THE DERIVE - NEWSLETTER #74 USER GROUP
Contents
1. yOrCyl 2 2 2 2 2 I 0 b 0 1 1 Oc 0 0 0 QO c 1 0 0 21 d0 1 Cal 0 0 1 2 deb 9 0 1 5 substituted Peter s special values and could confirm Peter s results This formula looks much better isn t it Many thanks to Peter for his inspiring contribution By the way gave this problem to my students at the Technical University Vienna and sur prisingly many of them tried the proof not using vector calculations but worked with trig func tions sine and cosine rule Website of GeometryExpressions www geometryexpressions com Everyone knows that mathematics offers an excellent opportunity to learn demonstrative reasoning but I contend also that there is no other subject in the usual curricula of the schools that affords a com parable opportunity to learn plausible reasoning I address my self to all interested students of mathematics of all grades and I say Cer tainly let us learn proving but also et us learn guessing G Polya 1954 Mathematics and Plausible Reasoning Princeton NJ Princeton Univer sity Press Josef Bohm Working with DEQME Differential Equations Made Easy Review and Description of some features Josef B hm DUG Member Nils Hahnfeld gathered a bundle of programs and functions into one application DEQME v 7 0 which can be downloaded and purchased from www ti89 com The application2. Plot the parallelagrams 11 patti pariB AJ 12 Y U W09 U 13 goaltriA B Flot one af the final triangles 14 POLYGOM FILLC A B C Plot to fill the initial triangle 15 goaltr B C goaltr C B3 p22 P L ke Rosendahl An Interesting Triangle Property 15 goaltriB C goaltr C B3 Plot the other teo final triangles Prove the properties of the final triangles 16 BD D 17 Ppt amp B C Al Ppt C A 0 E18 Ppt B C A B O 0 19 IPBECB C B C B 20 Ppt C A A 0 A B D N L 74 asked myself if this special triangle would have a nice result for its area and reproduced the construction using GeometryExpressions Josef 6 5350428 6 5350428 3 2 a b 2 8 c 2b tah Gab th tac 2a b c thc a c cca Jatb c ja bt c fatbhte 2 At the bottom you can see a part of the formula for the area with given sides a b and c of the initial triangle It is a nice formula indeed can export this formula to DERIVE MATHEMATICA and with the new GE version to TI NspireCAS too Abs CGOlbs a 2 C b 2 amp C220 D 5 13 Sin
3. 27082 42 ini This is the general solution but how to obtain the special solution The solution which is presented in the Prgm IO screen is stored as res and can be recalled in the Home screen In F7 Menu you can find the respective note Pi lAigebralcstclother rrsntolciesn ue Pi lAigebralcstclother rrsntolciesn ue res 2 05 2 05 Bsolwel4 5 105 2 CE ee PS x u 2 e E3 P qe cien 2 3 p 83 424 3 42 ES 2 22 85 4 7 62 83 1 2 22 8353 pes 25 solue a 7 2 2 85 1 2 22 853 z ez 3 2 3 2 42 5 24 Ed 3 lni RAD AUTO FUNC 2 20 3 1nt33 z Pee ligebralcaiclother Prantolciesn ue 3 x243o0 u 2 2 3 0 24 ui Tek 4 zu 24 us dx 8 x 80 45 942 6 9 30 u3 3 u2 24 40 882 B expandl x 3 x u 1e 3 42 2 ub x expand anst1 gt 2 HAIH RAD AUTO FUNC Josef B hm Working with DEQME Just for checking the result I load DERIVE and apply the LIN FRAC function ey ox G Aa a a d ae dd Lex 4 3 LNI 3 LH 1 1 6 2 2 2 After some manipulations I obtain the same solution Well done DEQME 3 2 2 2 3 2 cox Dex v 8 In case of non intersecting linear functions in numerator and denominator DERIVE provides a sp
4. Qa the C 1 172 v cx 1 22 2 172 a hb c 172 b 2 10 2 2 1 2 In D b c 172 be 1 C29 172 Qa c b cw 1 C1 72 Qa he c 172 2 103 Abs 2 1 2 2331 CGCa 4 7192 COD 2 v 2 2 42 amp 13 C 1 6 c 172 a c br 1 c 172 fa c 1 17 2 a h c 17 2 a 2 iiiar 7 1 172 bw 1 c 17 2 a fw 1 172 5 172 h 2 1 1 b c3 172 Go 1 172 lac 1 17 2 cpp 1 2 092 2 abs 2 Cb 2 193 2 2 amp 2 abs 2 4 2 DO 2 D 6 C39 41 2 Ca he 7 193 03 17 2 v Cache 1 172 lab c 172 a 4 CCh 2 a 2 2 CB 4 2 30 2 2 2 B 2 232 DO v 2 CCB 2 D 742 13 3 C172 fat be 1 C90 C717 2 Cac hc cw 13 Ct 1 2 Bb c CC 17 2 D C10 1 517 8 The MATHEMATICA output of this formu
5. Cabri or GeoGebra or any other DGS But we can do it on the handheld too using either the TI 92 Voyage 200 with their Cabri Application or the TI Nspire handheld device The next step could be verifying the conjecture using special data for three points and verifying the fact that the other two possibilities drawing H the triangle will give the same result TI NSpire the Graphs amp Geometry Application I will proceed with DERIVE using the slider bars keeping the coordinates of the triangle vertices gen eral and then continue proving the two properties 1 CaseMode Sensitive 2 Word 3 V CU V UJ 4 le VOU BS sexes Introduce sliders far xb xc and yc from 10 10 5 traiangletA B Plot the triangle P L ke Rosendahl An Interesting Triangle Property VJ z b e M il 6 square U Vw square over UV outside of the triangle neg direction of rotation ET square U Yi V V perpiV UJ U perptv UJ 5 square A squaretC square B Plot the squares P L ke Rosendahl An Interesting Triangle Property Vv vj paint of parallelogram with paint of triangle S as vertex 9 Ppt U V W perptw UJ perp VV parallelogram with vertex Vy 10 V W perpt PptiU V W perptw w
6. EC 5 re ie EG 15 1 21 24 25 30 60 90 24 2 30 32 36 40 3 40 80 120 33 38 39 44 48 Ge Ln 100 150 6 9 4 8 12 25 30 35 30 36 42 40 45 50 48 54 60 55 60 65 66 72 78 5 10 15 6 12 1 m DERIVE and CAS TI User Forum Tom Barbara Oregon USA Hi Josef Well your routine looks correct Studying your example will be a good lesson in programming for me Thanks so much Fred J Tydeman I notice that in a function definition one can have multiple statements on one line with a comma as the separa tor between statements However outside of a function when I try either a comma or a semi colon to separate multiple statements on one line they are considered errors So how does one do multiple statements on one line Danny Ross Lunsford Use a PROG statement to group them e g 10 SIN x DISPLAY Hello world This evaluates to whatever is returned by the last statement in the group drl Fred J Tydeman I created a file a mth with the contents of PROG X 3 4 a l b 5 I did Simplify Basic the x a line and got 3c 3e Johann Wiesenbauer Hi Fred Maybe I m wrong but it appears to me that you didn t understand the crucial difference between x 3 y 4 and prog x 3 y 4 In the first case x and y will have the corresponding values after INPUTTING that line in the second case x and y will have those values only after SIMPLIFYING th
7. Particular solution es 5 x u t ty o2 RAD AUTO FUNC NEEDS 2 2 tx y S 83 x 1 2 ansi iolss and 4 22 HAIN RAD AUTO FUNC Durr E3udation be made exact m tir vind through b an inkte3ratin3 Factor LF Heina DF if 22 cana AMA avn ony depends on Enter Ok E C CHMCEL AUTO FUNC only contains x sudx EDIE oge gds MultiPl throuah bu I F x DTaBxCAcTxC EEgT i3 HAIN RAD AUTO FUNC TETTE Cxru3dx x usduzt is homogeneous 1 Lilse Yeux Curr C Cu 1 ee de Ct 22 Simplify u 2 F ai edt Cu HAIN RAD AUTO FUNC Since UFU ln 2 KAD AUTO FUNC RAD AUTO FUNC Presses eked pata eR Ex Pixs Nonie bearen Ex Enter OkK ESC CANCEL RAD AUTO FUNC B 1 Cy Dax d ER ED 1 exact now and has Lhe general solutions 2 HAIN AUTO FUNC will be continued p32 Johann Wiesenbauer Titbits 37 Titbits 37 Factoring Integers with DERIVE 2 c Johann Wiesenbauer Vienna University of Technology This is a sequel of my last titbits where I started with the discussion of several algo rithms for the factorization of integers namely Pollard s two factoring methods rho and p 1 as well as Lenstra s ECM In the foll
