Manuale d’uso / di manutenzione del prodotto hp 49g+ del fabbricante HP (Hewlett-Packard)
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hp 49g+ graphing calculator user’s manual.
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Preface You have in your hands a compact symb olic and numerical computer that will facilitate calculation and mathematical analysis of problems in a variety of disciplines, from elementary mathem atics to advanced engine ering and science subjects.
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Page TOC-1 Table of Contents Chapter 1 – Getting Started , 1-1 Basic Operations , 1-1 Batteries, 1-1 Turning the calculator on and off, 1-2 Adjusting the display contrast, 1-2 Contents of the calcul.
Page TO C-2 Creating alge braic ex press ions , 2- 4 Using th e Equation Writer (EQW) to create expressions , 2-5 Crea ting arit hmetic exp ressions, 2-5 Creating alge braic ex press ions , 2- 8 Organ.
Page TO C-3 Unit conv ersion s, 3-14 Phys ical cons tants in the cal cul ator , 3-14 Defining and u s ing functio ns , 3-16 Re fere nce , 3-18 Chapter 4 – Calculati ons wit h complex numbers , 4-1 D.
Page TOC-4 The PARTFRAC function, 5-11 The FCOEF function, 5-11 The FROOTS function, 5-12 Step-by-step operations with polynomials and fractions , 5-12 Reference , 5-13 Chapter 6 – Solution to equat.
Page TO C-5 Chapter 8 – Vectors , 8-1 Ent e rin g vec t ors , 8-1 Typi ng v ector s in th e stack, 8-1 S tori ng v ector s in to v a riables in the stac k, 8- 2 Using the matrix writer (MTRW ) to en.
Page TOC-6 Solution with the inverse matrix, 9-10 Solution by “division” of matrices, 9-10 References , 9-10 Chapter 10 – Graphics , 10-1 Graphs options in the calculator , 10-1 Plotting an expr.
Page TOC-7 Chapter 14 – Differential Equations , 14-1 The CALC/DIFF menu , 14-1 Solution to linear an d non-linear equations , 14-1 Function LDEC, 14-2 Function DESOLVE, 14-3 The variable ODETYPE, 1.
Page TOC-8 Chapter 17 – Numbers in Different Bases , 17-1 The BASE menu , 17-1 Writing non-decimal numbers , 17-1 Reference , 17-2 Chapter 18 – Using SD cards , 18-1 Storing objects in the SD card.
Page 1-1 Chapter 1 Getting started This chapter is aimed at providing basic information in the operation of your calculator. The exercises are aimed at familiarizing yourself with the basic operations and settings before actually performing a calculation.
Page 1-2 b. Insert a new CR2032 lithium battery. Make sure its positive (+) side is facing up. c. Replace the plate and push it to the original place. After installing the batteries, press [ON] to turn the power on. Warning: When the low battery icon is displayed, you need to replace the batteries as soon as possible.
Page 1-3 For details on the meaning of these specifications see Chapter 2 in the calculator’s User’s Guide. The second line shows the characters { HOME } indicating that the HOME directory is the current file directory in the calculator’s memory.
Page 1-4 and Chapter 2 and Appendix L in the User’s Guide for more information on editing) @VIEW B VIEW the contents of a variable @@ RCL @@ C ReCaLl the contents of a variable @@STO@ D STOre the contents of a variable ! PURGE E PURGE a variable CLEAR F CLEAR the display or stack These six functions form the first page of the TOOL menu.
Page 1-5 the blue ALPHA key, key (7,1) , can be combined with some of the other keys to activate the alternative functions shown in the keyboard. For example, the P key, key(4,4) , has the following s.
Page 1-6 ~p ALPHA function, to enter the upper-case letter P ~„p ALPHA-Left-Shift function, to enter the lower-case letter p ~…p ALPHA-Right-Shift function, to enter the symbol π Of the six functions associated with a key only the first four are shown in the keyboard itself.
Page 1-7 Press the !!@@OK#@ F soft menu key to return to normal display. Examples of selecting different calculator modes are shown next. Operating Mode The calculator offers two operating modes: the Algebraic mode, and the Reverse Polish Notation ( RPN ) mode.
Page 1-8 1./3.*3. ——————— /23.Q3™™™+!¸2.5` After pressing ` the calculator displays the expression: √ (3.*(5.-1/(3.*3.))/(23.^3+EXP(2.
Page 1-9 different levels are referred to as the stack levels , i.e., stack level 1, stack level 2, etc. Basically, what RPN means is that, instea d of writing an operation such as 3 + 2, in the calculator by using 3+2` we write first the operands, in the proper order, and then the operator, i.
Page 1-10 5 . 2 3 23 3 3 1 5 3 e + ⋅ − ⋅ 3` Enter 3 in level 1 5` Enter 5 in level 1, 3 moves to level 2 3` Enter 3 in level 1, 5 moves to level 2, 3 to level 3 3* Place 3 and multiply, 9 appears in level 1 Y 1/(3 × 3), last value in lev.
Page 1-11 more about reals, see Chapter 2 in thi s Guide. To illustrate this and other number formats try the following exercises: • Standard format : This mode is the most used mode as it shows numbers in the most familiar notation. Press the !!@@OK#@ soft menu key, with the Number format set to Std , to return to the calculator display.
Page 1-12 Press the !!@@OK#@ soft menu key to complete the selection: Press the !!@@OK#@ soft menu key return to the calculator display. The number now is shown as: Notice how the number is rounded, not truncated. Thus, the number 123.4567890123456, for this setting, is displayed as 123.
Page 1-13 This result, 1.23E2, is the calculator’s version of powers-of-ten notation, i.e., 1.235 × 10 2 . In this, so-called, scientific notation, the number 3 in front of the Sci number format (shown earlier) represents the number of significant figures after the decimal point.
Page 1-14 • Decimal comma vs. decimal point Decimal points in floating-point numbers can be replaced by commas, if the user is more familiar with such notation.
Page 1-15 • Grades : There are 400 grades ( 400 g ) in a complete circumference. The angle measure affects the trig functions like SIN, COS, TAN and associated functions. To change the angle measure mode, use the following procedure: • Press the H button.
Page 1-16 Selecting CAS settings CAS stands for C omputer A lgebraic S ystem. This is the mathematical core of the calculator where the symbolic mathematical operations and functions are programmed. The CAS offers a number of settings can be adjusted according to the type of operation of interest.
