Benutzeranleitung / Produktwartung 48gII des Produzenten HP (Hewlett-Packard)
Zur Seite of 175
hp 48gII graphing calculator user’s ma nual K.
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Preface You have in y our hands a compact s ymbol ic and n umeri cal compute r that wil l facili tate calcul ation and mathe matical an alys is of prob lems in a variety of disciplines, from elementary mathematics to advanced engineering and science subjects.
Page TOC-1 Table of Contents Chapter 1 – Getting Started , 1-1 Basic Operat ions , 1- 1 Batteries, 1-1 Tu rning the cal cula tor on and off, 1-2 Adjust ing the dis play contra st, 1-2 Con tents of t.
Page TO C-2 Creating alge braic expr essio ns, 2-4 Usin g the Equ a ti on Wr iter (E QW) to cr ea te e xpr essi on s , 2 -5 Creat ing arith metic expressi on s, 2-5 Creating alge braic e xpressions, 2.
Page TOC-3 Operation s with un its, 3-12 Unit co nversion s, 3-14 Physical constants in t he calculator , 3-14 Defini ng an d using functio ns , 3-16 Referenc e , 3-18 Chapter 4 – Calcula tions with.
Page TOC-4 The PROPFRAC funct ion, 5-11 The P A RT FRA C f unc tio n, 5-11 The FCOEF function, 5- 11 The FROOTS function, 5-12 Step-by -step operatio ns with po lynomials and fractio ns , 5- 12 Refere.
Page TOC-5 The S EQ func tion , 7-6 The MAP functio n , 7-6 Referenc e , 7-6 Chapter 8 – Vectors , 8-1 Ente ring vect ors , 8-1 Typing vec tors in the st ack, 8-1 Stor ing vect ors into var iables i.
Page TOC-6 Function TRACE, 9-7 Solution of li near syste ms , 9-7 Using the numeric al solver fo r linear systems, 9-8 Solution w ith the inve rse matri x, 9-10 Solu t ion by “division” of m atric.
Page TOC-7 Curl , 13-2 Referenc e , 13-2 Chapt er 14 – Differe ntia l Equat ions , 14-1 The C ALC/DIFF menu , 14-1 Solu tion to linear and no n-linear equa tion s , 14-1 Function LDEC, 14-2 Function.
Page TOC-8 Chapt er 17 – Numb ers i n Di ffer ent Ba se s , 17 -1 The B ASE menu , 17-1 Writing non-decimal nu mbers , 17- 1 Referenc e , 17-2 Warranty – W- 1 Serv ice , W-2 Regul atory informatio.
Page 1-1 Chapter 1 Getting started This cha pter is aimed at providing ba sic information i n the operation of you r calcul ator. The exercises are aimed at famil iarizing yourself wi th the ba sic opera tion s and setting s befor e act ually perfo rmi ng a calc ulation .
Page 1-2 b. Insert a new CR2032 lit hium battery. Make sure it s positi ve (+) sid e is fac ing up. c. Repl ace the pl ate and pu sh it t o the o r iginal place. After installing the batteries, press [ ON] to turn the power on. Warnin g: When the low bat tery icon is displa yed, you need to repl ace the batte ries as soon as po ssible .
Page 1-3 RAD X YZ HEX R = 'X' For details on the meaning of these specific ations see Chapter 2 in the calculato r’s user's guide. The se cond line shows the characters { HO ME } ind icating tha t the HOME dir ectory is the current fil e director y in the calculator’s memo ry.
Page 1-4 @EDIT A EDIT the contents of a variable (see Chapter 2 in this Guide and Chapter 2 and Appendix L in the user's guide for more informat ion on edi ting) @ VIEW B VIEW the conten ts of a .
Page 1-5 th e bl ue ALPH A key , ke y (7 ,1) , can be combined w ith some of the other keys to ac tivat e the a lternati ve func tions show n in t he keybo ard.
Page 1-6 ~„p ALPHA-Le ft-Shif t functi on, to e nter th e lowe r-case l etter p ~…p ALPHA-Rig ht-Shift functi on, to enter the symbol π Of the six functio ns assoc iated w ith a key o nly the first four are sho wn i n the keyboard it self. The figure in next page shows these four labels for the P key.
Page 1-7 Press the !! @@OK# @ F s o f t m e n u k e y t o r e t u r n t o n o r m a l d i s p l a y . E x a m p l e s o f selecting di fferent calculator modes a re show n next. Op er at i ng M ode The calcul ator offer s two operatin g modes: t he Algebraic mode, and the Reve rse Pol ish Notati on ( RPN ) mode.
Page 1-8 1./3.*3. ————— /23.Q3™™ ™ + !¸2.5` After p ressing ` the calc ulator displays the expressio n: √ (3.*(5.-1 /(3.*3.))/23 .^3+EXP(2.
Page 1-9 dif feren t l evels ar e referr ed to as the stack leve ls , i.e., stack level 1, stack level 2, et c. Basic ally, what RPN means i s that , instead of writing an operat ion such as 3 + 2, in the calc ulator by using 3+2` we write first the operands, in t he proper order, and then the o perator, i.
Page 1-10 5 . 2 3 23 3 3 1 5 3 e + ⋅ − ⋅ 3` Ent e r 3 in le ve l 1 5` Ent e r 5 in le ve l 1 , 3 m ov e s t o le v el 2 3` Ent e r 3 in le ve l 1, 5 mo v es to le v el 2, 3 t o lev e l 3 3* Place 3 and mu ltipl y, 9 a ppears in level 1 Y 1/(3 × 3), last v alue in lev .
Page 1-11 m o r e a b o u t r e a l s , s e e C h a p t e r 2 i n t h i s G u i d e . T o i l l u s t r a t e t h i s a n d o t h e r number forma ts try the f ollowin g exercises: • Standard format : T his mo de is t he mo st u s ed mod e as it s ho ws nu mbe rs in th e m os t fa mi li ar nota tion.
