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THE DERIVE - NEWSLETTER #25 USER GROUP

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1. the intersection of the tangent and the y axis is the equation of the Tangent C 22x 4 t 1 Hp et us plot it C clrgraph the settings for the Graph screen C 212 xmi n 12 xmax 23 ymi n 7 y max C graph t1 Now let s approach from the left C f x f x h h froml x C expand froml 0 h 0 C limt ans 1 h 0 the derivatives are different gt not differentiable Wwe obtain another tangent C 2xt4ot 2 graph t2 Could you imagine the curve s shape C graph f x lt has a Cusp Do you have an idea how the derivative looks like Zeros of the derivative What s the value of the derivative for x 0 Let calculate the derivative and store under dgf x C T Ex x dqf x C graph dqf x C daf X x lt Tabelle der Sekantensteigungen C dqf x x gt 0 Feste Stelle xO 0 Can you explain the last results io iL i a ee 79 7 110 Dont forget house keeping mi bl 2511 si1 sl 5811 3812 i C delvar dgf fromr froml dqf t1 t2 13 1 35 1 2 1 1 1 l C setmode Split Screen Full ml 71 1 71 291 71 591 71 751 71 881 1 381 Back with Diamond HOME Zu viele Werte alter Taste HELEITUM D AUTO FUNC ri used this teaching unit to repeat the differentiability of a function using a View Screen The two prod ucts the script and the program supported each another in an excellent way Josef
2. 5 In 3D plot windows it is no longer required that an expression be plotted before issuing the Write to Acrospin command 6 If ACROSPIN EXE is in the DfW directory as it should be then it will automatically be found by the Write To Acrospin command 7 When multiple expressions are highlighted and copied to the clipboard by pressing Ctrl C the highlighted rectangle is no longer corrupted 8 In 2D and 3D plot windows allow pi to be entered in upper or lower case 9 Allow quoted strings and underscores to be used in function and variable names given in the De clare Function Definition Declare Variable Value and Declare Variable Domain commands 10 Ifa vector is highlighted and the solve icon is clicked on the Solve System instead of Solve AI gebraically command is called this was a serious deficiency in earlier versions of DfW p 10 DERIVE USER FORUM D N L 25 The following enhancements are in both DfW and DfD version 4 03 11 A new built in LOAD file name function makes it possible to have DMO files automatically load utility files 12 The new function DIF NUMERIC y x x0 h n defined in NUMERIC MTH numerically calcu lates the nth derivative of y x at x xO using a centered finite difference with a step of size h 13 The DSOLVEI function in ODEI MTH is now better able to solve separable ODEs 14 Some problems with the DIRECTION FIELD function defined in ODE APPR MTH have been corrected 15 Recognize
3. ld Get sz 5 5 t 15 For obtdining FLA 31 d 3 Z s 15 dt 25 lin ta il 215 In the case of continuous random variables it is best to use a probability density function di rectly The use of a direct integration in this way stresses the concept that a probability is represented by the area under the graph of a probability density function whether or not the function concerned is integrable analytically or not Expressions 18 20 below illustrate the calculation of P O lt X lt 2 where X 7 2 e 3 me ERF l 10 18 Fix ee See 0 3 AT 19 simplified above and the right side approximated be low 20 Fix dx 0 1508549639 CONCLUSION The point of using a computer algebra system should really be that any problem can be tack led This context has highlighted a class of integrals and series which unfortunately cannot be tackled successfully with DERIVE However given certain general results DERIVE can be used to compute means variances moments probabilities etc without algebraic difficul ties Furthermore all of these can involve some symbolic parameters so that functional rela tions can be explored This is not possible with a package which can only manipulate num bers The shortcomings of DERIVE can be turned to advantage if the student is constrained to study relevant techniques and concepts and hence to do some parts of the computation by
4. Given 1 SSS 2 1 te VO roa uu ker 2 2 E ae 2 N de E ro m e 3 SENS 4 giri fdr DERIVE evaluates g r INT f r and g R2 g R1 but doesn t simplify INT f r R1 R2 Best regards Leon Magiera A van der Meer Twente Netherlands A W J vanderMeer math utwente nl Consider the following DERIVE session 1 1 Fix tz dx a b SIN x Antiderrvative calculated by hand x a TAN b 2 2 Gx z AT AM amp Ei E E Ka b Ka b d al 3 6x dx b SINix a 4 Gimi Glo 0 Conclusion G is not the antiderivative on the whole real axis In this case G has discontinuities on 0 2 pi For this example Derive has a trick to remove these discontinuities by adding a step function Calculated by DERIVE Simp 1 2b COS x 2 AT AN 2 2B SINOO Ca b fifa bi Ka bja Ka b Which gives the correct results 5 Fix Zur 6 PLZT FED Nee eon once acd EE l eT ES dg z a b SINCx2 ia ob dac B g Ju Ha Fidem FCD 1 at Biss All screenshots are from DERIVE 6 10 D N L 25 DERIVE USER FORUM p 13 So DERIVE is cleverer then you expected In your example DERIVE has found that there is a possibility that certain combinations of values of the parameters may result in discontinuities in the antiderivative g
5. MEt ne ST 8 Failed Calculate mean and variance direct ly Peter Mitic Probability Distributions 2 m 9 mean iz x F x2 dx 10 mean i 5 In z i Ell variance z x Fix dx mean D 12 variance 10 The situation with respect to parameters is not at all consistent In the case of a Normal 0 1 2 X 2 random variable with probability density function N x DERIVE can immediately A simplify the expression for the moment generating function M t INT N x 4e x t x inf inf to obtain the result e t 2 2 with no apparent need to restrict the range of t This result is required for a proof of the Central Limit Theorem See the next DNL Josef We have already pointed out Mitic and Thomas 1994 that problems arise when the com puter fails to calculate a result and that consequently problems presented as student exer cises must be chosen so that they are do able This is the case here In order to teach mo ment generating functions one has to choose suitable probability density functions We have found that probability density functions of the form g x x x gt 0 with n a positive integer work provided that n is explicit General results may be conjectured by constructing a VECTOR of particular examples In the context of the class of probability density functions x e fun N VECTORCINT x m 4e6 x nlb seo O0 ant dX 0 10 xe R neZ the
6. VECTOR CITERATES 2 4 s S 1 15 3 2 ee k 16 3 3 1058285412302491481 3 1326286132812381971 3 1393502030468672071 3 1410319508905096377 3 1414524722854620752 3 1415576079118576454 3 141583892148318406 3 141590463228050057 3 1415921059992 13602 3 1415925166921566904 3 1415926193653861406 3 1415926450336881605 3 1415926514508704393 3 1415926530552116196 3 1415926534558914124 17 N 18 3 1415926535897932384 p 40 Neil Bibby A day in the life D N L 25 Example 2 Population growth and exponential modelling British Year 11 Grade 10 This work formed the basis of some GCSE coursework We started with the popula tion data for the major Italian cities as given in the following table Table 6 Population of the main Italian citi in thousands 1800 1980 t oo 1 185 71 1860 1 1870 1 1 88o 1 1900 71 t9ito t 910 1 1930 71 1940 1 1950 1 i960 t 1970 1 1980 1 e PET THER Ar GNU al A em S Genoa 91 110 129 130 180 13 272 316 608 635 648 784 8r2 760 Turin 78 135 178 208 254 336 427 02 97 619 71 1026 1178 rio Milan 135 242 242 161 j z 493 79 836 991 1116 i160 1583 1724 1635 Rome 163 174 184 144 300 463 42 692 1008 0156 1652 1 188 i800 2831 Naples 417 449 417 449 494 64 723 22 8339 866 rori 1183 12333 n Palermo 139 180 186 219 14 310 342 394 390 412 491 88 6 1 700 Aaaa I EISE DM ME E oo e e e RR P Nete the continued growth of ci
7. xi x4 fo 4 0 0 3 sun 3 3 Nat rlich ist die Bestimmung von a nicht nur durch Plotexperimente sondern auch rechnerisch leicht m glich Man unterstellt naheliegend x X cos 31 bzw x X cos 3t als partikul re L sung und berechnet f r X 0 f r alle 7 X gt a 4 3 Leo Klingen Schwingungen Oscillations OMMAND Center Delete Help Move Options Plot Quit Range Scale Transfer Window aXes Zoom Enter option ross x B8 6393 y 2 0625 Scale x 1 5707 yi2 Derive ZD plot Figur 4 F r die notwendigen zweiten Ableitungen und algebraischen Umformungen hilft DERIVE Rechen fehler zu vermeiden brigens ist auch ein Realexperiment ohne sonderlichen Aufwand f r Liebhaber oder im Rahmen eines Sch lerprojektes ohne weiteres m glich Ein elektrisches Analogon besteht in einem Antennen Schwingkreis mit mehreren von Sendern fremderregten Schwingungen darunter vielleicht eine besonders starke eines Lokalsenders Wenn diese unerw nscht ist kann man sie heraussaugen durch Ankopplung eines abgestimmten zweiten Schwingungskreises mit eventuell verst rkter Frequenz des Lokalsenders sodass das Frequenzge misch des Restes ungest rt in der blichen Weise weiterverarbeitet werden kann Man kann versu chen auch diese Situation zu simulieren An electronic analogon is an antenna oscillation circuit with some undesirable frequen cy ies local station You could suck it out by connecting a second circ
8. 0 005 given function casi Zx sin 3x DNL Plot 12 and obtain the interpolating polynomial 13 results in the nine given points 14 would plot the interpolating polynomial in form of a thick line point size medium or large and points con nected KLAR amp R amp W TURTLE GRAPHICS amp p 15 AN IMPLEMENTATION OF TURTLE GRAPHICS IN DERIVE 3 1 2 r 3 Josef Lechner Eugenio Roanes Lozano Eugenio Roanes Macias Johann Wiesenbauer Bundesgymnasium Amstetten Austria Dept Algebra Univ Complutense de Madrid Spain Institut f r Algebra Tech Univ Wien Austria Introduction If you have worked previously with any LOGO dialect you will know about the nice possibilities of the Turtle Graphics We had already developed implementations for Turbo Pascal Ro4 and Maple V Ro1 Ro2 Ro3 Now we have developed another one for DERIVE 3 that is presented below Two Remarks To follow the article a certain knowledge of the possibilities of Derive and Turtle Graphics is sup posed Ab is a very nice introduction to the later and A dS is an impressive collection of ideas for its use Note that in some early versions of Derive 3 evaluation is made in such a way that this code does not work The code has been checked in versions 3 13 Classic and XM and Derive for Windows Beta v 0 1 Getting Started We should begin by setting the graphics in Connected Mode and the Color in the non Auto Mode Then
9. 2 a a 7 User and expects a 10 but he obtains a a 7 and simplifying again and again 3 a v 7 Simp 2 4 a 14 Simp 3 m s c 21 Simp 4 So what can be done Let s try with the assignment NEW X u x u with u being an expression in x That works because expressions in function parameters are evaluated before passing That is an interesting alternative to the ITERATES command 7 NEWA u a u User 8 a 1 User i 2 9 NEWA fa User 2 a 10 1 Simp 9 11 a 1 5 User Simp User 12 1 41666 Simp 9 13 a 1 41666 User Simp User 14 1 41421 Simp 9 154 a 1 421471 User Simp User D N L 25 Comments on the TURTLE Graphics p 27 Obviously the variable has changed its value Let s try once more Set back the value for a Watch the annotations please lw seh User 108 1 5 Simp 9 19 1 41666 Simp 9 202 LATAAT Simp 9 21 1 41421 Simp 9 23 NEWX u x u NEWY u y u NEWA u a u User 24 FWD l w NEWX x l COS a w NEWY y l SIN a w O NEWA a w It seems to work 25 home x 0 y a 0 User 26 quadrat home FWD 2 0 FWD 2 90 FWD 2 90 FWD 2 90 User D 0 2 Q 218 Zz 2 Simp 26 Or lt 2 0 0 The big disillusion 28 home 0 0 User Simp User 2 2 0 29 VECTOR EWD 2Z 0 X Ly 4 User Simp User 2 0 Z iW 30 InputMode
