Instruction/ maintenance manual of the product 50g HP
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HP g gr aphing calc ulator user ’s guide H Ed i ti on 1 HP part number F2 2 2 9AA-9 0006.
Notice REG ISTER Y OUR PRODU CT A T: ww w .regis ter .hp.com TH IS MANUAL AND ANY E XAMPLE S CONT AINE D HEREIN ARE PR O VID E D “ AS IS” AND ARE SUB JECT T O CHANGE WITHOUT NOT ICE .
Pref ace Y ou ha ve in y our hands a compact s ymboli c and numer ical computer that w ill fac ilitate calc ulati on and mathematical anal ysis o f pr oblems in a var iety of disc iplines, f r om elementary mathematic s to adv anced engineer ing and sc ience subjec ts.
F or s ymboli c oper ati ons the calc ulator inc ludes a po werf ul Co mputer A lgebrai c S y ste m (CAS) that lets y ou select diff er ent modes o f oper ation , e .g ., complex number s vs . r eal numbers , or e x act (s y mbolic) v s . appr o x imate (numer ical) mode .
Pa g e TO C - 1 T abl e o f contents Chapter 1 - Getting started ,1-1 Basic Operations ,1-1 Batteries ,1-1 Turning the calculator on an d 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 MENU ,2-67 Selected CHOOSE boxes ,2-69 Chapter 3 - Calculation with real numbers ,3-1 Checking calculato rs settings ,3-1 Checking c.
Pa g e TO C - 4 Physical constants in the calc ulator ,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 su b-menu ,6-30 The TVM sub-menu ,6-30 Chapter 7 - Solving multiple equations ,7-1 R.
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 coordi nate system ,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 nu mber of lists ,10-14 Lists represent columns of the matrix ,10-15 Lists represent rows of .
Pa g e TO C - 1 1 Function TRAN ,11-15 Additional matrix operations (The matri x OPER menu) ,11-15 Function AXL ,11-16 Function AXM ,11-16 Function LCXM ,11-16 Solution of linear systems ,11-17 Using .
Pa g e TO C - 1 2 Function QXA ,11-53 Function SYLVESTER ,11-54 Function GAUSS ,11-54 Linear Applications ,11-54 Function IMAGE ,11-55 Function ISOM ,11- 55 Function KER ,11-56 Function MKISOM ,11-56 .
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 ,.
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 se ries ,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 visualizati on 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 keybo ard ,20-5 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 - 2 2 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 PIXO.
Pa g e TO C - 23 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 fo.
Pa g e TO C - 24 Storing objects on an SD ca rd ,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 G e t ting started T his chapte r pr ov ides basi c inf ormatio n about the oper ation of y our calculator . It is desi gned to familiar i z e y ou w ith the basic oper ations and se ttings b e fo r e y ou perfor m a calc ulation .
Pa g e 1 - 2 b . Insert a ne w CR203 2 lithium batter y . Make sur e its positi v e (+) side is f aci ng up . c. R eplace the plate and p u sh it to the ori ginal place.
Pa g e 1 - 3 At the top o f the display y ou will ha v e two lines o f infor mation that de sc ribe the settings o f the calculator . The f irst line sho ws the c har acte rs: R D XYZ HE X R= 'X' F or details on the meaning of the se s y mbols see C hapter 2 .
Pa g e 1 - 4 E ach gr oup of 6 entr i es is called a Menu page . The c ur r ent menu , know n as the T OOL menu (s ee belo w) , has e ight en tri es ar ranged in tw o pages. T he next page , containing the ne xt two entr ies o f the menu is av ailable b y pr essing the L (NeXT menu) k e y .
Pa g e 1 - 5 T his CHOOSE bo x is labeled B ASE MENU and pr o v ide s a list of n umber ed fu nct ion s, from 1 . H EX x to 6. B R. T his displa y wi ll constitute the f irs t page of this CHOOSE bo x menu sho w ing si x menu f uncti ons.
Pa g e 1 - 6 If y ou no w pr es s ‚ã , instead of the CHOO SE bo x that y ou sa w earli er , the displa y w ill no w show six s oft menu la bels as the f irst page of the S T A CK menu: T o na vi g.
Pa g e 1 - 7 The T OOL m enu T he soft menu k e y s for the men u c ur r ent ly displ ay ed , kno w n as t he T OOL men u , ar e assoc iat ed with oper ations r elated to manipulation o f var iables (.
Pa g e 1 - 8 9 k e y the T IME c hoose bo x is acti vated . T his operati on can also be r epr esented as ‚Ó . Th e TIM E ch oo se box i s sh o wn in th e figu re b el ow: As indicated abov e, the TIME men u pr o vi des f our differ ent options , number ed 1 thr ough 4.
Pa g e 1 - 9 Let ’s change the minute f ield to 2 5, by pr ess ing: 25 !!@@OK#@ . T he seco nds f ield is no w highli ghted . Suppose that y ou w ant to c hange the seconds fi eld to 4 5, u se: 45 !!@@OK #@ T he time for mat f ield is no w highlighted .
Pa g e 1 - 1 0 Setting th e date After s etting the time for mat option , the SET T IME AND D A TE input for m w ill look as f ollo w s: T o set the date , f irst set the date f ormat . The de fault f or mat is M/D/Y (month/ day/y ear). T o modif y this f or mat , pre ss the do w n arr o w ke y .
P age 1-11 Intr oduc ing the calc ulator ’s k e yboar d The f igur e below sh ow s a di agram of the calculator ’s k ey boar d w ith the number ing of its ro ws and columns. T h e f i g u r e s h o w s 1 0 r o w s o f k e y s c o m b i n e d w i t h 3 , 5 , o r 6 c o l u m n s .
P age 1-12 shift ke y , k e y (9 ,1 ) , and the ALPHA k e y , ke y ( 7 ,1) , can be combined w ith some of the other k e y s to acti vat e the alternati ve func tions sho w n in the k e yboar d .
Pa g e 1 - 1 3 Pr ess the !!@ @OK#@ soft men u k e y to r etur n to nor mal displa y . Example s of s electing diffe r ent calc ulator modes ar e show n next . Oper at ing Mode T he calculator o ffer s two oper a ting mode s: the Algebr aic mode , and the Re v ers e P olish Notati on ( RPN ) mode .
Pa g e 1 - 1 4 T o enter this e xpre ssion in the calc ulator w e w ill f irs t use the equation w r iter , ‚O . P lease identify the f ollo w ing k e y s in the k e yboar d , besi des the nume ri c k e y pad ke y s: !@.
Pa g e 1 - 1 5 Change the oper ating mode to RPN by f irst pr es sing the H butt on . Sele ct th e RPN oper ating mode b y either u sing the ke y , 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 t o complete the oper ation .
Pa g e 1 - 1 6 3.` Ent er 3 in le v el 1 5.` Ent er 5 in le v el 1, 3 mov es to y 3.` Ent er 3 in le v el 1, 5 mov es to lev el 2 , 3 t o lev el 3 3.* P lace 3 and multipl y , 9 appears in le v el 1 Y 1/(3 × 3), la st value in le v .
Pa g e 1 - 1 7 Notice ho w the e xp r essi on is placed in stac k le ve l 1 after pre ssing ` . Pr essing the EV AL ke y at this point w i ll ev aluate the numer ical value o f that e xpr es sion Note.
Pa g e 1 - 1 8 mor e about r e al s, see C hapter 2 . T o illustr ate this and other numbe r for mats try the f ollo w ing ex er c ises: Θ Standard f ormat : T his mode is the most us ed mode as it sho ws nu mbers in the mos t famili ar notation .
Pa g e 1 - 1 9 Notice that the Number F or mat mode is set t o Fix f ollo wed b y a z er o ( 0 ). T his number indicat es the number of dec imals to be sho w n af t er the dec imal point in the calc ulator’s displa y . Pr ess the !!@@OK#@ soft menu ke y to r eturn to the calc ulator displa y .
Pa g e 1 - 2 0 Pres s the !!@@OK#@ soft menu k ey to complete the sel ec tion: Pr ess the !!@@OK#@ soft menu k e y r eturn to the calc ulator displa y . The number no w is s h ow n as: Notice ho w the number is r ounded, not tr uncated . Th us , the number 12 3 .
Pa g e 1 - 2 1 same fa shion that w e c hanged the Fixe d number o f dec imals in the exa mp l e ab ove ) . Pr es s the !!@@OK#@ soft menu k ey r eturn to the calc ulator displa y . The number no w is s h ow n as: T his re sult , 1.2 3E2 , is the calculat or’s v ersio n of po w ers-o f- ten notatio n, i.
Pa g e 1 - 2 2 Pr es s the !!@@OK#@ soft menu k ey re turn to the calc ulator dis pla y . The n umber no w is s h ow n as: Becau se this number has thr ee fi gur es in the inte ger part, it is sho wn w ith fo ur signif icati v e fi gur es and a z ero po wer o f ten , while using the Engineer ing f ormat .
Pa g e 1 - 23 Θ Pr es s the !!@@OK#@ soft menu k ey re turn to the calc ulator dis pla y . The n umber 12 3 .4 5 6 7 8 9012 , enter ed earlier , no w is sho wn as: Angle M easur e T r igonometr i c func tions , for e xample , r equir e arguments r epr ese nting plane angles .
Pa g e 1 - 24 k e y . If u sing the lat t er appr oach , use u p and dow n arr ow k ey s , — ˜ , to se lect the pr ef err ed mode , and pr ess the !!@@OK#@ soft menu k e y to complete the ope r ation .
Pa g e 1 - 25 fr om the positi v e z ax is to the r adial dis tance ρ . T he Rec tangular and Spher ical coor dinate sy stems ar e re lated by the fo llo wi ng quantities: T o c hange the coordinat e s ys tem in y our calculat or , follo w these s teps: Θ Pr es s the H button.
Pa g e 1 - 26 _L ast S tac k : K eeps the conten ts of the last st ack en tr y f or us e with the f unct ions UNDO and ANS (see C hapter 2). Th e _Beep option can be us ef ul to adv ise the user a bout err ors . Y ou may w ant to des elect this option if u sing yo ur calc ulator in a cla ssr oom or libr ary .
Pa g e 1 - 27 Selec ting Displa y modes T he calculator dis play can be c ustomi z ed to y our pr ef er ence b y selecting dif f erent disp lay mod es . T o see the op tional di splay sett ings use the follow ing : Θ F i r st , pr es s the H button to acti v ate the CAL CULA T OR MODE S input f or m.
Pa g e 1 - 2 8 Pr essing the @ CHOOS so ft menu k e y w ill pr o vi de a list of a v ailable s y ste m fonts , as sho w n belo w: T he options a vaila ble ar e thr ee standar d Sys t e m Fo n t s (si z es 8, 7 , and 6 ) and a Br o wse .
Pa g e 1 - 2 9 displa y the DISPLA Y MODE S input f orm . Pr ess the do wn ar r ow k e y , ˜ , tw i ce , to get to the Stack line . This line show s two pr operties that can be modif ied . When thes e pr oper ti es ar e selec ted (chec k ed) the f ollo w ing eff ects ar e acti v ated: _Small Changes f ont si z e to small .
Pa g e 1 - 3 0 times , to ge t to the EQW (E quati on W rit er ) line . This line sho w s tw o pr oper ti es that can be modif ied . When thes e properti es ar e select ed (chec k ed) the fo llo w ing.
Pa g e 1 - 3 1 r ight arr ow k ey ( ™ ) to select the under line in fr ont of the opti ons _Clock or _Analog . T oggle the @ @CHK@@ soft men u k e y until the desir ed setting is ac hie v ed. If the _Cloc k option is se lected , the time of the da y and date w ill be sho wn in the upper ri ght corner of the display .
Pa g e 2- 1 Chapter 2 Intr oducing th e calculator In this c hapter we present a n umb er of basi c oper ations of the calculator inc luding the use of the E quation W r iter and the manipulation of data obj ects in the calc ulator .
Pa g e 2- 2 the CA S, it mi ght be a good i dea to s w itch dir ectl y into appr o x imate mode . R efe r to Appendi x C f or mor e details . Mi x ing integers and reals together or mi s takin g an integer for a r eal is a common occ urr ence .
Pa g e 2- 3 Binary integ ers , obje cts of t ype 10 , are used i n some computer sc ienc e applicati ons. Graphics objec ts , ob jec ts of type 11, st or e graphi cs pr oduced by the calc ulator . T agg ed objects , obj ects of t y pe 12 , ar e us ed in the output of man y pr ograms t o identify r esults .
Pa g e 2- 4 T he r esulting e xpr essi on is: 5.*(1.+1./7.5)/( √ 3.-2.^3). Press ` to get the expr essio n in the dis play as f ollow s: Notice that , if your CA S is s et to EXA CT (see Appe ndi x C) and you en ter y our e xpr es sion us ing integer number s for in teger v alues , the r esult is a s y mbolic quantity , e .
Pa g e 2- 5 T o e valuat e the e xpr essi on w e can us e the EV AL functi on , as f ollo ws: μ„î` As in the pr ev ious e xample , yo u wi ll be ask ed to appr ov e c hanging the CAS setti ng to Appr o x . Once this is done , y ou w ill get the same r esult as bef or e .
Pa g e 2- 6 T his lat t er r esult is pur el y numer ical , so that the two r esults in the stac k, although r epr esenting the same e xpr essi on, seem diff er ent .
Pa g e 2- 7 T he editing cur sor is sho wn a s a blinking left arr o w ov er the f irs t char acter in the line to be edited. Since the editing in this case consists of r emov ing some c har acte rs a.
Pa g e 2- 8 W e set the calc ulator oper ating mode to Algebr aic , the CA S to Exac t , and the displa y to T e xtbook . T o ente r this algebr aic e xpr es sion w e us e the foll ow ing keys tro kes.
Pa g e 2- 9 Θ Pr ess the r ight arr o w k e y , ™ , until the c ursor is to the r ight of the x Θ Ty p e Q2 to enter the po wer 2 f or the x Θ Pr ess the r ight arr o w k e y , ™ , until the c ursor is to the r ight of the y Θ Pr ess the de lete k ey , ƒ , once to era se the c har acter s y.
Pa g e 2- 1 0 Θ Pr es sing ` once more to r eturn to normal display . T o see the entir e e xpr essi on in the sc r een, w e can change the optio n _Small Stack Di sp in the DIS P L A Y MODE S input for m (see Chapter 1).
Pa g e 2- 1 1 T he six s oft menu k ey s f or the E quation W rit er acti vat e the follo wing f uncti ons: @EDIT : lets the u ser edit an entry in the line editor (see e x amples abo ve) @CURS : hi g.
Pa g e 2- 1 2 T he r esult is the e xpr essi on T he c ursor is sho w n as a left-fac ing ke y . T he c urso r indicat es the c ur ren t edition location . T yp ing a char act er , functi on name , or oper ation w ill enter the cor re sponding char acter or c har acter s in the cur sor location .
Pa g e 2- 1 3 Suppos e that no w y ou w ant to add the fr ac tion 1/3 to this entir e expr ession , i .e ., y ou wan t to en ter the e xpr es sion: F i r st , w e need to hi ghlight the entir e f ir s.
Pa g e 2- 1 4 Sho wing the expression in smaller -siz e T o sho w the expr es sion in a smaller -si z e font ( whi c h could be u sef ul if the e xpr essi on is long and con vo luted), simply pr ess the @BIG soft menu k ey .
Pa g e 2- 1 5 If y ou w ant a floating-po int (numer ical) e v aluation , us e the NUM fu nct ion (i .e ., …ï ) . T he r esult is as follo ws: Use the function UNDO ( …¯ ) on c e m ore t o r.
Pa g e 2- 1 6 A s ymboli c ev aluation once mor e. Suppo se that , at this point , w e want to e valuate the left-hand side fr acti on onl y . Pr ess the upper ar r o w ke y ( — ) thr ee times to se.
Pa g e 2- 1 7 Editing arithmetic e xpr essions W e w ill show s ome of the editing featur es in the E quation W riter as an e x erc ise . W e start b y enter ing the follo wi ng expr essi on used in t.
Pa g e 2- 1 8 Pr es s the do wn ar r o w k e y ( ˜ ) to tri gger the c lear editing cur sor . The sc r een no w looks like this: B y using the le f t ar r o w ke y ( š ) y ou can mov e the c ursor in the gener al left dir ecti on , but stopp ing at each indi vi dual component of the e xpr essi on .
Pa g e 2- 1 9 Ne xt , we ’ll conv ert the 2 in f r ont of the parenth eses in the denominator into a 2/3 by using: šƒƒ2/3 At this point the e xpr essi on looks as f ollo w s: T he final step is to r emo ve the 1/3 in the r i ght-hand side of the e xpr ession .
Pa g e 2- 2 0 Use t he fo llow ing k ey str ok es: 2 / R3 ™™ * ~‚n+ „¸ ~‚m ™™ * ‚¹ ~„x + 2 * ~‚m * ~‚c ~„y ——— / ~‚t Q1/3 T his re sults in the output: In this e xample w e us ed se ve ral lo we r -case English lett ers , e .
Pa g e 2- 2 1 Editing algebr aic ex pressions T he editing o f algebrai c equati ons follo ws the same r ules as the editing of algebr aic equati ons. Namely : Θ Use the ar r o w k e y s ( š™—˜ ) to highli ght e xpr essi ons Θ Use the do wn ar r o w ke y ( ˜ ) , r epeatedly , to tr igger the c lear editing c ursor .
Pa g e 2- 22 2. θ 3. Δ y 4. μ 5. 2 6. x 7. μ in the e xponential f unction 8. λ 9. 3 i n t h e √ 3 ter m 10. the 2 in the 2/ √ 3 fr acti on At an y po int we can c hange the clear editing c urs or into the insertio n cur sor b y pr essing the dele te k e y ( ƒ ) .
Pa g e 2- 23 Ev aluating a sub-e xpression Since w e alr eady ha ve the sub-e xpre ssi on highli ghted , let ’s pr ess the @EVAL soft menu k e y to ev aluate this sub-e xpr ession . T he r esult is: Some algebr aic ex pre ssions cannot be simplif ied an ymor e.
Pa g e 2- 24 3 in the f irst te rm of the numer ator . Then , pr ess the r ight ar r o w k e y , ™ , to nav igate thr ough the expr essi on. Simplifying an e x pr ession Pr ess the @ BIG soft menu k e y to get the sc r een to look as in the pre vi ous f igur e (see abo ve).
Pa g e 2- 2 5 Press ‚¯ to r ecov er the or iginal e xpre ssion . Ne xt , enter the f ollo w ing keys tro kes : ˜ ˜˜™™™™™™™———‚™ to sele c t the last two ter ms in the expr ession , i .e ., pr ess the @ FACTO soft menu k e y , to g e t Press ‚¯ to reco v er the ori ginal e xpre ssion .
Pa g e 2- 26 Ne xt , select the command DER VX (the deri vati ve w ith r espec t to the v ari able X, the c urr ent CAS indepe ndent var iable) b y using: ~d˜˜˜ .
Pa g e 2- 27 Detailed e xplanation on the use of the help fac i lity f or the CA S is pr esented in Chapter 1. T o r eturn to the E quation W rite r , pre ss the @EXIT s oft menu k ey .
Pa g e 2- 28 Ne xt , we ’ll copy the f r actio n 2/ √ 3 from t he lef tm ost fa ctor in th e exp r es sion, and place it in the numerat or of the ar gument fo r the LN function .
Pa g e 2- 2 9 W e can no w cop y this expr essi on and place it in the denominator o f the LN ar gument , as follo ws: ‚¨™™ … ( 2 7 times ) … ™ ƒƒ … (9 times) … ƒ ‚¬ T he line e.
Pa g e 2- 3 0 T o see the cor r esponding e xpr es sio n in the line editor , pr es s ‚— and the A soft menu k ey , to sho w : T his expr es sion sho w s the gener al form o f a summation typed di.
Pa g e 2- 3 1 and the v ari able of diff er entiati on . T o f ill these input locati ons, use the f ollo w ing keys tro kes : ~„t™~‚a*~„tQ2 ™™+~‚b*~„t+~‚d The r esulting scr een is .
Pa g e 2- 32 Definite integr als W e w ill use the E quati on W r iter to ent er the follo w ing def inite inte gral: . Pr es s ‚O to ac ti vat e the E quation W rite r .
Pa g e 2- 3 3 Double integr als ar e also pos sible . F or e x ample , w hich e v aluates to 3 6. P artial e valuati on is poss ible , for e x ample: T his integral e v aluates t o 3 6. Organi zing data in t he calculator Y ou can or gani z e data in yo ur calculator b y stor ing var iables in a dir ectory tr ee .
Pa g e 2- 3 4 @CHDIR : Change to s elected direct or y @CANCL : Cancel action @@OK@ @ : Appr ov e a selec tion F or ex ample , to c hange dir ectory to the CA SDI R , pr ess the do w n -arr o w k ey , ˜ , and pr ess @CHDIR . T his acti on clo ses the Fi l e M a n a g e r w indow and r eturns us to normal calc ulator dis play .
Pa g e 2- 3 5 T o mo ve betw een the differ ent so f t men u commands, y ou can u se not onl y the NEXT k e y ( L ), but also the PREV k ey ( „« ). T he user is in v ited to try these f uncti ons on his or her o w n. Their applicati ons ar e str aightf orwar d.
Pa g e 2- 36 T his time the CA SD IR is hi ghlighted in the scr een. T o s ee the contents of the dir ect or y pr ess the @@ OK@@ soft menu k e y or ` , to get the f ollo w ing sc r een: T he scr een sho w s a table des cr ibing the var iable s contained in the CA SD IR dir ect or y .
Pa g e 2- 3 7 Pr essing the L k e y sho ws one mor e var iable st ored in this dir ectory: • T o see the contents o f the var ia ble EPS , f or e xam ple , use ‚ @EPS@ . T his sho w s the v alue of EP S to be .0000000 001 • T o see the v alue of a numeri cal v ari able , w e need t o pre ss onl y the soft menu k ey f or the v ar iable .
Pa g e 2- 3 8 loc k the alphabetic k ey boar d tempor aril y and enter a f ull name bef or e unloc king it again. T he fo llo w ing combination s of k e y str ok es w ill lock the alphabeti c k e yboar d: ~~ locks the alpha betic k e y boar d in upper case .
Pa g e 2- 3 9 Creating subdir ec tor ies Subdir ector i es can be cr eated by using the FI LE S env ironme nt or by u sing the c om ma nd C RD I R. Th e t wo ap proa ch es for cr e at i ng su b- di r e cto ries a r e pr esen ted next .
Pa g e 2- 4 0 Th e Object input f i eld, the f irst input f ield in the f orm , is highlight ed by def ault . T his input fi eld can hold the conte nts of a ne w var ia ble that is being cr eated.
Pa g e 2- 4 1 T o mo v e into the MAN S dir ect ory , pr ess the co rr es ponding so ft menu k ey ( A in this case) , and ` if in algebr ai c mode. T he dir ectory tr ee w ill be sho wn in the second line o f the display as {HOME M N S} .
Pa g e 2- 42 Us e the do wn ar r o w k e y ( ˜ ) to selec t the option 2. M E M O RY … , or ju st press 2 . Then , pre ss @@OK@@ . T his will pr oduce the follo w ing pull-dow n menu: Us e the do wn ar r o w k ey ( ˜ ) t o select the 5 . DIRE CT OR Y option , or j ust press 5 .
Pa g e 2- 4 3 Pr ess the @ @OK@ soft menu k ey to ac tiv ate the comm and , to cr eate the sub- dir ectory: Mov ing among subdirectories T o mo ve do wn the dir ectory tr ee , y ou need to pr ess the so ft menu k ey cor r esponding to the sub-dir ect or y y ou wan t to mo v e to .
Pa g e 2- 4 4 T he ‘S2’ str ing in this f orm is the name of the sub-dir ectory that is being deleted . T he soft men u k ey s pro vi de the fo llo w ing options: @YES@ Pr oceed w ith deleting the.
Pa g e 2- 4 5 Us e the do wn ar r o w k e y ( ˜ ) to selec t the option 2. M E M O RY … T h e n , press @@OK@ @ . This w ill produ ce the fo llo w ing pull-do w n menu: Us e the dow n arr o w k e y ( ˜ ) to select the 5 . DIRE CT OR Y option . Then , press @@OK@ @ .
Pa g e 2- 4 6 Press @@OK@@ , to get: Then , pres s ) @@S3@@ to enter ‘S3 ’ as the ar gument to PGDI R . Press ` to delete the sub-dir ectory: Command PGDIR in RPN m o de T o us e the PGDIR in RPN mode y ou need to hav e the name o f the direc tory , between q uotes , alr eady a vaila ble in the stac k bef or e accessing the command .
Pa g e 2- 4 7 Using the PURGE command fr om the T OOL menu T he T OOL me nu is av ailable by pr essing the I ke y ( Algebr aic and RPN modes sho wn): T he PUR GE command is av ailable by pr essing the @PURGE s oft menu k e y .
Pa g e 2- 4 8 Using the FI LE S menu W e w ill 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 t o this sub-dir ectory , use the f ollo w ing: „¡ and sel ect the INTR O sub-dir ectory as sho w n in this scr een : Press @@OK@@ to ent er the dir ectory .
Pa g e 2- 49 T o enter v ari able A (see table abov e) , w e fir st enter its contents , na me ly , the number 12 . 5, and then its name, A, as follo ws: 12.5 @@OK@@ ~a @@OK@@ . Resulting in the f ollo wing sc r een: Press @@OK@@ once more to c reate the v ari able.
Pa g e 2- 5 0 Using the ST O command A simpler w ay to cr eate a v ar ia ble is by us ing the S T O command (i .e ., the K k e y) . W e pro vi de e xample s in both the Algebr ai c and RPN modes, .
Pa g e 2- 5 1 z1: 3+5*„¥ K~„z1` (if needed , accept c hange to Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ K~„p1` . T he scr een , at this point , will look as follo ws: Y ou w ill see si x of the se ven v ari ables lis ted at the bottom of the sc r een: p1, z1, R, Q, A12 , α .
Pa g e 2- 52 z1: ³3+5*„¥ ³~„z1 K (if needed , accept change to Comple x mode) p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . T he scr een , at this point , will look as follo ws: Y ou w ill see si x of the se v en var iables lis ted at the bottom of the sc reen: p1, z1, R, Q, A12 , α .
Pa g e 2-53 Pr essing the soft me nu k e y cor r esponding t o p1 will pr o v ide an er r or messa ge (tr y L @@@p1@@ ` ): Note: By pre ss i n g @@@p1@@ ` we ar e trying t o acti vate (r un) the p1 progr am . Ho w ev er , this pr ogr am e xpects a numer ical input .
Pa g e 2- 5 4 At this point , the scr een looks lik e 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 pr ogram in RPN mo de , yo u only need to enter the in put (5) and pr es s the corr es ponding soft menu k ey .
Pa ge 2- 55 Notice that this time the con tents of pr ogr am p1 are liste d in the scr ee n . T o see the r emaining v ari able s in this direc tory , pr ess L : Listing the con tents of all v ariables in the s c r een Use the k e y str ok e combinati on ‚˜ to list the cont ents of all v ar iable s in the sc r een .
Pa g e 2- 5 6 fo llow ed by the var iable ’s soft menu k e y . F or e xample , in RPN , if w e want to c ha nge the conten ts of var iable z1 to ‘ a+b ⋅ i ’, u s e : ³~„a+~„b*„¥` T his wil l place the algebrai c e xpr essi on ‘ a+b ⋅ i ’ in le v el 1: i n t h e st a ck .
Pa g e 2- 57 Use t he up ar r o w k ey — t o select the sub-dir ect or y MAN S and pres s @@O K@@ . If y ou no w press „§ , the scr een will sho w the contents of sub-direc tory MANS (notice that.
Pa g e 2- 5 8 Ne xt , use the delet e k ey thr ee times, to r emo ve the la st thr ee lines in the displa y : ƒ ƒ ƒ . At this po int , the stac k is r eady t o e xec ute the command ANS( 1) z1. Pr es s ` to e xec ute this command . Then , use ‚ @@z1@ , to ve rify the contents of the v ar iable .
Pa g e 2- 59 Cop ying two or mor e v ariables using the stac k in RPN mode T he follo wing is an e xer cis e to de monstr ate ho w to copy tw o or mor e var iable s using the st ack w hen the calc ulator is in RPN mode.
Pa g e 2- 6 0 T he sc r een no w sho w s the new o rde ring o f the var ia bles: RPN mode In RPN mode, the lis t of r e -or der ed var iables is list ed in the s tack be for e appl y ing the command ORDER. Su ppose that w e start fr om the same situati on as abo ve , but in RPN mode, i .
Pa g e 2- 6 1 Notice that v ar iable A12 is no longer ther e . If yo u no w pr ess 㤠, the sc r een w ill sho w the contents of sub-dir ectory MANS , including v ari able A12 : Deleting va riables V ar iables can be deleted using functi on P URGE .
Pa g e 2- 6 2 va riab le p1 . Pr ess I @PURGE@ J @@p1@@ ` . The sc reen w ill no w s ho w va riab le p1 rem ove d : Y ou can us e the P URGE command to er as e mor e than one var iable b y plac ing their name s in a list in the ar gument of P URGE .
Pa g e 2- 6 3 the HIS T k ey : UNDO r esults f r om the k e ys tr ok e seq uence ‚¯ , w hile CMD r esults f r om the k e y str ok e seq uence „® . T o illus trat e the us e of UNDO , try the follo w ing ex er c ise in algebr aic (A L G) mode: 5*4/3` .
Pa g e 2- 6 4 As y ou can see , the number s 3, 2 , and 5, u sed in the fi rst calc ulation abo ve , ar e listed in the s electi on bo x , as w ell as the algebr aic ‘S IN(5x2)’ , but not the SIN f uncti on enter e d pr ev io us to the algebr aic .
Pa g e 2- 65 Ex ample of flag setting: general solutions v s. pr incipal value F or e xample , the def ault v alue f or s y ste m flag 01 is Gener al solu tions . What this means is that , if an equation has m ultiple soluti ons, all the s olutions w ill be r eturned b y the calculato r , most lik el y in a list .