Page 1-12 Press the !! @@OK#@ soft menu key return t o the calculat or di splay. The number now is shown as : Notic e ho w th e number is roun ded, not tr unca ted. Thus, t he number 123.4 567890123 456, for this sett i ng, is displayed as 123.457, and not as 123.
Page 1-13 This result, 1. 23E2 , is th e cal culat or’s versi on of pow ers-o f-ten not ation, i.e., 1.235 × 10 2 . In thi s, so-calle d, scientif ic nota tion, th e numbe r 3 in front of the Sci number format (shown earlier) represents the number of signifi cant figures afte r the deci mal point.
Page 1-14 • Decimal comm a vs. decimal point Decimal po ints in floating -point numbers can b e replaced by co m mas, if the user is more famili ar with s uch notation.
Page 1-15 • Grades : There are 400 gr ades ( 400 g ) i n a comp let e circum ference. The angle measure affect s the trig f unctions like SIN, COS, TAN and associ ated fun ction s. To change the ang le measure m ode, use the following procedure: • Pr ess the H button .
Page 1-16 Selecting CAS settings CAS stands for C ompu ter A lgebraic S y s t em . T h i s is th e m a t h e m at i c a l c o re o f the calcul ator where the symbolic mathematical operations and functions are prog ramm ed. The CA S offer s a n umber of settin gs ca n be adjust ed ac co rdin g to the type of operation of interest.
Page 1-17 opt ions above ). Unsel ected op tions will show no che ck mark in t he underline preceding the option of interest (e.g., the _Numer ic, _Appr ox, _Compl ex, _Verbose, _S tep/S tep, _Incr Pow options above).
Page 1-18 The calcul ator displ ay can be cu stom ized t o you r prefe r ence by s electing differ ent display mod es. To see th e option al display set tings use the fo llowin g: • Fi rst, press th e H bu tton to acti vate the CAL C ULATOR MODE S input form.
Page 1-19 ( D ) to dis play the DISPLAY MO DES inpu t form. The Font: field is highl ighted, and th e option Ft8_0 :system 8 is selec ted. This is the default value of t he display fon t.
Page 1-20 Selectin g properties of the S t ack First, press the H button to activ ate the CA LCULA TOR M ODES i nput fo rm. Within t he CALC ULA TOR MOD ES input form, press the @@DISP@ soft menu key ( D ) to di splay the D ISPLAY MO DES inp ut form. P ress the dow n arrow key, ˜ , twic e, to get to the Sta c k line .
Page 1-21 Selectin g properties of th e equation writer (E QW ) First, press the H button to activ ate the CA LCULA TOR M ODES i nput fo rm. Within t he CA LCULA TOR MO DES input f orm, press th e @@DISP@ soft menu ke y ( D ) to di splay the D ISPLAY MO DES inp ut form.
Page 1-22.
Page 2-1 Chapter 2 Introducing the calculator In this chapter we present a number of basic operations of the calcula tor including the use of the Equati on Writer an d the mani pulation of data obj ects in the calcu lat or. Study th e exampl es in this chapt er to get a good gras p of the capabi lities of the calcu lator for future applicat i ons.
Page 2-2 Notice that, if you r CAS is set to EXACT (s ee Appendix C in us er's gu ide) and you enter your expression using integer numbers for int eger values, the result is a sy m boli c quantity , e. g., 5*„Ü1+1/7.5 ™/ „ÜR3-2Q3 Before producing a resu lt, you will be asked to cha nge to Approx imate mode.
Page 2-3 To evaluate the expression we ca n use the EVAL function, as follows: µ„î If the C AS is set to Exact , you will be as ked to approve changi ng the CAS settin g to Ap pro x .
Page 2-4 This expression is semi -symbolic in the sense that t here are floating-point comp onents to the resu lt, as we ll as a √ 3. Next, we sw itch stac k locations and eval uate using fu nction Æ NUM: ™…ï .
Page 2-5 Entering this expression w hen the calculator is set in the RPN mo de is exactly the same as this Algebr aic mode exercise. For a dditional information o n editing al gebr aic ex pressio ns in the ca lcul ator’s display or stack see Chapter 2 in the calculator’s user' s guide.
Page 2-6 The curso r is shown as a le ft-facing key. The cursor ind icates the cu rrent edition location. For example, for the cursor in th e location indicated above, type now : *„Ü5+1/3 The edited expression looks as fo llo ws: Su ppose that you want to repl ace the quantity between pare ntheses in the denominat or (i.
Page 2-7 The expression now looks as follow s: Su ppose t hat now you want to add the fr action 1 /3 t o this e ntire ex pression , i.e., you wa nt to enter the expression : 3 1 ) 2 5 ( 2 5 5 2 + + .
Page 2-8 Creating algebrai c expression s An algebrai c expression is very sim ilar to an arithmet ic expression, except that Engl ish and Greek letters may be included.
Page 2-9 Also , you can alway s copy special characters by using t he CHARS menu ( …± ) if y ou do n’t want to memo rize the keystroke combinatio n that produces it. A listing of c ommonly used ~‚ keystroke combinations was listed i n an ear lier section.
Page 2-10 Variables Variables are similar to files on a computer hard drive. One v ariable can stor e one obje ct (numeri cal val ues, alg ebraic expr ession s, lists, vectors, matrices, progr ams, etc).
Page 2-11 To unlock the upper-ca se locked keyboa rd, pr ess ~ Try the following exercises: ³~~math1~` ³~~m„ a„t„h ~` ³~~m„ ~at„h~` The calculator di splay wi ll show the fo llowing (left-hand sid e is Algebrai c mode, right-hand side is RPN mode): Creat ing varia bl es The simp lest way to crea te a va riable is by using the K .
Page 2-12 Press ` t o create the variable. The variable is now sho wn in the so ft men u key l abe l s: The following are the keystrokes required to enter the remaini ng varia bles: A12: 3V5K~a12` Q: ³~„r/„Ü ~„m+~„r™™ K~q` R: „Ô3‚í2‚í1™ K~r` z1: 3+5*„¥ K~„z1` (Accept chan ge to Comp lex mode if asked).
