UCSB Math TI-85 Tutorials: Basics

Transcription

1 3 UCSB Math TI-85 Tutorials: Basis If your alulator sreen doesn t show anything, try adjusting the ontrast aording to the instrutions on page 3, or page I-3, of the alulator manual You should read the Getting Started hapter of your alulator manual through page 9, and the pages I-1 to I-14 Chapter 1, Operating the TI- 85 But if you haven t mastered those setions, don t worry about it; read this page, and then go bak to the manual Most of the stuff there and here beomes ompletely lear only after you ve read it more than one and tried it out on your alulator These tutorial sheets are only about topis of diret use in UCSB Math ourses There s a lot more to the TI-85 than is dealt with here This sheet onerns some basi points on using your alulator Other sheets are about running and editing alulator programs Your TA may provide some programs for you in disussion you will download them into your alulator, so bring the wire that ame with it Your TA will also go over the use of the program editor Names: Variables, programs, lists, and sundry other things on the TI-85 have names, whih are strings of letters and numbers with the first being a letter To type a letter, first press the blue key, then the letter key (the letters are in blue above the keys) To type a lower-ase letter, press, then, then the letter Two s in suession put the alulator in alpha lok mode, useful for typing a sequene of letters When in alpha lok mode, and swith bak and forth between upper and lower ase when in upper-ase alpha lok mode takes it out of alpha mode You an tell when the alulator is in alpha mode beause the ursor beomes a white-on-blak The key is a shortut for a lower-ase Also inluded with the letters are the equals sign and the blank spae (symbolized on the keypad by ) Equations: Let s enter a simple equation into the alulator variable Type this The atual sequene of keypresses to do this is Now press You should see the word on the sreen Next, type This time, the atual key sequene is! Pressing stores the number 3 into variable Type and press The ontents of are a formula that involves ; the value stored in is used to evaluate the formula and the result of the evaluation is displayed You should see 26, ie, 3 3 " 1 Now store 4 into, and evaluate again you ll get 63, whih is 4 3 " 1 When you type and press, you get the result of evaluating whatever is stored in But if you want to see literally what is stored in, you an use the # key (seondary funtion of the key) Type $ #, then, then The olon % The olon % ats as a ommand separator With it, you an put two or more ommands on the same line In the example above, we ould have typed the single line %& %& pressed, and gotten the same result Useful keys: $ ' always holds the last value displayed on the sreen If you omit an initial operand, eg, type ($, the alulator will insert ' ) always holds the last line YOU typed It s useful for orreting mistakes: you an get bak the last line, then edit it using the arrow keys,, and the insert key, +* (seondary funtion of ) Menus: Many keys produe menus of further operations, whih $, are displayed at the bottom of the sreen For example,,* brings up a menu One of the menu items is # ; selet it by pressing the button immediately below (the one labelled - ) The result is a seondary menu A small arrowhead at the right, indiates that still more menu items are available; press to bring these into view Let s use 0/, the one at the 1 key: it simply puts that symbol on the alulator sreen Type this line 20/ When you press, the alulator will display 3 2 One press of 4 * suppresses the seondary menu, and another removes the menu altogether When a seondary menu is visible, you an still selet an item from the primary menu by pressing before the menu key Brakets 5 and 6 signal menu hoies in these tutorials We just did $, 5,* # 6 Some speial haraters are aessed through menus; eg, use $ * to get braes 7 and 8, to get 9 Reserved names: The TI-85 has some reserved names, used for its internal funtions or for some built-in onstants Page 8-2 of the alulator manual lists the built-in onstants You an refer to these names, but you an t hange them Unfortunately, some of the names, suh as :, ;, <, =, and >, are ommon letters that you re liable to try to use for other purposes You an t, and you ll get an error message if you try Other kinds '?& of reserved names, eg, alulator funtion names like, are listed in Appendix A On page A-22 are also listed variables used by the TI-85 These are things that you an hange, but many of the alulator funtions also hange them as side effets For example, are used by the graphing funtions of the alulator $, Modes: The alulator has a variety of modes Type and you ll see a sreenful When you position the ursor on one of these and press, that mode will be set For example, if you move the ursor to the 4 on the line beginning with A& B C and press, the alulator will now display numbers using only 4 digits to the right of the deimal point (It will still ompute with all 12 digits, however) Press 4 * to get bak to the home sreen, and try it out Put the mode bak to A& B C to get all digits available 1998 John E oner doner@mathusbedu

