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THE DERIVE - NEWSLETTER #24 USER GROUP

1. ZQMT is an exterior angle of the triangle QOM hence equals 2 APMQ is another isosceles triangle with ZQMP ZQPM 90 p 7 ZTMP o 4 90 and ZTMP ZQMT ZQMP 29 90 90 4 lt 39 o and to 2 2 3 2 So if we bisect the angle ZMOQ o we obtain a third of the given angle a This leads obviously to Anmerkung zum WorldWideWeb Interesting WWW pages Highly recommended The MacTutor History of Mathematics archive http turnbull mcs st and ac uk history Index of Biographies http turnbull mcs st and ac uk history Bioglndex html History Topic Index http turnbull mcs st and ac uk history Indexes HistoryTopics html Famous Curves Index http turnbull mcs st and ac uk history Curves Curves html Mathmeticians of the Day http turnbull mcs st and ac uk history Day files Now html Additional Matrials htto turnbull mcs st and ac uk history Indexes Extras index html 2K 2K ok 2 OK ok 2k OK ok 2k OK ok ok 2K ok ok 2 ok ok 2 OK OK ok 2K OK ok 2K OK ok 2K Ke ok OK OK OK Ok 2K OK OK OK OK OK OK OK OK Ok 2K OK Ok OK OK OK OK OK OK ok OK OK Ok OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK OK KOK OK KK OK OK KKK Two screen shots of PLOMAT house from the TI 92 to plot matrices of points left side Points discrete and large right side Points Connected c lz onitrscelRe rsenltstnlorsulr 1 ME Esn icelkesraenhsrtnorss e J J p44 AC DC 3 D N L 24 Two other
2. circumference and length equal to RS Finally BE BM and X such that EX is parallel to CD What is the relation between BX and n REFERENCES 1 Boyer C B Historia de la matem tica Alianza Editorial Madrid 1987 English version A History of Mathematics John Wiley and Sons Inc 2 Gardner M Nuevos Pasatiempos Matem ticos Alianza Editorial Madrid 1982 English version New Mathematical Diversions Scientific American N Y 3 Kline M El pensamiento matem tico desde la Antig edad a nuestros dias Alianza Edi torial Madrid 1992 English version Mathematical thought from ancient to modern times Oxford University Press Inc 4 Proceedings of TEMU 95 Servicio de Publicaciones de la Universitat Polit cnica de Catalunya Barcelona 1994 5 Rich A D and Stoutemyer D R Inside the DERIVE Computer Algebra System The International DERIVE Journal vol 1 number 1 April 1994 6 Steinhaus H Instant neas Matem ticas Salvat Ediciones Barcelona 1989 English version Mathematical Snapshots Oxford University Press Inc N Y Probability Distributions Proof and Computations 1 Peter Mitic Medstead UK ABSTRACT The use of DERIVE to perform relevant algebraic computations in the context of probability distributions is discussed with particular reference to some common distributions Problems with using DERIVE in this way are noted and some partial solutions are suggested INTRODUCTIO
3. 1 File LODES V amp MTH 3 2 2 RCP a b c SOLUTIONS x ax bx c x Tifa b e x EXPCCRCP a b cl x 3 al Teta b c x EXPCCRCP a b cl x 4 2 D N L 24 J L Rodriguez amp M J Fern ndez 3rd order ODEs p37 D y 2y 5y 6y 0 Check the solution jx x dX 33 SGH 2 5 6 c3 e clea c 2 ETF x fu 34 vx is che cl cz2 e 35 yr ix Bela Bey ix 8 vix 0 2 y y 2y Check the solution ce x x SaH 1 0 2 e ecz2 COSC X c3 SIN XJ cl e x x 38 yla z e Cc2 COSfx c3 SIN x cl e 39 rl vi Ex 2 v x y Sy 5y 11 2 41 GH 5 5 11 cle e c2 COSCU2 x c3 SINCJ2 x2 4 y 4y Ty 65 y 0 1 y 0 y 0 0 Find the solution and check the result DX x 43 SPH 4 7 6 x 0 1 0 D e 42 e SINCJ2 x 2x x 44 yix is 2 Fre SINCI2 x 45 y tx dey x Perle 6 v x 0 46 y Q y CO y 02 1 0 0 5 y 3y 4y xe cosx Check the result 2M 48 Sall 3 0 4 x xea COStx DX 3 2 a G x S x G x 2 c2 1 162 cl 2 X 0056x SIN x 49 HE E 152 50 gO fox 3 2 e Bey S x G x C27 c2 1 162 cl 2 X 7 COSCx SINX 50 yx i c ir St 152 gO 50 1 yi ted Sey Go dere 2x 2 2x 2 a 9 x 15 x 2 27 c2 2
4. ity generating function for a Binomial n p random variable The motivation for introducing the moment generating function is that it is required to prove the Central Limit Theorem which we aim to discuss in a later paper The same problem is encountered in computing a sum in which there is a symbolic limit In the case of the moment generating function the relevant sum is M p n t we p p GC x 0 Unfortunately DERIVE cannot simplify it to obtain the result M p n t p e 1 p This illustrates a general inability of DERIVE to handle such series which have proved to be trouble some in other computer algebra packages as well However once it has been determined by hand DERIVE can very easily perform the necessary calculus by calculating M 0 and o M 0 j A particularly compact way of doing this is illustrated in the DERIVE session below 1 BR DISTL MTN User t n 2 M p n t p 1 p User d 3 mean lim M p n t User E gt 0 dt 4 np Simp 3 d 42 2 5 Variance i Lim M p 1 mean User t gt 0 tdt 6 n p 1 p Simp 5 In the case of a Geometric random variable the same problems are encountered There is an additi onal problem in that given a particular case DERIVE is only able to obtain a result for the moment generating function M t of such a random variable if the range of the parameter t is restricted so that the series is convergent This problem
5. z iL w L 1 Z L W L 2 Z Complex Solution performed with DERIVE 6 3 w Complex 4 2 2 2 zc w L J 72 Br Azul 1 4i z 273 4 SOLVE 0 wl w w Z L W L 2 2 L Z 4 2 2 2 e B ZI a Aa 2 19 bz Se ML ea 2 2 i Z 4 2 2 2 zei w Jt w 60 Tludruwxtw Dy se L w 1 5 SOLVE 0 Z z v Z tL W L 2 2 L W 2 It wos Bw 1 ie 19 btw 1 2 2 L W Gert von Morz Hannover Germany As a Power User of DERIVE I came across this bug in Version 3 02 SUM COMB n k k 0 n Can you tell me why DNL As have learned form the DERIVE manual DERIVE uses antidifferences to calculate sums D N L 24 DERIVE USER FORUM p11 As with antiderivatives closed form antidifferences may not exist in terms of the op erators and functions known to DERIVE Even when such an antidifference exists there is no known method that is guaranteed to find it DERIVE manual page 202 It could be a little consolation that in this case unlike to the problem above even the TI 92 gives up n n 6 gt coMB n k 2 Solution performed with DERIVE 6 k 0 Ian D Souza Montreal Canada ian vir com DERIVE Users Help I need answer for this one I m running WINDOWSO95 on a Pentium 120 with 32Mb RAM DERIVE for WINDOWS can
6. cedure provided on the German WIKI page Wm s ls ie zu es nk n lk l Dr E n l 2 1 b q Pu with Loo f a f b 2 X f ati h and h Then Zan will deliver the n approximation The next diagram explains the calculation scheme I BN h E I h gt I gt 13 3 I E I I I 4 1 4 2 4 3 44 a 7 77700 The program is displayed on the next page Now I wanted to know if my implementation is working and how the calculation times are perform ing compared with DERIVE s numerical integration and with Terence s and Steven s procedures I am starting with Terence s integral from page 3 Let at first do DERIVE its job needing nearly half of a second for performing the integration BI http de wikipedia org wiki Romberg Integration D N L 24 Josef B hm Comments on the ROMBERG Method p15 4 SIN 3 dr x 0 4 1 75820313894905 0 469 sec This is my DERIVE Romberg program rom prog f x a b n eps st rtab n k r_ i Prog st VECTOR Cb a 2 n 1 2 CLIMCf x a LIMCf x b 2 I LIMC f x a i b a 24 n_ 1 i 1 2 n_ 1 DD n n rtab APPEND st 1 VECTORC j n 125 o 18 72 Loop If n gt n exit K 9x2 r_ st n_ Loop If k_ gt n_ exit i rtab n 1 k_ 1 rtab n_ D k_L 1 FIRSTCREVERSE r_ 2 C2 2 k 1 r_ APPEND r_ i k_ 1 rtab APPEND rtab APPEND r VECTORC j
7. dey x COSQ2 xd i t 75 SINCZ x 3 3 3 SINLZ x SIN COS x J SIN xJ COS x D N L 24 J L Rodriguez amp M J Fern ndez 3rd order ODEs p39 76 Simplify the difference of both right sides 4 B 2 1 4 CS A t l ll 77 SIN Z x 3j 3 3 TAN C2 303 SIN Z x SIN x COsck SIN x COS x J a COSC2 x 2 1 E m 78 3 3 3 3 SINC2 x SIM x COS 0x J SIM x COS x J 79 Trigonometry Collect 80 Trigpower Sines 81 E 52 Result from 1996 COSA x COS 2 xJ 2 c2 dx 83 CUSC2 xJ LNCTAN CE 33 LNCSIN Z x33 COSt4 x SIMCZ xJ 1 o4 c c3 SINE cl 4 2 4 2 4 84 simplified LNCSIMCZ xl J COSC2 x d LNCTAN CE 13 1 85 A c2 O0502 x c3 SIMC2 x cl z 2 2 e fe 9 y 3y 2y l e a Find a particular solution b Find the solution with y 1 22 y 1 23 y 1 24 aX e 87 SPI 3 2 O x x l e x x 2x LNfe 1 3 x x 1 LNle 1 850 e e LNle 1 2 4 2 2 ax e 89 SPLI 3 2 0 x cd ee x l e x x e 1 e 1 LH LH 2 x 2 90 e l 2x e l e 3 x e 1 1 1 e e 1 e e LH e fe 2 2 2 4 e 1 2 4 2 2 5 REFERENCES 1 Bronson R 1989 2500 Solved Problems in Differential Equations McGraw Hill 2
8. 3 Assuming w l o g that y z we are left with 0 il N5 1 V5 1 2 Er A eae 3 SOLVE2 x y 1 y y 1 x 2 2 2 2 N5 1 V5 1 D N L 24 DERIVE USER FORUM The other two cases x y and x z respectively are obtained by simple permutations of the solutions above They yield two additional solutions namely 1 0 1 and 1 1 0 T TBITS 9 deals also with this problem SOLUTIONS of DERIVE 6 and solve of the TI 92 have no problems solving this non linear system Josef 2010 SOLUTIONS x y 1 zax z 1l vyayz 1 x x v z l o 1 c R sebralestclbiherlPranzoleieen ue 1 0 1 1 alt 0 J5 1 5 1 J5 zk Bsolve x y i z and x z 1 y and uzb eee EEE cie z J5 1 Jc 3 J5 1 2 2 2 2 2 2 eet ads ooo BEE are 5 1 J5 1 J5 1 d z 1 or x 0 and v 1 and z 1 Ep cr E dg Mcr me MAI RAD EXACT FUR 1 50 2 e e e Heinz Rainer Geyer St Katharinen Germany Vielen Dank f r Deine Materialien Bei meinen Bem hungen um Teilermengen habe ich eine Funk tion SET und die entsprechenden Mengenoperationen in DERIVE vermisst Umst ndlich war auch dass die IF Funktion keine echte Leerausf hrung zul sst Das ist schwer zu bearbeiten TTD M glicherweise l sst sich die direkte Farbwahl hnlich wie in BASIC direkt implementieren In meiner 6 Klasse bin ich gerade bei den Teilern und Vielfachen Ich habe etwas herumexperimen tiert Hier findest Du die Ergebnisse meiner Zusammenfassung 1
9. Kent R Saff E B 1992 Fundamentos de Ecuaciones Diferenciales Addison Wesley Iberoamericana 3 Soft Warehouse 1992 DERIVE User Manual This file shows once more the power of a CAS in the hands of an experienced user and it raises once more the question about the necessity of so many drill examples in the text books for the future And the future may have still begun Josef p40 Thomas Weth A Lexicon of Curves 9 D N L 24 Ebene Algebraische und Transzendente Kurven 9 Thomas Weth Wurzburg Germany Pascalsche Schnecken Snails of Pascal Wenn alle die Kr fte die bei der Entwickelung einer Pflanze mitwirken mathematisch erkannt w ren und ebenso der innere Mechanismus ihrer Organe so w rde man im stande sein die ganze Lebensentwi ckelung durch Formeln darzustellen insbesondere w rde man die Gleichungen derjenigen Kurven erhal ten k nnen welche den Umri ihrer Bl tter darstellen Aber umgekehrt wenn man auch diese Gleichun gen kennte w rde man dennoch nicht das Leben jener Pflanze durch Formeln darstellen k nnen doch auch von diesem Ziele ist man noch weit entfernt indem man sich begn gen mu die Blattumrisse durch Gleichungen darzustellen die nicht exakt sondern nur in einfacher Weise angen hert diese wieder geben Loria 1902 S 307 Bodo Habenicht versuchte ausgangs des 19 Jhdts dem Geist der Zeit wis senschaftlicher und technischer H chstleistungen R ntgenstrahlen Eiffelturm entsprec
10. MS 158 43 25 au 45 50 69 3 105 136 E30 250 3458 575 690 1150 1725 3450 26 DSETCG25 1 5 75 125 G25 Ich bin sicher es gibt bessere L sungen Jedenfalls ist mir der Unterschied zwischen Simplify und approX deutlich geworden Heinz Rainer complained that there are no set operations in DERIVE he also would like to have a true not execution the is not very comfortable to work with He also has the idea to choose the plot colour directly by a command similar to BASIC Working with divi sors and multiples in his 6 form he experimented with sets of divisors See Rainer s results He is sure that there are better solutions but he has learned the difference between Simplify and approX In DERIVE 4 x you will find Set operations And the DIVISORS are also imple mented DEAMEIUEICGERI 1 2 3 5 6 10 15 23 25 30 46 50 69 75 115 138 150 230 345 575 690 1150 1725 3450 George Freeman and Al K pf Kuweit City Dear Josef we have come across this bug in DERIVE 3 02 x x Fix 34 G x is z 2 Jl x JU x x 1 2 G F XxIJ 2 3 Fial x SIGN x 1 Can you tell us why D N L 24 DERIVE USER FORUM p 7 DNL Try the following 4 x Real 1 1 5 G F x 1 x 5 FiG xJ x believe know George and Al Kopf seems not to be an Arabic name Yussuf Rudiger Baumann Celle Germany Mit ihre
11. Rollt ein Kreis au en auf einem festen Kreis ab so beschreibt ein markierter Punkt auf dem rol lenden Kreis eine Pascalsche Schnecke e als Inversionskurve Bildet man einen Kegelschnitt durch eine Inver sion an einem Kreis ab dessen Mittelpunkt mit dem Brennpunkt des Kegelschnitts zusammen f llt so erh lt man Pascalsche Schnecken e qals Ortslinie merkw rdiger Dreieckspunkte Betrachtet man zu einem gegebenen Kreis Seh nendreiecke deren eine Ecke A festliegt und bei dem der zugeh rige Winkel a konstant ist so ist der Ort der In und Ankreismittelpunkte aller derartigen Dreiecke eine Pascalsche Schnecke e qals Ortslinie eines Winkelscheitels Bewegt sich ein konstanter Winkel so dass seine Schenkel zwei feste Kreise st ndig ber h ren so beschreibt sein Scheitelpunkt eine Pas calsche Schnecke D N L 24 Other ways to obtain a Cardioid It is worth to be mentioned that we can ob tain a Snail of Pascal in some other ways e as Trochoid A point on a circle rolling outside on a fixed other circle describes a Snail of Pascal e asa Curve of Inversion If we map a conic by an inversion at a circle with its centre lying in a focal point of the conic then we again receive a Snail of Pascal e as locus of remarkable points of a tri angle Observing triangles inscribed in a given circle with a fixed vertex A and the accom panying constant angle o then the locus of the centres of all the incircles and excircles
