THE DERIVE - NEWSLETTER #13 USER GROUP
Contents
1. 2 a SIN t 11 4 a COS t a 4a SINC COSCE COS t x a 4 x a 12 2 3 m a a SOLVE x 4 a COS t a t t ASIN t ACOS _ t 2 2 x a x a x a xX a a a 3 m a a ACOS t ASIN __ v t ACOS t ACOS 2 2 2 2 2 xX a 13 a t ACOS 2 Substitute for fin the second component of expression 11 a SIN t 14 y 4 a SIN Ct COS t COS t x a yy X a yy a a 5IN ACOS a a l l 2 15 y 4 a SIN ACOS _ cos Acos COS ACOS and with little manipulation you can find the algebraic form mm 16 17 Ysa 2 x 3 a x 18 y xX a The area 20 o COS a 3 p18 Thomas Weth A Lexicon of Curves 3 Kurzreferenz f r DERIVE User D N L 13 Die Trisektrix von Maclaurin VECTOR 4 a COS phi a COS phi a 0 5 2 5 0 5 Polar form Parameter form Algebraic form Ausblick In der n chsten Folge des DERIVE Kurven Lexikons werden einige Konstruktionen vorgestellt mit welchen die bisher vorgestellten Kurven einheitlich erzeugt werden k nnen Besondere Beachtung werden dabei einige Konstruktionen erfahren die erst n den2. Hh M s dr EI Sm eee EE 4 Seren euren a a T The DERIVE Owl of Wisdom
3. 100 1 45 24 APPROX LOG 10 478 4971966 100 1 45 25 APPROX LOG 10 1 672275938 480 1694726 2 The actual number of equations in the Euclidean algorithm is lower than n by 2 though For example 26 DIM euclid fib2 480 fab2 479 478 In other words log s X with 1 5 2 is approximately the upper bound for the number of divisions in euclid a b with a b lt s we are looking for So far so good But what about the average number of divisions needed It is not very much lower as one can show namely 12 LN 2 LN s 37 M In our example where s 10 100 we get 28 2 12 LN 2 100 APPROX _ LN 10 s D 194 0540228 WH Finally I also showed in my very first Titbits how the so called extended Euclidean algorithm EEA can be implemented in DERIVE The original version looked like this p34 Johann Wiesenbauer Titbits 1 D N Li 13 eea a b dellast ITERATESCIF MOD x x 0 x k MOD x x x x FLOOR x x x x x FLOOR 29 1 2 2 1 2 4 3 1 2 4 6 5 1 x x p x a b 1 0 0 1 2 6 30 eea 1234567 234567 1234567 234567 1 0 0 1 234567 61732 0 1 1 5 61732 49371 1 3 5 16 49371 12361 3 4 16 21 31 12361 12288 4 15 21 79 12288 73 15 19 79 100 73 24 19 3207 100 16879 24 1 3207 9640 16879 50737 Well what should I say more It still works which shouldn t be taken for granted
4. icosqc Maybe there is anybody among the DERIVE Users and specialists all over the world who knows the solution Many thanks for your help David Sj strand CAS and Spreadsheets Computer Algebra and Spreadsheets e g Excel David Sj strand Onsala Sweden In this paper we present some examples of how to use expressions generated with a computer algebra system DERIVE in a spreadsheet program MS Excel These examples will appear in a coming book Mathematics with Excel Studentlitteratur Sweden 1 Systems of linear equations The system of linear equations ax by cz d a x b y c 2 d ax b y c z d can easily be solved in DERIVE The solutions will be expressions in a1 a d3 which is suitable if you take into account that Al A2 and so on are cell references in a spreadsheet Steps to be taken in DERIVE DERIVE should be in Word Input Mode This can be achieved via Options gt Mode Settings gt Input Edit the expression alex blxy cl z d1 a2 x b2 y c2 z d2 a3 x b3 xy c3 z d3 A system of linear equations is a vector of equations Solve the system using the Solve button Looking Glass in the menu bar highlight x y and z The solution is given as a vector list Take care that the formulae for x y and z will be separate expressions You can do in the following way bl e2 d3 c3 d2 b2 c3 dl cl d3 b3 cl d2 c2 dl 4 AY SS al b2 c3 b3 c2 a2 b3 cl bl c3
5. 1 L n x d x x a LIM n x a LIM d x a Simplify the following two examples amp note how DERIVE returns a conditional expression when it cannot determine that a predicate is always true or always false 12 L SIN x SINH x x 0 3 L SIN a x SINH b x x 0 Then use F2 and F3 to redefine L to have a 4th unknown clause identical to its else clause and resimplity the second example First Example 4 L SIN x SINH x x 0 1 Second Example 5 L SIN a x SINH b x x 0 b x d d b e IFlaz20Abz20 L a COS a x 5 d 6 x dx 2 Highlight the third expression and simplify 6 once more b x b x d d b e b e a 7 Tha 0 A be 0 Dia x 0 dx dx 2 2 b lim a COS a x a 8 x30 b x b x b e b e 9 Tim b x30 2 2 The recommended redefinition of Lin dxa L n d x a If LIM n x a 0 a LIM d x a 0 10 L 3 n x A d x x a LIM n x a LIM d x a LIM n x a LIM d x a 11 L SIN a x SINH b x x 0 recommend using DERIVE s stepwise simplification feature Josef D N L 13 Selections from the DERIVE BBS p 9 Message 2397 From SOFT WAREHOUSE to PUBLIC about MOCK CHAOS When plotted using the default range pi t lt pi the following elementary expression closely mimics the usual phase plot of the chaotic Lorenz strange attractor ATAN t L SIN E
6. 75262139 495424665 2 81204193 048573916 2 75262139 49542466 281204188 048573916 2 75262139 49542466 2 81204188 3 2 1 E x 2 3 4 5 2 Just for fun solved the system using the 113 3 93 73 Wa X3 75 53 33 i iii Yr Besrberunc Excel Solver and then using the tools M alee tN provided by DERIVE _ Josef 3 4 p00000T 4 EETA 381204188 1 1042E 14 NEWTONSQ f g xl yl 3 2 5 3 2 6 536942383 3 579251930 FIRST REVERSE NEWTONS f g xl yl 3 2 5 288418538 3 061774538 0 4857391618 2 752621386 4 981228080 2 839802628 4 954551640 2 812383378 4 954246648 2 812041929 J M C Lopes A Flat Function Minimization of a Flat Function A Classroom Experiment with DERIVE Jos M Cardia Lopes Porto Portugal Edgar and Himmelblau present a formulation problem where the operation cost C of a hypothetical chemical plant is a function of the amount of product per batch P When the problem is solved we obtain the following function 400 000 800P 15 000 000 14 2 P P With this function we can motivate our students for th need and particularities of some common numerical approaches And DERIVE is a good help to do it in the classroom Suppose we want to claculate the value of P that minimizes the operating cost C The first idea is to determine the derivative dC dP and solve it to zero C 0 4 0 7 15000000 14 2 p 1 c 400000 800 p p In 1994 C Lopes ask
7. But I am 82 years old retired and don t have time to learn the language necessary to run MAPLE Ialso have MathCad and like it very much but if I want to take a quick look at the graph of a polynomial or do some symbolic linear algebra I turn to DERIVE Let those who need a more sophisticated program use MAPLE but don t spoil for us less gifted mathematically the joy ofusing DERIVE Heinz Message 2860 From JERRY GLYNN to PUBLIC about WEB FUNCTION On my recent trip to Seattle to the National Council of Teachers of Mathematics Meeting I had a chance to visit Janet Ray and Mike Pepe at Seattle Central Comm College They are long time users of Derive and most other good math software Janet and I had discussed how to use Derive to plot some functions in Derive using the iterates command and this lead to discussions with Mike this was about 15 months ago and on this visit I saw the result of this work It represents to me the best example of the use of Derive in a graphical manner I strongly suggest that you try this out and let all of us know what you find f x pug oT xt o T TET vs 1j next m tri element m 3 1 web a n iterates next m m tri a n f x 2 6x 1 x web 3 5 approXimate the last line Open a 2D graphing window and do Option State Connected and plot the result from approXimating web 3 5 You also should plot x and f x to see the full effect 0 1 0 2 0 3 0 4 0 5 06 dhg De 1 0 1 I set up a lis
8. fex f7 x F8 x f9 f10 x 10 fl x f2 f1G0 f3 f2 f100 f4 f3 f2 f100 F5 F4CF3 F2 F1 x FECTS CF4 3 2 5100 f7 f6 f5 4 3 F2 F1 x F8 F7 FECES F4 F3 F2 F1 x f9 f8 f7 F6 F5 F4 F3 F2 F1 x F1O FOCF8 F7 FECTS 4 3 F2 F1 x I checked but in case of 10 nested functions the auxiliary version is not necessary now See some examples This is the example given in DNL 12 2 x 1 GENERATE SIN x 1 x 4 2 x 2 SIN x 1 2 SIN x SIN x 1 SIN x 2 2 2 SIN x 2 SIN x 1 2 1 x 2 list x 1 2x 1 3 x 1 4x 14 5 x 1 6x x 1 x 3 GENERATE list DIMENSION 1ist 2 2 2 1 1 2 x 1 4 20 x X 25 5 26 157 2 2 2 2 2 l 2 x 3 1 2 x 2x 1 2x 1 2x 1 2 120 240 314 x 217 vn 24648 2 2 2 2x 1 2 x 1 A C Robin Colchester England I note that my difficulty with defining the derivative of x 3 using Derive has provoked some interest Dr Schumm s solution using LIMIT in DNL 12 is a more satisfactory solution than the use of DCUBE u x in DNL 11 as whilst this gave DCUBE x z x correctly it did not give DCUBE x 243 x as 3 x 243 2 as of course I wished p 6 DERIVE USER FORUM D N L 13 I have another similar problem involving integrals and the integrals I have been using often involve INT 1
9. x i x FLOOR x x x 6 4 2 4 6 x MOD x x ae x a FLOOR a b 6 5 4 6 b MOD a b 4 euclid 1234567891 234567891 1234567891 5 23456780 61728436 234567891 3 61728436 49382583 61728436 1 49382583 12345853 49382583 3 12345853 12345024 12345853 1 12345024 829 12345024 14891 829 385 5 829 2 385 59 385 6 59 31 59 1 31 28 3l 1 28 3 28 9 3 1 3 E 3 1 0 p32 Johann Wiesenbauer Titbits 1 D N L 13 euclid a b q_ r_ s_ t Loop If b 0 RETURN REVERSE s_ q FLOOR a b r_ MOD a b 6 If r_ 0 t la a a aet b a E la age q suco b Sp r_ s_ ADJOIN t_ s a bir 7 euc11d 1234567891 234567891 1734567801 5 234567891 61723436 734567801 3 61728436 40382583 61728436 1 49382583 12345853 40387583 3 012345853 12345024 12345853 1 12345024 820 12345024 14301 820 385 8 820 2 385 50 fibpair s returns the largest pair of adjacent Fibonacci numbers not greater than s Here the update is rather an alternative than an improvement fibpair s ITERATECIF x x gt s x x x x x 1 0 9 1 2 1 2 1 100 10 fibpair 10 11 9216345717656874712980450562726202415567360565930794777111390850331644813674856981646960226192287360 5696323922575865414847 061494575 945648081290145228607189038829076215134884313127297923138542545712321 fibpair s f_ 1 0 Loop 12 If if