2 erivatives erivatives: The TI-85 an generate estimates of the derivative of a funtion at a point; ie, the slope of its graph there There are several possibilities for the definition of the derivative; we ll use fe F xgih lim HJ 0 ff x K HG " ff x " HG 2H With small values of H, the quotient in this formula should be a good approximation to f E F xg Notie we don t alulate ff xg at all, but rather ff x K HG and ff x " HG The TI-85 has a feature to help in situations like this: the L BA& funtion, whih is used to evaluate an expression with a temporary hange to a variable ouring in the expression, and has the syntax L BAMN expression, variable, temporary valueo For example, L BAMN P P&-O yields 125, but doesn t hange whatever value may be in (For onveniene, you an get L BA&N as the first menu hoie after $# # ) Let s put the formula for approximating the derivative into a alulator variable: N L BA&N&PMP& ( O& L BA&N&!P&P& Ö O&QN O So if we put an expression for a funtion f into, the value of x into, and some small number into, just evaluating will give us the orresponding approximation to fe F xg 1 We ll illustrate the proedure by finding f E FSR 7G where ff xg2h osf 3xG First, we put the defining expression for the funtion f into the alulator variable, and 7 in the alulator variable : ; ' N& O T&U Then we give the value 1 and evaluate : & If you ve done everthing orretly, you should see V T &WMX U U Y whih is ffsr 7 K 1G " ffsr 7" 1G 2 If you don t, look for a mistake 2 If you see orret digits, but not all of them, read the setion on Modes in the Basis tutorial This is far from a good estimate of fe FSR 7G, beause is muh too large Type the line 1 The equation is long and hard to type orretly Wath those parentheses! You an look at $ # to hek it by doing (RCL is the seondary funtion of the! key) To edit it, type $ #, then make your hanges and finally press One you get it right, you ll never have to hange again, beause estimating other derivatives only requires hanges to,, and 2 If you get Z\[ , it s beause your alulator is in egree mode instead of Radian mode Q %M and press (The olon % ats as a ommand separator; with it, you an put two or more ommands on the same line Here, the first ommand divides by 2, and the seond evaluates ) You should see the result of re-evaluating with a value of just one-half what it was before in this ase, the new value of is 05 Press again, and this time you ll get the result obtained with H]R 25 Continue pressing until you re satisfied you have a suffiiently aurate answer This works beause when you just press without typing anything else, the alulator simply re-exeutes the previous line (ie, it exeutes the ontents of ) ) Of ourse, if you type anything else, and then want to ontinue, you ll have to retype Q %& Auray: But how do you tell when your answer is suffiiently aurate? eide how many digits you want aurately, and re-evaluate with suessively smaller stepsizes until that number of digits appears to have stabilized the ones of interest aren t hanging any more as you redue the step-size further Note that for this to work, it s best to redue the step size geometrially, eg, by dividing it by 2 or 10 eah time To illustrate, here is a table of values obtained by evaluating with a geometrially dereasing step size: H Result Change igits are underlined one it is apparent that they are stable In the last line, " 2R 589 are all stable digits however, if this result is to be reported to 4 signifiant digits, you should round it off, noting that the next digit is at least a 5, and say " 2R 590 Stability isn t quite the same thing as auray to a ertain number of digits An important advantage arises from looking at a sequene of answers generated from a geometri sequene of parameter values If you just make a single alulation, you really have little idea how many digits are orret Of ourse, the basi funtions of the alulator suh as ; ' are programmed to produe all 12 digits orretly But when you re exeuting an approximate algorithm, suh as the one represents, you don t get an indiation of auray from a single evaluation with a single parameter value On the other hand, if you have a whole series of evaluations with a geometrially hanging parameter, and you an see how some of the digits in the answer are stabilizing, you may infer that the stable digits are atually orret digits Stabilization in this manner is not a sure thing, but it is usually a good indiation of orretness 1998 John E oner doner@mathusbedu