Pa g e 2- 6 6 ` (k eeping a s econd copy in the RPN stack) ³~ „t` Use the follo w ing k ey str oke sequence to enter the Q U AD command: ‚N~q (use the u p and dow n arr ow k ey s , —˜ , to se lect command QU AD) , pr ess @@OK@@ .
Pa g e 2- 6 7 CHOO SE bo x es vs . Soft MENU In some of the ex er c ises pr es ented in this chapter w e hav e seen menu lists of commands dis play ed in the scr een.
Pa g e 2- 6 8 T he sc r een sho w s flag 117 not se t ( CHOO SE box es ) , as sho wn her e: Pr es s the @ @CHK @@ s oft menu k e y to set f lag 117 to soft MENU . The s cr een w ill r ef lect that c hange: Press @@OK@@ t w ice to retur n to normal calc ulator displa y .
Pa g e 2- 69 Note: mos t of the e xam ples in this user guide a ssume that the cur r ent s et ting o f flag 117 is its default setting (that is, not se t) . If y ou ha ve s et the flag but w ant to str i ctl y follo w the e xam ples in this guide , y ou should c lear the flag bef or e con tinuing .
Pa g e 2- 70 • T he CMDS (CoMmanD S) menu , acti v ated w ithin the Eq uation W rit er , i. e. , ‚O L @CMDS.
Pa g e 3 - 1 Chapter 3 Calculation with re al numbers T his chapte r demonstr ates the us e of the calc ulator f or oper ations and f uncti ons r elated to r eal numbers . Oper ations along the se lines ar e use ful f or mos t common calc ulati ons in the ph ysi cal sc iences and engineer ing.
Pa g e 3 - 2 2 . Coordinate sy stem spe c ification (X Y Z , R ∠ Z, R ∠∠ ). T h e s y m b o l ∠ stands f or an angular coor dinate . XYZ: Carte sian or r ect angular (x,y ,z) R ∠ Z: cylindr ic a l P olar co or dinates (r , θ ,z ) R ∠∠ : Spher i cal coor dinates ( ρ,θ,φ ) 3 .
Pa g e 3 - 3 R eal number calc ulations w ill be demonstr ated in both the Algebr ai c (AL G) and R ev er se P olish Notati on (RPN) modes . Changing sign of a number , var iabl e , or e xpression Use the k ey . In AL G mode , y ou can pr ess be fo re e nter ing the number , e.
Pa g e 3 - 4 Alte rnati v el y , in RPN mode, y ou can separ ate the oper ands with a space ( # ) bef or e pr essing the oper ator k e y . Example s: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / Using parentheses P ar entheses can be used to gr oup operati ons, as w ell as to enclose ar guments of f unctions .
Pa g e 3 - 5 Squares and squar e roots T he squar e func tion , S Q, is av ailable thr ough the k e y str ok e combinati on: „º . When calc ulating in the st ack in AL G mode , e nter the fu ncti on bef or e the argument , e.g ., „º2.3` In RPN mode , enter the numbe r fir st , then the f uncti on, e .
Pa g e 3 - 6 Using po wers o f 10 in entering data P owe rs of te n, i.e. , nu mb e rs of t he fo rm - 4 .5 ´ 10 -2 , etc., ar e enter e d b y using the V k e y . F or e x ample , in AL G mode: 4.5V2` Or , in RPN mode: 4.5V2` Natural logar ithms and e xponential func tion Natur al logar ithms (i .
Pa g e 3 - 7 the in ver se tr igonometr i c functi ons r e present angles , the ans w er fr om these func tions w ill be gi v en in the select ed angular measur e (DEG , R AD , GRD) . Some e xamples ar e show n ne xt: In AL G mode: „¼0.25` „¾0.85` „À1.
Pa g e 3 - 8 comb ination „´ . W ith the defa ult setting of CHOO SE bo xe s fo r syst em flag 117 (see C hapter 2) , the MTH menu is sho wn as the f ollo w ing menu list: As the y ar e a gr eat number of mathematic f uncti ons a vailable in the calc ulator , the MTH menu is s orted b y the t y pe of ob ject the f uncti ons appl y on .
Pa g e 3 - 9 Hy perbolic functions and th eir in verses Selecting Option 4. HYP ERBOLIC.. , in the MTH men u , and pr es sing @@OK@@ , pr oduces the h yper bolic f unction men u: The h y perbolic f un.
Pa g e 3 - 1 0 T he r esult is: T he oper ations sho wn abo ve as sume that yo u are u sing the defa ult setting f or s y stem f lag 117 ( CHOO SE box es ).
Pa g e 3 - 1 1 F or ex ample , to calculat e tanh( 2 . 5), in the AL G mode , when u sing SO FT m e nu s ove r CHOO SE bo xe s , f ollo w this pr ocedur e: „´ Sele c t MTH menu ) @@HYP@ Selec t the HYP ERBOLIC.. menu @@TANH@ Selec t the TA N H fun cti on 2.
Pa g e 3 - 1 2 Option 19 . MA TH.. r eturns the u ser to the MTH men u . The r emaining functi ons ar e gr ouped in to si x diffe r ent gr oups des cr ibed be low .
Pa g e 3 - 1 3 T he r esult is sho wn ne xt: In RPN mode , recall that ar gument y is located in the second le v el of the st ack , w hile argument x is located in the f i r st le vel o f the stac k . T his means , y ou should enter x f irst , and then , y , jus t 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 sho uld be us ed as y MOD x , and not as MOD (y,x) .
Pa g e 3 - 1 5 G AMM A: T he G amma f unction Γ ( α ) P SI: N -th deri vati v e of the digamma f uncti on P si: Digamma f uncti on, de ri vati v e of the ln(Gamma) T he Gamma functi on is def ined b y . T his functi on has appli cations in applied mathematic s fo r sc ience and engineer ing , as well a s in pr obab ility and statis tic s.
Pa g e 3 - 1 6 Ex amples of thes e spec ial f unctions ar e sho w n her e using both the AL G and RPN modes. As an e x er c ise , v er if y that G AMMA(2 . 3) = 1.166 711…, PSI(1 . 5 , 3) = 1 .40909 .. , and P s i ( 1 .5) = 3. 6489 9 7 39 . . E- 2 .
Pa g e 3 - 1 7 Selec ting an y of thes e entr ies w ill place the value s elected , w hether a sy mbol (e .g ., e , i , π , MINR , or MAXR ) or a v alue ( 2 .71. ., (0,1) , 3 .14 .., 1E - 4 99 , 9. 9 9. . E 4 9 9 ) in the s tac k. P lease notice that e is a v ailable fr om the k e y board as exp ( 1) , i .
Pa g e 3 - 1 8 T he user w ill recogni z e most o f these units (some , e.g ., dy ne , are not u sed v ery often no w aday s) fr om his or her ph ysi cs c lasse s: N = newto ns, dyn = dyne s, gf = gr .
Pa g e 3 - 1 9 A vailable units T he follo w ing is a l ist of the units av ailable in the UNI T S menu . T he unit sy mbol is sho wn f irs t follo wed b y the unit name in parenth eses: LENG TH m (me.
Pa g e 3 - 2 0 SPEED m/s (meter per s econd), cm/s (centimeter per second), f t/s (f eet per s econd) , kph (kilometer per ho ur ) , mph (mile per hour), knot (nautical mile s per hour), c (speed of l.
Pa g e 3 - 2 1 ANGLE (planar and soli d angle measur ements) o (se x agesimal degree), r (radi an) , gr ad (gr ade) , ar cmin (minute of ar c) , ar cs (second of ar c) , sr (ster adian) LIGHT (Illumin.
Pa g e 3 - 2 2 Conv erting to base units T o conv er t an y of these units to the def ault units in the SI s yst em, u se the functi on UB A SE . F or e xample , to find out what is the v alue of 1 po.
Pa g e 3 - 23 ` Con vert the units In RPN mode , s y stem f lag 117 set to SO FT m e nu s : 1 Enter 1 (n o underline) ‚Û Select the UNIT S menu „« @ ) VISC Select the VIS C OS ITY opti on @@@P@@.
Pa g e 3 - 24 Notice that the under scor e is ente r ed automati call y when the RPN mode is acti v e . The r esult is the follo w ing sc r een: As indicated ear lier , if s ys tem flag 117 is s et to SOF T m en u s , then the UNI T S menu w ill sho w up as labels f or the soft menu k e ys .
Pa g e 3 - 25 Yy o t t a + 2 4 d d e c i - 1 Z z etta + 21 c c enti - 2 E e x a +18 m milli -3 P peta +15 μ mic r o -6 T ter a +12 n nano - 9 Gg i g a + 9 p p i c o - 1 2 Mm e g a + 6 f f e m t o - 1.
Pa g e 3 - 26 whi ch sho ws as 6 5_(m ⋅ yd). T o conv ert to units of the SI s y stem , use f uncti on UB A SE: T o calc ulate a di visi on , say , 3 2 50 mi / 5 0 h , ent er it as (3 2 50_mi)/(5 0_.
Pa g e 3 - 27 St ack calc ulations 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 ` / T hese oper ati ons pr .
Pa g e 3 - 28 UF A CT(x ,y) : fac tor s a unit y fr om unit obj ect x UNI T(x ,y) : combines v alue of x w ith units o f y T he UB A SE func tion w as disc ussed in detail in an earli er sec tio n in this cha pter . T o access any o f these f unctions f ollow the e xamples pro vided ear lier f or UB A SE .
Pa g e 3 - 2 9 Ex amples of UNI T UNIT( 2 5,1_m) ` UNI T(11. 3,1_mph) ` Ph y sical constants in t he calculator F ollow ing a l ong the treatment o f units, w e disc uss the u se of ph ysi cal constants that ar e av ailable in the calc ulato r’s memory .
Pa g e 3 - 3 0 T he soft menu k ey s cor r esponding t o this CONS T A NT S LIBRAR Y sc r een inc lude the f ollo w ing func tions: SI w hen selec ted , constants v alues ar e sho wn in S I units ENGL.
Pa g e 3 - 3 1 T o see the v alues of the const ants in the English (or Imperi al) s ys tem , pre ss the @ENGL opti on: If w e de -select the UNIT S opti on (pr ess @UNITS ) onl y the v alues ar e sho.
Pa g e 3 - 32 Special ph ysical functions Menu 117 , tr igge r ed by u sing MENU(117) in AL G mode, or 117 ` MENU in RPN mode , pr oduces the fo llo w ing menu (labels lis ted in the displa y b y usin.
Pa g e 3 - 3 3 ZF A C T OR(x T , y P ) , w her e x T is the r educed t emper atur e , i .e ., the r atio of ac tual temper ature t o pseudo -c ri tical temper ature , and y P is the r educed pr essur e, i .e., the r atio of the ac tual pr essur e t o the pseudo -c r itical pr es sur e .
Pa g e 3 - 3 4 Function T I NC F uncti on T INC(T 0 , Δ T) calculat es T 0 +D T . The ope rati on of this f uncti on is similar to that of f uncti on TDEL T A in the sense that it r eturns a r esult in the units of T 0 . Otherwise , it re turns a simple additi on of value s, e .
Pa g e 3 - 3 5 Pr ess the J k ey , and y ou will noti ce that ther e is a new v ar iable in y our soft menu k ey ( @@@H@@ ) . T o see the contents of this v ar iable pr ess ‚ @ @@H@@ .
Pa g e 3 - 3 6 T he contents of the v ar iable K ar e: << α β ‘ α+β ’ >>. Functions defined b y mor e than one e xpression In this secti on w e disc us s the treatme nt of f uncti ons that are de fi ned by tw o or mor e e xpre ssio ns.
Pa g e 3 - 37 Combined IFTE functions T o pr ogr am a mor e compli cated f uncti on such as y ou can combine se v er al le ve ls of the IFTE func tion , i .
Pa g e 4 - 1 Chapter 4 Calculations with compl e x numbers T his chapte r show s e xam ples of calc ulations and a pplication o f functi ons to comp lex n umbers . Definitions A comple x number z is a number w r itten as z = x + iy , wher e 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 . Enterin g comple x numbers Comple x numbers in the calc ulator can be enter ed in either of the tw o Car tesian repr esenta tions, nam el y , x+iy , or (x ,y) . T he r esults in t he calc ulator w ill be show n in the or der ed-pair format , i.
Pa g e 4 - 3 Notice that the las t entr y sho ws a comple x number in the f orm x+iy . T his is so becaus e the number w as enter ed bet w een single quot es, w hic h r epr ese nts an algebr aic e xpr essi on . T o ev aluate this number u se the EV AL k e y( μ ).
Pa g e 4 - 4 On the other hand , if the coor dinate s yst em is set to c ylindr ical coor dinates (us e C YLIN) , ent ering a com plex n umber (x,y), wher e x and y are r eal numbers , will pr oduce a polar repr esentati on . F or e x ample , in c y lindr ical coor dinates , enter the number (3 .
Pa g e 4 - 5 Changing sign of a complex number Changing the si gn of a comple x number can be accomplish ed by u sing the k e y , e .g ., -(5-3i) = -5 + 3i Entering the unit imaginary number T o ent er the unit imaginar y number ty pe : „¥ Notice that the n umber i is enter ed as the order ed pair (0,1) if the CA S is set to AP PR O X mode .
Pa g e 4 - 6 CMP LX menu through the MTH menu Assuming that s y st em flag 117 is se t to CHOOSE bo x es (s ee Chapter 2), the CMPLX sub-men u w ithin the MTH menu is acc essed by using: „´9 @@OK@ @ .
Pa g e 4 - 7 T his fir st sc r een sho ws f uncti ons RE , IM, and C R . Noti ce that the last f uncti on r eturns a list {3 . 5 .} re pre senting the r eal and imaginar y compone nts of the comp lex n umber : T he follo wing s cr een sho ws func tions R C, AB S , and ARG .
Pa g e 4 - 8 T he re sulting menu inc lude some of the f uncti ons alread y intr oduced in the pr e vi ou s secti on , namely , AR G , AB S, C ONJ , IM, NEG , RE , and SIGN . It also inc ludes fu nctio n i whi c h serve s the same pur pos e as the k e y str ok e comb ination „¥ , i .
Pa g e 4 - 9 Functions fr om th e MTH menu T he h yper bolic f uncti ons and their in v ers es , as w ell as the Gamma, P SI , and P si func tions (spec ial f uncti ons) we re introduced and appli ed to r eal numbers in Chapte r 3 .
Pa g e 4 - 1 0 F uncti on DROI TE is f ound in the command catalog ( ‚N ). Using E V AL(AN S(1)) simplif ies the r esult to:.
Pa g e 5 - 1 Chapter 5 Algebraic and ar it hmetic oper ations An algebr aic ob ject , o r simpl y , algebr aic , is an y number , var i able name or algebr aic e xpr essi on that can be operat ed upon , manipulated, and comb ined accor ding to the rule s of algebr a .
Pa g e 5 - 2 (e xponential , logar ithmic , tr igonometry , h yper bolic , etc .) , as y ou would an y r eal or comple x number . T o demons trat e basic oper ations w ith algebr aic obj ects , let’.
Pa g e 5 - 3 ‚¹ @@A1@ @ „¸ @@A2@ @ T he same r esults ar e obtained in RPN mode if using the fo llo w ing ke ys tr ok es: @@A1@ @ @@A2@ @ +μ @@A1@ @ @@A2@ @ -μ @@A1@ @ @@A2@ @ *μ @@A1@@ @@A2@.
Pa g e 5 - 4 W e notice that , at the bottom of the sc r een , the line See: EXP AND F A CT OR suggests links t o other help f ac ility entr ies , the f unctions E XP AND and F A CT OR. T o mo ve dir ectly t o those entr ie s, pr ess the soft men u k ey @SEE1! for E XP AND , and @SEE2! for F A CT OR.
Pa g e 5 - 5 F A CT OR: LNCOLLE CT : LIN: P ARTFRA C: S OL VE: S UBS T : TEXP AND: Not e: R ecall that , to use these , or any othe r functi ons in the RPN mode, y ou mus t enter the ar gument f irst , and then the func tion .
Pa g e 5 - 6 Other forms o f substitution in alg ebr aic e xpressions F uncti ons SUB S T , sho wn abo v e , is used to subs titute a var ia ble in an expr ession . A second f orm of substituti on can b e accomplished b y using the ‚¦ (ass oc iated w ith the I k e y) .
Pa g e 5 - 7 A differ ent appr oach to subs titution consis ts in def ining the substituti on e xpr essi ons in calc ulator v ar iables and plac ing the name of the var iables in the or iginal e xpr essi on .
Pa g e 5 - 8 LNCOLLE CT , and TEXP AND ar e also contained in the AL G menu pr es ented earli er . F uncti ons LNP1 and EXP M wer e intr oduced in menu HYP ERBOLIC, under the MTH men u (See Chapt er 2) .
Pa g e 5 - 9 Functions in the ARITHME T I C menu T he ARITHME T IC menu cont ains a number of sub-menu s for s pec ifi c appli cations in n umber theory (int egers , poly nomials , etc .) , as w ell as a nu mber of f uncti ons that apply to ge ner al arithme tic ope rati ons .
Pa g e 5 - 1 0 L GCD (Greatest C ommon Denominator): P ROPFRA C (pr oper fr action) SI MP 2 : T he functi ons ass oci ated w ith the ARITHME T IC submenus: INTE GER , P OL YNOMIAL , MODUL O , and PERM.
Pa g e 5 - 1 1 F A CT OR F act ori z es an integer n umber or a poly nomial FCOEF Gener ates f rac tio n giv en r oots and multipli c ity FR OO T S R eturns r oots and multipli c ity giv en a fr actio.
Pa g e 5 - 1 2 Applications of the ARI THME T I C menu T his s ectio n is intended to pr es ent some of the back ground neces sar y f or appli cation of the ARI THMET IC menu f unctions . Def initions ar e pr esen ted ne xt r egarding the su bj ects of pol ynomials , pol ynomi al fr acti ons and modular ar ithmetic .
Pa g e 5 - 1 3 multipl y ing j times k in modulus n arithmeti c is, in essence , the integer r emainder o f j ⋅ k / n in inf inite arithmeti c , if j ⋅ k>n . F or e xample , in modulus 12 ar ithmetic w e hav e 7 ⋅ 3 = 21 = 12 + 9 , (or , 7 ⋅ 3/12 = 21/12 = 1 + 9/12 , i .
Pa g e 5 - 1 4 Notice that , whene v er a r esult in the ri ght -hand si de of the “ congruence ” s ymbol pr oduces a r esult that is lar ger than the modulo (in this case , n = 6), you can alw ay s subtr act a multiple of the modulo fr om that re sult and simplif y it to a number smaller than the modulo .
Pa g e 5 - 1 5 [SP C] entry , and then pr es s the corr esponding modular arithme tic f uncti on . F or e x ample , using a modulus o f 12 , tr y the f ollo wing oper ations: ADDTMOD e xamples 6+5 ≡.
Pa g e 5 - 1 6 oper ating on them. Y ou can also conv er t an y number into a r ing number b y using the f uncti on EXP ANDMOD . F or ex ample, EXP A NDMO D(1 2 5) ≡ 5 (mod 12) EXP A NDMO D(17 ) ≡.
Pa g e 5 - 1 7 P ol ynomials P oly nomials ar e algebrai c expr essi ons consisting of one or mor e ter ms cont aining decr easing po we rs of a gi v en v ari able . F or e xample , ‘X^3+2*X^2 - 3*X+2’ is a thir d-or der poly nomi al in X, while ‘S IN(X)^2 - 2’ is a second-or der poly nomial in SI N(X) .
Pa g e 5 - 1 8 number s (func tion ICHINREM) . T he input consis ts of tw o v ector s [e xpr essi on_1, modulo_1] and [e xpr es si on_2 , modulo_2] . The o utput is a v ector containing [e xpr essi on_3, modulo_3] , wher e modulo_3 i s r elated to the pr oduct (modulo_1) ⋅ (modulo_2) .
Pa g e 5 - 1 9 An alter nate def initi on of the Hermite pol yn omials is wher e d n /dx n = n- th der i vati ve w ith r espec t to x . This is the def inition u sed in the calc ulator . Ex amples: The Her mite pol ynomi als of or ders 3 and 5 ar e giv en b y: HERMITE( 3) = ‘8*X^3-12*X’ , And HER MI TE(5) = ‘3 2*x^5-160*X^3+120*X’ .
Pa g e 5 - 2 0 F or ex ample , for n = 2 , w e w ill w rit e: Chec k this r esult w ith yo ur calculator : L A GR ANGE([[ x1,x2],[y1,y2] ]) = ‘((y1-y2)*X+(y2*x1-y1*x2))/(x1- x2)’ . Other e x ample s: LA GR ANGE([[1, 2 , 3][2 , 8 , 15]]) = ‘(X^2+9* X-6)/2’ L A GRANGE([[0.
Pa g e 5 - 2 1 T he PCOEF function Gi ven an ar r ay co ntaining the r oots of a pol y nomial , the fu nction PC OEF gener ates an ar r ay containing the coe ffi c ients o f the corr esponding poly nomial . T he coeffi c ients cor r espond t o decr easing or der o f the independent var ia ble.
Pa g e 5 - 2 2 T he EPSX0 function and t he CAS v ariable EPS Th e va riab le ε (epsilon) is typ icall y used in mathemati cal te xtbooks to r epr esen t a v ery small number . T he calc ulator’s CAS cr eate s a v ari able EP S , w ith def ault v alue 0.
Pa g e 5 - 23 Fra c ti on s F r acti ons can be expanded and fact or ed b y using func tions EXP AND and F A CT OR, f r om the AL G menu (‚×) . F or ex ample: EXP AND(‘(1+X)^3/((X-1) *(X+3))’) .
Pa g e 5 - 24 If y ou hav e the C omple x mode acti v e , the r esult w ill be: ‘2*X+(1/2/(X+i)+1/2/(X- 2 )+5/(X -5)+1/2/X+1/2/(X- i))’ T he FCOEF func tion T he function FC OEF is used to obta in a r a ti onal fr action , giv en the r oots and poles of the f r action .
Pa g e 5 - 25 mode selec ted, then the r esults w ould 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 ith poly nomials and fractio.
Pa g e 5 - 26 T he CONVERT M enu and algebr aic oper ations T he CONVER T menu is acti vated b y u sing „Ú ke y (the 6 key ) . T hi s menu summar i z es all con ver sion men us in the calc ulator . The lis t of these men us is sho wn next: T he functi ons a vaila ble in each o f the sub-menu s ar e sho w n next .
Pa g e 5 - 27 B ASE con vert menu (Option 2) T his menu is the same as the UNI T S menu obtained b y u sing ‚ã . The appli cations of this menu ar e discu sse d in det ail in Chapter 19 . TRIGONOMETRIC conv er t menu (Option 3) T his menu is the same as the TRIG men u obtained b y using ‚Ñ .
Pa g e 5 - 28 Fu n c ti o n NUM has the same eff ect as the k ey str ok e combination ‚ï (ass oc iated w ith the ` key) . Fun ct io n NU M co nver ts a s ym bo lic res ul t i nt o its floating-po int value . Func tion Q conv er ts a floating-po int v alue into a fr acti on .
Pa g e 5 - 2 9 LIN LNCOLLE CT P O WEREXP AND S IMPLIF Y.
Pa g e 6 - 1 Chapter 6 Solution to single equations In this c hapter w e featur e thos e functi ons that the calc ulator pr o vi des f or sol v ing single equations o f the for m f(X) = 0.
Pa g e 6 - 2 Using the RPN mode, the s olution is accomplished b y enter ing the equation in the stac k , f ollo we d by the v ar ia ble , bef or e enter ing f uncti on IS OL. R ight bef or e the e xec ution of I SOL , the RPN st ack should look as in the f igur e to the left.
Pa g e 6 - 3 The s cr e e n shot sho wn abo v e displa ys tw o solutions . In the firs t one , β 4 -5 β =12 5, SOL VE produce s no solu tions { }. In the s econd one , β 4 - 5 β = 6, S O L VE pr oduces f our soluti ons, sho w n in the last output line .
Pa g e 6 - 4 In the f irst case S OL VEVX could not find a s olution . In the second case , S OL VEVX f ound a single solu tion , X = 2 . The fol low i ng scr e ens sh o w the RP N stack for solving t.
Pa g e 6 - 5 The S ymbolic S olv er functi ons pre sented abo ve pr oduce soluti ons to rati onal equati ons (mainly , poly nomial equations). If the equation to be s ol ved f or has all numer i cal coeffi c ients , a numer ical soluti on is pos sible thr ough the use of the Numer ical S olv er f eatur es of the calc ulator .
Pa g e 6 - 6 P ol ynomial Equations Using the Sol ve poly… option in the calc ulator’s SOL V E en vir onment y ou can: (1) f ind the solu tions to a pol yn omial equati on; (2) obtain the coeff ic ie nts of the pol y nomial ha v ing a number of gi ven r oots; (3) obtain an algebr aic e xpr essi on f or the p o ly nomial a s a functi on of X.
Pa g e 6 - 7 All the s olutions ar e complex n umbers: (0.4 3 2 ,-0. 38 9) , (0.4 3 2 , 0. 38 9) , (-0.7 6 6, 0.6 3 2) , (-0.7 66 , -0.6 3 2) . Gene r ating poly nomial coefficients gi ven the polyn omial's roots Suppos e y ou w ant to gener ate the poly nomial w hose r oots are the n umbers [1, 5, - 2 , 4].
Pa g e 6 - 8 Press ˜ to tr igger the line editor to see all the coeff i c ients . Gene r ating an algebraic e xpression f or the polynomial Y ou can use the calc ulator to gener ate an algebr aic e xpr es sion f or a poly nomial giv en the coe ffi c ients or the r o o ts of the pol y nomial .
Pa g e 6 - 9 T o e xpand the pr oducts , y ou can us e the EXP AND command . The r esulting e xpr es si on is: ' X^4+-3*X^3+ -3*X^2 +11*X-6' .
Pa g e 6 - 1 0 Ex ample 1 – Calculating pa yment on a loan If $2 milli on ar e borr o w ed at an annual int er est rat e of 6 . 5% to be r epaid in 60 monthly pa y ments , what should be the monthl y pay ment? F or the debt to be totall y r epaid in 6 0 months, the f utur e value s of the loan should be z er o.
Pa g e 6 - 1 1 pay m ents . Suppo se that w e use 2 4 peri ods in the firs t line of the amorti z ation scr een, i.e ., 24 @@OK @@ . T hen , pr ess @@AMOR@@ . Y ou w ill get the f ollo w ing re su l t : T his scr een is interpr eted as indi cating that after 2 4 months of pa y ing back the debt , the borr ow er has paid up US $ 7 2 3,211.
Pa g e 6 - 1 2 ˜ Skip P MT , since w e w ill be sol v ing fo r it 0 @@OK@@ Enter FV = 0, the opti on End is highlight ed @@CHOOS ! — @@OK@@ Change pa yme nt option t o Begin — š @@S OLVE! Highl ight P MT and so lv e f or it T he scr een no w sho ws the v alue of P MT as –38 , 9 21.
Pa g e 6 - 1 3 ™ ‚í Enter a comma ³ ‚ @@PYR@ @ Enter name o f var iable P YR ™ ‚í Enter a comma ³ ‚ @@FV@ @ . Enter name o f var iable FV ` Ex ec ute P URG E command T he fo llo w ing two s cr een shots sho w the P URGE co mmand for purging all the v ari ables in the dir ectory , and the r esul t af t er e x ecu ting 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 w ly c r eated E Q v ari able: Then , enter the SOL VE env ironment and select S olv e equation… , by using: ‚Ï @@OK@@ .
Pa g e 6 - 1 5 This , ho w ev er , is not the only po ssible soluti on fo r this equation . T o obtain a negati ve s olutio n, f or e xampl e, ent er a negati v e number in the X: f ield be for e sol ving the equati on. T r y 3 @@@OK@@ ˜ @SOLVE@ . T he soluti on is no w X: - 3.
Pa g e 6 - 1 6 T he equation is her e e xx is the unit str ain in the x -directi on , σ xx , σ yy , and σ zz , are the nor mal str esses on the partic le in the dir ecti ons of the x -, y-, and z -.
Pa g e 6 - 1 7 W ith the ex : fi eld highli ghted , pr ess @SOLVE @ to solve f or ex : T he soluti on can be seen fr om within the S OL VE EQU A TION in put for m b y pr essing @EDI T wh il e t he ex: f ield is highli ghted. T he r esulting value is 2.
Pa g e 6 - 1 8 Spec ifi c ener gy in an open c hannel is def ined as the ener g y per unit w eight measur ed w ith r es pect to the c hannel bottom . Let E = s pec if ic ener g y , y = c hannel depth .
Pa g e 6 - 1 9 Θ Solv e for y . T he r esult is 0.14 9 8 3 6.., i .e., y = 0.14 9 8 3 6 . Θ It is kno w n, how ev er , that ther e are ac tuall y two s oluti ons av ailable f or y in the spec if ic ener gy equatio n.
Pa g e 6 - 2 0 In the ne xt e x ample w e w ill use the D ARC Y functi on f or f inding fr ic tion fact ors in pipeline s. T hus , w e def ine the f unctio n in the follo wing f r ame .
Pa g e 6 - 2 1 Ex ample 3 – F low in a pipe Y ou may w a nt t o cr eate a separ ate sub-dir ectory (PIPE S) to tr y this e x ample . T he main equation go v ernin g flo w in a p ipe is, o f course , the D ar cy- W eisbac h equati on.
Pa g e 6 - 2 2 T he combined equation has pr imitiv e v ari ables: h f , Q , L, g, D, ε , and Nu . Lau nch t he num erical solver ( ‚Ï @@ OK@@ ) to s ee the primiti ve v ari ables listed in the S OL VE E QU A TION in put f orm: Suppo se that w e us e the value s hf = 2 m, ε = 0.
Pa g e 6 - 23 Ex ample 4 – Uni versal gr av itation Ne wton ’s la w of uni v ersal gr av itati on indicat es that the magnitude of the attr acti v e f or ce betw een tw o bodi es o f mas ses m 1 a.