10. 0710678 8 9440037 11 8385B5 X 5 9884235 5 9815232 5 9808847 5 9965756 5 878887 5 8180333 5 7444495 5 6585691 55609219 5 4521095 x Coord hJ 0 y 2 4218906 2 5975156 2 720579 2 9444412 3 1136029 3 2704999 3 4381157 3 991466 3 7376056 3 8756333 y Coord zn 2 23042313 David Sj strand CAS and Spreadsheets 4 Systems of two non linear equations with two unknown A system of equations canbe solved numerically with the method of Newton Raphson For the system x y 20 g x y 0 consisting of two equations with two unknown x and y and the solution a P we get according to Newton Raphson s method that xn yn gt a D for n gt oo if the point xi yi 1s sufficiently close to a D Xn and yn n gt 1 are given by the formulae fy By Sp Vy SY MV SY f OS y SM 2 Xp Yn Sy Xn Yn ESA em qu 2 T f Ug d Jem x9 P 5 n l n Vt gs Steps to be taken in DERIVE Enter the expressions d d f g g f d yl d yl xl d d d d sued d xl d yl d xl d yl d d g pe g d xl d xl yl A d d d d ENIM TL PA d xl d yl d xl d yl d You can enter the expression simply by typing DIF f x1 x Suppose that we want to find approximate solutions to the system User Manual DERIVE page 143 3x y 2x 4siny 6 0 3x 2
11. 9 04992154 47752015 025132741 0 99752305 5 00969663 031415927 895051021 5 2403591 037699112 0 04914539 5 46655056 043992297 0 9375049 5 650600317 0 50265452 o64470201 5 20002297 0 56545668 85224505 6 10757304 0 62531553 638741792 b 30623773 0569115035 624019769 6 4960529 O 75399224 6001355065 667620952 051651409 791150364 6 64605652 107364594 F 3132 02 7 00490355 09444770 7 541559105 7 1521238 1 00530965 7 54206716 720713619 1 0681415 7 135118589 7 40940709 30 1 1308 335 692209729 7 51045638 We can now reap the fruits of our efforts You can plot a new triangle and its circumscribed circle just by entering new coordinates in the cells in the range C2 D4 12 14 16 18 20 The DERIVE slider bar version from 2007 David Sj strand CAS and Spreadsheets 3 The incircle of a given triangle What we did in 2 with a given triangle and its circumscribed circle you can do in a corresponding way with a given triangle and its inscribed circle Let xi y1 x y2 and x3 y3 be three given points in the plane Let xM yM the center of the inscribed circle of the triangle defined by the given points xM and yM can be regarded as functions of the 6 variables xi y x y x3 and y3 Steps to be taken in DERIVE In DERIVE we can derive the expressions for xM and y M x1 y1 x2 y2 The side opposite to point xi y1 has length fi 1 1 2 and 3 The intersection point between the bisecto
12. Gr nden sehr hilfreich da DERIVE auch hierbei eine undokumentierte Parameterdarstellung erlaubt Bei der Einstellung des Darstellungsmodus Polar wird n mlich der erste Term des geordneten Paares a t b t als Parameterdarstellung des Leitstrahls Radiusvektors r interpretiert der zweite als Parameter dar stellung des Polarwinkels So wird beispielsweise durch t 1 eine unter dem Winkel 1 rad vom Ursprung ausgehende Strecke dargestellt deren Lange durch den Laufbereich des Parameters t gegeben ist Demgegen ber wird 1 t als Kreisbogenst ck vom Radius 1 interpretiert und t t nat rlich als Spirale Soll nun beispielsweise f r die in Polarkoordinaten gegebene Funktion r sin 2 eine Unterteilung des Definitionsbereichs 0 27 in vier Teilintervalle erfolgen so l t sich dies erreichen durch VECTOR SIN 2 k n 2 t k n 2 t k 0 3 anschlie end Simplify und Plot wobei der Parameter t jeweils von 0 bis 7 2 laufen mu einmalige Eingabe mit Ctrl Enter abschlie en Sofern die Farbeinstellung f r die Darstellung der Graphen auf Auto steht werden nun die vier Teilabschnitte des Vierblatts in unterschiedlichen Farben dargestellt und die Zuordnung dieser Teile zu den Abschnitten des Definitionsbereichs l t sich besser erkennen DERIVE allows an undocumented change color of new plots and makes clear the parameter form for functions in polar form This assignment of these parts to the parts of the is very helpful
13. Group It 1s published at least four times a year with a contents of 30 pages minimum The goals of the D N L are to enable the exchange of experiences made with DERIVE as well as to create a group to discuss the possi bilities of new methodical and didactical manners in teaching mathematics D N L 13 Contributions Please send all contributions to the Editor Non English speakers are encouraged to write their contributions in English to reinforce the international touch of the D N L It must be said though that non English articles will be warmly welcomed nonetheless Your contributions will be edited but not assessed By submitting articles the author gives his permission for reprinting it in D N L The more contri butions you will send the more lively and richer in contents the DERIVE Newsletter will be Editor Mag Josef Bohm A 3042 Wurmla D Lust 1 Austria Phone Fax 43 0 2275 8207 Preview Contributions for the next issues Fluid flow in DERIVE Reuther a o BRA Applications in Electrical Engineering Scheuermann GER Stability od systems of ODEs Kozubik SLK Los desplazamientos en las finciones elementales Ramos ESP Algebraic Operations on Polynomials in DERIVE Roanes ESP Prime Iterating Number Generators Wild UK Graphic Integration Probability Theory Linear Programming B hm AUS DERIVE in Austrian Schools some examples Lechner amp Eisler AUS Tilgung fremderregter Schwingungen Kling
14. Matematik published by the Malaysian Mathematical Society Ob diese Aufgabe zu Fu gel st werden kann Mit DERIVE geht es in einer Zeile allerdings wird MISC MTH gebraucht und hat bei mir mit DERIVE XM etwa 6 5 Minuten gedauert Tats chlich hatte ich die L sung zunachst durch Expandieren Speichern auf eine Datei und Anschauen gefunden 13 MISC MTH Loaded with LOAD UTILITY 2 MAX VECTOR POLY COEFF 142 x43 x 244 x 3 20 x i 1i 60 3 88425110877597693745 4 387 2 sec on a 486 33MHz with DERIVE XM Is there anybody who knows how to solve this problem by reasoning My first attempt was to EXPAND to SAVE the expression and look for the largest coefficient Derive 6 needs only 2 seconds without loading the utility file Paul Drijvers Utrecht Netherlands Can you draw this picture in DERIVE Do not look at the back DNL did And thanks Paul for your nice tree 1 F 2 t E FLOOR 10 t Qe 2 G t 10 10 t Jt X 1 Je T 3 H t t A 2 uu b 4 K t 5 ASIN SIN n t p Row F t GCE Z m 85 F t G t M d u H t Kt 6 Plot 5 with t from 0 to 1 Dr Wolfram Koepf Berlin Germany Josef Lechner wants a fifth argument of the ITERATES function to construct e g the expression x E EG Qs 8 F F Bulletin of the DERIVE User Group 12 page 4 I agree absolutely with this wish However I like to show how one is able to solve the given problem with the current version
15. a3 bl c2 b2 cl al c2 d3 c3 d2 a2 c3 dl1 cl d3 a3 cl d2 c2 dl1 al b2 c3 b3 c2 a2 b3 c1l bl c3 a3 bl c2 b2 cl1 al b2 d3 b3 d2 a2 b3 dl bl d3 a3 bl d2 al b2 c3 b3 c2 a2 b3 cl bl c3 a3 bl c2 bl c2 d3 c3 d2 b2 c3 dl cl d3 b3 eled2 c2 d1 diis al b2 c3 b3 c2 a2 b3 c1 bl c3 a3 bl c2 b2 cl al e2 d3 c3 d2 a2 c3 dl cl d3 a3 el d2 c2 dl1 al b2 c3 b3 c2 a2 b3 cl bl c3 a3 bl c2 b2 cl1 al b2 d3 b3 d2 a2 b3 dl bl d3 a3 bl d2 b2 dl z al b2 c3 b3 c2 a2 b3 cl bl c3 a3 blL c2 b2 cl D N L 13 David Sj strand CAS and Spreadsheets p21 The expressions for x y and z can be saved in BASIC format and then be used in a spreadsheet Save the formulae under the name LINEQU BAS File gt Write gt Basic file Metzwerkumgeb Dateiname linequ bad v ung Dateityp Basic file bas Abbrechen Expressions i All F wt t Selected Steps to be taken in MS Excel a Open the file LINEQU BAS Cells Al A3 and A5 should now contain the expressions for x y and z as strings fe x bl c2 d3 c3 d2 b2 c3 d1 c1 d3 9b3 01 d2 y a1 c2 d3 c3 d2 a2 c3 d1 c1 d3 4 a3 c1 d2 c2 d 1 a1 b2 c3 b3 c2 z a1 b2 d3 b3 d2 a2 b3 d 1 b1 d3 4 a3 b1 d2 b2 d V a1 b2 c3 b3 c2 o b Move these expressions to the cells B7 B8 and B9
16. and displays the percentage of free memory statistic Note that this is the only time a meaningful statistic can be calculated since only then is the size of the active data structures known p36 BBS amp DERIVE USER FORUM D N L 13 Therefore if a lot of data structures are being produced many garbage collections will occur But if most or all of the data structures being produced are temporary the garbage collection will free up most of memory and the percent free will remain high And it should remain high since the system is not in danger of running out of memory Aloha Al Rich Soft Warehouse Inc DNL Answers to inquiries of Mr Herdt and Mr Lopes In one of the later messages in the BBS I found the similar problem Message 3374 From SOFT WAREHOUSE to KEITH WILLIAMS about TRANPOSE OPERATOR BUG Thank you for your fax letter dated December 24 1993 concerning a problem with DERIVE s transpose operator The problem arose because DERIVE version 2 59 improperly transposes ragged arrays Ragged arrays are vectors of vectors not all having the same number of elements The problem did not arise in earlier versions of DERIVE because those versions did not even attempt to transpose ragged arrays The problem has been resolved and will be included in the next version of DERIVE Please send me your mailing address and I will send you a complementary update as soon as it 1s available Aloha Al Rich Soft Warehouse Inc I sent a FAX to SWHH