10. 24 hours which is required in my work and wish to see where I am up to I can t because interrupting loses all the information Similary having continous output e g to look at convergence as the function runs and producing dynamic plots of results that change with each new calculation is not possible I am aware that a great deal of time and effort has been invested in the Windows version and as fan and power user of Derive I would ask plead that a programming window be added in the next ver sion I think the programming window in the TI 92 is a good model on which to base a DERIVE pro gramming window I am sure but again I don t know that all the programming functions such as FOR WHILE OUTPUT IF THEN PLOT are already there in muLisp but the DERIVE user can t get at them or utilise them I know that we users are always asking for additions and improvements and after such a mammoth effort as DfW more wishes must seem as a pain in the butt but that 1s the nature of the beast To end a little question is a UNIX version of DERIVE a realistic possibility Please don t swear All my hopes D N L 25 DERIVE USER FORUM p 9 Humberto M Pereira Silva Rio Tinto Portugal Dear friend Reading newsletter 23 n which V Hermans from Holland blamed for the lack of any incentive to pupils of High School to work with electronic machines in the classroom will it be Texas Casio HP or else In Portugal there are some new idea
11. REPEAT command We couldn t maintain the standard syntax but the idea 1s similar REPEATCk IF k 1 to rep APPEND REPEAT k_ 1 to rep I tried successfully REPEATCk zVECTOR to rep i k Finally the variables are initialised xcor 0 ycor 0 heading 0 About REPEAT What has to be repeated has to be stored in the to_rep variable list and the only argument of REPEAT is how many times has to be repeated what is stored in that list Therefore to execute repeatedly a list of commands just type to_rep List of commands REPEAT number and Plot Observe that the way it is defined a REPEAT can not be nested inside another REPEAT Defining a function REPEAT2 depending on a variable to_rep2 does not always work p 18 X L amp R amp R amp W TURTLE GRAPHICS D N L 25 Some Simple Examples Remember 1 To Simplify home and to choose Delete All in the Plot menu in between executing the exam ples 11 Simplify and Plot each new expression which is not an assignation Example 1 Draw an L 2 home 3 xcor t 0 ycor i 0 heading 0 4 LLT 90 FOC2 RT SO FD 32 Example 2 Draw a flag and return to the origin using only the Cartesian commands 5 xcor 0 yeor cs 0 heading 0 5 SETY 2 SETAC2 SETY L SETACO SETY O Example 3 Draw a regular heptagon 7 xcor 0 ycor 0 heading 0 260 8 tn rep E 7 9 REPEAT 7 Example
12. See section 5 2 4 in the DERIVE manual for more details and hints D N L 25 DERIVE USER FORUM Problems Duff s awk can run out of memory with large files more than a 10x10 grid say depending on how much low memory you have available Classic DERIVE does not leave enough available memory to oad mawk through Options Execute use Duff s awk or run ACRO BAT outside DERIVE DERIVE XM is OK Other awk interpreters may or may not have enough memory to run within DERIVE gawk does not The old large memory model compilation of mawk 1 0 bmawk does slightly better that the new version but does not seem to be currently available from the net Additional notes You can save a few keystrokes by faking ACROSPIN EXE through the following C miniprogram include lt stdlib h gt void main system acro bat Compile as ACROSPIN EXE and place in your DERIVE directory Just use acro as the save file name the 3dv data file can be used for producing high quality hardcopy on a variety of devices Look into 3DVKIT1 ZIP found whereever you found 3DV25 ZIP The memory problems could be solved if somebody with access to an awk compiler would post a compiled version of acro awk Or write it in C or whatever but be careful to allow for multiple superimposed plots in the file Happy spinning DNL followed Oscar Garcia s hints and tips And it worked Fortunately some times ago R Schorn had sent 3dv exe but a
13. and then plot 39 p 20 L amp R amp R amp W TURTLE GRAPHICS D N L 25 depth 2 and 4 40 scor 0 ycor ss 0 heading 0 4 RTSO 42 0 vcor 0 43 KOCH l 2 44 xcor c 0 vcar 0 heading 0 45 RT S0 46 0 vcar 0 47 KOCH 1 4 KOCH can be used to draw stars snow flakes 48 xcor 0 ycor i 0 heading 0 49 to rep KOCHC l 4 RT 120 amp 50 REPEAT 3 In DERIVE 6 we can write a short pro gram Josef gt kochstar len n star n_ t a Prog star to rep pisses 4 Cc 51 Loop 1 5 Le fh 3 RETURN star star APPEND star to rep a n 1 52 xcor 0 ycor 0 heading 0 Hos 52 SETFOSC1 1 54 to_rep KOCH L 3 RT 120 55 kochstar 33 os i as i D N L 25 Comments on the TURTLE Graphics p 21 This is the original DOS DERIVE screen from 1997 Conclusions Many geometric designs and drawings can be produced in a more convenient way using Turtle Graphics instead of Cartesian coordinates Therefore we think that this can be another useful tool for DERIVErs References A dS H Abelson A diSessa Turtle Geometry The Computer as a Medium for exploring Mathe matics M I T 1981 Ab H Abelson Apple Logo Byte Books McGraw Hill 1992 Rol E Roanes M E Roanes L An Implementation of Turtle Graphics in Maple V 2 CAN Nieuwsbrief 12 March 1994 pages 43 48 Ro2 E Roanes M E
14. auch leise Klagen die Themen und die zugeh rigen L sungen sind hin und wieder von einem anderen Stern sehr weit weg von meinen t glichen Problemchen mit der Mathema tik D Blum Dazu mochte ich bemerken da ja Gott sei Dank nicht nur Lehrer zu unseren Mitgliedern z hlen und da es auch den Lehrern ges tattet sein sollte so dann und wann nach den Sternen zu greifen Ich m chte aber allen die sich auch betrof fen f hlen ermuntern ber Ihre Problemchen zu berichten Da kann keine Zuschrift und kein Beitrag zu wenig aufregend oder zu einfach sein Sie alle gestal ten den DNL Und glauben Sie mir bitte auch ich habe meine Mathe Problemchen mit den 15 19j hrigen und das oft nicht zu knapp Ich habe mich aber bem ht gerade diesen DNL besonders f r den Lehreralltag nutzbar zu machen Im n chsten DNL werden wir uns auch der Geometrie widmen Abbildungen von Objekten im R mit ver deckten Kanten Fraktale usw Auch eine ganze Schachtel voll mit TI Fragen und Antworten steht f r Sie bereit In vielen F llen k nnen DERIVE Beitr ge Anre gungen f r den Einsatz mit dem TI geben und umge kehrt Wir zeigen das auch in diesem Heft Ich hoffe dass Sie bereits ab dem n chsten DNL DERIVE und Tl Dateien von einer eigenen Homepa ge abholen k nnen Bis zum n chsten Mal Josef LETTER OF THE EDITOR pl Dear DERIVE and TI friends Certainly you will be astonished at the photo gr
15. definition Have you found an easier way is there Al or Dave of producing a vector of undefined variables without hav ing to type them all in Something like VARIABLES 30 a which would simplify to a b c d X y Z aa bb cc dd p 14 DERIVE USER FORUM D N L 25 You will also notice that on line 9 the LIM function variables vector needs an EXPAND command it doesn t work without t Any ideas why If nobody has come up with a function that will do this before don t remember seeing one in any previous DNL you might like to use it a forth coming DNL 1 det vars sache cresce ce qq decine ence 2 FUNCTION x sx xvalues r l 3 FOLYNOMITAL_AUALd IMS vecron der var X le sim vecaa r dam r sd POLYNOMIAL Cx is POLYNOMIAL AUXCDIMENSION xvalues poc 5 POLYNOMIAL AUX x dam vecron der var X EC DAS Hin vecrona r dam r 6 POLYNOMIAL x i POLYNOMIAL AUX x DIMENSION xvalues vars vect ix VECTOR defvars r 1 DIMENMSION xvalues Br r co eff SOLUTIONMSCVECTORCFUNCTION xvalues 3 POLYNOMIAL xvalues 3 r 1 DIMENSION xvalues 3 wars vect 5 r r r interpolate yam POLYNOMIAL x2 9 EXPANDivars_vectijsco_eff Example Interpolate cost sin 3x by a polynomial at x 0 1 2 3 4 5 5 7 8 10 FUNCTION x23 sx COSC2 x SINGS sx 11 xvalues 0 1 2 3 4 5 5 7 amp 9 12 interpolate 13 TABLECFUNCTION x2 x xvalues 14 TABLECinterpolate x 0 9
16. expression can be used to check that the areas under the graphs of the first 11 members of this class are all 1 We would prefer to see the general result obtained using MATHEMATICA 2 2 which does not simplify the result to 1 7 n X I 1 E e n n n 0 into which we can substitute n 0 1 2 The need to consider particular examples in this way is regrettable but it is unavoidable with DERIVE Peter Mitic Probability Distributions 2 CALCULATING PROBABILITIES Calculating probabilities presents less of a problem technically and also reinforces concepts in a way that a statistical package cannot A frequent task with discrete distributions is to use the probability generating function G r to calculate probabilities by isolating a coefficient or coefficients of t in a series expansion of G t This often causes computational problems which overshadow the point of the exercise We illustrate 14 16 below how this task is very simple using DERIVE The geometric distribution Geom 1 6 has the probability generating function given by 14 The coefficient of t in G t is obtained by computing d G t dt t 0 No formal expansion of G t is needed so the student can concentrate on the precise mean ing of terms in a probability generating function If the substitution t O is programmed as below it is necessary to use the limit construct shown despite its technical incorrectness t
17. h F x 7 ITERATE A s x h s F x 10 Are you surprised that we get 8 F x 10 h 10 F x 9 h 45 F x 8 h 120 F x 7 h 210 F x 6 h 252 FCx 5 h 210 F x 4 h 120 F x 3 h 45 F x 2 h 10 F x h F x As you can see we can use now the A Josef D N L 25 DERIVE USER FORUM p 7 Mr Schmidt sent a wonderful MATHEMATICA graphic of a snailhouse together with the functions to produce it I tried with DERIVE and brought the snail into life using ACD and ACROSPIN Many thanks Mr Schmidt for your idea The snail is a new object in my collec tion of animated graphs The 1997 snail can be found on page 36 Notation Decimal NotationDigits 3 n p 2 pil0 r 0 0972 3 0 125 10 20 o z0 0 04s f6 eon 20 p 9 BE i ESL 1F Foon gt 1 1 15 2 2 HAUS 0 VECTOR VECTOR COS Q9 pil0 r 1 1 COS 0 SIN pil0 r 1 1 COS 0 r SIN 0 z0 o 0 57 0 0 2 n pil0 ZACK a Z VECTOR VECTOR COS pil0 r 1 1 az COS z pil0 SIN pil0 r 1 1 ag C0S 72 0110 frac SINCe O11 20 6 D 57 a L 12 0 2 HAUS 8 VECTOR ZACK a U u 2 3 The respective Graphs produced in the 3D Plot Window of DERIVE 6 p 8 DERIVE USER FORUM D N L 25 Terence Etchells Liverpool UK t a etchells livjm ac uk DERIVE The Next Generation You may or may not be aware that I am studying for a PhD in Neural N
18. hand REFERENCES Etchells T 1992 Investigating Probability Distributions with DERIVE In Teach ing Mathematics with DERIVE Proceedings of the International School on the Didactics of Computer Algebra Krems Chartwell Bratt Mitic P and Thomas P 1994 Limitations and Pitfalls of Computer Algebra In Com puters and Education 22 4 Pergamon Peter has forwarded two more papers of interest Exploiting New Features in DERIVE3 Multiple Decisions and Whole Structure Programming and The Normal Distribution Two Proofs and a Simulation Josef Kurt Schmidt s MATHEMATICA snail from 1997 D N L 25 The 25 DERIVE Newsletter p 37 A Toast Celebrating the 25th Issue of the DERIVE Newsletter Josef and Noor we in the DERIVE community are all so very grateful for your inspi ration to initiate the independent DERIVE User Group and its bulletin and for your talent and dedication in making it work so well Many of us are surely thinking Can it really be 25 issues How quickly time passes when we are having fun That is surely the secret of teaching mathematics make it fun I have taken the occasion to review the previous issues and I am awed by the total volume of great unique and still relevant ideas in those issues They are collector s treasures and I cherish each and every one We all look forward to each issue with great anticipation It is astounding to learn such ingenious and effective ways that others have u