Pa g e 6 - 24 Sol v e for F , and pr ess to r eturn to norm al calculator dis play . T he sol ution 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 ype an equati on, sa y X^2 - 12 5 = 0, dir ectl y on the stac k, and pr es s @@@OK@@@ . At this point the equati on is r eady f or solu tion . Alte rnati v ely , y ou can acti vate the equati on wr iter after pr essing @E DIT to enter y our equation .
Pa g e 6 - 26 The S OL VE soft menu The SOL V E soft menu a llow s a ccess to some of the numerical solv er fu nctions thr ough the soft men u k e ys . T o access this men u 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 v ate the S OL VE soft men u .
Pa g e 6 - 27 Ex ample 1 - Sol v ing the equati on t 2 -5t = - 4 F or ex ample , if you s tor e the equati on ‘t^2 -5*t=- 4’ into E Q, and pr ess @) SOLVR , it w ill acti v ate the f ollo wing menu: T his result indi cates that y ou can sol v e for a v alue of t f or the equation lis ted at the top of the displa y .
Pa g e 6 - 2 8 Y ou can also sol ve mor e than one equati on by s olv ing one equation at a time , and r epeating the pr ocess until a soluti on is fo und .
Pa g e 6 - 2 9 Using units with the SOL VR sub-menu T hese ar e some rule s on the us e of units w ith the S OL VR sub-men u: Θ Enter ing a guess w ith units fo r a gi ven v ar ia ble , w ill intr oduce the u se of thos e units in the soluti on.
Pa g e 6 - 3 0 T his functi on 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 or o f i t s ro o ts [r 1 , r 2 , …, r n ].
Pa g e 6 - 3 1 Press J to e x it the S OL VR en vi r onment . F ind y our w ay bac k to the TVM su b- menu w i thin the S OL VE sub-menu to try the other functi ons a vailable . Function T VMROO T This fun c tion requires as argument t he na me of one of the v ariables in t he T VM pr oblem .
Pa g e 7- 1 Chapter 7 Solv ing multiple equations Man y pr oblems of sc i ence and engineer ing req uir e the simultaneous solu tions of mor e than one equation . The calc ulator pro v ides s ev er al pr ocedur es f or solv ing multiple equations as pr esented belo w .
Pa g e 7- 2 Use co mmand S OL VE at this point (f r om the S . SL V menu: „Î ) A fter about 40 s econds, may be more , yo u get as re sult a list: { ‘t = (x -x0)/(C OS( θ 0)*v0)’ ‘ y0 = (2*C.
Pa g e 7- 3 the cont ents of T1 and T2 to the stac k and adding and subtr acting them . Here is ho w to do it w ith the eq uation w r iter : Enter and s tor e ter m T1: Enter and st or e ter m T2: Notice that w e ar e using the RPN mode in this ex ample, ho we v er , the pr ocedur e in the AL G mode should be v ery simi lar .
Pa g e 7- 4 Notice that the r esult include s a vec tor [ ] contained w ithin a list { }. T o r emo ve the list s y mbol , use μ . F inally , to decompose the v ector , use f uncti on OB J .
Pa g e 7- 5 Ex ample 1 - Ex ample from the help facilit y As w ith all functi on entr ie s in the help fac ility , ther e is an e x ample at t ached to the MSL V entr y as sho wn abo v e . Notice that f uncti on MSL V r equir es thr ee ar guments: 1. A v ector co ntaining the equati ons, i .
Pa g e 7- 6 disc har ge (m 3 /s or ft 3 /s) , A is the c ro ss-sec tional ar ea (m 2 or ft 2 ), C u is a coeff ic ient that depends on the s ys tem of units (C u = 1. 0 fo r the SI, C u = 1.4 8 6 fo r the English s ys tem o f units) , n is the Manning’s coe ff ic ient , a measure o f the c ha nnel surf ace r oughness (e .
Pa g e 7- 7 μ @@@EQ1@@ μ @@@EQ2@ @ . T he equatio ns ar e listed in the s tack a s follo ws (small fo nt option s elected): W e can see that these eq uations ar e indeed giv en in ter ms of the pr imitiv e var iable s b, m , y , g, S o , n, C u , Q, and H o .
Pa g e 7- 8 Ne xt , we ’ll enter var iable E QS: LL @@EQS@ , fo llow ed by v ector [y ,Q]: ‚í„Ô~„y‚í~q™ and b y t he in itial gu esses ‚í„Ô5‚í 10 . Bef or e pre ssing ` , the sc r een w ill look like this: Press ` to sol v e the sy stem of equatio ns.
Pa g e 7- 9 T he re sult is a list of thr ee v ector s. T he fir st vec tor in the list w ill be the equations sol ved . The second v ector is the list of unkno wns. The thir d v ector r epres ents the solu tion . T o be able to see the se v ector s, pr es s the do wn-a r r o w k e y ˜ to acti v ate the line editor .
Pa g e 7- 1 0 T he cosine la w indicat es 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 ⋅ co s γ . In or der to sol v e an y tr iangle , you need to kno w at least thr ee of the fo llo w ing si x v ar iable s: a, b, c, α, β, γ .
Pa g e 7- 1 1 ‘SIN( α )/a = SIN( 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*CO S( 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 i ables me nu . Y our men u should sho w the va riab le s @LVARI! ! @TITLE @@EQ@ @ . Preparing to run t he ME S T he next s tep is to acti vate the ME S and tr y one s ample soluti on.
Pa g e 7- 1 3 Let ’s tr y a sim ple soluti on of Cas e I, using a = 5, b = 3, c = 5 . Us e the fo llo w ing entr ies: 5 [ a ] a:5 is listed in the top left corner of the displa y . 3 [ b ] b:3 is listed in the top left corner of the displa y . 5 [ c ] c:5 is listed in the top left corner of the display .
Pa g e 7- 1 4 Pr es sing „ @@ALL@@ will s olv e for all the v ar iables , tempor aril y sho w ing the intermediate r esults. Pr ess ‚ @@ALL @@ to see t he solu tions: When done , pres s $ to r eturn to the ME S env ironment . Pre ss J to ex it t he ME S en v ir onment and r eturn to the nor mal calc ulator displa y .
Pa g e 7- 1 5 Progr amming the MES triangle solution using User RPL T o fac ilitate acti vating the ME S for f utur e soluti ons , w e will c r eate a pr ogram that w ill load the ME S wi th a single ke y str oke .
Pa g e 7- 1 6 Use a = 3, b = 4, c = 6 . The soluti on pr ocedure us ed her e consists of so lv ing fo r all var ia bles at once , and then r ecalling the soluti ons to the st ack: J @TRISO T o clear up data and r e -start ME S 3 [ a ] 4 [ b ] 6 [ c ] T o en ter data L T o mo ve t o the next v ar iable s menu .
Pa g e 7- 1 7 Adding an INFO but ton to your dir ec tory An inf ormati on button can be us ef ul for y our dir ectory to help y ou r emember the oper ation o f the func tions in the dir ectory . In this dir ecto r y , all w e need to r emember is to pr ess @TRISO to get a tr iangle s olution s tarted.
Pa g e 7- 1 8 An e xplanatio n of the v ari ables f ollo ws : SOL VEP = a progr am that tr iggers the multiple equati on sol v er fo r the partic ular s et of equations s tor ed in var iable PEQ ; NAME = a v ari able stor ing the name of the multiple equati on sol ve r , namely , "v el .
Pa g e 7- 1 9 Notice that afte r y ou enter a partic ular value , the calc ulator displa y s the v ari able and its value in the upper le f t co rner o f the displa y . W e hav e no w enter ed the kno wn v a r iables . T o calc ulate the unkno w ns w e can pr oceed in two 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 o f calculat or’s ob ject that can be u sef ul f or data pr oces sing and in pr ogr amming.
Pa g e 8 - 2 T he fi gur e belo w sho w s the RPN stac k befo r e pre ssing the K key: Composing and decomposing lists Compo sing and decompo sing lists mak es sense in RPN mode onl y . Under such oper ating mode , decomposing a list is ac hie v ed by u sing functi on OB J .
Pa g e 8 - 3 In RPN mode , the follo wi ng scr een show s the thr ee lists and their name s read y to be stor ed. T o stor e the lis ts in this case you need to pr ess K three times . Changing sign T he sign - change k e y ( ) , w hen applied to a list of number s, w i ll c hange the sign o f all elements in the list .
Pa g e 8 - 4 Subtr actio n, multiplicati on, and di v ision o f lists of numbers o f the same length pr oduce a list of the s ame length with ter m-b y- te rm oper ations .
Pa g e 8 - 5 AB S EXP and LN L OG and ANTIL OG S Q and squar e root SIN, ASIN COS, ACOS T AN, A T AN INVER SE (1/x) Real number functions from the MTH menu F uncti ons of inter est fr om the MTH me nu.
Pa g e 8 - 6 T ANH, A T ANH S IGN, MANT , XP ON IP , FP FL OOR, CEIL D R, R D Ex amples of functions t hat use tw o arguments T he scr een shots belo w show appli cations o f the functi on % to list ar guments . F unction % r e quir es t w o ar guments.
Pa g e 8 - 7 %({10,20, 30},{ 1,2 , 3}) = {%(10,1),%(20,2),%(3 0, 3)} T his desc r iption o f func tion % f or list ar guments sh o ws the gener al pattern of e valuati on of an y f uncti on w ith two ar guments when one or both ar guments ar e lists .
Pa g e 8 - 8 T he follo w ing ex ampl e sho w s applicati ons of the f uncti ons RE(R eal part) , IM(imaginar y part) , AB S(magnitude) , and AR G(argument) o f comple x numbers .
Pa g e 8 - 9 T his menu cont ains the fo llo w ing func tio ns: Δ LIS T : C alc ulate incr ement among consec uti ve elements in list Σ LIS T : Ca lculat e summation o f elemen ts in the list Π LIS.
Pa g e 8 - 1 0 M anipulating elements of a list T he PR G (pr ogr amming) menu inc ludes a LI S T sub-m enu w ith a n umber of func tions t o manipulate ele ments of a list .
Pa g e 8 - 1 1 F uncti ons GET I and PUT I , also av ailable in sub-me nu PR G/ ELEMENT S/, ca n also be us ed to ext rac t and place elements in a list .
Pa g e 8 - 1 2 SE Q is use ful t o pr oduce a list of v alues gi ve n a par ti c ular expr essi on and is desc r ibed in mor e detail her e . T he SEQ f uncti on tak es as ar guments an e xpr essi on .
Pa g e 8 - 1 3 In both case s, y ou can ei ther t y pe out the MAP command (as in the e x amples abo v e) or select the command fr om the CA T menu . T he follo w ing call to func tion MAP us es a pr .
Pa g e 8 - 1 4 to r eplace the plus sign (+) w ith ADD: Ne xt , we s tor e the edited e xpres sion in to v ari able @@@G@@@ : Ev aluating G(L1,L2) now pr oduces the f ollo w ing r esult: As an alter nati ve , y ou can define the f uncti on w ith ADD rathe r than the plus sign (+), fr om the s tart, i .
Pa g e 8 - 1 5 Applications of lists T his sectio n show s a couple of appli cations o f lists to the calc ulation o f statisti cs of a sa mple. B y a samp le w e u nderstand a list of v alu es , sa y , {s 1 , s 2 , …, s n }.
Pa g e 8 - 1 6 3 . Di v ide the r esult abo ve by n = 10: 4. Appl y the INV() func tion to the lat est r esult: T hus , the harmonic mean o f list S is s h = 1.63 4 8… Geometric mean of a list T he geometri c mean of a sample is def ined as T o f ind the geometri c mean of the list stor ed in S , w e can use the f ollo w ing pr ocedur e: 1.
Pa g e 8 - 1 7 T hus , the geometri c mean of list S is s g = 1.00 3 203… W eighted aver age Suppo se that the data in list S , def ined abo ve , namel y : S = {1,5,3,1 ,2,1,3,4,2,1 } is affec ted b.
Pa g e 8 - 1 8 3. U se f un c t io n Σ LIS T , once mo r e , to calc ulate the denominat or of s w : 4. Use the e xpre ssi on ANS( 2)/ANS(1) to cal culat e the w ei ghted a v er age: Th us, the w ei ghted av er age of list S w ith w eights in lis t W is s w = 2 .
Pa g e 8 - 1 9 T he clas s mar k data can be st ored in v ari able S , while the fr equency coun t can be stor ed in var iable W , as follo ws: Gi ven the list of class marks S = {s 1 , s 2 , …, s n.
Pa g e 8 - 2 0 T o calc ulate this las t r esult , w e can us e the fo llow ing: T he standar d dev iati on of the gr ouped data is the sq uar e r oot of the var iance: N s s w w s s w V n k k k n k k.
Pa g e 9 - 1 Chapter 9 V ec tors T his Chapter pr o v ides e x amples o f enter ing and operating w ith vect ors , both mathematical v ector s of man y elements, as w ell as ph y sical v ectors of 2 and 3 components .
Pa g e 9 - 2 wher e θ is the angle betw een the tw o vec tors . The c r os s pr oduct pr oduces a vec tor A × B who se magnitude is | A × B | = | A || B |sin( θ ) , and its dir ecti on is gi v en by the s o -ca lled r ight -hand r ule (consult a textbook on Math , Ph y sics , or Mechani cs to s ee this oper ation illu str ated gr aphicall y) .
Pa g e 9 - 3 Stor ing vectors int o var iables V ector s can b e s tor ed into var iables . The sc r een shots belo w show the v ectors u 2 = [1, 2] , u 3 = [-3, 2, -2] , v 2 = [3 ,-1] , v 3 = [1, -5, 2 ] stor ed into v ariables @ @@u2@@ , @@@u3@@ , @@@v2@@ , and @@@v3@@ , r especti v el y .
Pa g e 9 - 4 Th e ← WID k ey is u sed to decr ease the w idth of the columns in the spr eadsheet . Pr ess this k ey a couple o f times to see the column w idth dec r ease in y our Matr i x W r iter . Th e @ W I D → k ey is used to incr ease the w idth of the columns in the spr eadsheet .
Pa g e 9 - 5 Th e @+ROW@ k e y w ill add a r o w full o f z er os at the location o f the select ed cell o f the spr eadsheet . Th e @-ROW k ey w ill delete the r o w corr esponding t o the selec ted cell of the spr eadsheet . Th e @+COL@ k e y w ill add a column full o f z er os at the location o f the selec ted cell of the spr eadsheet .
Pa g e 9 - 6 Building a vector with ARR Y Th e fu nct ion → ARR Y , av ailable in the func tion catalog ( ‚N‚é , us e —˜ to locat e the functi on), ca n als o be u sed to build a v ect or or ar r ay in the f ollo w ing wa y . In AL G mode , enter ARR Y( v ector el ements, numb er of elements ) , e .
Pa g e 9 - 7 In RPN mode , the functi on [ → ARR Y] take s the obj ects fr om stac k lev els n+1, n, n- 1 , …, dow n to st ack le ve ls 3 and 2 , and con v erts them into a v ector of n elements .
Pa g e 9 - 8 Highli ghting the entir e e xpr essio n and using the @EV AL@ soft men u ke y , w e get the re su l t : -1 5 . T o r eplace an e lement in an arr a y use f uncti on PUT (y ou can find it in the func tion cat alog ‚N , or i n the P RG/LI S T/ELEMENT S sub-men u – the later w as intr oduced in Chapter 8).
Pa g e 9 - 9 Simple oper ations with vectors T o illus tr ate oper atio ns w ith vec tor s we w ill use the v ector s A, u2 , u3, v2 , and v3, sto r ed in an ear lier e xe r c ise . Changing sign T o change the si gn of a v ect or use the k e y , e .
Pa g e 9 - 1 0 Absolute value func tion T he absolute v alue func tion ( ABS), when appli ed to a vec tor , pr oduces the magnitude of the v ector . F or a v ector A = [ A 1 ,A 2 ,…,A n ], the magnitude is def ined as . In the AL G mode , enter the functi on name f ollo we d by the v ector ar gument .
Pa g e 9 - 1 1 Dot pr oduct F uncti on DO T is used to calc ula t e the dot pr oduct of tw o vec tors o f the same length. S ome e xample s of applicati on of f uncti on DO T , using the v ecto rs A, u2 , u3, v2 , and v3, stor ed ear lie r , ar e show n next in AL G mode .
Pa g e 9 - 1 2 In the RPN mode , appli cation o f func tion V w ill list the components o f a ve ctor in the st ack , e .g., V (A ) will pr oduce the f ollo w ing outpu t in the RPN stack (vector A is li sted in stack lev el 6: ).
Pa g e 9 - 1 3 When the r ect angular , or Cartesi an, coor dinate s yst em is select ed, the t op line of the displa y w ill show an XY Z fi eld , and any 2 -D or 3-D v ector ent er ed in the calc ula t or is r eproduced as the (x ,y ,z) components o f the vecto r .
Pa g e 9 - 1 4 T he fi gur e belo w sho w s the tr ansfor mation o f the v ector f r om spher ical to Cartesi an coor dinates , w ith x = ρ sin( φ ) cos( θ ), y = ρ sin ( φ ) cos ( θ ), z = ρ co s( φ ) . F or this case , x = 3 .204 , y = 1.4 9 4, and z = 3 .
Pa g e 9 - 1 5 equi vale nt (r , θ ,z) with r = ρ sin φ , θ = θ , z = ρ co s φ . F or e x ample , the f ollo w ing f igur e sho ws the v ector ent er ed in spher ical coor d i nates, and tr ansf ormed to polar coor dinates . F or this case , ρ = 5, θ = 2 5 o , and φ = 4 5 o , while the tr ansfor mation sho ws tha t r = 3 .
Pa g e 9 - 1 6 Suppos e that yo u want t o find the angle between v e c tors A = 3 i -5 j +6 k , B = 2 i + j -3 k , y ou could tr y the f ollo w ing oper ati on (angular measur e set to degr ees) in AL G mode: 1 - Enter vec tors [3,-5, 6], pr ess ` , [2 ,1,-3], pr ess ` .
Pa g e 9 - 1 7 Thu s, M = (10 i +2 6 j +2 5 k ) m ⋅ N. W e kno w that the magnitud e o f M is such that | M | = | r || F |sin( θ ), wher e θ is the angle between r and F .
Pa g e 9 - 1 8 Ne xt , we calc ulate v ector P 0 P = r as ANS(1) – AN S(2), i .e ., F inally , w e tak e the dot pr oduct o f ANS(1) and ANS( 4) and mak e it equal t o z er o to complete the oper at.
Pa g e 9 - 1 9 In this secti on w e w ill show ing yo u wa y s to transf or m: a column vec tor into a r o w vec tor , a r o w ve ctor int o a column vec tor , a list into a vec tor , and a vec tor (or matr i x) into a list . W e f irst demons tr ate thes e transf ormations u sing the RPN mode.
Pa g e 9 - 2 0 If w e no w appl y func tion OB J once mor e , the list in st ack le vel 1:, {3 .}, w ill be decomposed as follo ws: Function LIS T T his functi on is used to c r eate a list gi ven the eleme nts of the list and the list length or si z e.
Pa g e 9 - 2 1 3 - Use f uncti on ARR Y to build the column vec tor T hese thr ee steps can be put t ogether into a U serRP L pr ogr am, e nter ed as fo llo ws (in RPN mode , still) : ‚å„° @.
Pa g e 9 - 2 2 2 - Use f uncti on OB J to d ecompose the l ist in stac k le vel 1: 3 - Pr ess the de lete k e y ƒ (also kno wn a s functi on DROP) t o eliminate the number in st ack lev el 1: 4 -.
Pa g e 9 - 23 Th is variab le, @@CXR@@ , can no w be used t o dir ectl y transf orm a column v ector to a r o w vec tor . In RPN mode , enter the column v ector , and then pre ss @@CXR@ @ .
Pa g e 9 - 24 A ne w v ar iabl e , @@LXV@@ , will be av ailable in the soft menu labels after pr es sing J : Press ‚ @@LXV@@ to see the pr ogram co ntained in the v ari able LXV : << OBJ 1 LIST RRY >> Th is va r i ab le, @@LXV@@ , can no w be used to dir ectly tr ansfor m a list into a v ector .
Pa g e 1 0 - 1 Chapter 10 ! Cr eating and manipulating matrices T his chapte r show s a number of e xamples aimed at c reating matr ices in the calc ulator and demons tr ating manipulation of matr i x elements . Definitions A matr i x is simpl y a re ctangular arr ay of ob jec ts (e.
Pa g e 1 0 - 2 Entering matr ices in the stac k In this secti on w e pr esent tw o diffe r ent methods to enter matr ices in the calc ulator st ack: (1) using the Matr i x W rit er , and (2) typ ing the matri x dir ectl y i nto th e sta ck.
Pa g e 1 0 - 3 If y ou ha ve s elected the t extbook displa y option (using H @) DISP! and c hecking off Textbook ) , the matr ix w ill look lik e the one sho w n abo ve . O the r w ise , the displa y w ill sho w: T he displa y in RPN mode w ill look very similar to these .
Pa g e 1 0 - 4 or in the MA TRICE S/CREA TE menu a v ailable thr ough „Ø : T he MTH/M A TRI X/MAKE sub menu (let’s call it the MAKE menu) contains the fo llo w ing fu ncti ons: w hile the MA TRIC.
Pa g e 1 0 - 5 As y ou can see f r om e xploring the se menu s (MAKE and CREA TE) , the y both hav e the same f uncti ons GET , GET I, P UT , P U T I , SUB , REP L , RDM, R ANM, HILBERT , V ANDERMONDE , IDN, CON , → DIA G , and DI A G → .
Pa g e 1 0 - 6 Functions GET and P UT F uncti ons GET , GE TI , P UT , and P UT I, oper ate w ith matri ces in a similar manner as w ith lists or v ectors , i .
Pa g e 1 0 - 7 Notice that the s cr een is prepar ed fo r a subseq uent appli cation o f GET I or GE T , b y incr easing the column index o f the ori ginal re fer ence b y 1, (i .e., f r om {2 ,2} to {2 , 3}) , w hile sho w ing the ex trac ted v alue , namel y A(2 ,2) = 1.
Pa g e 1 0 - 8 If the ar gument is a r eal matri x, TRN simpl y pr oduces the tr anspos e of the r eal matr i x. T r y , f or e xample , TRN(A ) , and compar e it w ith TRAN(A ) . In RPN mode , the tr ansconjugat e of matri x A is calc ulated by using @@@A@@@ TRN .
Pa g e 1 0 - 9 In RPN mode this is accomplished b y using {4 ,3} ` 1.5 ` CON . Function IDN F uncti on ID N (IDeNtity matri x) cr eates an identity matri x giv en its si z e. R ecall that an identity matr i x has to be a squar e matri x , ther ef or e , only one v alue is r equir ed to des cr i be it completel y .
Pa g e 1 0 - 1 0 v ector ’s dimensi on, in the latte r the number of r o ws and columns of the matr ix . T he follo wing e x amples illus tr ate the use o f functi on RDM: Re -dimensioning a vector .
Pa g e 1 0 - 1 1 If using RPN mode , we a ssume that the matr i x is in the stac k and us e {6} ` RDM . Function R ANM F uncti on RANM (RANdom Matr i x) w ill gener ate a matr i x w ith 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 or iginal 2 × 3 matr i x is alread y in the stac k, u se {1,2} ` {2,3} ` SUB . Function REP L F uncti on REPL r e place s or inserts a sub-matr i x into a lar ger one .
Pa g e 1 0 - 1 3 In RPN mode , wi th the 3 × 3 matri x in the stac k, w e simply hav e to acti v ate fu nct ion DI G to obtain the same r esult as abo ve .
Pa g e 1 0 - 1 4 F or ex ample, the f ollo wing command in AL G mode for the list {1,2 , 3, 4}: In RPN mode, ente r {1,2,3,4} ` V NDER MONDE . Function HIL BER T F uncti on HI LBER T cr eates the Hilbert matri x cor re sponding to a dimensi on n.
Pa g e 1 0 - 1 5 enter ed in the displa y as you perf or m those k ey str ok es . F irs t , we pr esent the steps ne cessar y to produce p r og r am C RMC. Lists r epr esent columns of the matri x Th e p r o gra m @CRMC allo w s yo u to put together a p × n matr i x (i .
Pa g e 1 0 - 1 6 ~„n # n „´ @) MATRX! @ ) COL! @COL! COL ` Pr ogr am is display ed in le v 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 pr ogram u se J ‚ @CRMC .
Pa g e 1 0 - 1 7 Lists r epr esent ro w s of the matrix T he pre vi ous pr ogram can be easil y modif ied to c r eate a matri x when the input lists w ill become the r o ws o f the r esulting matr i x. T he only ch ange to be perfor med is to c h ange COL → for ROW → in the pr ogr am listing.
Pa g e 1 0 - 1 8 Both appr oa c hes w ill show the same func tions: When s y stem f lag 117 is set to S OFT menus , the COL menu is accessible thr ough „´ !) MATRX ) !)@@COL@ , or thr ough „Ø !) @CREAT@ ! ) @ @COL@ . Both appr oache s w ill sho w the same s et of f uncti ons: The op er ation of these functions is presented b elo w .
Pa g e 1 0 - 1 9 In this r esult , the fir st column occ upi es the highe st stac k lev el af t er decompositi on , and stac k lev el 1 is occu pi ed by the n umber of co lumns of the or iginal matr ix . T he matr i x does not surv i v e decompositi on, i .
Pa g e 1 0 - 2 0 In RPN mode , enter the matr i x fir st , then the v ector , and the column n umber , bef or e appl y ing fu nction C OL+. T he fi gur e belo w sho w s the RPN stac k bef or e and after appl y ing functi on COL+.
Pa g e 1 0 - 2 1 In RPN mode , functi on CS WP lets y ou s wap the columns of a matri x list ed in stac k le vel 3, who se indi ces ar e listed in s tac k lev els 1 and 2 .
Pa g e 1 0 - 22 When s y st em flag 117 is set to S OFT menus , the R O W menu is acces sible thr ough „´ !) MATRX ! )@@ROW@ , or thr ough „Ø !) @CREAT@ ! ) @@ ROW@ . Both appr oache s w ill sho w the same s et of f uncti ons: The op er ation of these functions is presented b elo w .
Pa g e 1 0 - 2 3 matr i x does not survi ve decompo sition , i .e ., it is no longer a vailable in the stack. Function RO W → Fu n c ti o n ROW → has the opposite e ffec t of the f unctio n → R O W , i .
Pa g e 1 0 - 24 Function RO W- F uncti on RO W - tak es as ar gument a matri x and an integer number r epr esenting the positi on of a r o w in the matri x . The f unctio n re turns the or iginal matr i x , minu s a ro w , as w ell as the e xtr acted r o w sho w n as a v ector .
Pa g e 1 0 - 2 5 As y ou can see , the r ow s that or iginally occ up ied po sitions 2 and 3 ha ve been s wa pped. Function RCI F uncti on R CI stands f or m ultiply ing R ow I by a C onst ant value and r eplace the r esulting r o w at the same location .
Pa g e 1 0 - 26 In RPN mode , enter the matr i x fir st , fo llo w ed by the constant v alue , then b y the r o w to be multiplied b y the co nstant v alue , and f inally ente r the ro w that will be r eplaced.
P age 11-1 Chapter 11 M atr ix Oper ations and Linear Algebra In Chapte r 10 we intr oduced the concept of a matri x and pr esen ted a number of f uncti ons f or enter ing, c r eating, o r manipulating matri ces . In this Chapt er w e pr esen t ex a m ples of matr i x oper ations and appli cations t o problems of linear algebr a.
P age 11-2 Addition and subtr ac tion Consi der a pair of matr ices A = [a ij ] m × n and B = [b ij ] m × n . Addition and subtr action of the se tw o matri ces is onl y pos sible if they ha v e the same number of r ow s and columns . The r esulting matr i x , C = A ± B = [c ij ] m × n has elements c ij = a ij ± b ij .
P age 11-3 B y combining addition and subtr acti on w ith multiplicati on b y a scalar w e can fo rm linear combinati ons of matr ices of the same dimensions , e .g., In a linear combinati on of matr i ces, w e can multipl y a matr i x by an imaginary number to obtain a matri x of comple x n umbers, e .
P age 11-4 Matrix multiplication Matri x multipli cation is def ined 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 . Noti ce that matr i x multiplicati on is onl y possible if the number of columns in the f ir st oper and is equal to the number o f r o ws of the second oper and .
P age 11-5 (another r o w vect or). F or the calculator to identify a r o w vector , y ou must us e double br ack ets to enter it: T erm-b y-term multiplication T erm-b y- t erm multiplicati on of two matr ice s of the same dimensions is pos sible thr ough the us e of f unction HAD AMARD .
P age 11-6 In algebr aic mode , the k e y str ok es ar e: [enter or se lect the matr i x] Q [enter the po w er] ` . In RPN mode , the k ey str ok es ar e: [ent er or select the matr i x] † [ent er the po we r] Q` . Matri ces can be r aised to negati ve po wer s.
P age 11-7 T o v er if y the pr operties of the in v erse matr ix , consider the f ollo w ing multiplicati ons: Characteri zing a matr ix (T he matri x NORM menu) T he matri x NORM (NORMALIZE) menu is.
P age 11-8 Function ABS F uncti on ABS calc ulate s what is kno w n as the F r obenius nor m of a matr i x. F or a matr i x A = [a ij ] m × n , the F robeniu s norm of the matr ix is def ined as If t.
P age 11-9 Functions RNRM and CNRM F uncti on RNRM r etur ns the Ro w NoRM o f a matr i x , whil e functi on CNRM r eturns the C olumn NoRM of a matr i x. Ex amples, Singular value decomposition T o unders tand the oper ation o f F uncti on SNRM, w e need to intr oduce the concept of matr i x decompositi on.
P age 11-10 Function SR AD F uncti on SRAD de termine s the Spectr al R ADius o f a matri x, de fined as the lar gest of the a bsolute v alues of its e igen v alues .
P age 11-11 T ry the follo wing ex erc ise f or matri x condition n umber on matr i x A3 3 . The conditi on number is COND( A3 3) , ro w norm , and column norm f or A3 3 ar e sho w n to the le ft .