17. for introduction and explication of domain functions in polar form Having set the coordinate system to polar the expression a t b t is interpreted as parameter form 5 vector d on r t o t So t 1 is a segment of legth t which 0 t lt nl2 forms an angle of 1 rad with the x axis 1 t is a 0 t a2 part of the circumference of a circle with radius 1 and t t is a spiral 2 vecron k 0 i Expression 1 gives a demonstration of a quadrifolium r sin 2 in polar form which 0 t lt nid shows the four parts of the leave in different ken ken colours Option gt Display gt Color gt vcr mid ip el Ri dh Automatically D N L 13 Selections from the DERIVE BBS p 7 ae T Weniger erfreulich sind die von Version zu Version vorgenommenen undokumentierten kleineren Anderungen bei bereits vorhandenen Funktionen und Operatoren von DERIVE Bis zur Version 2 58 wurde streng unterschieden zwischen den drei Objekten Vektor 7 geordnete Liste von Elementen einzeilige Matrix und einspaltige Matrix vergl auch Handbuch Vers 2 8 135 ff Verst ndlicherweise lie sich daher ein Vektor im Unterschied zu den beiden anderen Objekten nicht transponieren Ab Version 2 59 l t sich nun pl tzlich auch ein Vektor transponieren und es entsteht eine einspaltige Matrix die konsequenterweise bei erneutem Transponieren in eine einzeilige Matrix
18. gt s RETURN f f2 1 1 1 Ulf 100 13 f bpair 10 14 9216845717656874712980450562726202415567360565980794777111390850331644813674856981646960226192287360 5696323922575865414847061494575945648081290145228607189038829076215134884313127297923138542545712321 In spite of the now built in function fibonacci n the following two original ways of computing Fibonacci numbers are still interesting for didactic reasons as they show how the golden section number X 1 5 2 comes into play 1 45 n 1 45 n FT f bl n 5 1 1 45 n 16 fiab2 n FLOOR 4 0 5 45 5 17 f b1 100 354224848179261915075 18 f b2 100 354224848179261915075 19 FIBONACCI 100 354224848179261915075 D N L1 13 Johann Wiesenbauer Titbits 1 p33 Now we are able to answer the following important question What is the index n of the bigger Fibonacci number Fn in fibpair s For example if s 10 100 then due to 1 1 45 n 100 20 NSOLVE m 10 n O 1000 n 480 1694726 5 2 n 480 should be the correct index Indeed we have 100 eal 21 Perr gt 10 fib2 480 10 true true Solving the equation above for general s algebraically leads to 2 LN s LN C5 1 1 45 n SOLVE s n Real n 22 45 2 J 45 1 E which can approximately also be written as l4 45 23 n Locs 1 6 22 5938 2 Let s check this in our example
19. in view of all the versions of Derive we have seen since version 2 56 used then Needless to say that this piece of code would look completely different now maybe like this eea a b q_ r_ s_ Prog a a 1 0 b b 0 11 Loop If FIRST b O 32 RETURN REVERSE s_ ADJOIN FIRST a FIRST b FIRSTCREST a FIRSTCREST b s FLOOR FIRST a FIRST b a qb b rr ui Ya c 33 eea 1234567 234567 1234567 234567 1 O 1 234567 61732 0 1 1 5 61732 49371 1 5 3 16 49371 12361 3 16 4 21 34 12361 12288 4 21 15 79 12288 73 15 79 19 100 73 24 19 100 3207 16879 24 1 3207 16879 9640 50737 You will certainly have noticed that we introduced a new order of the colums as well as some pairings of numbers This has the following meaning At any time numbers in the first and second columns are the inner product x y a b where x y can be seen in the third and fourth columns respectively In particular the gcd a b which is 1 in this example can be written as 35 9640 50737 1234567 234567 1 By the way if you are only interested in this vector 9640 50737 which is quite often the case in the applications then you could also use the built in function extended_gcd a b It should also work for Gaussian integers but sometimes it fails This is a different story though which I will tell you another time 36 EXTENDE
20. letzten Jahren unter Mithilfe der Computerprogramme DERIVE und Cabri Geometre entdeckt wurden z lAigebralcsiclother Pramto ciesn uel _ mucx tan ip rayi J an pi x wu x Z2 aj tanti p ray2 uy tan i3 pi x z2 a Bsolvetrayl and ray2 x yt 2 a cosipi zinti pj sinipi cos 3 pl costpi sint3 po 2 a cosipi zinti pj x anc LI x Ez lpigebralcalc other Pramto ciesn uel ui uode a sin 2Z t sin t3 z sinit a cos 2 t3 cos t3 uper nu Sint m Expandi Za costi a mgzaoluerx 2 2 3 cosit a ti re 4 amp i3 1 mn gand y_ faa sa b _ sigala x 2 3 lal x_ un 3 3 20a x xf de x sigala x 2 3 lal x_ Zaft 26a x x2 MAIN RAD AUTO FURC 15 z a q 3 a x x J af 2 a x x Sai 2 a x XL r 4 a COS phi a COS phi x x SORT 3a x SORT x a Y Lr ATXI X 2 oF ax U Outlook In the next part some constructions will be shown which enable to produce the curves introduced till now in a homogenous way We will give special attention to those constructions which have been found out in the last years with help of computerprograms as DERIVE and Cabri Geometre x m sebra cate jother Prsm1o ciesn url __ c2 acsin p sin ga Sinipi cosl p cospi sin i p sintp cos 3 p cospi siniz p a sind pP siniz pj sin iz pj adcos 4 p costz py m ibcollect x x m LCcollect u 4 y 1 Fer Fi Fur FE Far c largebralcstelother Pranto cies
21. nun aus den Kurvengleichungen durch Aufl sen der algebraischen Kurvengleichung nach y eine Parameterdarstellung der Kurve gewinnen PLOT Befehl oder eine andere Parameterform finden die Asymptotik der Kurve untersuchen x a Nullstellen berechnen 0 0 und 3a 0 mit Hilfe von S r dq der von der Trisektrix eingeschlossene Fl cheninhalt berechnen S 3a 3 Supported by DERIVE you may gain the parametric form of the curve solve the equation for y or find another one investigate the asymptote x a find the zeros 0 0 and 3a 0 calculate the area enclosed by the curve A DERIVE Trisectrix Session Expressions 1 through 8 are accompanied by a plot with slider bars for p and a 1 rayl y x TAN p I 5 2 a 0 3 ray y x 2 a TAN 3 p SOLUTIONS rayl a ray2 x y 1 4 2 COS p G a 4 a SIN p a SIN p 2 COS 2 p 1 5 re COS p COS 2 p SIN p SIN 2 p COS p COS 2 p SINCp SIN 2 p 2 COS t 3 a 4 a SIN t a SIN t 2 COS 2 t 1 T6 COS t COS 2 t SIN Ct SINC2 t COS t COS 2 t SINCt SINC2 t 7 Trigonometry Collect a COS 3 t a SIN 3 t ta 20 cos Ct COS t This is another fine parameter form for the trisectrix Now let s derive the analytic form by eliminating parameter f D N L 13 Thomas Weth A Lexicon of Curves 3 p17 9 Trigonometry Expand 10 Trigpower Cosines
22. the formula SQRT C2 A3 2 D2 B3 2 in cell C8 The German version of MS Excel needs WURZEL instead of SQRT Enter Circumscribed Circle in cell A10 and t x and y in the cells All B11 and C11 respectively Then enter 0 in cell A12 and the formula PIQ 50 A12 in cell A13 and copy it to the range A13 A112 2n Then enter the formulae A 3 C 8 COS A12 and B 3 C 8 SIN A12 in cells B12 and C12 respectively Copy the contents of B12 C12 to the range B112 C112 and select range B12 C112 Move the cursor with the Ctrl key pressed to the top right corner of the selected range The cursor changes to a little Now you can drag the table into your diagram and add the points of the circle If you change any of the coordinates then the triangle and subsequently the circle will change accordingly You might find a superfluous segment connecting the first point of the circle with the last point of the triangle Then select this segment and make it invisible by reformatting Now you should have something like this p24 David Sj strand CAS and Spreadsheets D N L 13 Adjust scales manually in order to receive a figure where the circle really looks like a circle Select the axes and give the command Format Scale B12 3443 5049 COS A12 302352941 E R E a Circumscribed Circle t x y ES Op 2117647081 405882353 006283155 91101022 4 29590495 O 12566371 908749739 453003593 018949556
23. they will work it out For now do not trust it Message 2822 From HARALD LANG to PUBLIC about DERIVE AND EDUCATION There seems to be a discussion going on between Gary Ingram and Jerry Glynn of which only parts are readable for the public so maybe I haven t got the 1ssue properly I don t quite see why DERIVE should be viewed as a pedagogical tool only I teach mathematics at the Royal Institute of Technology in Stockholm but I use DERIVE there only to prepare examples and exams but I also do research in economics at Stockholm University and there I use DERIVE also for my research To me DERIVE is a MATHEMATICAL ASSISTANT i e I don t use it for anything I couldn t in principle do on paper myself but if I instruct my assistant it does it much faster and with fewer hopefully no errors It 1s like a pocket calculator it is useful both in education and research To me a useful upgrade should contain better and more efficient algorithms and I agree with Gary Ingram that the latest upgrade was a bit thin in that respect For instance one thing that has irritated me is that if I solve a problem and DERIVE is unable to simplify the answer I don t get any answer at all To fix the ideas consider solving x from the equation ax b c Ifa b and c are messy expressions then DERIVE tries to simplify the answer x c b a by cancelling common factors in the denominator and numerator This can easily be too complicated for DERIVE which is natu