19. mentioned that the DERIVIAN Turtle is one result of the many fruitful talks at the DERIVE Conference 1996 in Bonn The most important are the breaks After this conference began a very busy exchange of ideas via e mail Following the TURTLE MTH file you might ask yourself what is that there are two assigments in one expression So did I Fortunately Josef Lechner is very conscientious and he collects many things concerning Computer Algebra in general and concerning DERIVE in special When I asked Josef to give an additional explanation for this strange expression he took two sheets of paper from his huge desk and asked me Will that be enough summarized my e mails with Johann Wiesenbauer Many thanks Josef think that will do it So find here instead of his TITBITS J Wiesenbauers double assignation Comment of 2011 The following problems are from 1997 Later versions of DERIVE don t behave the same recommend experimenting with a expression a applying the various possibilities of Simplify from the menu and the edit bar as well Is it able to implement TURTLE graphics in DERIVE Not completely The reason is that DERIVE does not support self assignments In DERIVE we meet a strict functional programming style Especially users which are not accustomed with that way or programming might be frustrated by the result of the assignment a expression a One example The user enters tLe Aa lt 3 User
20. pare times for multiplying some non special numbers having 100 200 400 digits and time the multiplica tion Times for the smallest examples are likely to masked by the timer resolution unless done using ITERATE Es Ul AU a ia d f we A 1 WEBER sag A is we J ES d eene MAC MTS ri E sii Here they are at the occasion of Josef s and Josef s daughter s birthday same day 1 row Constantin Dominic Yvonne 24 row Moritz Naomi and Maxime D N L 25 Carl s and Marvin s Laboratory 4 p 49 Parametric Plots In previous sections we saw that DERIVE will plot any vector of functions say Lio fases f as n different plots on the two dimensional Cartesian Plane There is an exception to this rule It is when n 2 In this case DERIVE regards the pair as the parametric desription and assumes x t fi t Y t fo t for some parameter t We will use this feature to solve the following problem You and a fried decide to play cath with as baseball To make things a little more interesting your friend decides to ride a ferris wheel during the game The seat of the ferris wheel is situated so that when a passenger boards it the seat is five feet above the ground and it is 20 feet from the center of the wheel The angular velocity of the wheel is 0 5 radians per second You are standing 28 feet from the ground center of the wheel You can throw the ball with a velocity of 64 feet per second and release it
21. position of the ball Let Obe the angle that the ball is pitched angle with the negative x axis Then P t 2 28 64 cos 0 t Y t 254 64 sin 0 t 161 A nice question for the non English or non American students Where does the number 16 come from Josef We need to determine the choice of that will cause the curve determined by P t Y t to intersect the path of our friend when t 1 2 We will determine the time delay later Author 28 64 CosS 8 t 5 64 SIN 8 t 16 t 2 In DERIVE for DOS ist typed as Alt h We can not draw any graphs at this point We must make a choice for either 0 or t This is the beauty of having a tool like DERIVE We can experiment Let s make a naive choice 0 1 4 Use Manage Substitute to draw replace and then Plot the resulting expression for going from 0 to 1 As you see on the graph below this overshoots the mark and is too long a time Now go back to the Algebra Window and use Manage Substitute to replace in our original expression with 7 8 This time when you Plot let run from 0 to 0 7 As we see this under shoots the mark and is still too long a time Pe zoon Trsce Re rsenmsinraule t n 4 M The Ferris Wheel li oan the TI 92 IN 3 t mw Line of sight 16 friend s path You continue the investigation and find out the value for that will work for this problem Hint 7 8 isn t off by much Once you have the correct angle you can u
22. setting out exams in which the students are given a figure to find the movements needed to be formed So our students would love and enjoy mathematics much more or perhaps they hate us even more If someone dares to put it in practice tell us 1 Gardner M Viajes por el tiempo otras perplejidades matem ticas Capitulos 3 y 4 Ed Labor S A 1988 English version Time travel and other mathematical bewilderments was really fascinated by Alfonso s article It combines fun and creative phantasy with mathematics in a wonderful way imagined wooden tans to play with So I tried to pro duce shaded tans just for fun to present the solutions or the problems in a nice form That is the wooden Yacht If you want to work with the shaded tans then use the new fans The complete solution for the Yacht will be found on the diskette of the year I hope to be able to present the actual files to be downloaded on the homepage of my school in the near future Josef p 48 DERIVE USER FORUM D N L 25 Al Rich s interesting answer to Nigel Backhouse s comparison of calculation times The following is in response to Nigel Backhouse comments dated 27 January 1997 to the DERIVE News mail ing list concerning integer factoring of 2101 1 To factor integers DERIVE uses trial division followed by a Monte Carlo Primality test followed by the Pol lard Rho algorithm as detailed in Knuth The Art of Computer Programming Volume 2 For
23. that SUM ax b n x c inf can be expressed in terms of the Riemann Zeta function even if n is symbolic provided that n gt 1 Horst Scheppelmann Hameln German T Ist Ihnen ein Buch zu DERIVE bekannt das sich berwiegend mit vektorieller analytischer Geo metrie befa t wie sie z B mit dem Lehrbuch von LAMBACHER SCHWEIZER aus dem Klett Verlag in der Sekundarstufe II Klassen 11 13 unterrichtet wird Mit freundlichen Gr en und W nschen f r das Neue Jahr Ihr H S DNL The question is directed to our German friends but maybe there is anybody among you who does know a book dealing with vectorial analytical geometry for Secondary Schools forms 11 13 Michael S Mullen Austin TX USA msmullen tenet edu Why can DERIVE XM 3 01 not solve 2 x x 2 The plot shows all three solutions including the obvious one x 2 I tried to declare x as a complex variable and I tried to expand the equation prior to solving but DERIVE just yields 2 x x 2 0 Any assistance is appreciated DNL There were several answers all of them with similar contents One example had the same experience on the TI 22 until changed the mode to Ap proximate X Fa 4 amp 1 SOLUTIONS 2 x 0 x 5 10 DERIVE 6 2 is the simplified result and 3 the approximated one 2 2 4 3 0 7666646959 Mike Hammet Greenville SC USA Hammet_Mike furman furman edu I have just received and started using DfW I wan
24. the Turtle file turtles ut mth should be loaded normally or as a utility file in the background Example 0 For instance to draw the square of vertices 0 0 0 1 1 1 1 0 it is enough to Plot LFD 1 RTC90 FD 1 RTC90 FD 1 RT 90 FD 1 RT 90 M Works now also for DERIVE 6 Josef p 16 L amp R amp R amp W TURTLE GRAPHICS D N L 25 Or even more simple to make the assignation to rep z FD 1 RT 90 and to Simplify and Plot REPEAT 4 Note that the syntax adopted for the REPEAT command is not the usual in Logo Observe that unfortunately unlike in Logo there is no visible cursor for the turtle in the graphics screen About the implementation the content of the turtles ut mth file The first two lines of the turtles ut mth file are Angle Degree Precision Approximate in order to increase the speed nevertheless this can should be omitted in some cases see Ro3 The main problems we had to produce this implementation were related to the difficulties to change from a programme assignations made to variables The following auxiliary functions are to be used NEWX u xcor u NEWY Cu ycor u NEWD u Cheading u The turtle is at any moment at point xcor ycor and heading towards heading counting clockwise from the usual halfaxis y The user can check the value of these variables by simplifying them Advancing a length 1 can be implemented easily FDC1_ Lxcor ycor NE
25. the most part these algorithms are implemented using loops rather than recursion This combination of factoring methods is usually the most efficient until the size of the second largest prime factor exceeds about 12 digits Glossing over details the more complicated elliptic curve method is then usually the most efficient until the size of the second largest prime factor exceeds about 25 digits Quadratic residue methods are usually the most effi cient beyond that The times Backhouse reports measure the relative speed of the algorithms used by the various computer algebra systems to factor 2 101 1 more than they measure the speed of the underlying arithmetic To try to determine the underlying combination of algorithms being used by the various systems construct a family of integers hav ing n digits in its second largest prime and then compare the times required to factor the family of integers by the various systems The relative rankings on different CPUs are likely to be the same and more or less proportional to sequential memory fetch and store so testing on additional CPUs is not likely to yield much additional information The bignum arithmetic in DERIVE uses the same nonrecursive algorithms also described by Knuth as your other systems but DERIVE s is implemented in Intel 32 bit assembly language Therefore the DERIVE big num arithmetic should be faster than bignum arithmetic compiled from languages such as C To test this com
26. 4 Draw regular polygons of 5 7 and 9 sides of length 1 10 xcor 0 ycor 0 heading s 0 x 11 n 12 to_rep roo ul pps Me 14 REPEATS Fla m s 16 REPEAT C7 lr qd 19 18 REPEAT S Example 5 Draw the 36 wires of a chart wheel 19 xcar 0 ycor 0 heading 0 20 to rep FD 1 BKOL RTG10 21 REPEAT 36 PenUp and PenDown There is no PenUp PenDown command in our implementation Instead the user has to play with only simplifying that only changes the position of the turtle or Simplify and Plotting that changes the position of the turtle and draws the correspondent segments Example 6 Draw two segments the first one with endpoints 0 1 and 0 3 and the second one with endpoints 1 3 and 3 3 28 xcor cs 0 ycor ss 0 heading 0 29 FOCLI 2 3 aw y 0 g 30 car 0 veer l lt 31 UEBE2 1 32 RTCSO FD 1 0 3 33 LU vecors xp exc 34 FDC2 29 and 32 must be simplified only 31 and 34 must be plotted A Recursive Example Example 7 Function KOCH below constructs the side of a koch star with length len and depth n KOCHClen n IF n gt 0 KOCHClen 3 n 1 LT 60 KOCH len 3 n 1 RT 120 KOCH len 3 n 1 LT 60 KOCHClen 3 n 1 FDClen 36 xcor 0 ycor 0 heading 0 37 RTCSO 38 0 ycor 0 39 KOCH l 1 simplify 37 in order to have the x axis as base
27. 9 xcor 0 ycor iz 0 heading 0 70 pat2 2 5 0 6 D N L 25 Comments on the TURTLE Graphics p 23 71 xcor 0 ycor 0 heading 0 72 RICSO L0 veor D 73 VECTIORC patzt2 5 LTCS03 1 4 twig len n IF n gt 0 twig len 3 n 1 LT 45 twig len 3 n 1 BK len 3 RT 45 twig len 3 n 1 RT 45 twig len 3 n 1 BKClen 3 RT 315 twig len 3 n 1 FDClen 3 5 xcor 0 ycor 0 heading 0 76 twig 3 1 Wf xcor 0 yeor 0 heading 0 78 twigl3 5 More References Lindenmayer Systems DNL 51 and DNL 52 Snowflake a o DNL 33 and DNL 34 Contribution from Matia Koth LOGO in DERIVE DNL 238 Josef Lechner p 24 Comments on the TURTLE Graphics D N L 25 2 pas twig2Clen n IF n gt 0 twigClen 3 n 1 LT 40 twig len 3 n 1 BK len 3 RT 45 twig len 3 n 1 RT 40 twig len 3 n 1 BKClen 3 RT 310 twig len 3 n 1 FDClen 80 xcor lt 0 ycor 0 heading 0 81 twig2 3 5 twig3Clen n IF n gt 0 twigClen 3 n 1 LT C30 twig len 3 n 1 BK len 3 RT 45 twig len 3 n 1 RT 30 twig Clen 3 n 1 BKClen 3 RT 250 twig len 3 n 1 FDClen D N L 25 Comments on the TURTLE Graphics p 25 83 xcor 0 ycor 0 heading 0 84 tw1g3 3 5 Two more examples on page 28 p 26 Comments on the TURTLE Graphics D N L 25 In an earlier Letter of the Editor