P age 11-12 F or ex ample , try finding the r ank for the matr i x: Y ou w ill f ind tha t the rank is 2 . That is becaus e the second r o w [2 , 4, 6] is equal to the f irs t r ow [1,2 , 3] multiplied b y 2 , thu s, r o w tw o is linearl y dependent of r o w 1 and the max imum number o f linearl y independent r o ws is 2 .
P age 11-13 The determinant of a matr ix T he deter minant of a 2x2 and o r a 3x3 matr i x ar e r epr esen ted b y the same arr angement of elemen ts of the matr ices , but enc losed be t w een verti cal lines, i.
P age 11-14 Function TR A CE F uncti on TRA CE calc ulates the tr ace of sq uare matr ix , def ined as the sum of the elements in its main diagonal , or . Ex amples: F or squar e matr i ces of hi gher or der de terminants can be calc ulated by u sing smaller or der deter minant called co fact ors .
P age 11-15 Function TR AN F uncti on TRAN re turns the tr anspo se of a r eal or the conj ugate tr anspo se of a comple x matri x. TRAN is equi v alent t o TRN.
P age 11-16 MAD and RSD ar e relat ed to the soluti on of s y ste ms of linear equati ons and wil l be pr esen ted in a subseq uent secti on in this Chapt er .
P age 11-17 T he implementati on of func tion L CXM fo r this case r equir es y ou to ente r: 2`3`‚ @@P1@@ LCXM ` T he follo w ing fi gur e show s the RPN s tack be fo r e and af t er appl y ing func tion LC X M : In AL G mode , this e x ample can be obtained b y using: T he progr am P1 must still ha ve been c r eated and stor ed in RPN mode .
P age 11-18 , , Using the num er ical solv er f or linear s ystems Ther e are man y w a ys to so lv e a s y stem of linear equatio ns w ith the calculator . One pos sibility is thr ough the numer i cal sol v er ‚Ï . F r om the numer ical sol v er scr een, sho w n belo w (left) , selec t the option 4.
P age 11-19 T his sy st em has the same number o f equations as o f unknow ns , and will be r efer r ed to as a sq uare s ys tem. In gener al, ther e should be a unique solu tion to the s y stem .
P age 11-20 T o chec k that the solu tion is cor r ect , enter the matr i x A and multiply times this solu tion v ector (e xample in algebr aic mode): Under-det ermined s ystem T he sy stem of li near.
P age 11-21 T o see the details of the so lutio n vect or , if needed, pr ess the @EDIT! butt on. T his w ill acti vat e the Matri x W r iter . W ithin this env ir onment , use the r ight- and left - arr o w k e y s to mo v e about the v ector : T hus , the solution is x = [15 .
P age 11-2 2 Let ’ s store the latest r esult in a var iable X, and the matr i x into var iable A, as fo llo w s: Press K~x` to stor e the solution v ector into var iable X Press ƒ ƒ ƒ to clear t.
P age 11-2 3 can be w ritten as the matri x equati on A ⋅ x = b , if This s yst em has mor e equations than unkno w ns (an ov er -determined s yste m) .
P age 11-2 4 Press ` to r eturn to the numer ical so lv er en v iro nment . T o check that the solu tion is corr ect , try the follo wing: • Press —— , to highlight the A: field . • Press L @CALC@ ` , to copy matri x A onto the stack . • Press @@@OK@@@ to r eturn to the n umer ical sol v er en vir onment .
P age 11-2 5 • If A is a squar e matr i x and A is non- singul ar (i .e ., it’s in ver se matr i x e xis t , or its determinant is non - z er o) , LS Q r etur ns the ex act so lution to the linear s y stem .
P age 11-2 6 Under-det ermined s ystem Consi der the s ys tem 2x 1 + 3x 2 –5x 3 = -10, x 1 – 3x 2 + 8x 3 = 85, wi th T he soluti on using LS Q is sho wn ne xt: Over-determin ed s ystem Consi der the s ys tem x 1 + 3x 2 = 15, 2x 1 – 5x 2 = 5, -x 1 + x 2 = 2 2 , wi th T he soluti on using LS Q is sho wn ne xt: .
P age 11-2 7 Compar e these thr ee solu tions w ith the ones calc ulated wi th the numer ical solv er . Solution with the in v erse matri x T he soluti on to the s ys tem A ⋅ x = b , wher e A is a squar e matri x is x = A -1 ⋅ b . T his re sults fr om multiply ing the f irst eq uation b y A -1 , i .
Pa g e 1 1 - 2 8 T he pr ocedure f or the cas e of “ di v iding ” b by A is illustr ated belo w for the case 2x 1 + 3x 2 –5x 3 = 13, x 1 – 3x 2 + 8x 3 = -13, 2x 1 – 2x 2 + 4x 3 = -6 , The pr ocedu r e is sho wn in the follo wing s cr een shots: T he same solu tion as f ound abo ve w ith the in ver se 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]] `/ T he re sult of this oper ation is: Gaussian and Gauss-Jordan elimination Gaus sian elimination is a pr ocedure b y w hi.
P age 11-30 T o start the pr ocess o f forw ar d elimination , w e di vi de the f irst equati on (E1) b y 2 , and s tor e it in E1, an d sho w the thr ee equ ation s again to pr oduce: Ne xt, w e r eplac e the s econd equation E2 b y (equ ati on 2 – 3 × equation 1, i .
P age 11-31 an e xpre ssi on = 0. Thu s, the las t set of equati ons is interpr eted to be the fo llo w ing equiv alent set of equatio ns: X +2Y+3Z = 7 , Y+ Z = 3, - 7Z = -14.
P age 11-3 2 T o obtain a solution t o the sy stem matr i x equation us ing Gaussi an elimination , we f i r st c re a t e w h a t i s k n ow n a s t h e augmente d matr i x cor re sponding to A , i .e ., T he matri x A aug is the same as the or iginal matr i x A w ith a ne w r o w , cor re sponding to the elements o f the vec tor b , added (i.
P age 11-3 3 Multiply r o w 2 by –1/8: 8Y2 @RCI! Multiply r ow 2 b y 6 add it to ro w 3, r eplacing it: 6#2#3 @RCIJ! If y ou w er e perfor ming these oper ati ons by hand , y ou w ould wr ite the fo.
P age 11-34 Multiply r o w 3 by –1/7 : 7Y 3 @ RCI! Multiply r ow 3 b y –1, add it to r o w 2 , r eplac ing it: 1 # 3 #2 @RCIJ! Multiply r ow 3 b y –3, add it to ro w 1, r eplacing it: 3#3#1 @RCI.
Pa g e 1 1 - 3 5 While perf orming p iv oting in a matr i x elimination pr ocedure , yo u can impro ve the numer i cal soluti on ev en mor e b y selecting a s the pi vo t the element w ith the lar gest ab solut e value in the column and r o w of inte r est .
Pa g e 1 1 - 3 6 No w we are r eady to st ar t the Ga uss-Jor dan elimination w ith full p i vo ting . W e w ill need to k eep trac k of the permutati on matr ix b y hand , so tak e yo ur notebook and w r ite the P m a trix s h own ab ove. F i r st , w e chec k the pi vo t a 11 .
P age 11-3 7 Hav ing f illed up w ith z er os the elements of column 1 below the p i v ot , now w e pr oceed to c heck the pi vot at po sition (2 ,2) .
P age 11-38 2 Y #3#1 @RCIJ F i nall y , w e eliminate the –1/16 f r om positi on (1,2) by using: 16 Y # 2#1 @RCIJ W e no w hav e an identity matri x in the por ti on of the augmented matr i x cor re.
P age 11-3 9 T hen, f or this partic ular e x ample , in RPN mode , use: [2,-1,41] ` [[1,2,3 ],[2,0,3],[8 ,16,-1]] `/ T he calculat or sho ws an a ugmented matr i x consisting o f the coeff ic ients m.
P age 11-40 T o see the in ter mediate s teps in calc ulating and inv er se , j ust e nter the matr ix A fr om abov e, and pr ess Y , w hile keep ing the step-b y-st ep op ti on acti v e in the calc ulator’s CA S .
P age 11-41 T he r esult ( A -1 ) n × n = C n × n / det ( A n × n ) , is a gener al r esult that appli es to an y non -singular matr i x A . A gener al for m for the ele ments of C can be w r it te n 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 s oluti on: [ X=-1,Y=2,Z = - 3]. F uncti on LINS OL VE w or ks wi th sy mb o lic e xpr es sions . F uncti ons REF , rr ef , and RREF , w ork w ith the au gmented matr i x in a Gaus sian eliminati on appr oach .
P age 11-4 3 T he diagonal matr i x that r esults f r om a Gaus s-Jor dan elimination is called a r o w-r educed echelon f or m. F unction RREF ( R ow-R educed E chelon F or m) The r esult of this functi on call is to pr oduce the r o w-r educed echelon f orm so that the matr i x of coeff ic ients is r educed to an identity matri x.
P age 11-44 T he re sult is the augmented matr i x corr esponding to the s yst em of equations: X+Y = 0 X- Y =2 Residual er rors in linear s ystem solutions (F unc tion RSD) F uncti on R SD calculate s the Re SiDuals or err ors in the soluti on of the matr i x equation A ⋅ x = b , repr esenting a s y stem o f n linear equations in n unkno w ns.
P age 11-45 Eigenv alues and eigenv ectors Gi v en a sq uar e matri x A , we can wr ite the eige nv alue equation A ⋅ x = λ⋅ x , w here the v alues of λ that satisfy the equation ar e know n as the ei gen value s of matr i x A .
Pa g e 1 1 - 4 6 Using the var iable λ to r epre sent e igen value s, this c har acter isti c pol ynomi al is to be interpr eted as λ 3 -2 λ 2 -2 2 λ +21=0. Function EG VL F uncti on E G VL (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 matr i x , while the corr esponding ei gen values ar e the components of a vec tor . F or ex ample , in AL G mode , the ei gen vect ors and e igen v alues of the matr i x listed be lo w ar e found by a pply ing functi on E G V : T he re sult sho ws the e igen v alues as the columns of the matr i x in the re sult list .
P age 11-48 • A list w ith the eigen v ect ors cor r es ponding to eac h ei gen v alue of matr i x A (stac k le ve l 2) • A v ector w ith the eige n vec tor s of matr i x A (st ack lev el 4) F or .
P age 11-4 9 Notice that the equati on ( x ⋅ I - A ) ⋅ p( x )=m ( x ) ⋅ I is similar , in f orm , to the ei gen value equati on A ⋅ x = λ⋅ x . As an e x ample , in RPN mode , try: [[4,1,-2] [ 1,2,-1][-2,- 1,0]] M D T he r esult is: 4: -8. 3: [[ 0.
P age 11-50 Function L U F uncti on L U tak es as input a s quar e matr ix A , and r eturns a lo wer - tr iangular matr i x L , an upper tr i angular matri x U , and a p e rmut ation matr i x P , in st ack le vels 3, 2 , and 1, re specti v el y . The r esult s L , U , and P , satisf y the equati on P ⋅ A = L ⋅ U .
P age 11-51 decompositi on, w hile the v ector s r epr esents the main diagonal of the matr i x S used earli er . F or ex ample, in RPN mode: [[ 5,4,-1],[2,- 3,5],[7,2,8] ] SVD 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 produces the QR fa ctoriz a tio n of a ma tr ix 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 permu tation matr i x, in stac k le vels 3, 2 , and 1.
Pa g e 1 1 - 5 3 T his menu includes f uncti ons AXQ, CHOLE SKY , G A US S, QX A, and S YL VE S TER. Function AX Q In RPN mode , f unction AXQ pr oduces the quadr ati c f orm cor r esponding to a matr i x A n × n in stac k le ve l 2 using the n var iable s in a vec tor placed in stac k le vel 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 Inf ormati on on the func tions list ed in this menu is pr esen ted belo w b y using the calc ulator ’s o w n help fac ility . The f igur es sho w the he lp fac ility entry and the attached e xamples .
P age 11-5 6 Function KER Function MKISOM.
Pa g e 1 2 - 1 Chapter 12 Gr a phi cs In this c hapter w e intr oduce some o f the gr aphic s capabiliti es of the calculat or . W e w ill pr esent gr aphics of f uncti ons in Cartesian coor dinates and polar coor dinates , par ametr ic plots , gr aphic s of coni cs , bar plots, scatter plots , and a v ari ety of thr ee -dimensi onal gr aphs .
Pa g e 1 2 - 2 T hese gr aph opti ons ar e desc ri bed bri ef ly ne xt . Fu n c ti o n : f or equations o f the for m y = f(x) in plane Cartesi an coordinates P olar : for equati ons of the fr om r = .
Pa g e 1 2 - 3 Θ Ente r the PL O T en v ir onment b y pr es sing „ñ (pr ess them simultaneou sly if in RPN mode). Pr ess @ADD to get y ou into the equati on w riter . Y ou will be pr ompted to f ill the r ight-hand side of an equati on Y1(x) = .
Pa g e 1 2 - 4 Θ Enter the P L O T WINDO W env ironme nt by enter ing „ò (pr ess them simultaneousl y if in RPN mode). Us e a range of –4 to 4 f or H- VI EW , then p r ess @AUT O to generate the V - VIEW automaticall y .
Pa g e 1 2 - 5 Some useful PL O T operations f or FUNCTION plots In or der to disc u ss these P L O T options , w e'll modif y the func tion to f or ce it to hav e some r eal r oots (Since the cur r ent curve is totall y contained abov e the x ax is, it has no r e al r oots.
Pa g e 1 2 - 6 R OO T : 1.66 3 5 ... The calc ulator indicated , bef or e sho w ing the r oot , that it w as found thr ough SIGN REVER S A L . Press L to r ecov er the menu . Θ Pr es sing @ISE CT w ill gi ve y ou the int ersecti on of the c urve w ith the x -ax is, w hic h is esse ntiall y the roo t .
Pa g e 1 2 - 7 Θ Enter the PL O T env ir onment by pr essing , simultaneously if in RPN mode , „ñ . Noti ce that the highli ghted fi eld in the PL O T en vir onment no w contain s the der i vati ve of Y1(X) . Pr ess L @@@OK@@@ to r eturn to return to nor mal calculat or displa y .
Pa g e 1 2 - 8 T o r etur n to nor mal calc ulator f uncti on , pr ess @) PICT @ CANCL . Graphics of tr anscendental functions In this secti on w e us e some of the gr aphics f eatur es of the calc ulator to sho w the typi cal beha vi or of the natur al log, e xponential , tri gonometr ic and h yper bolic func tions .
Pa g e 1 2 - 9 10 by u si n g 1 @@@OK@@ 10 @@@OK@@@ . Ne xt , pr ess the s oft k e y labeled @AUTO to let the calc ulator det ermine the cor r esponding v ertical r ange . After a cou ple of seconds this r ange w ill be sho wn in the P L O T WINDOW -FUNCTION w indo w .
Pa g e 1 2 - 1 0 Graph of the e xponential function F irst , load the f uncti on e xp(X) , b y pr essing , simultaneous ly if in RPN mode , the left-shif t k e y „ and the ñ ( V ) k e y to acces s the PL O T -FUNCTI ON w indo w .
Pa g e 1 2 - 1 1 T he PP AR var iable Press J to r ecov er y our v ari ables menu , if needed. In y our v ari ables menu y ou should ha v e a v ar iable labe led PP AR .
Pa g e 1 2 - 1 2 As indicated ear lier , the l n(x) and exp(x) f uncti ons ar e in v ers e of each othe r , i .e ., ln(e xp(x)) = x , and e xp(ln(x)) = x. T his can be v er if ied in the calc ulator b y typing and e v aluating the follo wi ng expr essi ons in the Eq uation W rit er: LN(EXP(X)) and EXP(LN( X)) .
Pa g e 1 2 - 1 3 Summary of FUNCT I ON plot oper ation In this secti on w e pr esent inf ormati on r egar ding the PL O T SETUP , P L O T- FUNCT ION, and P L O T WINDO W sc r eens accessible thr ough the left-shif t k ey comb ined w ith the soft-menu k e y s A thr ough D .
Pa g e 1 2 - 1 4 Θ Use @CANCL t o cancel an y c hanges to the P L O T SE TUP w indo w and r eturn to nor mal calc ulator dis play . Θ Press @@@OK@@@ to sav e changes to the options in the P L O T SE TUP w indo w an d r etur n to normal calc ulator displa y .
Pa g e 1 2 - 1 5 Θ Ente r lo w er and uppe r limits fo r hor i z ont al v ie w (H- V ie w) , and pr ess @AUTO , w hile the c urso r is in one of the V - Vi e w f ields , to ge ner ate the v ertical v i e w (V - Vie w) range automaticall y .
Pa g e 1 2 - 1 6 „ó , simult aneousl y if in RPN mode: Plots the gr aph based on the setting s stor ed in var ia ble PP AR and the cur r ent f unctions de fined in the PL O T – FUNCT ION scr een.
Pa g e 1 2 - 1 7 Generating a table of v alues for a function T he combinati ons „õ ( E ) and „ö ( F ), pressed simultaneousl y if in RPN mode , let’s the us er pr oduce a table o f values o f functi ons .
Pa g e 1 2 - 1 8 the corr esponding value s of f(x) , listed as Y1 b y de fault . Y ou can use the up and do wn ar r o w k ey s to mov e about in the t able . Y ou w ill notice that w e did not ha ve to indicate an ending value f or the independent v ar iable x .
Pa g e 1 2 - 1 9 W e w ill tr y to plot the f uncti on f( θ ) = 2(1-sin( θ )), as follo w s: Θ F irs t , mak e sur e that y our calc ulator ’s angle measur e is set t o r adians. Θ Press „ô , simultaneousl y if in RPN mode , to access to the PL O T SE TUP wi ndo w .
Pa g e 1 2 - 2 0 Θ Press L @ CANCL to ret urn to th e PL O T WI ND OW s cr e en. Press L @@@OK@@@ to r etur n to normal calc ulator displa y . In this e xe r c ise w e enter ed the eq uation to be plotted dir ectl y in the PL O T SETUP w indo w . W e can also enter equati ons f or plotting using the P L O T wi ndow , i .
Pa g e 1 2 - 2 1 T he calculator ha s the ability of plotting one or more coni c c ur v es b y selecting Con ic as the functi on TYPE in the PL O T e nv ir onment .
Pa g e 1 2 - 2 2 Θ T o see labels: @EDI T L @) LABEL @MENU Θ T o r eco ver the men u: LL @) PICT Θ T o es timate the coor dinates of the po int of inter secti on, pr ess the @ ( X,Y ) @ menu k ey and mo v e the cur sor as c lose as po ssible to thos e points using the arr ow k ey s .
Pa g e 1 2 - 23 whi ch in vol ve constant values x 0 , y 0 , v 0 , and θ 0 , we need to st ore the v alues of those par ameters in v ar iables . T o de velop this e xample , cr eate a sub-dir ect or .
Pa g e 1 2 - 24 Θ Press @AUTO . This w ill gener ate autom ati c values of the H-V ie w and V - Vi e w r anges based on the v alues of the independent var iable t and the def initi ons of X(t) and Y(t) u sed . The r esult w ill be: Θ Press @ERASE @DR AW to dr a w the paramet ri c plot .
Pa g e 1 2 - 2 5 par ameters . The other v ar iables contain the v a lues o f constants us ed in the def initions o f X(t) and Y(t) . Y ou can stor e differ ent v alues in the v ari ables and pr oduce ne w par ametr ic plots of the pr o jectile eq uations us ed in this e xample .
Pa g e 1 2 - 26 P lotting th e solution to simple differ ential equations T he plot of a simple differ ential equati on can be obtained by selec ting Diff Eq in the TYPE f ield o f the PL O T SETUP en.
Pa g e 1 2 - 27 Θ Press L to reco v er th e menu . Pr ess L @) PICT to reco ver the or igin al gr aphics menu . Θ When w e obs erved the gr aph being plo tted, y ou'll noti ce that the gr aph is not v ery smooth . T hat is becaus e the plotter is using a time step that 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 egio ns that satisfy a certain mathemati cal condition that can be e ither true or f alse .
Pa g e 1 2 - 2 9 Θ Press „ô , simultaneou sly if in RPN mode , to access t o the PL O T SETUP wi n dow . Θ Press ˜ and ty pe ‘(X^2/3 6+Y^2/9 < 1) ⋅ (X^2/16+Y^2/9 > 1)’ @@@OK@@@ to def ine the conditions t o be plot t ed. Θ Press @ERASE @DRAW t o dra w the tr uth 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 uncti on S T O Σ (av ailable in the functi on catalog, ‚N ) . Pr ess V AR to reco v er y our var iable s menu . A soft menu k ey labeled Σ D A T should be a v ailable in the stac k.
Pa g e 1 2 - 3 1 accommodate the max imum v alue in column 1 of Σ D A T . Bar plots ar e usef ul when plotting categori cal (i .e ., non -numeri cal) data. Suppo se that y ou w ant to plot the data in co lumn 2 o f the Σ DA T m a t rix : Θ Press „ô , simultaneou sly if in RPN mode , to access t o the PL O T SETUP wi n dow .
Pa g e 1 2 - 32 Θ Press @ERASE @ DRAW to dr aw the bar plot . Pre ss @EDIT L @LA BEL @MENU to see the plot unenc umber ed b y the menu and w ith identify ing la bels (the c ursor w ill be in the middle of the plot , ho w e ver ) : Θ Press LL @) PICT to l eav e the EDI T en v ir onment .
Pa g e 1 2 - 3 3 Slope fields Slope fi elds ar e used to v isuali z e the solutio ns to a differ ential equati on of the fo rm y’ = f(x ,y) . Basi call y , what 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) . Th us, slope f ie lds are u sef ul tools f or v isuali zing par ti c ularl y diffi cult equations t o sol v e .
Pa g e 1 2 - 3 5 Θ Press @ERASE @ DRAW to dr a w the thr ee -dimensional surf ace . The r esult is a w ir ef rame p ictur e of the surface w ith the re fer ence coor dinate sy stem sho w n at the lo w er left corner of the s cr e e n. B y using the arr ow k ey s ( š™— ˜ ) y ou can change the or ient ation of the surf ace.
Pa g e 1 2 - 3 6 Θ Press „ô , simul taneousl y if in RPN mode , to access the P L O T SETUP wi n dow . Θ Press ˜ and type ‘S IN(X^2+Y^2) ’ @@@OK@@@ . Θ Press @ERASE @DR AW to dr aw the plot . Θ When done , pres s @ EXIT . Θ Press @CANCL to r etur n to P L O T W INDO W .
Pa g e 1 2 - 37 Θ Press @EDIT L @LABEL @ MENU to s ee the gr aph w ith labels and r anges . This partic ular v ersi on of the gr aph is limited to the lo wer part of the dis play . W e can change the v ie wpoint to see a differ ent versi on of the gr aph.
Pa g e 1 2 - 3 8 T ry also a Wir ef r ame plot f or the surface z = f(x,y) = x 2 +y 2 Θ Press „ô , simul taneousl y if in RPN mode , to access the P L O T SETUP wi n dow . Θ Press ˜ and t y pe ‘X^2+Y^2’ @@@OK@@@ . Θ Press @ERASE @DRAW to dr aw the slope fie ld plot .
Pa g e 1 2 - 3 9 Θ Press @EDIT ! L @ LABEL @MENU to see the gr aph w ith labels and r anges. Θ Press LL @) PICT@CANCL to retur n to the PL O T WINDOW en v ir onment . Θ Press $ , or L @@@OK@@@ , to retur n to normal calculator dis play . T ry als o a P s-Conto ur plot for the surf ace z = f(x,y) = sin x cos y .
Pa g e 1 2 - 4 0 Θ Mak e sur e that ‘X’ is s elected as the Indep: and ‘ Y’ as the Depnd: variab le s. Θ Press L @@@OK@@@ to r eturn to nor mal calculat or displa y . Θ Press „ò , simultaneou sl y if in RPN mode , to acce ss the P L O T WINDO W sc r een.
Pa g e 1 2 - 4 1 Θ Press „ô , sim ultaneo usl y if in RPN mode , to acces s to the P L O T SETUP w indow . Θ Cha ng e TYPE to Gr idma p . Θ Press ˜ and t y pe ‘SIN(X+i*Y )’ @@@OK@@@ . Θ Mak e sur e that ‘X’ is s elected as the Indep: and ‘ Y’ as the Depnd: variab le s.
Pa g e 1 2 - 4 2 F or ex ample, t o pr oduce a Pr- Surface plot f or the surface x = x(X,Y ) = X sin Y , y = y(X,Y) = x cos Y , z=z(X,Y)=X, us e the fo llo w ing: Θ Press „ô , sim ultaneo usl y if in RPN mode , to acces s to the P L O T SETUP w indow .
Pa g e 1 2 - 4 3 Inter ac ti ve dr a wing Whene v er w e pr oduce a tw o -dimensi onal gr aph , w e f ind in the gr aphic s sc r een a so ft men u k e y labe led @) EDIT .
Pa g e 1 2 - 4 4 Ne xt , we illus tr ate the use o f the differ ent dr a w ing functi ons on the r esulting gr aphi cs sc r een . The y req uir e use of the c ursor and the ar r o w k ey s ( š™— ˜ ) to mo v e the c ursor about the gr aphic s scr een.
Pa g e 1 2 - 4 5 should ha ve a s tr aight angle tr aced b y a hori z ontal and a v ertical segme nts. T he cur sor is still acti ve . T o deacti vat e it , w ithout mov ing it at all, pr ess @LINE . T he cu rsor r eturns to its n ormal sha pe (a cr o ss) and the LINE func tion is no longer acti ve .
Pa g e 1 2 - 4 6 DEL T his command is u sed to r emov e parts of the gr aph betw een two MARK positi ons. Mo v e the cur sor to a po int in the gr aph, and pr ess @MARK . Mov e the c ursor t o a diff er ent point , pres s @M ARK again . Then , pr ess @@DEL@ .
Pa g e 1 2 - 47 X,Y T his command copies the coor dinates o f the cur r ent cur sor positi on, in us er coor dinates , in the stac k . Z ooming in and out in th e gr aphics display Whene v er y ou.
Pa g e 1 2 - 4 8 Y ou can alw a ys r etu r n to the v er y last z oom wi ndow b y u sing @ ZLAST . BO XZ Z ooming in and out of a gi v en gr aph can be pe rfor med by u sing the soft-menu k ey B O XZ . W ith BO XZ you s elect the re ctangular s ector (the “bo x”) that y ou want to z oom in into .
Pa g e 1 2 - 4 9 c ursor at the cent er of the sc reen , the w indo w gets z oomed so that the x -ax is e xtends fr om –64. 5 to 6 5 . 5 . ZSQR Z ooms the gra ph so that the plotting scale is maintained at 1:1 b y adjus ting the x scale , keep ing the y scale f i xe d, if the w indow is w ider than tall er .
Pa g e 1 2 - 5 0 S OL VER.. „Î (the 7 key ) C h. 6 TRIGONO ME TRIC. . ‚Ñ (the 8 key ) Ch. 5 EXP &LN.. „Ð (the 8 key ) C h. 5 T he S YMB/GRAPH menu T he GR AP H su b-menu w ithin the S YMB.
Pa g e 1 2 - 5 1 T AB V AL(X^2 -1,{1, 3}) pr oduces a list of {min max} v alues o f the functi on in the interv al {1, 3}, w hile SIGNT AB(X^2 -1) show s the sign o f the func tion in the interv al (- ∞ ,+) , 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 . T he question marks indicates uncer tainty or non-d ef inition. F or ex ample, fo r X<0, LN(X) is not defined , thu s the X lines sho ws a que stion mar k in that interv al. R ight at z er o (0+0) F is inf inite , for X = e , F = 1/e .
P age 13-1 Chapter 13 Calculus Applications In this Chapte r we dis cu ss appli cations of the calc ulator ’s functi ons to oper ations r elated to Calc ulus, e .
P age 13-2 Function lim T he calculat or pr ov ides f uncti on lim t o c a l cu l at e l i m i t s of fu n c t io n s. Th i s f un c t io n use s as input an e xpre ssi on re pr esenting a func tion and the v alue wher e the limit is to be calc ulated.
P age 13-3 T o calc ulate one -sided limits, add +0 or -0 t o the value to the v ari 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 . Re call that in RPN mode the ar guments must be e nter ed bef ore the func tion is appli ed. T he DERIV&INTEG menu T he functi ons a vailable in this sub-me nu ar e listed be low : Out of the se func tions DERIV and DER VX ar e used f or deri vati v es.
P age 13-5 be differ entiated . T hus , to calc ulate the deri vati v e d(sin(r ) ,r ) , us e , in AL G mode: ‚¿~„r„ÜS~„r` In RPN mode , this expr essi on must be enc los ed in quot es befo r e enter ing it into t he sta ck.
P age 13-6 T o e valuate the der iv ati v e in the E quation W r iter , pr es s the up-arr o w k e y — , fo ur times, t o selec t the entir e e xpr essi on , then, pr ess @ EVAL .
P age 13-7 Deri v ativ es of equations Y ou can use the calc ulator to calc ulate der i v ativ es o f equations , i .e ., e xpr essi ons in w hic h deri vati v es w ill ex ist in both sides o f the equal sign.
P age 13-8 Analyzing gr aphics of func tions In Chapter 11 w e pre sented some f unctions that ar e av ailable in the graphic s sc r een f or anal yzing gr aphi cs of func tions of the f orm y = f(x). The se fu nctio ns inc lude (X,Y) and TR A CE f or determining po ints on the gr aph , as w ell as func tions in the Z OOM and FCN menu .
P age 13-9 Θ Press L @PICT @CANCL $ to r eturn to normal calc ulator displa y . Notice that the slope and tangent line that y ou r eques ted ar e listed in the stac k .
P age 13-10 T his re sult indicat es that the r ange of the f uncti on cor r esponding to the domain D = { -1,5 } is R = . Function SIGNT AB F uncti on SIGNT AB, a v ailable thr ough the command catalog ( ‚N ), pro v ides inf orma tion on th e sign of a function th r o ugh it s domai n .