3 H UCSB Math TI-85 Tutorials: ifferential Equations ifferential Equations and Euler s method: Here s a very simple initial value problem, onsisting of a differential equation for an unknown funtion y and a value for that funtion at a single point: ye H " R 2y^ yf 0GIH 1 This problem an be solved numerially using Euler s method Euler s method is based on the approximation yf t K`_ tgih yf tgak`_ t ye F tg (Important: we ustomarily use t for the independent variable in our differential equations problems instead of x) The numerial solution via Euler s method is approximate, and usually gets more aurate as the step size _ t is redued In numerial work, the step size is often denoted by h or H we ll use H, beause in the TI-85, : is reserved for a speial onstant Let s apply Euler s method to the problem above Suppose we d like to know the value of yf 2G, and we think a step ) size of 1 will be good enough On the TI-85, we ll use instead beause the alulator sometimes internally What we need to do ) is store 0 intoc ), 1 into, 1 into, and then repeat the steps ( b N T ) O& ) and C ( C 20 times Then C ) will have 2 in it, and will have our estimate for yf 2G For the sake of generality isolating those parts of this proess that are speifi to eah problem from those whih stay the same we will put the expression for ye in a alulator equation, instead of typing it diretly into the alulation So do these steps, pressing after eah line: V T b ) T & & ) d C What we want to do next is set up a single line that will do all the rest, so we an repeat it by pressing Type this: ) ( b ) %&C ( C% 7&CP ) 8 You an put more than one ommand into an entry by separating them with a % (The 7&CP ) 8 at the end is a trik to get the alulator to display two things on the same line you ould also use 50CP ) 6, or N&CP ) O ) Now press 20 times, until the value in C ) is 2 The orresponding is the answer If you did it orretly, you ll see 7 T X X U X d U Y U U This number is not far from the exat answer, eeif 4 R , but it isn t very lose either A better answer would be obtained with a smaller and orrespondingly more steps Try repeating the whole proess, but with T d and pressing 100 times Lists: The TI-85 operates with lists of values as well as with single values A list is entered with 7 and 8 symbols, whih are found by first typing *, and then using the two menu keys to get the brakets For example 7 PM1P&X 8, stores the indiated list of three numbers into the variable, Then, displays the list of the squares of these numbers, '?& $, displays the list of their sin s, and so on (When not all of the list fits on the sreen, the TI-85 signals this by showing three dots where the missing part is; the arrow keys are used to sroll those parts into view) You must use ommas to separate list elements when you enter them, but the TI-85 uses spaes instead when it displays a list Try entering b 7&P&1 8, then 7 P& 8 b 7&1P&- 8, and finally 7 P& 8 b 7M1P&-P&X 8 The last gives an error message You an refer to the individual elements of lists: with, as above,, N O is 2,, N O is 4, and, N&O is 6 oing, N O hanges, to 7 P&!P&X 8 Systems of equations: What if you have a system of differential equations to deal with? For example, ye H " x^ yf 0GIH 1 xe H y^ xf 0GIH 0 is a system with solution yf tggh os t^ xf tggh sin t You ould use two alulator equations named, say, and h, two variables ) and 4, and a longer ommand to repeat This would probably be ok for two equations, but the method rapidly gets umbersome as we go to problems with three or four simultaneous equations Instead, we an put everything in lists ) will be a list of two omponents, the first orresponding to y, and the seond to x Think of it as adopting this translation from mathematial symbols to alulator variables: y i j ) N O x i j ) N O To solve our example, first do this to initialize everything: 7 V ) N O P ) N O&8 T & 7 P d 8 ) d C Then type 3 : ) ( b ) %&C ( C%&?&' k C% ) Press repeatedly C and ) are displayed on separate lines, and when C is 1, we ll see the solution for yf 1GS^ xf 1G Obtaining greater auray is mainly a matter of using a smaller value for, and pressing orrespondingly more times to reah a given value for C 3 This line is too long to fit on one physial line aross the sreen, but don t worry about it, a logial line an be longer than a physial line, and the alulator just wraps around We use?&' k C beause the trik of using 7&CP ) 8 to display bothc and ) on the same line won t work if ) is a list The required spae in?&' k C is the -mode of the N&O key, the symbol, seond from the right on the bottom row 1998 John E oner doner@mathusbedu