28. CAS The cause 1s of course the algorithm of the respective CAS But even without knowing this one can understand why the above solution is not recognized For this one merely needs to determine the solution step by step by means of elementary row manipulations This can be carried out quite easily with the commands of the TI 92 By means of the commands mRow factor matrix n one can multiply the row n of matrix by factor The command mRowAdd factor matrix n n allows the row n of matrix to be multiplied by factor and to be added to the row nz p 52 The TI 92 Corner ES nisse a sre otner ern ro e1ean a z 2 6 3 m mEOowHdd 2 4 3 3 4 3 39 4 3 9 Ek 611 4 3 3 61 4 3 9 k11 1 2 gt HAIN RAD AUTO FUME 2 30 m sebralcaTc other Prsmto ctear 2 2 d x E 2 6 gee m mEOwHdd 2 80 9 9 18 1 3 LES E 2 6 Ta sm 76 ee ue cim H 9 1s O 15 15 k t12 II PE PEPREGIEPEIEPEPLTEPRPEP IT n nm nm nm m Ead n HAM KAD AUTO FUR S730 ee rigebralcatc otherPranto ciear a z E 15 15 kri2 26 ZI e nRouRdd 15 8 1 es gus Ge eho dod sepa AE eS s EEE Pe E 0 0 k i8 we 1 2100 715 15 k 12 1 2 3 gt HAIN RAD AUTO FUNC ESO re lrigebralcatc other Prantolciear a z 1 2 gu dg 1 JI0O 1 1 2110 0 0 5 1811 3 gt Male FUNC B20 1 a2 2l Si mult Equations D N L 25 Firstly row 1 is multiplied by 2 and is then added to row 2 Then row 1 is multiplied
29. Etchells Liverpool UK t a etchells livjm ac uk Hi Josef AAGH Did you not notice something wrong with the output of the ROMBERG AXU TABLE function on page 3 of the last newsletter It is clearly rubbish The problem arises with an incorrect transcription of the auxiliary functions a has been replaced with a in the function ROMBERG AUX TABLE It should read ROMBERG AUX TABLE v f x a b n ITERATES VECTOR 2 k v SUB r 1 v SUB r Z k JI f l n O RT2 C 0ULl I v 4k clLBROMBERG START D Xx a bjn 4 11 n 1 Also the example should have been set to 13 digits precision to show the convergence of the method in the table With 6 digits precision Simpson s rule gives correct result with 8 strips for this example Could you please publish a correction in the DNL 25 please DNL Of course do Please excuse my mistake There were some problems in reading your email because of a special encoding of some characters e g Author ROMBERG FBsinx x 0 E3 3 and approX eu 19 ROMBERG_TABLE 1 75804691 28 0139169 1792 45321 458861 105 469873330 6 1 758195 28 0140294 1792 45365 458861 112 n 20 1 75820265 28 0140362 1792 45367 n n 1 7582031 28 0140366 n n un 1 75820313 n n n i Steven Schonefeld Angola IN USA schonefelds alpha tristate edu Hi Josef I just got DNL 24 and have a couple of comments on the utility file ROMBERG MTH by Terence Etchells 1 There was a problem with the integral of SIN x x fro
30. It applies to any vector matrix size shape and to number symbolic computations In fact you can 1m prove RK and EULER in ODE APR john Terence Etchells Liverpool UK t a etchells livjm ac uk Hi ALL I ve done a fair bit of programming in my time using DERIVE and to my knowledge there are no books on it at the moment However I have a book on Numerical Analysis published this month There are many activities in this book that deal with programming DERIVE for example program ming the Newton Raphson Method Cobweb diagrams Secant method Lagrange interpolating poly nomials Simpson s rule Richardson s method Euler and Runge Kutta methods for ODEs etc etc Contact Philip Yorke at Philip chartwel demon co uk for further details Incidently I have been playing around with operators recently with DFW 4 03 and I discovered for myself some interesting mathematics Try this The operator D on f x we define as Df x f x h f x I would prefer to use capital delta for this but I doubt whether the character will transmit through e mail 1 FOX Ato x hs ct Tam p p 2 xx h Simplify A F x x h 3 ACFCx x h FCx h F x Now I want the operator A to act on F again i e A 2 F x No Problem 4 ACACFCx x h F x 2 h 2 F x h F x What about A 3 F x or even 10 F x We can use the ITERATE command 5 TTERATELAIS x H3 s FOX 33 What do we get 6 F x 3 h 3 F x 2 h 3 FCx
31. N v 13 SUBTRACT ELEMENTS m i j s User 2 G 3 b 14 gls 4 3 3 6 User 4 3 2 k 2 6 23 s 2 6 3 15 SUBTRACT ELEMENTS 4 3 Or Most i gt sse ge Se 4 3 9 k 4 3 9 k 2 Gr ar 6 2 6 Ed 6 23 SUBTRACT ELEMENTS 0 9 9 18 3 1 2 0 9 9 18 4 3 9 k I le SEES ep 2 6 E 6 2 G 3 6 24 SUBTRACT ELEMENTS Q 9 9 18 FT 0 9 9 18 OQ wL5 25 R 12 0 0 Q deed8 20 or with a TI like MROWADD User 21 MROWADD faktor matrix nl n2 SUBTRACT ELEMENTS matrix nl n2 faktor 2 6 E 6 2 6 cx 6 RD 22 MROWADD 0 9 9 19 pope ar seg 9 18 9 D s ds A O O O kei p 54 The TI 92 Corner Script amp Program D N L 25 Example of a script in combination with W Propper s program ABLEIT find it among the files We investigate the differential quotient of a special function for x 0 We set up the screen C setmode Split Screen Top Bottom Split 1 App Home Split 2 App Text EdTtor Split Screen Ratio 2 1 clrhome C abs x 2 2 2 f x f x 0 CERRI we define the rate of change approaching from right hand side fromr x C f xth f x h fromr x Now special for x 0 1 For Fir F4 FE cues Toe Fin From right limes mt h 0 C limt fromr 0 h 0 Why that h mus be positive Hence C fromr 0 h 0 Failed Help the TI and try Expand C expand fromr 0 h 0 the limit is Cod mcos Dp bod dos 0 the slope of the tangent is
32. Roanes L An implementation of Turtle Graphics in Maple V Maple Tech Special Issue 1994 pages 82 85 reprint of the article above R03 E Roanes L E Roanes M Turtle Graphics in Maple V 2 In Robert J Lopez Editor Ma ple V Mathematics and Its Application Birkh user 1994 pages 3 12 Ro4 E Roanes L Automatizaci n e implementaci n de algunos problemas algebraicos y geom tricos Tesis Doctoral Univ Polit cnica de Madrid 1993 RS A Rich J Rich T Shelby D Stoutemyer DERIVE User Manual SoftWarehouse 1994 Now is 2011 and we have DERIVE 6 10 which is much more powerful than the DOS Version from 1997 remember my LINDENMAYER Systems contribution from DNL 51 and DNL 32 This has a tight connection to Turtle Graphics use turtles ut mth for reproducing some examples p 22 Comments on the TURTLE Graphics D N L 25 hilb len n IF n gt 0 hilb len 3 n 1 LT 90 hilb len 3 n 1 RT 90 hilb 1len 3 n 1 RT 90 hilb len 3 n 1 LT 90 hilb len 3 n 1 FD len 58 xcor 0 yeor D heading i 0 59 RT CS0 F60 0 ycor s 0 61 omadBbs 1 62 xcor 0 ycor 0 heading 0 63 0 ycor 0 64 hilbte 4 pat2 len n IF n gt 0 pat2 len 2 n 1 LT 90 pat2 len 3 n 1 RT 180 pat2 len 3 n 1 LT 90 pat2 len 2 n 1 FD len 67 xcor 0 ycor i 0 heading 0 68 pat2 2 1 6
33. THE DERIVE NEWSLETTER 25 ISSN 1990 7079 THE BULLETIN OF THE OCON IMS DLI MVL Z USER GROUP TI 02 Contents 1 Letter of the Editor 2 Editorial Preview 3 DERIVE User Forum J Lecher E Roanes L E Roanes M J Wiesenbauer 15 TURTLE GRAPHIC in DERIVE Leo Klingen 29 Tilgung fremderregter Schwingungen durch abgestimmte Ankopplung Oscillations Peter Mitic 33 Probability Distributions 2 37 The 25th DERIVE Newsletter Neil Bibby 38 A day in the life DERIVE as a Demonstration Tool in Upper Secondary 44 AG DC A Tangrams with DERIVE Carl Leinbach amp Marvin Brubaker 49 Carl and Marvin s Laboratory 4 Parametric Plots 51 The TI 92 Corner K H Keunecke W Pr pper J B hm revised 2011 March 1997 D N L 25 Learning Numerical Analysis through DERIVE T Etchells J Berry This book covers the major numerical methods and their analysis for first courses at college and under graduate level The relative merits of each method are covered both analyt cally providing a thorough grounding in the algebraic approach and practically through the tried and tested computer lab based ac tivities DERIVE provides a platform on EDS which to quickly and accurately perform many complicated numeri cal calculations Also DERIVE s ability to algebraically manipulate expressions and perform calculus operations enhances the investiga tion of the convergence of numeri cal method
34. WX xcor 1_COS 90 heading NEWY ycor 1_SIN 90 heading and moving backwards the same distance can be obtained from the previous function BK 1_ FD 1_ When the turtle turns only the direction heading s altered The new value is reduced modulo 360 in order not to get high values for the heading variable after many rotations RTCa_ xcor NEWY ycor 0 NEWDCMOD heading a_ 360 LT a_ RT a_ The last of the typical turtle commands is home return to the centre of the screen facing upwards home NEWX 0 NEWYCO NEWD O But there are some more standard turtle commands mixing turtle ideas and Cartesian coordinates To go to a certain position of known coordinates can be done using SETPOS x_ y_ Lxcor ycor xcor x_ Yycor y_ 0 D N L 25 OG L amp R amp R amp W TURTLE GRAPHICS p 17 The actual position 1s given by pos xcor ycor And to change only one of the coordinates of the current position can be done using SETX x_ L xcor ycor xcor x_ ycor SETY y_ L xcor ycor xcor ycor y_ It is also possible to force the turtle to look in direction a_ with SETHCa_ Lxcor NEWYCycor O heading a_ or to look towards a certain point x y with SETHTOWARDS x y 9 z xcor NEWY Cycor 0 heading ATANCx xcor y ycor Note that to clear the screen there is no ClearScreen command The Delete All in Derive s Plot menu should be used instead Another standard feature of Logo is the
35. Word User 0 0 2 Q 31 ITERATES FWD 2 0 turtlepos home 4 2 g User Simp User 2 0 p 28 Comments on the TURTLE Graphics D N L 25 What s possible in a list does not work in a VECTOR 335 d ge l User 1 2 1 2 1 2 34 nen E NEWA E NEWA E User 2 a 2 a 2 a Hoos 1 5 1 41666 L 41421 Simp 34 toG a wei User 1 2 37 VECTOR NERA fa t bw d 3 1 5 1 5 1 5 User Simp User 2 a Conclusion To implement a Turtle graphic in DERIVE it would be necessary for the program to sup port a self assigment of the variables and or the the support of composition of functions would be im proved So I will close And yet it works I tried to translate Josef L s comments hope that he will recognize the sense of his comments in my words Josef B J Wiesenbauer has promised to check the DERIVE utility files for improvements using the new built in DfW and DfD4 functions and capabilities Do you have ideas D N L 25 Leo Klingen Schwingungen Oscillations p 29 Tilgung fremderregter Schwingungen durch abgestimmte Ankopplung Cancellation of Separately Excited Oscillations by a Tuned Connection Leo H Klingen Bonn GER Eine rotierende Maschine mit exzentrischen beweglichen Teilen steht auf einem Fundament das beim Lauf der Maschine m glichst in Ruhe bleiben soll Nicht immer stellt eine m glichst gro e Masse des Fundaments eine optimale L sung dar Bei
36. again by 2 and the added to row 3 below left Now row 2 is di vided by 9 below right 7 F r1sebralestelotherPrantolciear a z E Slo elo dRd2 2 6 3S 6 wmEOow 124 9 0 93 9 18 2 B 15 15 k 12 2 5 3 B 1 z Ae B 15 15 kt 12 we 207 18 1 0 15 10 k 1211 2 gt KAD AUTO FUNC 4 50 If row 2 is multiplied by 15 and then added to row 3 the result shown on the left 1s obtained The last element of the bottom row is k 18 Therefore it is necessary to differentiate between the two cases k 18 or k 18 In the first case the set 1 of equations is solu ble but not in the second The solution for the case A 18 can after further transformations be read directly from the display k 18 x 2283 y z 2 zeRe The CAS have not made the distinction between the cases but rather have erroneously divided the last row by k 18 Even with the TI commands this s consistently possible as can be seen in the display on the left This result is identical to the result of the command rref at the beginning of this report The lack of distinction in cases and illegal division respectively can occur only if in the row echelon form of the augmented coefficient matrix in any single row only zeroes and to the far right an ex pression appear The set of solutions 1s then either empty or it has a dimension greater than or equal to l as if shown in example 1 Remarks l The reported problem arose in the cou