24. to draw your attention to Dr Wiesenbauer s Algebra and Number Theory contributions promi sed in the last DNL I m very glad that Dr Wiesen bauer has fulfilled his promise It is really 1mpossi ble for me to particularly point out one or the other contribution in the next issues each one of them is original and unique The extracts from the BBS will be continued Some pages of computer paper 60 are waiting to be evaluated whether they could be of interest for you I can promise some useful electronic discussions One week ago I received a wonderful DERIVE animation from Norway Halvor Devold produced a show Keplers World System in 3 D Many thanks dear Halvor Participants of the Krems Conference 1993 will certainly remember your im pressive demonstration As you have given your permission to distribute the file I feel free to incorporate your show into the 1994 diskette at the end of the year I m sure to act in your sense Thanks again Three administrative notes at last You can reach us by FAX now If it does not work the first time please give it another try See Informationpage Togetherwith the next DNL you will receive a receipt about your membership fees for 1994 If you haven t settled your dues for 1994 then please use the reminder Thanks in advance I am looking forward to the 4th DUG year with you Sincerely yours My P2 E DI TORIA ILL The DERIVE NEWSLETTER is the Bulletin of the DERIVE User
25. x n x O inf This integral should simplify to n Derive does not do this simplification but it does verify it for actual values of n What I wish to know is whether there is any way of teaching Derive how to do integrals with which it 1s not familiar thinking of this one in particular so that when it encounters it in a more complicated expression it can be simplified correctly PS Has anyone thought of having a DERIVE Bulletin Board in the UK J M M Cardia Lopes Porto Portugal I have tried to run DERIVE 2 03 at a HP 9000 750 33 workstation by means of SOFT PC a MS DOS emulator for UNIX DERIVE works but the first screen stays over the next screens The same with the graphics generated by PLOT It looks bad and is very deficient Do you know how to solve this bug Do you know anyone who has the same problem K Winkelhausen Arch Switzerland Ich w re froh wenn ich einen Ausdruck s mtlicher DERIVE Anwender der Schweiz erhalten k nnte um einen regen Kontaktaustausch innerhalb des Landes vornehmen zu k nnen DNL Lieber Herr Winkelhausen Ich mochte grundsatzlich keine Adressen weiterleiten Ich nehme aber an da Sie mit der Angabe Ihrer Anschrift einverstanden sind so k nnen sich alle Schweizer DERIVE Anwender mit Ihnen in Verbindung setzen Solothurnstr 15 CH 3296 Arch K Herdt Osnabr ck Germany F r die Einf hrung und Erlauterung der Darstellung von Funktionen in Polarkoordinaten finde ich es aus didaktischen
26. 3 ATIAN t COS t 3 Message 2399 From SOFT WAREHOUSE to PUBLIC about A DERIVE DATA BASE PROGRAM File DATABASE MTH implements a simple data base program and gives examples of its use File DataBase MTH Example of data base table look up lst establish some unassigned multi character vars even in character mode age height eye_color brown blue Establish some data records darth_vader age 84 weight 240 veronica age 30 weight 130 eye_color brown GET AUX decrements k until it is 0 or the key is found Note how we must use 3 valued logic true false unknown because Derive usually can t determine if two distinct nonnumeric keys are identically equal as math vars or never equal as math vars For example the equation age weight could be identically true or never true GET_AUX rec key k IF k 0 IF kKey ELEMENT rec k 1 ELEMENT rec k 2 G GET_AUX rec key k 1 GET_AUX rec key k 1 Use Manage Substitute to replace d with the appropriate subexpression So that the subsequent two test cases simplify to brown and GET rec key GET_AUX rec key d GET veronica eye_color GET darth_vader eye_color d 3 tried d 3 GET veronica eye color brown GETCdarth vader eye color P10 Selections from the DERIVE BBS D N L 13 Message 2538 From WILLIAMSON to SOFT WAREHOUSE about 2399 DATABASE MTH COMMENTS The instructions with DAT
27. ABASE MTH could be clearer It should be pointed out that the k parameter in GET AUX represents the number of fields in a record that can contain an undetermined number of fields Dennis Message 2583 From SOFT WAREHOUSE to PUBLIC about MATRIX KRONECKER File KRONPROD MTH contains a function that returns the Kronecker Product of two matrices NCOLS a DIM a 1 KRON PROD AUX a b m n pg VECTOR VECTOR aj 14 FPLOOR 1 PB I IF FLOOR JG BL IF MOBIL LT BJ MOD 7G 349446 1 14 0ym p L KRON_PROD a b KRON PROD AUX a b DIM a NCOLS a DIM b NCOLS b See one example 2 3 4 0 mL z 5 6 7 8 0 0 0 0 0 0 0 0 0 KRON_PROD m1 m2 8 16 24 35 40 45 42 48 54 49 56 63j 56 64 72 A short explanation for those few of you who like me don t know the definition of the Kronecker product of two matrices all elements of m2 are multiplied by the elements of m1 and these products replace the elements of m1 The black rectangle is the product of 2 and m2 and the red one is the product of 7 and m2 Definition found in the fine book of Karsten Schmidt amp G tz Trenkler Einf hrung in die Moderne Matrix Algebra Springer Message 2607 From SOFT WAREHOUSE to PUBLIC about SORTING File SORT MTH is an interactive lesson that helps teach recursive Derive programming The goal is a function that sorts a vector of rational numbers into nondescending order To take the lesson begin with Transfer Load Derive SORT t
28. D_GCD 1234567 234567 1 9640 50737 D N L 13 BBS S DERIVE USER FORUM p35 Many many years ago I found an interesting article in the Scientific American about Public Key Cryptology Specially the RSA and the Merkle Hellman Algorithm were described I was fascinated of this number theory application and wrote a GWBASIC program That was 1978 or 79 and I had time enough for programming because there was no DERIVE and no DUG For Merkle Hellman s method I needed the EEA algorithm applied on two relative prime numbers Having attended Dr Wiesenbauer s lecture I tried to find my program KRYPTO BAS Here you can see some lines BASIC Code the Berlekamp The full program is very nice and demonstrates in an easy to understand way the method how to produce a public key how the system works how to encrypt and to decrypt a message If anybody is interested in this program then please let me know Josef REM A bit adapted for QBASIC DEFLNG A Z INPUT Two numbers with GCD 1 Z1 Z2 XO ls YL e li A Z1 B Z2 starti Q INT A B R INT A Q B 5 X2 XU 0 c Xls ne YO Q XL IF Yl 0 THEN O1S ELSE 015 LE Y2 gt 0 THEN 025 pe ELSE O20 t LE R lt 2 THEN PRINT Bz 2 OIS ABS X2 x sAl3029 ABS Y2 5 x 422 END A B B R X0 Xl YO Yl Xl 22 Yl Y2 GOTO start Two numbers with GCD 1 1234567 234567 1 1234567 x 9640 234567 x 50737 I I I 1 Fe
29. THE DERIVE NEWSLETTER 13 ISSN 1990 7079 THE BULLETIN OF THE L7L I T V C A USER GROUP Con tent s Letter of the Editor Editorial Preview DERIVE User Forum Bulletin Board Service Thomas Weth A Lexicon of Curves 3 The Trisektrix David Sjostrand CAS and Spreadsheets MS Excel J M Cardia Lopes Minimization of a Flat Function J Wiesenbauer Titbits in Algebra and Numbertheory revised reprint 2007 March 1994 D N L 13 INFORMATION Book Shelf D N L 13 1 ANALYSIS Bilder und Filme mit DERIVE U Keusen und H J Kayser MBU 3 93 1 Bergmoser Holler Verlag GmbH Aachen 2 LINEARE ALGEBRA Matrizen mit DERIVE B Barzel und P Drijvers MBU 5 93 1 Bergmoser Holler Verlag GmbH Aachen 3 Ein elementarer Zugang zu Potenzreihen W Koepf Didaktik der Mathematik 21 1993 292 299 Bayerischer Schulbuchverlag 4 MATEMATICA I Consorzio Nettuno S Cappuccio amp G C Barozzi Pitagora Editrice Bologna EC ul C DA The Seventh Annual International Conference on Technology in Collegiate Mathematics November 17 20 1994 Hosted by Valencia Community College Sponsored by Adison Wesley Publishing Company Founded by Franklin Demana and Bernd Waits Conference Chair Judith Jones Walt Disney World Dolphin Hotel Orlando Florida Mail or Fax to Addison Wesley Publishing Company Attn Beth Sheehan 1 Jacob Way Reading MA 01867 Fax 617 944 8964 LC Ii We have a FAX now We have a FAX no
30. angle with the trisectrix The angle to be divided into three equal parts 1s drawn with its vertex in S and one ray on the x axis The other ray then intersects the trisectrix in P ZSOP then is the third of the given angle 0 Der Winkel ZSOP ist dann das gesuchte Drittel des urspr nglichen Winkels Herleitung der Kurvengleichung Zur Herleitung der Kurvengleichung betrachtet man AOSP Da Au enwinkel von AOSP ist und o 3y gilt ZSPO 2y Nach dem Sinussatz gilt also in AOSP Sine rule r sin y sin y a a Lo 4156 Te 2a sin2y 2siny cosy cosy 9 Ersetzt man r durch p gem p sin y r sin 3 y Sinussatz Sine rule und ber cksichtigt dass 2 a sin 3y 3 sin y 4 sin y so erh lt man psiny 3siny 4sin y also COS V a iO a 2 p 3 4sin y 4cos y 1 COS V COS V Damit erhalt man die Polargleichung mit S als Pol die in Formelsammlungen bliche Darstellung a p 4a cosy cos W p16 Thomas Weth A Lexicon of Curves 3 D N L 13 W hlt man S als Ursprung eines kartesischen Koordinatensystems s Zeichnung so folgt mit X X p x y und cosy die algebraische Gleichung f r die p x y Koordinaten des Kurvenpunktes P x y 2 2 4ax X Ty 2 2 X y a also DERIVE y a x x 3a x 0 Ix y Damit ist die Trisektrix von MacLaurin eine algebraische Kurve dritter Ordnung Mit DERIVE lassen sich