4 Programs Running programs: To run a program, simply type its name on the main sreen, and press When a program is running, a group of pixels in the upper right orner of the sreen will flash on and off, appearing to be moving To abort a program that is taking too long (maybe it s in an endless loop?), press the key (Some programs are designed to run indefinitely, so aborting with the key is the only way to stop them) The key will also abort other operations, eg, graph plotting Typing h, and hoosing 50, 6 produes a menu of available programs Choosing one puts its name on the main sreen Then you an press to run it Program Editing: You will need to onsult the manual for a omplete treatment of this subjet; this tutorial only overs the major points and some helpful tips To edit an existing program or reate a new one, press h, There are two menu hoies 50, 6 just produes a menu of existing program names; it s only useful for getting the name of a program onto the sreen 50 * 6 puts the alulator into program edit mode, and displays the menu of exiting program names You an type a name of a program, or selet a name from the menu, and then press If there already exists a program of that name, its text will be read into to the edit buffer, but if it s an altogether new name, the buffer will be empty and you ll simply see a single line with a olon % you an then begin to type program ode Program ommands are just like the ommands you an type diretly on the main sreen After you type a line, press and the ursor will move to a new line To move from line to line for editing, or to move the ursor within a line, use the arrow keys +* The key deletes the harater the ursor is on, and allows you to insert additional haraters The 5 * Q 6 menu hoie presents a list of input-output ommands; hoosing one simply types that ommand into the program at the urrent ursor position Similarly, the 5 # 6 menu hoie gives a list of ontrol ommands 5 * ;6 inserts a blank line ahead of the line the ursor is on 50 ;6 deletes the line the ursor is on 5ml 6 inserts the last line deleted ahead of the line the ursor is on So if you want to move a line from one plae to another, you first delete it with 50 ;6, then you move the ursor to the desired plae and hoose 5ml 6 The funtions of the various input-output and ontrol ommands are disussed in the alulator manual They re not unlike those in the noqpsrst programming language A olon % is a ommand separator; with it, you an have two or more ommands on the same line (eg, so that you an fit all the ommands of interest in the tiny window) If you position the ursor at the end of a line and press, you ll delete the newline and thus ombine two lines You an separate the ommands by inserting a olon with +* +% To break a line in two, position the ursor where you want the break, and type +* It s easy to get one or more blank lines, with just a olon, at the end of your program They don t usually do any harm, but here s how to delete them: put the ursor at the end of the previous line, and press until all the extra lines are gone Making opies: Usually, if you want to hange a program your teaher supplied, it s better to edit a opy of the program instead of the original That way, if you make a mistake, you an always go bak to the original Here s how to make a opy of a program: press the h, key, then hoose * Type an altogether new name for the program, and press You ll be in Edit mode, with the # ursor on, and a line beginning with a olon Now type then type the name of the program you want to opy (or press h, to see the list of programs and pik one with a menu key) When you press, the entire text of the program you named will be opied into your new program Edit this as you please, and when you 4 *, you ll have a new program with the name you hose Remember: # in program Edit mode, you an use to retrieve the text of any other program into the one you re editing Standard istribution: The Math epartment has a standard distribution of TI-85 programs These inlude C l k, l,, l, and *& C l k, all of whih are desribed in these tutorial sheets Your teaher or TA will provide the programs, and show you how to opy programs from one alulator to another Edit the programs as you please, but edit opies, not the originals! 1998 John E oner doner@mathusbedu