37. aph No I haven t fallen into any fountain of youth May I introduce one of the youngest members of the DERIVE and TI community Kimberly our first granddaughter in the tender age of one hour This picture will answer many questions Hello Barbel amp friends Here is now our 25th issue and on this occasion I ve asked SWHH and SWHE for a contribution They have both sent greeting addresses page 37 many thanks It is a pleas ure for me to print them but I d like to share their appreciation with all of you Sometimes I receive minor complaints now and then the problems and the accompany ng solutions seem to be from another star very far away from my daily Problemchens with mathematics D Blum Please consider that we fortu nately have a big number of non teachers in our group and that even the teachers should be allowed to reach for the stars now and then I d like to encour age all of you who are feeling similar to report about their Problemchens small problems No ques tion no letter no contribution can be too simple or too unexciting You all produce the DNL And be lieve me I have the same maths Problemchens with the 15 19 years students as you do I have tried to make this DNL especially useful for the teachers workaday routine In the next DNL you will again find geometric items mappings of objects in R hidden lines fractals etc There is a box full of TI questions wit
38. chwingungen Oscillations Mit den Substitutionen x x4 und X x4 kann man Runge Kutta ansetzen 6 RK x3 x4 2 x1 1 5 x2 2 COS 3 t 3 x1 3 x2 t x1 x2 X3 x4 LO 0 ad 0 0 0435 50 L1 2 First of all we have to find the initial values for a that is the constant amplitude of the con nected oscillation We plot for two values a 2 3 and a 4 3 negative because oppo site to the spurious oscillation Using a we recognize a remaining vibration whereas using a2 the effect of the additional oscillator causes a complete extinction of the basement s vi brations Zuvor muss der Anfangswert a das ist bei vernachl ssigter D mpfung die konstante Amplitude der angekoppelten Schwingung bestimmt werden Wir plotten f r zwei Werte a 2 3 und a 4 3 negativ wegen der Gegenphasigkeit zur St rschwingung Beim Wert a bleibt wie die Grafik verr t eine Restschwingung des Fundaments brig Beim Wert a ist die Gegenwirkung des Zusatzschwin gers so gro dass sich eine vollst ndige Ausl schung der fremderregten Schwingung des Maschinen fundaments ergibt x1 for a1 2 3 27 Ct x4 2 xl pe See 4 2 008 13 3X1 3 x2 ft xe x3 34 o 0 2 0 0 3 s hun 2 3 x1 for a2 4 3 8 CS x4 2x1 1 5 x2 2 COSC 3 t 3 x1 3 x2 t xl x2 x3 x4 o 0 4 0 Ux hun 2 3 x2 9 Ct MAS 2D LE 2 ODS BEE 3x1 300 t xL x2
39. d reliable persons I ever met Josef and his wonderful wife Noor are absolutely charming people If you haven t met them yet per sonally come to one of the various international conferences Josef lists them in the DNL and you will make friends with them very quickly There is no way not to become very friendly with them DERIVE and the DERIVE User Group The first would not be the same without the second Josef and Noor thank you Bernhard Kutzler Managing Director of Soft Warehouse Europe p 38 Neil Bibby A day in the life D N L 25 A Day in the Life DERIVE as a Demonstration Tool in Upper Secondary Mathematics Neil Bibby Lancaster UK During recent years I have developed an integrated use of the DERIVE package into my work at all levels of upper secondary mathematics This has been primarily as a demonstration tool by which I mean that the students have not had direct hands on use of the package There are teo good reasons for this 1 DERIVE is a sophisticated piece of software which requires a thorough knowl edge of mathematics on the part of the user including a strong grasp of notation and notational variants Although there is some evidence that it has been used suc cessfully as a hands on tool for students equally other work has cast doubt on this I think of DERIVE as a professional tool which in professional hands can provide a rich learning environment 2 The availability of graphic calulator
40. e a e 0 5 This gives an el cos lipse or hypObola as I sometimes call them and is easily plotted using DERIVE 6 2 H966 Scale x H z y B z Derive ZD plot Neil Bibby A day in the life It is now instructive to ask what is the cartesian equation of this curve We use x the substitutions r Jx y and cos y x y lent 2 x 1 3y 1 The static graph above cannot do justice to the dynamic effect to obtain the cartesian equiva of seeing this plotted remember to switch back to rectangulars in a new colour exactly on top of the polar verson The monochrome calculator displays cannot compete here even if they could plot implicitly The cartesian form provides us with new information we can see from the equation that x is a line of symmetry and that the lines y are tangent lines in the follwing diagram we confirm 3 these ES 1 x y X 60 P 2 2 x y T EIS 2 2 x y S51 x 4v 2 Ee P E EC 2 2 2 2 x y 2 2 BEZ eye EN 93 3 2 2 7 M 2 2 2 67 4 x y x 1 x y 2 Jx Ey X 2 s sx 2 2 2 2 2 2 2 2 2 4 x y x 1 x y 63 Cx aW TEZE 43v WX ry 72 e 2 2 2 2 2 2 2 2 2 2 X y 64 xx y 4 25x 3 fe Zy Yen 4 2 2 2 2 2 2 2 69 4 x y x 1 65 2 x 42 y JG y 2 9 1 2 2 2 2 E 70 x 3 y 1 66 2 x 2 y 1 x J x y 4 I hope that these examp
41. en Schwingungen Oscillations Oder man wendet nach Substitution x x3 die Runge Kutta Methode an 4 RKC x3 0 5 x1 2 C0OS 3 t t x1 x3 0 0 0 0 3 50 yy 1 2 485 4 COS 3 t Das Ergebnis zeigt Figur 2 mit einer erheblichen Schwingungsantwort des Fundaments We can find an exact solution using DSOLVE2 or an approximative solution using the Runge Kutta Method The result in figure 2 shows a serious vibration response of the base ment We can achieve an interesting extinction of the resulting oscillation by connecting an addi tional mass m using a second spring spring constant c See figure 3 Figur 2 Eine interessante Ausl schung der resultierenden Schwingung kann man durch Ankopplung einer weiteren Masse m ber eine zweite Feder mit der Federkonstanten c erreichen Vgl die Prinzipskizze Figur 3 er ES Mit den zus tzlichen Daten m 1 c 3 die so gew hlt sind dass sich als Eigenfrequenz die Frequenz des St rers ergibt erh lt man das System unter Ber cksichtigung richtiger Vorzeichen nach dem Prin Zip actio reactio The additional data mz 1 c 3 are chosen in such a way that the proper frequency results from the frequency of the dis turber principle of actio reactio az Then we use again Runge Kutta X1 0 5 x1 Xe 15 x X5 Ca X9 3 x5 x1 oder x x Re X 2x 1 5x E cos 4 3 t Figur 3 X5 3x 3x0 0 Leo Klingen S
42. etworks As an aside I am writing an article for the DNL on how I programmed Derive to train a neural network My experi ences lead to the following request As you know DERIVE is my number one program If I were to have only one program on my desert island PC it would be DERIVE version 4 03 As a teacher of 16 19 mathematics it has always been the best and the Windows version is superb However now as a user at the University level I can see why some universities prefer other CAS such as MAPLE MATLAB and MATHEMATICA Granted they are more difficult to learn and to use and are not very intuitive But they have an important facility that DERIVE does not That is what I call Vertical Programming i e For while If Then End While as opposed to what I call horizontal progamm ng vector 1f then r I have attatched a MTH file that performs Neural network training The effort required to program Derive to do this was considerable I really enjoyed doing it but thats by the by and most of the effort was in eliminating repetition of calculation through auxiliary functions Something that I am sure although I don t know would be made easier with vertical programming Another aspect which I find a little frustrating 1s the way DERIVE works as a Turing machine i e all calculations are done before any ouptut is given Which means that 1f I run a function that requires a long compute time say 48 hours and after
43. f Simpson s Rule is that it is exact for cubics This two for the price of one aspect needs some explanation A formal proof is possible of course but an informal graphical approach might be appropriate and convincing initially The diagram shows the graphs of a cubic its quadratic interpolator and the differ ence of the two This difference is a new cubic whose zeros must be a a h and b a 2h p 42 Neil Bibby A day in the life D N L 25 So the point where x a h must be a point of inflexion i e the symmetry point of the graph Hence the two lobes of this cubic are of equal area and thus Simpsons Rule is exact The visual plausibility of this is compelling but should not detract from the details of the argument once again the two aspects corroborate each other Example 4 Polar and Cartesian Coordinates British Year 13 Grade 12 DERIVE version 3 has the facility to plot implicitly This is naturally slower than the explicit plotting but can be used to good effect especially when explicit plotting is simply not possible e g y y x It is especially effective when the equivalence of polar forms and cartesian rectangular forms of an equation are being consid ered One approach to the conic sections is to use the focus directrix definitions to arrive atr nn It is instructive to consider the lima ons r a l e cos 1 e cos l l T first As an example we consider r DR cau wher
44. feel it is worth it please feel free to share it with other users DNL Some answers M S Mullen Your test was a great idea 2 101 1 really throws DERIVE into a tissy After a little brute force I found that 2 93 1 1s a fairly good test with a much shorter duration On my 486DX100 the expand factor sequence took 14 sec for DERIVE XM and 11 sec for DERIVE loaded low On my old 386 SX16 XM took 63 sec Have fun M S Mullen has a nice question in his mail Who is General Failure and why is he reading Drive C Scott Guth Hello DERIVE enthusiasts I thought I d offer my results to FACTOR 2 101 1 Rational Using DERIVE for WINDOWS Pentium 90M Hz 32MB RAM w WIN95 13 6 minutes Using Maple V Release 3 same computer as above 4 02 minutes That sure says something about the Maple kernel Dr N B Backhouse Dear Derivers Let me give you some more timings of factorisations of 2 101 1 All done on a 133 Mhz Pentium 32 Megs of RAM which may be or may not be relevant DERIVE 3 00 13 2 minutes Macsyma 2 10 9 0 minutes Mathematica 3 0 2 5 minutes Maple V5 Release 4 1 75 minutes Ubasic 0 75 minutes Didn t they do it well Now can someone tell me if it is significant that both DERIVE and Macsyma are Lisp based whereas Maple and Mathematica are written in C Read Al Rich s interesting answer on page 46 Josef p 12 DERIVE USER FORUM D N L 25 Leon Magiera Wroclaw Poland magiera rainbow if pwr wroc pl