P age 13-11 Θ Le v el 3: the f uncti on f(VX) Θ T w o lists, the f irs t one indicate s the var iati on of the f unction (i .e., w her e it inc reas es or dec reas es) in ter ms o f the independent var iable VX, the second one indicate s the var iati on of the f uncti on in term s of the dependent v a r iable .
P age 13-12 The interpr etation of the v ariati on table show n abov e is as follo ws: the functi on F(X) incr eases f or X in the int erval (- ∞ , -1), reac hing a max imum equal to 3 6 at X = -1. Then , F(X) decr eas es until X = 11/3, r eaching a minimum of –4 00/2 7 .
P age 13-13 W e fi nd two c r itical po ints, one at x = 11/3 and one at x = -1. T o ev aluate the second der i vati ve at eac h point use: T he last s cr een show s that f”(11/3) = 14 , thus , x = 11/3 is a r elati v e minimum .
P age 13-14 Anti-deri v ati ves and integr als An anti-der iv ati ve o f a func tion f(x) is a func tion F(x) su ch that f(x) = dF/dx . F or e x ample , since d(x 3 ) /dx = 3x 2 , an anti-der i v ati ve o f f(x) = 3x 2 is F(x) = x 3 + C, w here C is a constant .
P age 13-15 abo v e . The ir re sult is the so -called discr ete der i vati ve , i .e . , one de fined f or integer n umbers onl y . Definite integr als In a def inite integr al of a f uncti on, the r esulting anti-der i vati ve is e valuated at the upper and lo wer limit o f an int erval (a ,b) and the ev a l uated value s subtr acted .
P age 13-16 T his is the gener al for mat for the de finit e integral w hen typed dir ectly into the stac k, i .e ., ∫ (lo w er limit , upper limit , in tegr and , var iable of in tegr ation) Pr es .
P age 13-17 T he follo w ing ex ample sh o ws the e v aluation of a defi nite integr al in the E quation W riter , step-b y-step: ʳʳʳʳʳ Notice that the st ep-by-s tep pr ocess pr ov ide s infor mation on the inter mediate step s follo wed b y the CAS to solv e this integr al .
P age 13-18 T ec hniques o f integr ation Se v er al techni ques of int egr ation can be im plemented in the calc ulators , as sho w n in the f ollo w ing e x amples . Substitution or chang e o f var iables Suppose w e want to calc ulate the integr al .
P age 13-19 Integration b y par ts and differentials A differ ential o f a functi on y = f(x) , is de fined a s dy = f’(x) dx , w her e f’(x) is the der i vati v e of f(x). Differ enti als ar e used to r epr esen t small incr ements in the var iables .
P age 13-20 Integration b y par tial fr actions F unction P A R TFRA C, pr esented in Chap te r 5, pr ov ides the decomposition of a fr action int o par ti al fr acti ons. T his techni que is us eful t o r educe a complicated fr action into a sum of simple f r actio ns that can then be integrated t erm b y ter m.
P age 13-21 Using the calc ulator , w e pr oceed as f ollo ws: Alternati ve ly , y ou can ev aluate the i n tegra l to inf inity fr om the start, e .g .
P age 13-2 2 Some n otes in the u se of units in the limits of int egrati ons: 1 – T he units of the low er limit of integr ation w i ll be the ones u sed in the f inal r esult , as illu str ated in the tw o e x amples belo w : 2 - Upper limit units mu st be consisten t w ith low er limit units.
Pa g e 1 3 - 23 T a ylor and Mac laur in’s series A fu nction f( x) can be expanded in to an inf inite ser ie s ar ound a point x=x 0 by using a T a y lor’s ser ie s, namel y , , wher e f (n) (x) repr esen ts the n - th der i vati ve of f(x) w ith respect to x , f (0) (x) = f(x) .
P age 13-2 4 wher e ξ is a n umber near x = x 0 . Since ξ is ty pi cally unkn o wn , inst ead of an estimat e of the r esidual , w e pr ov ide an es timate of the or der of the r esi dual in re fe ren c e t o h, i. e. , we s ay t h a t R k (x) ha s an err or of orde r h n+1 , or R ≈ O(h k+1 ).
P age 13-2 5 inc reme nt h. T he list r etur ned as the fir st output ob ject inc ludes the fo llo w ing items: 1 - Bi-dir ecti onal limit of the func tio n at point of e xpansion , i .
Pa g e 1 4 - 1 Chapter 14 M ulti-v ariate Calculus Applications Multi- v ar iate calculus r ef ers to functi ons of two or mor e v ar iables . In this Chapte r we dis c uss the basi c concepts of multi-v ari ate calc ulus including partial der i vati v es and multiple int egrals .
Pa g e 1 4 - 2 . Similarl y , . W e w ill use the multi-var i ate functi ons def ined earli er to calc ulate partial der i vati v es using thes e def initions .
Pa g e 1 4 - 3 ther ef or e , w ith DERVX y ou can onl y calculat e deri vati v es w ith r espect to X . Some e xamples o f fir st-order partial der iv ati ve s are sho wn ne xt: ʳʳʳʳʳ Hi gh er -.
Pa g e 1 4 - 4 T hir d-, fourth-, and higher or der der i vati ves ar e def ined in a similar manner . T o calc ulate higher o r der der i vati ves in the calculator , simply r e peat the der i vati v e functi on as man y times as needed .
Pa g e 1 4 - 5 A diff er ent v ersi on of the c hain rule appli es to the cas e 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 r itical points at (X,Y ) = (1, 0) , and (X,Y) = (-1, 0 ). T o c alc ulate the disc r iminant , we pr oceed t o calculate the second der i v ati ves , 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 Appli cations of f uncti on HE S S ar e easier to v i suali z e in the RPN mode . Consi der as an ex ample the f uncti on φ (X,Y ,Z) = X 2 + XY + XZ , w e ’ll appl y fu nct ion H E S S to fu nct ion φ i n t h e f ol l owi n g e xa m p l e.
Pa g e 1 4 - 8 T he re sulting 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. T he disc r iminant , f or this cr itical 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 coor dinate transf ormation Consi der the coordinat e tr ansfor mation x = x(u ,v) , y = y(u ,v) . T he Jacobi an of this tr ansf ormati on is def i ned as . When calc ulating an int egr al using suc h transf ormati on , the expr ession to u se is , w her e R’ is the r egi on R e xpre ssed in (u ,v ) coor dina te s.
Pa g e 1 4 - 1 0 w here the r egion R’ in polar coor dinates is R’ = { α < θ < β , f( θ ) < r < g( θ )}. Double integr als in polar coor dinates can be enter ed in the ca lc ulator , making sur e that the Jacobi an |J| = r is includ ed in the integr and .
P age 15-1 Chapter 15 V ec tor Anal y sis Applications In this Chapt er we pr esent a number of f unctio ns fr om the CAL C menu that appl y to the analy sis of scalar and ve ctor f iel ds.
P age 15-2 At an y partic ular point , the maximum r a t e of change o f the functi on occ urs in the dir ecti on of the gr adien 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 -v ar ia bles , grad f = [ ∂φ / ∂ x 1 , ∂φ / ∂ x 2 , … ∂φ / ∂ x n ], and the list of va riab le s [ ‘ x 1 ’ ‘ x 2 ’…’x n ’].
P age 15-4 not hav e a potential func tion assoc iated w ith it , since, ∂ f/ ∂ z ≠∂ h/ ∂ x. The cal c ula tor r espon se in th is case is sho wn bel o w: Di ver gence T he div er gence of a.
P age 15-5 Cur l The c url of a v ector fi eld 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 “ cr oss-pr oduct” of the del oper ator w ith the vec tor f ield, i .e ., T he cur l of v ect or fi eld can be calculat ed with f uncti on CURL .
P age 15-6 As an e xample , in an earlie r ex ample w e attempted to f ind a potenti al func tion for th e ve ctor f ie ld F (x,y ,z) = (x+y) i + (x-y+z) j + xz k , and got an e rr or message back f r om func tion P O TENT IAL. T o ve rify that this is a r otati onal f ield (i .
P age 15-7 pr oduces the v e c tor potenti al func tion Φ 2 = [0, ZYX- 2YX, Y -( 2ZX-X)], w hic h is diffe r ent fr om Φ 1 . T he last command in the sc reen shot sho w s that indeed F = ∇× Φ 2 . Th us, a v ector potenti al functi on is not uniquel y determined .
Pa g e 1 6 - 1 Chapter 16 Differ ential Equations In this Chapte r we pr esent e xample s of so lv ing or dinar y diff er ential equati ons (ODE) using calc ulator f uncti ons. A differ ential equatio n is an equati on in vol v ing der i vati ves of the independen t var iable .
Pa g e 1 6 - 2 ( H @) DISP ) is not se lected . Pr ess ˜ to see the equati on in the E quati on Wr i t e r. An alter nati v e notatio n for der iv ati v es typed dir ectl y in the st ack is to u se .
Pa g e 1 6 - 3 EV AL(AN S(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 e xample , y ou could also us e: ‘ ∂ t( ∂ t(u(t))))+ ω 0^2*u (t) = 0’ to enter the diffe r ential equation .
Pa g e 1 6 - 4 T hese f unctions ar e brie fl y desc r ibed next . T he y w ill be desc r ibed in mor e detail in later parts of this Chapte r . DE S OL VE: Differ enti al E quati on S OL VEr , pro vi.
Pa g e 1 6 - 5 Both of thes e inputs must be gi ven in ter ms of the def ault independent v ar iable fo r the calculator ’s CAS (ty pi cally ‘X’) . T he output fr om the functi on is the gener a l soluti on of the ODE . The f unction LDE C is a v ailable thr ough in the CAL C/DI FF men u .
Pa g e 1 6 - 6 T he soluti on, sho w n par ti ally he re in the E quation W r iter , is: R eplac ing the combinatio n of constants accompan y ing the e xponenti al terms w ith simpler values , the e xpr essi on can be simplifi ed to y = K 1 ⋅ e –3x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + ( 4 5 0 ⋅ x 2 +3 3 0 ⋅ x+2 41)/13 500.
Pa g e 1 6 - 7 2x 1 ’(t) + x 2 ’(t) = 0. In algebr aic f orm , this is wr itten as : A ⋅ x ’(t) = 0, wher e . T he s y stem can be s olv ed b y using func tion LDE C w ith argume nts [0, 0] an.
Pa g e 1 6 - 8 Ex ample 2 -- So lv e the second-o rde r ODE: d 2 y/dx 2 + x (dy/dx) = e xp(x) . In the calc ulator use: ‘ d1d1y(x)+x *d1y(x) = EXP( x) ’ ` ‘ y(x) ’ ` DESOLVE T he r esult is an.
Pa g e 1 6 - 9 P er f or ming the integr ation by hand, w e can only ge t it as far as: becaus e the integr al of exp(x)/x is no t av ailable in c losed f or m. Ex ample 3 – Sol v ing an equati on w ith initial co nditions . Sol ve d 2 y/dt 2 + 5y = 2 cos(t/2) , w ith initial conditi ons 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 ty pe in this case . Laplace T r ansfor ms T he Laplace tr ansform o f a func tion f(t) pr oduces a f unction.
Pa g e 1 6 - 1 1 Laplace tr ansfor m and inv erses in the calc ulator T he calculat or pr o vi des the f uncti ons L AP and ILAP to calc ulate the L aplace tr ansfor m and the in v erse L aplace tr ansfor m, r especti v ely , of a func tion f(VX) , w here VX is the CA S def ault independent v ar iable , whi ch y ou should set t o ‘X’ .
Pa g e 1 6 - 1 2 Ex ample 3 – Deter mine the in ve rse L aplace tr ansfor m of F(s) = sin(s) . Use: ‘SIN(X)’ ` IL AP . The calc ulator tak es a fe w seconds to r eturn the r esul t: ‘IL AP( SIN(X))’ , meaning that ther e is no c los ed-fo rm e xpr es sion f(t), such that f(t ) = L -1 {sin(s)}.
Pa g e 1 6 - 1 3 Θ Differ entiati on theor em for the n- th der iv ati 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 L inear it y theor em .
Pa g e 1 6 - 1 4 Θ Shift theor em fo r a shif t t o the ri ght . Le t F(s) = L{f(t)}, then L{f(t-a)}=e –as ⋅ L{f(t)} = e –as ⋅ F(s) . Θ Shift theor em f or a shift to the left . Le t F(s) = L{f(t)}, and a >0, then Θ Similar ity theor em .
Pa g e 1 6 - 1 5 Dir ac’s d elta function and Heav isid e’s step function In the analy sis of contr ol s y stems it is cu stomary to utili z e a t y pe of f uncti ons that r epr esent certain ph y.
Pa g e 1 6 - 1 6 Y ou can pr o v e that L{H(t)} = 1/s , from wh ich it fol lows th a t L { U o ⋅ H(t)} = U o /s , wher e U o is a cons tant . 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 L aplace transf orm in the solution of linear ODEs At the beginning of the s ectio n on Laplace tr ansfor ms we indi cated that y ou could us e these tr ansfor ms to con v ert a linear ODE in the time do main into an algebr aic eq uation in the image domain .
Pa g e 1 6 - 1 8 T he r esult is ‘H=( (X+1)*h0+a)/(X^2+(k +1)*X+k)’ . T o f ind the soluti on to the ODE , h(t) , w e need to us e the inv erse L aplace tr ansfor m, as f ollo w s: OB J ƒ ƒ Isolat es ri ght -hand si de of las t expr essi on ILAP μ Obtains the in ver se L aplace tr ansfor m T he r esult is .
Pa g e 1 6 - 1 9 W ith 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 tr ansfor med equati on is s 2 ⋅ Y(s) – s ⋅ y o – y 1 + 2 ⋅ Y(s) = 3/(s 2 +9) .
Pa g e 1 6 - 2 0 Ex ample 3 – Consider the equati on d 2 y/dt 2 +y = δ (t-3) , wher e δ (t) is Dir ac’s d e lta func tion . Using La place transf orms , w e 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)}. Wi th ‘ Delta(X-3) ’ ` L AP , the calc ulator pr oduces EXP(-3*X) , i .
Pa g e 1 6 - 2 1 Chec k what the s olution t o the OD E w ould be if y ou us e the functi on LDEC: ‘Delta(X- 3)’ ` ‘X^2+1’ ` LDE C μ Note s : [1].
Pa g e 1 6 - 22 T he re sult is: ‘S IN(X-3)*Heav isi de(X-3) + cC1*S IN(X) + cC0*CO S(X)’ . P lease notice that the v ari able X in this expr essi on actuall y r e p r esen ts the v ari able t in the or iginal ODE .
Pa g e 1 6 - 2 3 Use o f the f unction H(X) w ith LD E C, L AP , or IL AP , is not allo wed in the calc ulator . Y ou hav e to us e the main results pr ov ided earlier w hen dealing w ith the Heav iside step f uncti on , i .
Pa g e 1 6 - 24 w here H(t) is Hea v iside ’s step f uncti on. Us ing Laplace tr ansfor ms, w e can wri te : L {d 2 y/dt 2 +y} = L{H(t- 3)}, L{d 2 y/dt 2 } + L{y(t)} = L{H(t- 3)} . The la st ter m in this e xpr essi on is: L{H(t -3)} = (1/s) ⋅ e –3s .
Pa g e 1 6 - 2 5 Ex ample 4 – P lot the so lution to Ex ample 3 using the same v alues of y o and y 1 used in the plot of Ex ample 1, abov e . W e now plot the f unction y(t) = 0.
Pa g e 1 6 - 26 f(t) = U o ⋅ [1-(t-a)/(b-1)] ⋅ [H(t-a) -H(t -b)]. Ex amples of the plots gener ated by the se func tions , fo r Uo = 1, a = 2 , b = 3, c = 4 , hori z ontal r ange = (0,5 ) , and v ertical r ange = (-1, 1.
Pa g e 1 6 - 2 7 T he follo w ing ex erc ises ar e in AL G mode , with CA S mode s et to Ex act . ( W hen y ou pr oduce a gr aph , the CAS mode w ill be re set to Appr o x. Mak e sur e to se t it back t o Exact afte r pr oduc ing the gra ph.) Suppo se , f or ex ample , that the func tion f(t) = t 2 +t is per iodi c w ith per iod T = 2 .
Pa g e 1 6 - 2 8 Function FOURIER An alter nati ve w a y to def ine a F our ier ser ies is by using comple x number s as fo llo w s: wh ere F uncti on FOURIER pr ov i des the coeff ic ient c n o f the complex -for m of the F ouri er ser i es giv en the functi on f(t) and the v alue of n.
Pa g e 1 6 - 2 9 Ne xt, w e mo ve to the CA SDI R sub-dir ector y under HOME to c hange the value of var iable PERIOD , e.g ., „ (hold ) §`J @) CASDI `2 K @PERIOD ` R eturn to the su b-dir ectory w.
Pa g e 1 6 - 3 0 T he fitting is some what accepta ble for 0<t<2 , although not as good as in the pr ev ious e xample . A general e xpression for c n T he functi on F OURIER can pro v ide a gener al e xpr essi on f or the coeff ic ient 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 F ouri er ser ies Hav ing deter mined the gener al expr ession for c n , we can put toge ther a.
Pa g e 1 6 - 32 Or , in the calculator entry line as: DEFINE(‘F(X,k,c0) = c0+ Σ (n=1,k ,c(n)*EXP(2*i* π *n*X/T)+ c(-n)*EXP(-( 2*i* π *n*X/T))’) , w here T is the per iod , T = 2 .
Pa g e 1 6 - 33 Accept c hange to Approx mode if r eques ted . The r esult is the v alue –0.40 46 7…. The ac tual value of the f uncti on g(0. 5) is g(0.
Pa g e 1 6 - 3 4 per iodi c ity in the gr aph o f the ser ies . This per i odic it y is eas y to v isuali z e by e xpa nding the hor i z ontal range of the plot to (-0.5, 4) : F ourier series f or a triangular w av e Consi der the functi on w hich w e assume to be per i odic w ith peri od T = 2 .
Pa g e 1 6 - 3 5 T he calculat or r eturns an int egr al that cannot be e valuat ed numer icall y becaus e it depends on the par ameter n . The coeff ic ient can s till be calc ulated by typing its de finiti on in the calc ulator , i .e ., w here T = 2 is the per i od.
Pa g e 1 6 - 3 6 Press `` to cop y this re sult to the scr een. T hen , r eacti vat e the E quation W r iter to calc ulate the second integr al defi ning the coeffi c ie nt c n , namel y , Once again, r eplac ing e in π = (-1) n , and using e 2in π = 1, we get: Press `` to cop y this second r esult to the sc r een .
Pa g e 1 6 - 37 T his re sult is used to de fine the f unction c(n) as f ollo ws: DEFINE(‘ c(n) = - (((-1)^n-1)/(n^2* π ^2*(-1)^n)’) i. e. , Ne xt, w e def ine function F(X,k ,c0) to calc ulate t.
Pa g e 1 6 - 3 8 F r om the plot it is very diffi c ult to distinguish the or iginal functi on fr om the F ourier s eri es appr o ximati on. U sing k = 2 , or 5 ter ms in the ser ies, sho ws not so go.
Pa g e 1 6 - 3 9 In th is case , th e per iod T , is 4. Mak e s ur e to chang e the value of v ari abl e @@@T@@@ to 4 (use: 4K @@@T@@ ` ) . F unction g(X) can be de fined in the calc ulator by us in g.
Pa g e 1 6 - 4 0 Th e si m pl i fic at io n of th e rig h t -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) . T hen , r et y pe the expr es sion f or c(n) as sho wn in the f igur e to the left abo v e , to def ine func tion c(n).
Pa g e 1 6 - 4 1 W e can use this r esult as the f irs t input to the f uncti on LD E C w hen us ed to obtain a soluti on to the s y ste m d 2 y/dX 2 + 0.
Pa g e 1 6 - 42 T he soluti on is sho wn belo w: F ourier T r ansf orms Befor e pr esen ting the concept of F our ier tr ansf orms , we ’ll d i scus s the gener al def initio n of an integr al tr ansf orm .
Pa g e 1 6 - 4 3 T he amplitudes A n w ill be r ef er red t o as the spectr um of the f uncti on and w ill be a measur e of the magnitude of the component of f(x) w ith fr equency f n = n/T . T he basic or f undamental fr equency in the F ouri er ser ies is f 0 = 1/T , thu s, all other fr equenc ies ar e multiple s of this basi c f req uency , i .
Pa g e 1 6 - 4 4 and The continuous spectrum is giv en by Th e fu nct ion s C ( ω ), S ( ω ), and A( ω ) ar e continuous functi ons of a v ari able ω , w hich beco mes the tr ansfor m v ari able fo r the F our ier tr ansfor ms def ined belo w .
Pa g e 1 6 - 4 5 Def ine this e xpr essio n as a f unction by u sing func tion DEFINE ( „à ) . Then , plot the continuo us spectr um, in the r ange 0 < ω < 10 , as: Definition o f Four ier transf orms Diffe r ent t y p e s of F ourie r transf or ms can be defined .
Pa g e 1 6 - 4 6 The continuous spect r um, F( ω ) , is calculated w ith the integral: T his re sult can be r ationali z ed b y multipl y ing numer ator and denominator b y the conjugat e of the denominator , namel y , 1-i ω . T he r esult is now : which is a co mp lex fu nct ion.
Pa g e 1 6 - 4 7 Pr oper ties o f th e F ourier transfor m L inearity : If a and b are co nstants , 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 u sing the FFT is r e du ced by a f act or of 10000/6 64 ≈ 15 . The FFT op er ates on t he sequenc e {x j } b y par titi oning it int o a number o f shorter seque nces . The DFT ’s of the shorter seq uences ar e calc ulated and later comb ined together in a highl y eff ic ient manner .
Pa g e 1 6 - 49 T he fi gur e belo w is a box plot o f the data pr oduced. T o obtain the gra ph, f irs t cop y the ar r ay j ust c r eated, then tr ansfor m it into a column v ector b y using: OB J 1 + ARR Y (F uncti ons OB J and ARR Y ar e av ailable in the command cat alog, ‚N ) .
Pa g e 1 6 - 50 Ex ample 2 – T o pr oduce the signal gi ven the s pectr um, w e modif y the pr ogr 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 Ex cept for a lar ge peak at t = 0, the signal is mo stl y nois e . A smaller v er ti cal scale (-0. 5 to 0. 5) sho ws the si gnal as f ollo ws: Solution to specific second-or der dif.
Pa g e 1 6 - 52 w here M = n/2 or (n-1)/2 , whi che v er is an integer . Legendr e’s pol y nomials ar e pr e -pr ogr ammed in the calculator and can be r ecalled by u sing the func tion LE GENDRE gi v en the or der of the pol ynomi al , n.
Pa g e 1 6 - 5 3 wher e ν is not an integer , and the func tion Gamma Γ ( α ) is def ined in Chapter 3. If ν = n , an integer , the Bessel f uncti ons of the f ir st kind for n = intege r ar e def.
Pa g e 1 6 - 5 4 Y ν (x) = [J ν (x) cos νπ – J −ν ( x)]/sin νπ , fo r non -int eger ν , and f or n integer , w ith n > 0, by wher e γ is the Euler cons tant , def ined by and h m r epr.
Pa g e 1 6 - 5 5 T he modifi ed Bessel f unctions o f the second kind , K ν (x) = ( π /2) ⋅ [I - ν (x) − I ν (x)]/sin νπ , ar e also so lutions o f this OD E .
Pa g e 1 6 - 5 6 Laguerr e’s equation Lague rr e ’s equation is the s econd-orde r , linear OD E of the f orm x ⋅ (d 2 y/dx 2 ) +(1 − x) ⋅ (d y/dx) + n ⋅ y = 0. L aguerr e pol ynomi als, de fined as , ar e soluti ons to L aguerr e ’s equation .
Pa g e 1 6 - 57 L 2 (x) = 1- 2x+ 0.5x 2 L 3 (x) = 1-3x+1. 5x 2 - 0 . 16 666… x 3 . W eber ’s equation and H er mite poly nomials W eber’s eq uation is def ined as d 2 y/dx 2 +(n+1/2 - x 2 /4)y =.
Pa g e 1 6 - 5 8 F i r st , c r eate the e xpr es sion de fining the de ri vati v e and stor e it into var i able E Q. T he fi gur e to the left sho ws the AL G mode command, w hile the ri ght-hand side f igur e 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 t o 0.5, and f inal v alue of t to 0.7 5, sol v e f or v(0.7 5) = 2 . 066…) @@OK@ @ @INIT+ — 1 @@OK@@ ™ ™ @SOLVE (wa it ) @EDIT (Changes initi al value o f t to 0.
Pa g e 1 6 - 6 0 Θ „ô (simultaneousl y , if in RPN mode) to enter P L O T env i r onment Θ Hi ghligh t the f ield in f r ont o f TYPE , using the —˜ k ey s. T hen , pres s @CHOOS , and highlight Diff Eq , u sing the —˜ k ey s. Pr ess @@OK@@ .
Pa g e 1 6 - 6 1 LL @) PICT T o re c over m e nu a n d re t u rn to PI C T envi ro n me n t. @ ( X,Y ) @ T o determine coor dina t es of an y point on the gr aph . Use the š™ k ey s to mo ve the cursor ar oun d the plot a r ea . At th e bottom of the sc r een y ou w ill see the coor dinates of the c urs or as (X,Y) , i .
Pa g e 1 6 - 62 time t = 2 , the input for m fo r the differ ential equation s olv er should look a s fo llo w s (notice that the Init: v alue f or the Soln: is a v ect or [0, 6]) : Press @SOLVE (wai t) @EDIT to s ol ve f or w(t=2) . The so lution r eads [.
Pa g e 1 6 - 6 3 (Changes initi al value of t to 0.7 5, and f inal value o f t to 1, sol ve again f or w(1) = [-0.4 6 9 -0.6 0 7]) R epeat for t = 1.2 5, 1.5 0, 1.7 5, 2 .0 0. Pre ss @@OK@@ after v ie w ing the last r esult in @EDIT . T o r eturn to nor mal calculator displa y , pr ess $ or L @@OK@@ .
Pa g e 1 6 - 6 4 Notice that the opti on V - V ar : is set to 1, indicating that the f irst ele ment in the v ector s oluti on, namel y , x ’ , is to be plotted against the independent v ar iable t . Accept c hanges to P L O T SETUP b y pr essing L @@OK@@ .
Pa g e 1 6 - 65 Press LL @PICT @CANCL $ to r etur n to nor mal calc ulator dis play . Numerical solution for stiff first-or d er ODE Consi der the ODE: d y/dt = -100y+100t+ 101, sub jec t to the initial conditi on y(0) = 1.
Pa g e 1 6 - 6 6 Her e w e are try ing to obtain the v alue of y( 2) giv en y(0) = 1. W ith the Soln: Final f ield highli ghted, pr ess @SOLVE . Y ou can chec k that a soluti on tak es abo ut 6 sec on ds, wh il e i n t he previou s fi rst - orde r exa mp le th e s ol ut ion was alm os t instantaneou s.
Pa g e 1 6 - 67 Note: T he opti on Stiff is also a vailable f or gr aphical s oluti ons of differ ential equati ons. Numerical solution to ODEs w it h th e S O L VE/DIFF menu T he S OL VE soft men u is acti va ted b y using 7 4 MENU in RPN mode . T his menu is pr esent ed in detail in Cha pter 6 .
Pa g e 1 6 - 6 8 T he value of the so lution , y fin a l , w ill be av ailable in v ar iable @@@y@@@ . This f uncti on is appr opr iate f or pr ogramming since it lea v es the diff er ential eq uation spec if icati ons and the toler ance in the st ack r eady f or a new s olution .
Pa g e 1 6 - 69 contain only the v alue of ε , and the step Δ x w ill be tak en as a small default value . After running f unction @@RKF@ @ , the s tack w ill show the lines: 2 : {‘ x ’ , ‘ y.
Pa g e 1 6 - 70 T hese r esults indi cate that ( Δ x) ne xt = 0. 34 04 9… Function RRKS TEP T his f uncti on use s an input list similar to that of func tion RRK , as well as the toler ance for the.
Pa g e 1 6 - 7 1 T hese r esults indi cate that ( Δ x) ne xt = 0. 005 5 8… and that the RKF method (CURRENT = 1) should be used. Function RKFERR T his functi on r etur ns the abso lute er r or estimate f or a gi ven s tep whe n sol v ing a pr oblem as that des cr ibed f or func tion RKF .
Pa g e 1 6 - 72 T he follo w ing scr een shots sho w the RPN st ack bef ore and af t er applicati on of func tion R SBERR: T hese r esults indi cate that Δ y = 4.1514… and err or = 2 .7 6 2 ..., fo r Dx = 0.1. Chec k that , if Dx is redu ced to 0. 01, Δ y = -0.
Pa g e 1 7- 1 Chapter 17 Pr obability Applications In this Chapte r w e pr ov ide e xample s of applicati ons of calc ulator’s func tions to pr obab ility distr ibutions . T he MTH/PR OB ABILITY .. sub-m enu - part 1 T he MTH/PR OB ABILITY .. sub-men u is accessible thr ough the ke ys tr ok e sequence „´ .
Pa g e 1 7- 2 T o simplify notation , use P(n ,r) f or per mutati ons, and C(n ,r) f or combinations . W e can calculat e combinations , perm utations , and factor i als with f uncti ons CO MB, P ERM, and ! fr om the MT H/P R OBA BILITY .
Pa g e 1 7- 3 R andom number gener ators , in gener al, oper ate b y taking a v alue , called the “ seed” of the gener ator , and per f or ming some mathematical algor ithm on that “ seed” that gener ates a ne w (ps eudo)r andom number .
Pa g e 1 7- 4 fu nct ion (pmf) is r epr esented by f (x) = P[X=x], i .e ., the pr obability that the ra nd om va riab le X ta kes th e val ue x. T he mass distr ibuti on functi on mus t satisf y the c.
Pa g e 1 7- 5 P oisson distribution The probabilit y mass f unction of the P oisson di str ibut ion is giv en by . In this e xpre ssi on, if the r andom var i able X r epre sents the n umber of occ urr ences o f an e ven t or observati on per unit time , length , area , volume , etc.
Pa g e 1 7- 6 Continuous pr obabilit y distr ibutions T he proba bility distributi on f or a continuou s r andom var ia ble , X, is c harac ter i z e d b y a f uncti on f(x) know n as the pr obab ilit y density functi on (pdf) .