5 Programming Euler s method Please read the ifferential Equations tutorial before this one Program l : This program implements Euler s method for numerially solving differential equations, in partiular initial value problems of the form: yemh ff t^ ygs^ yf agih b Here, f is a given funtion of t and y, where t is the independent variable and y the dependent The design of the l program is based on the disussion at the beginning of the ifferential Equations tutorial, where an equation was solved by setting up some initializations, and then repeatedly exeuting ) ( b ) %&C ( C% 7&CP ) 8 Program l ontains the u :? A loop ontrol onstrution, so the program itself ontrols the number of steps The ommands between ü :? A and will be exeuted until the ondition on the ü :? A line beomes false Here is the text of the l program: l vw x < x B y % )* ) %&C * C % N&C C * O&Q %&u :? A C z C % ) ( b ) %&C ( C %& % ) It s neessary to set more things initially, but you aren t in the position of pressing hundreds or thousands of times; it only takes one press to run the program The l program presumes the following are already set: { the equation for the derivative of ) in, { initial values for ) and C in )* and C *, { a final value for C in C, { a desired number of steps in The program alulates a step size so that exatly steps will over the distane from C * to C Starting from C H C *, it then proeeds to do as many Euler s method steps as neessary to make C }~C (that s steps, of ourse), and finally displays ) Example: The logisti equation arises in modeling of population growth Here is a partiular problem involving a logisti equation: yeh R 1y 1" y 1000 ^ yf 0GIH 100ƒ Find yf 25G It s often more onvenient to put initialization lines in a program of their own This makes it onvenient to re-run them, with hanges if neessary The following program sets things up for l to solve the logisti equation problem: C l k vw x < x B y %M T ) N M ) Q d d d O % d C * % - C % d d )* %M- d Run C l k, then run l If you did it orretly, you should see - X W T&Y X W U X X Y Y For other differential equations, just edit a opy of C l k appropriately you should only need to hange the definition of and some of the numbers 4 If you want more auray, just type b %& l and press one or more times Eah time, l is run with doubled, and therefore the step size ut in half Eah time gives better auray, but takes twie as long Program l an also be used to solve systems of differential equations, in the manner explained in the last setion of the ifferential Equations tutorial Example: To solve the same system as in the ifferential Equations tutorial, ye H " x^ yf 0GIH 1 xe H y^ xf 0GIH 0 with the same translation to alulator variables: y i j ) N O x i j ) N O we first do these initializations 7 V ) N O P ) N O&8 7 P d 8 )* & C - then type b %& l and press repeatedly You ll see values for ) obtained with H 10^ 20^ 40^ RmR R If you did everything orretly, the answers will onverge toward os 1 and sin 1 You may find it onvenient to reate a set-up program, similar to C l k, to handle the initializations 4 To make a opy: press h, 5m * 6, type an unused name for the opy $ #, perhaps press, and you ll see a line with just a olon o, and then type C l k, or press h, and hoose 50 C l k 6 from the menu When you press, program C l k is opied to your new program Make your hanges, and press 4 * when done 1998 John E oner doner@mathusbedu

6 Improving on Euler Program, l : Euler s method an be improved upon onsiderably at very little omputational expense The Modified Euler s Method uses Euler s method to obtain an initial estimate for yf t K`_ tg, uses that to get an initial estimate of ye F t Kˆ_ tg, and uses the average of this and the originally alulated ye F tg to update y Where Euler s Method for an equation ye H f F t^ yg is based on the equation yf t K`_ tgih yf tgak _ t ff t^ ygs^ the Modified Euler s Method is based on y guess H yf tgšk`_ t ff t^ ygs^ yf t K`_ tg H yf tgšk`_ 1Œ t 2F ff t^ yf tgsgšk ff t K _ t^ y guess GSG, Program l implements this method, l v x < x B y % )* ) %&C * C % N&C C * O&Q %&u :? A C z C % ) )!A %& ) k % ) ( b ) k ) %&C ( C % N ) k ( O&Q ) k % ) A ( b ) k ) %& % ) Program, l is interhangeable with l, but produes aurate answers more quikly For a given, it takes twie as long to run, but still saves time when you re after a ertain level of auray For a given value of,, l does about twie as many alulations as l Try this experiment: using a simple system, suh as the sin and os one given in the Programming Euler s Method tutorial, run, l with some, and l with 2Ž About the same amount of omputational labor, but how do they ompare on auray? ouble and repeat the experiment Example: The SIR model, featured in Calulus in Context and many other books, represents the spread of disease through a population S stands for the number of suseptibles, I for the infeteds, and R for the reovereds In this ase, we re onsidering a population of 50000, 2100 of whom are infeted at time 0, and 2500 already reovered (and therefore immune) SE H " R 0001SI IE H R 00001SI " IŒ 14 RE H IŒ 14 SF 0G H 45400^ I F 0GIH 2100^ RF 0GIH 2500R We an use, l (or l ) to solve this system We ll use the obvious translation of the mathematial symbols to alulator variables S išj ) N O I išj ) N O R išj ) N&O With that, we an use the following initializations to find, say, the solution for t H 10: 7 V T d d d d b ) N O b ) N O P T d d d d b ) N O b ) N OM ) N O&QM1P ) N O&Q!