8. 000 of x 2 000 or x 3 _ 4 pego x 1 500 1 956 or 1 500 1 936 i LL A 4 ld Git 4 2 311 RAD EXACT Now check for a very small imaginary part this is what you proposed to do Multiple Eigen values are presented Fo FE FT i FiF Fur FE Pet c Risebra caTe other Prsmro e1esm ue _ di 1 508 1 936 or x 1 5808 1 935 ij te 21 19 2 n a 1 pe msio auanei real 7 1 B 5 2 3 6 819 9809 1 898 i 9892 pela sec adie sts 2 311 RAD EXACT DE in zu 1 1 dimcl abitCimaatl lil S er cS B up eigul Els augment C1 TI SEES Pm DERIVE and CAS TI User Forum 1 00 2 000 5 0007 2 21 500 1 956 1 500 1 936 i 3 00 4 LL 2 72 21 L4 5 01L 4 2 311 AIH DE H RAD EXACT Pi Aigebralcstclother rsntolciesn ue Pi lAigebralcstclother rsntolciesn _ 11 00 7 000 2 0002 m 21911 4 22200 2 i eigvs 4 4 2 1 ar x t D XE E 52525202 2 2 19 4 4 2 B mu eigvl NE e 1 2 Z3 2 1 7 culc a 2 MAIN RAD EXACT DE _ 15730 must admit that don t have any idea why the TI delivers so strange Eigenvalues for your 2x2 matrix Ti NspireCAS shows the same behaviour To your other questions The language cannot be checked I
9. 15 available for TI 89 TI 92 and Voyage 200 Nils and I had an extended exchange of emails and files In some details he could refer to earlier DNL contributions I will give an overview about the many features of DEQME First of all see screen shots which show all options offered in the various menus reaching from Basics of 1 order DEs over PDEs to special DEs Laplace Transforms and Eigenvalues Differential Equations Made Easy itr Mare TE HEI E Jut er DE Checker TRL Circuits Bernoulli tClairaut Equation Lagrange Equation Linear Fractions Slope Field sPartic Solution Euler s Metho Te 000 7 k Made Easy ol we ap Differential Ey Tec Inverse Laplace Laplace STEP Made Easu FourierSeries STEP Fartial Fractions 1 Order Et M de TRAC able Homogeneous Exac Hon Exact Linear in x Linear in 7 His n me E a did h Or DE Ae kor PIA OI Dro Cd Jn Fi Fu FE Far 2 Order Homogeneous Han Hamaaeneous Z Drder Checker 5oluer xlzxz lariation of Para Ilndetermined Coeff Bezzel Equation Feductiaon af Order Legendre hu Euler FiF Fir Fi Fur FE For JFF 1 drder 2 drder Order 09 1 Basics Lar lace Differential 5 lave Equation 4 asHelmhaltz Equation 6 Poisson E ti Made Easu prr For For 134 PO
10. 200 in Sequence Mode 2 M uil zr uz ul in 13 13 Axes aF LOT ul uzin 1 uil zr 4 5 1 zZ ugin 13 uidzjz2 1 5 Pisces 0 ance Se etae mean B E p 2 ai ef RHzt 3 240 Ps ef is saa Serea is 3 Serea 17 122 298 0 41 ef 9 7 2 D7 C6 2xD6 RAD EXACT Josef Guido Herweyers Tutorials for the NSpireCAS P9 Tutorials for the NSpireCAS Distributions Guido Herweyers T3 Vlaanderen amp T3 Wallonie Belgium 1 Sampling a normal distribution Press Con to turn on the handheld Press for the home menu m Press 35 to open a new Lists and m T Spreadsheet page A Motes S Data amp Stan G New Das Fs B D Info Hints Consider a population that has a normal distribution with mean 175 and standard deviation 10 We want to take a sample of this population with sample size Wariables n amp Reslze Enter n in cell A1 and 50 in B1 Press Cot for the context menu Choose 7 Variables 1 Store Var Type n in the place of var From now on the value of this cell is linked to the variable n Go to the gray position of column C second line to define the column and type randnorm 175 10 n This will generate the sample in column C First you are asked
11. 5 2 Ch 2 1 2 CC 10 1 c 172 bvr 1 c 172 a h c 1 1 2 a hb c 1 2 1 1 ArcTan t tt a 2 C 13 b 2 2 2 Clar D 172 fas br i 19 C932 172 v 1 41 2 lah c 1 2 3131 7145 13 6 C102 CCBO2 2 4 3 4 2 3 6 2 4 3 2 2 a 2 2 CCC C2 20 4 22 1 2 C103 CC C6 C10 CCCCa v D 6 c3 172 Ca br 1 C9 172 C 1 17 2 ach 17 2 a 1 r 1 1 B e c2 172 bs 1 172 c 1 172 17 2 hb 2 a 2 amp 2 CGU 1 6 c 172 4 Qa c br 1 c9 172 ach 1 1 2 a h c 172 b 4 1 iiiar 7 1 Bb c 172 bw 1 c 1 2 C 1 17 2 c 17 2 190 2 iibh 2 199 9931 19 b C32 1 2 Qa the 1 172 v c0 1 3 1 2 Olathe cy 172 2 2 C B 2 C 1 2 In 1 b C32 1 2
12. 6 ADJOIN O VECTOR 1 j DIM f 13 Loon 1_ 1 Loop HEXT PRIME p 1 JACOBI n p 1 exit p SQUARE ROOT n b 2 INVERSE MOD 2 b p MODS b a b_ nia INVERSE_MODCp_ n ooo ES Ty u u l een Pon ax 2 b x c_ d_ GCD y d If 4 If lt m_ Prag This 1 not the complete code of the program getrel Please refer to the DERIVE file Josef User Simp User 5 getrel true User Simp User 22789 1 0 2 1 23065 0 4 3 0 6 x rel E rel 23052 1002 23039 1 4 2 0 LL 23026 1 LO 0 1 211 User Simp User 2 0 2 1 7 SOLVECMOD 22789 2 3 5 n true User Simp User 2 4 3 0 8 SOLVECMOD 23065 n MOD 2 3 5 n true User Simp User 2 0 0 2 9 SOLVECMOD 23052 n MOD 2 3 5 n true User Simp User 2 4 2 0 10 SOLVECMOD 23039 M0D 2 3 5 n true User Simp User 2 0 1 2 11 SOLVECMOD 23026 MOD 2 3 5 n true Johann Wiesenbauer Titbits 37 p35 The next point on our agenda is to find subset s of all the v i such that their product is a perfect square i e has a vector representation with only even numbers which is the null vector O of suitable size when considered mod 2 In other words we must find vec tors x such that x E rel mod 2 Those vectors form a subspace of the correspond ing vector space which can be given
13. by a basis User kernel c e i_ j 0 p t_ V W_ Prog c MOD C 2 t DIM c e IDENTITY MATRIX DIM C u_ ix 41 DIM c Loop Loop Qe Lg Prog SELECT 0 k_ 1 DIM c RETURN VECTORCe_jk_ k_ d 12 t SELECT Ce 1l Ke Ue If v exit _ FIRST v_ V cV p_ Loop If v 1 exit 1_ FIRST v 1_ _ 2 CL We w ji e Ip 2 e l1 W_ v_ REST v User Simp User 0 0 1 1 0 13 kernel E rel Lo 1 0 0 1 User Simp User 1 0 2 1 001 1 0 2 4 2 2 14 1 0 0 2 0o 100 1 0 4 4 2 1420 reer The computation above shows that we are very close to our ultimate goal of finding solu tions of x 2zy 2 mod n by simply combining the components of x rel and the row vec tors of E rel in the way indicated above by the 1 s in the vectors of the kernel k as well as of the the matrix rel 2 See the examples below for details p36 Johann Wiesenbauer Titbits 37 User Simp User 2 2 X 2 15 m Xx rel n d 2 3 5 isl true L 4 3 4 J J Simp User 2 2 16 SOLVECMOD 24792 MOD 60 n true User Simp User 17 GCD 24792 60 n GCD 24792 60 n 229 109 User Simp User 2 2 2 222 18 SOLVE MOD MOD x rel x_rel n MOD 2 3 5 true V 2 5 J J Simp User 2 2 19 SOLVECMOD 24454
14. n the number v to be factored b upper bound for the primes in f if this parameter is omitted it will be chosen by the program in a reasonable way f factorbasis consisting of 1 2 and the odd primes p below b such that n is a quadratic residue mod p m ceiling b 2 which is chosen in such way that the interval length of I m m is about b fp product of all primes in f which will be needed later for technical reasons User setup v 0 Prog n z V b gt 0 CEILINGCEXP 0 43 CLN n LNCLN n 1 f APPEND 1 2 SELECTCPRIME q 2 g_ 1 3 b m CEILING O 3 b fp true User Simp User 2 setup 24961 true User Simp User 3 n b f m fp 24961 9 1 2 3 5 3 30 The next routine getrel is supposed to collect a sufficiently large number k of rela tions i e pairs u_i v_i such that u_i 2 i mod n i 1 2 k p34 Johann Wiesenbauer Titbits 37 and v i is b smooth where u_i and v i are obtained by using polynomials ax b 2 n for various values of a and b as outlined above The output of getrel is a vector x rel con taining the u i and a matrix E rel whose row vectors represent the exponents of the prime factorizations of the v_i over the factor basis f gerrek kos dod cs cse rq egere x pus tco Catone Prag Dd k DIMCT 1 p FLOORCC2 1 43 m m MIN E 2 10
15. of the day Additionally it was nice to program this feature The next page shows the utility file Josef B hm What s the Time Grandie 17 2 3 xb 6 5 9 10 days Mon Tue Wed Thu Fri Sat Sun tume res t res iz ex res start dummw Frog dummy RANDOM OJ DISPLAY Trainer for calculating tomes DISPLAY DISPLAYC titnj gives n problems to add and subtract times and DISPLAYC time_ex_resz shows the correct answers 9 DISPLAYGC 3 DISPLAYC time gives a day of the week together with the time of the day DISPLAYC and a time to add or to subtract DISPLAY res gives the correct answer j DISPLAYC 3 DISPLAY tome offers n examples in form of a table and DISPLAY time_ex_resz shows the correct answers J DISPLAY start start i days time2 Frog tl RANDOM Z 1 hl RANDOM 24 hice BL EPIS BI spss IRChb 12 tpm ean 2 ml 803 timel APPEMD days tl STRINGCh1 2 hrs STRING mIO min sp hz RANDOME 247 m2 i RANDOMCSD time APFENDCSTRING HZ hrs STRINGEm2Z min op z JL CRANDOMCZ 12 APPEND timel op time zeit 17 pes psc d sepu qub hre Ch SDN Prog st tl 24 50 1 60 ml h2 60 2 erg IFfop st d st d mr MODC