12. Set of divisors 1st attempt gt Jn n n 2 DIVISOR1 n vecron 1r ron 1 o Inch in i i n n 3 DIVISOR2 n vecronl ar rvoor 1 o i FLOOR n 1 n 4 PART1 n SELECT k z 0 k DIVISOR1 n 5 PART2 n SELECT k 0 k DIVISOR2 n 6 DIVSET n APPENDCPART1 n PART2 n 7 DIVSET 3450 8 11 2 37 5 5 10 15 23 25 304 400 50 B9 732 LIS 138 130 7230 345 53 5 BEU 10 1725 3450 9 0 11 sec in 1996 10 8 seconds only by approXimating 7 10 Attempt 2 Why not counting until n n n 11 DIVISORO n vecronl ar roor T o Til n 12 PARTO n SELECTCk 0 k DIVISORO n 13 PART0 3450 P6 DERIVE USER FORUM D N L 24 14 211 2 3 5 5 10 1 23 25 30 48 30 BB 75 1215 138 150 230 3453 575 590 1150 1725 3450 15 Again 0 11 sec in 1996 it needed 10 3 sec 16 Attempt 3 it should work faster if working with half of the divisors n 17 PART3 n vecron k mao FIS PART3L3450 3450 1775 1250 G80 5 5 345 230 150 138 115 75 58 19 D SET n APPENDCPART1 n REVERSE PART3 n SAUL ESET saa 117 2 53 553 00 IU 715 2735 25 30s 45 750 BS CES LCTIS LOCIS 150 7230 18 575 690 1150 1725 3450 21 seems to work but 22 DESETER2SI 1 5 275 75 175 625 23 hence DSET n If Jn FLOOR n 24 D_SET n DELETE ELEMENT D SET n DIM D_SET n 2 23 CDSE A450 I1 4 3 5 B
13. The routine is on the next page It is only of historical interest because in the meanwhile the SORT routine has been implemented Josef D N L 24 Josef B hm Comments on the ROMBERG Method p13 SWAP ELEMENTS v i j VECTOR IF m i ELEMENT v j IF m j ELEMENT v i ELEMENT v m m DIMENSION v FIND MIN v k m IF k gt DIMENSION v m IF ELEMENT v k lt ELEMENT v m FIND MIN v k 1 k FIND MIN v k 1 m SORT AUX v i IF i DIMENSION v v SORT AUX SWAP ELEMENTS v i FIND MIN v i i i 1 SORT Vi SORT AUX V1 SORT 7 4 3 795 9 5 ir 3 4 2 7 9 forwarded the other question to one of our ITERATES RECURSION specialists Josef Lechner and hope to receive an answer Some Comments on the ROMBERG Method Josef B hm The Romberg Method Werner Romberg 1909 2003 is an improvement of the trapezoidal method for numerical integration applying the Richardson Acceleration or Richardson Ex trapolation in order to improve the convergence of the method In Steven Schonefeld s book I found The plan is to calculate the trapezoidal rule approximation for h b a h hy 2 h hy2 and then apply Richardson s improvement several times to increase the accuracy of the approximation to the integral Of course the first Richardson im provement on the trapezoidal rule results in Simpson s rule Steven provides a DERIVE implementation for DERIVE for DOS of cours
14. User Forum Terence Etchells You will find W Pr pper s wonderful DIRA for discussing curves in a German and in an English version DNL 23 I also have the pleasure to enclose D Stoutemyer s Program Library together with a printable docu mentation Many thanks David DIRA and the library are not on a Word document because they are too big At last want to point out that there is lot of useful Tl stuff to be downloaded from the Internet Web pages of TI SWH At last I m glad to announce a lot of Tl contributions for the next year Most of them will be interesting for DERIVE Users too So I think that E Laughbaum s contribution for the TI on the next pages can easily be transferred on PC DERIVE see page 47 There will be among others Dynamic System Permutations Numbers of Bernoulli Continued Fractions R Schorn Mortgage Ta bles Tania Koller The Simplex Method on the TI Bruce Chaffee Diophantine Equations amp The Gauss Seidel Method L Tortosa amp J Santacruz Endpoint vs Interior Extrema White amp Leinbach Investigations on GCD and LCM with the TI Griebel First experiences with a Tl in the class room B hm SOLSYST A FUNCTION TO SOLVE SIMULTANEOUS EQUATIONS BY W PR PPER N RNBERG GERMANY Solsyst sys var Func Local i j n rowDim sys gt n newMat n n 1 For i 1 n For j 1 n d left sys i 1 var j 1 gt mtx i j right sys i 1 gt mtx i n 1 EndFor EndFor det subMat mtx 1 1 n n gt
15. der letzte DNL des Jahres 1996 ist fertig W hrend er nun zum Drucken geht wird noch rasch die Diskette des Jahres 96 randvoll gepackt und berpr ft ob auch alle Dateien und Weihnachtsgeschen ke drauf sind Dann werden meine Frau Noor und dieses Mal auch meine Tochter Astrid einige Hundert Kopien ziehen die Newsletter mit Diskette 3D Brille und Renewal Form in ein gro en Kuvert stecken berall die notwendi gen Stempel anbringen und sie dann auf die weite Reise schicken Bitte beachten Sie meine neue email Adresse am Ende dieser Seite Endlich habe ich meinen eigenen Internetzugang an der Schule Dass dieses Medium bereits gen tzt wird zeigt das reiche User Forum Es bietet sich auch ein neues Service der DUG an Falls Sie den einen oder anderen Artikel aus einem DNL auch von fr he ren als Textfile brauchen k nnten kann ich Ihnen diesen gerne ber email schi cken Heute habe ich wieder in meinem elektronischen Postkasten gefischt und hatte Anglergl ck es gibt eine Antwort von Al Rich auf das Matrizenproblem im User Forum und Terence Etchells machte eine aufregende Ank ndigung f r einen m glichen Beitrag im n chsten Jahr Ich m chte Sie auch nochmals auf meine Partnersuche f r ein EU Projekt im Rahmen des COMENIUS Programms auf der Infoseite aufmerksam machen Ich konnte heuer zwei Klassen mit TI 92 ausstatten daher w re ich an einem Austausch an Unterrichtsmaterialien f r den TI 92 aber nach wie
16. e Gex 12 x 2 27 c2 21 E52 DE po LOSE n af 53 needs resimpliftying 52 54 XE E COS x p38 J L Rodriguez amp M J Fern ndez 3rd order ODEs D N L 24 6 y 2y 5y 24e y 0 4 y 0 2 1 y 0 25 Check the initial conditions Ei 56 SPL 2 5 D x 24 2 2 5 cup 40x Ei x 57 SPCC 2 5 D x d e cue scis CS Se a 2 SINC2 x 2 COSC2 x 7 3x x 58 vx e e 2 5IN x 2 COS02 x 7 59 y O y D y 00 4 1 5 y 2y 3y 10y e 34x 16 10x 6x 34 7 y 0 23 y 0 2 y 0 20 qQ X 2 51 SPC 2 3 10 x e 34 x 16 10 x o 6 x 34 0 3 0 2 gx 2 52 X g x 3 26y 150y 8 Find the general solutions 26y 150y A tan 2x KH 64 55C 5 26 150 x 20 a x j x h e 65 cl e aes e n e2 COSC34 3 oc3 SINCA34 x 3 43 3 66 SuaC 5 26 150 x 600 x 3 2 3 X d x 4 140625 x 2125 x 53475 x 19765 67 cl e 2 er COSC 34x 3 lt SINC 34 3 140625 3 68 SaCc 5 26 150 x 600 x Xx 658 SaC 5 26 150 x 20 2 70 Memory Exhausted A 72 c0 4 D x TANCZ x LNCSIN CZ x COSC2 xJ LNCTAN Cx J l 3 a r 00502 x c3 SIMC2 x cl 2 2 2 LNCSINCZ x J COSC2xJ LNCTAN CX 1 2 1 74 X IS 1 Ke t cz 005 2 x c3 5IN 2 x cl 2 2 4 B 2 1 y s
17. n n_ If ABS rtab n_ n_ rtabj n 1 n_ 1 eps ABS rtab n_ 221 n 2 rtab n_ 1 Cn_ 1 lt eps RETURN rtab n_ n_ n 1 rtab Now let s compare Terence Etchells 0 078 sec 5 8 12 1 75804591 765892 28 0139169547487 1 92 453219 4834 4 558561105625941 10 4 6987232052835 10 1 92460104930111 10 8 1 75819500516204 28 0140294546536 1 92 453651 833 4 588611124725655 10 4 59873331065501 10 5 5 1 75820205343736 28 01403623300B4 1792 453578528557 4 5886111 290047 10 1 75820310595234 28 0140366526776 17972 45364019761 1 75820313707965 28 0140366788448 1 7582031388323 The first column contains the initial values which are found by Simpson s approximation According to Steven s explanation these values are the 2 step values in his procedure starting with approxima tions obtained by applying the trapezoidal rule Try finding the Simpson values in the next tables p16 Josef B hm Comments on the ROMBERG Method D N L 24 Steven Schonefeld 0 047 sec l 4 1 521598 75234603 tit tit ttt 2 2 1 7 2008680299869 1 75292948654958 tit ttt 4 1 1 f46559S8899586 1 75804691765892 1 755295 07973288 ttt 5 0 5 1 755678610112 1 75815500515204 1 75820487706225 1 75820196969287 ls 0 25 1 75759851535802 1 75820265343736 1 75820315332238 1 75820312611053 32 0 125 1 75805196055376 1 75820310895234 1 75820313932001 1 75820313893902 64 0 0525 1 75819534294818 1 75820313707965 1 75820313895481 1 7582
18. the function F a b c has been defined to control the different kinds of roots that the charac teristic polynomial can have Thus if F a b c gt 0 the polynomial has a real root and two conjugate complex ones If F a b c lt 0 the polynomial presents three different real roots and if F a b c 0 it has only one real root with multiplicity three if and only if 3b a When the polynomial presents two real roots one simple and the other of double multiplicity a function P a b c has been defined due to the fact that the SOLVE function gives a vector with two components one of them being the double multiplicity root The sign of P a b c together with the relative magnitude of the vector components determine the position of the multiple simple root in the vector Taking into account the previous considerations about the roots the function SGH a b c x c1 c2 c3 calculates the general solution of the homogeneous equation p36 J L Rodriguez amp M J Fern ndez 3rd order ODEs D N L 24 As Yl a b c x Y2 a b c x and Y3 a b c x functions defined the program are linear independ ent solutions of the homogeneous linear differential equation it can be assured that if the parameter variation methods for finding a particular solution of the non homogeneous equation has been used then the corresponding equation system has only one solution which can be obtained by means of Cramer s rule Then a function SGC a b c x f c1 c2 c3 has been
19. 4 3 1 COS 9 3 1 COS 9 il 3 1 COS 0 COS SIN 6 4 3 COS 0 2 63 IL SRM I 1 COS 4 1 COS 3 COS 1 p32 Josef Bohm DERIVE ACD ACROSPIN D N L 24 4 3 COS 0 2 64 sorve E e 3 COS 6 1 2 II 2 OUS H 2 65 E asin asin y posu asin 3 2 3 2 2 3 66 6 0 841 5 44 0 841 COS SIN 6 4 3 COS 0 2 67 vector RAS o 0 841 55 44 1 COS f 1 COS 6 3 COS 6 1 5 44 0 841 40 4 2 2 W w V 2 w 1 SIGN w 1 3 84 VECTOR xo Xx sue Z2 0 05 4 2 w W V 2 w 1 SIGN w 1 3 In line 3 cone is the parameter form of the double cone it is intersected by the plane pp in 52 Approximating 4 returns the family of lines building the cone approximating 55 gives the grid of the plane 54 is necessary to obtain a scaling factor to have a 1 by 1 grid We intersect cone and plane in line 57 substitute for t in the cone s parameter form and find a space curve in 63 which is our conic section a parabola As we don t want to plot a parabola coming from anywhere and leaving for anywhere we have to find the parameter values which represent the intersection points of the parabola with the base The shading should run normal to the parabola s axis so we reparametrize the curve to end with the shading s vector in 84 We then ap
20. ACROSPIN files for space curves and for polyhedrons from the DERIVE environment I must admit that I am a fan of ACROSPIN and I would like to recommend strongly to buy this piece of software Nevertheless you may find this con tribution also useful if you don t want to use ACROSPIN Using the functions ISOMETRIC and COPROJECTION from DERIVE s utility file GRAPHICS MTH you can also produce impressive plots See the end of this paper Fortunately there are a lot of other DERIVIANS which like geometry so I am glad to announce for the next DNLs some contributions to produce several mappings of ob jects H K mmel a o and a hidden line algorithm for DERIVE from our friend Hubert Weller It is easy analysing the format of an ACD file and then to edit a few 3D points for simple objects for one s own ACD file To do so for objects consisting of many points this will be a boring work So I remembered my programming past and wrote a tool ACD EXE which can help I am sure that DERIVE and ACROSPIN as well may benefit Now you are able to animate space curves polyhedrons and surfaces given in any parameter form in any combination of objects colours and layers Using an algorithm of Richard Schorn from Kauf beuren you can produce analglyphs red green pictures to obtain a stereographic presentation of the object At this place I want to thank Mr Schorn for his comments and support Many letters were ex changed between Kaufbeuren and W rmla to share
21. Derive for Windows Hope to meet you all next year again Please settle your membership fee for 1997 Sincerely yours Josef email nojo boehm pgyv at P2 EDITORIAL The DERIVE NEWSLETTER is the Bulle tin of the DERIVE User Group It 1s pub lished at least four times a year with a con tents of 40 pages minimum The goals of the DNL are to enable the exchange of ex perience 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 Editor Mag Josef B hm A 3042 W rmla D Lust 1 Austria Phone 43 0 660 3136365 e mail nojo boehm a pgv at Preview Contributions for the next issues D N L 24 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 though 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 D N L The more contributions you will send the more lively and richer in contents the DERIVE Newsletter will be LOGO in DERIVE Lechner amp Roanes Lozano AUT amp ESP 3D Geometry Reichel AUT Algebra at A Level Goldstein UK Graphic Integration Linear Programmi