Page 2-13 • RPN mod e (Use I @@ OK@@ to change to RPN mode). Use the following key strokes to store the val ue of –0. 25 into variabl e α : 0.25 ` ~‚a` . At this point, the screen will look as fol lows: This expr ession means that the value –0.
Page 2-14 p1: ‚å‚é~„r³„ì* ~„rQ2™™™ ³ ~„p1™` K . The scree n, at this point, will look as fol lows: You will see six of the seven variables listed at the bottom of the screen: p1, z 1, R, Q, A12, α .
Page 2-15 Us ing the right-s hift key ‚ foll owed by s oft menu key labels This app r oach for viewing the cont ents of a variabl e works th e same in both Algebrai c and RPN modes.
Page 2-16 Deleting variables The simplest way of deleting variables is by using fun ction PURGE. This funct ion can be acce ssed dir ectly by using the TOOLS menu ( I ), or by usi n g th e FI L ES me n u „¡ @ @OK@@ . Using f unction PURGE in the stack in Algebraic mode Our vari able l ist contains variab les p1, z1, Q, R, and α .
Page 2-17 To delete two variab les simultaneously, s ay vari ables R and Q , f irst create a list (in RPN mode, the elements of the list need not be separated by commas as in Alge brai c mo de) : J „ä³ @@@R !@ @ ™ ³ @@@!@ @ ` The n, pr es s I @ PURGE@ use to purge the variables.
Page 2-18 @@ OK@@ ˜˜˜˜ Sho w the MEMORY menu list and sel ect DIRECTORY @@ OK@@ —— S ho w the D IRECTORY menu list and select ORD ER @@ OK@@ activate the ORDER command There is an al ternative way to acces s these menu s as soft M ENU keys, by setting system flag 117 .
Page 2-19 Press t he @CHECK! soft men u key to set f lag 117 to soft ME NU . The screen w ill refl ect that chan ge: Press tw ice to return to normal ca lculator disp lay. Now , we’ll try to find th e ORDER c ommand usin g similar keystr okes to th ose used above, i.
Page 2-20 To activate the ORDER co mmand we press the C ( @ORDER ) sof t menu key. References For a dditional information on e ntering and mani pulating e xpressions in the display or in the Equation Writer see C hapter 2 of the calculator’s user's guid e.
Page 3-1 Chapter 3 Calculations with real numbers This chapter demonstrates the use of the calculator for operations and functi ons relat ed to real nu mbers. The user should be ac quainted with the key board to identif y certain fu nctions avai lable in the keyboard ( e.
Page 3-2 6.3` 8.5 - 4.2` 2.5 * 2.3` 4.5 / Alternat ively, in RPN mode, you can sepa rate the operan ds with a space ( # ) befo re pressing the operator key. Examples: 3.7#5.2 + 6.3#8.5 - 4.2#2.5 * 2.3#4.5 / • Paren theses ( „Ü ) can be used to grou p operations, as well as to enc los e argu men ts o f fun cti on s.
Page 3-3 „Ê 2.32` Exam ple in RPN mode: 2.32„Ê • The squar e function, S Q, is availabl e through „º . Exam ple in ALG mode: „º2.3` Exam ple in RPN mode: 2.3„º The square ro ot function , √ , is availa ble through the R key. When calcul ating in the stack in ALG mode, enter the functio n before the argum ent , e.
Page 3-4 ente r the fu nction XROOT foll owed by the argume nts ( y,x ), separated by co mmas , e. g., ‚»3‚í 27` In RPN mo de, ente r the argu ment y , first, then, x , and finall y the function call, e.
Page 3-5 2.45` ‚¹ 2.3` „¸ • Three trigonomet r ic functi ons are readil y availabl e i n the keyboa r d: sine ( S ), cosi ne ( T ), and tangen t ( U ) .
Page 3-6 Real number functions in the MTH m enu The MT H ( „´ ) menu include a number of mathematic al functions m ostly applic able to re al number s. With t he d efault sett ing of C HOOSE box es for system flag 1 17 (se e Chapter 2), the MTH menu show s the followi ng functions: The funct ions are g rouped b y the type of argu ment (1.
Page 3-7 For exam ple, in A LG mode, t he keystroke sequence t o calculate, sa y, tan h(2 .5), is the fol lowin g: „´4 @@ OK@@ 5 @ @OK@@ 2.5` In the RPN mo de, the keystrokes to perf orm this ca lcul ation are the following: 2.
Page 3-8 Finall y, in order to select, for exampl e, the hype rbolic tangent (tanh) func tion, simply press @@TAH@ . Note: To see addition al options in these soft menus, press the L key or the „« keystro ke sequence. For ex ample, to calculate tan h(2 .
Page 3-9 Option 1. Tool s.. conta ins func tions used to oper ate on un its (di scussed later). Options 3. Length.. thro ugh 17. Vis cos ity .. contain menu s with a number of units for each of the quant ities described. For exam ple, selec ting option 8.
Page 3-10 Pressing on t he appropriate soft menu key will open the su b-menu of units for that part i cul ar select i on. For exampl e, for the @) SPEED sub-menu, the followi ng units ar e ava ilable: Pressing th e soft menu key @) UITS will take you back to the U NITS menu.
Page 3-11 Attaching u nits to n umbers To attach a unit object to a number, the number must be followed by an underscore ( ‚Ý , key(8,5)). Thus, a force of 5 N will be entered as 5_N.
Page 3-12 _________________ __________________________________ _ Pre fix Nam e x Pr efi x Nam e x _________________ __________________________________ _ Y yo tta +24 d dec i -1 Z zetta +2 1 c cent i -.
Page 3-13 whic h shows as 65_ (m ⋅ yd). To convert to units of the SI system, use function UBAS E (fi nd it u sing the com mand catal og, ‚N ): Note: Recall tha t the ANS(1) variable is ava ilabl e through the keystroke com binatio n „î (asso ciated wi th the ` key).