16. p38 09 The DERIVE Session These are the abstracts of the special session Applications and Libraries development in Derive G Aguilera J L Gal n M Gal n Y Padilla P Rodr guez University of Mal ga Generating random Samples from continuous and discrete distributions In this talk we will introduce the utility file RandombDistributions mth This file has been developed for generating random values from main continuous and discrete distributions The programs contained in the file can be grouped within the following blocks Random values from uniform distribution the program RandomUniform returns an uniform random sample between 0 and 1 This program is the base of the other generations Different algorithms have been used to develop this program in order to improve the Derive s build on function to generate samples from a continue uniform random distribution Random values from discrete distributions some generic algorithms for discrete distribution have been implemented as well as specific algorithms for some discrete distributions Uniform Poisson Binomial Geometric Negative Binomial Random values from continuous distributions specific algorithms for main con tinuous distributions have been implemented Uniform Exponential Normal Log normal Cauchy Chi square Student s t F Gamma Beta Graphical approach a program to plot the obtained samples together with the densi
17. subjects of their curriculum or in their careers This toolbox will contain instructions which may either be suggested by the teacher or developed by the student as being interesting and convenient so as to explore the mathematical concepts associated with them There are two examples of possible toolboxes In the first one instructions are related to geometrical aspects of the plane which are studied in high school The instructions are concerning with drawings equations of different geometric objects and some dis tance between the quoted objects In the second one a toolbox is built for the subject Calculus of a single variable in engineering studies In this toolbox there can be instructions about complex numbers to express them in their different expressions calculate their power roots etc It also in corporates the instructions that enable to analyze the concepts of differential calculus such as the tangent the study of increasing and convexity etc About integral calculus several tools can be included for the geometric applications Also procedures for the trapeze method and the Simpson method are implemented In the study of approximate methods of solving equations we can encourage students touse algorithms NEWTON and FIXED POINT which DERIVE has incorporated and de sign a procedure for the method of the bisection of the interval 40 09 The DERIVE Session D N L 74 E Roanes Lozano F A Gonzales Redondo Uni
18. 22 y2 p mod n with y1 and y2 both B smooth numbers and some common prime p B then combining them like this z1 z2 p 2 yl y2 mod p yields a representation where now the right side is perfectly B smooth This situation will occur surprisingly often which is also known as birthday paradox As for the choice of a there are several possibilities but we choose here one of the simplest where q 2 sqrt 2n m for some prime 4 Furthermore we choose b to be a solution of 2 mod a with b a 2 These choices of a will guarantee that the coefficient c of f x is an integer indeed and that the extremal values of f x located at the endpoints and the center of have about the same absolute size Johann Wiesenbauer Titbits 37 p33 Since Q x is a square mod n and a fortiori also mod p for any prime divisor p of n we need only consider primes p for which n is a square mod p This is always fulfilled for 2 and for an odd prime p equivalent to jacobi n p 1 where jacobi is the so called Jacobi symbol which exists as a library function in DERIVE Hence if we consider the set FB 1 2 u p lt B p is an odd prime and jacobi p n 1 then we want to factor f x using only numbers in FB For this reason FB is also called a factorbase w r t B denoted by f in the program Lets start our implementation with the fundamental routine setup v P that sets the values of a number of important global variables namely
19. E Tranzsf T rab Differential Esau ln Tank 1 Annihilator Made Easy sFPicard z Method Error Function Line Josef B hm Working with DEQME I open the first menu point under F1 1 Order 1 Basics and want to inform about General Sol ution DPE g pey n ale teu onn o Ginger solution Diff Eguat ions Order of a DE The goal is to solve Diff Equation Implicit Sol general solution satisfies Explicit Sol a given Diff Equation 1 ar du dx 5 has the aen sol 5 Homogeneous Exz duz dx au Hon Homog DE has gen sol yeCe taxa Enter k ESC CHHCEL Then I switch back to F1 2 Any 1 Order DE and would like to solve the differential equation 2 5 LE 2 4 ea c zu x uors6zx umtdo Et ED General solution Exl z 2 2 2 rl wisst Indep var OPTIONAL Initial condition at x or t gp or Enter k ESC CHH EL I don t receive the solution because the TI built in desolve cannot solve this kind of DE Under F1 I can find the option G Linear Fractions So I try again Solution In _ 3 1 1 DE u sC2g x Bm 7 y utg oxi u st 2x j Z340 x Q 232 2 Enterz K ESC CANCEL 3 2 EE 243 222
20. MOD 180 n true User Simp User 20 GCD 24454 180 n GCD 24454 180 n 109 229 Ok it s high time we put all this together into one single routine that will do all the steps above for us Here o is an offset as to the number of relations with regard to the size dim f of the factor base usually the default value o 1 will do and the memory consuming matrices x_rel and E_rel will be purged by default after their use This may help to keep the file small if you want to save the session after applying qs to a huge number n as global variables are also stored I also included in a textbox the output of all the divisiors of n found by this routine trivial as well as nontrivial ones so you might get a better feeling for the overall success rate of this routine User qs n 1 dispose true k_ x Prog setup n getrel DIM f o k_ kernel E rel If k_ RETURN Too small number of relations _ FIRSTCEXP k LOG x rel 21 _ EXPCk_ E_rel 2 LOG f If dispose x rel E rel d VECTORCGCD z n 2 X y 2 DISPLAY d 1 2 SELECICL Z nZ Ifd RETURN 1 MIN d d 229 109 User Simp User 22 qs 24961 109 Johann Wiesenbauer Titibits 37 p37 Well this was a rather small number hence let s conclude by trying a bigger one viz the notorious Mersenne number 2767 1 which Mersenne himself thought mistakingly to be prime Needless to s
21. THE DERIVE NEWSLETTER 74 ISSN 1990 7079 THE BULLETIN OF THE OCON MNS DLI MVL Z LI Contents 1 Letter of the Editor 2 Editorial Preview 3 User Forum Roland Schr der 5 The Common Measure Guido Herweyers 9 Tutorials for the TI Nspire Distributions Josef B hm 15 Whats the Time Grandie Peter L ke Rosendahl 19 Interesting Property of a Triangle Josef B hm 26 Working with DEQME Johann Wiesenbauer 32 Titbits 37 38 ACAO 9 The DERIVE Session July 2009 Interesting websites http www fachgruppe computeralgebra de cms tiki index php page Rundbrief You can download the Rundbriefe mostly in German starting in December 1987 docu ments are in pdf format This provides a very interesting history of development and spreading of CAS The Rundbriefe contain plenty of information about the various computer algebra systems The next sites are recommended by Prof de Villiers Many thanks for his wonderful and inspiring Newsletter I ll give a selection of his recommendations You can find much more starting with his personal website http mysite mweb co za residents profmd homepage4 html You can download 2 volumes together nearly 600 pages containing the complete proceedings of the ICMI conference Tatwan May 2009 ICMI Study 19 Proof and Proving in Mathematics Education from http ocs library utoronto ca index php icmi 8 Goto the website of the Mathematical Association of Am