45. from 5 feet above the ground At what angle and time after your friend is directly above the ground center of the wheel must you throw the ball so that it can be caught when your friend s angle of elevation with the ground is 45 You may assume that the horizontal component of the velocity of the ball remains constant Let s draw a picture eU ft Newton s Laws tell us that the height of the ball above the ground at time f is given by Y t Y v t 161 where Y 5ft voy initial velocity in the y direction The vertical component of the velocity of the ball is vy voy 32t First we look at the path followed by your friend on the ferris wheel Place a coordinate system at the ground center of the wheel It is easy to figure that for your friend the position f seconds after the wheel starts is x t 20 y t 25 20 co p 50 Carl s and Marvin s Laboratory 4 D N L 25 From the picture we can see that your friend will be in position to catch the ball when 1 2 We will draw your friend s path from the time the ferris wheel starts until the ball is supposed to arrive In DERIVE author 20 mL ZI 25 20 ost 4 To plot this enter the Plot Window and choose the Plot option DERIVE will prompt you for the range of the parameter in this case enter 0 for Min and 7 2 for Max An eighth of a circle will be drawn What about the ball Let P t and Y t be the horizontal and vertical components of the
46. h answers for you waiting to be opened I think that in many cases DERIVE contribu tions might inspire you for Tl applications and vice versa You will find examples in this issue I hope that I will be able to offer the facility to download DERIVES and TI files from my school s home page in the near future Until the next DNL Josef If you are interested about the change of appearance of Kim between 1997 and now then have a look on page 48 There you can find a picture of Kim together with her 3 sisters and 3 cousins who accomplished our unique set of seven grandchildren Noor and Josef proud grandparents P2 E DI TORIA HL The DERIVE NEWSLETTER is the Bulle tin of the DERIVE User Group It 1s pub lished at least four times a year with con tents of 40 pages minimum The goals of the DNL are to enable the exchange of ex periences made with DERIVE as well as to create a group to discuss the possibilities of new methodical and didactical manners in teaching mathematics We include now a section dealing with the use of the TI 92 and we try to combine these modern technologies Editor Mag Josef B hm A 3042 W rmla D Lust 1 Austria Phone 43 0 660 31 36 365 e mail nojo boehm pev at D N L 25 Contributions Please send all contributions to the Editor Non English speakers are encouraged to write their contributions in English to rein force the international touch of the DNL It must be said tho
47. ld cover most of the pro gramming facilities of DERIVE Also the problems should be simple to grasp for my students and should be practical That means that it should be utterly clear that such a utility or subprogram can be of great use during the rest of course Can you give me some suggestions I intend to share my experiences in the International DERIVE Journal if interesting Thanks in advance Yours sincerely Drs Jos C M Verhoosel Fontys PTH Dept of Mathematics Eindhoven The Netherlands DNL could imagine that the TURTLE GRAPHICS article in this issue could contribute for your course and Terence s nwe book of course And now once more Terence Etchells Terence Etchells Liverpool UK t a etchells livjm ac uk Hi Josef I have written a little set of functions that will produce an interpolating polynomial to approximate a function at any given points on that function So for example you may wish to find a polynomial that passes through the points 0 sin 0 p1 2 sin pi 2 and pi sin pi So our function is sin x and the x values of the desired points are 0 pV 2 p l As we have 3 points the polynomial will be a quadratic You will find attatched a MTH file INTERPOL MTH that will do this automatically In fact it will work for up to 20 points i e a polynomial up to order 19 this can be increased by increasing the available variables in the def vars and the line above that resets all variables we are to use to empty
48. les will convey the favour of this approach I now find DERIVE an indispensible tool in my day to day teaching Please don t hesitate to contact me if you need further advice or assistance Neil Bibby Department of Mathematics University College of St Martin Lancaster LAl 3JD UK p 44 AC DC four D N L 25 DERITANS TANGRAMS with DERIVE Alfonso J Poblacion Valladolid Spain INTRODUCTION One of the oldest parts of Recreational Mathematics concerns Dissection problems Among them the ones derived from seven pieces called tans known as tangrams are very popular see figure 1 Arranging these tans it is possible to form a wide variety of figures The rules are simple the seven tans must be used and it is not allowed to overlap any of them although we can let holes among the pieces Martin Gardener in 1 classifies in three types the questions we can deal with tangrams 1 to find how to build a given tangram or to prove its impossibility 2 to find how to form several real figures such as animals objects or persons in the most artis tic or amazing way 3 to solve problems about Combinatorial Geometry that the seven tans set forth In the mentioned reference we can find more about these categories of problems and the history of tangrams HOW TO USE DERIVE We will try to play with the tans using DERIVE and our imagination First of all we will define the seven pieces lines 3 and 4 from its original place in the initial sq
49. life p 41 0 LN 322 population 1500 m LN 493 N LN 579 1800 UJ LN 836 LN 992 N S FLI Ix 88 b 5 LN 1116 6 LN 1260 7 LN 1583 8 LN 1724 9 LN 1635 26 0 180389 x 6 02224 time 1800 10 0 180389 x 6 02224 27 e The visual immediacy of this treatment combined with the use of TI 82 albeit in a blackbox mode proved pedagogically Example 3 Simpson s Rule British Year 12 Grade 11 My preference with this algorithm is to emphasise the fitting of a quadratic function by three points of the graph of the function whose integral is required A nice exam ple to start with is INT cos x x n S3 n 3 The interpolating quadratic is easily 9x 2g The graph of this gives the following together with that of cosine omitting here but also worth including is the graph of the second degree Taylor approximation of co sine at O to emphasise that this approximates better at O but doesn t interpolate at all seen to be y 1 A comparison of the corre sponding integrals is then illu minating the exact value is v3 1 732 the Simpson value is 1 745 and the Taylor value is 1 711 Students should appreciate that Simpson s Rule actually due to Newton originally who else is exact for quadratics this seems obvious but a few corroborative cal culations using DERIVE push home the point However one of the most intriguing features o
50. lso could find it on the net together with KIT ZIP and MAWK I have contact to Oscar and it seems that he is trying making other DERIVE files of 3D objects suitable for his 3dviewer The files produced by my ACD EXE DNL 24 have another format than the DERIVE produced ACD files Oscar sent also a patch to include automatic scaling If you are interested in that then please call or write I ll send it to you add a screen shot of 3dviewer m De X KK s Qe pem piat n N P T We HL TN A M Me A 3dv is still working the DOS environment Josef John Alexiou USA ja72 prism gatech edu I am a Mechanical Engineering graduate student at Georgia Tech We are using DERIVE for WINDOWS for Multibody Dynamics and I was wondering if there are other people out there doing the same I was interested in exchanging some information and maybe MTH files I have noticed that in many packages people use a LIM f x x a fora substitution unfortu nately this is SLOW and cannot handle vectors very well I have a solution to this problem You can use the ITERATE function to iterate once all the variables with their values For example P6 DERIVE USER FORUM D N L 25 1 SUBEQ eq expr ITERATE eq LHS expr RHS expr 1 User X uus NC N 1 en e e P ve d e T5 2 2 2 User Simp User X y X N 1 sid i x s atb y a ui 3 2 2 User X Tow a b 2 2 V a b 1 4 2 2 Simp 3 2 a b
51. m zero to four I think the problem involves evaluating the function at x 0 it is undefined there Perhaps you should define the function F x IF x 0 1 SIN x x and integrate this function from zero to four 2 The interested reader might wish to see the treatment of Romberg integration in my book NUMERICAL ANALYSIS via DERIVE MathWare Urbana IL 61801 USA phone 800 255 2468 ISBN 0 9623629 2 1 Keep up the good work Steven Schonefeld DNL I ll show Terence s example treated with Steven s procedure 4 5 ROMB MTH SIN Cx Eos su 2 d X 24 ROMB O 4 5 VOR 1 4 2 I XXxXU xXx xU Txxxr p 5 2 kx XxU WaEAKM 4 1 gt D WeEEKW e 8 0 5 2 5 5 B Approx 24 16 ES 2 2 2 32 WIS oe Y p 4 DERIVE USER FORUM D N L 25 SIN X 260 E x tr x 0 1 User x 214 ROMB O 4p 5 User 1 4 1 62159 YY XX ox YY XX ox YY XXX 2 2 te 720097 31275232 DONORUM WERTEN 4 ily Ie WAS oe 1 758042 115838 NR 28 Approx 27 8 AES Le Tooo 12581 Suy5920 Id 16 ou tee oro 157153820 T5920 5820 32 125 214575805 b225920 75820 1275820 It is quite interesting that Steven s algorithm seems to need the special definition of SIN x x for x 0 while Terence s procedure does not Oscar Garcia Frederiksberg Danmark Oscar Garcia flec kvl dk 3D plot animation with 3dv If you don t have AcroSpin or even if you do you can still animate DERIVE 3D plots using 3D Viewer This is a freeware program that allows r
52. ns Plot Quit Range Scale Transfer Window aXes Zoom indou aXes Zoom Derive XM i Derive XH y 1 Scale x H 5 y B 5 2D plot ross x BH 5H78 y 8 8943 Scale x B H1 y B H1 zD plot Neil Bibby A day in the life The screen shots above are from DERIVE for DOS from 1997 In fact the left diagram shows a hexagon and a 360 gon The zoom facility of DERIVE allows us to easily see this in the right diagram But how can we calculate x Ludolphus van Ceulen born late 16th century in The Netherlands devised a simple iterative method based on the Archimedian approach to this day x continues to be known as the Ludolphine number die Ludolph sche Zahl in German speaking Europe By applying Pythagoras to two triangles in the following figure we can simply relate the length s of the side of a regular 2n gon to the length s of the side of a regular n gon The relation is that l S 42 44 s Starting with a hexagon of unit side i e n 6 and so 1 we can approximate x as the semi perimeter of the polygon by 3 2 s The ITERATES and ITERATE functions of DERIVE make this very easy to do and we can easily ad just the accuracy of the calculation to avoid cumulative rounding errors occuring 1 0 8 0 6 0 4 0 2 2 k 1 11 vecron crrensres iC itd 853584 1 0 353 2 E PENNE uj k 812 2903 73 1553 E373 E 73 105313 3 D 03 79703 PESCA SII 3 DTI 13 PrecisionDigits 20 14 NotationDigits 20 2 k 1 15
53. otating 3 d wire frames in real time under mouse control What you need 1 3DV EXE Available in 3DV25 ZIP from many archive sites and bulletin boards For example in SimTel s directory MSDOS GRAPHICS on the Web you can look in WWW SIMTEL NET MSDOS GRAPHICS for example 2 An AWK interpreter You can get MAWK by ftp from ftp cdrom com V simtelnet gnw gnuish mawkl22x zip or Duff s awk from SimTel s directory msdos awk etc 3 The following awk scrip to vonvert an AcroSpin file generated by DERIVE to 3DV format POINTLIST NF 0 if NF 4 point 1 2 3 S4 npoints SET COLOR color 3 LINELIST END if NF 2 line nlines S0O color END print npoints POEL 20 Beine VDEIDGODORSUneponmtlmbrerp Ltp print nlines 2 for i in line Lb Sar Cs rre ay 9 Print Pore LLO Pr IN CDOT Save it as ACRO AWK in your DERIVE directory 4 Save the following as ACRO BAT in your DERIVE directory mawk f acro awk acro acd acro 3d CrX3qvNesdv acro 3d This assumes that you have MAWK EXE somewhere in your path and that 3DV EXE is 1 C 3DV Otherwise alter name s and or path s as needed To use from DERIVE 1 Highlight the expression and switch to a 3D plot window Plot on the screen if you want 2 Do Transfer Acrospin Save Use acro for the file name 3 Do Options Execute Enter acro and press ENTER 4 In 3dviewer move around the object with the mouse To exit click a mouse button