31. be regarded as functions of the 6 variables x y X2 y x3 and y3 x1 y1 AT Ox2 y2 Steps to be taken in DERIVE In DERIVE we can derive the expressions for xM and y M As the three vertices of the triangle have equal distances to point xM yM we see that we can get the desired position if we solve the following system of equations which is in fact linear in xM and y M 2 2 2 2 10 equl xl xu yl ym x2 xn Cy yn 2 2 2 2 11 equ xl xu yl ym x3 xn y3 yn SOLUTIONS equl A equ2 xm yn 12 1 Use SOLVE or SOLUTIONS to solve the system with respect to xM and yM You will receive two bulky expressions for xM and yM If you apply the EXPAND command on the numerator and denominator they will look like this 2 2 2 2 2 2 2 2 2 2 2 2 xl y2 xl y3 x2 yl x2 y3 x3 yl x3 y2 yl y2 yl y3 yl y2 yl y3 y2 y3 y2 y3 xm 2 xl y2 2 x1l y3 2 x2 yl 2 x2 y3 2 x3 yl 2 x3 y2 2 2 2 2 2 2 2 2 2 2 2 2 xl x2 xl x3 xl x2 xl x3 xl y2 xley3 x2 x3 x2 x3 x2 yl x2 y3 x3 yl x3 y2 2 x1l y2 2 xl y3 2 x2 yl 2 x2 y3 2 x3 yl 2 x3 y2 David Sj strand CAS and Spreadsheets Select Highlight express
32. bergeht F r jeden vom Nullvektor verschiedenen Vektor v ergibt sich also die sicherlich nicht sonderlich befriedigende Ungleichung v v 40 Da die Autoren von DERIVE auch mit einem bestimmten p dagogischen Anspruch angetreten sind ist dies eigentlich bedauerlich DNL Answers to inquiries of Mr Herdt and Mr Lopes In one of the later messages in the BBS I found the similar problem Message 3374 From SOFT WAREHOUSE to KEITH WILLIAMS about TRANPOSE OPERATOR BUG Thank you for your fax letter dated December 24 1993 concerning a problem with DERIVE s transpose operator The problem arose because DERIVE version 2 59 improperly transposes ragged arrays Ragged arrays are vectors of vectors not all having the same number of elements The problem did not arise in earlier versions of DERIVE because those versions did not even attempt to transpose ragged arrays The problem has been resolved and will be included in the next version of DERIVE Please send me your mailing address and I will send you a complementary update as soon as it is available Aloha Al Rich Soft Warehouse Inc sent a FAX to SWHH in the evening and thanks modern times same night at 03 22 am received an answer from Al Rich In response to requests by users beginning with version 2 59 DERIVE transposes a vector to an n by 1 column matrix The transpose of an n by 1 column matrix is a n by 1 row matrix Since a by n row matrix is not equivalent to an n element vector th
33. e Integralkritertum f r die Konvergenz unendlicher Reihen Auf der nachsten Seite k nnen Sie die GeoGebra Konstruktion sehen Au erdem habe ich die Trisektrix mit DERIVE und den Schiebereglern nach der Konstruktionsvorschrift hergestellt Mit dem Schieberegler f r a verandert man die GroBe der Schlinge und der Schieberegler f r p lasst den Punkt P auf der Ortslinie wandern Maclaurin s Trisectrix In the first part of the DERIVE lexicon of curves the Cissoid has been introduced This curve was used for doubling a die Beside doubling a die dividing an arbitrary angle into three equal parts trisection was another classic problem During the centuries innumerable mathematicians have been busy to solve trisection of an angle using only ruler and compasses By methods of algebra easily can be shown that this is impossible Nevertheless there are ideas from nonprof mathematicians how to solve the problem There were so many very complicated ideas and constructions that in 1983 Dudley published a manual for the examiners What do do when trisector comes Among the nonelementar solutions using other tools beside ruler and compasses you can find a lot which are using algebraic curves One of the best known is the curve discovered by the Scot Maclaurin 1698 1746 professor for mathematics in Aberdeen with age 19 since 1726 in Edinburgh In his most important opus A treatise of fluxions 1742 is to be found the socal
34. ed for zeroing the derivative in MIXED amp EXACT Mode d 3 c dp 17 10 2 5 80 7 p 225000 p 2625000 3 4 Precision Exact 5 Notation Rational 17 10 2 5 amp 0 7 p 225000 p 2625000 6 SOLVE p 2 p 17 10 2 5 7 ip 225000 p 2625000 v p unit circ le 8 Precision Mixed 9 Notation Mixed 17 10 2 5 80 7 p 225000 p 2625000 10 SOLVE p 2 p 11 p unit circle o v p 3856 931625 17 10 80 7 p 225000 p 2625000 12 SOLVE p Real 2 p 17 10 2 5 13 Tp 225000 p 2625000 v p 1 Edgar amp Himmelblau Optimization of a Chemical Process McGraw Hill Intl Edition p28 problem 1 7 p30 J M C Lopes A Flat Function D N L 13 Surprise DERIVE is unable to 1 2310 mE solve this equation at the Exact or at the FOREN PT Mixed Mode But the function really has un a minimum In fact if we graph this a 10 function and the first problem for most students is to choose an appropriate scale we can see that the minimum 4 10 5 exists and it is close to P 3850 Then Why is it impossible to determine its mini mum To answer this question it is a good idea to plot dC dP 1000 2000 3000 4000 5000 l 2 10 5 2 10 5 Derivate dC dP As we can see in the figure the graph of dC dP
35. en Vor einer Woche erhielt ich eine wundersch ne DERIVE Animation aus Norwegen Halvor Devold erzeugte eine Show Keplers World System in 3 D mit DERIVE und GRASP Herzlichen Dank lieber Halvor Die Teilnehmer von Krems 93 wer den sich gerne an Deine beeindruckende Demon stration erinnern Da Du mir die Weitergabe freige stellt hast werde ich Deine Show in die Jahresdis kette 1994 aufnehmen Ich bin sicher damit in Deinem Sinn zu handeln Zum Schlu noch drei administrative Hinweise Wir sind endlich mit einem FAX erreichbar Siehe Informations Sie erhalten mit dem n chsten DNL eine Quit tung ber Ihren Mitgliedsbeitrag 1994 Falls Sie den Jahresbeitrag noch nicht geleistet haben dann ben tzen Sie bitte das beiliegende Erinnerungsschreiben Danke im voraus Ich freue mich auf ein DUG Jahr 4 mit Ihnen EUG Ihr LETTER OF THE EDITOR pl Dear DUG Member I m really glad to meet you all again in the DUG I would like to take the opportunity to thank for all your nice letters and for your appreciation You will find a little selection in the User Forum Please don t see this as self praise because your approval especially goes to the numerous DERIVIANS who submit papers and give suggestions One glance at the preview shows that we can expect an interesting and contentful DNL year I m sorry to say that I don t have time not yet to extend the content of the DNL I have a sideline too I want
36. en GER Der Fermat Punkt im Dreieck Geyer GER Continued Fractions in DERIVE Cordob a o ESP Life Game Turtle Commands in DERIVE Lechner AUS Dreieck MTH Wadsack AUS and others At this position in the last DNL you could find my Christmas tree In the letter of the Editor I expressed my hopes that I wouldn t have to draw a DERIVE Easter Bunny But look Mr Klingen from Bonn sent a letter containing a wonderful rabbit Because you have wished an Easter Bunny in your last issue here is one It is built from 28 points with 5 parametric splines consisting each of 6 or 7 points Impressum Medieninhaber DERIVE User Group A 3042 W rmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef Bohm Herstellung Selbstverlag D N L 13 DERIVE USER FORUM p 3 Dott Mariarosa Castelletti Segrate Italy I introduced the DERIVE User Group and the DERIVE Newsletter at a meeting of Milanese section of Mathesis Mathesis founded in 30 years by Federigo Enriques important Italian mathematician is an association of Medium High School amp University Math teachers Mathesis is present all over Italy through autonomous local sections The activities of those sections are training updating and co ordination of the members In most cases they rely upon University Institutions for example in Milan that is my section they rely upon Mathematics Faculty The next issue of the Milanese section s bull
37. etin will advertise the existence of the DERIVE User Group Many Mathesis members other than me either know or have heard about DERIVE and they would like to share knowledge and experiences If possible it might be interesting to present some of the DERIVE Newsletter contributions on the Mathesis Bulletin of Milan Some of the members and by myself I would like to translate the articles into Italian to override the difficulties of English and German to many of the potential readers While I am waiting for your reply also if negative I renew to you my appreciation for your work and hope to hear from you soon again Yours sincerely DNL Thanks for the flowers You were the first tried my FAX hope it worked I m glad for your PR work in Italy Here is another letter from Italy Sebastiano Cappuccio Forli Italy Progetto Nettuno is a project of the Politecnico di Torino Politecnico di Milano and Universit di Napoli Federico II for a triennial course of Informatica Computer Science with TV lessons broadcasted by satel lite There are about 500 students in the Academic year 1993 94 all over Italy Students can attend their lessons at home There are 6 centri di ascolto with computers library tutors where students can also see recorded les sons and sit for examinations Lessons are also broadcasted by State TV from 3 00 to 6 00 a m and students can use their own videorecor ders Users are not only students but a