Page 1-17 options above). Unselected options will show no check mark in the underline preceding the option of interest (e.g., the _Numeric, _Approx, _Complex, _Verbose, _Step/Step, _Incr Pow options above). • After having selected and unselected all the options that you want in the CAS MODES input form, press the @@@OK@@@ soft menu key.
Page 1-18 The calculator display can be customized to your preference by selecting different display modes. To see the opti onal display settings use the following: • First, press the H button to activate the CALCULATOR MODES input form.
Page 1-19 ( D ) to display the DISPLAY MODES input form. The Font: field is highlighted, and the option Ft8_0:system 8 is selected. This is the default value of the display font.
Page 1-20 Selecting properties of the Stack First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , once, to get to the Edit line.
Page 1-21 Selecting properties of the equation writer (EQW) First, press the H button to activate the CALCULATOR MODES input form. Within the CALCULATOR MODES input form, press the @@DISP@ soft menu key ( D ) to display the DISPLAY MODES input form. Press the down arrow key, ˜ , three times, to get to the EQW (Equation Writer) line.
Pag e 2-1 Chapt er 2 Intro duci ng the calcu lator In this chapte r we pre sent a nu mber o f bas ic ope rations of the cal cu lato r incl uding the us e o f the Equ atio n Write r and the manipu l ation o f data o bjec ts in the cal cu lato r.
Pag e 2-2 Notice that, if your CAS is set to EX AC T (see App endix C in User’s G uide) and you enter your expression using int eger n umbers for integ er va lues, the result is a symbolic quantity, e.g., Bef ore produ cing a res ul t, yo u wil l be as ke d to change to Appro ximate mode.
Pag e 2-3 To evaluate t he expr ession we can use the EV AL function , as follo ws: If the CAS is set to Exact , yo u wi ll be as ked to approve changing the CAS setting to Appro x .
Pag e 2-4 This expression is semi-sym bolic in th e sense tha t th ere are floati ng-p oint compone nts to the re sul t, as wel l as a √ 3. Next, we switch stack locations and evalu ate u sing f u nctio n NUM: .
Pag e 2-5 Entering this express ion when the cal cul ator is s et in the R PN mode is e xactly the same as this Alge braic mo de e xerc ise . For additio nal info rmation o n edit ing alge braic expre ss ions in the cal cu lato r’s displa y or stack see C hap ter 2 in t he calculator’ s User’s G uide.
Pag e 2-6 The cu rso r is s hown as a le ft- faci ng key . The c urso r indicate s the c urrent editio n lo cation. For ex ample , fo r the cu rso r in the l ocati on indicate d above , type now: The edit ed expression looks as follows: Suppos e that yo u want to repl ace the quantity be twee n parenthes es in the denominator (i.
Pag e 2-7 The expr ession now looks as follows: Suppos e that now you want to add the frac tion 1/ 3 to this entire e xpres sio n, i.e., you want to ent er the exp ression: 3 1 ) 2 5 ( 2 5 5 2 + + ⋅.
Pag e 2-8 Creating algebraic express ions An alge braic expre ss ion i s very simil ar to an ari thmetic e xpres sio n, e xcept that Englis h and Greek le tters may be incl u ded.
Pag e 2-9 Als o, y ou can al ways c opy s peci al charac ters by us ing the CHAR S menu ( ) if yo u don’t want to memo rize the k eystro ke co mbination that produces it. A listing of commonly used keystroke co mbinations was listed i n an earlier sect ion.
Pag e 2-10 Variables Variable s are s imil ar to f ile s o n a compu ter hard dri ve. One variabl e c an store one ob ject (numer ical v alues, alg ebra ic expr essions, list s, vect ors, matrices , programs, etc).
Pag e 2-11 To unl ock the u pper-case locked keybo ard, press Try the following exercises: The c alc ulat or d isp lay will s how t he followin g (left -ha nd si de i s A lgeb ra ic mode, right-hand s ide is RPN mode ): Creating variables The simp lest way to create a var iable is by using the .
Pag e 2-12 Press to creat e the v aria ble. The va riab le is now shown in t he soft menu key labels: The follo wing a re the keyst rokes required t o enter th e remain ing variable s: A12: Q: R: z1: (Accept change to Compl ex mode if aske d).
Pag e 2-13 • RPN m od e (Use to change to R PN mode). U se the f ol lo wing keystrokes to stor e the v alue of –0.25 into va riab le α : . At this p oint, the screen will look a s follows: This e xpres sio n means that the valu e –0 .25 i s ready to be s tore d into α .
Pag e 2-14 p1: . The screen, at th is point, will loo k as fol lows: You will see six of the sev en v aria bles listed a t th e bottom of the screen: p1, z1, R, Q , A12, α . Checking variables content s The simp lest way to check a v ar iable cont ent is by pressing the soft m enu key labe l f or the variable .
Pag e 2-15 Using the right-s hift k ey foll owed by s oft menu key labe ls This approach f or vi ewing the conte nts o f a variabl e w ork s the s ame in bo th Alge braic and RPN mo des .
Pag e 2-16 Deleti ng variabl es The simplest way of deleting v ariables is by using function PURGE. This fu nctio n can be acc ess ed dire ctly by u sing the TOOLS menu ( ), or by usin g th e FI LE S m enu . Usi ng funct ion PURG E in t he st ack in Algebrai c mode Our variabl e l ist contains variabl es p1, z1, Q, R, and α .
Pag e 2-17 To dele te two variabl es s imu ltane ou sl y, s ay variabl es R and Q , firs t create a list (in RP N mode, t he element s of the list need not b e separa ted b y comm as as in Al gebraic mode ): Then, p ress use to purge the v ari ables.
Pag e 2-18 Show the ME MORY menu list and select DIRE CTORY Show the DIR ECTORY me nu l is t and sel ect ORDER activate the ORDER command There is an alterna tiv e way to access th ese menus as soft ME NU keys, b y settin g system flag 117 . (For inf ormati on o n Flags s ee Chapte rs 2 and 24 in the calcu lato r’s Use r’s Guide).
Pag e 2-19 Press th e soft menu key to set flag 117 to sof t ME NU . The screen will refl ect that change: Press twice to retu rn to no rmal calc ulato r display. Now, we’l l try to find the OR DER command using similar k eystrok es to those used above, i.