22. St Louis l Recognize that the structure of the model is h x a x 21 Dlx e x e d x e ex es f where there are 5 corners 2 Find the corner parameters el e2 They are given as 0 30 60 70 and 175 Thus the initial model is h x a x bx 30 ex 60 d x 70 ex 175 f where x is time in minutes 3 Simplify the model for each rate interval using the TI 92 1 Fir Fir Fur FE FE ER Conditions a a x tb x 3 c x 60 d s Fol eF x20andx 30 a b c d ej x 30 b 60 c 70 d i1ip ad x bjx 30 cols eil d x rol eb x gt 30 and x lt 60 atb c d e x 30 b 60 c 7O d 1p Ba xl b x 30 Fc x 60 d s Fol ep x gt 60 and x lt 70 atbto d e x 30 b 60 c 70 d 1p etahs x 1759 f1x lt 70 and x260 KAD AUTO FUNC 3 730 Bax b x 30 cels eil rd s Po te atbt eo d el x 30 b 60 c 7O d lk Bax b x 30 cels eil d x Po te atbteotd el x 30 b 60 c 7O d lk Bax b x 30 cels eil d x Po te atbteotdt el x 30 b 60 c 7O d 1h x gt 70andx lt 175 x2175 p56 THE TI 92 CORNER SUMS amp ABS 4 Setthe coefficients of x equal to each rate of change that is each piece of the model simplifies to a linear function as shown on the TI 92 screens Thus the coefficient of x 1s the rate of change for each piece a b c d e 1100 a b c d e 0 a b c d e 400 a b c d e 0 a b c d e 1233 5 Solve the syste
23. User X 0 SIN x 16 ROMBERG E UBRO 5 buta User Simp User X SIN x 17 ROMBERG TABLE x du Or uis 5 User x 1 75804 3 29705 11 0441 122 461 15013 0 1 125819 3 299734 1120451 122 473 n 18 1 75820 3 29736 11 0451 n n Simp 17 1 75820 3 29736 n n an 1 75820 ww ww ww ww p 4 DERIVE USER FORUM D N L 24 DNL Some days later there was another message from Terence which might be interesting for pure DERIVIANS and TI Users as well How are you It was nice to meet up with you again in Bonn enjoyed our talks too Whilst discussing the TI 92 with David Stoutemyer at Bonn I was bemoaning the fact that the T1 did not have my favourite function ITERATES He said no problem we ll write one Dave then gave me a brief tutorial on writing functions in the TI 92 function programming language Time was short so he scribbled a few 1deas on paper On the trip back from Bonn and over Summer I set to write a series of DERIVE functions for the TI 92 such as ITERATESO ITERATE ELEMENTO DEL ELEMY This is DELETE ELEMENT but the TI 92 restricts function names to 8 charac ters SWP ELEM REV ELEM RHS LHS Also DERIVE will easily plot a 2 x n matrix in a 2D plot window I could not find an easy way to do this with the TI 92 so I wrote a program PLOT MAT that plots a 2 x n matrix I will very shortly be putting these functions on my Web page see the signature file below for the URL if you should wish
24. cos w SIN u v cos sm en v sin 2 2 LLESEIS mb Ue Wye y 2m 07 053 03 5 d d T1124 bim Mb e cross 1im mb lim mb v 0 v 0 du Tea dv Pil oe E COS Qum ESSEN IE U BEIN EI bem ae xL FE l14 v gives the circle in the middle of the strip 115 0 COS u SIN 0 5 u COS u O SIN u SIN 0 5 u SIN u SOP COST Oy 5 413 116 t 0 gt pedal points of the normals 117 COS u SIN u 0 118 t 0 5 gt end points of the normals FII l0O5s009 00 9SIN OS5u ACOs Gil 0 5 SIN u SIN 0 5 u SINE 5 COS On 5m COS u SIN u 1120 vector 0 5 COS S EN O0 5 vu COS u 0 5 SIN u SIN 0 5 u SEIN U 0 II I u 0 ZT 5 5008 0 25 1 20 p28 Josef Bohm DERIVE ACD ACROSPIN D N L 26 As you can imagine you can use this interaction between DERIVE and ACROSPIN not only to pro duce nice pictures In a comfortable way you can make visible results from differential geometry So you see on the other Moebius strip the normals You could add the tangents the normals and binor mals to space curves the tangent planes and and Approximate expressions 111 and 120 and save them as you have done before Then apply option 3 for the strip and option 2 for family of normals try option 3 then you will obtain a second strip There is another idea to use this interaction make clear the meaning of the parameters Let me dem onstra
25. defined It calculates the general solution of non homogeneous differential equations Finally the functions SPH a b c x x0 y01 y02 y03 and SPC a b c x f x0 1y01 y02 y03 have been defined to calculate a particular solution to both homogeneous and non homogeneous equations respectively It is just necessary to remember the names and arguments of the functions SGH SGC SPH and SPC in their right order 3 FILE LISTING File LODE3 MTH RPC a b c 99S0LVE x 3ta x Z4b xTtc x RCP a b c teVECTOR URHS ELBEMENT RPC a D GO k k L 3 T113 6 6 EXP ELEMENT REP 8 b amp 1 x T2 a b c x EXP ELEMENT RCP a b c 2 x As the file takes more than two pages and fortunately can add the Diskette of the Year to this issue you can find the file LODES MTH on the diskette in subdirectory MTH24 So we will go on immediately to the examples The paper submitted had included 8 pages with 81 examples I ll try to give a selection of some typical examples Josef 4 EXAMPLES A 486 DX4 75 has been used in the solutions of the differential equations below The file has been optimized in order to get a the solution to all third order linear differential equations with constants of 1 2 and b its minimum execution time changed the original file for its use with later DERIVE versions The first two expressions had to be adapted because of distinguishing between SOLVE and SOLUTIONS Josef 2010 The first expressions read now
26. email address at the end of the page am happy to have my own email access now at my school And I like to use this media as you can see in the rich User Forum So l d like to offer another DUG service if you would like to have the text file of any DNL article even from earlier DNLs then I could email it to you Today I was fishing again in my elec tronic mailbox and was lucky Among others found Al Rich s answer to the matrix problem in the User Forum and my friend Terence Etchells made an exciting announcement for a possible contribution for 1997 I want to focus your attention once more on my call for partners for an EU project within the COMENIUS program on the Information page I was able to equip two of my classes with TI 92s and so am very inter ested to exchange teaching materials for the TI but also for DERIVE In these courses deal mainly with precalculus stuff don t want to finish my last letter of 96 without thank ing you all for your enthusiastic cooperation Without this there would be no DNL any longer But we find much more valuable the warm hearted friendship which connects the DERIVIANS from all over the world Each meeting each conference is a witness of this fact Many personal friendships and meetings have not only enriched our lives Josef and Noor but am sure those of many of us In that sense wish you all a Merry Christmas and a healthy happy and successful 1997 year 1 after
27. for the tube Er eed ts ICH SEARS az e Po lr LI I er I d pec l l m 1 P od rd Josef Bohm DERIVE ACD e 5 Lr dra ard NECS Wy TY EXAM r s i a uen 2 mn R DERIVE 6 graphs are below EA Od NI CES TOES x I II UTD VERSEHEN SER i HF E LI PISESA iy ie X Kk A as R Msc ain NR AS en RK ARTE OE Sie s ee EEE VEN ER PP RN T E cer NIS EV AGE MEE ETUDES Aa VM iH ril i a XS CHITI ACROSPIN D N L 26 BEAT ER III TEN IER i eer SEAT EP eee Ms ee D EN Ur PENY Uu y us RAT D i Js r7 D N L 24 Josef B hm DERIVE ACD ACROSPIN p27 If you know the parameter form of a surface it is very easy to produce the family of parameter lines with a VECTOR VECTOR function and then to apply option 2 or 3 from ACD EXE to create the animation of the surface You all will know the famous Moebius Strip Its parameter form is given by F u v a cos afi v cos Sin u t v cos v sin With a 1 we obtain the DERIVE representation VECTOR VECTOR COS u v COS u 2 COS u SIN u v COS u 2 SIN u v SIN u 2 u 0 2 pi pi 20 v 0 3 0 3 0 1 instead of VECTOR VECTOR you can use my FIG function FIG COS u v COS u 2 COS u SIN u v COS u 2 SIN u v SIN u 2 u v Ura pts 40 0 93 0294 190 n ss u DENN u u u 110 mb cos u v cos
28. results in a Snail of Pascal e aS locus of a vertex of an angle A constant angle moving with its sides touching two fixed circles gives a Snail of Pascal as the locus of its vertex Dreiteilung des Winkels mit Pascalschen Schnecken Angle trisection using S of P Gegeben sei eine spezielle Pascalsche Schnecke mit a 5 2 Den zu drittelnden Winkel ZSMP tr gt man wie in nebenstehender Zeichnung an P sei der Schnitt punkt des einen Schenkels von c mit der Pascalschen Schnecke Dann gilt Da das Dreieck QOM gleichschenklig ist LMOQ ZMQO g Der Winkel ZQMT ist Au enwinkel zum Dreieck QOM hat also die Gr e 29 Im gleichschenkligen Dreieck PMQ gilt f r die Basiswinkel ZQPM ZQMP 180 0 5 90 32 F r den Winkel ZTMP gilt also ZTMP w 90 und weiters ZTMP ZQMT ZQMP 29 oo 3 4 90 m Daraus folgt 2 bzw p bg 2 3 2 2 Im folgenden wird nicht unterschieden zwischen Winkel und Winkelma Die jeweilige Bedeutung ergibt sich aus dem Kontext In the following we will not differ between angle and its measure D N L 24 Thomas Weth A Lexicon of Curves 9 p43 Halbiert man also den Winkel LMOQ 9 so erh lt man ein Drittel des gegebenen Winkels c Let s take a special Snail of Pascal with a 5 2 ZSMP is the angle to be trisected P is the intersection point of one of its leg with the curve The following can be deduced As AQOM is isosceles we find ZMOQ MQO
29. to do Z Transforms with DERIVE For example the Z Transform of E s s 5 s 2 s 1 for T 0 01 sec DNL Answer from Al Rich SWH It isn t built in but as recall the Z transform is simply an infinite series You might try that using the DERIVE sum function and perhaps approximating infinity when an analytic result can t be determined Another answer from znmeb plaza ds adp com If you can do Laplace Transforms you can do Z transforms forget the formula but there is a simple substitution that turns any Laplace transform into a Z transform Check books on signal processing sampled data systems etc The two sided Z transform is a Laurent series think p10 DERIVE u USER FORUM D N L 24 Harald Lang Stockholm Sweden lans math kth se My version of DERIVE Classic 3 13 has a bug If I try to solve z i z fi w ly w fi 0 for w it says memory full after a few seconds However solving for z 1s fine Both z and w are de clared Complex Cheers Harald Lang Answer from Alain Pomirol Langon France Pomirol aol com Same problem with 3 11 version of DERIVE XM The given equation is simplified to zl z 38 1 w w 38 stil z i but is not factorized The solution is 1 Factorize 2 Solve equation w It s a problem of the employed method Note The TI 92 csolves this equation without any problems Yours truly Alain Pomirol DNL Same problem in DfW Josef
30. x zx 3x x 0 x 0 1 2 2 2 2 2 14 SOLVECSUBST x y x v l x DJ v v 1 2 15 SOLVECSUBST x y x y 1 x 1 v true 2 15 SOLVECSUBST x y 1 y 21 x D vl v 1v v Q0 2 5 5 1 17 S0LWE SUBEST x y x vw 1 x m y y 2 2 2 2 2 5 1 JS 1 18 SOLVE SUBST x y x 1 x y y 2 2 2 2 20 POWERMOD S 1 5 2 21 POWER MOD S 1 5 2 P P_ 22 CRT a mJ Terate woof a vecror ower ade M gt p M m m mi 1 m_ ri se ee el ree ent nan er 24 CRT MOD S3 5 MODCKS 7 MOD BS 183 5 7 16 83 FE MERIT 2 mW 541 93x57 26 CRTC MOD OB2 5 MOD 83 7 MOD 83 1521 5 7 15 83 POWERMOD from page 51 vs implemented POWER MOD and CRT from page 51 now CRT vs the implemented CRT function 157 is the solution of the congruencies x 1 mod 4 x 2 mod 5 and x 3 mod 11 D N L 24 THE TI 92 CORNER SUMS amp ABS p53 As I don t want to print all the Tl programs and functions which have received for the DNL s TI Corner many thanks to you all I include the files in a special Tl directory on the Diskette of the Year am sure that not all TI users have Tl Graph LINK at their disposal so they will not be able to use the programs tried to find another way I include a document in Word for WINDOWS 6 format contain ing many programs and functions in readable form
31. 015 sec 12 rom p zl 2 TT 0 046 sac 5 5 ERP Ca le ee ces meld 0 842 7007927949508 J This looks pretty good doesn t it D N L 24 Reviews p17 POLIEDROS 1 0 P Familar Ramos POLIEDROS Version 1 0 s a PC program that represents graphically regular polyhedra Platonic and kaplep roiniut Tia ee iub qs Archimedean solids Language Kepler Poinsot solids and semi catalan Sor as USUS regular polyhedra and their duals Archimedian and Catalan solids through AcroSpin The program includes a study of the metric prop erties of each polyhedron inform ing about the number of faces ver tices and edges the vertex configu ration the dihedral angles the cir cumradius midradius and inradius the surface area and volume The calculations of these metric relationships have been avaluated with DERIVE The user is allowed to select the program language English or Spanish Do you know a Disdyakistriacontahedron And if so do you also know that it 1s dual to a Rhombi truncated Icosidodecahedron And at last do you how it looks like POLIEDROS gives an answer Dual disdyakistriacontahedron Faces 62 12118 2816 3 3814 Uertices 128 Edges 188 Uertex configuration 4 6 18 Dihedral angles 4 6 159 5 41 43 1180 148 16 57 6 108 142 377 21 Circumradius 3 88239 44998 51293 58476 Midradius 3 76937 71279 21716 68267 Inradii 3 73686 79774 99789 69648 4 3 66854 24886 725
32. 0313894901 And this is my rom_prog SINX 7 rom proag x 0 4 7 x 1 621598 75254603 1 72009680299869 1 75292948054958 S Sen 1 74855938899386 1 75804691765892 1 75838807973286 ze Je 1 755786010112 1 5819500515204 1 75820487766225 1 75820196969287 1 75759851535802 1 75820265343736 1 75820316332238 1 75820313511053 1 75820314068482 1 75805156055376 1 75820210895234 1 75820313932001 1 75820313893902 1 75820213895011 1 75820313894841 1 75819534294818 1 75820313707905 1 75820312895481 1 75820313894501 1 75820313894905 1 75820312894905 My rom prog 0 047 sec SIN x 8 TS cera koe Hy oue oy d 1 75820313894905 x Including considering the required accuracy 0 047 sec The last column of the matrix in 8 is not displayed One can enter the required accuracy as last parameter and then the output is only the value of the inte gral if the number of steps is sufficient enough Finally I ask rom_prog to calculate the ERF 1 value 2 2 10 ron ro tinc x ee are s s fir 0 771743332258053 0 625262955596 749 0 538367777441205 0 8431028 30042941 0 842726051189 356 0842711599479115 m 0 54151822124476 D 542703035845955 0 84270034800728 0 84270066294196 0 542430505490232 O 842 700933572054 0 84270079342046 O 842 700 92 763484 0 84270079326867 0 842533227681257 0 8427000744531 0 542700792555457 0 842700752549091 0842700792945819 0 84270079294950 0