54. r on the integration interval but did not have a trick to remove these I expect that the definite integral will be correctly calculated if you use realistic numerical values for the parameters Klaus Fischer Darmstadt Germany Klaus Fischer suggests to include the most important and interesting WWW sites for DERIVE and TI 92 users into the information page Excellent idea we will do that Any recommendations are ap preciated See the Information page Jos Verhoosel Eindhoven Netherland J C M Verhoosel pth nl Dear DERIVE User I teach mathematics for students who will become teacher in mathematics themselves As part of their education they invest in their 2nd year 80 hours of programming programming structures recurrent programming proce dures functions subranges etc Recently I was asked if I was willing to replace this course with a new one Programming in DERIVE I think this 1s a fantastic opportunity for them to extend their and my knowledge of DERIVE Do you know if there exists such a book or course To my knowledge there is none Forced by circumstances the course should start 3th february 1997 and concerns 2nd years students in mathe matics and I really would like to give the course I already found some interesting examples in some utility files of DERIVE and in some Titibits files from Johann Wiesenbauer My question for you Can you give me some more programming problems in DERIVE The problems shou
55. rograms http www studli se chartwell html Publisher http www kolleg nuernberg de ti92 htm W Pr pper s Tl programs ableit galton binvert http www tech plym ac uk maths C TMHOME CTM HTML John Berry in Plymouth http www cms livjm ac uk www homepage cmstetch index htm Terence Etchells DERIVE and TI All these sites are now 2011 only of historical interest Josef Do you know other sites Share your knowledge with us We will visit some DUG members home pages in the future Jan Vermeylen sent a valuable collection of addresses Wait for the next issue Thanks to Jan from Kapellen in Belgium Here is one of his goodies http archives math utk edu Mathematical Archives with among others a lot of historical information D N L 25 Liebe DERIVE und Tl Freunde Sie werden sicher ber das Bild erstaunt sein Ich bin in keinen Jungbrunnen gefallen Das ist eines der j ngsten Mitglieder der DERIVE Gemeinde Kim berly unsere erste Enkelin im zarten Alter von einer Stunde Damit sind nun viele Fragen beantwortet Hallo B rbel amp Co Hier ist nun unsere 25 Ausgabe und ich habe aus Anla dieses kleinen Jubi l ums SWHH und SWHE um einen Beitrag f r den DNL 25 gebeten Es sind zwei Gru worte geworden Seite 37 f r die ich mich herzlich bedanke Ich dru cke sie gerne ab m chte aber die viele Anerkennung die darin ausgedr ckt wird gerne mit Ih nen allen teilen Manchmal kommen aller dings
56. rom which the solution of the linear equation can be read directly This is shown in the following example 2X Fy 3z22z4 gt 3 4 1002 x2 x y z 0 gt 1 1 1 0 S 0 1 0 3 yz3 3 1 l1 2 0 0 1 1 xz This method can be applied to equations with parameters and is shown in the example below in which Sy cy emen tis an arbitrary real number 3x 2y tz 2 3 2 21 100 41 x t l 5x 4dy z22 gt 5 4 I 2 Hec Otel SS yel ey zei 3 2 21 6 0 0 1 t 1 PF hhigebralcats ptherlrantolciear 2 2 However if one selects the following equations with B E 2 a ium Ed adi 3 6 4 a 4 3939 k 4 3 WEGE mid 2x 6y 3z 6 1 4x 3y 4 3z 6 4x 3y 9z k qc cs dod o p 0 sd GIEEPEMJIEPEFEPE3SPE RIN RAD AUTO FUNC 17734 then neither DERIVE nor the TI 92 give the correct 9 amp complete solution In the display the command rref 4 3 3 ES was applied to the augmented matrix of the system of rre Are equations From the result t can be concluded that for allk e Re the system of equations has no solution The bottom row may be interpreted as Ox 0y 0z 1 and gives accordingly a false statement This result is obviously incorrect Because for when k 18 you get a different solution as can be seen from the adjacent display of the screen The solution is read as 3 l k 18 x gt 3 y z 2 zeRe where z is an arbitrary parameter The question is then why the solution for k 18 is not recognized by the
57. rse of vector algebra The students should find out how the intersection of the planes E 2x 6y 3z 6 E2 4x 3y 3z 6 E3 Ax 3y 9z k depends on the real number X 2 The commands mRow und mRowAdd are very well suited to helping students master the method of solving systems of equations They can concentrate fully upon the method since the calculator relieves them of the numerical work D N L 25 The TI 92 Corner Simult Equations p 53 As a devoted DERIVIAN I tried to reproduce Karl Heinz s ideas using DERIVE Josef found two ways to do so Using the elimination addition method by defining the singular equations or using a special function from the Utility file VECTOR MTH I m sure that this file doesn t need any additional explanation 1 The system y User 22 SOLVE Z Xt6O y 3 2 296 4 Xvt3 y43 220 42 x 3 y 49 z kl DXx y z User 3 Simp 2 4 gl 2 x 6 y 3 z 6 92 4 x 3 yt3 z 6 93 4 x 3 yt9 z k User 5 Define two equations hl and h2 User 6 hl 292 2 3G01 h2 793 2 g1 User 2 x 6 y 3 z 6 7 gl hl h2 9 z 9 y 18 Expd User NN Area el aS ee OR 8 ll 23 h2 25 hl User 2 x 6 y 3 z 6 9 gl hl 11 9 z y 18 User Simp User O 3 k 18 PLOs File VECTOR MTH Copyright c 1990 1994 by Soft Warehouse Ino User 11 defaultl 1 User SUBTRACT ELEMENTS v Tl deraultlk VECTORGLE m i Vo defaultl v ELEMENT v 12 i J m m DIMENSIO
58. s Each chapter includes the development and algebraic analysis of the methods lab based activities 1deas for coursework case studies exercises and solutions Free supporting utility files are downloadable via Chartwell Bratt s web server Chapter 1 introduces the basic tool of numerical methods which is recurrence relations their solution and ill conditioning problems In chapter 2 we use recurrence relations methods that are used in solving equations Chapter 3 deals with the approximation of functions by polynomials and in particular the Taylor Polynomial which is then used extensively in chapter 4 to analyse the errors associated with numerical methods Chapters 5 and 6 deal with numerical ap proaches to the calculus of differentiation and inte gration In chapter 7 we introduce and analyse nu merical methods of solving differential equations ISBN 0 86238 468 0 239 pages Chartwell Bratt February 1997 INFORMATION T dn Book Shelf D N L 25 Mathematical Activities with Computer Algebra a photocopiable resource book from Etchells T Hunter M Monaghan J Pozzi S and Rothery A This photocopiable resource book is the first of a new generation of support materials for the educational use of computer algebra Designed to be used with any computer algebra system the authors go beyond mere button pressing and show how to harness the power of computer algebra systems for educational purposes Concepts are illustra
59. s enables students to have their own graph ics system in the palm of their hand so my preferred way of working was to inte grate the use of the two where this was possible The students can create their own version of the DERIVE screen display to reinforce what they have seen on the DERIVE display It will become apparent that my examples for the most part use the graphic capa bilities of DERIVE I do not claim that DERIVE is uniquely suited to this no doubt much of what I present is equally implementable on other graphics systems How ever the graphics system of DERIVE is flexible and easy to use once one has mas tered a few basic techniques Example 1 x and Van Ceulen s algorithm British Year 9 Grade 8 It is surprising that the treatment of the topic of x is so little changed by the avail ability of microtechnology Apart from encounters with recurring decimals it is the only place in elementary mathematics where the infinite process has to be really faced up to Now that we have the computational tools there seems little to stop us from doing just that The Archimedian approach of starting with a regular hexagon of unit radius and then doubling the number of sides is now eminently feasible on DERIVE It s easy to draw regular n gons as the following diagram shows EN E 1 5 g 5 1 5 2 B 5 1 5 OMMAND frame Center Delete Help Move Options Plot Quit Range Scale Transfer OMMAND Center Delete Help Move Optio
60. s two years ago when the use of electronic cal culators were introduced in the last three terminal years 10 11 12 but unlikely were forbidden in the final examinations I personally think that would be examination to test the skill of pupils in Calculus because I think it will be more important in their future life than some subjects that has been taught in school benches I m an electronic engineer who works and teaches the last form in High School and the first two years in University in Maths Analysis and Linear Algebra matrices At the other hand we have to congratulate for the excellent quality achieved with the new DERIVE for WINDOWS that I have been using for a month or so it comes with some new fine fea tures and is even easier to work quicker and powerful Hoping to hear from you soon in meantime I wish you a nice and profitabel 1997 Yours H M P S Soft Warehouse Hawaii swh aloha com Hello Power Users of DERIVE To keep you up to date the following is a summary of the enhancements made in DERIVE for Win dows version 4 03 1 The quality of printed screen images of 2D and 3D plots has been improved by the elimination of spurious points 2 The date time strings on printouts now use the format dd mm yy or mm dd yy appropriate for the user s own country Comments in the DMO files are now handled like they were in the DOS version of DERIVE 4 A File Change Directory command is also included in the 2D and 3D plot windows
61. schwingungsf higem Fundament m haben wir den wohl bekannten Fall der Federschwingung Federkonstante c mit periodischer Fremderregung x die hier beim prinzipiellen Ansatz von Figur 1 am Aufh ngepunkt der Feder angreift one Xe A rotating machine with eccentric moveable parts is placed on a basement which should keep as motionless as possible when the machine is running Sometimes the optimal solution is not to have the mass of the foundation as big as possible With an oscilllating vibrating base dn ment m4 we recognize the well known case of a spring oscillation spring constant c with periodical separate excitation which acts in the spring s suspension point figure 1 Due to Hooke s Law and Newton s Dynamic Fundamen f Xi tal Law we obtain the motion equation when neglecting damping m Figur 1 Mit den Daten c 1 m 2 und x 4 cos V3 t und der realen Federdehnung x x ergibt sich die Bewegungsgleichung nach dem Hookeschen Gesetz und Newtons dynamischen Grundgesetz wenn D mpfung vernachl ssigt wird zu 2X mex Xx oder or x T0 5x 2cos V3 r Man kann sie exakt l sen mit 3 DSOLVEZ TUCU 0 5 2 DDR C3 EX b 0 0 0 4 2 6 2 J2 Tx 2 6 2 P cos e Jes SLE 5 5 2 2 5 5 J2 Jat 2 6 2 J2 Jovr 2 6 suf Jes e 5 3 est 2 2 5 5 2 2 5 Fast i es 2 2 J2 t 2 sn e s J ee E 5 2 2 5 Leo Kling