Pag e 2-20 To activate the OR DER command we pres s the ( ) soft menu key. Ref erences For additio nal i nfo rmation o n enteri ng and manipul ating ex press ions in the displa y or in th e Eq uation Wr iter see Ch apt er 2 of the calculator ’s User’s Guide .
Pag e 3-1 Chapt er 3 Calculat ions w ith real numbers This c hapter demo nstrate s the u se o f the cal cul ator f or o peratio ns and fu nctio ns rel ated to re al nu mbers . The us er sho ul d be acqu ainted w ith the keyboard to identify certain functions ava ilable in the keyboard (e.
Pag e 3-2 Alternati vely , in RPN mo de, y ou can se parate the o perands w ith a space ( ) bef o re pres sing the oper ator k ey. Ex ample s: • Parenthes es ( ) can be us ed to grou p operati ons, as we ll as to enclose a rg umen ts of func tion s.
Pag e 3-3 Example in R PN mode : • The s quare fu nctio n, SQ, is avail abl e throu gh . Exampl e in AL G mo de: Example in R PN mode : The sq uare root fun ct ion, √ , is avai labl e thro ugh the R ke y. Whe n calcu lating in the s tack in ALG mode, e nter the f unctio n befo re the argument , e.
Pag e 3-4 enter the f unctio n XROOT fo ll owed by the arguments ( y,x ), s eparated by commas, e.g ., In RPN mode , enter the argu ment y , first, then, x , and f inall y the function call , e.
Pag e 3-5 • Three trigono metric f unc tions are readil y avail able in the k eybo ard: sine ( ), cosine ( ), and tangent ( ). Arg uments of th ese fu nctio ns are angl es in ei ther degre es, radians, grade s.
Pag e 3-6 Real num ber f uncti ons i n the MTH me nu The MTH ( ) me nu i nclu de a nu mber o f mathe matical f u nctio ns mos tly applicabl e to real numbe rs. W ith the def au lt se tting of CHOOSE boxe s for system flag 117 (see Cha pter 2), the MTH men u shows the fol lowing func tion s: The f u nctions are gro upe d by the type of argu ment (1.
Pag e 3-7 For examp le, in ALG mode, th e keystroke sequence to ca lculate, say, tanh(2.5), is the follo wing: In the RPN mo de, the ke ystrok es to perfo rm this cal cul ation are the fo ll owing: The o peratio ns s how n above as su me that yo u are us ing the de fau lt s etting f or system flag 117 ( CHOOSE boxes ).
Pag e 3-8 Finally, in order to sel ect, for example, the hyperbo lic tangent (tanh) f unctio n, simp ly press . Note: To see additional opti ons i n these sof t menu s, press the key or the keystroke sequence.
Pag e 3-9 Option 1. Tools. . conta in s function s used to oper at e on unit s ( disc ussed later ). Options 3. Length.. through 17.Viscosity .. contain menu s w ith a numbe r of units fo r each o f the quantitie s des cribed. For ex ample , sel ect ing optio n 8.
Pag e 3-10 Press ing on the appropriate sof t menu ke y wil l o pen the sub- menu of uni ts f or that particul ar sel ecti on. Fo r exampl e, f or the sub-me nu, the f ol lo wing unit s are avai labl e: Pressin g th e soft menu key will take you back to the U NITS menu.
Pag e 3-11 Attach ing un its to n umb er s To attach a uni t obje ct to a nu mber , the nu mber mu st be fo ll o wed by an unders core ( , key (8,5)). T hus , a fo rce o f 5 N w ill be ente red as 5_N.
Pag e 3-12 ____________________________________________________ Prefix Name x Prefix Name x ____________________________________________________ Y yotta +24 d deci -1 Z zetta +21 c centi -2 E exa +18 .
Pag e 3-13 which sh ows as 65_(m ⋅ yd). To convert to units of the SI system, use function UBASE (find it using the command catalog, ): Note: Rec all that the ANS(1) variabl e is avail able thro u gh the key stro ke combination (as sociate d with the key).
Pag e 3-14 These oper ation s prod uce the following output : Unit conversions The U NITS menu contains a TOOLS su b-me nu, w hich provides the fo ll ow ing func tion s: CONVERT(x,y): convert unit obj.
Pag e 3-15 The soft menu keys corresponding to this CONSTA NTS LIBRA RY screen include the following functions: SI when selected, constant s va lues are shown in S I units (*) ENG L when selected, con.
Pag e 3-16 To copy the valu e o f Vm to the s tack, sel ec t the variabl e name , and pres s ! , then, pr ess . For the cal cul ator se t to the ALG, the s creen will lo ok like this: The di spl ay sho ws w hat is c all ed a tagge d val ue , . In here, Vm, is the tag of this resul t.
Pag e 3-17 and get the resu lt yo u want witho ut having to type the express ion in the right- hand side for each sep arat e value. In th e foll owing exam ple, we assume you hav e set your calcu lator to ALG mode.
Pag e 3-18 betwee n quo tes that contai n that lo cal variabl e, and show the eval uate d expression . To activate the f unctio n in ALG mode, type the name of the f unctio n fol lo wed by the a rgumen t between par entheses, e.
Pag e 4-1 Chapt er 4 Calculations w ith complex numbers This chapter s hows exampl es of calcu l ations and applic ation o f f u nctions to compl ex numbe rs. Def ini ti on s A co mple x nu mber z is written as z = x + iy , (Cart esian r epresen tati on) wher e x and y are re al nu mbers , and i is the imaginary unit define d by i 2 = - 1.
Pag e 4-2 Entering complex numbers Complex numb ers in th e calculator can be ent ered in eith er of the two Cart esian r epresent ation s, nam ely, x+iy , or (x,y) . The r esults in th e calculat or wil l be s hown i n the or dered- pair fo rmat, i.e .
Pag e 4-3 The re su lt show n above repres ents a magnitu de, 3. 7, and an angl e 0.33029…. The ang le symbol ( ∠ ) i s show n in fron t of th e an gle m eas ure. Retu rn to Cartes ian or rectangu lar coo rdinates by u sing fu nction RECT (available in the catalog, ).