38. ey cannot be subtracted and their difference is not the zero vector Perhaps DERIVE should transform the vector into a row matrix so the subtraction can occur However this is not valid if the vector has symbolic elements that couls themselves be replaced by vectors The screen problem that occurs when running DERIVE under Soft PC must be a bug in Soft PC not DERIVE Apparently Soft PC does not correctly emulate the screen services provided by the IBM PC BIOS I recommend Mr Lopes contact the authors of Soft PC to see if they can resolve the problem Sicerely Al Rich p 8 Selections from the DERIVE BBS D N L 13 Message 2392 From SOFT WAREHOUSE to PUBLIC about L HOPITAL S RULE The Derive LIM function is quite powerful but it is educational to implement your own function for computing limits using L Hopital s rule particularly if you want the rule to be automatically repeated if necessary Consequently file LHOPITAL MTH is an interactive lesson that helps teach recursive Derive programming To take the lesson begin with Transfer Load Derive LHOPITAL then follow the directions on the screen Note that the use of LIM within the definition in the file is merely to SUBSTITUTE the limit abscissa for the limit variable in the numerator and in the denominator so it is not cheating by using the internal implementation of L Hopital s rule File LHOPITAL DFW Recursive L Hopital s rule for Q O L n d x a If LIM n x a 0 a LIM d x a 0
39. following summarizes how much memory is available for each of the various DERIVE data structures Segments Kilobytes Data Structure 1 64K muLISP machine code and compiled DERIVE code 2 1288 DERIVE code stored as muLISP pseudo code 2 128K Mathematical expression storage 1 64K symbols and number storage 1 64K ASCII strings for symbols 1 64K Binary value for numbers 2 128K Control and variable stacks 10 640K TOTAL Note that the space allocated to the first two structures above is fixed Space is dynamically allocated for the remaining structures on a demand basis The space allocated to these structures can be less than the maximums listed above If while running DERIVE the space required for any one of these structures exceeds the above limits a Memory Full error message occurs Message 2972 From SOFT WAREHOUSE to PUBLIC about DOMAIN DECLARATION SYNTAX Currently DERIVE provides no way to declare variable domains from the Author line or in MTH files A number of users on and off this BBS have requested this capability The following examples shows the syntax I am considering for domain declarations for the next version of DERIVE Declaration Meaning xe up X iS a real x Xu X 1S an integer sop X is a complex K ar es 104 X is a real in the half open interval 5 10 eT pou ata X is an integer in the infinite interval 0 inf Note the similarity of the above with the current syntax used to assign variable values x de X 1S an una
40. hen follow the directions on the screen Your rewards will be a greater problem solving ability and a useful function to add to your personal utility library renamed SORT MTH to SORT MTH and the function SORT to SORT because SORT has been implemented into DERIVE since long Josef Pale SORT MTH Selection sort elements of numeric vector v into nondescending order Find minimum element of v from element k with element m min so far FIND MIN v k m IF k DIM v m IF ELEMENT v k ELEMENT v m FIND MIN v k 1 k FIND_MIN v k 1 m D N L 13 Selections from the DERIVE BBS p11 Sort elements of v into nondescending order from element i on SORT AUX v i IF i DIM v v SORT AUX SWAF ELEMENTS v 1 EIND_ MINV 1 2 1 L SORT_ v Define SORT_ v then test amp debug it on the following examples SORT l SORT 42 SORT 415 72 17 2 31 Solution and check using built in SORT function SORT v SORT AUX v 1 SORT_ SORT 42 42 1 1 SORT_ 5 2 3 2 3 5 2 2 401 SORT z Ja 0 100 25 100 e J2 0 n 100 e 4 401 SORT C nz 2 0 100 25 100 e J2 0 n 100 e 4 Message 2699 From JERRY GLYNN to PUBLIC about OFFICIAL BUG REPORT FLOOR MOD and IF functions do not work correctly in Derive with Limits This is a known problem to Al and Dave who write the program but they do not have a fix for it in hand Hopefully in time
41. in the evening and thanks modern times same night at 0322 am I received an answer from Al Rich In response to requests by users beginning with version 2 59 DERIVE transposes a vector to an n by 1 column matrix The transpose of an n by 1 column matrix is a n by 1 row matrix Since a 1 by n row matrix is not equivalent to an n element vector they cannot be subtracted and their difference is not the zero vector Perhaps DERIVE should transform the vector into a row matrix so the subtraction can occur However this 1s not valid if the vector has symbolic elements that couls themselves be replaced by vectors The screen problem that occurs when running DERIVE under Soft PC must be a bug in Soft PC not DERIVE Apparently Soft PC does not correctly emulate the screen services provided by the IBM PC BIOS I recommend Mr Lopes contact the authors of Soft PC to see if they can resolve the problem Sicerely Al Rich Albert Floch lay Dirinon France TU Je profite de la pr sente lettre pour vous remercier d avoir envoyez ma lettre du 6 Juillet 1993 au Dr Kutzler Celui ci a tr s bien compris l objet de ma mauvaise humeur mai la prix excessif de la mise jour n etant pas d SOFT WAREHOUSE mais l importeur francais je continue faire confiance DERIVE tout en changeant fournisseur Avec mes meilleurs voeux pour l nne 1994 E em m 4 zz e a tt cA A B NM c A Sa ee al ET a A
42. ions 18 and 19 and save them in BASIC format under the name say CIRCUMS BAS Steps to be taken in MS Excel Open the file CIRCUMS BAS Cells A1 and A3 are containing the expressions for xM and yM as strings These strings can again easily be converted to an MS Excel formula simply by removing the xm and ym EE Be KIKIKI KI KI DPD DMT PY Then you will receive an error message A Y342 33 Y 1 23 Y2 because you have nothing but empty cells in the range X1 Y3 Now you can Create a worksheet to plot a triangle with given vertices together with its circumcircle l Move the expression for yM to cell B3 and the expression for xM to cell A3 You can drag with the mouse or use the commands Edit Cut and Edit Paste Enter xM and yM in the cells A2 and B2 respectively 2 Plota triangle with vertices 3 1 9 4 and 2 6 Enter the coordinates in the range X1 Y3 and the formula X1 and Y1 in the cells X4 and Y4 to close the triangle similar as in DERIVE Then move the contents of range X1 Y4 to the range C2 D5 by again applying Cut and Paste Select the range C2 D5 and plot the triangle as a XY diagram 3 We plot the circle as another XY diagram and have to provide the points forming the circumference The points will be defined by the parameter representation of the circle The equation of the circle 1s x xM R cost 2 2 with R 2 4 xl xM yl yM y yM Rsint Enter R in cell 7 and
43. is nearly coincident with the axis that is why the intersection is so difficult to find And it is the right time to speak a little about ill conditioned problems Now we can try to use an interval 17 10 2 5 80 7 p 225000 p 2625000 method solve us equation biz near p Real we work in approximate mode or 2 we use NSOLVE P 15 p 3856 931625 Now the students are probaby motivated to discuss the particularities of the two classes of numerical methods for zeroing a function the iterative and the intervalar methods We can also try another approach to approximate the function by a Taylor polynomial which is the right degree and the right base point determine the derivative and find its zero s 20 TAYLOR c p 3850 2 2 6 21 0 008704314509 p 67 14356204 p 1 054558048 10 d 2 6 122 0 008704314509 p 67 14356204 p 1 054558048 10 dp 23 SOLVE 0 01740862901 p 67 14356203 p Real VDEELHEHET E 24 TAYLOR c p 3850 3 P 25 sorve TAYLOR c p 3850 3 dp 26 p 3856 931669 v p 6373 566236 1 4 196 1 2 10 6 2nd order Taylor Bc N eb polynomial 105 ieee amp 1045 3rd order Taylor polynomial 6 10 5 4 10 5 2 10 5 Derivative dC dP g 1000 2000 3000 4000 5000 6000 7000 8000 2 1095 Perhaps the Taylor approximation is not the best approach to this problem but it is always a possible approach and a new and enriching topic to discu
44. kons wurde die Kissoide vorgestellt Sie diente zur Verdoppelung des W rfels Neben der Wirfel verdoppelung war die Dreiteilung eines beliebigen Winkels ein weiteres klassisches Problem Im Laufe der Zeit besch ftigten sich ungez hlte Mathematiker und Laien damit die Winkeldrittelung mit Zirkel und Lineal zu l sen Mit den Methoden der Algebra kann mittlerweile leicht gezeigt werden vgl Kunz da sich mit Zirkel und Lineal alleine ein beliebiger Winkel nicht in drei kongruente Teile zerlegen l t Trotzdem gingen und gehen immer noch Vorschlage von Laien an den Universitaten ein die dieses Problem scheinbar elementar mit Zirkel und Lineal gel st haben Unter diesen Vorschlagen fanden sich mitunter so komplizierte und genaue N herungsl sungen da 1983 Dudley ein Handbuch f r die gepeinigten berpr fer derar tiger Beweise unter dem Titel What to do when the trisector comes herausgab Unter den nichtelementaren L sungen der Winkel dreiteilung also denen welche au er Zirkel und Lineal andere Hilfsmittel verwenden finden sich viele welche algebraische Kurven zu Hilfe nehmen Eine der bekanntesten ist die Kurve die von Maclaurin erfunden wurde Der Schotte Colin Maclaurin 1698 1746 wurde mit 19 Jahren Professor f r Mathematik in Aberdeen ab 1726 in Edinburgh In seinem wichtigsten Werk A treatise of fluxions 1742 findet sich u a die nach ihm be nannte Reihe sowie das gew hnlich Cauchy zuge schrieben