22. UN e eee nS at x t or 7 particular solution iz Y or s or i 3 6 3 2143 Enterz K ESC CANCEL e E a ar 3 Fart 5071060 Enter BE and ixan duzdzz Sete tee ety HOTEL Use variables x and Ex Ex x us A y erc ceu M rj Enter z k ESC CAHCEL Runge Kutta is implemented in order to find numerical solutions l 2 Rund3e Eutta Methodi 00 VIT 0 er DE Pe psc HpProximates coordinates cds P t using iat circu 3 Llairaut Equation Z hef xEk amp n The 2 9 n 2Tk 1 422 Lagrange Equation kashf isini thr uini k 242 Linear Fractions k_d hef ox ch th uyini E 32 Slope Field Then 2 Partic Hethod tok 1te4k_ Tk 43 6 Euler Metra AE SULT Enter k ESC CANCEL Josef Bohm Working with DEQME mi ye P ee P D w beet Enter Information Sete tee 2 2 4 Step Size ax 05 Humber of pointzszn Entaer k RAD AUTO Hori Ex act Linear in x LLinear in RAD AUTO FUNC du dtzEkci 8 Ee lows enter udlus For B Use F1 l if op end is RAD AUTO FUNC Fey Fur FB ore Eulzr Exit 1 5hou Steps USE 514 TENTER CESC sinis Int laggy 1 53 Solve for e s sini
23. and can be considered a sort of Golden Ratio of exponentials since Pm DERIVE CAS TI User Forum checked Jim s procedure calculating the Omega Constant 12 fSQLNHGCLO 13 1 0 567143290408 7838 72899888568662210355549875381578718651250813513107527 x 14 NSOLYEC1L x 15 x 0 567145 290409 7s 35 725999666060 271055556 714697465514 204615655s1054259 0 567143790409 7838 7299996866 2710355549 753815 76718651 26081 351310292 15 e 17 0 567145 29 0409 76 72999968662 210555549 7 5 3815 76 18551250813513107587 ET In MATHEMATICA is this function im plemented as ProductLog z ra Inl Product Log 1 as Wl ProductLog 1 0 VW functian Out z 0 567143 cu m H ProductLog 1 20 0 5 0 567143290409 fase 300 M ix The graph shows f W and its inverse Lambert s W function Among many other papers you can download another of D Jeffrey s papers from the above mentioned website Corless H Ding N J Higham D J Jeffrey The solution of S exp S A is not always the Lam bert W function of A In ISSAC 2007 Editor C W Brown pp 116 121 ACM Press 2007 Nils Hahnfeld Virgin Islands Hallo Josef How can I convince the TI 89 that the 2x2 matrix 3 2 2 1 does not have complex Eigenvalues but twice the Eigenvalue 1 Do you know any tricks Or 15 it necessary to take the imaginary part 0 000001
24. ands We do not know of any other similar implementations in CASs apart from 1 that uses special facilities for inserting graphics and the similar 2 by the same authors of this paper but written in Maple We believe this is an interesting example of synergy among different branches of knowl edge Mathematics History of Mathematics and Computer Science that can increase the interest of the students for different topics K Schmidt Schmalkalden University of Applied Sciences Making Life in an Introductory Linear Algebra Curse Easier with DERIVE In teaching linear algebra we have to deal with the following problem while the level of the mathematical skills which are required to work with examples is generally low stu dents only need to add subtract and multiply the number of calculations is usually large Therefore working with examples is time consuming and error prone if done by hand Students get tired quickly and lose their interest in this increasingly important area of mathematics The faculty therefore decided to move the introductory linear algebra course from the classroom to the PC lab and acquired a special DERIVE license that al lows its use on all the PCs the faculty owns and also on the private PCs of the students A utility file was then developed to facilitate teaching by providing functions for the com putation of zero matrices and vectors matrices and vectors of ones as well as idempo tent and orthogonal
25. as 0 Would be nice if the next OS would perform better Three more questions Isit possible to check the language setting of the device within a program 5 it possible to present an Integral in pretty print without performing an integration When will it be possible to have a dialogue in TI Nspire programs Best regards Nils DERIVE and CAS TI User Forum p43 These are the matrices which were sent The first one gives the strange Eigenvalues sS ul ein 1 00 5 292 78 1 000 5 292 78 4 z 1 000 2 000 25 0002 B 1 3364915231032 1 3 RAD EXACT 1 000 2 000 3 000 4 X1 1 5 1 23654916731037 x1 3 7 Male RAD EXACT DE ir DERIVE shows the correct Eigenvalue but does not consider that 1 is a double Eigenvalue try to solve the characteristic equation This seems to work similar to the DERIVE proce dure receive the Eigenvalue 1 but not its double occurency ram Fier FE For 1 FiF Fu FE For ee lcontroiftosrlFind Mode vi AlsebralcalclotherPramto clesn ue re 1 500 1 9364 1 500 1 956 4 3 00 identitucdimt amp m23 llokx 20 x 42 zu x 1 ndF unc A z 2 B eigu z 1 Lg om x 9 9 1 890 i or 909 1 290 i my eigVcEE3 2 31I 2 1 015 RAD EXACT 17 Fir Fu FE FEF J Pi lAlsebralcsiclother Pranto ciean ue B mu eigu t 2 zT z 2 J 1
26. at line After all prog x 3 4 is a program as its name says and programs will never be started automati cally in input mode Hope this helps Johann Germain Labont grmlabonte yahoo ca Hello I ve come accross your contact information from the Derive User Group web site had some interest in muLISP on which Derive was programmed From what can gather the latest version of muLISP which was sold up to the first half of 2005 along with Derive for DOS in cluded both the 16 bit and 32 bit kernels Is this correct If so what would be the version number I m hoping that one day a copy of this ver sion of muLISP will come on the used software market e g eBay and would like to know what questions to ask the seller Sincere regards Germain Labonte Mississauga Ontario Canada Is there anybody who can help More User Forum on page 41 Roland Schroder The Common Measure The Common Measure Roland Schroder Celle Germany Two wooden sticks of lengths a 27 cm and b 21 cm shall be sawed up without using any measuring instrument to as many pieces of equal length as possible We can do this by putting the sticks flush together separate the overhang a b from the longer stick Now we choose the two shortest remaining sticks and repeat the procedure This describes a recursive algorithm which ends if we cannot choose the two shorter sticks because they all are of the same length
27. atistics of the sample data in the spreadsheet window The sample mean x and the sample standard deviation s are estimations for 175 and 0 10 Observe The variability from sample to sample with the same sample size 2 The estimations are getting better with increasing sample size Guido Herweyers Tutorials for the NSpireCAS 2 Binomial distributions Forty per cent of a sweet assortment are soft centres and the remainder hard centres A handful of ten sweets may be regarded as a random sample from a huge pile of the sweets a What is the probability that there are three of ten soft centres in a handful b What is the probability that there are at most four of ten soft centres in a handful Press Con to turn on the handheld Press for the home menu Press d to open a calculator page 1 alrulata dibrapha amp 3iListe E Ep A Males C Dala Ba nc ee Docu SiSyatem Into amp Hintz Add pace with Calculabar application ba the apen document Press rene then 7 Statistics 5 Distributions D Biniomial Pdf tat Caleulatione Tat Results The number of trials is n 10 The success probability is p 0 4 The X value is 3 If a random variable X has a binomial distribution with parameters n and p then binomParl 10 43 sda BUT P X x binompdf n p x binomCdf 10 4 4 632103 3 21 5 x binomcdf n p x