&1 8 7&1-1 d d P d d P - d d 8 )* d C * d C d Type b %, l (or b %& l ) and press With, l, you should see 7 & 1 YT W W W d - X The three dots indiate that part of the answer is off sreen; use the arrow keys to sroll it into view This answer was obtained with 1 / 2 day steps, orresponding to H 20 Press again to repeat the alulation with a smaller step size The equation for was long and rather lumsy to type, and perhaps hard to understand later We an use a trik to improve readability: establish the symbol translations as alulator equations Putting it all in a set-up program, we get *& C l k vw x < x B y %M ) N O % * ) N O % ) N&O %& 7 V T d d d d b b* P T d d d d b b* * Q&1P * Q!&1 8 % 7&1-1 d d P d d P - d d 8 )* % d C * % d C % d Run *& C l k, then type b %, l and press 1998 John E oner doner@mathusbedu

7 Graphing E solutions (sin os equations) Graphing Solutions of ifferential Equations: The TI-85 has a built-in module for graphing the (numerial) solutions of differential equations We ould atually write programs that would do this, but the built-in module is faster, more aurate, and more flexible than anything we ould easily reate The built-in module uses a numerial method muh better than Euler s; unfortunately, its mathematial sophistiation preludes disussion of it in this tutorial A downside is that the built-in module uses a fixed set of variable names, P P T T T P Y, so you must forego using the names natural to your problem The big advantage of this module is that you don t have to write programs, and you get both graphial and numeri solutions Example: We ll illustrate the use of the module by solving the system ye H " x^ yf 0GIH 1 xe H y^ xf 0GIH 0 The exat solution for this system is y H osf tg x H sinf tg The first step is to plae the alulator in its differential equation graphing mode From the home sreen, use the, ommand (the seondary funtion of the, key), then use the arrow keys to move the ursor to the line > ; A B x B y?& Move the ursor over to the?& item and press Use 4 * to return to the home sreen Press h As you see, the menus are different from those in ordinary graphing mode Selet the menu item 2 NMCO& If you see equations already there, use the 5m 6 menu item repeatedly to delete them Now type a new set of equations 2 & 2 0 Pressing at the end of a line, or moving the ursor below the last line, auses the alulator to begin a new equation You don t have to press after the last equation, but it doesn t hurt if you do As you have probably inferred, we replaed y by, and x by Next, you must set the initial onditions 5 * * # 6 menu item, and set up * & *& d Choose the You also need to set the ranges hoose the 5 h 6 item, and set these values: C,?M d C, B d C C k T C AM C d,?m d, B d ;!A& T?& A& T d d Now you re ready hoose the 50h 6 menu item and wath it go When it finishes, you an press # to remove the menu bar from the sreen, and h or 4 * (one) to bring it bak Try the 50 # 6 item; you an move the ursor forward from the beginning with the right arrow, and you an use the up and down arrows to move it from one graph to another (the number in the upper right of the sreen tells you whih urve it s on) C,?&, C, B govern the range of C -values for whih the alulations are made; C C k determines the C -values for whih points are alulated, and thus influenes the smoothness of the graph; B speify the range of values on the axes; in our example, C is plotted on the horizontal axis, so we used,?& H C,?& and, B H C, B ; ;A ontrol plaement of the tik marks, just as in ordinary graphing; C A& C determines the C -value at whih plotting begins; if for example, you were only interested in the part of the solution for 2 š Ĉ š 5, you ould set C A& CœH 2,,?& H 2, and, B `H 5 (if you set C AM C signifiantly greater than C,?&, be patient even though it only plots from C A& C on, it still has to ompute all the way from C,!?& );?& A governs auray of the omputations, and you shouldn t need to hange it The 5m 6 menu hoie lets you get the values of the solution for any partiular C Use the up and down arrow keys to selet the urve you want a value for lets you determine what goes on the horizontal (x) and vertial (y) axes of the graph The default is to show all the solutions versus C ; but you an hange that For example, you an plot 2 on the vertial axis and on the horizontal (remember to hange,?&,, B, et, to some appropriate values) The SIR model, desribed in the Improving on Euler tutorial, is a good seond example on whih you might like to try the differential equations module 5 The atual graph is made of straight line segments between the alulated points C C k does not affet auray, only plotting You would probably get the best looking graph by hoosing C C k to span exatly one sreen pixel: The sreen has 128 pixels aross, so there are 127 steps from left edge to right edge ChoosingC C kž Ÿ, B Z,?& 127 will result in one step per pixel 1998 John E oner doner@mathusbedu