Pa g e 1 7- 7 , w hile its cdf is giv en b y F(x) = 1 - e xp(- x/ β ) , f or x>0, β >0. T he beta distribution T he pdf for the gamma dis tr ibution is gi v en b y As in the case of the gamma dis tribut ion , the corr esponding cdf for the bet a distr ibuti on is also gi v en b y an integr al w ith no c losed-f orm solu tion .
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 def ine all these f unctions .
Pa g e 1 7- 9 Continuous distributions f or statistical infer ence In this sec tion w e disc uss f our contin uous pr obability distr ibutions that ar e commonl y used f or pr oblems r elated to statis tical inf er ence .
Pa g e 1 7- 1 0 wher e μ is the mean , and σ 2 is the v ari ance of the dis tributi on . T o calc ulate the val ue of f( μ , σ 2 ,x) fo r the normal distr ibution , us e functi on NDIS T with the fo llo w ing ar guments: the mean , μ , the var iance , σ 2 , and, the v alue x , i.
Pa g e 1 7- 1 1 wher e Γ ( α ) = ( α -1)! is the G AMMA functi on def ined in Chapter 3 . T he calculator pr ov ides f or values o f the upper - t ail (cumulati v e) distr ibution func tion f or the t-distr ibution , f uncti on UTPT , gi ve n the paramet er ν and the value of t , i .
Pa g e 1 7- 1 2 T he calculator pr ov ides f or values o f the upper - t ail (cumulati v e) distr ibution fu nct ion fo r th e χ 2 -distr ibutio n using [UTP C] gi v en the v alue of x and the par ameter ν .
Pa g e 1 7- 1 3 T he calculator pr ov ides f or values o f the upper - t ail (cumulati v e) distr ibution func tion f or the F distr ibuti on, f uncti on UTPF , gi ven the par ameter s ν N and ν D, and the value of F . The definition of th is 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 ibuti ons the e xpr essi ons to s olv e w ill be mor e compli cated due to the pr esence o f integr als, i .e ., • Gamma, • Beta , A numer ical soluti on w ith the numer i cal sol ver w ill not be feasible beca use of the integr al sign in v olv ed in the e xpre ssi on.
Pa g e 1 7- 1 5 Ther e are tw o r oots of this functi on f ound by using f unction @ROOT w i thin the plo t en vi r onment . Because o f the integr al in the equatio n, the r oot is appro ximat ed and w ill not be sho wn in the plot s cr een . Y ou w ill only get the me ssage Cons tant? Sho wn in the sc r een.
Pa g e 1 7- 1 6 Notice that the second par amet er in the UTPN functi on is σ 2, n o t σ 2 , r epr esenting the v ar iance of the distr ibuti on. A lso , the s ymbol ν (the low er-case Gr eek letter no) is not a v ailable in the calc ulator . Y ou can us e , for e xample , γ (gamma) instead o f ν .
Pa g e 1 7- 1 7 Th us, at this point , you w ill hav e the four equati ons av ailable for so lution . Y ou needs ju st load one of the equati ons into the E Q f ie ld in the nume ri cal solv er and pr oceed w ith sol v ing for one o f the var ia bles .
Pa g e 1 7- 1 8 W ith these four equati ons, w henev er y ou launch the numer i cal s olv er y ou ha ve the f ollo w ing cho i ces: Ex amples of s olution o f equations E QNA, E QT A, E QCA, and E QF .
P age 18-1 Chapter 18 Statistical Applications In this Chapte r we intr oduce statisti cal applicati ons of the calc ulator including statis tic s of a sample , f r equency dis tributi on of data , simple r egre ssi on, conf i dence int ervals , and h ypothe sis te sting .
P age 18-2 St or e the pr ogram in a v ar iable called LX C. After st or ing this pr ogram in RPN mode y ou can also us e it in AL G mode. T o sto r e a column vec tor into v ar iable Σ D A T use functi on S T O Σ , av ailable thr ough the catalog ( ‚N ) , e .
P age 18-3 Ex ample 1 -- F or the data st or ed in the pr ev ious e x ample , the single -var iable statis tic s r esults ar e the f ollo w ing: M e a n : 2. 1 3333333333 , S t d D e v: 0 . 9 6 42 0 7 9 49 4 0 6, Va r i a n c e : 0 . 9 2969696969 7 T otal: 2 5 .
P age 18-4 Ex amples of calc ulation of these measur es, using lis ts, ar e a vailable in C hapter 8. T he median is the value that s plits the data set in the mi ddle when the e lements ar e placed in incr easing orde r . If y ou hav e an odd number , n, of or dered elements , the median of this sam ple is the value located in positi on (n+1)/2 .
P age 18-5 Th e ran g e of the sample is the differ ence betw een the maximum and minim um v alues of the sample . Since the calc ulator , thr ough the pr e -pr ogr ammed statis tical f uncti ons pr o v ides the max imum and minimum values o f the sample , y ou can easily calc ulate the range .
P age 18-6 Definition s T o unders tand the meaning of thes e par ameters w e pr esent the follo w ing def initions : Gi ven a se t of n data values: {x 1 , x 2 , …, x n } lis ted in no partic ular .
P age 18-7 Θ Gener ate the list o f 200 number b y using RDLIS T(200) in AL G mode , or 200 ` @ RDLIST@ in RPN mode . Θ Us e pr ogram LXC (s ee abo ve) to con vert the list th us gener ated into a column vec tor . Θ St ore the column v ector into Σ DA T , by u s i n g f un c t i o n ST O Σ .
P age 18-8 to calc ulate for unif orm-si z e c lasses (or bins) , and the class mar k is just the a ver age of the clas s boundari es f or eac h cla ss.
P age 18-9 « DUP S I ZE 1 GET fr eq k « {k 1} 0 CON cfr eq « ‘fr eq(1,1)’ EV AL ‘ cfr eq(1,1)’ S T O 2 k FOR j ‘ cfr eq(j-1,1) +fr eq(j,1)’ EV AL ‘ cfr eq (j,1)’ S T O NEXT cfr e q » » » Sa ve it un der the name CFRE Q.
P age 18-10 Θ Press @CAN CEL to r etur n to the pre vi ous s cr een. Change the V - v ie w and Bar W idth once mor e , now to r ead V- Vi e w: 0 3 0, Bar Wi dth: 10. The ne w histogr am, bas ed on the same data set , no w looks lik e this: A plot of f r equency count , f i , v s.
P age 18-11 Θ F i r st , enter the two r ow s of data into column in the var iable Σ DA T by u s i n g the matri x wr iter , and func tion S T O Σ . Θ T o access the pr ogr am 3. Fit data.. , us e the follo wi ng k ey str ok es: ‚Ù˜˜ @@@OK@@@ T he input fo rm w ill sho w the c urr ent Σ D A T , alread y loaded.
P age 18-12 Wher e s x , s y ar e the standar d de v iati ons of x and y , re spec ti vel y , i .e . Th e va lu es s xy and r xy ar e the "Co var iance" and "Cor r elati on," r es pecti v ely , obtained b y using the "F it data" featur e of the calc ulator .
P age 18-13 T he gener al fo rm of the r egr essi on equati on is η = A + B ξ . Best data fitting T he calculat or can deter mine whi ch one of its linear or linear i z ed r elatio nship off ers the bes t fitting f or a set of (x ,y) data points . W e w ill illustr ate the u se of this featur e w ith an e x ample .
P age 18-14 X-Col, Y -Co l: these options a pply onl y w hen yo u hav e mor e than t w o columns in the matr ix Σ D A T . B y def ault , the x column is column 1, and the y col umn is column 2 .
P age 18-15 B. I f n ⋅ p is an integer , say k , calc ulate the mean of the k - th and (k -1) th or der ed observ ations . T his algorithm can be implemented in the f ollo w ing pr ogr am typed in R.
P age 18-16 T he D A T A sub-menu T he D A T A sub-menu cont ains functi ons us ed to manipulate the statis tic s matri x Σ DA TA : The ope rati on of thes e func tions is as f ollo w s: Σ + : add r o w in lev el 1 to bottom of Σ DA T A ma t rix. Σ - : r emo ve s last r o w in Σ D A T A matri x and place s it in le vel o f 1 of the s tack .
P age 18-17 Σ P AR: show s statis tical par ameter s. RE SET : r eset par ameter s to default v alues INFO: sh o ws s tatist ical par ameter s The MODL sub-menu w ithin Σ PA R T his sub-menu con tai.
P age 18-18 T he functi ons inc luded ar e: B ARP L: produce s a bar plot with data in Xcol column of the Σ D ATA m a t r i x . HIS TP: pr oduces his togr am of the data in Xcol co lumn in the Σ DA .
P age 18-19 Σ X^2 : pr o v ides the sum of s quar es of v alues in Xcol column . Σ Y^2 : pr ov ides the sum of squar es of value s in Ycol column . Σ X*Y : pr ov ides the sum o f x ⋅ y , i .e., the pr oducts o f data in columns Xcol and Ycol. N Σ : pr o v ides the number of col umns in the Σ DA T A m a tr ix.
P age 18-20 @) STAT @ ) £PAR @ RESET r esets statis tical par ameters L @ ) STAT @PLOT @SCATR pr oduces s catter plot @STATL dr aw s data fit as a s trai ght line @CANCL r eturns t o main display Θ Deter mine the f itting equation and so me of its statis tic s: @) STAT @ ) FIT@ @£LINE pr oduces '1.
P age 18-21 Θ F it data using columns 1 (x) and 3 (y) using a logar ithmic f itting: L @ ) STAT @ ) £ PAR 3 @YCO L sel ect Y col = 3, and @) MODL @LOGFI sele ct Mod el = Log f it L @ ) STAT @PLOT @ SCATR pr oduce scatter gr am of y v s. x @STATL sho w line for log f itting Ob v iou sly , the log-f it is not a good ch oi ce .
P age 18-2 2 L @ ) STAT @PLOT @ SCATR pr oduce scatter gr am of y v s. x @STATL sho w line for log f itting Θ T o r eturn to S T A T menu use: L @) STAT Θ T o get y our var iable menu bac k use: J . Confidence inter v als St atistical inf er ence is the proce ss of making conc lusi ons about a populati on based on info rmati on fr om sample data.
P age 18-2 3 Θ P oint es timation: w hen a single value of the par amet er θ is pro v ided . Θ Co nfi dence interval: a n umer ical interv al that contains the par ameter θ at a gi ven le v el of pr obability . Θ E stimator : rule o r method of estimati on of the par ameter θ .
P age 18-2 4 Θ The par ameter α is kno wn as the si gnif icance le v el . T y pi cal v alues of α ar e 0. 01, 0. 05, 0.1, cor re sponding to conf idence le v els of 0.
P age 18-2 5 Small samples and large sampl es T he behav i or of the Student’s t distr ibution is suc h that for n>3 0, the distr ibution is indistinguishable fr om the standar d nor mal distribu tion .
P age 18-2 6 E stimator s for the mean and s tandar d dev iation o f the diff er ence and sum of the statis tics S 1 and S 2 ar e gi v en b y: In t hese expressions, ⎯ X 1 and ⎯ X 2 ar e the v alu.
P age 18-2 7 In this case , the cente red conf idence intervals f or the sum and diff er ence of the mean v alues of the populations , i .e ., μ 1 ±μ 2 , are gi ven b y : wher e ν = n 1 +n 2 - 2 is the number of degrees o f fr eedom in the Student’s t distr ibuti on.
P age 18-2 8 These options ar e to be i nterpr eted as follow s : 1. Z -INT : 1 μ .: Single sample conf idence in te r v al fo r the population mean , μ , w ith kno wn populati on var iance , or for lar ge s amples w ith unkno wn populatio n v ari ance .
P age 18-29 Press @HELP to obtain a sc r een e xplaining the meaning of the confi dence interval in terms o f r andom numbers gener ated by a calc ulator . T o scr oll do wn the r esulting sc r een use the do w n -ar r o w k ey ˜ . Pr ess @@@OK@@@ whe n done with the help sc r een.
P age 18-30 Ex ample 2 -- Data f r om two s amples (s amples 1 and 2) indicat e 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 w n that the populations ’ standar d dev iati ons ar e σ 1 = 3 .
P age 18-31 When done , pre ss @@@OK@@@ . The r esults, as te xt and gr aph, ar e sho wn be lo w: Ex ample 4 -- Determine a 90% conf idence inter v al f or the differ ence between two pr oportions if sample 1 sho ws 20 su ccess es out of 120 tr ials , and sample 2 s ho ws 15 s uccesses out of 1 00 trial s .
P age 18-3 2 Ex ample 5 – Determine a 9 5% conf idence in terval f or the mean of the populatio n if a s ample of 50 elements has a mean of 15 . 5 and a st andard de vi atio n of 5 . The popul ation ’s standar d dev iation is unkno wn . Press ‚Ù— @@@OK@@@ to access the confi dence inter v al f eatur e in the calc ulator .
P age 18-3 3 T hese r esults assume that the v alues s 1 and s 2 ar e the population st andar d de vi ations . If these v alues actuall y r epr esent the s amples ’ standar d de v iatio ns, y ou should enter the s ame values as be for e, bu t wi th the option _pooled selected .
P age 18-34 T he confi dence interv al fo r the population v ari ance σ 2 i s therefor e , [(n -1) ⋅ S 2 / χ 2 n-1 , α /2 ; (n-1) ⋅ S 2 / χ 2 n-1,1- α /2 ].
P age 18-35 Hy pot hesis testing A h ypo thesis is a declar ation made about a populati on (for ins tance , w ith r espect to its mean) . A cceptance of the h y pothesis is based o n a statisti cal test on a sample tak en fr om the population . The consequent acti on and dec isi on - making ar e called h y pothesis te sting .
Pa g e 1 8 - 3 6 Err ors in h ypothesis testing In h ypothe sis testing w e use the ter ms err ors of T y pe I and T y pe II to def ine the case s in w hich a tr ue h ypothe sis is re jec ted or a fals e h ypothe sis is accepted (not r ejected) , respect i vel y .
P age 18-3 7 Th e va lu e of β , i .e ., the pr obability of making an er r or of T ype II , depends on α , the sample si z e n, and on the tr ue value o f the paramet er tes ted . Th us, the val ue of β is deter mined af t er the hy pothesis testing is perf ormed .
P age 18-38 T he cr ite ri a to use f or h y pothesis te sting is: Θ Rej ec t H o if P -value < α Θ Do not r ej ect H o if P -value > α . T he P -v alue fo r a two -si ded tes t can be calc .
P age 18-3 9 Ne xt, w e u se the P - v alue assoc iated w ith either z ο or t ο , and compar e it to α to dec ide w hether or no t to r ej ect the n ull hy pothesis. T he P - v alue f or a tw o -sided tes t is defined as e ither 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 st andard de vi ations s 1 and s 2 . If the populations standar d dev iati ons cor r esponding to the samples , σ 1 and σ 2 , ar e kno wn , or if n 1 &.
P age 18-41 T he cr ite ri a to use f or h y pothesis te sting is: Θ Rej ec t 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 w ith tw o sample.
P age 18-4 2 wher e Φ (z) is the c umulativ e distributi on fu nctio n (CD F ) of the st andar d normal distr ibuti on (see Cha pter 17). R ejec t the null h ypothe sis, H 0 , if z 0 >z α /2 , or if z 0 < - z α /2 .
P age 18-4 3 T wo - tail ed test If using a two -tailed test w e w ill find the v alue 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 fu nctio n (CD F ) of the st andar d normal distr ibuti on.
P age 18-44 1. Z - T est : 1 μ .: Single s ample hy pothesis testing f or the population mean , μ , w ith kno w n population v ar iance , or f or lar ge samples w ith unknow n populatio n v ari ance .
P age 18-45 Then , w e r ej ect H 0 : μ = 150 , against H 1 : μ ≠ 150 . The test z v alue is z 0 = 5. 656854 . T he P- va l u e i s 1. 54 × 10 -8 . Th e cri ti ca l va lu es of ± z α /2 = ± 1.9 5 9 9 64 , corr esponding to cr iti cal ⎯ x r ange of {14 7 .
P age 18-46 W e r ej ect the null h ypothe sis, H 0 : μ 0 = 15 0, against the alter nati v e h ypo thesis , H 1 : μ > 15 0. The t est t v alue is t 0 = 5. 6 5 68 5 4, w ith a P -v alue = 0. 000000 3 9 3 5 2 5 . The c r itical v alue of t is t α = 1.
P age 18-4 7 T hus , w e accept (mor e acc urat ely , we do no t r ejec t) the h y pothesis: H 0 : μ 1 −μ 2 = 0 , or H 0 : μ 1 =μ 2 , against the alternati ve h y pothesis 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 T he test c r iter ia ar e the same as in h y pothesis te sting of means , namely , Θ Rej ec t H o if P -value < α Θ Do not r ej ect H o if P -value > α . P lease noti ce that this pr ocedur e is valid onl y if the populati on fr om w hic h the sample w as tak en is a Normal populati on .
P age 18-4 9 T he follo w ing table sho ws h ow to select the nu merat or and denominator f or F o depending on the alter nati ve h ypothe sis cho sen: ___________ _____________________ ______________.
P age 18-50 Ther efor e , the F test stati stics is F o = s M 2 /s m 2 =0. 3 6/0.2 5=1. 44 T he P -v alue is P -value = P(F>F o ) = P(F>1.44) = UTPF( ν N , ν D ,F o ) = UTPF( 20, 30,1.
P age 18-51 W e get the , so -called, nor mal equations: T his is a s y stem o f linear equati ons w ith a and b a s the unkno w ns, whi c h can be sol v ed using the linear equation f eature s of the calculator . T her e is, ho w ev er , no need to bother wi th these calc ulations becau se y ou can use the 3.
Pa g e 1 8 - 52 F r om w hic h it fo llow s that the standar d dev iations o f x and y , and the co var iance of x ,y ar e giv en , r espec tiv el y , by , , and Also , the sample corr elation coeff i.
Pa g e 1 8 - 5 3 Θ Co nfi dence limits for r egres sion coeff i c ients: F or the slope ( Β ): b − (t n- 2 , α /2 ) ⋅ s e / √ S xx < Β < b + (t n- 2 , α /2 ) ⋅ s e / √ S xx , F or.
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 . Pr ocedure f or inference statistics f or linear regression using the calculator 1) Ent er (x ,y) as columns of data in the st atistical matr ix Σ D AT.
Pa g e 1 8 - 5 5 1: Covariance: 2.025 T hese r esults ar e interpr eted as a = -0.8 6 , b = 3 .2 4, r xy = 0.9 8 9 7 20 2 2 9 7 4 9 , and s xy = 2 . 0 2 5 . T he corr elati on coeff ic ient is c los e enough to 1. 0 to co nfir m the linear tr end obs erved in the gr aph .
P age 18-5 6 Ex ample 2 -- Suppos e that the y-data used in Ex ample 1 r e pr esent the elongation (in h undr edths of an inc h) of a me tal w ir e w hen sub jec ted to a f or ce x (in tens o f pounds) . The ph y sical phe nomenon is suc h that w e e xpect t he inter cept , A, to be z er o .
P age 18-5 7 Multiple lin ear fitting Consi der a data set of the fo rm Suppo se that w e sear c h for a data f itting of the for m y = b 0 + b 1 ⋅ x 1 + b 2 ⋅ x 2 + b 3 ⋅ x 3 + … + b n ⋅ x n .
P age 18-5 8 W ith the calculat or , in RPN mode , yo u can pr oceed as fo llo ws: F irst , w ithin y our HOME dir ect ory , cr eate a sub-dir ect or y to be called MPFI T (Multiple linear and P oly nomial data FI Tting) , and ent er the M P FIT su b- dir ectory .
P age 18-5 9 Compar e these f itted value s with the or iginal data as sho w n in the table belo w: P oly nomial fitting Consi der the x -y data set {(x 1 ,y 1 ), ( x 2 ,y 2 ), … , ( x n ,y n )}. Suppos e that we w ant to f it a poly nomial or or der p to this data set .
P age 18-60 If p > n -1 , then add columns n+1, …, p-1, p+1 , to V n to f or m matri x X . In st ep 3 fr om this lis t , we hav e to be a war e that column i ( i = n+1, n+2 , …, p+1 ) is the v ector [x 1 i x 2 i … x n i ]. If w e w er e to use a list o f data values f or x r ather than a v ector , i .
P age 18-61 « Open pr ogram x y p Enter l ists x and y , and p (le v els 3,2 ,1) « Open subpr ogram 1 x SI ZE n Deter mine siz e of x list « Open subpr ogram 2 x V ANDERMOND E P lace x in s.
P age 18-6 2 Becau se w e w ill be using the same x -y data for f itting poly nomi als of diff er ent or ders , it is adv isable to s av e the lists of data v alues x and y into v ari ables xx and yy , re specti vel y . This w a y , we w ill not ha ve to t y pe them all o v er again in eac h applicati on of the pr ogr am P OL Y .
P age 18-63 Θ T he corr elation coe ff ic ient , r . T h is value is constr ained to the r a nge –1 < r < 1. T he cl os er r is to +1 or –1, the better the data f itting. Θ T he sum of squar ed er ro rs, S SE . T his is the quantity tha t is to be minimi z ed by lea st-squar e appr oac h.
P age 18-64 x V ANDERMOND E P lace x in stac k, obtain V n I F ‘ p<n -1’ THEN T his I F is s tep 3 in algor ithm n P lace n in stac k p 2 + Calc ulate p+1 FOR j S tar t loop , j = n-1 to p+1, s.
P age 18-6 5 “SSE” T A G T ag r esult as S SE » Close sub-progr am 4 » Clo se sub-pr ogram 3 » C lose su b-pr ogr am 2 » Clo se sub-pr ogr am 1 » Clo se main pr ogram Sa ve this pr ogr am under the name P OL YR , to emph asi z e calculati on of the correlation coeffic ient r .
P age 19-1 Chapter 19 Numbers in Differ ent Bases In this Chapt er w e pre sent e x amples of calculati ons of number in bases other than the dec imal basis .
P age 19-2 W ith sy st em flag 117 set to S OFT menus, the B A SE menu sho ws the f ollo w ing: W ith this for mat , it is ev ident that the L OGIC, BIT , and B YTE entri es w ithin the B ASE menu ar e th emselv es sub-menus. These menus are discussed later in this Chapter .
P age 19-3 As the dec imal (D E C) sy stem has 10 digits (0,1,2 , 3, 4,5, 6, 7 , 8 , 9) , the he xadec imal (HEX) sy stem 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 binar y (BIN) s ys tem has only 2 di gits (0,1) .
P age 19-4 T he only e ffec t of selecting the DE C imal s y stem is that dec imal numbers , whe n started w ith the s ymbol #, ar e wr itten with the suff ix d . W ordsi ze T he wor dsi z e is the number of b its in a b inar y obj ect . B y defa ult , the w ordsi z e is 64 bites .
P age 19-5 The L OGIC m enu T he L OGIC menu , av ailable thr ough the B A SE ( ‚ã ) pr ov ides the f ollo wing fu nct ions : T he functi ons AND , OR, X OR (e x c lusi v e OR) , and NO T ar e logical f uncti ons.
P age 19-6 AND (BIN) OR (BIN) X OR (BIN) NO T (HEX) T he B I T menu T he BIT men u , av ailable thr ough the B ASE ( ‚ã ) pr ov ide s the follo w ing fu nct ions : F uncti ons RL, SL , A SR, SR, RR , contained in the BIT menu , ar e used to manipulate b its in a binary integer .
P age 19-7 T he B Y TE menu T he B YTE menu , av ailable thr ough the B A SE ( ‚ã ) pr o v ides the f ollo w ing fu nct ions : F uncti ons RLB, SLB , SRB , RRB, cont ained in the BIT menu , ar e used to manipulate b its in a binary integer . T he def initi on of thes e func tions are sho wn belo w : RLB: R otate Left one byte , e.
Pa g e 2 0 - 1 Chapter 20 Customi zing menus and k e yboar d T hro ugh the use of the man y calc ulator menus y ou hav e become famili ar w ith the oper ati on of men us f or a v ar iety of appli catio ns.
Pa g e 2 0 - 2 M enu numbers (R CLMENU and MENU func tions) E ac h pre-defined men u has a number attac hed to it . F or ex ample, su ppose that y ou acti vate the MTH menu ( „´ ). Then , using the f uncti on catalog ( ‚N ) f ind functi on R CLMENU and ac ti vate it.
Pa g e 2 0 - 3 T o acti vate an y of those functi ons y ou simply need to enter the f unction ar gument (a number ) , and then pr es s the corr es ponding soft menu k ey .
Pa g e 2 0 - 4 Y ou can try using this list w ith TMENU or MENU in RPN mode to ve rify that y ou get the same menu a s obtained ear lier in AL G mode .
Pa g e 2 0 - 5 Customizing the k e y board E ach k ey in the k e yboar d can be i dentif ied by tw o numbers r e pr esenting their r o w and column. F or e xam ple , the V AR ke y ( J ) is located in r o w 3 of column 1, and w ill be r ef er red t o as k ey 31.
Pa g e 2 0 - 6 T he functi ons av ailable ar e: A SN: Assigns an obj ect to a k e y spec ified b y XY .Z S T O KE Y S : Stor es user -d ef ined k e y l ist RC LK EY S : Ret urn s curren t use r-de fin.
Pa g e 2 0 - 7 Operating user-defined ke ys T o oper a t e this user -def ined k e y , enter „Ì be fo re pre ssing the C key . Notice that afte r pre ssing „Ì the sc r een sho w s the spec ifi cation 1US R in the second displa y line .
Pa g e 2 0 - 8 T o un -assign all user -defined k ey s use: AL G mode: DELKE YS(0) RPN mode: 0 DELKEYS Chec k that the use r -k e y def initions w er e r emov ed b y using f unction R C LKE Y S.
P age 21-1 Chapter 21 Pr ogr amming in User RP L language Use r RP L language is the pr ogramming language mo st commonl y used to pr ogr am the calculator . The pr ogram components can be put t ogether in the line editor by inc luding them betw een pr ogram container s « » in the appr opr iat e orde r .
P age 21-2 „´ @LIST @ADD@ AD D Calc ulate (1+x 2 ), / / then di v ide ['] ~„x™ 'x' „° @) @MEM@@ @ ) @DIR@@ @ PURGE PURGE P u rge varia b le x ` Pr ogr am in lev el 1 _________.
P age 21-3 use a local v ar iable w ithin the pr ogram that is only de fi ned for that pr ogr am and w ill not be a v ailable fo r use afte r pr ogr am e xec ution .
P age 21-4 Global V ariable Scope An y var iable that y ou def i ne in the HO ME dir ectory or an y other dir ecto r y or sub-dir ectory w ill be consider ed a global var iable fr om the point o f vi ew of pr ogr am dev elopment . Ho we v er , the sco pe of suc h v ari able , i .
P age 21-5 Local V ariable Scope L ocal var iable s are ac tiv e only w ithin a pr ogr am or sub-pr ogram . The r ef or e , their scope is limited t o the pr ogr am or sub-pr ogram w her e the y’r e def ined .
P age 21-6 S T ART : S T AR T -NEXT -S TEP constru ct f or br anching FOR: F O R - NE XT- S TEP constr uct f or loops DO: DO-UNT IL -END constr uct f or loops WHILE: WHILE-REP EA T-END co nstru ct f o.
P age 21-7 Functions listed b y sub-menu T he follo wing is a lis ting of the func tions w ithin the P RG sub-menus lis ted b y sub- menu . ST A CK MEM/DIR BR CH/IF BRCH/WHILE TYP E DUP P URGE IF WHIL.
P age 21-8 LIS T/ELEM GROB CHARS MODES/FLAG MO DES/MISC GE T GROB S UB SF BEEP GE TI BL ANK REP L CF CLK PU T GO R POS F S ? S Y M PU TI G X O R SIZE F C ? S T K S IZE SUB NUM F S?C ARG P O S REPL.
P age 21-9 Shortc uts in the PR G menu Man y of the func tions lis ted abo ve f or the P RG menu ar e r eadily a v ailable thr ough other means: Θ Compar ison oper ators ( ≠ , ≤ , <, ≥ , >) ar e a vailable in the k ey boar d.
P age 21-10 „ @ ) @IF@ @ „ @CASE@ „ @ ) @IF@ @ „ @CASE@ „ @ ) START „ @) @ FOR@ „ @ ) START „ @) @ FOR@ „ @ ) @@DO@@ „ @ WHILE Notice that the ins ert pr ompt ( ) is av ailabl e after the k e y w or d fo r each constr uct s o yo u can start t y ping at the r ight locatio n.
P age 21-11 @) STACK DUP „° @) STACK @ @DUP@@ SW A P „° @) STACK @SWAP@ DR OP „° @) STACK @DROP@ @) @MEM@@ @ ) @DIR@@ PU RG E „° @) @MEM@@ @ ) @ DIR@@ @PUR GE ORDER „° @) @MEM@@ @ ) @DI.
P age 21-12 @) @BRCH@ @ ) WHILE@ WHILE „° @) @B RCH@ @ ) WHILE @ @WHILE REP EA T „° ) @BRCH@ @ ) WHILE@ @REPEA END „° ) @BRCH@ @ ) WHILE@ @ @END@ @ ) TEST@ == „° @ ) TEST@ @ @@ ≠ @@@ AND.
P age 21-13 @) LIST@ @ ) PROC@ REVLI S T „° @) LIST@ @ ) PROC@ @REVLI@ SO RT „° @) LIST@ @ ) PROC@ L @SORT@ SE Q „° @) LIST@ @ ) P ROC@ L @@SEQ@@ @) MODES @ ) ANG LE@ DE G „°L @) MODES @ ).
P age 21-14 fu nctio ns from th e M TH m enu . Spe c ifica lly , you ca n use fun ction s for li st oper ations suc h as S ORT , Σ LIS T , etc ., a vail able thr ough the MTH/LI S T menu .
P age 21-15 Ex amples of sequential pr ogramming In gener al , a pr ogr am is an y sequence o f calc ulato r instruc tions enc lo sed between the pr ogram container s and ». Subpr ograms can be inc luded as part o f a pr ogr am. The e xamples pr esented pr e v iou sly in this guide (e .