62. se the trace to determine how long it takes the ball to reach your friend s path Now you can decide how long to wait If you want DERIVE to solve for analytically you can also do that once you have the value for 9 Go back to the Algebra Window and substitute 7 2 into the parametric equations for your friend s position and then set P t with given as the value you found from your graphical exploration equal to the x coordinate of your friend s position This example shows that it 1s possible for students with only a knowledge of trigonometric rela tions to solve a very sophisticated problem using that knowledge and some good graphics We grant that some analytical steps were bypassed and the answer may be an approximation to the correct answer but was the student any less involved with the mathematics We leave that question for your decide The TI 92 Corner Simult Equations p 51 THE TI 92 CORNER EDITED BY B WAITS F DEMANA B KUTZLER amp J B HM DERIVE and T1 92 do not always yield the complete set of solutions to systems of linear equations containing parameters Karl Heinz Keunecke Kiel Germany kh KeuKiel NetzService de At school in the field of linear algebra it is often helpful to make use of CAS to solve systems of linear equations After the input of the augmented matrix students can calculate the row echelon form of the matrix by means of the DERIVE command Row Reduce or with the command rref of the TI 92 f
63. sed this technology So this toast of appreciation is not only to Josef and Noor but also to all of the past and future authors Your ideas are enriching mathematics for students educators and mathematical hobbyists all over the world DERIVE Aloha and Mahalo IN TI 92 The 25th issue of the DERIVE Newsletter This is a wonderful moment The 25th DNL This means that I have reveived 24 issues before There are 4 issues a year 24 divided by 4 gives wait a moment I can do this without DERIVE just another second 6 I think this gives 6 What were we talking about Oh yes 6 years Yes it is true The DERIVE User Group serves a growing community of DERIVE users and DERIVE enthusi asts for more than 6 years now The DNL 1s the major vehicle for communication among DUG members And it serves its purpose very well It is informative challenging and critical The DNL is a valuable source of hints for the newcomers it s a highly appreciated source of know how for the experts and it is a very serious and important means of feedback for the authors of DERIVE I have seen both David Stoutemyer and AI bert Rich sitting in their office in Honolulu studying the newest issue of the DNL very carefully sometimes with deep wrinkles on their foreheads I cannot remember a single one of the 24 issues having been published late Knowing Josef personally for more than 8 years now this is no surprise He is one of the most diligent an
64. t to show you something that is good for comparing its speed with other versions of DERIVE and for checking the speed of your computer also It is the following math problem 2 101 1 which expands quickly gives 2535301200456458802992406410751 which factored slowly 7432339208719 341117531003194129 D N L 25 DERIVE USER FORUM p 11 Several years ago I ran this on a 286 8 MHz machine and it factored the number in 12 hours 20 min utes cpu time That was with DERIVE 1 6 I wondered if it could even do the problem before I finally let it run overnight and returned the next day to find that it worked Since then I have tried it on some much faster computers with newer versions of DERIVE Here are the cpu times for some other computers and versions of DERIVE DERIVE for WINDOWS 486 50 22 3 minutes 486 100 12 1 minutes DERIVE 3 486 50 47 0 minutes 486 100 25 4 minutes DERIVE XM 486 50 24 0 minutes 486 100 13 6 minutes Here DERIVE 3 and DERIVE XM were running under DOS only with WINDOWS off By the way running DERIVE 3 under WINDOWS slows it down by a factor of only about 1 or 2 percent but DERIVE XM is a disaster in WINDOWS which slows it down by about a factor of five It takes XM over two hours to factor the number of running under WINDOWS on a 486 50 So it seems DfW is even a little faster than XM This is a fun problem to run on DERIVE I would like to see the speed of a Pentium 200 on it If you
65. ted techniques and methods presented and model ling and applications are explained Appendices give overviews of DERIVE Maple Mathematica Theorist MathPlus and the new TI 92 calculator Activity Worksheets Help Sheets and Teaching Notes cover a wide range of mathe matical topics at school and college level Topics covered include functions and graphs differ entiation integration sequences and series vectors and matrices mechanics trigonometry numerical methods Activities include Multiplying factors Equation of a tangent Taxing functions The tile factory Function and derivative visualisation The approximate derivative function Sketching graphs Pollution and population Max cone Optimising transport costs Area under a curve Enclosed areas A function whose derivative is itself Wine glass design The limit of a sequence Visualising Taylor approximations Visualising matrix transformations Blood groups Circular motion Swing safety No turning back Modelling the sine function Solving equations with tangents 20 Pounds 96 pp 1996 ISBN 0 86238 405 2 Both books are available in 2011 e g AMAZON Josef In the last DNL I gave a wrong ISBN for An Introduction to the Mathematics of Biology Here is the correct one ISBN 3 7643 3809 1 Interesting WEB sites http www swp co at swhe swhe html Soft Warehouse Europe http derive com Soft Warehouse Hawaii http ti com calc for Tl 92 p
66. ties in the inter war years despite fascist attempts co encourage rurality and che surge in the period 1950 70 Naples was taly s largest city until che First World War Source B R Mitchell International Historical Statistics Europe 1750 1988 N Y 1991 We decided to concentrate on Milan from 1880 to 1980 not necessarily the easiest choice why the data can be plotted as follows 1800 population time 1800 10 The problem now was to try to fit a curve of the form y a b to this data with y as the population in thousands and x as the number of decades from 1880 Initial suggestions usually were a 322 and for the choice of b to give the visual best fit to the other points Since the doubling time appeared from the table to be about 4 decades by the rule of 70 this gave a growth rate of about 18 since 70 4 18 and hence b x 1 18 Further discussion led to the method of visual best fit being questioned and the TI 82 s statistical ExpReg function being used to provide a least squares fit I hasten to add that the intricacies of the least squares method were not discussed the TI 82 function was therefore treated as reliable and trusted blackbox which would give an appropriate result DERIVE s FIT function could also have been used for this purpose The values obtained were a 412 5 b 1 198 4 s f and plotting y a b with these values gave the following result Neil Bibby A day in the
67. uare 1 Notation Decimal NotationDigits 3 2 PJ J2 0 0 MN v 2 17 2 29 2p EP P J2 J2 3 a z b E d ee 3 2 0 2 15 2 272 2 2 2 0 0 UD 2 2 02 to 2 2 0 3 J2 3 2 20a 2 2 J2 0 J2 2 eS eee 3 32 2 3 J2 J2 m 0 0 J2 J2 2 2 SEES J2 0 2 2 J2 0 2 J2 4 d You might wonder about the colours with the DERIVE plot This is easy to perform with DERIVE 6 Transfer by applying Mark and Copy the figure into the Algebra Window Mark the graph and activate Convert Picture Object in the Edit Menu Then you can switch to Paintbrush and treat the graph as you like See also Tania Koller s note in DNL 63 3 page 14 5 Two basic functions to play 6 Tx y amp U v t COSCa x SINCa y u SINCa x COS a y v MOVEMENT w u v VECTOR T Cw QW amp u V 1 DIMENSION w 7 eae 1 2 The first one moves a point x y by a rotation of angle a followed by a translation of vector u v The last applies a movement T to a whole fan w As I said before we can add REFLECTIONS or any other affine motions we desire Now the goal is to find the movements needed to bring the tans from its initial coordinates to the new ones required to form a figure Let me explain it with an example We desire to sketch the Yacht in figure 2 After discovering where the tans must be put this can also be tried using DERIVE but it can make us spend a lot of time
68. ugh that non English articles will be warmly welcomed nonethe less Your contributions will be edited but not assessed By submitting articles the author gives his consent for reprinting it in DNL The more contributions you will send the more lively and richer in contents the DERIVE Newsletter will be Preview Contributions for the next issues 3D Geometry Reichel AUT Algebra at A Level Goldstein UK Graphic Integration Linear Programming Various Projections B hm AUT A Utility file for complex dynamic systems Lechner AUT Examples for Statistics Roeloffs NL Linear Mappings and Computer Graphics K mmel GER Solving Word problems Textaufgaben with DERIVE B hm AUT Line Searching with DERIVE Collie UK About the Cesaro Glove Osculant Halprin AUS Hidden lines Weller GER Fractals and other Graphics Koth AUT Experimenting with GRAM SCHMIDT Schonefeld USA Implicit Multivalue Bivariate Function 3D Plots Biryukov RUS The TI 92 Section Waits a o and Setif FRA Vermeylen BEL Leinbach USA Halprin AUS Speck NZL Weth GER Wiesenbauer AUT Aue GER Pr pper GER Koller AUT Stahl USA Mitic UK Tortosa ESP Santonja ESP Wadsack AUT Schorn GER Chaffee USA and Impressum Medieninhaber DERIVE User Group A 3042 W rmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef B hm Herstellung Selbstverlag D N L 25 DERIVE USER FORUM p 3 Terence
69. uit with possibly amplified frequency You could try to simulate this situation D N L 25 Peter Mitic Probability Distributions 2 p 33 Probability Distributions Proof and Computations 2 Peter Mitic Medstead UK MEANS AND VARIANCES OF CONTINUOUS RANDOM VARIABLES Similar problems are encountered with continuous distributions for which the moment gen erating function is M t Je fo dx where f x is a probability density function When cal culating the mean and variance of some continuous random variables it appears that DERIVE is more successful in evaluating integrals directly rather than via a moment generat ing function The DERIVE session below shows that integrals which do not involve the pa rameter r can be evaluated easily Integrals which contain this parameter cannot always be evaluated even if the range of the parameter is restricted such that the integral is convergent t 1 2 in the expressions 5 7 below In particular it is easy to check that the area under the graph of a probability density function is 1 expressions 2 to 4 below The probability density function used F x in expression 2 is that of a y 5 random variable xf2 3 2 Jere X 1 Eug ss SER 2 Check that the area under the graph is 1 m 3 Fol de 1 xat 42 MCR EFE ner 09 x t 1 2 3 2 Jere X 5 Met 0 G aiT 6 t e Real e 1 2 wet d s 372 fore x E
70. we can find the movements by testing or calculate them exactly I got the Yacht with these Simplify and plot the result in Connected Mode N 8 woven 0 9 MOVEMENT c 0 O PS EDD au woven NM 10 MOVEMENT 4 n 1 12 MOVEMENT a 2 5 0 Fi 1 p 46 ac DC four D N L 25 3 0 T 1 1 5 1 13 MOVEMENT B 2 42 t108 4 2 J42 SIN x 2 2 00S 3 12 12 2 2 12 12 io concen aqati l 5 1 2i2 sin ee Ne es 2 12 MENU xz Of course the last ones cannot be obtained at first sight The complete procedure can be found in file DERITANS MTH The next two functions can be helpful to find angles and distances S EE S EET 2 1 2 1 EINES 2 V DIST x_ y_ o x_ 1 1 D N L 25 AC DC Four p 47 You can find much more about tans in the mentioned reference To practise I propose you two different problems 1 Try to find the movements to compose these figures 2 What is the biggest area that a tangram can contain In 1 it is given a solution but as far as I know it is not still proved that this is the biggest one This is an open problem how it goes REFERENCES Finally two crazy suggestions It could be funny to play with DERIVE and the tans among several players trying to find the movements before the opponents Or perhaps anyone could think about

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