Pag e 4-4 (3+5i) + (6-3i) = (9,2); (5-2i) - (3+4i) = (2,-6) (3-i)·(2-4i) = (2,-14); (5-2i)/(3+4i) = (0.28,-1.04) 1/(3+4i) = (0.12, -0.16) ; -(5-3i) = -5 + 3i The CMPLX me nus There a re two CMP LX (CoMPLeX n umber s) men us ava ilable in the ca lculator.
Page 4-5 CONJ(z): Produces the complex conjugate of z Examples of applications of these functions are shown next. Recall that, for ALG mode, the function must precede the argument, while in RPN mode, you enter the argument first, and then select the function.
Pag e 4-6 Func tions applied to c omplex number s Many o f the key board-bas ed f unc tions and MTH menu fu nctio ns def ined i n Chapter 3 for real numbers (e.g., S Q, ,LN, e x , etc.), can be applie d to com plex num be rs. Th e r esult i s an oth er com plex num be r, a s illus tra t ed i n t he fo ll o wing e xample s.
Pag e 4-7 Functio n DROITE is fo und in the co mmand catalo g ( ). Ref erence Additional inf ormatio n on co mple x nu mber ope ratio ns is pre se nted in Chapter 4 of the calculator’ s User’s G uide.
Pag e 5-1 Chapt er 5 Algebraic and arit hmetic operations An alge braic o bject, or s impl y, al gebraic , is any numbe r, variabl e name or alge braic expre ss ion that c an be ope rated u pon, manipul ated, and combine d acco rding to the ru l es o f al gebra.
Pag e 5-2 Afte r buil ding the o bject , pres s to s how it in the s tack (AL G and RPN mode s shown b elow): Simple operations w ith algebr aic objec ts Alge braic o bjects can be adde d, su btracte .
Pag e 5-3 In ALG m ode, the foll owing keystrokes will show a number of operations with the al gebraics contai ned in variabl es and ( press to recover variable menu ): The sam e results are obta ined.
Pag e 5-4 Functions in t he ALG menu The ALG (Alg ebra ic) menu is av ai lable by using the keystroke seq uence (asso ciated with the key). With system flag 117 set to CHOOSE boxes , the AL G menu s h.
Pag e 5-5 Copy the exampl es provided o nto y ou r stack by pres sing . For exampl e, for the EXP AND entr y shown a bove, p ress the s of t menu k ey to get the foll owing exa mple copi ed to th e stack (p ress to execute the comm and ): Thu s, we leave fo r the u ser to ex plo re the appl icatio ns o f the fu nctio ns in the ALG (or ALG B) menu.
Pag e 5-6 Operation s with tr ansc enden tal func tions The cal cu lato r of fe rs a nu mber o f f uncti ons that c an be us ed to replac e expres sio ns containing l ogarith mic and e xpo nential fu nctio ns ( ), as well a s tri gon om etri c func ti ons ( ).
Pag e 5-7 These function s allow to simp lify expressi ons by replac ing some ca teg ory of tri gon omet ric funct ions for an oth er one. For ex am ple, t he fun cti on A COS 2S all ows to replace the fu nction arccosine (acos(x)) wit h its exp ression i n term s of arcsine (as in(x)).
Pag e 5-8 FACTORS: SIMP2: The function s associat ed with the A RITHMETIC submenus: IN TEG ER, POLYNOMIAL, MOD ULO, and PE RMUTATION, are th e foll owing: Additional inf ormatio n o n applicati ons of the ARIT HMET IC menu functio ns are present ed in Ch apter 5 in the ca lculator’s User’s G uide.
Pag e 5-9 The variable VX A variable call ed VX exis ts in the calcu l ator’s {HOME CASDIR} dire ctory that takes, b y default, t he v alue of ‘X’. This is t he na me of the pr eferred indepe ndent variabl e f or al gebraic and calc ul u s appl icatio ns.
Pag e 5-10 Note : you cou ld get the l atter resu lt by u sing PARTFRAC: PARTFRA C(‘(X^3-2*X +2)/(X-1)’) = ‘ X^2+X-1 + 1/(X-1)’. Th e PEVA L fun ct ion The fun cti ons PE V AL ( Polyn omi al E.
Pag e 5-11 The PROPFRAC functi on The f unc tion PR OPFRAC conve rts a ratio nal f ractio n into a “proper” fractio n, i.e. , an intege r part added to a f ractio nal part, if su ch decompos itio n is poss ibl e.
Pag e 5-12 The FROOT S functi on The f u nction FR OOTS obtains the ro ots and pol es of a fracti on. As an exampl e, appl ying f unc tion FR OOTS to the resu l t produ ced abo ve, wi ll resu l t in: [1 –2 –3 –5 0 3 2 1 –5 2].
Pag e 5-13 Ref erence Additional inf ormati on, de fi nitions , and ex ample s o f al gebraic and ari thmetic opera tions ar e present ed in C hap ter 5 of the ca lculator’s User’ s Guide.
Pag e 6-1 Chapt er 6 Solu tio n to e q uati ons Associated with the key there a re tw o men us of equation -solvin g fu nctions, the Symbo lic SOLVer ( ), and the NU Me rical SoL Ver ( ). Following, we pr esent som e of the functions con tain ed in t hese menu s.
Pag e 6-2 Using t he RPN m ode, the solution is accomp lished b y enteri ng t he equat ion in the stac k, f ol lo wed by the variabl e, bef ore enteri ng fu nctio n ISOL. Right before the execution of ISOL, t he RPN st ack sh ould look as in the fig ure to the lef t.
Pag e 6-3 The following examp les show the use of function S OLV E in ALG and RPN mode s: The screen shot sh own a bove d ispla ys two solutions. I n th e first one, β 4 -5 β =125, SOLV E pr oduces no solu tions { }. In the second one, β 4 - 5 β = 6, SOLV E pr oduces fou r solu tions, shown in the last output line.
Pag e 6-4 Func ti on SOLV EVX The functi on S OLVE VX solv es an equati on for th e default CA S va ria ble containe d in the re serve d variable name VX. By de fau lt, thi s variabl e is set to ‘X’. Ex ample s, u si ng the ALG mo de w ith VX = ‘X’, are sho wn bel o w: In the firs t case SOLVEVX cou ld no t fi nd a solu tion.
Pag e 6-5 To use function ZE ROS in RPN m ode, enter first th e polynomia l expression, then the var iab le to solve for, an d th en function ZERO S. Th e following screen shots sho w the R PN stack b.