33. 23 The left picture is an analglyph of a C60 molecule model see the contribution submitted by Richard Schorn in DNL 21 You have to imagine red and green lines and viewing them through a red green glass ME S RR i l V SA RIALAK TN h VR D ET 4 li The right picture is composed from a sphere and a space curve a loxodrome You can make visible the single points applying option and then produce the closed curve using option 2 I ll lead you the way how to do it At first we have to produce the list of points using DERIVE 28 c 0 2 r1 e c phi 29 r0 1 SORT 1 r1 r1 w ATAN rl r r0 SIN w 30 VECTOR r COS phi r SIN phi rO COS w phi 20 0 0 25 31 VECTOR r COS phi r SIN phi r0 COS w phi 0 20 0 25 32 VECTOR VECTOR 0 5 SIN phi COS theta 0 5 SIN phi SIN theta 0 5 COS phi 0 5 phi 0 2 pi pi 10 theta 0 2 pi pi 10 33 FIG 0 5 SIN phi COS theta 0 5 SIN phi SIN theta 0 5 COS phi 0 5 theta phi 0 2 pi 20 0 2 pi 20 30 and 31 give the points of the loxodrome I had to divide into two parts because ACD can work only with vectors consisting of 100 elements each of which can contain again 100 components p24 Josef Bohm DERIVE ACD ACROSPIN D N L 24 Expression 33 is the parameter form of a sphere Using my FIG function you can avoid the bulky VECTORUCVEGCTOERT 5 command ApproXimate 30 31 and 33
34. 3 for any consumption over 2000 kWh Sliding scale commissions are business examples of multi constant rates of change and are used to provide incentive for higher employee productivity For example a business may pay the sales staff 3 commission on sales from 0 to 10 000 5 on sales from 10 001 to 15 000 and 8 commission on sales 15 001 and over The pay scale for piece work by a telemarketing company 1s 0 35 for the first 300 calls in the week 0 42 for the next 200 calls and 0 65 for any call over 500 calls A new long distance phone company has the following rate schedule 12 per minute for the first 15 minutes 9 per minute for the next 10 minutes and 6 per minute for any time over 25 minutes The 1994 or any year United States federal income tax form 1040 schedule for single filers had rates of taxation of 15 on the first 22 750 of taxable income 28 on the next 32 350 31 on the next 59 900 of taxable income etc That is the taxable income brackets are at 22 750 55 100 and 115 000 There were two more brackets that will be ignored for the sake of brevity A commercial airline flight from Reno Nevada to St Louis Missouri ascends at a constant rate after initial take off of 1100 feet per minute for 30 minutes until it reaches a cruising altitude of 33 000 feet It then levels off until it is 60 minutes into the flight At 60 minutes into the flight it has burned off enough fuel to ascend to 3
35. 7 000 feet at a rate of 400 feet per minute This takes 10 minutes The plane remains at 37 000 feet until 175 minutes of flight time when it descends at a rate of 1233 feet per minute and then lands in St Louis D N L 24 THE TI 92 CORNER SUMS amp ABS p55 The Algorithm for Finding the Model The algorithm assumes a knowledge of behavior of sums of absolute value functions Laughbaum 1996 Recognize that the structure of the model for all of the above situations 1s M ax e Pix e boss dx e f where there are n corners l Find the corners ei e corners are normally known 2 Simplify the model for each rate interval using the TI 92 3 Setthe coefficients of x equal to each rate of change rates of change are normally known 4 Solve the system 5 Find fusing a geometric transformation of a vertical shift An Example The on board computer on a commercial airline flight from Reno Nevada to St Louis Missouri commands the plane to ascend at a constant rate after initial take off of 1100 feet per minute for 30 minutes until it reaches a cruising alti tude of 33 000 feet It then levels off until it is 60 minutes into the flight At 60 minutes into the flight it has burned off enough fuel to ascend to 37 000 feet at a rate of 400 feet per minute This takes 10 minutes The plane remains at 37 000 feet until 175 minutes of flight time when it descends at a rate of 1233 feet per minute and then lands in
36. 85 73611 6 3 44895 48811 77933 84551 18 firea 174 29283 83423 23928 88293 Uolume 286 88339 88749 89484 82845 Press any key to continue Very interesting for you is the fact that POLIEDROS includes the ACROSPIN EXE file Math Ware has granted AWR Software a non transferrable license agreement subject to renegotiation to use AcroSpin in POLIEDROS Available at AWR Software Huertos 21 46500 SAGUNTO Valencia Spain FAX 96 266 34 07 3700 Ptas An Introduction to the Mathematics of Biology by Yeagers Shonkwiler and Herod The authors of this textbook have adopted the philosophy that mathematical biology is not merely the intrusion of one science into another but has a unity of its own The biology and mathematics are equal they are complete and flow smoothly into and out of one another The book has several important features that the authors have developed from their classroom ex perience A unique feature is the use of a CAS Maple in parts of every chapter The models can eas ily be transferred to other CA systems as DERIVE Graphic visualizations are provided for all the mathematical results The chapters are Biology Mathematics and a Mathematical Biology Laboratory Some Mathematical Tools Reproduction and the Drive for Survival Interactions between Organisms and their Environment Age Dependent Population Structures Random Movements in Space and Time The Biological Disposition of Drugs and Inorganic Toxins Neuroph
37. E 6 commands APPEND COPROJECTIONCFIG O 5 SINCp COSC D 5 SINCp SINE O 5 COS5tp 0 5 8 mp O Zer 20 O 2 7 20022 APPENDCFIG C O 5 SINCp COS 8 O 5 SINMGpI SINE D 5 COSOp 0 5 8 mp O Z m 20 D 2m 207 VECTOR r COSp r SINGBp rO COSQw p 20 20 0 25 VECTOR C r COSQpl r SINCp rO COSQw mp 20 20 0 25 Inspired by books dealing with other CAS packages I tried to produce a more sophisticated anima tion an elliptic torus with a torus knot line on it And to make the space curve more impressive I su perimposed its tube Using the utility file GRAPHICS MTH from SWHH FLUSETIKN a b Cy p dq t ie a b COS a t COS or a D COS q t 9lN D t OSSIN OQ E LOZEIKN Op oy Dk 2 Sg PLOS FCDA E COD E EEE NZ SEIN SEE TI FIN Zee Jt oLN Oc I 106 VECTOR TKN 8 3s Do 2 de dg ty QUE 40 TIOJTELL TOR ay D amp O dq t la f b COS5 00 Te D COSTP SSIN O9 c SIN FLUBTEIG ERL TORI 3r Sy Oly Oy 0 2 H 20 U 27 20 1109 SPACE TUBE TAN 32 Jr Oy 2 Dy Dr Ua Ze o9 Tilo TR IG SPACE TUBE IKNi 6 23 Se 27 Oy Dr Ge dl 2 D Oy Oy 27 00 95 2 n 20 p26 ge rra ea You have to approximate the expressions 106 108 and 110 save them in three different BAS files and then use ACD Please be patient approximating expression 110 Using ACD I chose colour 1 for the space curve colour 4 for the torus and colour 14
38. FF u x n 2 QUOTIENT REMAINDER U x n 1 X n x Furthermore for those with older versions of DERIVE the following implementation of PREM might still be of interest 1f is not available in your DERIVE version then use any other unused variable instead of it o9WAP f X y ITERATE LIM f amp O y f LIM f x y G0 X 1 REM p 9 V ITERATE SWAP REMAINDER SWAP p v v J SWAP q v v vV v JV VARIABLES p qa SUB 1 1 PREM p q V REM LIM TERMS EXPAND q v SUB 1 v 1 1 LIM v DIF f v f f TERMS EXPAND p v SUB 1 LIM v DIF f vjf f TERMS EXPAND q v SUB 1 p q v You might wonder by now what this mysterious PREM routine is all about Sorry I should have told you that earlier Above all it is a very useful tool to solve a system of polynomial equations by trans forming it into a solvable system of equations without losing any solutions of the original system To be more precise if f and g are polynomials in x x where we may assume that deg f x deg g x w l o g then either deg g x 20 or we can shift to polynomials q r in X X Which are defined by the following equation reflecting the polynomial division of m f by g m f g g r with deg r x deg f x Here m is a polynomial in x x that is chosen in such a way that fractions are avoided see the formula MUL FAC p q v above for its exact definition and the pseudo remainder r is the output of PREM f g x1 Again ei
39. Hello dear DERIVERS Try to plot the function x 1 2 I did using DERIVE 2 56 and 3 0 You can see that obviously this function takes 0 1nf gt 0 inf Now plot ABS x 1 2 and surprisingly you will have a function defined in the whole real line How can it be Then if you Simplify ABS x 1 2 DERIVE gives you ABS x 1 2 So if you Simplify ABS 4 1 2 the result is 2 and this is not right And the same happens for every even root I think this happens because DERIVE has implemented this simplification for odd roots for which it is valid But I think this must be modified for even ones don t you If I did something wrong or anyone has a further explanation please let me know Thanks p 8 DERIVE USER FORUM D N L 24 DNL There were some explanations in DERIVE News Group In general all had the same content so I ll combine the different answers michel gosse magic fr Al Rich A van der Meer Josef a o The problem with DERIVE is that it calculates with complex numbers The absolute value of 2i then becomes 2 Like virtually all CAS DERIVE works in the complex domain not just in the real domain You can define a function that will plot correctly but that is a little tricky f x s if x lt 0 0 absS x 1 2 Alfonso J Poblaci n Valladolid Spain Thank you very much about your message I have got a lot of tangrams I will try to send explaining what DERIVE does with the ABS them by e mail
40. I wrote the first column in this series using DERIVE 2 56 then In the meantime numerous new versions of DERIVE have come out taking ac count of the various users wishes The present culmination of this development is DERIVE for Win dows 4 0 called DfW in the following for short which has become available these days What I think about DfW Well despite a lot of apparent improvements such as the lightning fast graphics the new file management system or the enhanced scrolling capabilities it isn t exactly true that it made me feel all warm inside from the very beginning you can t teach an old dog new tricks as the saying goes What I found most irritating was the input window that pops up every few seconds Why on earth didn t they use the main window for this purpose like every other CAS I know of but after minimizing it 1 e increasing the screen resolution and maximizing the main window at the same time I finally got used to it Currently I am discovering new nice features of DfW every day and it is no longer inconceivable that I end up in saying one day I like it One of those nice features I was talking about is the option to include the main variable as a third pa rameter in the functions REMAINDER and QUOTIENT Up to now the main variable could only be adjusted by using Manage Order from the main menu To the best of my knowledge this was sug gested by Eugenio Roanes Lozano who had to apply in 3 a lot of tricker
41. N Certain components of courses in probability and elementary statistics can benefit from the algebraic manipulation facilities offered by DERIVE The way in which the DERIVE interface is used i e select an expression and operate on it makes it particularly suitable for computations which involve sequen tial operations Consequently the user can concentrate on wider aspects of problem solving and is free to appreciate the overall strategy used In this paper we use DERIVE to prove certain results in probability theory and show how some shortcomings of DERIVE may be overcome We consider how DERIVE may be used to obtain some standard results in probability theory and assess its efficacy in doing so The initial discussion centres on the ability of DERIVE to calculate means and variances given some standard probability distributions This involves summing series and evaluating integrals It is an advantage from a didactic point of view to be able to do such com putations directly Looking up standard results is not meaningful unless it is accompanied by a good conceptual understanding Routine computations can help to provide this MEANS AND VARIANCES OF DISCRETE RANDOM VARIABLES We consider the cases of the binomial and geometric distributions because the principal pedagogic and technical points are covered by these distributions For a discrete random variable X defined in a domain S the definitions below may be used to calcu late the m
