . T hen, pr ess the r ight arr ow 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 r eturn to the Eq uation W r iter , pr ess the @EXIT so f t 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 lef tm ost fa ctor in th e 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 , pres s ‚— 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_v alue , ending_value , summation e xpres sion ) Press ` to re turn to the E quation W riter .
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 ac tiv ate 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 o u can organi z e data in your calc ulator by stor ing var iables in a dir ectory tr ee .
Pa g e 2 - 3 4 @CHDIR : Change to s elected d ir e ct ory @CANCL : Cancel action @@OK@@ : Appr ov e a selecti on F or ex ample, to c hange directory to the CA SD IR, pr ess the do wn-arro w ke y , ˜ , and pre ss @CH DIR . This acti on close s the Fi l e M a n a g e r w indo w 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@@ soft m enu key or ` , to get the follo wing 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 sho ws one mor e 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@ . This sho ws the va lue of EP S to be .00 00000001 • T o see the value of a numer ical var iable , we need to pre ss onl y the so ft 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 algebr aic 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 ( ˜ ) to s elect the 5 . DIRE CT OR Y opti on, or ju st 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 w ith deleting th.
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 opti on. T hen, 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-direc tor y: 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 k ey (A lgebraic 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 , u se 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 o u w ill see six o f the sev en var iables listed at the bottom of the scr een: 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 o u w ill see six o f the se ven v ari ables list ed at the bottom of the s cr een: 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 message (try L @@@p1 @@ ` ): Note: By pre ss i ng @@@p1@@ ` we ar e trying to acti vate (run) the p1 pr ogram . Ho we ver , this progr a m e xpects a numeri 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 . F or e xample , in RPN, if w e 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@ ` (Algebr aic mode), or $ @INTRO@ (RPN mode) to re turn 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 ex ecute this command . Then , use ‚ @@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 no w show vari ab le p1 re m ove d : Y o u can use the P URGE command to er ase mor e than one var iable b y placing 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 h e UNDO command ( ‚¯ ) w ill 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 , mo st lik ely in a lis t.
Pa g e 2- 6 6 ` (keepi ng a second cop y in the RPN stac k) 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 s et flag 117 to s oft MENU . T he scr een will 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 • T he CMDS (CoMmanD S) menu , acti vated w ithin the E quation W r iter , 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 . Co ordinate s ystem 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 o lish Notation (RPN) mode s. Changing sign of a number , v ariable, or e xpression Use the ke y . In AL G mode , you can pr ess bef ore enter ing the number , e .
Pa g e 3 - 4 Alter nativ ely , in RPN mode , y ou can separat e the operands w ith 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 ke y . F or ex ample, 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 repr 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 def ault setting 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@ Select the HYP ERBOLIC.. menu @@TANH@ Select the TA N H fu nct ion 2.
Pa g e 3 - 1 2 Option 19 . MA TH.. r eturns the user to the MTH men u . T he r emaining func tions 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 ve l of the stac k, while ar gument x is located in the f irst le vel of the s tack . T his means, y ou should enter x firs t , 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 functi on Γ ( α ) P SI: N- th der iv ati ve o f the digamma functi on 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 , o r MAXR ) or a v alue ( 2 .71.., (0,1) , 3 . 14.., 1E-4 99 , 9. 9 9. . E 4 9 9 ) in the st ack . P lease notice that e is a vailable f r om the k eyboar d as ex p (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 = newtons, dyn = dyn es, gf = gr ams.
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 parentheses: 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 VISC OS ITY option @@@P@@ Se.
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 s yste m flag 117 is set to SO F T m en u s , then the UNI T 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 centi - 2 E ex a +18 m milli -3 P pe ta +15 μ mi cr o - 6 T ter a +12 n n ano - 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 .
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 A CT(x ,y): f actor s a unit y fr om unit objec t x UNIT(x ,y): combines v alue 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 when selec ted, constants v alues are sho wn in SI units ENGL w hen s.
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 , trigge r ed by u sing 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 A CT OR(x T , y P ) , w here x T is the reduced te mper ature , i . e ., the rati o of 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) calc ulates T 0 +D T . The oper ation of this f unction is similar to that of f uncti on TDEL T A in the se nse that it r eturns a r esult 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 yo u will noti ce that there is a ne w var iable in y our soft 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 nu mber wr itten as z = x + iy , w here 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 k e 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 ke y , e .g., -(5-3 i) = -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 last 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 same pur pose as the k ey strok e combinati on „¥ , 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 re sult to:.
Pa g e 5 - 1 Chapter 5 Algebraic and arithmetic operations An algebr aic obj ect , or simply , algebr aic , is any number , v ari 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 , h yper bolic, 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.
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 AR 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 , sho wn abo ve , is us ed to substitute a v ariable in an e xpressi on. 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 ar e also contained in the AL G menu pr esented earli er . Func tions 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, M ODUL O, and PERMUT A T I.
Pa g e 5 - 1 1 F A CT OR Fact ori z es an integer number or a poly nomial FCOEF Gener ates fr action gi ven r oots and multipli city FROO T S Retur ns root s and multiplic ity giv en a fr action GCD G.
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 arithmetic is , in essence, the integer r emainder of j ⋅ k / n in infinite ar ithmetic , 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 “ congr uence” s ymbol 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 o u can also conv er t an y number into a r i ng number b y using the func tion EXP ANDM OD . For e xample , EXP AN DMO D(1 2 5) ≡ 5 (mod 12) EXP AN DMOD (17 ) .
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 deri vati ve w ith res pect 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 calculat or’s CA S cr eates a v ari able EP S, w ith default 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 A ND a nd F A CT OR, fr om the AL G menu (‚×) . F or ex ample: EXP A ND(‘(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 fract.