28. ax for the normal density function with mean 175 and standard deviation 0 10 is normpdf x 175 10 Confirm with 155 165 175 185 sample Select the spreadsheet window with Change the sample size into 100 1 Plot Type HD AUTO REAL Press then dar BS TIE Actions 121 Select All Points 3 Actions gh 4 4 2 Add Movable Line 9 Plot Function 3 Remove Selected 4 Lock Intercept at Zero 5 Regression Residual Squares Normal POF FS Plot value se Flot Function Shade Under Functiar sample Guido Herweyers Tutorials for the NSpireCAS The histogram width is too small Go to the x axis until appears press to get a closed 23 Press or 4 to change the histogram width Observe the influence of the histogram width To change the window settings press E 1 ype k 3 Actlons then WindenrZoom 9 4 Window Zoom n 1 Window settings NH choose Xmin 145 and Xmax 205 Displace the function label Select the spreadsheet window with To take a new sample press Take a few samples the changing 150 165 190 195 sample histograms illustrate the variability of the data Change the sample size into 200 1000 The histograms are getting closer and closer to the density function 150 165 195 sample Task find out how you can produce the one Variable st
29. ay that for the numbers if this size the overhead of the quadratic sieve is still to big to be a real competitor to the other factoring methods we have been discussing so far All the same if you have a look at the huge matrix below you might still wonder how Derive can be so fast namely less than 3s on my As al ways have fun when trying out examples of your own User Simp User 67 23 setup 2 1 true User Simp User 24 b DIM f m 147573952589676412927 21 312 31 94 UserzSimp User 25 getrel true User 26 x rel E rel 1 26 85138228182631575566 Ley E ee 0 254320008912668516464 2 34 Gr 0 27131959020989732516 l 052 292 0 eq 0 91834232634956827117 0 21205539502124754815 quoque ab 138631399311054 240574 Bp Ge ae oy sme pede 2 134120451326901583108 0 17080547131876123141 D qoo spo 0 61343871582884083679 q 15 33 o o Gai 0 This 1s again only a part of the output User 28 1 rel 28 00041 1 01 001 01 1 1 0 1 0 29 0 0 1010101011101 1 0 0 1 O0 1 0 1 1 1 0 1 O O 1 1 0 1 0 Please refer to the file Josef 11010000 00000011 761838257287 193707721 147573952589676412927 User Simp User 30 qs 2 67 1 193707721 0 0 0 1 1 1 01 0 1 O 1 1 1 0 0 0 1 0 1 0
30. aybe that some of you will remember DNL 45 where I introduced proudly our first granddaughter Kim Later I presented a picture with Kim and the TI 89 with Grandpa s eyeglasses of course Now Kim is attending the grammar school in Tulln Lower Austria and sometimes it is not too often she asks her grandpa for support in mathematics Below 15 Kim together with her sister Yvonne sitting on my desk some years ago At the right you can see Kim and Yvonne both left together with their younger sisters and her mother Astrid our daughter It was last year when Kim needed some exercising for calculations with times I took the occasion to extent my collection of training programs and wrote a DERIVE utility for adding and subtracting times and for adding times to certain day times in the week D zc Ed e lim f XII 4 X 1 LOADCD DOKUS SCHULE skil1s time_US mth Trainer for calculating times ti n gives n problems to add and subtract times and time ex res shows the correct answers time gives a day of the week together with the time of the day and a time to add or to subtract res gives the correct answer time ex n offers n examples in form of a table and time ex res shows the correct answers 16 Josef B hm What s the Time Grandie my training programs show the same structure except one which treats exercising with Venn diagrams for visualising set opera
31. cost cost x p and AUTO FUNC Particular Solution is 1Inrlniu lnrilni20 m i F 1 2 cos6x 1 HAIN AUTO FUNC Fir Fer FE For Togs lathr 3 C31 Fr3mlD rTean lr and 5197751 d Ju let 7 lntln un In tar 2 2 AUTO FUNC deSolue u xu FUNC 2020 os Ep 1 Janis 2 oo sanr and u 1 z 1 580 y 5 044 Press Enter to continue RAD AUTO FUNC 2 20 Lau of Conlin de has analytic Solution vzHeLE CEE its 3rarh 10065 as Follows EnterzDk E C CHMCEL RAD AUTO FUNC Male AUTO FUNC With 21553219 051 t 18 5 415 3549 RAD AUTO FUNC Male RAD AUTO FUNC SS cce Enter Separable Differential Ean dejdx ve Into _ uw lny x c3 dx 12 Separate Wariables Exi xed 51 Enter Ok ESC CANCEL RAD AUTO FUR Male RAD AUTO FUNC 22 Particular Solution is lnrln 2 0 4 53 Solve far cost J and signtsi 42 Find c usina 13271 false gt COSL 1 RAD AUTO FUR i Male RAD AUTO FUNC Compare with the deSolve result It might be a nice problem for studen
32. e time ex res 12 Loop Ifn gt exit table APPENDCtable tome 453119 time res ex res p Rx RETURN table t res Among the files accompanying this DNL you can find zeit mth which provides a German version without the am pm notation It should be no problem for non German users to adapt the file for their own language if necessary tage Mo 50 must be changed Do 8h Omin 3h So 2h 2min 12h 59min 12 zeitb 5 Do 9h Smin 15h 18min Fr 20h 43min 12h 13min Do 10h 29min 11h 2min Do Sa 13 res 11 Omin 13h 3min Oh 23min 8h 56min Do 21 31min P L ke Rosendahl An Interesting Triangle Property An Interesting Property of a Triangle Peter L ke Rosendahl Germany In 1968 the following theorem was presented as a problem in the journal American Mathematical Monthly Given is a triangle ABC We raise squares CBED ACFG and BAHK over the sides of the triangle outwards Then we draw the parallelograms FCDQ and EBKP Tri angle seems to be right and isosceles as well Prove this It might be nice to pose the problem to the students changing the last sentence What can you say about triangle PAQ Prove your conjecture Josef The first step finding a conjecture 1 a typical task for working with a dynamic geometry pro gram We can do it on the PC using TI NspireCAS or
33. e re cursion formula 1 u 2 After division by aq 1 we get q 1 2q which is a quadratic equation with solutions q 42 V2 1 As all elements of the sequence are positive and they shall remain positive we accept only the positive solution Using the positive quotient we can define a geometric sequence 1 1 2 5502209 85 2 7 E n 0 8 4 17 12 2 5 29 9 41 6 99 70 2 169 239 8 577 408 40 The right column of this table corresponds with the left column of the result of the recursion from above pe Roland Schroder The Common Measure So it seems to be clear that alternative removement starting with two sticks of lengths 1 and V2 is leading to a geometric sequence of stick lengths Because of q lt 1 it will cnverge to wards zero but never reach zero The procedure for finding a common measure will never come to an end There good reasons to suspect that 1 and V2 don t have a common meas ure We can name 1 and 2 as incommensureable The procedure can be used to create approximating representations as fractions together with a sound error estimation DERIVE gives N2 1 0 0000000221 Of 22619537 15994428 V2 0 0000000221 Solving for N2 results in the difference of a fraction approximation for 2 represented as a fraction and a number less 10 error estimation The follwing screen shots show the procedure performed on the Voyage