Page 3-14 These operations produce the following output: Unit convers ions The UNITS menu conta ins a TOOLS sub- menu, which provides the fol lowing function s: CONVERT( x,y): conv ert unit o bject x .
Page 3-15 The soft menu keys corresponding to this CONSTAN TS LIBRARY scr een include the follo w ing functions: SI when selected, constants v alues are shown in SI units (*) ENGL when selected, const.
Page 3-16 To copy th e value of Vm to the stack, sel ect the variable nam e, and press ! STK , then, press @UIT@ . For the calculator set to the ALG, t he screen will look l i ke this: The display sh ows w hat is called a tagg ed val ue , .
Page 3-17 and ge t the re sul t you want withou t having to type the ex press ion in the righ t- hand side f or eac h separate value. In the fo llowing exampl e, we assume you have set yo ur calculator to A LG mode.
Page 3-18 betw een quot es tha t cont ain that loca l va riable , and show the ev aluated expression. To ac tiva te the fun ction in A LG m ode, typ e the n ame of the func tio n followe d by the argument between parent heses, e.
Page 4-1 Chapter 4 Calculations with c omple x numbers This chap ter shows examples of calc ulati o ns and app licatio n of function s to complex numb ers. Definitions A complex number z is wri tten as z = x + iy , (Cartesian representation ) where x and y are real num bers, and i is t he imaginary unit defi ned by i 2 = - 1.
Page 4-2 Entering compl ex numbers Complex numbers in the calcul ator can be entered in either o f the two Cartesian representations, namely, x+iy , or (x,y ) . The results in the calculator will be show n in t he or dered -pair fo rmat , i .e., (x,y) .
Page 4-3 The resul t shown above represents a m agnitude, 3.7, and an ang le 0.33029… . The angle sy m bol ( ∠ ) is shown in fro nt of the angle measure. Return to C artesian or rectan gular coordin ates by usin g functi on RECT (ava ilabl e in the cat alog, ‚N ).
Page 4-4 (3+5i) + (6-3i) = ( 9,2); (5- 2i) - (3 +4i) = (2,-6 ) (3- i)·( 2-4i) = (2 ,-1 4); (5-2i)/(3+4 i) = (0.28,-1.04) 1/(3+4i) = (0.12, -0.1 6) ; -(5-3i) = -5 + 3i The C MP L X me n u s There are two CM PLX (CoMPLeX numbers) menus available in the calculator.
Page 4-5 CONJ(z): Pro duces the complex conjugate of z Ex ample s of app lications of the s e f unctions are s hown nex t. Recal l t hat, for ALG mode, the function must precede the argument, w hile in RPN mode, you enter the argument first, and then select the function.
Page 4-6 Functi ons applie d to complex n umbers Many of the keyboard-b ased functi ons and MTH menu f uncti ons defined in Chapter 3 for real numbers (e.g., SQ, , LN, e x , etc .), can be appli ed to complex number s. The result is ano ther co mplex number, as illustrated in the following examples.
Page 4-7 Functi on DROITE is found in th e command catal og ( ‚N ). Reference Additional information o n complex number operations is presented in Chapter 4 of the calculator’s us er’s guide.
Page 5-1 Chapter 5 Algebraic and arithmetic operations An algeb r aic obje ct, or simply , alge braic, is any num ber, variabl e name or algebr aic express ion that can be opera ted upon, manip ulated , and combined according t o the rul es of algebra.
Page 5-2 After bui lding the object, press to s how it in the stack (ALG and RPN modes show n below): Simple ope rations with algebraic objects Algebr aic objects can be added , subtracte d, mul tipli.
Page 5-3 In ALG mo de, the foll owing key strokes wil l show a num ber of operati ons with the al gebraics contained in variab les @@ A@@ and @@A@ @ (pr ess J to recove r varia ble menu): @@ A.
Page 5-4 F un cti o ns in th e ALG men u The ALG (Algebraic) menu i s availabl e by using the keystroke sequence ‚× (associa ted with the ‚ key). With system flag 117 set to CHOOSE boxes , the AL.
Page 5-5 Copy the exampl es provided onto your s tack by pressing @ECHO ! . For exampl e, for the EXPAN D entry sho wn ab ove, pr ess the @ECHO! soft menu key to get t he following example copied to the stack (press ` to execute the c ommand): Thus, we leave for the user to explore the applications of the functions in the ALG (or A LGB) menu.
Page 5-6 Operations wit h transcen dental f unction s The calcu lator o f fers a nu m ber o f functi ons th at can be u sed to r eplace expr essions containing l ogarithmic and expo nential functions ( „Ð ), as wel l as tr igonome tric functions ( ‚Ñ ).
Page 5-7 These function s allow to si mplify expression s by replacing some cat egory of trig onometric fu nctions for an other one . For exam ple , the fu nction ACOS2 S allows to r eplace t he fu ncti on ar c cos in e (a cos(x)) w ith i ts expressio n in terms of arcsine (asi n(x )).
Page 5-8 FA CTORS: SIMP2: The functio ns assoc iated w ith the A RITHMETI C submenus: INTEGER, POLY NO MIAL, MODUL O , and PERMUT ATION, are the fol lowin g: Addi tiona l informat ion on applicati ons of the A RITH METIC menu functi ons are presente d in Chapter 5 in the cal culat or’s user' s guide.
Page 5-9 The var ia ble VX A variable called VX exists in the calculator’s {HOM E CASDIR} direc tory that takes, by default, the value of ‘X’. This is the name of the preferred inde pendent var i able for algeb r aic and cal culus appli cations.
Page 5-10 Note : you could get the latter result by using PARTFRAC: PARTFRAC(‘(X^3-2*X +2)/(X-1) ’) = ‘X^2+X-1 + 1/ ( X-1)’. The P EVA L function The functi ons PEVAL (Poly nomial EVALu ation).
Page 5-11 The PROPFRA C funct ion The fun ction PROPFRAC conve r ts a r ational fraction i nto a “proper ” fraction , i.e., an int eger part added to a fraction al part, if su ch decomposition is possi ble.