42. Save each of the results as a single BAS file with different names eg LOX1 LOX2 and SPH respec tively Quit DERIVE Take care that ACD EXE and ACROSPIN EXE are in the same directory Then call ACD Choose option 1 because we want to see the points which build up the space curve LOX1 ENTER the name of the first BAS file LOXO ENTER that s the name of the ACD file to be created now 15 ENTER the points should be white colour code 15 1 ENTER the Ist layer y because we have not finished repeat with LOXI you will not be asked for the ACD file s name once more If you would finish now you would obtain only the points next figure but we will continue 1 ENTER we will see the space curve LOX1 ENTER the same point list will be used 4 ENTER we want to have a red curve 2 ENTER it should be another layer y ENTER because we are not yet ready repeat the two steps with LOX2 If we would finish now we could see the points and the curve in two layers which could be toggled on and off using the ENTER key in combination with the 1 and 2 key As we want to add the sphere we go on once more and proceed 3 ENTER the whole surface of the sphere SPH ENTER the name of the corresponding BAS file 14 ENTER let s have a yellow sphere 3 ENTERI in the 3rd layer n we have finished Now start typing ACROSPIN LOXO ENTER D N L 24 Josef B hm DERIVE ACD ACROSPIN p25 These are the DERIV
43. THE DERIVE NEWSLETTER 24 ISSN 1990 7079 IHE BULLETIN OF IHE USER GROUP Contents Letter of the Editor Editorial Preview DERIVE User Forum including ROMBERG MTH and Comments Some REVIEWS Josef B hm DERIVE ACD ACROSPIN R Contreras and F Guti rrez 3rd Order DIFFERENTIAL EQUATIONS Thomas Weth Lexicon of Curves 9 Snail of Pascal ACG DC 3 by Alfonso Poblaci n Peter Mitic Probability Distributions Proof and Computations 1 Johann Wiesenbauer Titbits 9 The TI 92 Corner W Pr pper SOLSYST for systems of linear equations E Laughbaum Sums of Absolute Value Functions T Etchells My favourite DERIVE functions for the TI 92 revised version 2010 December 1996 D N L 24 INFORMATION Book Shelf D N L 24 1 2 3 4 5 6 Mathematikunterricht mit Computeralgebra Systemen H Heugl W Klinger und J Lechner 307 Seiten DM 59 90 S 443 00 sFr 48 00 Addison Wesley 1996 ISBN 3 8273 1082 2 Ein didaktisches Lehrerbuch mit Erfahrungen aus dem sterreichischen DERIVE Projekt Eine Kurzbesprechung folgt im n chsten DNL TI 92 du lyc e la pr pa Henri Lemberg DUNOD Texas Instruments France Paris Mai 1996 ISBN 2 10 003039 6 This book recapitulates on 314 pages the mathematics curriculum of French gymnasiums and shows up all the concepts and algorithms which a French student needs to know for the entrance examination for French univ
44. am glad to announce another highlight for the next DNL Our friend Sergey Biryukow from Moscow has produced a DERIVE tool to produce plots of implicit 3D functions He saw ACD at Bonn so he took care that his output is ACD compatible Last question Which object is hidden in this star It is the Bottle of Klein u ae NN aie Eae Sy LU PUER aan P A aa ean an GE EEE an uis Pas ETT oem DEZE Zig LAL LF CHUL 7 HZ LL NI NZ P References 1 3D Programmierung mit BASIC Glaeser hpt 1986 ISBN 3 209 00626 1 2 Atlas mathematischer Bilder Leo H Klingen Addison Wesley 1996 ISBN 3 89319 947 0 3 Differentialgeometrie Alfred Gray Spektrum Akademischer Verlag ISBN 3 86025 141 4 4 Modern Differential Geometry of Curves and Surfaces A Gray CRC Press Inc Boca Raton 5 Computer Graphics F S Hill Jr Macmillan Publishing Company 1990 ISBN 0 02 354860 6 6 DERIVE Days D sseldorf Tagungsband B rbel Barzel ed Landesmediententrum Rheinland Pfalz 7 DERIVE News Letter 21 C60 The Buckyball Richard Schorn D N L 24 J L Rodriguez amp M J Fern ndez 3rd order ODEs p35 solving third order linear differential equations with constant coefficients Rodriguez Contreras J L Department of Mathematics University of Norte and University of Atlantico Baranquilla Colombia Fern ndez Guti rrez M J Department of Mathematics Universoty of Oviedo 33071 Oviedo Spain E mail mjfg pi
45. approximations to square the circle Alfonso J Poblacion Saez Valladolid Spain 3 A method from Poland This third method is due to the Polish Jesuit Reverend Adam Kochanski in 1685 He was the first to use a steel spring in the suspension of a clock s pendulum Kochanski s con struction with a circle of radius r 7 This reminded me P 7L on my school time KAT on the screen of a t TI 92 Josef EY F x KOCHANSKI DEG EXACT F HL Start with a circle centre O and radius r Proceed with a circle of radius OA and centre A on the diameter BOA in order to find C With centre in C draw one more circle having the same radius and obtain D Consider the segment OD and its intersection with the tangent to the initial circle passing through A This gives you point E Then F is on this tangent and verifies that EF 3 OA The length BF is approximately OA r What is here the value of 1 4 Ramanujan s contribution Finally I chose the construction of a great Hindu mathematician Srinivasa Ramanujan He gave us a lot of interesting formulae one of which is implemented in DERIVE to approximate m see 5 From a given circle consider M the midpoint of OA and T such that OT OB Then P is on the circumference such that TP 1s perpen dicular to AB and Q is such that BQ TP Take S as the midpoint of AQ and D satisfies AD AS The segments TR BQ and OS must be parallel Now draw AC being tangent to the
46. ark will be output if the numbers m m m are not pairwise coprime Number theory abounds with applica tions of this important routine but space is running out Therefore bye for now email j wiesenbauer tuwien ac at References 1 Wilfried N bauer and Johann Wiesenbauer Zahlentheorie Prugg Verlag Eisenstadt 1981 2 Alessandro Perotti Gr bner bases w th DERIVE The International DERIVE Journal 1996 Vol 3 No 2 83 98 3 Eugenio Roanes Lozano and Eugenio Roanes Macias Automatic theorem proving n elementary geometry with DERIVE The International DERIVE Journal 1996 Vol 3 No 2 67 82 1 leseff p v3 POLY COEFF p v POLY DEGREE Dp wi arb var i SOLUTIOMS O O x13 1 1 3 swap aux f x t yi SUBSTCSUBSTCSUBSTCF x tJ v x t y 4 swap f x yY iz swap aux f x arb var y 5 remainder v aux p 9 v vO swap REMAIMDER swap p v vO swap q v ON v vU remainder wip 9 v remainder v aux p q v VARIBBLESC p a 1J 3 5 1 1 POLY DEGREE p w POLY_DEGREE g wJ 7 mul fac p q v leoeffiq vw SB prem p gq t remainder v mul fac p 9 vi o a vi 2 9 premixey 1 2 x 2 l y 27 x v x y 1 2 10 prem x y 1I z z l1e x z x x v 1 y 1 2 2 2 11 nrem x vw x w 1 x y 11 vr 1 vl aw v x vw 1l 2 2 5 4 3 2 12 prem x w 1 y 1 x v x v c l wvl 2 x x zx Gee X 5 4 j 2 J5 A J5 l 13 SOLUTIOMS x
47. ase we finally arrive at the five triples 0 0 0 10 05 351 38 1 748 1 48 1 Se All these triples are really solutions of the original system Note that we have actually used 1 2a 2b 3 to compute them It is easy to see that both triangulated systems 1 2a 3 and 1 2b 3 on their own have more solutions than the original system Check it One could solve this system of polynomial equations also by means of my SOLVE2 function from my last Titbits but it is very tricky and cannot be generalized I hate to admit it but Eugenio s PREM function is streets ahead of my RED function the kernel of SOLVE2 when it comes to solving sys tems of polynomial equations It must be said though that it was invented for quite a different pur pose cf DNL 20 p38 If you are interested in learning more about applications of the PREM function I can recommend to you reading Eugenio s wonderful paper 3 where he used it extensively to prove theorems of elementary geometry by the so called Wu s method At any rate I don t want to leave this topic without thanking him for his cooperation and many fruitful discussions via email D N L 24 J Wiesenbauer s Titbits 9 p51 Back to DfW What else could be done to enhance it Well a lot of little things come to my mind e g that there 1s no calculation time shown after the instant simplification of expressions by or the fact that there 1s no Cartesian product for lists and sets an
48. ccessive Romberg approximations com puted by ROMBERG You may be interested to know that the function ROMBERG is significantly faster than classic DERIVE s internal approximation on certain integrals Try INT SIN x x x 0 4 with precision set to 10 digits and ROMBERG SIN X x x 0 4 5 Some thing funny happens in DERIVEXM 3 10 as the functions take longer to evaluate Best wishes Terence This file performs Romberg xntegratron on the Integral INT ft x a b n M f x a b n sSUM b a n LIM f x at 2 r 1 b a 7 2 n r l m Tf xrar b0 i b74 240 DIM th xa 2 SUM LIM t x oder ba m 14 nel LLM Eek by Sti sey De ZAM Cee ea bin ER 39 ROMBERG START f x a b n VECTOR S f x a b 2 r r 1 n ROMBERG AUX f x a b n ITERATES VECTOR 2 k v SUB r 1 v SUB r 2 k 1 r 1l n c k 2 c 1ll v k c ROMBERG START f x a b n 4 1 n 1 ROMBERG ADD v n VECTOR APPEND v SUB c VECTOR r 1 c 1 c 1 n ROMBERG EXTRACT v n v SUB n SUB 1 SUB 1 ROMBERG f x a b n ROMBERG EXTRACT ROMBERG AUX f x a b n n ROMBERG EXTRACT COLUMN v n VECTOR v SUB r SUB 1 r 1 n ROMBERG AUX TABLE v f x a b n ITERATES VECTOR 2 k v SUB r 1 v SUB r 2 k 1 r 1l n c k 2 c 1ll v k c ROMBERG START f x a b n 4 1 n 1 ROMBERG TABLE f x a b n ROMBERG ADD ROMBERG EXTRACT COLUMN ROMBERG AUX TABLE v Farb hn pn 14 Precision Approximate User 4 SIN x 15 ea 1 75820 User Simp
49. d what not But from a number theoretic point of view there is one thing that is on the top of my wish list and once more it concerns the notorious pow ermod function It should also work for negative exponents At present if you ask e g for the inverse of 3 mod 5 i e mod 3 1 5 you will get the answer mod 3 1 5 1 3 which is not exactly what you want It could be programmed along the following lines cp DNL 717 p35 POWERMODY a n m IF n2 0 MOD a n m IF GCD a m 1 MOD MODY ITERATE IF MOD a b 0 a b c d J b MObD a b j d c FLOOR a b d a b c d j a m 1 0 SUB 4 m n m If the exponent n is negative and gcd a m gt 1 the output will be a question mark as of course power mod a n m is undefined in this case I will use this function to implement another very important formula which is based on the so called Chinese Remainder Theorem CRT If m m m is a sequence of pairwise coprime integers then for arbitrary integers a a a the system of congruencies x a mod m x a modm x a modm r r has a solution which is unique mod m m m Fortunately there exists also an explicit formula for the solution which can be written in DERIVE notation as follows cf 1 p37 CRI a m zITERATE MOD a VECTOR p m POWERMODY p rn 1 m m mjp p PRODUCT m m m j 1 Here a denotes the vector a a a and m the vector m m m Again a question m
50. do the following integration without problems gt Declare a b as Real positive gt then Simplify INT x 4 EXP 2a 2 b 2 x 2 a42 3 b 2 x 0 inf gt works as expected gt Now try to Simplify this one INT x 4 EXP 2a 2 b 2 x 2 a 2 3 b 2 a 2 3 b 2 7 2 x 0 1nf gt jt gets stuck gt change the exponent of x from 4 to 5 no change to the power of x inside the exponential func tion Now it can do it It gets stuck for all even powers of x 2 4 and works for all odd powers of x gt Substitute the denominator in the exponential function a 2 3 5 2 with G declared Real positive and Simplify again now it works no problem Also seems to work when substitut ing for the denominator of the expression itself The previous revision DERIVE v4 00 gets stuck for all powers of x that are 2 4 even OR odd My old DERIVE v2 01 can do all these types of integrals with ease Integrals of the form x et dx with k positive are well defined 0 What s going on Why does the substitution with G work p12 DERIVE USER FORUM D N L 24 2 2 v 4 2 a Z b x X 8 3 m 1 dx 2 2 2 Tio 5 a 3 b J 256 Ca 3 b ab 0 oo Solution performed with DERIVE 6 2 2 G 2 a geb ek x 8 1 2 dx sa EF O n5 2 2 a 3 b J Bd a b a 3 6 J 0 Some of my other experiences with v4 00 point to a bug in the
51. dtm If getType dtm NUM and dtmz or getType dtm EXPR Then rref mtx omtx Return subMat mtx 1 n 1 n n 1 Else Return Nicht eindeutig l sbar EndIf EndFunc t4y z 6 X 1 NE ME Er ca solsyst we m BI Eger u d z u z ig j u z B ig z 4 u x 18 z 3 4 b271Tb ROLE utd x u L x a ub 223 t 1z2 x 8 u 214 x pus psu Bl me solsust T ce Ly Hicht eindeutig l sbar Lx a yih 2 y at h 0l hb al gt zolzustct D 7 12x 8y 14 6x 4v 561 T FUNC 2730 FUNC 2 730 With this nice function you can solve systems of linear equations in a DERIVE like way Non German speaking users have to change one string in the third line from the bottom Nicht eindeutig l sbar to the equivalent of No unique solution p54 THE TI 92 CORNER SUMS amp ABS D N L 24 Sums of Absolute Value Functions An Application Edward D Laughbaum Columbus Ohio elaughba math ohio state edu Consider the following a b c d g The level of the drug Imipramine in the blood of a patient rises at a constant rate for example 60 nanograms per week until the patient is at the prescribed level and then the rate of change re mains at 0 until the patient is taken off the drug at a constant rate for example 90 nanograms per week For users of electricity who also use a heat pump many power companies charge for example 0 08 per kWh used for the first 1000 kWh and 0 05 for the next 1000 and finally 0 0
52. e based on several con nected functions There was no programming possible in these times 1996 Among many resources describing this famous numerical method I found one in Wikipedia show ing the recursive procedure of this method R 0 0 5 6 a f a F R n 0 R n 1 0 An gt flat 2k 1 A k 1 R n m R n m 1 Rm 1 R n 1 m 1 1 R n m ga 14 Rin m 1 R n 1 m 1 where n gt 1 m gt i b a hn am It is followed by a table for calculating ERF 1 with an accuracy of 10 which I will use to check my Romberg program together with Terence Etchell s results of course LU Numerical Analysis via Derive Steven Schonefeld Mathware 1994 7 http enwikipedia org wiji Romberg s method p14 Josef Bohm Comments on the ROMBERG Method D N L 24 The ERF 1 table WIKI aan ses 1 02526296 0 545310253 1 03036770 O 642 753605 0 5042 71160 1 54161922 O 6842 705304 O 642 70065 O 642 70066 1 54245051 0 54270095 U 8423270079 O 642 70079 0 84z 70079 ERF 1 p X 0 542700792949 714 ef ERF x e dt 1 pi 2 2 0 EXP t J dt IT 0 0 642 700792549 714 I found also a German Wiki information but according to my understanding there are some typos in it and following the instructions there the algorithm should not work wrong subscripts I wrote a DERIVE program following this recursive procedure which is based on the corrected pro