Pag e 6-6 Fol lo wing, w e pres ent appli cations of items 3. S olv e p oly. . , 5. Solve f inance , and 1. Solve equat ion.. , in th at order. Ap pendix 1-A, in the calculator’s User’s Guide, con tain s instructions on h ow to use input forms with exam ples fo r the nu merical sol ver appl icatio ns.
Pag e 6-7 Press to return to stack. Th e stack will show th e follo wing r esults in ALG mode (the same re su lt w ou ld be show n in RPN mo de): All the solu tions are complex numbers: (0.432,-0.389), (0.432,0.389), (- 0.766, 0.632), (-0.766, -0.632).
Pag e 6-8 Gener ating an alge braic e xpre ss ion for the pol ynomial You can u se the calc ul ator to gene rate an alge braic ex press ion fo r a polynomia l given the coefficients or t he root s of the polynomi al. The resulting expre ssi on wi ll be give n in terms of the def au lt CAS variabl e X .
Pag e 6-9 Financial calculati ons The calcul ations in item 5. Sol ve finance .. in t he N um eri ca l S olve r ( NUM.SLV ) are used for calcu lations of time value of money of interest in the discip line of engine ering e cono mics and othe r financ ial appl icati ons.
Pag e 6-10 Then, en ter th e SOLV E en vir onment and select Solv e equat ion… , b y using: . The correspondin g screen will be shown as: The e quatio n we s to red in variabl e EQ is alre ady l oaded i n the Eq fiel d in th e SOLVE EQUAT ION inpu t fo rm.
Pag e 6-11 Notice that f unctio n MSLV requ ires three arguments: 1. A vector contai ning th e equations, i.e., ‘[S IN(X)+Y,X+SIN(Y)=1] ’ 2. A vector contain ing the v ariab les to so lve for, i.e., ‘[X,Y]’ 3. A vector co ntaining initial val ues fo r the sol ution, i.
Pag e 6-12 by MSLV is numerical, the inf ormatio n in the up per left corner sh ows the results of the iter ativ e process used to obta in a solution. The fina l solution is X = 1.
Pag e 7-1 Chapt er 7 Ope ra tio ns with l ists List s are a type of cal cul ator’s obje ct that can be u se fu l f o r data proces sing. This chapte r prese nts ex amples of ope rations with l ists. To get starte d with the exampl es i n this Chapter , we u se the Approx imate mo de (See Chapte r 1).
Pag e 7-2 Addi ti on, s ubt ract i on, mul ti pli cat i on, di vis i on Mu lti plic ation and divi sio n of a li st by a s ingl e nu mber is distribu ted ac ros s the lis t, f or ex ample : Subtractio.
Pag e 7-3 Note : If we had ent ered th e elements in lists L4 a nd L3 a s integer s, the infinit e symbol would be sh own when ever a di vision by zero occurs.
Pag e 7-4 A B S I N V E R S E ( 1 / x ) Lists of complex n umber s You can create a complex number list, say, L5 = L1 A DD i ⋅ L2 (typ e the instru ctio n as indicate d here), as fo ll ow s: Functi ons s u ch as LN , EXP, SQ, etc., can als o be applie d to a l ist of compl ex numbers, e.
Pag e 7-5 With system flag 117 set to SO FT menus, the MTH/LIST menu shows the followin g func tion s: The operation of the MTH/LIST menu is as follo ws: ∆ LIST : Calcul ate incr ement a mong consec.
Pag e 7-6 The SEQ f unctio n The SEQ fu nction, availabl e thro ugh the co mmand catalo g ( ), takes as argu ments an e xpres sio n in terms of an index , the name of the inde x, and starting, e nding.
Pag e 8-1 Chapt er 8 Vect ors This Chapter pro vides e xample s o f e ntering and o peratin g with vect ors , both mathematical vec tors of many e le ments , as w el l as physic al ve cto rs o f 2 and 3 components.
Pag e 8-2 ( ) or s paces ( ). Notice that af ter pres sing , in either mode, the calculator shows th e vector elements sep arat ed by spaces. Stori ng ve ctors i nt o variable s i n t he s t ack Vect ors can be stored into v ar iab les.
Pag e 8-3 The key is u sed to edit th e conten ts of a selected cell in the matrix writer. The key, w hen se le cte d, wil l produ ce a ve cto r, as oppo sed to a matrix of o ne row and many co lu mns. The ← key is u sed to decrea se the wid th of the column s in the sprea dsheet.
Pag e 8-4 The key w ill add a row fu ll of ze ros at the locatio n of the selected cell of the sprea dsheet. The key will de lete th e row corr esp ond in g t o th e se lect ed c ell of the spr eadsh eet. The key will add a co lu mn fu l l o f z ero s at the loc ation of the selected cell of the sprea dsheet.
Pag e 8-5 (3) Mo ve the curs or u p two posi tions by u sing . Then press . The s eco nd row w ill dis appear. (4) Press . A row o f three ze roe s appears in the s eco nd row. (5) Press . The f irs t col umn w ill disappe ar. (6) Press . A col u mn of two z ero es appe ars in the f irs t col umn.
Pag e 8-6 Attempting to add or s ubtract ve ctors of dif fe rent le ngth produ ces an erro r messag e: Mul ti pli cati on by a sc alar, and di vis i on by a scal ar Mu lti plicati on by a s cal ar or .
Pag e 8-7 The MTH/VECTOR menu The MTH m enu ( ) contains a me nu o f f unctio ns that spe cifical ly to vec tor ob jec ts: The VE CTOR menu contains th e fol lowing functions (system flag 117 set to CHOOSE boxes ): Magnitud e The magni tude of a vecto r, as discu ss ed e arlie r, can be f o und wi th fu nctio n ABS.
Pag e 8-8 Cr oss p ro d uct Functio n CR OSS (optio n 3 in the MTH/VECTOR menu) is use d to cal cul ate the cross p rod uct of t wo 2-D v ect ors, of t wo 3-D v ector s, or of one 2-D an d one 3- D v ect or.
P a g e 9 - 1 Chapt er 9 Matrices and linear algebra This chapter s hows exampl es o f creating matric es and o peratio ns wi th matrices , incl u ding li near al gebra appl icatio ns.