8 Graphing E solutions (SIR model) Graphing Solutions of ifferential Equations: The TI-85 has a built-in module for graphing the (numerial) solutions of differential equations We ould atually write programs that would do this, but the built-in module is faster, more aurate, and more flexible than anything we ould easily reate The built-in module uses a numerial method muh better than Euler s; unfortunately, its mathematial sophistiation preludes disussion of it in this tutorial A downside is that the built-in module uses a fixed set of variable names, P P T T T P Y, so you must forego using the names natural to your problem The big advantage of this module is that you don t have to write programs, and you get both graphial and numeri solutions Example: We ll illustrate the use of the module by doing the SIR model (f the desription in the Improving on Euler tutorial): SE H " R 0001SI IE H R 00001SI " IŒ 14 RE H IŒ 14 SF 0G H 45400^ I F 0GIH 2100^ RF 0GIH 2500R This system has no symboli solution the typial situation The first step is to plae the alulator in its differential equation graphing mode From the home sreen, use the, ommand (the seondary funtion of the, key), then use the arrow keys to move the ursor to the line > ; A B x B y?& Move the ursor over to the?& item and press Use 4 * to return to the home sreen Press h As you see, the menus are different from 2 those in ordinary graphing mode Selet the menu item NMCO& If you see equations already there, use the 5m 6 menu item repeatedly to delete them Now type a new set of equations 2 & T d d d d b b 2 0 T d d d d b b 2 QM1 Q!&1 Pressing at the end of a line, or moving the ursor below the last line, auses the alulator to begin a new equation You don t have to press after the last equation, but it doesn t hurt if you do As you have probably inferred, we replaed S by, I by, and R by Next, you must set the initial onditions Choose the 5 * * # 6 menu item, and set up * & 1-1 d d *& d d * - d d You also need to set the ranges hoose the 5 h 6 item, and set these values: C,?M d C, B 1 d C C k T - C AM C d,?m d, B 1 d ;!A& B 1 X d d ;!A& d d d d?& A& T d d Now you re ready hoose the 50h 6 menu item and wath it go When it finishes, you an press # to remove the menu bar from the sreen, and h or 4 * (one) to bring it bak Try the 50 # 6 item; you an move the ursor forward from the beginning with the right arrow, and you an use the up and down arrows to move it from one graph to another (the number in the upper right of the sreen tells you whih urve it s on) C,?&, C, B govern the range of C -values for whih the alulations are made; C C k determines the C -values for whih points are alulated, and thus influenes the smoothness of the graph; B speify the range of values on the axes; in our example, C is plotted on the horizontal axis, so we used,?& H C,?& and, B H C, B ; ;A ontrol plaement of the tik marks, just as in ordinary graphing; C A& C determines the C -value at whih plotting begins; if for example, you were only interested in the part of the solution for 2 š Ĉ š 5, you ould set C A& CœH 2,,?& H 2, and, B `H 5 (if you set C AM C signifiantly greater than C,?&, be patient even though it only plots from C A& C on, it still has to ompute all the way from C,!?& );?& A governs auray of the omputations, and you shouldn t need to hange it The 5m 6 menu hoie lets you get the values of the solution for any partiular C Use the up and down arrow keys to selet the urve you want a value for lets you determine what goes on the horizontal (x) and vertial (y) axes of the graph The default is to show all the solutions versus C ; but you an hange that For example, you an plot 2 on the vertial axis and on the horizontal (remember to hange,?&,, B, et, to some appropriate values) The sin os equations, desribed in the ifferential Equations tutorial, is a another good example on whih you might like to try the differential equations module 6 The atual graph is made of straight line segments between the alulated points C C k does not affet auray, only plotting You would probably get the best looking graph by hoosing C C k to span exatly one sreen pixel: The sreen has 128 pixels aross, so there are 127 steps from left edge to right edge ChoosingC C kž Ÿ, B Z,?& 127 will result in one step per pixel 1998 John E oner doner@mathusbedu