53. e mathematical language The Analytical Form of Leaves Quedlinburg 1895 One of the sim plest curves which can be found among Habenicht s Leave Curves is the Snail of Pascal which goes back to Etienne Pascal father of the well known Blaise Pascal It appears that spe cial Snails of Pascal can be used for the trisection of an angle and they play a role in several mechanic problems For that reason they were called Sauveur s and de l Hospital s Draw bridge Before that time Jacob and Johann Bernoulli had dealt with this curve too D N L 24 Thomas Weth A Lexicon of Curves 9 p4l Konstruktion Construction Genauso wie die Konchoiden des Nikomedes vgl Folge 5 lassen sich Pascalsche Schnecken kon struieren nur verwendet man nicht wie dort eine Gerade sondern einen Kreis als Leitlinie Kreiskonchoiden ergeben sich nach folgender Konstruktionsvorschrift Gegeben sind ein Kreis mit Mittelpunkt M und Durchmesser b und ein Punkt O auf der Kreis linie Von einem Kreispunkt Q aus tr gt man auf der Geraden OQ in die beiden m glichen Rich tungen jeweils eine Strecke konstanter L nge a ab die Endpunkte P und P dieser Strecken sind dann Konchoidenpunkte zum gegebenen Kreis Pascalsche Schnecken als Kreiskonchoiden f r drei verschiedene Abst nde a erstellt mit dem TI 92 The snails can be constructed similar to the Conchoids of Nikomedes Lexicon 5 with a circle as di rectrix instead of a line Given is a circle
54. ean and variance u and of respectively of X p46 P Mitic Probability Distributions 1 D N L 24 w I xP X x and o x P X x yw wherexeS Applying these results to a Binomial n p random variable we use the DERIVE construct COMB n x pX 1 p for P X x and sum over x from 0 to n The random variable X might re present the sum of the scores obtained in n independent tosses of a die which has the probability p of landing heads on any one toss In the discussion by Etchells Etchells 1992 these computations are done by considering n 1 then n 2 and then n 3 with results 14 P od p l p I 2p 62 2p 1 p and in 9p 03 3p 1 p respectively DERIVE has no problems in evaluating these sums and the general results 4 n p and o np 1 p may then be conjectured However DERIVE cannot perform the summations when a limit for the summation is non numeric so that a general proof is not possible This is a problem because omis sion of a formal proof gives the impression that a conjecture based on a few numerical results consti tutes a proof in its own right We suggest that if a general proof is not be given there should be as a minimum a statement that the proof of the general case is missing This proof may be approached by either by attempting to calculate the moment generating function M t De P X x where the range of t is such that the series is convergent or the probabil
55. ems occur In the binomial case the series cannot be summed unless the limits for the sum are explicitly numeric However the general result G t p t 1 p is relatively easy to obtain by considering binomial expansions In the geometric case the series cannot be summed unless the probability that an event happens is explicitly numeric Given that G t has been determined use of the formulae 4 7 G 1 and of G 1 G 1 GN for the mean and the variance of a random variable proceeds smoothly The figure shows the DERIVE realization of Ed Laughbaum s TI 92 paper page 54 how to combine piecewise defined func ee ee RE tions in a sum of absolute values function i N ee imos M E dte Sq qM SEM e ang Declare x gt 175 x iE Real 175 133 1233 Until now l ve always investigated these a 568 c 208 a 208 e functions from the other direction decom pose such a function in its linear compo nents I find it very challenging to reverse that investigation Josef Comment of the Editor leave this contribution in its original form from 1996 Many func tions which Johann developed have been implemented since then You can find the DERIVE 6 suitable file at the end of Titbits 9 titbits9 new mth is the respective file Titbits from Algebra and Number Theory 9 by Johann Wiesenbauer Vienna Time is flying isn t it It doesn t seem so long ago that
56. erated with Josef Lechner s DERIVE tool INTEGRAPH Proceedings of the DERIVE Days D sseldorf The following system of differential equations is the base of this attractor At last I want to give an impression how to work without ACROSPIN I produce a static representa tion of the parabolic intersection of the double cone As you can learn from the screen shot the trick is to to include ISOMETRIC and COPROJECTION from GRAPHICS MTH at the appropriate place Please compare with the according file on pages 31 and 32 Don t forget to use COPROJECTION other wise you would see only one family of parameter lines 55 VECTOR vec TUR isonr TRICCconeJ 9 97 COPROJECTION vec TUR Ivec TOR isonr TR 394 VECTOR vecron ISOMETRIC eb3 u B 98 COPROJECTION vec TUR vec TUR isonr TR C05 5 93 vECTOR ISOMETRIC 1 COSC 1 95 ECTOR ISOMETRIC lu J Z2 uw 1 p34 Josef Bohm DERIVE ACD ACROSPIN D N L 24 Among the downloadable files you can find in lt ACD gt all the MTH files mentioned in this contribu tion accompanied by ACD EXE and a self extracting compressed file ACDZIP EXE containing a lot of ready made ACD files And you will also find ACROSPIN I would like to ask you to produce your own ACDs And to enforce this CALL for ACDs I invite you for a competition The ACD of the Year The ACD of the Year will win one year free DUG membership Deadline is 31 May 1997 Much luck I
57. ersities TI 92 les programmes Jean Michel Ferrard DUNOD Texas Instruments France Paris Mai 1996 ISBN 2 10 003104 X This is a real treasury of 480 pages full with TI 92 functions and programs All the programs can be used for their own but they also can be assembled within a library divided in different folders containing their own menus which allow an easy approach to the programs The sections co vered are Arithmetique et Trigonometrie Polynomes Matrices Geometrie Developpements Li mites Analyse Geometrie Differentielle Fonctions Speciales Probabilites Even if you only have a very poor knowledge of French like me it is easy to follow Fortunately the language of maths is international If you want to learn and to train programming with the TI 92 then l d strongly rec ommend this book TI 92 le top des jeux Vincent Bastid et Emmanuel Neuville DUNOD Texas Instruments France Paris Mai 1996 ISBN 2 10 003040 X Transformez votre calculatrice TI 92 en console de jeux A collection of games containing TI tris D mineur Minesweeper Bataille Navale for two TIs TI Invaders Tic Tac Toe 3D and four more games have to thank Mathias Makowsky from Marbach Germany who faxed the titles of the three French books He saw them in a bookstore in Brittany during his holidays Many thanks the books you recommended are very useful Josef An Introduction to the Mathematics of Biology With Computer Algebra Models Yea
58. es of parameter lines with either u or v constant Save the two expressions as dif ferent BASIC files leave DERIVE and then call ACD The desire to produce ACROSPIN demonstrations of solids of revolutions was the inspiring idea to create a tool like ACD to make that possible Executing ACD you will find a menu in which you are asked to enter the type of object which you want to prepare for ACROSPIN You are able to combine several different objects in one ACD file with both individual colours and layers See now some examples D N L 24 Josef B hm DERIVE ACD ACROSPIN p21 From DERIVE to ACROSPIN Which kind of object do you want to display Make your choice space Curve or Polyhedron from a list of points in Stereo Vision surface show the families of parameter lines in Stereo Vision or show the complete surface in Stereo Vision Polyhedron from a list of edges in Stereo Uision Discrete points in Stereo Vision Your choice please From my point of view the most interesting thing is the fact that I can interact between the wonderful capabilities of DERIVE and the demonstrating facilities of ACROSPIN We could use this tool to train the students imagination of 3D space Give them the task to design 3D objects by the coordinates of the vertices and the edges Let them construct well known geometric solids like tetrahedrons octahe drons pyramids parts of a cube and so on They immediately can see
59. experiences First you have to produce the list of 3D points of the object in DERIVE Work in Approximate Mode and use 3 digits in Notation from the Options submenu If you connect the points then they will form a space curve or a polyhedron or only a polygon in space Save the DERIVE expression as a BASIC file So the most important thing is to create lists of points in the right order These lists of points can be used for many other projections parallel or cen tral perspective or the projection process can be done by another tool like ACROSPIN You can also use this lists in connection with ISOMETRIC So I will start with a toolbox for producing this lists of points for 3D objects TP a ee Ws InputMode Word Precision Approximate PrecisionDigits 4 Notation Decimal NotationDigits 3 The next 7 functions are from ELLE GRAPHICS MEM Gopyrignt 2 1990 by SOR Warehouse Ines COPROJECTION v VECTOR VECTOR u SUB n u v n DIMENSION v SUB 1 SPHERE r theta phl r TISIN DpHi COS theta SIN phi SIN theta COS phi p20 Josef Bohm DERIVE ACD ACROSPIN D N L 24 CYLINDER r theta z r COS theta r SIN theta z CONE alpha theta z z SIN alpha COS theta z SIN alpha SIN theta z NORMAL VECTOR v t SIGN DIF v t 2 BINORMAL v t SIGN CROSS DIF v t DIF v t 2 SPACE TUBE Vl ty Dh SeVEES SINODhIL NORMAN VECTOR V t ROOS phr BINORMAL v t The following is no
60. function All the answers pointed out in the same direction including one that I discovered later in the DNL 14 page 6 messages 2708 and 2710 I knew but I forgot that DERIVE works with complex numbers that was a terri ble oblivion but what I did not know was that it plots in the way it does with these functions A lot of our students usually work with DERIVE alone and they can be confused about these behaviours in case they detect them Imagine a complicated function that involves ABS and SQRT and its plot they believe in what they are seeing so we must advise them also most of them do not know so much about complex numbers In their first course they only deal with real numbers DNL Thanks for the tangrams I m sure they will show together with your contribution a new and still unknown facette of DERIVE and they combine in an ideal way mathematic tuition with entertainment Sergey Biryukow Moscow Russia Dear Josef I have written functions for Implicit 3D Plots in DERIVE and I am writing DOC amp DMO files now Functions for Implicit 3D Plots in ACROSPIN are also written and return a vector of lines 2 x 2 numeric matrices Is this format compatible with the tool you are going to present in the next DNL I shall try to polish my IMPLICIT PLOTS utility as fast as possible in order to include it in the 1996 DNL diskette Sincerely Sergey x y o y z x zi za D N L 24 DERIVE USER FORUM p 9 DNL Yes
61. gers Shonkwiler and Herod Birkh user Boston 1996 ISBN 0 8176 3809 1 You can find a short review on page 17 AGNESI to ZENO Over 100 Vignettes from the History of Math Sanderson Smith Key Curriculum Press Berkeley 1996 ISBN 1 55953 107 X This is not a Computer Algebra Book but it is a wonderful book to motivate teachers and stu dents as well for investigations and projects and presents a lot of facts concerning history of mathematics and the men and women who wrote this history The book is available from Jan Vermeylen s Rhombus Shop See the address on page 34 Call for partners At the Information Day about the European Union s Educational Programs was asked eventually to start a transnational project in the frame of the COMENIUS Pro gram So l d like to ask for partners who could imagine cooperating We have to be at least three participants from different European Union countries have two ideas In mind but maybe there are better ones oet up a transnational structure for math teacher s training in modern technolo gies Exchange evaluate improve and customize teaching materials for teaching maths using modern technologies If you are interested then please contact me as soon as possible Please notice my new email address Josef Boehm bboard at is not valid since a couple of years D N L 24 Liebe DUG Mitglieder Einige recht arbeitsreiche Tage und N chte liegen hinter mir Aber es ist wieder gelungen