34. ecial utility function FUN LIN CFF GEN Nils implemented this special case as you can see below Solution S 2x u 2 4 Indid ex Fey 22 DE 49 u zi2xtg 32z7C0x u922 4 Irndl4 x 7 y 22 7 x 3 uc rT 84 6 x h ER Enter anv 1 Order DE E DE xu tkt z zs z 2 sky sy Indep Dep var c Using Initial condition Initial condition Particular solution iz at x or Lt of 2 5 5 on Enter Ok ESC CANCEL TLinear ln x Enter DE Eo Grarh slote Field Linear in duz dxz Ciy RA k id E es DE E sspe HOTEL Use variables x and y FL Circuitz M TEZ Only on left side Bernoulli due dz x u Clairaut Equation dy dx x y pande Enter K FSC CANCEL Fir Zoom Josef B hm Working with DEQME e POE Tranzsf Fark Solution Enter DE and point xor 2 er DE Chis dusdxz Cltu z32z46Cxu Tg HOTEL Use variables x and Bernoulli i Equation c dads Lagrange Equation Linear ipie x 0 42 y Enterz k ESC CAHCEL cl nq qp ae aera eet a ot E M M Be AEN em General solution z s z Ex2 sory St Indep Dep Bo OPTIONAL Initial condition pM
35. er CAS as well to create a group to discuss the possibilities of new methodical and didactical manners in teaching mathematics Editor Mag Josef B hm D Lust 1 A 3042 W rmla Austria Phone 43 06604070480 e mail nojo boehm a pgv at N L 74 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 the DNL The more contributions you will send the more lively and richer in contents the DERIVE amp CAS TI Newsletter will be Next issue Deadline September 2009 15 August 2009 Preview Contributions waiting to be published Some simulations of Random Experiments J Bohm AUT Lorenz Kopp GER Wonderful World of Pedal Curves J Bohm Tools for 3D Problems P Luke Rosendahl GER Financial Mathematics 4 M R Phillips Hill Encription J Bohm Simulating a Graphing Calculator in DERIVE J Bohm Henon Mira Gumowski amp Co J Bohm Do you know this Cabri amp CAS on PC and Handheld W Wegscheider AUT Steiner Point P Luke Rosendahl GER Overcoming Branch amp Bound by Simulation J Bohm AUT Diophantine Polynomials D E McDougall Canada Graphics World C
36. erg 60 erg erg mr3 60 hr MODCerg 24 erg MOD terg hr 24 7 Si Sele Chip ame erg is IFterg 0 7 erg Bec EE Chr dE Shr See hp APPEND daysperg STRING hr2 hrs STRINGOmr2 min sp res ix res time exin n table Prog tasses time ex Loop Ifn gt exit table AFFEND table tome time ex res ex res res posee X RETURN tab le Josef B hm What s the Time Grandie The next expression is expression 11 I removed the expression number to have a better screen shot of the extended function time secs min hrs 17112 toum3 signs 51 tsk prj Prog Loop or 1111 signs 2 RANDOMC22 RANDOMC2 1 tsk 60 24 1 5 1 signs i 1 1 signs secs iz signs tsk t ores Secs timl MODGsecs 60 t res t res 1 11 60 time res 60 7113 Ct res tim23 60 t res APPENDCSTRING tims If secs gt 0 Loop If DIM tsk 0 RETURN RESTCREST CREST pr222 secs MOD tskjl 60 tskyl tskjl secs 60 main MODCtskjl hrs tskyl min3 50 S022 pros APPEND pr 51 STRING hrs2 hrs STRING Omon 2 min STRING secs sec 73 tsk RESTCtsk signs REST signs hrs STRINGCtAm22 min STRIMGCtAml2 sec n table Prog m cL tabl
37. erica and download among others an article on Teaching and Learning Differential Equations by Chris Rasmussen and Karen White head http www maa org t and l index html http www maa org t and l sampler rs 7 html The next website recommended offers plenty of geometric models and how to create them with paper and scissors http www korthalsaltes com http www korthalsaltes com three pyramides in cube htm Download contributions published in the Journal of Mathematics Education most of them sub mitted by Chinese researchers http educationforatoz com journalandmagazines html errer THE EDITOR Dear DUG Members I apologize for being late with DNL 74 At the end of June and begin of July I attended some interesting conferences ACAO9 CADGME and others which needed some extra preparations Then some clarifications were necessary to finalize the DEQME contribution of this DNL And as I wrote in my short info our webmaster Walter Wegscheider enjoys his very well deserved holidays When he will be back from his cruise on the Mediterranean Sea he will upload this issue I hope that it was worth to wait one month 09 Applications of Computer Algebra 2009 was an excellent organized con ference which was held in Montreal Many thanks to Kathleen Pineau Michel Beaudin and Gilles Picard who made the conference a full success There were among others a rich Educational Session and a special DERIVE
38. la does not improve its appearance P L ke Rosendahl An Interesting Triangle Property Peter delivered a formula for the area IA LE UL ESSI d Ppt A B C b d Ppt B C d c b Ppt C A B b d c Area of the initial Triangle 1 b d CROSS b 0 0 01 d 0 0 2 2 Area of the goal triangle 2 2 2 1 b 2 b ed c d CROSS c b d 0 0 b d c 0 0 2 2 2 2 2 1 b 2 b ed c d CROSS c b d b d c b d c 0 0 repeated the construction using GeometryExpressions with generalized coordinates of the edges A B and C and hoped for a nicer result 2 2 2 2 i ra TX ys ty 2 Kp Ua Q7 4 7975297 79 4 1772 7032 74 92 90 32 5 Cut Copy Content MathML Paste Presentation MathML Delete Derive Input Peter L ke Rosendal An Interesting Triangle Property Then exported this result to DERIVE other possibilities are to MAPLE MATHEMATICA TI Nspire Maxima 1 InputMode Word 2 2 XL ad d 2 0 0 1 2 2 xO 0 1 yO xl 1 yO x2 yO 1 0 ylex2 1 yl y0 1 2 2 yl 1 y2 1 2 0 1 y2 xl y2 y0 1 2 2 2 2 2 2 1 x2 2 yl 2 3 xO 0 x2 yl y2 1 2 yO 2 0 yl
39. matrices just in time whenever they are needed during the course The utility file also contains functions that test if a given matrix is symmetric idempotent or orthogonal DERIVE and CAS TI User Forum 41 Jim FitzSimons DERIVE or Maple not factor this 5x 12 I have been told a radical root exits for this polynomial Richard Schmitt schmittrichard yahoo com Suggestion The multi valued nature of inverse functions by Jeffrey Available at http www apmaths uwo ca djeffrey offprints html I have a question too Is the Lambert W function available in Derive 6 Thank you Jim FitzSimons Here 1s all I have 1 Prectsiantigits 64 otationDigi ts 64 InputhMode Word 4 coy f x uses iteration x 5 flx ITERATECe apy fatx uses Newton s method y CLNY x 1 6 x ITERATE __ v yO 32 1 gt Lh SS er eS et 1 fix IFLRE x lt 1 0 31 wrightomegalz Test the results al VECTOR x ixi x 4 4 0 05 VECTORCTl y v y al 10 2 1 11 az VELTOR C x x 4 4 0 05 must admit that have not heard about Lambert s W function before found some useful information in CRC Concise Encyclopedia of Mathematics Lambert s W function is the inverse of the function AW We W 1 is called Omega Con stant
40. nal quantum theory as applied to Nuclear Magnetic Resonance Back then I lobbied the Derive folks to include a function for taking the Kronecker Product of matrices And they were happy to comply However I now find that this feature is no longer supplied with the software I believe this came as an additional operator that one could load from a library and I guess it feel off the list of those that were included Perhaps some of the users recall this function and still have a copy of the library it came in Or per haps someone has invented their own operator or knows how to use the operators that are supplied to write a Kronecker Product function Back then it did not appear to be possible with the current tools and that is why I suggested to Derive that they supply one Thanks Tom B DNL Dear Tom didn t find the Kronecker product among my many DERIVE files And must admit that didn t know about this special matrix product but could find its defi nition So took the challenge trying to produce the respective function see the attached file hope that this works properly Then include a respective note in the next DERIVE News letter kron prod a b APPEND VECTOR APPEND COLUMNS VECTOR VECTOR a b DIM a 1 i k 1 DIM a Ti 71 1 5 d 5 a 10 2 kron prod 3 4 kron prod 10 20 30 4 G L 1 2 3 vo ow XD ae 10 16 1 20