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 ) . Thi s menu summar iz es all con ver sion menus in the calc ulator . T he list of thes e menus 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 effect a s the ke ys tr ok e combinati on ‚ï (assoc iated w ith the ` key) . Fun ct io n NU M conve r ts a symbo lic res ul t i nt o 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 e y 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 n o soluti ons { }. In the second 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 VE VX f ound a single solu tion , X = 2 . The foll owing screen s sh ow th e R PN sta ck for solvin g t.
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 8 9) , (-0.7 66 , 0.6 3 2) , (-0.7 6 6 , -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 gger the line editor to see all the coeff ic ients. Generating an algebraic expr ession for the poly nomial Y o u can use the calc ulator to gener ate an algebr aic e x pr 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 A ND command. T he resul ting 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@@ . T hen, pr ess @@AMOR@@ . Y ou w ill get the f ollo wing 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 we w ill be sol v ing for it 0 @@OK@@ Enter FV = 0, the opti on End is highlight ed @@CHOOS ! — @@OK@@ Change pa yment opti on to Begin — š @@SOLVE! H ighlight P MT and sol ve f or it The s cr een now sho ws the v alue of P MT as –38 , 9 2 1.
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@@ . En ter name of v ar 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 wl y cr eated E Q vari able: Then , enter the SOL VE en vir onm ent and select Solv e equation… , by using: ‚Ï @@OK@@ .
Pa g e 6 - 1 5 This , ho we ver , is not the only pos sible soluti on for this equation . 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 , ar e the normal str esses on the particle 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 for ex : The s oluti on can be seen fr om within the S OL VE E QUA T ION input f orm by pr essing @EDI T whil e th e ex : field is hi ghlighted. 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 o u may w ant to creat e a separat e sub-dir ectory (PIP E S) to tr y 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 - 22 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 E QU A TION in put fo rm: 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 pre ss to r eturn to normal calc ulator display . The soluti on 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 - 26 The S OL VE so ft 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). Alter nativ ely , 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 - 28 Y o u can also solv e more than one equation b y 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 su b-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 , g ive n a ve ct o r o f i t s roo t s [r 1 , r 2 , …, r n ].
Pa g e 6 - 3 1 Press J to ex it the S OL VR en vir onment . Find y our wa y back 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 po int (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 decompo se the vec tor , use f unction 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 entr y as sho wn abo ve . Notice that f uncti on MSL V r equir es three argume nts: 1. A v ector cont aining the equati ons, i .
Pa g e 7- 6 disc harge (m 3 /s or 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 SI , C u = 1.4 8 6 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@@ . The equati ons ar e listed in the stac k 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, C u, 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 r een will look lik e this: Press ` to solv e the sy stem 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*C OS( 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 cor ner of the display . 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 sol ve f or a ll the v ariable s, te mpor aril y show ing the intermediate re sults. Press ‚ @@ALL@@ to see t he sol utions: When done , pres s $ to retur n to the MES en vir onment. Pr ess J to e xit 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 , al l 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 stor ing the name of the multiple equati on solv er , namely , "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 have no w 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 times . Changing sign The si gn -change k ey ( ) , whe n applied to a lis t of number s, w ill change 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 E XP 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 i.
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),%(3 0, 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 incr ement among consecu tiv 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 ogr amming.
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 men u . The f ollow ing call to func tion MAP us es a pr ogra.
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 vi de the r esult abov e b y n = 10: 4. A pply the INV() functi on to the latest r esult: Thu s, the harmonic mean of lis t S is s h = 1.6 34 8… 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. 003 20 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 y.
Pa g e 8 - 1 8 3. U se f u n ct i on Σ LIS T , once more , to calc ulate the denominator of s w : 4. Use the expr essi on ANS( 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 follow 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 whose 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 ke y is used to dec r ease the w idth 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 ey is used to inc rease 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 selec ted 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 , a vailable in the f unction catalog ( ‚N‚é , us e —˜ 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 elem ents, number of elements ), e.
Pa g e 9 - 7 In RPN mode, the f unction [ → ARR Y] tak es the objec ts 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 unctio n PUT (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 , pr oduces the magnitude of the vec tor . F or a vector 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 , are 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 , o r Cartesian , coordinate s yst em is select ed, 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 ) cos( θ ), y = ρ sin ( φ ) cos ( θ ), z = ρ cos( φ ). F or this cas e , x = 3 .204 , y = 1.4 9 4 , and z = 3.
Pa g e 9 - 1 5 equi valent (r , θ ,z) with r = ρ sin φ , θ = θ , z = ρ cos φ . F or ex ample, the f ollo wi ng 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 , while 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 ollo wing oper ation (angular measur e set to degr ees) in AL G mode: 1 - Enter vect ors [3,-5, 6], press ` , [2 ,1,-3], pres s ` .
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 suc h that | M | = | r || F |sin( θ ) , w here 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) – AN S(2), i.e ., F inally , w e tak e the dot pr oduct of AN S(1) and ANS( 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 r o w vec tor into a co lumn vect or , a lis t into a vect or , and a v ector (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 stac k lev el 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 (also 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 ectly tr ansfor m a column v ector to a r ow v ector . In RPN mode , enter the column vec tor , and then pre ss @@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@@ t o see the pr ogram con tained 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 matri x will look lik e the one sho wn abo ve . Other w ise, 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 TR ICE S/CREA TE me nu av ailable thr ough „Ø : The MTH/MA TR IX/MAKE sub menu (let’s call it the MAKE menu) contains the fo llo w ing func tio ns: while the MA TR ICE.
Pa g e 1 0 - 5 As yo u can see f rom e xploring these men us (MAKE and CREA TE), the y both hav e the same functi ons GET , GE TI , PUT , P U T I, S UB, REPL , RDM, R ANM, HILBERT , V A NDERMONDE , IDN, CON, → DIA G , and DIA G → .