Page 5-12 The FR OOTS function The fu nction FROOTS obt ains the r oots and pole s of a fracti on. As an exam ple, appl ying function FROOTS to the resul t produced above, will resul t in: [1 –2 –3 –5 0 3 2 1 –5 2].
Page 5-13 Reference Additional information, definiti ons, and examples of alge braic and arithmeti c operations are presented in Chapter 5 of the calculator’s user’s guide.
Page 6-1 Chapter 6 Solution to equations Associated w ith the 7 key the re are two menus of e quation -sol ving functio ns, th e Symbolic SO LVer ( „Î ), and the NU Merical SoLVer ( ‚Ï ). Following, w e present some of the functions c ontained in these menus.
Page 6-2 Usin g the RPN mode, the s olution is accompl ished b y enteri ng th e equat ion in the stac k, fo llowed by the v ariab le, befo re enteri ng func tion I SOL. Right before the execution of ISOL, the RPN stack sho ul d look as in the figure to the lef t.
Page 6-3 The following examples show the use of function SOLV E in ALG and RPN modes. [ No te : not all lines will be visi ble when done with the exercises in t he following fi gures.] The screen shot sho wn ab ove di splays tw o solution s. In t he first one, β 4 -5 β =125, SOLVE produces no solutions { }.
Page 6-4 Function SOLVEVX The function SOLVEVX solves an equation for the default CAS variable containe d in the reserve d variable name VX . By defaul t, this variable is se t to ‘X’. Examples, using the ALG mode w ith VX = ‘X’, are shown below : In the first case SOL VEVX could not find a solution.
Page 6-5 To us e fu nctio n ZER OS in R PN mode , ent er firs t the p oly nomi al e xpre ss ion , then th e vari able to solve fo r, and then functi on ZEROS.
Page 6-6 Following, we present applications of items 3. So lve po ly.. , 5. So lve f in anc e , and 1. Solve equation.. , in that order. Appendix 1-A, in t he calcul ator’s user 's gu ide , c ont ai ns in st ruct io ns o n ho w to u se i nput fo rms w ith exam ples fo r the numerical solver applications.
Page 6-7 ‚í 1‚í1 @ @OK@@ @SOLVE@ S o l v e e q u a t i o n The screen will show the s olution as follows: Press ` to return to stack. The stack wi ll show the follo wing resu lts in A LG mode (the sa me result would be sho wn i n RPN mo de): All t he solutions are c omplex numbers: (0.
Page 6-8 Press ˜ t o trigger the line editor to see all the coeff icients. Generating an al gebraic express ion for the polyno mial You c an us e th e ca lculat or to genera te a n alg ebr aic expr ession fo r a polynomial given the coefficients or th e roots of the polynomial.
Page 6-9 ' (X-1)*(X-3)*(X+2)*(X-1) '. To e xpand t he prod ucts, y ou can use the EXPAND command. The resulting expression is: ' X^4+-3 *X^3+ - 3*X^2+11*X-6' . Fi na nc ia l ca l cu la t i ons Th e cal cu lat ions in it em 5. S olve finance.
Page 6-10 In RPN mode, ente r the equation between apos trophes and activat e command S T EQ . T hu s , f u n ct io n ST E Q c a n b e u s e d a s a s h or tc u t t o s t or e an e x pr e s s io n into variable EQ.
Page 6-11 Solution to simultaneous eq uations with M SLV Funct ion MSLV is ava ilable in th e ‚Ï menu. The help- fac ility entry for functi on MSLV is shown next: Notice that fu nction MSLV requ ires three argume nts: 1. A vector containi ng the equat ions, i.
Page 6-12 You may hav e noti ced th at, w hile prod ucing the solution, the scr een show s interme d iate informati o n on the up per left corner . Since the solut ion provided by MSLV is numerical, the information i n the upper left corner shows t he results of the iterative process used to obtain a solution.
Page 7-1 Chapter 7 Operations with lists Lists ar e a type of calcul ator’s object that can be usefu l for data proce ssing. This chapter presents examples of operations w ith lists. To get started wi th the example s in this Chapter, w e use the Approximate mode (See Chapter 1).
Page 7-2 Additi on , subtra ction , multiplication , division Mul tiplication and divis ion of a list by a singl e number is distrib uted across the li st, for ex ample: Subt raction o f a single numb.
Page 7-3 Note : If w e had entered the elements in lists L4 and L3 as integers, the in finite symbol wo uld be show n w henev er a di visi on by zero occ urs.
Page 7-4 A B S I N V E R S E ( 1 / x ) Lists of com plex n umbers You can c reate a co mplex number list, say, L5 = L1 A DD i ⋅ L2 (type th e inst ruction as ind icated here ), as foll ows: Functi ons such as LN, EXP, SQ, etc., can also be appl ied to a list of complex numbers, e.
Page 7-5 With system f lag 117 set to SOFT m enus, the M TH/LIST m enu shows th e following fun ctions : The operation of the MTH/LIST menu is as follows: ∆ LIST : Calculate increment among consecut.
Page 7-6 The S E Q fu n ct io n The SEQ function, available through th e c ommand catalog ( ‚N ), ta kes as argu m ents an ex pression in terms of an index, the name of the index, and starti ng, ending, and increment values for the index, and returns a l ist consisting of the ev alu ation of the expression for all possible values of the ind ex.
Page 8-1 Chapter 8 Ve cto rs This Chapter p rovides exa m ples of e ntering and operat i ng with vectors, both mathe matica l vector s of many ele m ents , as wel l as physi cal vect ors of 2 an d 3 com pone nts.
Page 8-2 ( ‚í ) or sp aces ( # ). Notice tha t after pr essing ` , in either mode, the calcul at or shows the vec tor elements se parated by spaces.
Page 8-3 The @EDIT k e y i s u s e d t o e d i t t h e c o n t e n t s o f a s e l e c t e d c e l l i n t h e mat rix writ er. The @VEC@@ key, when sele cted, will produce a vect or, as op posed to a matr ix of one row and ma ny colu mns. The ← WID key is used to decrease the w idth of the columns in the spreadsheet.