62. hend auch die Botanik durch die Sprache der Mathematik zu erfassen vgl Die analytische Form der Bl tter Quedlin burg 1895 Eine der einfachsten Kurven die sich in modifizierter Form unter den von Habenicht ange gebenen Blattkurven findet ist die Pascalsche Schnecke die auf Etienne Pascal den Vater des bekann ten Blaise Pascal zur ckgeht Wie sich herausstellt k nnen spezielle Pascalsche Schnecken zur Dreitei lung des Winkels verwendet werden und spielen bei mechanischen Problemen ein Rolle weswegen sie urspr nglich auch unter dem Namen Sauveur s und de l Hospital s Zugbr cke bekannt waren Bereits vorher hatten sich auch Johann und Jacob Bernoulli mit dieser Kurve besch ftigt If all the forces which are contributing to the evolution of a plant would be recognized mathe matically and also their internal mechanism of their organs then we were enabled to represent the whole process of its life by formulae We specially would obtain the equations for the curves which form the contour of their leaves But reversely if we even knew the equations we would yet be unable to express the plant s life by formulae This goal is very far as we have to be sat isfied with equations which are reproducing the leaves contours only in a very approximative way Loria 1902 p 307 At the end of the 19 century Bodo v Habenicht inspired by scientific and technical supreme achievements X rays Eiffel tower tried to describe botany by th
63. if they are right or not Rotating in ACROSPIN you will obtain top front and side view of your object You have to imagine all the following pictures in different colours and layers So you can switch off and on different parts of the objects See a composition of the objects axes house and tower p22 Josef Bohm DERIVE ACD ACROSPIN D N L 24 In DERIVE 6 you have to Insert F4 the objects separately CFIGUR W axes axesw FIGUR W tow toww and FIGUR W house housew respectively Then change the Scheme in the Color Plot Window to Custom and choose the color of your choice for the Grid I called the first combination of objects VILLAGE In the second figure you can see the eighth of a diamond Rotating this part seven times using DERIVE the whole figure will emerge You can find the objects in P 3D MTH The Diamond the girls best friend dram e lp25 0 090yd4 42 5xD25950515221 5 3975 20 59 0 145 0540 L25 05522213 O04 7 1250233 1 7121887 155355 0 dramwi 7 l 2 7 2 95 4 9 2 4 9 0 429549 9 9 FIGUR W diam diamw RO FIGUR _W diam diamw 8 Save both results as different BAS files then run ACD For the first part apply option 1 and then ap pend the whole object in another colour using option 2 ROCFIGUR W diam diamw 8 results in the left plot in DERIVE 6 you have to APPENDCROCFIGUR W diam diamw 8 ER AES TOR CR ED D N L 24 Josef B hm DERIVE ACD ACROSPIN p
64. is not unique to the Geometric distribution but it is easy to see what the restriction should be in this case In the example below in which the moment generating function for a Geometric 0 25 distribution is calculated we must restrict the range of t so that O 0 75 el 1 i e 0 lt t lt In 4 3 Unfortunately if Declare Variable is used with the upper limit In 4 3 DERIVE substitutes the rational form 1817 6316 for In 4 3 This approximation is slightly larger than In 4 3 and the series will not converge A smaller approximation such as In 4 3 0 28768 allows convergence to the correct result 1 4 3et DERIVE changes this decimal to the rational form 899 3125 The payback for DERIVE s inability to sum this series unaided is therefore some very positive work on the geometric series eo x pest fee Mpy dec t 2 012505275 6 User x 0 8 Simp 7 9 t amp Real 0 899 3125 User 1 10 fe Simp 7 4 3 amp The general case of computing the moment generating function for a Geometric p random variable also cannot be done at all The result pe 1 p e has to be obtained on paper but as with the Binomial n p distribution DERIVE can easily calculate and use M 0 and M 0 to find the mean 1 p and the variance 1 p p M t If instead of using the moment generating function we use the probability generating function of the distribution G t b p the same general probl
65. m othe 1 Fer Fir Fur FE Fh gt F isebralsiclbiherlranzolciear az a SL ek F x 1 os EM 1 1 1 i fly 1 1 1 21 1 1 1 1 1100 ri a 400 me ri 12535 L1i1005 0 400 0 12331 me L1100 0 400 05 FESER BAL IHRIN RAD AUTO IH RAD AUTO FUNC fst Ma Le 1 1 1 _ 133 2 Do zu 200 lz33 2 Male RAD AUTO FUNC 2 30 The model for the height control function is 1233 ho 550 x 30 200 x 60 200 x 70 x df 6 Find f using a geometric transformation of a vertical shift When x is O for example the function is 126387 5 Add 126387 5 as the parameter f E c eoon tracelRe raennsinbraule e c e on traceRe raennsinbraule e 1 1 The final model needed by the computer is h x 550 x 30 200 x 60 200 x 70 x 175 1263875 References Laughbaum E D 1996 Modeling data exhibiting multi constant rates of change The AMATYC Review 17 2 27 34 You can find the equivalent DERIVE application as SUM_ABS MTH Josef
66. mniscate three quarters of a pseudosphere and another ro tated curve There exists a simple formula for producing a solid of revolution using any curve given in parameter form x t y t F t o x t cos 9 x t sin o y t fr ME N E ERE Sd cu SC me zi s Et e e p ka 5 LI Mam LM yi Y Pd NI Wi Mass f A ep ub Dum me T _ a sae cet esie i Ny E E fee s aF 7 AN Lo bte koe e ELI i zoQ4U jn N A D rl me T er wo m 1 xm Eom a ELM j jj The next example uses the zooming abilities of ACROSPIN I produced the Henneberg Surface in different scales I took another scale and zoomed in Observe the dark spot in the centre of the graph at the right and what is hidden in it 2 2 SINH s COS t SINH 3 s COS 3 t 3 2 2 SINH s SIN t SINH 3 s SIN 3 t 3 2 COSH 2 s COS 2 t D N L 24 Josef B hm DERIVE ACD ACROSPIN p31 It is obvious that it cannot be too difficult to produce presentations EM Chapa of intersecting surfaces together with the intersection curves tan Nu 7X gents normals binormals osculating planes You can also add Tf 7 labels as you will see in one of the next examples I teach in a secondary school and I have some models of the conics sections But using a tool like DERIVE it 1s convenient to produce a double cone the intersecting plane then t
67. ng Various Projections B hm AUT Tilgung fremderregter Schwingungen Klingen GER 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 Bohm AUT Line Searching with DERIVE Collie UK About the Cesaro Glove Osculant Halprin AUS Tangrams with DERIVE Poblaci n ESP Hidden Lines Weller GER Fractals and other Graphics Koth AUS Experimenting with GRAM SCHMIDT Schonefeld USA The TI 92 Section Waits a o and Setif FRA Vermeylen Belgium Leinbach USA Halprin AUS Biryukow RUS Weth GER Wiesenbauer AUT Keunecke GER Aue GER Stahl USA Mitic UK Sirota RUS 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 24 DERIVE USER FORUM p 3 Terence Etchells Liverpool UK As you may be aware I have a new job lecturing in mathematics at the John Moores University I am sending an attached file called ROMBERG MTH This is a file that I recently wrote to perform Rom berg integration and produce the table of successive Romberg approximations ROMBERG f x a b n estimates INT f x a b n via Romberg s method starting with Simpson s rule estimates for 2 4 8 2 n strips ROMBERG TABLE f x a b n produces a table of the su
68. non ccu uniovi es Abstract In this work the general solution or a particular one to linear ordinary differential equa tions homogeneous or non homogeneous of third order with constant coefficients is given using DERIVE A file with the appropriate DERIVE functions s included 1 INTRODUCTION DERIVE cannot be considered a programming language However new functions that make use of the operations and functions included in the program can be defined In the program handbook it is said that DERIVE 1s not more than a collection of mutually recursive functions We believe that with the available information differential equations have not been explored adequately In the hand book we are informed that DERIVE contains three files ODEI MTH ODE2 MTH and ODE APPR MTH to solve respectively ordinary differential equations of first second order and numerically We have designed a file LODE3 MTH to find the general solution or a particular one of an homogeneous or non homogeneous linear differential equation of third order with constant coeffi cients VED by E eye OX 2 DESCRIPTION OF THE FILE Firstly the roots of the characteristic polynomial x ax bx c are determined Considering that these roots appear in vectorial form where each element is an equation in the form x root the RHS function that selects the right member of the equation has been used This function RHS is only present in DERIVE 2 58 and later Next
69. o calculate the intersecting curve as a space curve and a bit more ambitious to shade the inter secting surface All this 1s hard to do by hands only but let DERIVE NN do the calculations The students have to know the strategy to obtain NS results which can be used to be represented by ACROSPIN Wy yy DIR FFRFEER fee p ih You can turn the model round and observe it from all directions The different colours enforce the imagination Switch off and on the layers Compare the hyperbolic section with the parabolic and the other ones I want to represent a double cone the intersecting planes the inter section curves and I will try to add a shading for the intersection ET xus figure to make the picture more impressive N You can find the whole calculation in CONICS MTH The various Sn CONIC ACD files show the different conic sections X ES I want to show the start of the parabolic part to give some com pc ments on it Pe Hot done Sete COS 0 S t SIN S 44 2 AAt II 4 vector vector cone D Or 23 E xd 2 J 20 moas prea el 04 21 337 3 25 2 ae Gy E 89 1 mode Bee S CES D 4 Wer Oy c Se 3 55 VECTOR VECTOR pp Bu qe hy v 3 3 0 5 2 57 SOLVE pp cone t u v 1 i SIN 0 58 uM v 3 1 COS 9 3 1 COS 9 1 COS 60 3 t COS 3 t SIN 4 4 t fi 1 61 3 cos ie 34 SIN 9
70. parser Version 4 01 fixes some but others remain DERIVE 4 00 and DERIVE XM have problems doing some integrals with very large but constant expressions of the form INT BigConstant 0 0 27 where BigConstant is a very large expression NOT a function of x Where my old DERIVE v 2 01 correctly returns 2 x Bigconstant DERIVEXM and v4 00 seem to hang I haven t fully experimented with v4 01 but it can now do some trigonometric integrals that v2 01 can do but that XM and v4 00 can t do I ve played with all the Manage Trig settings One more thing Note that v4 01 will not read a MTH file containing multicharacter Greek letters created by v4 00 The extended ASCII characters used to represent concatenated Greek letters seems to have changed Thanks in advance for any answers Ian Llorens Fuster Valencia Spain llorens mat upv es Dear DERIVERS Let a function F u 1 where u is for example a matrix but that is not important I want to programme the iterative nested function F u 1 F F u 1 2 F F F u 1 2 3 F F F F u 1 2 3 n where n the number of iterations depends on u IS IT POSSIBLE DE ES DK BK B IN PAN ZW LAN ZW PA PAN YW IN YA CAN YW IN YA ZN YW IN NY ZN YW IN WY ZN YW IN YA ZN YW IN YA ZN YW IN NY ZN Another question Is it possible to sort the elements of a vector Thank you DNL lcan only answer the second question In DNL 13 you can find a sort routine
71. proximate 4 55 67 and 84 save in different BAS files run ACD and so on The next plot shows one of the favourite examples of Bert Waits So it is an honour for me to dedicate this DERIVE ACD ACROSPIN product Bert and his Power of Visualization It is a real repre sentation of the complex roots of a 5th order equation I used DERIVE to find the modulo surface of Z 6z 25z 0 The peaks are in the complex plane at the positions of the five solutions I added the complex plane the axes and their names R and I both letters as special space curves given by a list of points Look at the file BERT MTH on the diskette The R can be seen in this position The other graph shows another artificial range a fractal landscape generated by a recursive algorithm in DERIVE which I will present in one of the next DNLs Bert s Complex Range The Fractal Range T arf es j as E iis IE D N L 24 Josef B hm DERIVE ACD ACROSPIN p33 From fractals it is not very far to chaotic behaviour Josef Lechner was the first besides Richard Schorn who checked ACD He immediately tried successfully to visualize and to animate dynamic systems of three variables During a phone call he mentioned the possibility now to animate the LORENZ attractor So I will finish my interaction between DERIVE and ACROSPIN showing the famous LORENZ attractor together with a zoom in from another look out The list of points was gen
72. r Punktfolge haben Carl Leinbach amp Marvin Brubaker DNL 22 rev page 29 ein h bsches Beispiel geliefert Verallgemeinert man die Konstruktion ein wenig gelangt man zu Edward Sawada s misguided missile DNL 22 rev page 8 MEOS 2 MERI H posz 0 0 plis 0 5 0 59 SORTP S Tpzs DL 09 Purus RKI t LE Ue DUIJBCEIDIUDBEQUE AIDZrCBEUPCSRE OJwSuPOESSKk 3 POLGE LIESSEN ES FVBECTORLP IL ESIK KR ON FODGEL 0790 1 15 Wegen der entsetzlich langsamen Rekursion empfiehlt sich eine iterative Fassung POMGE rS pak ERATE I Dreri 7358571 5 larpra PIE TED Der Aufruf von BEOlGEZI0 9 05L 125 liefert das omin se Sawad sche missile Interessant sind auch die F lle r s gt 1 und r s lt I bei spielsweise folge2 0 15 0 9 50 und folge2 0 05 0 9 80 Hier k nnen die Sch ler selbst ndig experi mentieren und Vermutungen hinsichtlich Konvergenz bzw Divergenz aufstellen R diger Baumann shows a link between Carl Leinbach s amp Marvin Brubaker s recursive se quence of points in DNL 22 rev p 29 and E Sawada s misguided missile on p 8 same issue Generalizing the construction presented in the first contribution we obtain the mis sile As the recursive construction is very slow R diger recommends the iterative proce dure Furthermore he points out that there are interesting cases with r s 1 and r s 1 Students could set up conjectures concerning convergence and divergence Alfonso J Poblaci n Valladolid Spain