Pag e 9-2 Press o nce mo re to pl ace the matrix o n the s tack. The AL G mode stack is show n next, befo re and af ter pres sing , once more: If you ha v e selected th e text book d ispla y op tion (usin g and check ing off Textbook ), the ma trix will look like the one sh own ab ove.
P a g e 9 - 3 Operation s with matrices Matrice s, l ike othe r mathematical obje cts, can be added and s ubtrac ted. They can be mu lt iplie d by a sc alar, o r amo ng themse lves . An impo rtant oper ation f or l inear al gebra appl icatio ns is the inve rse o f a matri x.
Pag e 9-4 In RPN m ode, try the following eight examp les: Multiplication There ar e different m ultiplication opera tions th at inv olve ma trices. These a re desc ribed nex t. T he ex ample s are s how n in alge braic mo de. Mu ltiplication by a scalar Some e xample s o f mu lt ipli cation o f a matrix by a s calar are sho wn bel ow .
P a g e 9 - 5 Matr ix mu ltipl ication Matrix multiplicatio n is defined by C m × n = A m × p ⋅ B p × n . Notice that matrix multiplicati on is only possib le if the number of columns in the first op eran d is equal to the n umber of rows of th e second op eran d.
Pag e 9-6 The ide ntity matr ix The identity matrix has the property that A ⋅ I = I ⋅ A = A . T o ve r i f y t h is p r o p e rt y we present the following examp les using th e matr ices stored ea rlier on.
Page 9 -7 Char acterizing a matr ix (Th e matrix NORM menu) The matrix NORM (NOR MALIZE) men u is accessed through the k eystroke sequ ence . This menu is described in detail in Chapter 1 0 o f the calc ulat or’s User’ s Guid e. S ome of thes e functi ons ar e d escrib ed next.
Page 9 -8 This system of l inear equations can be written as a matrix equ ation, A n × m ⋅ x m × 1 = b n × 1 , i f we define the following matrix and vectors: m n nm n n m m a 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 s ystem has th e same n umb er of equat i ons a s of un k nowns, an d wi ll b e ref erred t o as a s quare syste m.
Page 9 -10 Solu ti on w it h t he inverse matrix The s oluti on t o the system A ⋅ x = b , where A is a sq uare matrix is x = A -1 ⋅ b . For the ex am ple used earli er, we c an find the s olution.
Pag e 10-1 Chapt er 10 Graphics In this chapte r we intro duc e so me o f the graphi cs capabil itie s o f the calc ulat or. We will p resen t g rap hi cs of fun cti ons in Ca rt esian coor din at es and pol ar coo rdinates , parametri c plo ts, graphics o f c onics , bar pl ots , scatterpl ots , and f ast 3D pl ots.
Pag e 10-2 Plotting an expr ession of th e form y = f (x) As an ex ample , le t's pl ot the fu nctio n, ) 2 exp( 2 1 ) ( 2 x x f − = π • First, en ter the P LOT SE TUP env ironm ent by pr essing, . Make sure that the optio n Function is sel ected as the TYPE , and that ‘X’ is sel ect ed as the independe nt variabl e ( INDEP ).
Pag e 10-3 VIE W, then press to generate the V- VIEW auto maticall y. The PLOT WINDOW screen looks as f oll ows: • Plo t the graph: (wait till the calcul ator finishes the graphs) • To see labels: • To recover the f irst graphics me nu: • To trace t he curve: .
Pag e 10-4 • We will gen er at e v alue s of th e fun ct ion f(x) , d efin ed ab ov e, for v alue s of x f rom –5 to 5, in increments o f 0.5. First, we need to ens ure that the graph type is se t to in the PLOT S ETUP scr een ( , press them simultaneously, if in RPN m ode).
Pag e 10-5 • • The ke y si mply changes the fo nt in the table fro m smal l to big, and vice v ersa. Try it. • • The key, whe n press ed, pro duce s a menu with the optio ns: In , Ou t , Dec imal , I nte ger , and Trig . Tr y the following exercises: • • With the o ption In highlight ed, pre ss .
Pag e 10-6 • Pr ess , simu ltane ou sly if in RPN mo de, to access to the PLOT SETUP win dow. • Change TYPE to Fast3D. ( , find Fast3D , ). • Pr ess and typ e ‘X^2+Y^2’ . • Make su re that ‘X’ is sel ected as the Indep: and ‘Y’ as the Depnd: variable s.
Pag e 10-7 • When done , pres s . • Pr ess to re turn to the PLOT W INDOW environment. • Change the Step data to read: Step Indep: 20 Depnd: 16 • Pr ess t o see the surface p lot. Sa mple v iews: • When done , pres s . • Pr ess to retu rn to PLOT WINDOW.
Pag e 10-8 • Pr ess to leave t he ED IT env ironm ent. • Pr ess to re turn to the PLOT W INDOW environment. T hen, press , or , to retu rn to normal calcul ator display. Ref erence Additional inf ormati on o n graphics i s avail able in Chapters 12 and 2 2 in the calcu lato r’s Us er’s Guide .
Pag e 11-1 Chapt er 11 Calculus A pplications In this Chapter w e disc us s appli cations of the calcu lato r’s fu nctions to operatio ns relate d to Calcu lu s, e.
Pag e 11-2 where th e limit is to b e calculated. Funct ion li m is availabl e through the command catalog ( ) or t hr ough opt ion 2 . LIMI TS & SE RIES … of the CA LC m enu (see abov e). Function lim is ente red in AL G mode as to cal cul ate the lim it ) ( lim x f a x → .
Pag e 11-3 Anti- deriv atives an d integrals An anti-deri vative of a fu nction f (x) i s a fu nctio n F(x) su ch that f (x) = dF/dx . One way to repres ent an anti- derivative is as a indef inite i ntegral , i.e., C x F dx x f + = ) ( ) ( if and o nly if, f(x ) = dF/dx, and C = cons tant.
Pag e 11-4 Ple ase no tice that fu nctio ns SIGM AVX and SIGM A are des igned f or integrands that involve s ome s ort o f integer f unctio n like the facto rial (!) function shown above. Their result is the so-called discrete deriva tive, i.e., one de fi ned fo r intege r numbe rs o nly .
Pag e 11-5 ∞ = − ⋅ = 0 ) ( ) ( ! ) ( ) ( n n o o n x x n x f x f , where f (n) (x) represents the n-th derivative of f (x) with respect to x, f (0) (x) = f(x).