Page 8-4 The @ROW@ k e y wil l add a row fu ll of zeros at the locatio n of the selected cel l of the spreadsheet. The @ROW key w ill del ete the row corresp ondi ng to the selected cell o f the spreadsheet. The @COL@ key will add a colu m n full of zer os at the location o f the selected cel l of the spreadsheet.
Page 8-5 (3) Move the cu rsor up t wo positions by using —— . Then press @ROW . The second r ow will disap pear. (4) Press @ROW@ . A row of three zero es appears in the second row. (5) Press @COL@ . The first c olumn will disappear. (6) Press @COL@ .
Page 8-6 Attemp ting to ad d or subtract vectors of diffe rent len gth prod uces an e rror message: Multiplicat ion by a scalar, and division by a scalar Multip licatio n by a sca lar or divi sion by .
Page 8-7 The M TH /V E C TOR me nu The MT H m en u ( „´ ) contains a menu of function s that specifi c all y to vect or ob jects: The VECTOR m enu cont ains the f ollowing func tions (system flag 117 set to CHOOSE bo xes): Ma gn it ud e The magni tude of a vector, as di scuss ed earlie r, can be found with funct i on ABS .
Page 8-8 Cross product Functi on CROSS (optio n 3 in th e MTH/V ECTOR menu) is used to cal cul ate the cross pr oduct of two 2-D vect ors, of two 3-D vecto r s, or of one 2- D and one 3- D vec tor . For the p urpose of ca lculating a cross prod uct, a 2-D v ecto r of the form [A x , A y ], is treated as th e 3-D vect or [A x , A y , 0].
Page 9-1 Chapter 9 Matrices and linear algebra This chap ter shows ex amples of cre ating matrices and operations with matrices, i nclu ding line ar alge bra applicati o ns.
Page 9-2 Press ` o nce more to place the matrix on the stack. The ALG mode st ack is shown nex t, before and after pres sing , once more: If you ha ve select ed the t extboo k displa y optio n (usin g H @) DISP! and checking off 3 Textbook ), the matrix will look like the one shown above.
Page 9-3 Operations with m atrices Matrices , like other mathem atical objects, ca n be added and su btracted. They can be mu ltiplied by a scal ar, or among themselves. An important operation for l inear al gebra applications is the i nverse of a matrix.
Page 9-4 In RPN m ode, you ca n try a few more exerci ses: A ` `+ A ` `- A ` `+ A ` `- A ` `+ A ` `- A ` `+ A ` `- Mul t ip l ic at i on There are different multiplication operations that involve matrices.
Page 9-5 Vector-matrix multiplication, on the other hand, is not defined. This mul tiplication can be pe r forme d, however, as a special case of matr ix mul tiplication as def i ned nex t. Matr ix multipli cation Matri x mul tiplicati on is defined by C m × n = A m × p ⋅ B p × n .
Page 9-6 The i denti ty matrix The identi ty mat rix has th e property t hat A ⋅ I = I ⋅ A = A . To verify t his property we present the following examples using the matrices stored earlier on .
Page 9-7 Characte rizin g a matrix (The matrix NORM men u) The matrix NORM (NORMALIZ E) menu is ac cesse d throug h the keyst roke sequence „´ . This menu is described in detail in Chapter 10 of the calculat o r’s user's guide. Some of these functions are desc ribed next.
Page 9-8 This system o f linear equations can be writ ten as a m atrix equa tion, A n × m ⋅ x m × 1 = b n × 1 , if we define the foll owing ma trix and vect ors: m n nm n n m m a a a a a a a a a .
Page 9-9 . 6 13 13 , , 4 2 2 8 3 1 5 3 2 3 2 1 − − = = − − − = b x A and x x x This system ha s the sam e number of equations as of unknow ns, and will be referred to a s a square system .
Page 9-10 Sol ution with the inv erse matrix The solution to t he system A ⋅ x = b , w here A is a squar e matrix is x = A -1 ⋅ b . For t he example u sed earlier, we can find the solution in the .
Page 10-1 Chapter 10 Graphics In this chap ter w e introdu ce some of the graphics capabi lities of the cal c ulator. We will present graphi cs of fu nctions in Cartes i an coordin ates an d polar coordinat es, parametric plot s, graphics of conics, bar pl ots, scatterpl ots, and fast 3D plots.
Page 10-2 Plottin g an e xpress ion of the f orm y = f (x) As an exampl e, le t's plo t the funct ion, ) 2 exp( 2 1 ) ( 2 x x f − = π • First, enter the PLOT SETUP environment by pressing, „ô . Make sur e that the option Fu nction is s el ected as the TYPE , and t hat ‘X’ is selected as the inde pendent v ariable ( INDEP ).
Page 10-3 VIE W, then press @ AUTO t o g e n e r a t e t h e V - V I E W a u t o m a t i c a l l y . T h e PLOT WIN DOW sc reen looks as f ollows: • Plo t the graph : @ERASE @DRAW (wa it till the c .
Page 10-4 • We will gen erate value s of the funct ion f(x), de fined above, for value s of x f r o m – 5 t o 5 , i n i n c r e m e n t s o f 0 . 5 . F i r s t , w e n e e d t o e n s u r e t h a t t h e graph type i s set to FUN CTION in the PLOT SETUP screen ( „ô , press them simultaneously, if in RPN mode).
Page 10-5 • • The @@IG @ key si mply changes the font in the ta ble from small to big, and vice versa. Try it. • • The @OOM key, wh en pressed, produces a menu wit h the options: In , Ou t , Decimal, Integer , and Trig . Try the fo llowing exercises: • • Wit h the op tion In highlighte d, press @@@O K@ @@ .
Page 10-6 • Press „ô , simultaneo usly if in R PN mode, t o access to the PLOT SETUP window. • Change TYPE to Fast 3D. ( @CHOOS! , f ind Fast3D , @@OK@@ ). • Press ˜ and type ‘X^2+Y^2 ’ @@@O K@ @@ . • Make sure tha t ‘X’ is sele cted as the Indep: and ‘Y’ as the Depnd: varia bles.