Pa g e 1 0 - 6 Functions GET and P UT F unctions GET , GETI , PUT , and P UTI, ope rate w ith matrice s 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 c alc ulated 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 above .
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 ND ERMONDE . 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 pr ogram 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 n umber , 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 eff ect of the func tio 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 xtracted r ow sh o 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 . Addition 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 , wher 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 , y ou must 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 k ey str 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 r obenius nor m of the matr ix is de fined 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 o u w ill find that the r a nk is 2 . T hat is because the second r o w [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 epr esented b y the three equati ons.
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 , pres s the @EDIT! button . This 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 sol ver env ironment . T o check that the soluti on is correc t, try the follo wing: • Pr ess —— , to highlight the A: field . • Pr ess L @CALC@ ` , to cop y matri x A onto the stack. • Pr ess @@@OK@@@ to r eturn to the numer ical solv er env ir onment .
P age 11-2 5 • If A is a squar e matri x and A is non -singular (i .e ., it’s inv erse 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 = 8 5, 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 matri x A with a ne w ro w , 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 ll.
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! .
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 m at rix s h own ab 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 il.
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 .
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 , w hile keep ing 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 elements o f C can be wr itten 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 , rr ef , and RREF , work w ith 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 E che lon 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 sy stem of n linear equati ons 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 wr ite the eigen 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 . F or each value o f λ , w e can find , fr om the same equation , values of x that satisfy the ei genvalue equati on.
Pa g e 1 1 - 4 6 Using the var iable λ to r eprese nt eigen 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 lis t with the e igen vect ors cor r espo nding to each ei genv alue of matr ix A (stack lev el 2) • A v ector w ith the eige nv ector s of matr i x A (stack le ve l 4) F or ex amp.
P age 11-4 9 Notice that the equati on ( x ⋅ I - A ) ⋅ p( x )=m( x ) ⋅ I is similar , i n for m, 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 lo wer -triangular matri x L , an upper tr iangular matri x U , and a perm utation matri x P , in stac k lev els 3, 2 , and 1, respec ti ve ly . T he 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 , GA U S S, QX A, and S YL VE S TER. 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 le vel 2 using the n var iables in a v ector placed in stack lev el 1.
P age 11-54 suc h that x = P ⋅ y , b y 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 PL O T en vir onment by pr essing „ñ (pr ess th em simultaneou sly if in RPN mode). Pr ess @ADD to get y ou into the equati on wr iter . Y o u will be pr ompted to fill the ri ght -hand side of an equati on Y1(x) = .
Pa g e 1 2 - 4 Θ Enter the PL O T WINDO W env ir onment b y enter ing „ò (pr ess them simultaneously if in RPN mode). Use a r ange of –4 to 4 for H- VIEW , then pres s @AUTO 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 Pres sing @ ISECT w ill giv e y ou the intersecti on of the curve w ith the x-ax is, 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 return to r eturn 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 @CAN CL . 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 -FUNCTION 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 var iables 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 e xp(x) functi ons are in ver se of each 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 SE TUP , 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 PL O T SE TUP windo w and re turn to nor mal calc ulator displa y . Θ P r ess @@@OK@@@ to save changes to the options in the PL O T SETUP window and r eturn t o normal calc ulator display .
Pa g e 1 2 - 1 5 Θ Enter lo wer and u pper limits f or hor i zo ntal v ie w (H- Vi ew), 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 ) , pr essed simultaneously 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 follow s: 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 P r ess L @CANCL to re tu rn t o t he PL OT W IN DOW scree n. P ress 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 equati ons fo r 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 reco ver the menu: LL @) PICT Θ T o estimate the coor dinates of the point of in tersection , press 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 s tor 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 ca.
Pa g e 1 2 - 24 Θ P r ess @AUTO . This will gener ate automatic v alues of the H-Vi ew and V- Vie w 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 P r ess @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 o u can stor e differ ent values in the var iables and pr oduce new par ametri c 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 P r ess L to r ec o ver the menu . Press L @) PICT to r ecov er the original graphi cs menu. Θ When we ob served the gr aph being plotted, yo u'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 P r ess „ô , simult aneousl y if in RPN mode , to access to the P L O T SETUP wi nd ow . Θ P r ess ˜ and type ‘(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 P r ess @E RASE @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 (av ailable in the func tion catalog, ‚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 av ailable in the stac 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 r ix: Θ P r ess „ô , simult aneousl y if in RPN mode , to access to the P L O T SETUP wi nd ow .
Pa g e 1 2- 32 Θ P r ess @ERASE @DRAW t o dr a w the bar plot . Pr ess @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 P r ess LL @) PICT to lea ve the EDIT e nv iro nment .
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 P r ess @ERASE @DRAW t o dr aw the thr ee -dimensional surf ace . 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 or ientation of the surf ace.
Pa g e 1 2 - 36 Θ P r ess „ô , simultaneou sly if in RPN mode , to access the PL O T SE TUP wi nd ow . Θ P r ess ˜ and t y pe ‘SIN(X^2+Y^2)’ @@@OK@@@ . Θ P r ess @ERASE @DRAW to dr aw the plot. Θ When done, pr ess @ EXIT . Θ P r ess @CANCL to retur n to PL O T WINDO W .
Pa g e 1 2 - 37 Θ P r ess @EDIT L @LABEL @MENU t o see the gr aph with la bels and r anges . 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 P r ess „ô , simultaneou sly if in RPN mode , to access the PL O T SE TUP wi nd ow . Θ P r ess ˜ and t ype ‘X^2+Y^2’ @@@OK@@@ . Θ P r ess @ERASE @DRAW to dr aw the slope f ield plot .