Pag e 11-6 a Tayl or s eries , and the o rder o f the serie s to be produ ced. Fu nctio n SERIES returns two ou tput ite ms a l ist with f ou r ite ms, and an ex press ion f or h = x - a, if the second argument in the function call is ‘x=a’, i.e., an expression for the incremen t h.
Pag e 12-1 Chapt er 12 Multi-variate C alculus A pplications Mul ti-variate cal cul us re fers to f unctio ns of two o r more variable s. In this Chapter we di scu ss bas ic co ncepts of mul ti- variate cal cu lu s: parti al de rivatives and mul tipl e inte grals .
Pag e 12-2 Multiple in tegrals A physical interpre tation of the dou ble integral of a fu nction f (x,y) over a regio n R o n the x- y plane is the vol u me of the s ol id bo dy co ntained u nder the su rface f( x,y) abo ve the regio n R.
Pag e 13-1 Chapt er 13 Vector A nalysis Applic ations This c hapter des cribes the u se o f f unc tions HESS, DIV, and CURL, f o r calc ul ating o perati ons of vect or anal ys is.
Pag e 13-2 Alternativ ely, use f unction DERIV a s fol lows: Diverg ence The d iv erg ence of a v ect or func tion , F (x,y,z) = f(x,y,z) i +g (x,y,z) j +h(x,y,z) k , is def ine d by taking a “do t-pro duc t” of the del o perator with the f unc tion, i.
Pag e 14-1 Chapt er 14 Diffe re ntial E qu atio ns In th is Cha pter we present exam ples of solving ordin ary d ifferential equa tions (ODE ) using cal culator functions. A dif ferential equ ation is an equation involving deriva tives of the indepen dent v ariable.
Pag e 14-2 Function LDEC The calcu l ator provide s f unc tion LDEC (Line ar Dif fe rential Equatio n Command) to find the ge neral so lu tio n to a line ar ODE of any o rder w ith co nstant coeff icients, whether it is homogen eous o r not.
Pag e 14-3 The solution is: which can be simplified to y = K 1 ⋅ e –3x + K 2 ⋅ e 5x + K 3 ⋅ e 2x + (450 ⋅ x 2 +330 ⋅ x+241)/13500. Func ti on DESOLV E The cal cu lato r pro vides f u nctio n DESOLVE (Diffe rential Equatio n SOLVEr) to solve certain types of differential equations.
Pag e 14-4 Th e variable ODETYPE You wi ll noti ce in the s of t-men u k ey label s a new variable call ed (ODE TYPE). This variable is produced with the call to the DESOL function and holds a string showing the type o f ODE used as input for DES OLVE .
Pag e 14-5 Laplac e Tran sforms The Laplace tran sform of a f unction f(t) produces a fu nction F(s) in the imag e do m a in t ha t c an be u ti l i z e d t o f i n d t h e s o l u t i o n o f a l i ne a r di f f e re n t i al eq u a t io n involving f(t) through algebraic meth ods.
Pag e 14-6 and yo u wil l notice that the CAS def aul t vari able X in the e quatio n writer s c r e e n r e p l a c e s t h e v a r i a b l e s i n t h i s d e f i n i t i o n . T h e r e f o r e , w h e n u s i n g t h e f u n c t i o n L A P y o u g e t b a c k a f un c t i o n o f X , w h i c h i s t h e L a p l a c e t r a n s f or m o f f(X).
Pag e 14-7 Using the ca lcul ator in ALG mode, first we define functio ns f(t) and g(t): Next, we move to the CASD IR sub-directory u nder HOME to chang e the value of varia ble PERIOD, e.g., (hold) Retu rn to the s ub- direct ory where you de fine d fu nctio ns f and g, and cal cu late the coefficients.
Pag e 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 thr ee 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)].
Pag e 15-1 Chapt er 15 Probability D ist ributions In thi s Chapter w e provide exampl es of applic ations of the pre- def ined probabil ity dis tributi ons in the calc ul ator. The M TH /PR OBA BILIT Y.. sub-men u - part 1 The MTH/PROBA BILITY.. sub-m enu is accessible th rough th e keystroke sequence .
Pag e 15-2 We can cal cu late combinati ons , permu tations , and f actorial s w ith functio ns COMB, PERM, and ! from the MTH/PROBABILITY.. sub-menu. The operation of those fun ct ions i s desc rib e.
Pag e 15-3 The M TH /PR OB men u - par t 2 In this se ctio n we dis cus s f ou r conti nuo us probabil ity dis tributio ns that are commonl y us ed fo r probl ems re lated to statistical infe rence: the normal distributio n, the Student’s t dis tributio n, the Chi-squ are ( χ 2 ) distribu tion, and the F-dis tributi on.
Pag e 15-4 UTPT, giv en t he pa ram eter ν an d th e va lue of t, i.e., UTPT( ν ,t) = P(T>t) = 1- P(T<t). For example, UTPT(5,2.5) = 2.7245…E -2. The C hi-sq uar e d istri butio n The Ch i-square ( χ 2 ) distribu tion has o ne parameter ν , known as th e deg rees of freedom.
Pag e 16-1 Chapt er 16 Stat istic al Applicat ions The cal cu lato r provide s the f ol l owing pre -pro grammed statistical feature s accessi ble thro ugh the k eystro ke co mbination (the key): Enter ing data Applic ations numbe r 1, 2 , and 4 f rom the lis t above requ ire that the data be availabl e as c ol umns of the matri x Σ DAT.
Pag e 16-2 The f orm l ist s the data in Σ DAT, shows that col umn 1 is selected (ther e is only one column in t he current Σ D AT). Move abou t the f orm with the arrow keys, and press the sof t me.
Pag e 16-3 Obtain ing fr eque nc y distr ibutions The appl icatio n 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 fre quenc y distri butio ns f or a s et of data. T he data mus t be pre sent in the form of a colum n v ect or st ored i n v ar iab le Σ DAT.
Pag e 16-4 This i nfo rmation indi cates that o ur data range s f rom - 9 to 9 . To produ ce a fre que ncy dist ributi on we wil l u se the interval (-8 ,8) divi ding it into 8 bins o f width 2 ea ch.