Page 10-7 • When done, press @EIT . • Press @CACL to return to the PLOT WINDOW envi ronment. • Change the Step d at a to read: Step Indep: 20 Depn d: 16 • Press @ERASE @DR AW to see t he surface plot. Sample views: • When done, press @EIT .
Page 10-8 • Press LL @) PICT to leave the EDIT environment. • Press @CACL to return t o the PLOT WINDOW en vironmen t. Then, pre ss $ , or L @@@OK@@@ , to re tur n to norm al c alc ul ator di spl ay. Reference Additional informati o n on g raphics is avail able in Chapte rs 12 and 2 2 in the calculato r’s user’s guide.
Page 11-1 Chapter 11 Calculus Applic ations In this Chapter we discus s appl ications of the cal culator’s functions to operations related to Calcul us, e.
Page 11-2 command cata log ( ‚N~„l ) or through option 2. LIMITS & SERIES… of the CALC menu (see above). Funct ion li m is entered in ALG mode as = to c alculate the lim it ) ( lim x f a x → . In RPN mode, enter the function f irst, then the expression ‘x= a’, and finall y function lim.
Page 11-3 Anti-der ivative s and i ntegrals An anti-deri vative of a function f(x ) is a function F(x ) such that f(x) = dF/d x. One way to represent an anti-deriv ative is as a indefinite integ r al , i.e., C x F dx x f + = ∫ ) ( ) ( if and onl y if, f(x) = dF/ dx, and C = const ant.
Page 11-4 Ple ase notice tha t fu nctions S IGMAVX and S IGMA are de signe d for integra nds tha t inv olve some s ort of integer f unctio n like t he facto rial (!) funct ion shown above. Thei r result is the so-cal led dis crete derivati ve, i.e., one defined for integ er numbers onl y.
Page 11-5 Infinite se ries A function f(x) can be expanded into an infinite se ries around a point x=x 0 by u sing a Tay lor’ s se ri es, namel y, ∑ ∞ = − ⋅ = 0 ) ( ) ( ! ) ( ) ( n n o o n x x n x f x f , where f (n) (x) rep resents the n-th deriva tive o f f(x) with respect to x, f (0) (x) = f(x).
Page 11-6 Funct ion SERIES pr oduc es a T aylor polynom ial usi ng as argum ents t he fun cti on f(x) to be ex panded, a variabl e name al one (for Maclaurin’ s series) or an expr ession of the form ‘vari able = val ue’ indi cating the point of expansion of a Taylor seri es, a nd th e or der o f th e seri es to be pr oduc ed.
Page 11-7 In the right -hand sid e figure above , we are using the line editor to see the series expansion in detail. Reference Additional definiti ons and appl ications of cal culus oper ations are pres ented in Chapte r 13 in the cal cul ator’s us er’s gu ide.
Page 12-1 Chapter 12 Multi-variate Calculus Applications Multi-vari ate calc ulus refers to f unctions of two o r more variables. In this Chapter we discu ss basic conce pts of multi -variate cal c ulus: par tial der ivatives and multi ple inte grals.
Page 12-2 Multiple in tegral s A physi cal int erpretat ion of the doub le integ ral of a functi on f(x,y ) over a region R on the x-y plane is the vol ume of the solid bod y contained under the surface f (x,y) above the region R.
Page 13-1 Chapter 13 Vector A n alysis Applications This cha pter descri bes the use o f func tions H ESS, DIV, and CURL, fo r calcu latin g operations of vector anal ysis.
Page 13-2 Alternatively, use function DERIV as follows: Divergence The divergence of a vector functi on, F (x, y,z) = f(x,y,z) i +g(x,y,z) j +h(x,y,z) k , is define d by taking a “dot-product” of the del operator with the functi on, i.e., F divF • ∇ = .
Page 14-1 Chapter 14 Differential Equations In this Chapter we present examples of solving ordinary differential equations (ODE) using ca lculator f unctio ns. A diffe rentia l equation is an eq uation invol ving derivati ves of th e indepe ndent variab le.
Page 14-2 Function LDEC The calcu lator provid es funct ion LDEC (Linea r Differ ential Equati on Command) to find the gene r al s o lu tion to a linear ODE of any order with const ant coefficients , whether it is homogeneou s or not.
Page 14-3 'X' ` 'XX X' ` DE The sol uti on is : which can be simp lified to y = K 1 ⋅ e –3 x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + (450 ⋅ x 2 +330 ⋅ x+241)/13500 .
Page 14-4 The var ia ble ODETY PE You wi ll notic e in the soft-men u key labels a n ew v ariable called @ODET (ODETYPE). This v ariable is produced with the call to the DESOL function and holds a string showing the type of OD E used as input for DESOLVE.
Page 14-5 Laplace Trans forms The La place trans form of a function f(t) produ ces a function F(s) in the image domain that can be util i zed to f ind the sol ution of a l inear diffe rential equat i on involvi ng f(t) throu gh algebr aic m etho ds. The steps i nvolved in this appl i cation are th r ee: 1.
Page 14-6 and you will notice that th e CAS default variable X in the equation writer screen replaces the variable s in this definiti on. The refore, when using the funct ion LAP you get back a function of X, which is the L apl ace trans form of f(X).
Page 14-7 Next, we move t o the CASDIR su b-directory unde r HO ME to change the valu e of variable PERIOD, e.g. [ Note : not all lines will be vi sibl e when done with the exerci ses in the following figures.
Page 14-8 Thus, c 0 = 1/3 , c 1 = ( π⋅ i+2)/ π 2 , c 2 = ( π⋅ i+1) /(2 π 2 ). The Fourier series with t hree elements will be written as g(t ) ≈ Re[(1/3 ) + ( π⋅ i+2)/ π 2 ⋅ exp(i ⋅π⋅ t)+ ( π⋅ i+1)/(2 π 2 ) ⋅ exp(2 ⋅ i ⋅π⋅ t)].