Pa g e 1 2 - 3 9 Θ P r ess @EDIT ! L @LABEL @MENU to see the gr aph w ith labels and r anges. Θ P r ess LL @) PICT@CANCL to r eturn to the PL O T WINDOW env ironment . Θ P r ess $ , or L @@@OK@@@ , to retur n to normal calc ulator display . 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 select ed as the Indep: and ‘Y ’ as the Depnd: varia bl es. Θ P r ess L @@@O K@@@ to r eturn to normal calc ulator display . Θ P r ess „ò , simultaneousl y if in RPN mode , to access the P L O T WINDO W scr e en .
Pa g e 1 2 - 4 1 Θ P r ess „ô , simult aneousl y if in RPN mode , to access to the PL O T SETUP w indow . Θ Ch ang e TYPE to Gr idmap . Θ P r ess ˜ and t ype ‘S IN(X+i*Y)’ @@@OK@@@ . Θ Make sur e that ‘X’ is select ed as the Indep: and ‘Y ’ as the Depnd: varia bl es.
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, use the follo wing: Θ P r ess „ô , simult aneousl y if in RPN mode , to access to the PL 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 c ursor about the gr aphics s cr 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 o u can alw ay s return to the v er y 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) Ch . 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) sho ws the sign o f the func tion 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 marks indicates uncer tainty or non -definition. F or 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 , pr ess the u p-arr ow k ey — , 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 r iter 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 o u can use the calc ulator to calc ulate der i vati ves of eq uations , i .e., e xpre ssions 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 @PIC T @CANCL $ t o r eturn 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 thr ough the command catalog ( ‚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 w o lists, the f irst one indicates the v ariati on of the functi on (i .e., w here 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 maxim um equal to 36 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-de ri vati ve 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 , integrand , var iable 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 Chapter 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 , w e pr oceed as follo 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) repr esen 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 ty picall 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 o l l owi n g e xa m p l e. T h e s cr e e n s h o ts 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 , for this c riti 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 luded 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 gr adient o f the func tion w ith re spect t o 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. The calcula tor response in th is case is shown below : Div ergence The di ver gence of a vecto.
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 b y 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 mess age back f r om functi on PO TENT IAL. T o v erify 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 las t command in the scr een shot show s that indeed F = ∇× Φ 2 . Thu s, a v ector potential f unction is not uniquel y determined .
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 select ed. Pr ess ˜ to see the equation in the E quatio n 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 V E: Differ ential E quati on S OL VEr , pr o vi des .
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 x p(x). In the calculator u se: ‘ d1d1y(x)+x*d 1y(x) = EXP(x) ’ ` ‘ y(x) ’ ` DESO LVE The r esult is an e .
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 re turn the r esult: ‘ILAP(SIN(X))’ , meaning that ther e is no clos ed-form e xpres sion f(t), such that f(t) = L -1 {sin(s)}.
Pa g e 1 6 - 1 3 Θ Differ entiati on theore m for the n-th deri vati v e . 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 Linear it y theor em . L{af(t)+bg(t)} = a ⋅ L{f(t)} + b ⋅ L{g(t)}.
Pa g e 1 6 - 1 4 Θ Shift theorem f or a shift to the ri ght . Let F(s) = L{f(t)}, then L{f(t-a)}=e –as ⋅ L{f(t)} = e –as ⋅ F(s) . Θ Shift theorem f or a shif t to the left . Le t F(s) = L{f(t)}, and a >0, then Θ Similarity 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 o u can pr o 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 Obt ains the inv erse La place transf orm 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 , wher e 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) ’ ` L AP , the calculator pr oduces 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 presents 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 ILAP , is not allow ed in the calc ulator . Y o u hav e to use the main r esults pro vided ear lier 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 peri odic 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-f orm o f the F ourier 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 .
Pa g e 1 6 - 3 0 The f itting is somew hat acceptable for 0<t<2 , alt hough 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 entr y 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… . T he actual value o f the func tion 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 - 3 5 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 - 3 6 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 , namely , Once again, r eplacing e in π = (-1) n , and using 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 - 4 2 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 r efer red to as the spectr um of the f unction and w ill 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 ) are continuous f unctions 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 , namel y , 1-i ω . The result is no w: 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 partitio ning it into a number o f shorter 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 a vailable in the command catalog , ‚N ).
Pa g e 1 6 - 5 0 Example 2 – T o pr oduce the signal gi ven the s pectrum, w e 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 ’.
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 f unction Gamma Γ ( α ) is defined in Chapter 3. If ν = n, an int eger , the Bes sel functi ons of the f irst kind for n = intege r ar e def in.
Pa g e 1 6 - 5 4 Y ν (x) = [J ν (x) cos 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 r epr 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 fo rm 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. 1 6666 …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, f.
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 (wai t) @EDIT (Changes initial v alue of t to 0.5, and f inal value of t to 0.7 5, s olv e for v(0.7 5 ) = 2 . 066…) @@OK@@ @INIT+ — 1 @@OK@@ ™ ™ @SO LVE (wai t) @EDIT (Changes initial v alue of t to 0.
Pa g e 1 6 - 6 0 Θ „ô (simultaneously , if in RPN mode) to ente r PL O T en vir onment Θ Highligh t the fi eld in fr ont of TYPE , using the —˜ k ey s.
Pa g e 1 6 - 6 1 LL @) PICT T o rec ove r m en u an d re tu rn t o PI CT e nvi ron m en t. @ ( X,Y ) @ T o determine coor dinates of an y point on the gr aph . Use the š™ k eys to mov e th e cur sor around the p lot area . At the bot tom 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 - 6 3 (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 r eturn to normal calc ulator display , 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 SE TUP by pr essing L @@OK@@ .
Pa g e 1 6 - 65 Press LL @PICT @C ANCL $ to r etur n to nor mal calc ulator displ ay . 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 o u can check that a so lution tak 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 s oft menu is ac tiv ated by u sing 7 4 MENU in RPN mode . This menu is pre sented in detail in Cha pter 6 .