45. led Maclaurin series and the integral criterion for the convergence of infinite series which is usually named after Cauchy Given are the two points S and O with SO 2a Two rays are rotating round S and O with a start position OS in the x axis The ray round S is rotating three times faster than the other one The intersection P is one point of the curve wanted and d ZPOS 3 w with y ZPS and x axis The graphic is a Cabri Geometre plot D N L 13 Thomas Weth A Lexicon of Curves 3 pls Konstruktion der Kurve Gegeben sind zwei Punkte S und O mit der Entfernung 2a Um S und O drehen sich zwei Strahlen beide mit gleichem Drehsinn und beginnend aus der Ruhelage auf der Geraden OS Der Strahl um S dreht sich dabei dreimal so schnell wie der Strahl um O Die Schnittpunkte dieser Strahlen bilden die Trisektrix von MacLaurin d h o 3y Obiges Bild wurde mit Cabri Geometre mit Hilfe der Ortslinienfunktion erstellt The left figure shows the GeoGebra construction of the Trisectrix and the right one is the DERIVE construction running parallel to the analytic derivation of the equation using slider bars for a and parameter p Die Winkeldrittelung mit Hilfe der Trisektrix Zunachst zeichnet man eine Trisektrix z B nach obiger punktweiser Konstruktion Der zu drittelnde Winkel wird sodann mit dem Scheitel in S an der x Achse angetragen Der freie Schenkel des Winkels schneidet die Trisektrix 1m Punkt P How to trisect an
46. lso teachers and technicians who want to refresh their memory The course of Matematica 1 Analysis is held by Giulio Cesare Barozzi University of Bologna the Italian guru for symbolic computation The attending book is for exercises of Matematica 1 and the use of DERIVE is systematic With my best regards S C PS Congratulations for your nice interesting news letter The Bulletin of the DERIVE User Group is one of my favourite readings Thank you DNL Sebastiano s letter was accompanied by the book mentioned above Although don t speak Italian was able to follow the ideas You will find the title and publisher in the book shelf Thank you very much Sebastiano H Nieder Hamburg Germany Ich danke Ihnen f r Ihren Einsatz f r die DUG und die Zeitschrift DNL die ich mit Gewinn lese Nicht jedes Software Produkt das in Schulen eingesetzt werden kann wird so unterst tzt wie DERIVE Wir bleiben DERIVE und DUG treu OStD R Schorn Kaufbeuren Germany In the Journal of Recreational Mathematics Volume 25 Number 2 1993 I found the following problem 2052 A Coefficient Problem by Charles Ashbacher Cedar Rapids Iowa On page 473 of the unsolved problems section of Index to Mathematical Problems 1989 1984 by Stanley Ra binowitz there is the problem Menemui 5 2 1 Find the largest coefficient in the expansion of 1 2x 3x 4 4x3 29 P4 DERIVE USER FORUM D N L 13 Menemui is the abbreviation for Menemui
47. n ue my y t sin z zB 1 a A E a x b 3 a x Jj a x lal 2 a cos 2 sine 2 Algebra c ligebralcaiclother rsntolciean vel ec rM IE MEAE ues LUE AAT Bine 3 3 a3 x J5 x 7 2 x yu 2 jXa x 2Xx Z2 Catx 5 HIH RAD AUTO FUHC i716 D N L 13 Thomas Weth A Lexicon of Curves 3 p19 On page 19 you can find the handheld derivation of the parameter form which looks different as the others which have been found until now and the implicit form The next screen shots show parameter and polar form representations and the Tl Cabri produced trisectrix wee eoonedit Aii fstalels ses c eoe rece ES nat pese e v t aF LOT 4 Flot dike O xcz vu zt1 t pl d r costa PNE ee costa Literatur Bieberbach Theorie der geometrischen Konstruktionen Birkhauser Basel 1952 Baptist P Winkeldreiteilung und Trisektierer Praxis der Mathematik 29 1987 Nr 1 S 43 50 Dudley U What to do when the trisector comes Mathematical Intelligencer 5 1 1983 Kunz Algebra vieweg 1991 Another problem Dr Weth l m n 2 Given is the set M KERN U m n EC of 2 x 2 matrices n m Wanted are all solutions of X 1 0 with 1 the 2 x 2 identity matrix All my attempts to solve this equation with DERIVE failed Also failed the attempt to solve the equivalent equation x x1 0 icosa sind icOSQ X tisina Calculating by hand I found solutions 1 und sina icosQ tisina
48. r Fz Fur F5 For The BASIC program was my contribution in Zro r ur 1994 In these days when the CAS calculators are very welcome tools in math education 2031234567 234567 unfortunately not in all schools and all countries 1 3640 1224567 50737 22455673 can offer the Extended Euklid Algorithm on the eo ies CAS devices TI 92 Voyage 200 Josef gcc 1234568 234568 5 e 19 1234568 100 234568 5 19 x1234568 100x234568 CRTFTO RAD EXACT FUNC zn Message 2997 From SOFT WAREHOUSE to JERRY GLYNN about GARBAGE COLLECTIONS You asked why DERIVE recycles memory 1 e garbage collect even though the percentage of free memory displayed on the bottom of the DERIVE screen never goes below 99 for a particular problem When DERIVE is simplifying a mathematical expression it generates a lot of data structures that are used only temporarily Space is allocated for these structures as they are generated until the end of free memory is reached At that point a garbage collection occurs to collect these no longer needed data structures The two pass compacting garbage collector begins by marking all structures that are still being used by the program i e those pointed to by program variables or the stack Next the garbage collector makes a linear sweep through memory collecting all structures that are not marked 1 e garbage and makes the space available for reuse At this point after the garbage collection DERIVE computes
49. r from x1 y1 and the opposite side is x4 y4 and the intersection between the bisector from x2 y2 and the opposite side is x5 y5 The centre of the incircle is the intersection point of the bisectors of the triangle According to the theorm of bisectors point x4 y4 divides side 11 connecting x1 y2 with x2 y2 in the ration 13 12 Therefor we get the equations 25 12 x2 x4 13 x4 x3 12 y2 y4 13 y4 y3 They form a system of linear equations in x4 and y4 It can easily be solved 26 SOLVE 12 x2 x4 13 x4 x3 12 y2 y4 13 y4 y3 x4 y4 12 x2 4 13 x3 12 y2 13 y3 27 x4 A y Zoo 124 13 124 13 In a corresponding way we get a system of equations for x5 and y5 If you solve it you will get l xl 13 x3 amp y5 I1 yl 13 y3 30 x5 1l I3 11 I3 Enter the following assignments for the coordinates 12 x2 13 x3 12 y2 13 y3 11 x1 13 x3 31 5 z y4 o x5 i oo y5 12 13 12 13 13 11 13 H yl 13 y3 and set up the next system of equations formed by the equations of the two bisectors y4 yl y5 y 33 y yl x xl v y2 x x x4 xl x5 x p26 34 sovel yl 1l 12 13 y4 yl David Sj strand CAS and Spreadsheets yo y x xl y y2 x4 xl Rename the solutions as xM and yM 36 xm l
50. ral and then we don t get any answer at all It would be much better if DERIVE automatically gave up a simplification if it 1s too complicated and performed a less ambitious simplification e g in the example simplifying the numerator and denominator separately without trying to find common factors Maybe this is the reason why MAPLE according to Gary I I don t have any experience of MAPLE is capable of solving more advanced equations Harald Lang P12 Selections from the DERIVE BBS D N L 13 Message 2848 From HEINZ to PUBLIC about IN DEFENSE OF DERIVE I rise to speak in defense and in praise of DERIVE I belong to that silent minority of DERIVE users who are not professional mathematicians We were born genetically deficient in math talent but with a superabundance of appreciation for the beauty and power of mathematics and a burning desire to solve math problems simply for the joy of seeing math formulas and symbols come alive in two and three dimensional graphs DERIVE like no other program has done that for me and for that I am very grateful I am also grateful for the many excellent books that were published exploring and explaining DERIVE Each one of these books seven of them opened up new insights into the power and and enjoyment of math and DERIVE I am not completely ignorant of math I have a PH D in organic chemistry and some sixty publications in the fields of metallo organic complexes catalysis and coal chemistry
51. resepctively and remove the x y and z but leave the sign to declare the expressions as formulae c Ifwe now for example want to solve the system x 1l6y 3z 6 9x 9y 2z 4 21x 12y 35z 2 we have to enter the coefficients in the range Al D3 d You should find the result in cells B7 B9 in decimal form If you would like to have the result in exact form as fractions then reformat cells B7 B9 using the respective dialogue box Zellen formatieren Zahlen Ausrichtung Schrift Rahmen Muster Schutz Kategorie Beispiel Standard 2006 3447 zahl wahrung Typ Buchhaltung Datum Uhrzeit wissenschaft Text Sonderformat Benutzerdefiniert E zs to 20 un E A C ga RE Er ur GB DEE um 3E E000 E t g t000 EE 240 00 3 240 00 oo000 p22 David Sj strand CAS and Spreadsheets D N L 13 e The solution is 8 SOLVE x 16 y 3 2 6A 9 x 9 y 2 Z 4 Fi x 2006 3447 21 x 12 y 35 z 2 x y Z y 850 3447 qj 7 188 383 2006 850 188 9 X A y A z 3447 3447 383 If you vary the entries in range Al D3 then you can solve every non singular system of linear equations consisting of three equations with 3 unknown 2 The circumcircle of a given triangle Let xi y1 X y2 and x3 y3 be three given points in the plane Let xM yM the center of the circumscribed circle of the triangle defined by the given points xM and y M can