Page 15-1 Chapter 15 Probability Distributions In this Chapter we pr ovide ex amp les of appl icatio ns of the pre-def ined probab ility dis tribu tions in the cal culator. The MTH/PROBAB ILITY.. sub-menu - part 1 The MTH/PRO BABILITY.. sub-menu is a ccessi ble through the keystroke sequenc e „´ .
Page 15-2 We can cal c ulate combinations, per mutations, and fa ctorial s with fun ctions COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of those fun c tions is descri bed next:.
Page 15-3 start lists of random numbers is presented in detail in Chapter 17 of the user's gui de. The MTH/PROB menu - part 2 In this s ection we discuss four continuou s probabil ity distribut i.
Page 15-4 The S t uden t-t di stribution The Stu dent-t, or simp ly, th e t-, di strib ution h as one param eter ν , kno wn as t he degrees of freedom of the distribution.
Page 16-1 Chapter 16 Statistical Applications The calculat o r provides the fol low ing pre-programmed stat istical features accessi ble through t he keystroke c ombinatio n ‚Ù (the 5 key): Ente ring data Appli cations nu mber 1, 2, and 4 from th e list above requ ire that the data be availabl e as colu mns of th e matrix Σ DAT.
Page 16-2 The form list s the data in Σ D A T , s h o w s th a t co l u m n 1 i s s e l e c t ed ( t h e r e is on l y one column in the current Σ DAT).
Page 16-3 Obtaining frequency distribution s The ap plication 2. Frequ encies.. i n t h e S T A T m e n u c a n b e u s e d t o o b t a i n freque ncy distributions for a set of data. The data mus t be present in the form of a col umn vecto r st ored in va riabl e Σ DA T.
Page 16-4 This informatio n indicates that ou r data ranges f rom -9 to 9. To produce a freque ncy distribution we will use the interval (-8,8) dividing it into 8 bins of width 2 each. • Se lec t t he pr ogr am 2. Frequencies.. by using ‚Ù˜ @@ @O K@@@ .
Page 16-5 d a t a s e t s ( x , y ) , s t o r e d i n c o l u m n s o f t h e Σ DAT matri x. For thi s appl ication, you need to hav e at lea st two column s in your Σ DAT v aria ble . For exampl e, to fit a linear r elat ionship to the data shown in the tab le bel ow: xy 00 .
Page 16-6 Leve l 3 shows the form of t he equ ation. Le vel 2 sh ows the sampl e correlat ion coeffic ient, and lev el 1 sho ws th e cova riance of x-y. Fo r defini tions of these parameters see Chapter 18 in the user’s guide. For a dditiona l informat ion on the data -fit featu re of the ca lculat or see Chapte r 18 in the user’s guide.
Page 16-7 • Pr ess @@@OK@@@ to ob tain the fol lowing resu lts: Confidence intervals The app lication 6. Conf Inte r val c an be acces sed by using ‚Ù— @@@O K@ @@ . The appl ication offe rs th e foll owing options : These op tions are to b e inte rpreted a s follow s: 1.
Page 16-8 5. T-I N T: 1 µ .: Single sample confidence interval for the po pulation m ean, µ , for small sa mples wit h unknow n pop ulation vari ance . 6. T-I N T: µ1−µ2 .: Confidence interval for the difference of the popu lation means, µ 1 - µ 2 , for sm all samples w ith unkno wn po pulation v arian ces.
Page 16-9 The grap h shows the standard normal distribu tion pdf (probabil ity density functio n), th e location of the cr itica l points ± z α/ 2 , the mean val ue (23.
Page 16-10 1. Z-T est : 1 µ .: S ingle sampl e hypothesis t esting for the p opulat ion mean , µ , with known pop ulation va riance , or for large samp les with unkn own popul ation varian ce.
Page 16-11 Sele ct µ ≠ 150 . Then, press @@@OK@@@ . T he re sul t is: The n, w e re jec t H 0 : µ = 15 0 , agai nst H 1 : µ ≠ 150 . The test z value is z 0 = 5.656 854. The P -value is 1.54 × 10 -8 . T he critical values of ± z α /2 = ± 1.959 964, correspo nding to critical x range of {147.
Page 17-1 Chapter 17 Nu mbe rs in Dif fer ent B ases Besides our deci mal (base 10, digits = 0-9) number system, you can work with a binary syst em (base 2, di gits = 0,1), an o ctal system (b ase 8, digit s = 0-7), or a hexadecimal system (base 16, digits=0-9,A -F), among others.
Page 17-2 OCT(al), or BI N (ary) in the BASE menu. For exam ple, if @HE ! is selected, binary intege rs will be a hexadec imal numbers, e.g., #53, #A5B, etc. As differ ent systems are se lected, the numbers w ill be autom atica lly conver ted to the new current base.
Page W-1 Warranty hp 48gII Graphing Calcul ator; Warranty period: 12 months 1. HP warrants to you, the end-user cust omer, that HP hardw are, accesso ries and supp lie s will be fre e from d efects in materials an d workmanshi p after the date o f pur c hase, for t he period sp eci fied above.
Page W-2 7. TO THE EXTENT AL LOWED BY LOCAL LAW, THE REMED IES IN THIS WARRANTY STA TEMENT ARE YO UR SOLE AND EXCLUSIVE REMEDIES. EXCEPT A S INDICATED ABOV E, IN NO EVENT WILL HP OR ITS SUPPLIERS B E .
Page W-3 Sweden +46-85199 2065 Switzerland +41 -1-43953 58 (German) +41-22-8 278780 (French) +39-0422 -303069 (Itali an) Turkey +420-5-4 1422523 UK +44-20 7-4580161 Czech Republic +420-5 -4142252 3 So.
Page W-4 Regulatory in form ation This se ction contains infor mation that shows ho w the hp 48gII graphi ng calcu lator compl ies with regul ations in certain r egions. Any modifi cations t o the cal culator not e xpressly approved by Hewl ett -Packard cou ld void t he authority to operate the 48gII in t hese regions.
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