Pa g e 1 6 - 6 8 The value o f the solu tion , y fin a l , w i ll be a vailable in v ari able @@@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 - 6 9 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. 340 4 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 s.
Pa g e 1 6 - 7 1 The se r esults indicate that ( Δ x) ne xt = 0. 005 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 ..., f or 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-menu - par t 1 The MTH/P ROB ABI LI TY .. su b-menu is accessible thr ough the k ey str oke s equence „´ .
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 ) , fo r 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 var iance of the dis tributi on. T o calc ulate the val ue of f( μ , σ 2 ,x) for the nor mal distr ibution , use func tion NDIS T w ith the follo w ing arguments: the mean , μ , the v ari ance, σ 2 , and, the v alue x , i .
Pa g e 1 7- 1 1 wher e Γ ( α ) = ( α -1)! is the G AMM A func tion defined in Cha pter 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 , gi ven the par ameter ν and the value 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 -distr ibution usi ng [UTPC] gi ven the v alue 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 paramet ers ν N and ν D, and t he value of F . T h e definition of this function is, theref 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 x pr essions 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 o u will o nly get the mes sage 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 o u can us e , 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 D A T use f unction S T O Σ , a vaila ble 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 value that splits the dat a set in the middle 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 , thr ough the pr e -pr ogrammed 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 of n data v alues: {x 1 , x 2 , …, x n } listed in no partic ular or.
P age 18-7 Θ Generate the list of 200 numbe r by u sing RDLIS T(200) in AL G mode , or 200 ` @ RDLIST@ in RPN mode . Θ Use pr ogram LX C (see abov e) to conv ert the list thus gener ated into a column vec tor . Θ Stor e the column vector 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 cumulati ve fr equency is obtained b.
P age 18-9 « DUP S IZE 1 GET f req k « {k 1} 0 CON cfr eq « ‘freq(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 NE X T cfr eq » » » Sa ve it unde r the name CFREQ.
P age 18-10 Θ P r ess @CANCEL to r eturn to the pr ev ious 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 , vs .
P age 18-11 Θ Fir st , enter the two r ow s of data into column in the v ari able Σ DA T b y us i n g the matri x wr iter , and f unction S T O Σ . Θ T o access the progr am 3. Fit data.. , us e the follo w ing k ey strok es: ‚Ù˜˜ @@@OK@@@ The input f orm w ill show the c urr ent Σ DA T , already loaded.
P age 18-12 Wher e s x , s y ar e the standar d dev iations of x and y , resp ecti vel y , i .e . The va lu es s xy and r xy are the "C ovar iance" and "Corr elation ," respec tiv ely , 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 -C ol: these options appl y only whe n yo u have mor e than t w o columns in the matr ix Σ D A T . B y def ault, the x co lumn is column 1, and the y column is co lumn 2 .
P age 18-15 B. If n ⋅ p is an integer , s ay k, calc ulate 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 RPN.
P age 18-16 The D A T A sub-menu The D A T A su b-menu contains f unctions used t o manipulate the statis tics matri x 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 tr ix. Σ - : r emo ve s last r ow in Σ D A T A matr ix and places it in lev el of 1 of the s tac k.
P age 18-17 Σ P AR: 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 O GFIT , E XPFIT , P WRFIT or BE S TFIT 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 ov ides the sum of s quar es of values in Xcol column . 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 oducts of data in columns Xcol and Ycol. N Σ : pro vi des the number of column s in the Σ DAT A m a t rix.
P age 18-20 @) STAT @ ) £PAR @RESET re sets statis tical par ameters L @) STAT @PLOT @SCA TR pr oduce s scatter plot @STATL dr aws data f it as a strai ght line @CANCL r eturns to main display Θ Determine the f itting equati on and some of its s tatisti cs: @) STAT @ ) FIT@ @£LINE produces '1.
P age 18-21 Θ Fit dat a using columns 1 (x) and 3 (y) using a logar ithmic f it ting: L @) STAT @ ) £PAR 3 @YCOL select Ycol = 3, and @) MODL @ LOGFI select Model = Logfit L @) STAT @PLOT @ SCATR pr oduce scatter 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 pr oduce scatter gram o f y vs. x @STATL sho w line for log f itting Θ T o return to S T A T menu use: L @) STAT Θ T o get your v ari able 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 oint es timation: w hen a single value o f the par ameter θ is pro vided . Θ Conf idence interval: a numer ical interval that contains the par ameter θ at a giv en leve l of pr obability . Θ Estimato r: r ule or method of estimati on of the parameter θ .
P age 18-2 4 Θ The parameter α is know n as the signif icance le vel . T y pical v alues 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 gi v en b y: 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 , ar e giv en by : 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 -I NT : 1 μ .: Single sample conf idence interval f or the population 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 sc reen e xpla ining the meaning of the conf idence interval in terms o f random number s generated b y a calculator . T o s cr oll dow n the r esulting sc r een use the do wn-arr ow k ey ˜ . Pres s @@@OK@@@ whe n done with 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 = 4 5 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 standar d 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 heref ore , [(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 y pe I and T y pe I I 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 y pe 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 Re je ct H o if P -value < 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 u.
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 Re je ct H o if P -value < 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 s .
P age 18-4 2 wher e Φ (z) is the c umulativ e distributi on func tion (CDF) o f the standard nor mal 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 , fr 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) o f the standard nor mal distr ibution .
P age 18-44 1. Z - T est: 1 μ .: Single sample h ypothesis te sting f or the population 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 tes t z value is z 0 = 5. 656854. T he P- va l u e i s 1. 54 × 10 -8 . Th e crit ica l va l ues of ± z α /2 = ± 1.9 5 99 64 , corr esponding to c ritical ⎯ x range o f {14 7 .