52. s of DERIVE recursively The DERIVE function GENERATE list k IF k 1 ELEMENT list 1 APPEND GENERATE list k 1 LIM ELEMENT list k x ELEMENT GENERATE list k 1 k 1 does the Job required If we declare F1 F2 F3 and F4 as functions of one variable too and define list F1 x F2 x F3 x F4 x then simplification of the expression GENERATE list DIMENSION list leads to the result InputMede Word fl x f2 x f3 x f4 x GENERATE f1 x f2 x f3 x f4 x 4 f1 x f2Cf1 x f3CF2CF1Ux fACT3CT2CTICXO22 2 D N L 13 DERIVE USER FORUM p 5 If we declare however F5 F10 as functions of one variable too and redefine list Fl x F2 x FE3 x FA x F5 x F6 x F7 x F8 x F9 x Fl10 x then simplification of GENERATE list DIMENSION list leads to memory overflow In each step the recursion invokes the previous value twice which generates hyperexponential complexity and hyperexponential memory requirements This problem can be resolved using an auxiliary function We redefine GENERATE by the statements GENERATE AUX list k aux APPEND aux LIM ELEMENT list k x ELEMENT aux k 1 GENERATE list k IF k 1 ELEMENT list 1 GENERATE_AUX list k GENERATE list k 1 and simplification of the expression GENERATE list DIMENSION list leads to f5 x ss FE f700 F8 x f9 x s fl009 GENERATE f1 x f2 x f3 x f4 f5 x
53. ssigned variable en x is a variable assigned the value 7 I would appreciate comments from BBS users about this proposed syntax for domain declarations Message 2980 From SOFT WAREHOUSE to WKSARTORY about 2885 AUTO SIMPLIFICATION The following undocumented feature of DERIVE 1s relevant to your question The Transfer Load Derive and Transfer Merge commands prompt the user to enter the name of a file to load After typing the file name if you press Ctrl Enter or Ctrl J instead of Enter each expression in the file is simplified as it is read in Note this is analogous to the way Ctrl Enter and Ctrl J simplify expressions entered on the Author line see Section 3 2 of the DERIVE User Manual Remember you heard it here first on the DERIVE BBS Aloha Al Rich Soft Warehouse Inc Message 2993 From JERRY GLYNN to PUBLIC about USE OF COMPUTERS TO TEACH I recently returned from a regional meeting in Winnipeg Canada of teachers of mathematics I was struck by the fact that we now have decent computers and quite good software in our society but not much discussion about how to use this stuff to teach Does anyone else see this If you do and will says so maybe we can start a discussion about how to use computers to teach math P14 Thomas Weth A Lexicon of Curves 3 D N L 13 Ebene Algebraische und Transzendente Kurven 3 Thomas Weth Wurzburg Germany Die Trisektrix von Maclaurin In der ersten Folge des DERIVE Kurvenlexi
54. ssion in the classroom D N L1 13 Johann Wiesenbauer Titbits 1 p31 Titbits in Algebra and Number Theory with DERIVE 1 Johann Wiesenbauer Vienna Austria This is the first part of a lecture held by Dr JohannWiesenbauer from the Technical University Vienna in the frame of the 3rd International Conference on School Mathematics in Vienna 23 2 25 2 1994 Dr Wiesenbauer showed some examples of sophisticated programming with DERIVE The DERIVE functions should be loaded as Utility files In the next issue you will find the second part of this lecture accompanied by Dr Wiesenbauer s explanation and interpretation You will marvel how he is handling and investigating supported by DERIVE huge numbers with ease The word has Dr Wiesenbauer Titbits 1 then and now Johann Wiesenbauer Vienna University of technology February 2007 This is a revised version of my Titbits 1 in the DNL 13 Note that if there are two functions with the same name the first one is usually the old version sometimes with small notational changes which is then overwritten by an updated version that makes full use of all the features of DfW 6 10 along with other improvements dellast v returns a copy of the vector v without its last component dellast v VECTOR v k 1 DIM v 1 Fl k 3 dellast v DELETE v 1 euclid a b returns the list of all equations in the Euclidean algorithm euclid a b dellast ITERATES CIF x 0
55. t 11 12 13 37 ym 11l 12 13 11 x1 12 x2 13 x3 11 x1 12 x2 13 x3 I11 yl 12 y2 13 y3 x5 x7 D N L 13 x e x 11 yl 12 y2 13 y3 and finally substitute for the lengths of the sides 1 2 and 3 x3 x2 y3 y 38 xm 1l 12 13 2y I22 4 X3 xl y3 yy 32 x2 x1 02 yl 2 2 2 2 2 2 4CO3 x2 y3 y2 x1 Q3 x1 y3 yl x2 JCCxl x2 yl y2 x3 2 2 2 2 2 2 J4CG3 x2 y3 y2 CG3 x1 y3 yl C1 x22 Cyl y2 D 39 ym 2 2 2 2 2 2 4CO3 x2 y3 y2 yl J x3 x1 y3 yl y2 J x1l x2 Cyl y2 y3 2 2 2 2 2 2 4COG3 x2 y3 y2 C3 x1 v3 y1 CG1 x2 yl y2 Save the expressions in BASIC format as INCIRCLE BAS Steps to be taken in MS Excel Open the file INCIRCLE BAS and do as before SOR must be replaced by SQRT or WURZEL For the radius of the incircle we can use the formula where p p Ip 12 p 13 and A l2 13 2 is half the perimeter of the triangle 1 sje pls e la s 2o o o n fn 20 P 22 23 xM INCIRCLE t 0 0 0785398 0 1570796 0 2356194 0 3141593 0 3926991 0 4712389 0 5497787 0 6283185 0 7068583 ae WURZEL C4 C3 2 D4 D3 2 C2 WURZEL C4 C2 2 D4 D2 2 C3 4WVUI ERE WURZEL C4 C2y 2 D4 D2y2 WURZEL C2 C3Y 2 HD2 D3 2 yM 2 3 3 74999541 2 4218906 0 0622577 7
56. t x x web 3 5 and graph it to get all 3 at once In the function where I used 2 6 try 3 5 or 2 and see the differences that emerge As you get a feel for the situation and the time it takes your machine to do the calculations increase the second arguement of web On my 486 with Derivexm I run 50 or 100 quite easily Now you can immediately plot web 3 5 now without approximating but activate the Option Approximate Before plotting in the 2D Plot Window Message 2885 From WKSARTORY to PUBLIC about COMMANDS BY FILE I am interested in issuing commands such as SIMPLIFY by putting them in a MTH file or some other input file rather than issuing them directly through the command line interface Is there any reference that explains how this can be done lt lt lt Reply see message 2980 gt gt gt Message 2971 From SOFT WAREHOUSE to PUBLIC about DERIVE MEMORY USAGE There has been some discussion recently on this BBS concerning memory usage by the conventional memory version of DERIVE It is a complicated subject but for those that are interested perhaps I can shed some light on it DERIVE is written in muLISP 90 which in turn is written in 8086 assembly language The 8086 microprocessor is a nightmare for implementing pointer based languages like LISP The 20 bit one megabyte address space 1s segmented into small 64K segments and only 4 segment registers are available D N L 13 Selections from the DERIVE BBS p13 The
57. w We have a FAX now Our FAX number is our phone number 43 0 2275 8207 We have a FAX now We have a FAX now We have a FAX now COO O At this occasion a FAX number correction Herbert Appel s FAX number is 0 9264 8700 Germany Sorry Herbert for the mistake D N L 13 Liebes DUG Mitglied ich freue mich wirklich Sie alle wieder in der DUG begr Den zu d rfen Bei dieser Gelegenheit m chte ich mich f r die vielen lieben Briefe und f r Ihre Anerkennung f r unsere Arbeit recht herzlich bedanken Eine kleine Auswahl finden Sie im User Forum Verstehen Sie das bitte nicht als Eigenlob denn Ihre Zustimmung gilt vor allem den recht zahlreichen DERIVIANERN die immer wieder Anregungen und Beitr ge liefern Ein Blick auf die Themenvorschau l t ein interessantes und reichhaltiges DNL Jahr erwarten Leider l t es meine Zeit noch nicht zu den Umfang des DNL zu erweitern Ich habe auch noch einen Nebenberuf Ich m chte besonders auf die zuletzt versprochenen Algebra und Zahlentheoriebeitr ge von Dr Wiesenbauer hinweisen Dr Wiesenbauer hat er freulicherweise sein Versprechen in diesem DNL bereits eingel st Ich kann keinen der vor uns lie genden Beitrage hervorheben jeder ist f r sich ori ginell und einmalig Die Ausz ge aus dem BBS werden fortgesetzt Ein paar Seiten Computerlisting ca 60 liegen noch zur Auswertung bereit Ich kann Ihnen einige interessante elektronische Diskus sionen ank ndig
58. x y 3cosx 420 Enter the expressions 2 2 2 p 3 xl yl 2 x1 4 SIN yl 6 q 3x1 2 x1 yl 3 COS x1 4 and simplify the expressions from above You will get formulas for x and 3 COS x1 4 COS y1 3 x1 3 x1 SIN x1 4 COS y1 i poe a E IA Eis Ae y2 Save these two expressions as NEWRAPH BAS and start MS Excel 3 SIN x1 4 COS y1 3 x1 yl 9 12 2 yl 1 Part of the simplified value for x p28 David Sj strand CAS and Spreadsheets D N L 13 Steps to be taken in MS Excel Open the file NEWRAPH BAS Let x 3 and y 2 in cells X2 and Y2 Enter the expressions for x and yz in cells X2 and Y2 Don t forge to add the leading sign Calculate the sequence of x and y values by copying the contents of range X2 Y2 to the range X2 Y10 We can see that our initial values x 3 and y 2 yield an approximate solution x 4 9543 y 2 8120 If you vary the content of the cells in X1 and Y1 you can find other solutions if there are any copied the cells into the V and W columns EB e4 4 37 1 2 COS VT HT 181 SINOAT 71 52 871 8 and found a second solution B 1 9 391 2 l 1 E 3 2 Then plotted both equations together with 5225159871 2 20326553 6 53694238 3 57925193 the found solution points 0 757 31861 267973572 520941854 3 06177454 048793859 2 70733136 4 99122598 2 83880263 0 485636457 2 7530482 495455164 281238333 048573914 2
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