P age 21-16 wher e C u is a constant that depends on the sy st em of units used [C u = 1. 0 for units of the Internati onal S ys tem (S.I .) , and C u = 1.
P age 21-17 Y ou can also separ ate the in put data w ith spaces in a single stac k line r ather than using ` . Pr ograms that simulate a sequence of stack operations In this case , the terms to be in v olv ed in the sequence o f oper ations ar e as sumed to be pr es ent in the stac k .
P age 21-18 As y ou can see , y is used f i r st , then w e us e b, g , a n d Q, in that order . Ther efor e, for the pur pose of this calculatio n we need to enter the v ar iables in the in ve rse or der , i .e. , (do not t y pe the f ollo w ing) : Q ` g ` b ` y ` F or the spec if ic v alues under consider ation w e use: 23 ` 32.
P age 21-19 Sa ve the pr ogram int o a var iable called hv: ³~„h~„v K A ne w var iable @@@hv @@@ should be av ailable in y our soft k e y menu . (Pr ess J to see y our v ar iable lis t .) The pr ogram le ft in the stac k can be e valuat ed by u sing func tion EV AL.
P age 21-20 it is al wa y s pos sible to r ecall the pr ogr am def inition int o the stac k ( ‚ @@@q@@@ ) to see the or der in w hic h the v ari ables mu st be ent er ed , namely , → Cu n y0 S0 .
P age 21-21 w hich indi cates the positi on of the diff er ent stac k input le vels in the fo rmula . B y compar ing this r esult w ith the or iginal f ormula that w e pr ogr ammed , i .e ., w e find that w e mu st enter y in s tack le vel 1 (S1), b in stac k lev el 2 (S2), g in stac k le v el 3 (S3) , and Q in st ack le vel 4 (S4).
P age 21-2 2 T he re sult is a stac k pr ompting the user f or the value o f a and plac ing the cu rsor r ight in fr on t of the prompt :a: Ent er a value f or a , sa y 3 5, then pre ss ` .
P age 21-2 3 @SST ↓ @ R esult: em pt y s tack , e x ec uting → a @SST ↓ @ R esult: empty stac k, ente ring subpr ogr am « @SST ↓ @ R esult: ‘2*a^2+3’ @SST ↓ @ R esult: ‘2*a^2+3’ , l.
P age 21-2 4 F ixi ng th e pr ogram T he only pos sible explanati on f or the failur e of the pr ogr am to pr oduce a numer ical r esult seems to be the lac k of the command NUM after the algebr aic e xpr essi on ‘2*a^2+3’ . Let ’s edit the progr am by adding the mis sing EV AL functi on .
P age 21-2 5 Input string progr am for two input v alues T he input str ing pr ogr am fo r t w o input values , say a and b , looks as f ollo ws: « “ Enter a and b: “ { “ :a: :b: “ {2 0} V } INPUT OBJ → » T his progr am can be ea sily c r eated b y modif y ing the contents o f INPT a.
P age 21-2 6 ` . The r esult is 4 9 88 7 . 06_J /m^3 . The units of J/m^3 ar e equiv alent to P ascals (P a) , the pr ef err ed pres sur e unit in the S .
P age 21-2 7 Enter v alues o f V = 0. 01_m^3, T = 300_K , and n = 0.8_mol . Bef or e pr es sing ` , the stac k w ill look like this: Press ` to get the re sult 199 5 4 8.2 4_J/m^3, or 199 5 48.2 4_ P a = 199 .5 5 kP a. Input through input f orms F uncti on INFORM ( „°L @) @@IN@ @ @INFOR@ .
Pa g e 2 1 - 2 8 T he lists in items 4 and 5 can be em pty lists. Also , if no v alue is to be select ed for these opti ons y ou can use the NO V AL command ( „°L @) @@IN@ @ @NOVAL@ ).
P age 21-29 3 . F ield f ormat info rmation: { } (an empty lis t , thus , defa ult value s used) 4. L ist of r eset val ues: { 120 1 .0001} 5 . L ist of initial v alues: { 110 1.5 .00001} Save th e prog ram i nto va riab le IN F P1 . P ress @INFP 1 to run the pr ogram .
P age 21-30 T hus , we demonstr ated the u se of f uncti on INFORM. T o see h o w to use the se input v alues in a calc ulation modify the pr ogr am as follo ws: « “ CHEZY’S EQN” { { “ C:” “Chezy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { } { 120 1 .
P age 21-31 « “ CHEZY’S EQN” { { “C:” “Chezy’s coefficient” 0} { “R:” “Hydraulic radius” 0 } { “S:” “Channel bed slope” 0} } { 2 1 } { 120 1 .
P age 21-3 2 Ac tiv ati on of the CHOO SE func tion w ill re turn e ither a z er o, if a @CANCEL ac ti on is used , or , if a c hoi ce is made , the ch oi ce s elect ed (e .
P age 21-3 3 commands “Operation canc elled” MSGBOX w ill sho w a message bo x indicating that the oper ation w as cancelled. Identif y ing output in pr ograms T he simplest w ay to identify numer ical pr ogr am output is to “tag ” the pr ogr am r esults .
P age 21-34 Ex amples of tagged output Ex ample 1 – tagging output fr om function FUNC a Let ’s modif y the f uncti on FUNCa, de f ined earlier , to pr oduce a tagged output .
Pa g e 2 1 - 3 5 « “ Enter a: “ { “ :a: “ {2 0} V } INPUT OBJ →→ a « ‘ 2*a^2+3 ‘ EVAL ” F ” → TAG a SWAP »» (R ecall that the functi on S W AP is av ailable b y using „° @) STACK @SW AP@ ). Stor e the pr ogram bac k into FUNCa b y using „ @FUNCa .
Pa g e 2 1 - 3 6 Ex ample 3 – tagging input and outpu t fr om f uncti on p(V ,T) In this e xample w e modify the pr ogr am @@@p@@@ so that the o utput tagged input v alues and t agged r esult .
P age 21-3 7 Stor e the progr am back into var ia ble p by using „ @@@p@@@ . Ne xt , r un the pr ogr am by pr essing @@@p@@@ . Ent er value s of V = 0.
P age 21-38 T he r esult is the f ollo w ing message bo x: Press @@@OK@@@ to c ancel the mes sage bo x . Y ou could us e a message bo x for o utput fr om a progr am b y using a tagged output , con verted to a s tring , as the output str ing f or MS GBO X.
P age 21-3 9 Press @@@OK@@@ to cancel message b o x output . The stack w ill now look like this: Including input and output in a m essage bo x W e could modify the pr ogram so that not onl y the output , but also the input , is inc luded in a message bo x .
P age 21-40 Y ou w ill notice that after ty ping the k e ys tr ok e sequence ‚ë a ne w line is gener a t ed in the stac k. T he last modif icati on that needs to be included is to type in the plu s sign three times after the call t o the functi on at the v ery e nd of the sub-pr ogram .
P age 21-41 Incorpor ating units within a program As y ou ha ve bee n able to obse r v e fr om all the ex amples f or the diffe r ent vers ion s of pro gram @@@p@@@ pr es ented in this cha pter , attac hing units to input v alues may be a t ediou s pr ocess .
P age 21-4 2 2. ‘ 1_m^3 ’ : T he S .I. un its corr espo nding to V ar e then placed in stac k lev el 1, the tagged input f or V is mo v ed to stack lev el 2 . 3 . * : B y multiply ing the contents of st ack le vels 1 and 2 , w e gener ate a number w ith units (e .
P age 21-4 3 Press @@@OK@@@ to cancel me ssage bo x output . Me s sag e bo x output without units Let ’s modify the progr a m @@@p@@@ once mor e to eliminate the us e of units thr oughout it . The unit-less pr ogram w ill look like 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 t w o or mor e r eal numbers . Depending on the ac tual numbers us ed, su ch a st atement can be true (r epr es ented b y the numer i cal value o f 1. in the calc ulator ) , or fals e (r epr ese nted by the numer ical value of 0.
P age 21-45 Logical oper ators L ogical oper ator s ar e logical partic les that ar e used to jo in or modify simple logical s tatements . The logical ope rat ors a vaila ble in the calculat or can be easily acc essed thr ough the ke ys trok e sequence: „° @ ) TEST@ L .
Pa g e 2 1 - 4 6 T he calculat or include s also the logi cal oper ator S AME . This is a non-standar d logical ope rat or used t o deter mine if two ob jec ts ar e identi cal . If they are identi cal , a value o f 1 (true) is r eturned , if no t, a value of 0 (f alse) is r etur ned.
P age 21-4 7 Br anc hing w ith I F In this secti on w e pr esen ts e xample s using the constr ucts IF…THEN…END and IF…THEN…ELSE…END . T he I F…THEN…END construct T he IF…THEN…END is the simplest of the IF pr ogr am constr ucts . The gener al fo rmat of this co nstruc t is: IF logical_statement THEN program_statements END .
P age 21-48 W ith the cur sor in fr ont of the IF st atement pr ompting the user f or the logical stat ement that wi ll acti vate the I F cons truct when the pr ogr am is e xec ut ed.
P age 21-4 9 Ex ample : T y pe in the f ollo w i ng pr ogram: « → x « IF ‘ x<3 ’ THEN ‘ x^2 ‘ ELSE ‘ 1-x ’ END EVAL ” Done ” MSGBOX » » and sa v e it under the name ‘f2 ’ . Pre ss J and v er ify that var iable @@@f2@@@ is indeed av ailable in your var ia ble menu .
P age 21-50 IF x<3 THEN x 2 ELSE 1-x END While this simple cons truc t w orks f ine w hen y our f uncti on has onl y tw o br anche s, y ou ma y need to nes t IF…THEN…ELSE…END constru cts to deal w ith func tion w ith three or mor e branc hes .
P age 21-51 A comple x IF construc t like this is called a set o f n ested IF … THEN … EL SE … END constr ucts . A poss ible wa y to e valuate f3(x), based on the nested IF constr uct sho wn abo.
Pa g e 2 1 - 52 pr ogr am_stateme nts , and pa sses pr ogram f lo w to the statement f ollow ing the END state ment. T he CASE , THEN, and END st atements ar e a vailable f or selecti ve typ ing by using „° @) @ BRCH@ @ ) CASE@ . If y ou ar e in the BRCH menu , i .
Pa g e 2 1 - 5 3 5. 6 @@ f3c@ Re su l t : -0.6 312 6 6… (i .e ., sin(x) , w ith x in r adians) 12 @@f3c@ Res ul t : 16 2 7 54.7 91419 (i.e ., e xp(x)) 23 @@f3c@ Res ul t - 2 . (i .e ., - 2) As yo u can see , f3c produces e xactl y the same r esults as f3 .
P age 21-54 Commands in v ol ved in the S T AR T constru ct ar e av ailable thr ough: „° @) @BRCH@ @ ) START @ST ART W ithin the BRCH men u ( „° @) @BRCH@ ) the follo wi ng ke ys tr ok es ar e a.
Pa g e 2 1 - 5 5 1. T his pr ogr am needs an integer numbe r as inpu t . Th us , bef or e e xec utio n, that number (n) is in st ack le v el 1. The pr ogram is the n ex ec 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 ( « - s tart subpr ogr am) @SST ↓ @ SL1 = 0.
P age 21-5 7 @SST ↓ @ SL1 = 1. (S + k 2 ) [Stor es value o f SL2 = 2 , into SL1 = ‘k ’] @SST ↓ @ SL1 = ‘S’ , SL2 = 1. (S + k 2 ) @SST ↓ @ Empty st ack [S tor es value of SL2 = 1, int o SL1 = ‘S’] @SST ↓ @ Empty stac k (NEXT – end of loop) --- loop e xec ution n umber 3 f or k = 2 @SST ↓ @ SL1 = 2 .
P age 21-5 8 3 @@@S1@@ Re su lt : S:14 4 @@@S1@@ Res ul t: S:30 5 @@@S1@@ Re su lt : S:55 8 @@@S1@@ Res ul t: S:204 10 @@@S1@@ Res ul t: S:385 20 @@@S1@@ Res u lt : S:2870 30 @@@S1@@ Res ul t: S:9455 .
P age 21-5 9 J 1 # 1. 5 # 0.5 ` Enter par ame ters 1 1. 5 0. 5 [ ‘ ] @GLIST ` En ter the pr ogr am name in lev el 1 „°LL @) @RUN@ @@DBG@ S tart the debugger . Use @SST ↓ @ t o step into the pr ogr am and see the detailed ope rati on of eac h command .
P age 21-60 T o av oid an inf inite loop , mak e sur e that start_value < end_value . Ex ample – ca lc ulate the summation S using a F OR…NEXT construc t T he follo w ing pr ogram calc ulates t.
P age 21-61 Ex ample – gener ate a list of number s using a FOR…S TEP construc t T ype in the pr ogram: « → xs xe dx « xe xs – dx / ABS 1. + → n « xs xe FOR x x dx STEP n → LIST » » » and stor e it in var ia ble @GLIS2 . Θ Chec k out that the pr ogr am call 0.
P age 21-6 2 T he follo w ing pr ogram calc ulates 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 pr ogram in a v ar iable @@ S3@@ .
Pa g e 2 1 - 6 3 T he WHILE construct T he gener al str uctur e of this command is: WHILE logical_statement REPEAT program_statements END T he WHILE stateme nt w ill r epeat the program_statements wh il e logical_statement is tr ue (non z er o) . If not , pr ogram contr ol is pa ssed to the stat ement r ight afte r END .
P age 21-64 and stor e it in var ia ble @GLIS4 . Θ Chec k out that the pr ogr am call 0. 5 ` 2. 5 ` 0.5 ` @ GLIS4 pr oduces the list {0. 5 1. 1.5 2 . 2 . 5}. Θ T o see st ep-by-step oper ation u se the pr ogr am D B UG for a short list , for e xample: J 1 # 1.
P age 21-6 5 If y ou enter “ TR Y A G AIN” ` @ DOERR , p r oduc e s the follow ing m essage: TR Y AGA I N F inally , 0` @DOERR , pr oduces the messa ge: I nter rupted ERRN T his functi on r etur ns a number r epr es enting the most r ecent err or .
P age 21-66 T hese ar e the components of the IFERR … THEN … END construc t or of the IFERR … THEN … EL SE … END constr uct . Both logi cal constr ucts ar e used fo r tr appi ng err or s dur ing pr ogr am ex ec uti on .
P age 21-6 7 User RP L pr ogramming in algebraic mode While all the pr ogr ams pre sent ed earli er are pr oduced and run in RPN mode , y ou can al wa y s type a pr ogr am in Us er RPL w hen in algebrai c mode b y using func tion RP L>. T his functi on is a vaila ble thr ough the command catalog .
P age 21-6 8 Wher eas , using RP L, ther e is no proble m when loading this pr ogram in algebr aic mode:.
Pa g e 22 - 1 Chapter 2 2 Pr ogr ams for gr aphic s manipulation T his chapt er include s a number of e x amples sho w ing ho w to use the calculat or’s func tions f or manipulating gr aphics int er acti v el y or thr ough the us e of pr ogr ams. As in Cha pter 21 w e r ecommend u sing RPN mode and setting s ys tem f lag 117 to S OFT menu labels.
Pa g e 22 - 2 T o us er -def ine a k e y yo u need to add to this list a command or pr ogram fo llo w ed by a r efer ence to the k e y (see details in C hapter 20) .
Pa g e 22 - 3 LA BE L (10) T he functi on L ABEL is us ed to label the ax es in a plot including the v ar iable names and minimum and max imum value s of the axe s. T he var ia ble names ar e select ed fr om info rmatio n contained in the var ia ble PP AR.
Pa g e 22 - 4 EQ ( 3) T he var ia ble name EQ is r es er v ed by the calc ulator to stor e the c urr ent equatio n in plots or solut ion to eq uations (s ee chapt er …) . T he soft menu k ey la beled E Q in this menu can be us ed as it w ould be if y ou hav e y our v ar iable men u av ailable , e .
Pa g e 22 - 5 T he follo w ing diagr am illu str ates the f uncti ons av ailable in the P P AR menu . T he letter s attached to eac h f unction in the di agr am ar e used f or r ef er ence purpos es in the desc ripti on of the func tions sho wn belo w .
Pa g e 22 - 6 INDEP (a) T he command IND EP spec ifi es the independent v ar iable and its plotting r ange . T hese spec ifi cations ar e stor ed as the thir d paramet er in the v ar ia ble PP AR. T he def ault v alue is 'X'. T he v alues that can be assigned t o the independent var iable spec if icati on ar e: Θ A v ari able name , e .
Pa g e 22 - 7 CENTR (g) T he command CENTR tak es as ar gument an or der e d pair (x ,y) or a value x , and adju sts the fi rst tw o elements in the v ari able P P AR, i .e., (x min , y min ) and (x max , y max ) , s o that the center o f the plot is (x ,y) or (x , 0) , r especti v el y .
Pa g e 22 - 8 A list o f two b inar y intege rs {#n #m}: sets the ti c k annotations in the x - and y- ax es to #n and #m pi xels , r espec tiv el y . AXE S (k) T he input value f or the axes command consis ts of e ither an order ed pair (x,y) or a list {(x ,y) atic k "x-ax is label" "y-ax is label"}.
Pa g e 22 - 9 The PTYP E menu within 3D (IV) T he PTYP E menu under 3D cont ains the follo w ing functi ons: T hese f uncti ons corr espond to the gr aphi cs opti ons Slopef ield , Wir efr ame , Y - Slice , P s-Contour , Gri dmap and Pr -Sur f ace pre sented ear lie r in this chapt er .
Pa ge 22- 1 0 XV OL (N) , YV OL (O) , and ZV OL (P) T hese f unctions t ake as input a minimum and maxi mum value and ar e used to spec ify the extent o f the parallelep iped wher e the gr aph w ill be gener ated (the v ie w ing par allelepiped). Thes e values ar e s tor ed in the v ar iable VP AR .
Pa ge 22- 1 1 The S T A T menu within PL O T T he S T A T menu pr o v ide s access to plots r elated to st atistical anal y sis. W ithin this menu w e find the f ollo wing men us: T he diagr am belo w sho ws the br anc hing of the S T A T me nu w ithin P L O T .
Pa ge 22- 1 2 The P T YP E m enu wi thin ST A T (I) The P TYP E menu pr o v ides the f ollo wing f uncti ons: The se ke ys cor res pond to the p lot ty pes Bar (A ) , H istogr am (B) , and Scatter (C) , pr esented ear lier .
Pa ge 22- 1 3 X COL (H) T he command XC OL is used t o indicate w hi ch o f the columns of Σ D A T , if mor e than one , w ill be the x - column or independent var iable column . YC O L ( I ) T he command Y COL is used to indicate w hic h of the columns of Σ DA T , i f m o re than one , w ill be the y- column or dependent v ar iable column .
Pa ge 22- 1 4 Θ SIMU: w hen selec ted , and if more than one gr aph is to be plotted in the same set o f axe s, plots all the gr aphs simultaneousl y .
Pa ge 22- 1 5 T hree -dimensional gr aphics T he thr ee -dimensional gr aphi cs a vaila ble , namel y , options Slopef ield , Wir efr ame , Y -Sli ce , P s-Contour , G r idmap and Pr -Surface , use th.
Pa ge 22- 1 6 @) PPAR Sho w plot par ameters ~„r` @INDEP Def ine ‘ r’ as the indep . v ari able ~„s` @DEPND De fine ‘ s ’ as the depende nt v ari able 1 # 10 @XRNG De f ine (-1, 10) as the x -r ange 1 # 5 @YRN G L Def ine (-1, 5 ) as t he y-r ange { (0, 0) {.
Pa ge 22- 1 7 @) PPAR Sho w plot par ameters { θ 0 6 .2 9} ` @INDEP De f ine ‘ θ ’ as the indep . V ariable ~y` @DEPND De fine ‘ Y’ a s the dependent v ar iable 3 # 3 @XRNG De fine (-3, 3) as the x -r ange 0. 5 # 2. 5 @YRNG L Def ine (-0. 5,2 .
Pa ge 22- 1 8 « S tart pr ogram {PPAR EQ} PURGE P u r ge c urr ent P P AR and E Q ‘ √ r’ STEQ Sto r e ‘ √ r’ i nto E Q ‘r’ INDEP Set independent v ari able to ‘ r’ ‘s’ DEPND Set dependent v ar iable t o ‘ s ’ FUNCTION Selec t FUNCTION as the plot type { (0.
Pa ge 22- 1 9 Ex ample 3 – A polar plot . Enter the follo wing pr ogr am: «S t a r t p r o g r a m RAD {PPAR EQ} PURGE Change to r adians , pur ge vars .
Pa ge 22- 2 0 P I CT T his soft k e y re fer s to a var iable called PICT that stor es the cur r ent conten ts of the gr aphi cs w indo w . This v ar iable name , ho w ev er , cannot be placed within quot es, an d ca n only stor e graph i cs object s.
Pa ge 22- 2 1 BO X T his command tak es as in put 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 aw s the bo x wh ose di agonals ar e r epr esente d by the tw o pairs of coor dinates in the input .
Pa ge 22- 22 Θ PI X? Chec ks if pi xe l at location (x ,y) or {#n, #m} is on . Θ PI X OFF turns o ff pi x el at location (x ,y) or {#n , #m}. Θ PI X ON turns on p i xe l at location (x ,y) or {#n , #m}.
Pa g e 22 - 23 (5 0., 5 0.) 12 . –180. 180. AR C Dr aw a c ir cle cen ter (5 0,5 0) , r= 12 . 1 8 FOR j Dr aw 8 lines w ithin the c ir cle (50., 5 0 .
Pa g e 22 - 24 It is suggest ed that you c r eate a separ a t e sub-dir ectory to sto r e the progr ams. Y ou could call the sub-dir ectory RIVER , since w e ar e dealing w ith irr egular open c han nel c r os s-secti ons , t y pi cal of r i ver s . T o see the pr ogram XSE CT in acti on, use the f ollo wi ng data sets .
Pa g e 22 - 2 5 P ix el coordinates T he fi gur e belo w sho w s the gr aphic coor dinate s fo r the t y pi cal (minimum) scr een of 13 1 × 64 pi xels . P i x els coor dinates ar e measured f r om the top left corner of the screen {# 0 h # 0h}, w hich corresponds to user-defined coor din ates Data set 1 Data set 2 xy x y 0.
Pa ge 22- 26 (x min , y max ) . T he max imum coor dinates in terms of p i xels cor r espond to the lo w er ri ght corner of the sc r een {# 8 2h #3Fh}, w hic h in use r-coor d inate s is the point (x max , y min ).
Pa g e 22 - 27 Animating a collec tion o f graphics T he calc ulato r pr o v ide s the f unction ANIMA TE to animate a n umber o f gr aphi cs that hav e been placed in the st ack . Y ou can gener ate a gr aph in the gr aphic s sc r een b y using the commands in the PL O T and PICT men us .
Pa g e 22-2 8 ANIMA TE is av ailable b y us ing „°L @) GROB L @ ANIMA ) . T he animation w ill be r e -started. Pr ess $ to st op the animation once mor e. Noti ce that the number 11 w ill still be lis ted in stac k le v el 1. Pr ess ƒ to dr op it fr om the stack.
Pa g e 22 - 2 9 Ex ample 2 - Animating the plotting of diff er ent po w er f uncti ons Suppos e that yo u want t o animate the plotting of the functi ons f(x) = x n , n = 0, 1, 2 , 3, 4, in the same set o f axe s.
Pa ge 22- 3 0 pr oduced in the calc ulator’s sc reen . T her ef or e , when an image is con v er ted into a GROB , it becomes a s equence of binary digits ( b inary dig its = bit s ), i . e . , 0’s and 1’s . T o illustr ate GR OBs and con ve rsi on of image s to GR OBS consider the f ollo w ing e xe r c ise .
Pa ge 22- 3 1 1` „°L @) GROB @ GRO B . Y ou w ill no w ha ve in le v el 1 the GROB desc r ibed as: As a gr aphic ob ject this eq uation can no w be placed in the gr aphi cs displa y . T o r ecov er the gr aphics dis play pr ess š . Then , mo ve the c urso r to an empt y sect or in the graph , and pr ess @) EDIT LL @ REPL .
Pa g e 22 - 32 BLANK T he functi on BL ANK , w ith ar guments #n and #m, c r eates a blank gra phics obj ect of w i dth and height spec ifi ed by the v alues #n and #m, r es pecti v ely .
Pa g e 22 - 3 3 An e xample o f a progr am using GROB T he follo w ing pr ogram pr oduces the gr aph of the sine f unctio n including a fr ame – dra w n w ith the func tion B O X – and a GROB t o label the gr aph.
Pa g e 22 - 3 4 sho w s the state o f str es ses w hen the element is r otated b y an angle φ . In this case, the normal str esses are σ ’ xx and σ ’ yy , while the shear str esses ar e τ ’ xy and τ ’ yx .
Pa g e 22 - 35 The stress cond ition for whic h t he she ar stress , τ ’ xy , is z er o , ind i cated by segment D’E’ , produces the s o -called princ ipal str esses , σ P xx (at po int D’) and σ P yy (at point E’).
Pa g e 22-3 6 separ ate v ar iables in the calc ulator . Thes e sub-pr ogr ams are then link ed by a main pr ogr am, that w e w ill call MOHRCIRCL . W e will fir st c r eate a sub- dir ect or y called MOHR C w ithin the HOME dir ectory , and mov e into that dir ect or y t o type the pr ograms .
Pa g e 22 - 37 At this point the pr ogram MOHR CIRCL s tarts calling the su b-pr ograms t o pr oduce the fi gur e . Be pa ti ent . The r esulting Mohr ’s c ir cle w ill look as in the pic tur e to the le ft.
Pa g e 22 - 3 8 inf ormatio n tell us is that some w here 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, pr ess $ . Then type the list corr esponding to the v alues { σ x σ y τ xy}, for this case , it w ill be { 25 75 50 } [ENTER] Then , pres s @CC&r .
Pa g e 22 - 3 9 necess ar y to plot the c irc le . It is suggest that w e r e -or der the var iable s in the sub-dir ectory , so that the pr ogr ams @MOHRC and @PRNST ar e the two f ir st v ari ables in the soft-menu k e y labels.
Pa ge 22- 4 0 T o find the v alues o f the str ess es corr esponding to a r otatio n of 3 5 o in the angle of th e stressed p art i cle, w e use: $š Clea r sc reen, show PICT in graphics scr e en @TRACE @ ( x,y ) @ . T o mov e c ursor o v er the c irc le sho w ing φ and (x ,y) Ne xt , pr ess ™ until y ou r ead φ = 3 5 .
Pa ge 22- 4 1 Since pr ogr am IND A T is use d also f or pr ogram @PRNST (P R iNc ipal S T resses), running that partic ular pr ogr am w ill no w use an input f or m, f or e x ample , T he r esult , a.
Pa g e 23 - 1 Chapter 2 3 Character strings Char acter s tring s are calc ulator obj ects enc losed betw een double quotes . T hey ar e tr eated as te xt b y the calc ulator . F or e x ample , the str ing “SINE FUNCT ION” , can be transf or med into a GR OB (Gra phic s Objec t) , to la bel a gr aph , or can be us ed as output in a pr ogr am.
Pa g e 23 - 2 String concatenation Str ing s can be concatenated (j oined together ) b y using the plu s sign +, f or exa mp l e: Concat enating str ings is a pr actical w a y to cr ea t e output in pr ogr ams.
Pa g e 23 - 3 T he operati on of NUM, CHR , OB J , and S TR w as pr esen ted ear lier in this Chapt er . W e hav e also s een the functi ons S UB and REP L in r elation t o gr aphic s earli er in this chapte r .
Pa g e 23 - 4 sc r een the ke y str ok e sequence to get such c harac ter ( . fo r this case) and the numer ical code corr esponding to the c har acter (10 in this cas e) .
Pa g e 24 - 1 Chapter 2 4 Calculator objec ts and flags Numbers , lists, v ec tors, matri ces, algebr ai cs, etc ., ar e calc ulator objec ts. T hey ar e classif ied accor ding to its nature into 30 diff er ent t y pes , whic h ar e desc r ibed belo w .
Pa g e 24 - 2 Number T y pe Ex am ple ___________ _____________________ _____________________ _______________ 21 Ext ended Real Number Long Real 2 2 Extended C omple x Number Long Complex 2 3 Link ed .
Pa g e 24 - 3 Calculator flags A flag is a v ar iable that can e ither be set or uns et . The st atus of a f lag affec ts the behav ior of the calc ulator , if the f lag is a sy stem f lag, or o f a pr ogr am, if it is a user f lag . The y ar e desc r ibed in mor e detail ne xt .
Pa g e 24 - 4 T he functi ons contained w ithin the FL A G menu ar e the f ollow ing: The ope rati on of thes e func tions is as f ollo w s: SF Set a f lag CF C lear a flag F S? R eturns 1 if flag is .
Pa g e 2 5 - 1 Chapter 2 5 Date and T ime F unc tions In this Chapt er w e demonstr ate some o f the func tions and calc ulations using times and date s.
Pa g e 2 5 - 2 Br ow sing alarms Option 1. Br o ws e alarms ... in the T IME menu lets y ou r e v ie w y our cur r ent alarms . F or e x ample , after enter ing the alarm us ed in the ex ample a bov e.
Pa g e 2 5 - 3 T he applicati on of these f uncti ons is demonstr ated belo w . D A TE: P laces c urr ent date in the stac k D A TE: Set s y stem date to spec ifi ed value T IME: P laces c ur r ent time in 2 4 -hr HH.MM S S f ormat T IME: Set sy stem time to spec if ied v alue in 2 4 -hr HH.
Pa g e 2 5 - 4 Calculating with tim es Th e fu nct ion s HMS , HM S , HMS+, and HM S - ar e us ed to manipulate value s in the HH.MM S S for mat . This is the same f ormat us ed to calc ulate w ith angle measur es in degr ees, min utes , and seconds.