P age 18-46 W e re ject the null h ypothe sis, H 0 : μ 0 = 15 0, against the alter nativ e hy pothesis , H 1 : μ > 150. T he test t va lue is t 0 = 5 .6 5 68 54 , w ith a P -value = 0. 0000003 9 35 2 5 . The c riti cal value 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 , w e do not r ejec t) the hy pothesis: H 0 : μ 1 −μ 2 = 0 , or H 0 : μ 1 =μ 2 , against the alter nati 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 Re je ct H o if P -value < 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 6/0.25=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 8 8… Since 0.17 8 8… > 0.0 5, i .e ., P -value > α , ther efor e , we cannot r eject 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 . There is , ho we ver , no need to bother w ith these calc ulations because y ou 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 Confi d ence limits f or r egr essi on coeffi ci ents: 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 o.
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 2 02 2 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 want 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 for m 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 ere to u se 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 E nter lists x and y , and p (le vels 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 P lace x in sta.
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 have to ty pe them all o ver 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 range –1 < r < 1. The clo ser 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 st ack , obtain V n IF ‘ p<n -1’ THEN T his IF is step 3 in algor ithm n P lace n in stac k p 2 + C alculate p+1 FOR j Start loop, j = n-1 to p+1, step = .
P age 18-65 “SSE” T A G T ag r esult as S SE » Close sub-pr ogram 4 » Close sub-pr ogram 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 Y R , to emphasi z e calculati on 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 YTE 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 men u, a vaila ble thr ough the B ASE ( ‚ã ) pr ov ides the f ollow ing 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 YTE menu The B Y TE menu , av ailable thr ough the BA SE ( ‚ã ) pr ov ides the fo llo w ing 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 def inition of the se fu ncti ons ar e sho wn belo w: RLB: Rotate L eft 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 ncti on RCLMENU 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 o u can try using this list wi th 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 ro w 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: Assi gns an object to a k ey spec ifie d by XY .Z S T OK E Y S: Stores user -defined key list RCL KEYS: Ret urn s curren t use r-defi ne d key li st.
Pa g e 2 0 - 7 Operating user-defined ke ys T o operate this us er -defined k ey , enter „Ì bef ore pr essing 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 . T he 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 ), / / the n div ide ['] ~„x™ 'x' „° @) @MEM@@ @ ) @DIR@@ @ PURGE PURGE Purg e va riab 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 : ST AR T -NEXT-S TEP constru ct f or br anching FOR: FOR-NE XT- STEP constr uct for loops DO: DO-UNT IL -END constru ct f or loops WHILE: WHILE -REPEA T -END cons truc t f or loops.
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 BR CH/IF BRCH/WHILE TYP E DUP P UR GE IF WHILE.
P age 21-8 LIST/ELEM GROB CHARS MODES/FLAG MO DES/MISC GET GROB SU B SF BEEP GET I BLANK REPL CF CLK PUT GO R POS F S? S Y M PUTI GX OR SIZ E FC ? S T K SI ZE S UB NUM F S?C ARG PO S REPL CHR F S?.
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 ( ≠ , ≤ , <, ≥ , >) are a vailable in the k eyboar d.
P age 21-10 „ @) @IF@@ „ @CASE@ „ @) @IF@@ „ @CASE@ „ @) START „ @) @FOR@ „ @) START „ @) @FOR@ „ @)@@DO@@ „ @WHILE Notice that the inse rt prompt ( ) is a vaila ble after the k ey w ord f or each 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 @S WAP@ DRO P „° @) STACK @DROP@ @) @MEM@@ @ ) @DIR@@ PUR GE „° @) @MEM@@ @ ) @DIR@ @ @PURGE ORDER „° @) @MEM @@ @ ) @DI.
P age 21-12 @) @BRCH@ @ ) WHILE@ WHILE „° @) @BRCH@ @ ) WHILE@ @WHILE REPE A T „° ) @BRCH@ @ ) WHILE@ @REP EA END „° ) @BRCH@ @ ) WHILE@ @@ END@ @) TEST@ == „° @) TEST@ @@@ ≠ @@@ AND „.
P age 21-13 @) LIST@ @ ) PRO C@ REVLI S T „° @) LIST@ @ ) PROC@ @REVL I@ SO RT „° @) LIST@ @ ) PROC@ L @SORT@ SEQ „° @) LIST@ @ ) 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 ST , et c., a vaila ble throug h 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 for units of the International S ys tem (S.I .) , and C u = 1.
P age 21-17 Y o u can also separ ate the input 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 ↓ @ Result: e mpt y stac k, e xec uting → a @SST ↓ @ Result: empty stac k, ente ring subpr ogram « @SST ↓ @ Re sult: ‘2*a^2+3’ @SST ↓ @ Result: ‘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’ . L et’s edit the pr ogram 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 9 88 7 . 06_J/m^3 . The units of J/m^3 are equi valent to P ascals (P a), the pre fer red pr essur 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 stack w ill look like 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 for m at info rmati on: { } (an empty list , thus , defa ult value s used) 4. List of reset values: { 120 1 .0001} 5 . Lis t of initial v alues: { 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:” “Chezy’s coe fficient” 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 , t o produce a t agged 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 , run 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 o u could use a me ssage bo x for outpu t fr om a progr am by using 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 o u wi ll notice that after typ ing the ke ystr ok 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 input value s may be a tedi ous proce ss.
P age 21-4 2 2. ‘ 1_m^3 ’ : The S .I. units cor r esponding to V ar e then placed in stac k lev el 1, the tagged input f or V is mo ved to stack lev el 2 . 3 . * : By multipl y 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 accessed 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 y pe 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 ’ . Pr ess J and ver if y that v ari 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 passes 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@ Re s ul t : - 0.6 312 66… (i .e., sin(x), with x in r adians) 12 @@f3c@ Re su l t : 16 2 7 5 4.7 91419 (i.e ., exp(x)) 23 @@f3c@ Re s ul t - 2 . (i.e ., - 2) As yo u can see, f3c pr oduces ex actly the same r esults as f3.