41. onal lengths What about side and diagonal of the unit square a V2 b 1 Jy al 1 42 1 dp e 2 42 3 22 ITERATESC b a b b K2 1 8 3 2 2 2 1 Jos Beye sepu ae 3 242 5 40 7 6 2 7 10 7 Jj2 It is necessary to enter the number of iterations otherwise the calculations does not stop in exact mode Working in approximate mode we get after 50 iteration steps 5 5 1 B50101823965 10 Some expressions are occurring twice the same column We can conclude that they are cut off twice and we define a little bit daring the following recursion instruction diu 8 uam 52 21 Roland Schroder The Common Measure We transfer this recursion to DERIVE LLIERATESUDB Ea ble Bez and we receive 42 1 42 1 3 2 42 3 2 42 5 2 7 Gag 7 1 12 42 17 12 42 28 42 41 28 42 41 99 70 42 99 0 2 169 2 239 150 2 238 577 408 2 577 408 2 985 2 1393 expression is appearing twice Approximating the matrix shows that all expressions positive This confirms at least for the first 9 steps that each expression can be sub tracted twice from its predecessor We would like to find the explicit representation of this sequence For this purpose we use a method which can be taken as the standard one We conjecture that it might be a geometric sequence aq nen So we substitute for the respective elements of the sequence in th
42. owing I ll talk about another basic idea when it comes to factoring integers which deals with nontrivial solutions x y of the con gruence x 2 y 2 mod n where n is again the positive integer to be factored and non trivial means here that x y mod n Since due to these conditions n is a divisor of the product x y x y but not of its factors x y and x y it easily follows from this that both gcd x y n and gcd x y n will be nontrivial divisors of n This simple idea can be exploited in several ways but in the following I ll focus on the so called quadratic sieve only Here one considers polynomials Q x in Z x of the form Q x ax b 2 n af x with f x ax 2 2bx c and 2 an integer and their values in the interval I m m that contains both real zeros and has the prop erty that the extrema of Q x located at the endpoints of I and near its center have about the same absolute size This will guarantee that the values of f x for x in I are relatively small which is important as we want f x to be B smooth for a bound B O of moderate size that is we are only interested in values of x in I such that p zB for all prime divisors of f x Well actually we also consider also the cases where this is al most true in the sense thta it is true except for at most one not too divisor p exceed ing B not too much say B p lt 10 B The underlying idea is that if we find two represen tations z1 2 y1 p mod n and
43. session You can find the abstracts of the DERIVE related contributions DNL 75 will present the full papers By the way don t forget to mark July 2010 in your agenda TIME 2010 will be held in Malaga Spain from 6 10 July We will have four extraordinary keynote speakers B rbel Barzel Germany Michel Beaudin Canada Colette Laborde France Eugenio Roanes Spain The Conference website is www time2010 uma es This DNL contains a variety of articles You can find another of Schroder s pro jects for the classroom an interesting paper on the use of the TI NspireCAS one of my exercising programs originally written for my granddaughter a great trian gle problem where you can use various CASs the first part of a review of Nils Hahnfeld s Differential Equations package for the handheld and last but not least informs Johann Wiesenbauer in his Titibits 37 Il about other methods for factor izing integers So I do hope that everybody will find some useful interesting or just delighting pages I wish you all a pretty summer and I am looking forward to meeting you again in fall Download all DNL DERIVE and TI files from http www austromath at dug P2 E DI TORIA 1 The DERIVE NEWSLETTER 1s the Bulle tin of the DERIVE amp CAS TI User Group It 1s published at least four times a year with a contents of 40 pages minimum The goals of the DNL are to enable the ex change of experiences made with DERIVE TI CAS and oth
44. t is not possible to have the Integral character within a text We hope that future versions of Nspire will enable dialogues Josef Big Air in Summertime Tania Koller sent a model of a jumping snowboarder as background picture in DERIVE This is a very welcome occasion to cool off on hot summer days Many thanks to Tania and her students 0 5851689596 D 8264315376 x 2 754197724
45. that Derive 6 was Far too good just for students http www scientific computing com scwmaraprO4derive6 html ACAO9 The DERIVE Session p39 Josef Bohm ACDCA amp Technical University of Vienna CAS Tools for Exercising There is no doubt that even in times of CAS a certain amount of manipulating skills in various fields of math education is still necessary Students need more or less exercising for mastering expanding and factorizing expres sions finding GCD and LCM solving triangles applying differentiation and integration rules to name only some of the fields where training of skills might be useful We present a respective library developed in Derive reaching from set theory to calculus which can support the students and teachers as well offering random generated prob lems together with the respective solutions A Garc a F Garc a G Rodr guez A de la Villa Univ de Madrid Univ de Salamanca Univ Pontificia Comillas Toolboxes with DERIVE The European Area of Higher Education implies a profound change in the Spanish univer sity We are heading towards a competency based teaching and a learning model with greater autonomy for the student who becomes the centre of the educational model New technologies can play an important role in this new scenario This paper suggests a new possibility in the use of new technologies The design of a toolbox which could be used later on by the student when needed in other
46. the cumulative distribution function b P X lt 4 263 39 Guido Herweyers Tutorials for the NSpireCAS 3 Poisson distributions if a large grass lawn contains on average 1 weed per 600 what will be the distribution of the total number X of weeds in an area of 400 cm a Find 2 b Find lt 3 A suitable model for this situation is a Poisson distribution whose mean is Press Con to turn on the handheld Press for the home menu Press 15 to open a calculator page Press then 7 Statistics 5 Distributions H Poisson Pdf A 213 and 2 If arandom variable X has a Poisson distribution with mean 4 then poisspdf A x a 2 11 4 P X x poisscdf A X the cumulative distribution function 400 m m 600 Z Graphs Bie 2 4 Sta Beo Doc ZAMMDocu amp iSyatem nio SHI rite Adda new page with a Calculator application ta the oper docume rt tat Calculations tat Results ist Math D Elnomial Pdf ist Operations E Binornial Cat F eometric Por Confidence Intervals E 5eometric Cat Har Tests 1 posta 2 JE essei 9 You can find much more materials at http www t3vlaanderen be and http www t3wallonie be Josef B hm What s the Time Grandie 15 What s the Time Grandie Josef B hm W rmla Austria M
47. tions The files are MTH files and should be loaded as Utility files Then simplify the command start You will be presented the instructions how to use the file see page 15 The trick is to use the DISPLAY com mand for this purpose see the code of start on the next page start contains the simplification of dummy random 0 which makes sure that we will be offered new problems at every run of the utility A session could according to the instructions given above start and run as follows 2 3 4 5 6 5 9 10 5 hrs 53 min 18 sec 13 hrs 37 min 18 sec 163 17 hrs 26 11 sec 4 hrs 35 20 sec 20 hrs 9 min 39 amp hrs 17 min 32 sec 19 hrs 30 min 36 sec time ex res 12 hrs 50 min 51 sec 28 hrs 27 ll sec time Fra 4 hrs 30 min am 7 hrs 7 min res Thu 8 hrs 23 min pm time Thu 4 hrs 46 min pm 17 hrs 38 min res Fra 10 hrs 24 min am Thu 12 hrs 23 min am 16 hrs 0 min Thu amp hrs 35 min pm 5 hrs 44 time ex b Mon amp hrs 53 am amp hrs 55 Wed 3 hrs 49 min am 15 hrs 12 min Mon 3 hrs 55 am 6 hrs 24 min Fra 4 hrs 23 min am Thu 2 hrs 51 pm time ex res Sun 11 hrs 58 pm Wed hrs 1 pm Mon 10 hrs 18 min am Kim learns English and she likes English more than mathematics so I could not resist to include the am pm notation of the times