Pag e 16-5 dat a sets ( x,y), st ored i n columns of th e Σ DAT matrix. For this application, you ne ed to have at l east two co lu mns in you r Σ DAT variable . For ex ample, to f it a li near rel ations hip to the data s hown i n the table bel ow: x y 0 0.
Pag e 16-6 Level 3 shows the form of the eq uation . Level 2 shows t he sam ple corr elation coefficient, and level 1 shows the cova ria nce of x-y. For d efinition s of these para meter s see Chap ter 18 in the User’s G uide. For additio nal inf ormati on o n the data-f it f eature of the cal cu lato r see Chapter 18 in the User’ s Guide.
Pag e 16-7 • Press to obtain the f ol lo wing resul ts: Confiden ce in ter vals The appl icatio n 6. Conf Inter val can be acce ss ed by u si ng . The appl icatio n of f ers the fo ll ow ing optio ns: These options are to be inte rpreted as f ol lo ws: 1.
Pag e 16-8 4. Z-INT: p 1− p 2 .: C onfid enc e int erv al for th e differ enc e of two p ropor ti ons, p 1 -p 2 , for lar ge sam p les wi th unkn own p op ulat ion va ri an ces . 5. T-INT: 1 µ . : Single sample co nfide nce inte rval f or the popu latio n mean, µ , for sma ll sa mp les wi th unkn own pop ulat ion v ar ia nce .
Pag e 16-9 The graph s hows the s tandard normal distribu tio n pdf (pro babil ity dens ity fu nction), the l ocatio n of the critical po ints ± z α/2 , the mean val ue (23.
Pag e 16-10 1. Z-Test: 1 µ .: S ingle samp le hypothesis testing for the population mean , µ , wit h know n p opula tion v ari an ce, or for la rg e sam ples w ith unkn own popu latio n variance .
Pag e 16-11 Sel e ct µ ≠ 150 . Then, pr ess . The resul t is: Then, we rej ect H 0 : µ = 150 , agains t H 1 : µ ≠ 150 . The test z value is z 0 = 5.656854. The P-v alue is 1.54 × 10 -8 . The critical values of ± z α /2 = ± 1.959964, corresponding to critical x rang e of {147.
Pag e 17-1 Chapt er 17 Num be r s in Di ffer e nt Base s Bes ides ou r decimal (base 10, di gits = 0 -9) number sys tem, yo u can work with a binary s yst em (base 2, digits = 0, 1), an o ctal sys tem (base 8, digi ts = 0-7 ), o r a hexade cimal sys tem (bas e 16 , digits =0- 9,A- F), among o thers .
Pag e 17-2 base to be u sed f or bi nary intege rs, cho os e eithe r HEX(ade cimal) , DEC(imal), OCT(al ), or B IN(ary) in the B ASE menu. For example, if is selected, binary in tegers will be a hexadecima l numbers, e.g., #53, #A5B, etc. A s different systems are selected, the n umbers will be a utomatica lly converted to the new cu rrent bas e.
Page 18-1 Chapter 18 Using SD cards The calculator provides a memory card port where you can insert an SD flash card for backing up calculator objects, or for downloading objects from other sources. The SD card in the calculator will appear as port number 3.
Page 18-2 Enter object, type the name of the stored object using port 3 (e.g., :3:VAR1 ), press K . Recalling an object from the SD card To recall an object from the SD card onto the screen, use function RCL, as follows: • In algebraic mode: Press „© , type the name of the stored object using port 3 (e.
Page W-1 Limited Warranty hp 49g+ graphing calculator; Warranty period: 12 months 1. HP warrants to you, the end- user customer, that HP hardware, accessories and supplies will be fr ee from defects in materials and workmanship after the date of pu rchase, for the period specified ab ov e .
Page W-2 7. TO THE EXTENT ALLOWED BY LOCAL LAW, THE REMEDIES IN THIS WARRANTY STATEMENT ARE YOUR SOLE AND EXCLUSIVE REMEDIES. EXCEPT AS INDICATED ABOVE, IN NO EVENT WILL HP O R I T S S U P P L I E R S.
Page W-3 Switzerland +41-1-4395358 (German) +41-22-8278780 (Fr ench) +39-0422-303069 (Ital ian) Turkey +420-5-414 22523 UK +44-207-4580161 Czech Republic +420-5-4142 2523 South Africa +27-11-541 9573 .
Page W-4 R R e e g g u u l l a a t t o o r r y y i i n n f f o o r r m m a a t t i i o o n n This section contains info r m ation that s h ows how the hp 49g+ graphin g calculator complies with regulations in certain regions.
Un punto importante, dopo l’acquisto del dispositivo (o anche prima di acquisto) è quello di leggere il manuale. Dobbiamo farlo per diversi motivi semplici:
Se non hai ancora comprato il HP (Hewlett-Packard) hp 49g+ è un buon momento per familiarizzare con i dati di base del prodotto. Prime consultare le pagine iniziali del manuale d’uso, che si trova al di sopra. Dovresti trovare lì i dati tecnici più importanti del HP (Hewlett-Packard) hp 49g+ - in questo modo è possibile verificare se l’apparecchio soddisfa le tue esigenze. Esplorando le pagine segenti del manuali d’uso HP (Hewlett-Packard) hp 49g+ imparerai tutte le caratteristiche del prodotto e le informazioni sul suo funzionamento. Le informazioni sul HP (Hewlett-Packard) hp 49g+ ti aiuteranno sicuramente a prendere una decisione relativa all’acquisto.
In una situazione in cui hai già il HP (Hewlett-Packard) hp 49g+, ma non hai ancora letto il manuale d’uso, dovresti farlo per le ragioni sopra descritte. Saprai quindi se hai correttamente usato le funzioni disponibili, e se hai commesso errori che possono ridurre la durata di vita del HP (Hewlett-Packard) hp 49g+.
Tuttavia, uno dei ruoli più importanti per l’utente svolti dal manuale d’uso è quello di aiutare a risolvere i problemi con il HP (Hewlett-Packard) hp 49g+. Quasi sempre, ci troverai Troubleshooting, cioè i guasti più frequenti e malfunzionamenti del dispositivo HP (Hewlett-Packard) hp 49g+ insieme con le istruzioni su come risolverli. Anche se non si riesci a risolvere il problema, il manuale d’uso ti mostrerà il percorso di ulteriori procedimenti – il contatto con il centro servizio clienti o il servizio più vicino.