P age 21-54 Commands in volv ed in the ST AR T constru ct ar e available thr ough: „° @) @BRCH@ @ ) START @START Within the BRCH men u ( „° @) @BRCH@ ) the follo wing k ey str ok es are a vaila .
Pa g e 2 1 - 5 5 1. This pr ogr am needs an integer 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 su bpr ogr am) @SST ↓ @ SL1 = 0.
P age 21-5 7 @SST ↓ @ SL1 = 1. (S + k 2 ) [Sto re s value of SL2 = 2 , into SL1 = ‘k ’] @SST ↓ @ SL1 = ‘S’ , SL2 = 1. (S + k 2 ) @SST ↓ @ Empty stac k [St or es value o f SL2 = 1, into SL1 = ‘S’] @SST ↓ @ Empty stack (NE X T – end of loop) --- loop e xec ution nu mber 3 f or k = 2 @SST ↓ @ SL1 = 2 .
P age 21-5 8 3 @@@S1@@ Resul t: S:14 4 @@@S1@@ Res ul t: S:30 5 @@@S1@@ Resul t: S:55 8 @@@S1@@ Res ul t: S:204 10 @@@S1@@ Resu lt : S:385 20 @@@S1@@ Res ul t: S:2870 30 @@@S1 @@ Res ul t: S:9455 100 .
P age 21-5 9 J 1 # 1.5 # 0.5 ` E nter parameters 1 1. 5 0.5 [ ‘ ] @GLIST ` En ter the progr am name in leve l 1 „°LL @) @RUN@ @ @DBG@ St art the debugger . Use @SST ↓ @ to step into the pr ogram and see the detailed ope rati on of each 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 y pe in the progr am: « → 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 Check 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 Check out that the pr ogram call 0. 5 ` 2. 5 ` 0. 5 ` @ GLIS4 pr oduces the list {0. 5 1. 1.5 2 . 2 . 5}. Θ T o see step-by-step oper ation use the pr ogram DBUG fo r a short list, f or e xample: J 1 # 1.
P age 21-65 If y ou enter “ TR Y A GAIN” ` @DOER R , pr oduces the following message: TR Y AGA I N F inally , 0` @ DOERR , pr oduces the messa ge: In terrupted ERRN This f unction r eturns a number r epres enting the most r ecent err or . F or 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 w ithin PL O T The S T A T menu pr ov ides access t o plots re lated to st atistical anal ysis . Within 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 .
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 more than one , w ill be the x - column or independent var iable column. YC O L ( I ) The command Y C OL is us ed to indicate w hich of the columns o f Σ DA T , i f mo re than one , w ill be the y- column or dependent v ari able column.
Pag e 22- 1 4 Θ S IMU: w hen selec ted, and if mor e 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 , opti ons Slopef ield, Wir efr ame , Y -Slice , P s -Co ntour , Gr idmap and Pr- Sur face , use the V.
Pag e 22- 1 6 @) PPAR Show plot par ameters ~„r` @INDEP D ef i ne ‘ r’ as the indep . vari able ~„s` @DEPND D efine ‘ s ’ as the dependent v ari able 1 # 10 @XRNG De fine (- 1, 10) as the x -r ange 1 # 5 @YRNG L De fine (-1, 5) as the y-r ange { (0, 0) {.
Pag e 22- 1 7 @) PPAR Show plot par ameters { θ 0 6.2 9} ` @INDEP Def ine ‘ θ ’ as the indep. V ari ab le ~y` @DEPND Def ine ‘Y ’ as the depe ndent v ariable 3 # 3 @XRNG Def ine (-3, 3) as the x -range 0. 5 # 2.5 @YRNG L D ef ine (-0. 5,2 . 5) as the y-range { (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 Store ‘ √ r’ into EQ ‘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, pur ge vars .
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 var iable name , how 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 . –18 0. 180. ARC Dra w a c i r cle cente r (5 0,50), r= 12 . 1 8 FOR j D ra w 8 lines w ithin the cir cle (50., 5 0.) D UP L ines ar e centered as (5 0,5 0) ‘12*COS( .
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 o u could call the sub-dir ectory RIVER , since we ar e dealing w ith irr egular 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 Data set 2 xy x y 0.
Pag e 22- 26 (x min , y max ) . The max imum coordinate s in terms of pi xels cor r espond 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 coor dinates of the two other corners 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 o u can generate a gr aph in the gra phics sc r een by u sing the commands in the PL O T and PICT menu s.
Pag e 22- 28 ANIMA TE is a vailable b y using „°L @) GROB L @ANIMA ) . T he animation will be r e -started. Pr ess $ to stop the animati on once mor e. Noti ce 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 @ GRO B . Y o u w ill no w have 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’ , produ ces the so -called princ ipal stresses , 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 . T hese sub-pr ograms ar e then link ed by 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 hen type the list corr esponding 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 screen , s ho w PICT in graphics screen @TRACE @ ( x,y ) @ . T o mov e curs or ov er the cir cle show 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 . F or e xample , the str ing “SINE FUNCTION” , can be transf ormed into a GR OB (Gr aphics Ob jec t) , to labe l 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 also seen the f u ncti ons SUB and REP L in r elation t o gr aphics 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 . F lags ar e var iable s that can be used to contr ol the calculat or properties.
Pa g e 24 - 2 Number T y pe Ex ample _______________ ____________________ _____________________ ____________ 21 Extended R eal Number Long Real 2 2 Extended Comple x Number L ong Complex 2 3 Link ed A.