73. t part of GRAPHICS MTH FIG v pl p2 pla ple n p2a pZ2e m VECTOR VECTOR v p2 p2a p2e p2e p2a m pl pla ple ple pla n ROTODJns cVECDOR ODj x ROTATE A12 PI k fl dn FIGUR W p w APPEND VECTOR ELEMENT p ELEMENT w kl kl DIMENSION w FIGUR E p e APPEND VECTOR ELEMENT p ELEMENT e KE 1 ELEMENT p ELEMENT amp pkl 2 inf inf 1inf l kl DIMENSION e J Pl m zn VECTOR VECTOR ELEMENT m i 1 COS phi_ ELEMENT m i 1 SIN phi J ELEMENI m y3 2 phx 07 24 piy2 pis 2m y 1415 DIMENSION mi P2 m zn COPROJECTION Pl m zn REV m zn APPEND P1 m zn P2 m zn You can create a polyhedron with a list containing all the edges of the solid P1 P2 P2 P3 Pi Pj with Pi xi yi zi Save this file as a BASIC file I prepared some DERIVE functions to produce polyhedrons from a list of its points and either a list describing the way how to connect the points or a list matrix containing the edges P WAY points way and P EDG points edges lobte LI ORT 0 32e vo Ro r es We Ve 243 1g dud 22 edges 1 1521z 124313 5 213 4 217 13 21 B 7211 P WAY points way P EDG points edges Both functions will when Simplified return the points of a pyramid Or you have produced a family of parameter lines with parameters u and v using the powerful VECTOR VECTOR f v construction Using COPROJECTION from GRAPHICS MTH you will have both famili
74. te winter option ApproxtHi545 D DOKUS DNLSDNLZ24 Free 68 Derive Algebra Derive 3 14 re ae eg SO nO Ok al Ng lu py ae tp peo bea ty lee bees 32434 iso yy Sp Ue Uy Vos pte oe Oly 1229 0 ea ar 14707 Cl Es ee pH 57 Ul EP E ln 2 2 4 0 0 3 7 1 53 0 2 0 2 2 0 0 4 Derive 6 10 ee peg dey ep ey dee Fle epg ey dee 595 519 7 2 152 89 1319 22 8 3 13 9 yn Lea rOl ey Ue Ol plow dou aa Ze Eder lee le lose Leon zy oy 2 21 14 0 0 13 171 23 01 12 0 72 2 0 0 74 Compare 0 83 and 8 3E 1 believe that it would be wasted time to change my BASIC program ACD from 1996 and compile it again because we do have now 3D plots in DERIVE In the following contribution will add the respective DERIVE 6 plots You may compare Much fun reading or rereading DERIVE ACD ACROSPIN Josef MI have a little present for all of you who have still the DERIVE DOS version David Parker gave permission to distribute the original ACROSPIN to the DUG Members you can download it among the other DNL24 files Many thanks to David D N L 24 Josef B hm DERIVE ACD ACROSPIN p19 Interaction between DERIVE and ACROSPIN with ACD Josef B hm W rmla Austria Using DERIVE 3 x you can save surfaces as ACD files if the surface is defined as a function of two variables f x y For other 3D solids it is not possible to have a direct conversion to an ACD file for running ACROSPIN animations It is also impossible to produce
75. te my idea We create a spindle torus and then try to produce certain sections Which values for the parameters are responsible for the various torus parts Unfortunately I cannot print in colours maybe in some years we will have a colour print DNL Yes now we have As I have mentioned before I wanted to demonstrate solids of revolution I also wanted to make visi ble the two families of parameter lines So I produced a DERIVE function to create these surfaces from any given profile We can design a profile by a list of points or by a piecewise defined function I show one example for each possibility left and right plot windows Josef Bohm DERIVE ACD ACROSPIN p29 D N L 24 EET The next plots show the solids you can see the profile together with the parallel circles then the longitudinal intersections and at last the full grid It is very nice to overlay the different layers and then to rotate the glasses round their axes to zoom in and out to translate to accelerate 3 121 prof LIB H1 3 81 3 H 51 tt63 F t IF t 8 HD t 3 2 M 64 UECT RCIFCCO t1 t B 5 4 2 122 Pli prof 4H 65 SAE ee ee ee ee a 123 ea es ae ih i HET RR NEN ff P ES SEES m aims ST rA afl LL S meuo prism jJ zL Nc Se p30 Josef Bohm DERIVE ACD ACROSPIN D N L 26 In this collection you will find a rotated le
76. ther deg r x 0 or we can repeat this procedure with g and r until we finally arrive at polynomials f g s t g doesn t contain x any longer and it is clear from the construction that every solution of f g 0 is also a solution of f g 0 Given a system of polynomial equations this algorithm can be used to achieve a triangulated system in a way very similar to Gau elimination I will try to give you an idea how this works by means of a simple example which was also used by A Perotti in 2 We consider the system of polynomial equations x z l y yszelex By applying PREM three times PREM x y 1 4 zx z 14y z x 2 y x y 2a PREM x y 1 Z y Z 1 x Z x y 2 1 y 1 2b PREM x y 2 1 y lx 2 y x y y x 5 X 44 2 x N 3 3 x 2 x 3 we get the triangulated systems made up of 1 2a 3 or 1 2b 3 respectively Solving 3 w r t x yields SOLVE x 5 xX 44 2 x 8 3 x 2 X X x 0 X 1 X SQRT 5 2 1 2 x SQRT 5 2 1 2 Using 2a and 2b we get the corresponding values of y SOLVE LIM x 2 y x y 1 x 0 y y 1 SOLVE LIM x 2 y x y 1 x 1 y y 2 SOLVE LIM x y 2 1 y 1 x 1 y y O y 1 SOLVE LIM x 2 y x y 1 X SQRT 5 2 1 2 y y SQRIT 5 2 1 2 SOLVE LIM x 2 y x y 1 x SQRT 5 2 1 2 y y SQRT 5 2 1 2 Now for reasons of symmetry y z must hold except for x 1 where we only know that y z e 0 1 By using 1 to decide this ambiguous c
77. to browse it there are lots of my files to down load as well e mail t a etchells livjm ac uk web page http www cms livjm ac uk www homepage cmstetch index htm DNL It would take a lot of space to print all the functions You will find them on the Diskette of the Year in subdirectory Tl accompanied by W Pr pper s DIRA package for investigating functions and his SOLSYST function to solve simultaneous equations in a DERIVE like way That are the true DERIVIANS who implement their favourite DERIVE functions on the TI 92 See more in the TI 92 Corner and on page 43 a demo of PLOT MAT Johann Wiesenbauer Vienna Austria In the following the polynomial system of equations cf The International DERIVE Journal Vol 3 No 2 p 96 will be solved by means of my routines cf DNL 23 Titbits 8 Johann refers to an article dealing with the implementation of Groebner bases in DERIVE Josef Preload RED u v and SOLVE2 u v x y from TITBITS8 MTH First we prove that there doesn t exist a solution of the system above where x y z are all different The following equation 2 fie RED Geer I SE zZ IE aX ee see WES shows that either z 1 or y 1 z 1 For reasons of symmetry z 1 or x 1 z 1 holds as well Therefore our assertion is certainly true for z 1 But for z 1 the system above simplifies to x y 0 x y 1 which leads to x z or y z due to de eo 2 SOLVE2 x y x y 1 U ser Simp User
78. vor auch f r DERIVE sehr interessiert Ich arbeite heuer besonders im Precalculus Bereich Meinen letzten Brief dieses Jahres m chte ich aber nicht beenden ohne Ihnen allen f r die engagierte Mitarbeit zu danke Ohne diese g be es l ngst keinen DNL mehr Noch wertvoller aber finden wir die herzliche Freundschaft die die DERIVIANER weltumspannend untereinander verbin det Jedes Zusammentreffen jede Konferenz ist ein deutli ches Zeichen daf r Viele pers nliche Begegnungen und Freundschaften haben sicher nicht nur unser Noor und Josef Leben sondern das von vielen unter uns bereichert In diesem Sinne w nschen ich allen ein frohes Weihnachts fest und ein gesundes gl ckliches und erfolgreiches Jahr 1997 Jahr I nach DERIVE for Windows Ich hoffe Euch alle im n chsten Jahr wieder begr en zu d rfen Josef Bitte begleichen Sie Ihren Mitgliedsbeitrag 1997 LETTER OF THE EDITOR pl Dear DUG Members Some very busy days and nights are lying behind me But it has lucked again the last DNL of 96 is ready While it is going to be printed the Diskette of the Year has to be filled brimful and checked if all the files and the Christmas gifts are on it Then my wife Noor and my daughter Astrid will produce several hundreds of copies put them together with the DNL the 3D spectacles and the renewal form in big envelops affix the stamps and address labels and send them on their big journeys Please note my new
79. with centre M and diameter b and a point O on the circle line Take any other point Q on the circle line draw the line OQ and find the two points P and P with QP QP a const Then P and P are two points of the curve Herleitung der Kurvengleichung Dervivation of the equation of the curve Aus der Konstruktion ergibt sich f r P und P OP OQ QP bzw r b cos g a und OP OQ QP bzw r b cos o a L sst man f r r negative Werte zu vereinfacht sich die Darstellung zur allgemeinen Polardarstellung Pascalscher Schnecken r b cos o ta This is the polar form See the construction 1 r a b COS a i 2 Mit r 4x y berechnet man mit a soc sy a b costed DERIVE sofort UR 2 a x y E x y b xy 0 3 icu Damit sind Pascalsche Schnecken symmetri cc Z A A 5 dix y sche algebraische Kurven vierter Ordnung NND x y F r a b erh lt man aus der Kurvenschar wie 2 2 2 2 H6 x y sad y b x in der nebenstehenden Abbildung die Kardioi de vgl Folge 7 HY e eg 2 2 H8 x bx y a x y We substitute for r 4 x y then we can see that these curves are symmetric algebraic curves of order 4 For a b we obtain the Cardioid Compare with Lexicon 7 p42 Thomas Weth A Lexicon of Curves 9 Weitere Erzeugungsweisen Es sei noch erw hnt dass Pascalsche Schnecken sich auch durch andere Konstruktionen erhalten lassen e als Rollkurve
80. y to circumvent this restric tion when defining his function PREM p q v POLY COEFF u x n 1 n LIM DIF U x N x 0 POLY DEGREE AUX u x n IF u 0 n POLY DEGREE AUX DIF u X x n 1 POLY DEGREE AUX DIF u x x n 1 POLY DEGREE u x POLY DEGREE AUX u x 1 LCOEFF p v POLY COEFF p v POLY DEGREE p v arb var RHS SOLVE 0 0 x SUB 1 SUBST f x a LIM f x a SWAP AUX fx t y SUBST SUBST SUBST f x t y x t y oWAP f x y SWAP AUX f Xx aro var y REMAINDER V AUX p a v vO S WAP REMAINDER SWAP p v VO SWAP Ga v vO v vO REMAINDER V p q v REMAINDER V AUX p a v VARIABLES p q SUB 1 MUL FAC p a v LCOEFF q v 1 POLY DEGREE p v POLY DEGREE q v PREM p q v REMAINDER V MUL FAC p q v p a v D N L 24 J Wiesenbauer s Titbits 9 p49 In DfW the same function can be defined without any auxiliary functions as follows PREM p q v 2 REMAINDER LIM TERMS EXPAND q v SUB 1 v 1 1 LIM v DIF f_ v f_ f_ TERMS EXPAND p v SUB 1 LIM v DIF f vy f f TERMS EXPAND a v SUB 1 p q v On close inspection of this code you will find out that I used the following more efficient implementa tions POLY DEGREE u X 2LIM x DIF f x f f TERMS EXPAND u x SUB 1 LCOEFF u x LIM TERMS EXPAND u x SUB 1 x 1 Though I didn t need POLY COEFF u x n I would like to point out the following interesting alter native to the definition given above cp DNL 21 p37 POLY COE
81. your results are compatible with my ACD but it must be said that even without ACROSPIN you will obtain nice plots using DERIVE s GRAPHICS MTH Dear DERIVERs the DMO and DOC file are on my desk but I decided to include it in one of the next DNLs to not overload this issue with geometric contributions So look forward to learning how to produce implicit 3D plots add one screen dump to give an imagination of Sergey s IMP SURF MTH see above David Sj strand Sweden Monet However I have some ideas concerning computer algebra and computer geometry It is a chal lenge to do with DERIVE what you can easily do with Cabri II So your mail really inspires me to make a contribution to the DNL I am very fascinated by the TI 92 and some other TI graph calcula tors Excel and Cabri but I have the same feeling as you have deep in my heart DERIVE is the num ber one mathematical software We are thinking of organizing a DERIVE TI 92 Cabri Conference in Sweden in the end of sum mer 1997 It would be great to see you and Noor in Sweden then Cheers David DNL There is nothing to add except the fact that we are quite sure to enjoy visiting Sweden if there is any chance to do so kevans ns Sperry Sun COM Is it possible to configure Classic DERIVE V3 04 to print to another port say LPT2 which points to a network printer M Walkenhorst BUNDOORA Victoria Australia M Walkenhorst ee latrobe edu au I was wondering is it possible
82. ysiology The Biochemistry of Cells A Biomathematical Approach to HIV and AIDS Genetics The text has extensive exercises problems and examples along with references for further study Birkh user Boston 1996 ISBN 1 55953 107 X 434 pages Hardcover DM 118 AS 862 sFr 98 p18 Josef Bohm DERIVE ACD ACROSPIN D N L 24 This contribution from 1996 is of historical and nostalgic interest In 1996 there was no 3D plot possible from within DERIVE What we could do was producing 3D projections isometric and other parallel projections or perspective projec tions applying self made projection procedures And there was David Parker s inexpensive ACROSPIN We could call ACROSPIN from DERIVE but only for functions of two variables not for all other kinds of 3D objects What was demonstrating in 1996 can be reproduced in the DOS environment even today if you have ACROSPIN available But there is one obstacle Com pare the two files below Both files have the same source and are saved as BASIC files what can be done in DERIVE 6 via the Write option B ION Load d TI Handheld k C File Page Setup Fortran File PrinE Preview Pascal File Print Ctrl F Rich Text Format File Cri R This is how it looked like with DERIVE 3 14 154 FIGUR_4 diam diamw gt EEH L L2 H 83 1 71 2 H 83 1 71 2 9 H 1 21 I2 8 83 1 71 2 9 8 1 TRANSFER SAVE Derive C Fortran Pascal Options Sta

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