48. to choose whether n is the name of a column or a variable Press 7 and w to choose nnt Deter 75 13 n Column wanaha 7 Colunn Pereranpe xr ariabla Faferenca Column Reference Variable reference confirm with variable Referer Guido Herweyers Tutorials for the NSpireCAS The sample appears in column C Move to the top position to enter a name for column C type sample Then press to select the whole column Press Qer mene for the context menu Choose 9 Quick Graph The window is split into two parts by default a dot plot of the data RE appears in the data and statistics rang window ER EA 652 155 165 175 185 Sample 5 then 1 Plot type 3 Histogram 155 165 175 165 sample Press then 2 Plot Properties 2 Histogram properties bi Acid Oo Remove 1 Histogram scale E Add am i al 3 Density es 157 We now have a histogram with asa pi o 1 Jj density scale total area 1 17e 19 Guido Herweyers Tutorials for the NSpireCAS Press for the tools menu then 5 Page 1 Custom Split J SwlectApp Ciritk 4 Swap Application 3 Delete Pag 160 175 13 sample Press 4 for a wider histogram window and confirm with 27 155 165 175 185 sample The synt
49. ts to verify the identity of the trig ex pressions Fir Fer Fi FE FB Tools BT3ebra caTc Fr3ralD cTean UF 2 salve 1nilnium In tan 59 Pe e tan Woe and tan gt Solvetans lo Beal Male RAD AUTO FUNC eran RAD AUTO FUNC Bau XT y y 1 2 This is a homogeneous DE Josef Bohm Working with DEQME Again I d like to I follow the steps Then I will compare with deSolve ir esit 2 1 1 5 ble me Hon Exact Linear in x Linear in RAD AUTO FUNC Qu 2 zm ys ISO or _ u2 2 u 1 dx 1 53 Separate Wariables 1 dues sb MAIN RAD AUTO FUNC RAD AUTO FUNC Fir Fer FE For TooTz RT3 amp br 3 C31 Fr3mlD rTean lr m deSolvel 2 8 RAD AUTO FUNC 1 Basics z Hnu 1 0rder Separable Ex ac RAD AUTO FUNC QUE A T Testi 4 X RAD AUTO FUNC Initial atz3 Thus Particular solution z 3 tax 2 gt ox Done RAD AUTO FUNC 1nlu 2 u 1 Plex vade NC cas du 0 15 homo3 LE Ex OF TIOMAL RAD AUTO FUNC 53 Separate Wariables F TC6u 1234C u 2 zru 123duzjf1 k 4 Integrate 5 lns e HAIN RAC AUTO FUNC i Using Initial conditions Thus
50. ty function has been developed With these drawings we can check graphically if the generated samples fit the distributions The use of this utility file is useful for simulating any process which follows a specific dis tribution Michel Beaudin cole de technologie sup rieure Montr al Another Look at a Trusted Mathematical Assistant From the DERIVE user manual version 3 September 1994 we can read the following Making mathematics more exciting and enjoyable is the driving force behind the devel opment of the DERIVE program In this talk we will try to show how some mathemati cal concepts studied by engineering students at university level differential equations multiple variable calculus systems of non linear equations can be easily illustrated by DERIVE Some will object that any other CAS could do the same well this is probably true but according to us not as quickly and naturally To accomplish this DERIVE not only has to be a tireless powerful and knowledgeable mathematical assistant it must be an easy natural and convenient tool Consequently time can be spent to prove some theorem or formula and the computer algebra system helps to reinforce the mathemati cal concepts Our examples will also make use of new features added in the latest ver sion of DERIVE version 6 10 released in October 2004 features that were not exploited as should be DERIVE has never been enough documented But we are still convinced
51. urrency Change P Charland CAN Cubics Quartics interesting features T Koller amp J Bohm Logos of Companies as an Inspiration for Math Teaching Exciting Surfaces in the FAZ Pierre Charland s Graphics Gallery BooleanPlots mth P Schofield UK Old traditional examples for a CAS what s new J Bohm AUT Truth Tables on the M R Phillips Advanced Regression Routines for the Tls M Phillips Where oh Where is IT GPS with CAS C amp P Leinbach USA Embroidery Patterns H Ludwig GER Mandelbrot and Newton with DERIVE Roman Ha ek CZ Snail shells Piotr Trebisz GER A Conics Explorer J B hm AUT Coding Theory for the Classroom J B hm AUT Tutorials for the NSpireCAS G Herweyers BEL Some Projects with Students R Schr der GER Runge Kutta Unvealed J B hm AUT The Horror Octahedron W Alvermann GER and others Impressum Medieninhaber DERIVE User Group A 3042 W rmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef B hm DERIVE and CAS TI User Forum Tom Barbara Oregon USA Hi Josef Thanks for adding me to the group I am a long time user of Derive from Version 1 However I did not use it that much in recent years although I did have the first windows version of the software I recently purchased the latest version that is available Besides the general algebraic tools I used derive s matrix capabilities for problems in finite dimensio
52. v Complutense de Madrid An Implementation on the Mayan numbering system in DERIVE The Mayan number system is a base 20 positional to be read from top to bottom not from left to right system that makes use of a symbol representing zero It has slightly different variations when used for counting days in religious and astronomical contexts Therefore 20 symbols are needed to represent 0 1 2 19 Of these the positive ones were represented using dots the value of each dot is 1 and horizontal segments the value of each segment is 5 while the zero was denoted by a shell see figure below If a number is greater than 20 the symbols corresponding to units twentieths 400 s 8000 s are stacked from bottom units to top in pure base 20 while in our decimal system the different orders correspond to tenths hundreds thousands etc We can choose the input and output bases in DERIVE in Options Mode Settings from 2 8 10 and 16 Therefore we have implemented a procedure that allows to convert num bers between any bases and that returns the output in row vector style We have im plemented another procedure that builds the 20 Mayan symbols for 0 1 2 19 making use of the DISPLAY command Finally another procedure denoted Maya that uses the previously mentioned procedures converts any number from base 10 to base 20 and represents it in the Mayan numbering system These procedures only make use of the standard DERIVE comm
53. which is 3 in our case Common Measure 3 We see that 3 is the GCD Greatest Common Divisor of 27 and 21 and that this recursive procedure is nothing else than the Euclidean Algorithm recommend giving the pupils pairs of straws or paper stripes and letting them perform this activity They should find out that this is a way to find the GCD of two numbers Maybe that some of you prefer using the computer instead of a saw then you can write in DERIVE ef 21 21 6 15 ITERATES b a b b 27 21 7 9 3 3 D Calculation ends when comparing the two shortest sticks the overhang o 0 The common measure is the minimum positive number appearing in the procedure Of course we don t know that only 7 iteration steps are necessary What happens if we do not enter the number of steps Roland Schroder The Common Measure 27 21 21 6 6 15 15 9 5 ITERATESC b a b b 27 21D 3 3 3 D 3 3 3 We see that ITERATES works until discovering the second occurence of an element which is 3 3 in our case If our goal is only finding the common measure applying our function it will be sufficient to deliver the first element of the last row FIRST REVERSECITERATESC b a b b 21232 3 l FIRST REVERSECITERATESC b a b b 104 7413332 2 l Our story does not end here It is mathematically interesting to take sticks of irrati
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