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 C lear a flag 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 lace s cur rent date in the st ack D A TE: Set sy stem date to specif ied value TIME: Places c urr ent time in 2 4 -hr HH .MMS S for mat TIME: S et s y stem time to spec ifi ed value 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 o rt 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 o u can back up the contents o f the cu rr 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 will not appear in the menu w hen using 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 o u can spec if y an y number of nested subdir ector ies.
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 . F or each kno wn v ari able, ty pe its value and pr ess the corr esponding menu k ey . If a v ari able is not show n, pr ess L to display fur th er variables. 8. Opti onal: supply a gues s 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 ke y display s additional 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 , yo u spec if y a set o f one or mor e equati ons. Y o u can get the follo wing inf ormation abou t the equation s et from the E quatio n Libr ary catalogs: The equations themsel ves and the number of equations .
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 Press to s tor e the pi ctur e in PIC T , the graphi cs memory . T hen y ou can use © PIC T (or © PICTURE) to v iew the p ic tur e again after y ou hav e quit the Equati on Libr ar y . Press a menu k ey or to v iew other equatio n informati 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 curr ent set (a s the y ar e used w hen the Multiple -E quation Sol ver is launc hed) .
Pa g e 27- 1 1 Constant? The initi al value o f a var iable may 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 related . A v ari able may not be in vol ved in the soluti on (no mark in the label), so it is not compatible w ith the var iables that w ere in volv ed. W rong dir ection . T he initial value of a var iable may 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 t ype of object in highlighted field !CANCL Cancel operation @@OK@@ Ac cep t en tr y If y ou pre ss !RESE.
Pa g e A - 4 (In RPN mode , we w ould hav e used 113 6.2 2 ` 2 `/ ). Press @@OK@@ to enter this ne w value . The input f orm w ill no w look lik e this: Press !TYPES to see the type of data in the P MT f ield (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 thr ough F ke ys ar e assoc iated w ith the soft menu options 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 Th e left- shift ke y „ and the right-shift key … 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. Th e numeri cal ke ys ( 0 to 9 ) are us ed to enter the digits of the dec imal number sy stem.
P age B-5 the other three f unctions is ass oci ated with the left-shif t „ ( MT H ), right-shift … ( CA T ) , and ~ ( P ) k eys . Diagr am s sho w ing the function or c haracter r esulting fr om .
Pa g e B - 6 Th e CMD function sho ws the most r ecent commands, the PRG fu nc tion 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 th e calculator ’s ke yboard Th e CMD function sho ws the most r ecent commands.
Pa g e B - 7 Th e e x k ey cal cul ates the e xponential func tion of x . Th e x 2 ke y calculat es the squar e of x (this is ref err ed to as the SQ fun ctio n) . The AS IN, A CO S, and A T AN functi ons calculate 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 Th e CA T functi on is used to activ ate the command catalog. Th e CLEAR functi on clears the s cr een. Th e LN func tion calc ulates the natur al logar ithm. The functi on calculates the x – th r oot of y . Th e Σ functi on is used to ent er summations (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 through Z ) . T he numbers, mathematical s ymbols ( - , + ), decimal poin t ( . ), and the space ( SPC ) ar e the same as the main functions of the se 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 ) . T he numbers, mathe matical sym bo l s ( - , +, × ), dec imal point ( . ) , and the space ( SP C ) are the same as 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 w ith the right-shift ke y … .
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 harac ters 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 lgebraic S ys tem . This is the mathemati cal cor e of 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 reco ver the or iginal men u in the CAL CULA T OR MODE S input box , pres s the L ke y . Of inter est at this point is the c hanging of the CAS settings .
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 eferr 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) . Th is 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 simplifi ed to X .
Pa g e C - 1 1 Notice that , in this instance, s oft menu k ey s E and F ar e the only one w ith ass oci ated commands , namely : !!CANCL E CANCeL the help f ac ilit y !!@@OK#@ F OK to acti vate help .
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 produce s no additional menu it ems) .
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 : Opens a gra phics s creen w here the u ser can modify highlighted char acter . Use this opti on car ef ully , since it w ill alter the modified c haracte r 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 (m u) ~‚m ρ (rho) ~‚f σ (sigma) ~‚s τ (tau) ~‚u ω (.
Pa g e E - 1 Appendix E The Selec tion T r ee in t he 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 entheses ). 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 . Press the do wn arr ow k e y ˜ , continuously , until the c lear , editing c ursor is tri ggered ar ound 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 rs t key i n second r o w fr om the ke yboard’s top). The G ke y show s the follo 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 APP S menu opens the equation 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 launches the matr ix w r 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 presented i.
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 $ , re lease it after enter ing second or thir d k e y): o $ (hold) AF : “Cold” r estart - all memory eras ed o $ (hold) B : Cancels k ey strok 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 ` . T he fo llow ing scr een shots sho w the f irst menu page in 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 type two or more letters of t he comm and of interest , by 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 ds, 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 TH S menu , accessible thr ough the command MA THS (av ailable in the catalog N ), contains the follo wing sub-me nus: 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 are 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: Skip s char acters to beginning o f wor d.
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 o sition : mov e 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 entheses 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 lo w w ith Los ses (10, 17) 2 : Bernoulli E quation (10, 15) 4: Flo w in Full P ipes (8 , 19) 4 : Forces and Energy ( 3 1 , 3 6) .
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 ar ranty HP 50g gr aphing calculator ; W arr anty period: 12 months 1. HP warr ants to you , the end-user c ustomer , that HP hard war 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 T EMENT ARE Y OUR SO LE AND EX CL US IVE REMEDIES . 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.
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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 art 15 of the FCC Rules . O per ation is subject 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 ᬆ ᬡٍ¾ᬢ ᖱႎಣℂٍ¾╬ชᵄ්ኂȴਥۉද߿ ળ (V CCI.
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