Pa g e 26 - 1 Chapter 2 6 M anaging memory In Chapte r 2 w e intr oduced the basic co ncepts of , a nd ope rati ons fo r , cr eating and managing var i ables and dir ec tor ies . In this Chapt er w e disc uss the management of the cal culat or’s memory , including the partition of memo r y and tec hniques f or backing u p data.
Pa g e 26 - 2 P or t 1 (ERAM ) can contain up to 12 8 KB of data . P ort 1, together with P ort 0 and the HOME dir ectory , cons titute the calc ulator’s RAM (R andom Acce ss Memory) segment of calc ulator ’s memory . T he R AM memory segment r equir es contin uous elec tr ic po w er suppl y f r om the calculat or bat t er ies t o operat e.
Pa g e 26 - 3 Chec king objec ts in memor y T o see the ob jec ts stor ed in memor y y ou can use the FILE S func tio n ( „¡ ). Th e sc ree n b el ow sh ows t he H OM E d i rec to r y wi th five d i re cto ri es, n a m ely , TRIANG , MA TRX , MPFIT , GRPH S , and CASDIR .
Pa g e 26 - 4 Bac k up objec ts Bac ku p obj ects ar e used t o copy data f r om y our home dir ect or y int o a memor y port. The pur pose of bac king up obj ects in memory port is to pr eserve the contents of the objects f or f utur e usage .
Pa g e 26 - 5 Bac king up and r estoring HOME Y ou can back u p the cont ents of the c urr ent HOME dir ectory in a single bac k up obj ect . T his ob jec t w ill contain all var iables , k e y assi gnments , and alar ms c urr en tly def ined in the HO ME dir ectory .
Pa g e 26 - 6 Stor ing, deleting, and r estoring back up objects T o c r eate a bac k up obj ect us e one of the f ollow ing appr oaches: Θ Us e 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 back up obj ect w ill hav e the same name as the o ri ginal object .
Pa g e 26 - 7 Using data in backup objects Although y ou cannot dir ectl y modify the contents o f back up objec ts, y ou can use tho se cont ents in calculat or oper ations. F or e x ample , y ou can r un pr ogr ams stor ed as back up objec ts or us e data fr om back up obj ects t o run pr ograms .
Pa g e 26 - 8 T o r emo ve an SD car d , turn o ff the HP 50 g, pr ess ge ntly on the e xposed edge of the car d and push in . The car d should spring out o f the slot a small distance , allo w ing it now to be easil y r emo ved f r om the calculator .
Pa g e 26 - 9 Accessing objects on an SD card Acce ssing an obj ect f r om the SD car d is similar to whe n an objec t is located in ports 0, 1, or 2 . How ev er , P ort 3 wi ll not appear in the menu when using the LIB fu ncti on ( ‚á ) . T he SD file s can only be managed u sing the F iler , or F i le Manager ( „¡ ).
Pa g e 26 - 1 0 Note that if the name of the object y ou intend to st ore on an SD car d is longer than ei ght c harac ters , it will a ppear in 8. 3 DOS f or mat in por t 3 in the F iler once it is stor ed on the ca r d.
Pa g e 26 - 1 1 Note that in the case of objects w ith long file s names , yo u can spec ify the f ull name of the objec t , or its truncat ed 8. 3 name , when ev aluating an obj ect on an SD car d.
Pa g e 26 - 1 2 T his will s tor e the obj ect pr ev iousl y on the stac k onto the SD card int o the dir ect or y named P ROG S into an obj ect named P ROG1. Not e: If PR OGS doe s not e xis t, the dir ectory will be au tomaticall y cr eated. Y ou can spec ify an y number of nested subdir ector ies .
Pa g e 26 - 1 3 Libr ary numbers If y ou us e the LIB menu ( ‚á ) and pr ess the so ft menu k e y corr es ponding to port 0, 1 or 2 , yo u wi ll see libr ar y n umbers list ed in the soft menu k e y labels . E ac h library has a thr ee or f our -digit n umber assoc iated w ith it .
Pa g e 26 - 1 4 w ill indicat e when this battery needs r eplacement . The diagr am belo w sho ws the location o f the back up bat t er y in the top compartment at the back o f the calc ulat or .
Pa g e 27- 1 Chapter 2 7 T he Equation Libr ar y T he E quation L ibrary is a collection o f equations and commands that enable y ou to so lv e simple s c ience and e ngineer ing pr oblems. T he libr ary consists o f mor e than 300 equatio ns gr ouped int o 15 techni cal subj ects con taining mor e than 100 pr oblem titles .
Pa g e 27- 2 7 . F or eac h know n var iable , type its value and pr es s the corr espo nding menu k e y . If a v ari able is not show n , pre ss L to disp la y furt h er variables. 8. Optional: su pply a gues s f or an unkno wn v ar iable . This can speed up the soluti on pr ocess or help to f oc us on one of s ev er al soluti ons.
Pa g e 27- 3 Using the m enu k ey s T he actions o f the unshifted and shifted var iable menu k ey s f or both sol ver s ar e identi cal. No tice that the Multiple Eq uation S olv er us es two f orms o f menu labels: blac k and w hite . The L k e y displa y s additional menu la bels, if r equir ed .
Pa g e 27- 4 Br o wsing in the Equation L ibrary When y ou se lect a sub ject and title in the E quation L ibrary , y ou spec ify a set of one or mor e equati ons. Y ou can get the f ollo w ing infor mation a bout the equati on set fr om the E quatio n Li brary catalogs: The equati ons themsel ves and the number o f equa ti ons .
Pa g e 27- 5 Vie wing v ariables and sel ecting units After y ou select a sub jec t and title , y ou can vi e w the catalog of names , desc r iptions , and units for the v ari ables in the equati on set b y pre ssing #VARS# . T he table belo w summari z es the oper ations av a i lable to y ou in the V ari able catalogs .
Pa g e 27- 6 Pr ess to stor e the pic tur e in PIC T , the gra phics memory . T hen y ou can use © PIC T ( or © P ICTURE) to v ie w the pi ctur e again af t er y ou hav e quit the E q uation L ibr ar y . Pr ess a menu k ey or to v ie w other equati on infor mation .
Pa g e 27- 7 T he menu labels f or the var iable k ey s ar e w hite at fir st , but c hange during the solu tion pr ocess as des cr ibed belo w . Becau se a solu tion in v olv es man y equations and m.
Pa g e 27- 8 Mea nings of Menu Labe ls Defining a set o f equations When y ou design a s et of eq uations , y ou should do it w ith an under standing o f ho w the Multiple -E quation Sol ver use s the equations to sol ve pr oblems.
Pa g e 27- 9 F or ex ample , the f ollo w ing thr ee equati ons defi ne initial v eloc ity a nd acceler atio n based on tw o observed dis tances and times . T he fir st tw o equations alone ar e mathematicall y suff ic ient f or solv ing the problem , but eac h equati on contains tw o unkno w n var ia bles.
Pa g e 27- 1 0 6. P ress ! MSOLV! to launc h the sol ver w ith the ne w set of equati ons . T o c hange the title and menu for a set of equations 1. Mak e sur e that the set o f equati ons is the c urr ent set (a s the y are u sed w hen the Multiple -E quati on Sol ve r is launc hed) .
Pa g e 27- 1 1 Constant? The initi al value of a v ar iable ma y be leading the r oot - f inder in the wr ong direc tion . Suppl y a guess in the oppo site dir ectio n fr om a cr iti cal value .
Pa g e 27- 1 2 Not r elated . A var iable ma y not be in v olv ed in the s oluti on (no mark in the label) , s o it is not com patible wi th the var ia bles that w er e inv ol ved . W r ong dir ecti on . The initial v alue of a var iable ma y be leading the roo t - f inder in the wr ong direc tion .
Pa g e A - 1 Appendi x A Using input forms T his ex ample o f setting time and date illu str ates the use of input f orms in the calc ulator . Some gener al rules: Θ Use the ar ro w ke ys ( š™˜— ) to mov e fr om on e f ield to th e ne xt in the input f or m.
Pa g e A - 2 In this par ti c ular case w e can giv e v alues to all but one of the var iables, s ay , n = 10, I%YR = 8. 5, PV = 10000, FV = 1000, and sol ve fo r va ri able P MT (the meaning of thes e var iables w ill be pr esent ed later ) . T r y the f ollo w ing: 10 @@OK@@ Enter n = 10 8.
Pa g e A - 3 !CALC Pr es s to access the stac k f or calculati ons !TYPES Pr ess to determin e the t y pe of object in highlighte d f ield !CANCL Cancel operation @@OK@ @ Ac cep t en tr y If y ou pr e.
Pa g e A - 4 (In RPN mode , w e would ha v e used 113 6 .2 2 ` 2 `/ ). Press @ @OK@@ to en ter this ne w v alue. T he inpu t for m w ill no w look lik e this: Press !TYPES t o see the type o f data in the P MT fi eld (the highligh ted f ield).
Pa g e B - 1 Appendi x B T he calc ulator ’s k e y board T he fi gur e belo w sho w s a diagr am o f the calc ulato r’s k e y board w ith the number ing of its r o ws and columns . T he fi gure sho ws 10 r ow s of k e y s combined w ith 3, 5, or 6 columns.
Pa g e B - 2 f i ve f uncti ons. T he main k e y func tions ar e sho wn in the f igur e belo w . T o oper ate this main k e y func tions simpl y pr ess the cor r esponding k e y . W e w ill r ef er to the ke y s by the r o w and column wher e the y are located in the sk etc h abo v e , th us , k e y (10,1) is the ON key .
Pa g e B - 3 M ain k e y functions Ke ys A thr ough F k ey s ar e assoc iated w ith the soft men u options that appear at the bottom of the calc ulator’s dis play . T hus , thes e k ey s will ac tiv a t e a v ari ety of func tions that c hange acco rding t o the acti v e menu .
P age B-4 The l eft- shi ft k e y „ and the r ight-shift key … ar e combi ned with other k ey s to acti vat e menus, en ter char acters , or calc ulate functi ons as desc r ibed else wher e. The n umeri cal k ey s ( 0 to 9 ) ar e used to enter the digits of the dec imal number s ys tem.
P age B-5 the other thr ee functi ons is a ssoc iated w ith the le f t-shift „ ( MTH ) , r ight-shift … ( CA T ) , and ~ ( P ) ke y s. Diagr ams show ing the f uncti on or char acter r esulting fr.
Pa g e B - 6 Th e CMD fu nction sho ws the most r e cent commands , the PRG fun ctio n acti v ates the pr ogramming men us , the MTRW f uncti on acti vat es the Matri x Wr i t e r, Left-shift „ func tions of th e calculator ’s k e yboard Th e CMD fu nction sho w s the most r ecent commands.
Pa g e B - 7 Th e e x k e y calc ulates the e xponenti al func tion o f x . Th e x 2 k e y calc ulates the sq uar e of x (this is re fer red to as the SQ fu nct ion) . T he AS IN , A CO S, and A T AN fu ncti ons calc ulate the ar csine , ar ccosine , and ar c tangent f uncti ons, r especti vel y .
Pa g e B - 8 Rig ht-s hif t … func tions of the calculator ’s k ey board Right-shift functions The sk etch abo v e show s the functi ons , char acter s, or men us ass oci ated w ith the diffe r ent calculator k ey s w hen the r igh t -shift k e y … is acti vated .
Pa g e B - 9 Th e CA T functi on is used to ac tiv ate the command c atalog . Th e CLE AR functi on c lears the sc r een . Th e LN func tion calc ulates the natur al logarithm . T he functi on calc ulates the x – th r oot of y . Th e Σ f uncti on is used to ent er summations (or the upper case Gree k letter sigma).
Pa g e B - 1 0 is used mainl y to e nter the upper -case letter s of the English alpha bet ( A thr ough Z ) . T he numbers , mathematical s ymbols ( - , + ), dec imal poi nt ( . ) , and the s pace ( SP C ) ar e the same as the main functi ons of these k ey s.
Pa g e B - 1 1 Notice that the ~„ combinati on is used mainl y to enter the lo wer -c ase letters of the English alphabet ( A thr ough Z ) . T he numbe rs , mathematical sym bo l s ( - , +, × ) , dec imal p o int ( . ) , and the spac e ( SP C ) are the s ame as the main func tions of these k ey s.
Pa g e B - 1 2 Alpha-right-shift c har ac ters T he follo wing sk etch sho ws the c harac ter s assoc iated w ith the diffe r ent calc ulat or k e y s w hen the ALPH A ~ is combined w ith the ri ght -shift k e y … .
Pa g e B - 1 3 ~… combination inc lude Gr eek let ter s ( α, β, Δ, δ, ε, ρ, μ, λ, σ, θ, τ , ω , and Π ) , other c har acte rs gener ated b y the ~… combinati on ar e |, ‘ , ^, =, <, >, /, “ , , __, ~, !, ?, <<>>, and @.
Pa g e C - 1 Appendi x C CAS settings CA S stands f or C omputer A lge br aic S y stem . T his is the mathematical cor e of the calc ulator w her e the sy mbolic mathematical oper atio ns and functi ons ar e pr ogr ammed. T he CA S offe rs a number of settings can be adj ust ed according to the type of oper ation of inter est .
Pa g e C - 2 Θ T o r eco ver the or iginal menu in the CAL CUL A T OR MODE S input bo x , pr ess the L k e y . Of inter est at this point is the c hanging of the CA S settings .
Pa g e C - 3 A v ari able called VX ex ists in the calc ulator ’s {HOME CA SDI R} dir ect or y that tak es, b y def ault , the v alue of ‘X’ . T his is the name of the pr efer r ed independent v ar iable f or algebr aic and calc ulus a pplicati ons.
Pa g e C - 4 T he same e x ample , corr es ponding to the RPN oper ating mode, is sho wn ne xt: Appr o x imate v s. Ex ac t CA S mode When the _ A ppr ox is s elected , sy mbolic oper ati ons (e.g ., def inite integrals , squar e roots , etc .) , w ill be calc ulated numer i cally .
Pa g e C - 5 T he k ey str ok es nece ssary for ent er ing these v alues in Algebr ai c mode ar e the fo llow ing: …¹2` R5` T he same calc ulations can be pr oduced in RPN mode . Stac k lev els 3: and 4: sho w the case of Ex act CAS se tting (i .e .
Pa g e C - 6 It is r ecommended that y ou se lect EXA CT mode as def ault CA S mode , and c hange to APP R O X mode if r equest ed b y the calc ulator in the perf ormance of an oper ation . F or add iti onal inf ormati on on r eal and integer numbers , as w ell as other c alcul at or’s obje cts, r efe r to Cha pte r 2 .
Pa g e C - 7 If y ou pr ess the OK so ft menu ke y (), then the _Comple x optio n is for ced, and the r esult is the f ollo wing: T he k ey str ok es us ed abo ve ar e the follo w ing: R„Ü5„Q2+ 8„Q2` When ask ed to change to C OMP LEX mode , u se: F .
Pa g e C - 8 F or ex ample , hav ing selec ted the S tep/step opti on, the f ollo wing s cr eens sho w the step-b y-step di v ision of tw o poly nomials , namel y , (X 3 -5X 2 +3X - 2)/(X - 2) . T his is accomplished b y using f uncti on DIV2 a s sho w n belo w .
Pa g e C - 9 . Increasing-po w er CAS mode When the _Incr po w CA S option is selec ted , poly nomi als wi ll be listed so that the ter ms w ill hav e incr easing po we rs of the independent v ar iable .
Pa g e C - 1 0 Rigor ous CAS setting When the _Ri gorous CA S option is se lected , the algebrai c e xpr essi on |X|, i .e., the absolute v alue , is not simplified to X . If the _R igor ous CA S option is not selec ted , the algebrai c e xpr essi on |X| is simplif ied to X .
Pa g e C - 1 1 Notice that , in this ins tance , soft menu k ey s E and F are the o nly o ne w ith as soc iated commands , namel y: !!CANCL E CANCeL the help f ac ilit y !!@@OK#@ F OK to ac ti vate he.
Pa g e C - 1 2 Notice that the re ar e six co mmands assoc iated w ith the s oft menu k e y s in this case (y ou can chec k that ther e are onl y si x commands because pr essing the L pr oduces no additi onal menu items).
Pa g e C - 1 3 T o nav igate qui ckl y to a partic ular command in the help fac ility list w ithout ha ving to u se the arr o w k e ys all the time , we can us e a shortcu t consisting of typing the f irs t letter in the command’s name .
Pa g e C - 1 4 In no e vent unle ss r equir ed b y applicable la w w ill an y copy r ight holde r be liable t o yo u for damage s, inc luding an y general , speci al , inc ident al or cons equential d.
Pa g e D - 1 Appendi x D Additional c har acter set While y ou can us e an y of the u pper -case and lo w er -case English letter f r om the k e yboar d, ther e are 2 5 5 char acter s usable in the calc ulator . Including spec ial ch arac ter s l ike θ , λ , et c.
Pa g e D - 2 func tions assoc iated w ith the soft menu k e y s, f4 , f5, and f6. The se func tions ar e: @MODIF : Opens a graphi cs sc r een whe r e the user can modify highlight ed c harac ter . Use this opti on car ef ull y , since it w ill alter the modif ied c har acter u p to the ne xt r ese t of the calc ulator .
Pa g e D - 3 Gr ee k lett er s α (alpha) ~‚a β (beta) ~‚b δ (delta) ~‚d ε (epsilon) ~‚e θ (theta) ~‚t λ (lambda) ~‚n μ (mu) ~‚m ρ (r ho) ~‚f σ (sigma) ~‚s τ (tau) ~‚u ω .
Pa g e E - 1 Appendi x E T h e Selec tion T ree in the Equation W riter T he expr essi on tr ee is a diagr am sho w ing ho w the E quati on W r iter inte rpr ets an ex p r e 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 hi er ar 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 o f the hier ar ch y-of-oper ation r ules in this selecti on. F i r st the y (Step A1) . T hen, y-3 (S tep A2 , par enth eses) . Then , (y-3)x (Step A3, multiplicati on) .
Pa g e E - 3 Step B1 S te p B2 Step B3 St ep B4 = Step A5 St ep B5 = Step A6 W e can also fol lo w the ev aluation o f the expr essi on starting fr om the 4 in the ar gument of the S IN func tion in the denominat or . Pr ess the do wn ar r o w k e y ˜ , continuousl y , until the clear , editing cu rsor is tr igger ed around the y , once mor e .
Pa g e E - 4 Step C3 Step C 4 St ep C5 = St ep B5 = Step A6 The expr ession t r ee f or t he expr ession p r esente d abov e is s ho wn next: T he steps in the e v aluation of the thr ee terms ( A1 thr ough A6 , B1 thro ugh B5, and C1 thr ough C5) ar e sho w n ne xt to the c ir c le containing numbers , v ari able s, or oper ators .
Pa g e F - 1 Appendi x F T he Applications (APP S) menu T he Applicati ons ( APP S) menu is av ailable thr ough the G key ( fi rst key i n second r o w fr om the k e yboar d’s top) . T he G k ey sh o ws the f ollo w ing applicati ons: T he differ ent appli cations ar e desc ribed ne xt .
Pa g e F - 2 I/O func tions .. Selecting opti on 2 . I/O f uncti ons .. in the APP S menu w i ll pr oduce the f ollo w ing menu lis t of input/ou tput func tions T hese appli cations ar e desc r ibed .
Pa g e F - 3 T he Const ants Libr ar y is disc us sed in detail in C hapter 3 . Numeric sol ver .. Selec ting option 3 . Constan ts lib .. in the APP S menu pr oduces the nume ri cal solver me nu: This oper ation is equi valent to the k e y str ok e sequence ‚Ï .
Pa g e F - 4 Equation wr iter .. Selec ting option 6 .E quation w r iter .. in the APP S menu opens the equation wri ter: T his oper ation is eq ui val ent to the k ey str ok e seq uence ‚O . The equati on w rit er is intr oduced in det ail in Chapter 2 .
Pa g e F - 5 M atr ix W riter .. Selec ting option 8.Matr i x W r iter .. in the APP S menu launc hes the matr i x wr iter : T his oper ation is eq ui val ent to the k ey str ok e seq uence „² .The Matr i x W r iter is pr esen ted in detail in Chapter 10.
Pa g e F - 6 T his oper ation is eq ui val ent to the k ey str ok e seq uence „´ . T he MTH menu is intr oduced in Chapt er 3 (r eal numbers). Other func tions f r om the MTH menu ar e pr esented i.
Pa g e F - 7 Note that flag –117 should be se t if you ar e going to us e the E quatio n L ibrary . Note too that the E quation L ibr ary w ill only appear on the AP P S menu if the two E quation L ibrary files ar e stor ed on the calculator . T he E quation L ibrary is e xplained in de tail in chapt er 2 7 .
P age G-1 Appendi x G Useful shortc uts Pr esented her ein ar e a number o f k e yboar d shor tc uts commonl y used in the calc ulat or : Θ Adjust d isplay c ontrast: $ (hold) + , or $ (hold) - Θ T oggle between RPN and AL G modes: H @@@OK@@ or H` .
P age G-2 Θ Set/c lear sy stem flag 117 (CHOO SE bo xe s vs . S OFT menus): H @) FLAGS —„ —˜ @@CHK@ Θ In AL G mode , SF(-117) selects S O FT menus CF(-117) se lects CHOO SE BO XE S .
P age G-3 Θ S ystem-lev el o per ation (H old $ , r elease it after enter ing second or thir d k e y) : o $ (ho ld) AF : “C old” r estart - all memory er ased o $ (ho ld) B : Cancels k ey str ok .
P age H-1 Appendi x H T he CAS help facilit y T he CAS help f ac ilit y is a vaila ble thro ugh the k ey str ok e sequence I L @HELP ` . The f ollo w ing sc r een shots sh o w the fir st menu page in the listing of the CAS help fac i lity . T he commands ar e listed in alphabeti cal or der .
P age H-2 Θ Y ou c an type t w o or mor e let ters of the c ommand of inter est , by locking the alphabeti c k e y boar d. T his w ill tak e yo u to the command of int er est , or to its nei ghborhood. A fterwar d s, y ou need to unloc k the alpha k e yboar d, and u se the v ertical arr ow k ey s —˜ to locate the command , if needed.
Pa g e I - 1 Appendi x I Command catalog list T his is a l ist of all commands in the command catalog ( ‚N ) . Those commands that belong t o the CA S (C omput er Algebr aic S y stem) ar e lis ted also in Appendi x H.
Pa g e J - 1 Appendi x J T he MA THS me nu T he MA THS menu , accessible thr ough the command MA THS (a v ailable in the catalog N ), contains the fo llo w ing sub-menu s: T he CMPLX sub-menu T he CM P L X su b-menu contains fu nctions pertinent to oper ations w ith complex numbers: T hese f uncti ons are des cr ibed in Chapter 4.
Pa g e J - 2 T he HYPERBOLIC sub-menu T he HYPERB OLIC sub-menu co ntains the h y perboli c func tio ns and their in v ers es . T hese f unctions ar e descr ibed in Chapter 3 . T he I NTE GER sub-menu T he INTEGER su b-menu pr o v ides f uncti ons for manipulating integer number s and some pol ynomi als.
Pa g e J - 3 T he POL YNOM IAL sub-menu T he POL YNOMIAL sub-men u includes f uncti ons for ge ner ating and manipulating pol yno mials . The se func tions ar e pr es ented in Chapte r 5: T he TES T S sub-menu T he TE S TS su b-menu inc ludes r elati onal oper ator s (e .
Pa g e K - 1 Appendi x K Th e MA I N m en u T he MAIN menu is av ailable in the command catalog . This menu inc lude the fo llo w ing sub-menu s: T he CASCF G command T his is the f irs t entr y in the MAIN menu . T his command conf igur es the CA S .
Pa g e K - 2 T he DIFF sub-m enu T he DI FF sub-me nu contains the f ollo w ing f unctio ns: T hese f unctions ar e also av ailable thr ough the CAL C/DI FF sub-menu (s tart wi th „Ö ) .
Pa g e K - 3 T hese f uncti ons are als o av ailable in the TRIG menu ( ‚Ñ ) . Description of these f uncti ons is incl uded in C hapter 5 . T he SOL VER sub-m enu T he S OL VER menu include s the fo llo w ing func tions: T hese f uncti ons are a v ailable in the CAL C/S OL VE menu (st art with „Ö ).
Pa g e K - 4 T he sub-menus INTE GER , MODUL AR , and P OL YNOMIAL ar e pre sented in detail in Appe ndi x J. The E XP &LN sub-menu T he EXP&LN menu contains the follo w ing functions: T his menu is also acces sible thr ough the k e yboar d by using „Ð .
Pa g e K - 5 T hese f uncti ons ar e av ailable thr ough the CONVER T/REWR ITE me nu (start w ith „Ú ) . T he func tions ar e pr esent ed in Chapter 5, ex cept for f uncti ons XNUM and XQ , whi ch .
Pa g e L - 1 Appendi x L L ine editor commands When y ou tr igger the line editor b y u sing „˜ in the RPN stac k or in AL G mode , the follo wing s oft menu f unctions ar e pr ov ided (pr ess L to see the r emaining fu nctions): T he functi ons ar e br ief ly de sc ribed as follo ws: SKIP: Skip s char acters to beginning o f wor d.
Pa g e L - 2 T he items sho w in this scr e en are s elf-e xplanator y . F or e x ample , X and Y positi ons mean the po sition on a line (X) and the line number (Y ) . Stk Siz e means the number of ob jects in the AL G mode history or in the RPN stac k.
Pa g e L - 3 T he SEARCH sub-menu T he functi ons 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 v ided w ith this command is sho wn next: Rep l ac e : Use this co mmand to f ind and r eplace a s tr ing.
Pa g e L - 4 T he GO T O sub-menu T he functi ons in the GO T O sub-men u are t he follo w ing: Goto L ine: to mo ve to a spec ifi ed line. T he input fo rm pr o v ided w ith this command is: Goto P ositi on : mov e to a spec ifi ed position in the command line .
Pa g e L - 5.
Pa g e M - 1 Appendi x M T abl e o f Built-In Equations T he E quation Libr ar y consists o f 15 sub jects cor r esponding t o the secti ons in the table belo w) and mor e than 100 titles. T he n umbers in par e ntheses belo w indicat e the number of equati ons in the set and the number of v ari ables in the set .
Pa g e M - 2 3: Fluids ( 2 9 , 29) 1: Pr essur e a t D epth (1, 4) 3: F lo w w ith Lo ss es (10, 17) 2 : Bernoulli E quation (10, 15 ) 4: F lo w in F ull P ipes (8 , 19) 4 : F o r ces an d Energy (3 1.
Pa g e M - 3 9: Op ti cs ( 1 1 , 1 4) 1: La w of Ref r acti on (1, 4) 4: Spher i cal Ref lecti on (3, 5) 2 : Criti cal Angle (1, 3) 5: Spher i cal Ref r acti on (1, 5) 3: Br ew ster’s L a w (2 , 4) .
Pa g e N - 1 Appendi x 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 Alg.
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 testi ng 18-36 Errors in programming 21-64 EULER 5-10 Euler constant 16-54 Euler equation 16-51 Euler formula 4-1 EVAL .
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 m enu 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 STOALA RM 25-4 STOKEYS 20-6 STREAM 8-11 String 23-1 String concatenation 23-2 Student t distribut.
Pa g e N - 1 8 TINC 3-34 TITLE 7-1 4 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 L imited W arr ant y HP 5 0g graphing calc ulator ; W arr anty peri od: 12 months 1. HP w arr ants to y ou , the end-us er cu stomer , that HP hard w ar e, access or ies and suppli es w ill be fr ee fr om d e fec ts in mater ials and w orkmanship afte r the date of pur chas e , for the per iod s pecif ied abo v e .
Pa g e LW- 2 W ARR ANTY S T A TEMENT ARE Y OUR SOLE AND EX CL US IVE REMEDIE S . EX CEPT A S INDICA TED ABO VE , IN NO EVENT WILL HP OR I T S S UP PLIER S BE LIABLE FOR L OS S OF D A T A OR F OR DIRE .
Pa g e LW- 3 Swi t ze r l a n d +41-1- 4 3 9 5 3 5 8 (German) + 4 1 -2 2- 8 27878 0 ( F r e n c h ) +3 9-0 2 - 7 5419 7 8 2 (Italian) T urk e y +4 20 -5- 414 2 2 5 2 3 UK +44 - 20 7 - 4 5 80161 Cz ech.
Pa g e LW- 4 Regulat or y inf ormation F edera l C o mmunications Commission Notice T his equipment has bee n tes ted and fo und to compl y w ith the limits for a C lass B digital de vi ce , pursuant t o P art 15 of the FCC R ules .
Pa g e LW- 5 This de v ice complie s with P ar t 15 of the FCC R ules. Oper ation is sub ject to the follo wing tw o c ondi tions: (1) this dev ice may not caus e harmful interf er ence , and (2) this de vi ce must accept an y interfer ence rece iv ed , including interf er ence that may ca use undesir ed oper ation .
Pa g e LW- 6 This compli ance is indicated b y the follo w ing confor mit y marking placed on the pr oduc t: Japanese Notice 䈖 䈱ⵝ⟎䈲䇮 ᖱႎಣℂⵝ⟎╬㔚ᵄ㓚ኂ⥄ਥ ⷙද⼏ળ.
An important point after buying a device HP 50g (or even before the purchase) is to read its user manual. We should do this for several simple reasons:
If you have not bought HP 50g yet, this is a good time to familiarize yourself with the basic data on the product. First of all view first pages of the manual, you can find above. You should find there the most important technical data HP 50g - thus you can check whether the hardware meets your expectations. When delving into next pages of the user manual, HP 50g you will learn all the available features of the product, as well as information on its operation. The information that you get HP 50g will certainly help you make a decision on the purchase.
If you already are a holder of HP 50g, but have not read the manual yet, you should do it for the reasons described above. You will learn then if you properly used the available features, and whether you have not made any mistakes, which can shorten the lifetime HP 50g.
However, one of the most important roles played by the user manual is to help in solving problems with HP 50g. Almost always you will find there Troubleshooting, which are the most frequently occurring failures and malfunctions of the device HP 50g along with tips on how to solve them. Even if you fail to solve the problem, the manual will show you a further procedure – contact to the customer service center or the nearest service center
