THE DERIVE - NEWSLETTER #16 USER GROUP
Contents
1. Jose M Cardia Lopes Porto Portugal J Bohm 1993 has recently written in this magazine about the Newton method for solving non linear equations pointing out by means of an interesting example the crucial importance of the choice of the initial values The popularity of this method is due mostly to its quadratic convergence if the initial values are well chosen If it doesn t converge the divergence soon becomes obvious Nevertheless with ill conditioned problems the Newton method as any method can lead to a large number of iterations and to not very good solutions A particular class off ill conditioned problems which often appear in the classroom is the prob lem of solving polynomial equations with multiple roots In fact it can be demonstrated that if we have an equation f x 0 with a multiple root X if f x is the computer representation of f x with the perturbations due to the ill condition if A is an upper bound on the perturbations in f x and if X is the corresponding root of f x 0 then t IX X 7x Am FX where m is the multiplicity of the root With this expression we can evaluate the effect of a perturba tion in f x on the determination of the multiple root of the equation As a result of the power 1 m the determination of multiple roots may be an ill conditioned problem Since the ill conditioning is a consequence of the problem structure we can only avoid it if we change the problem2. EDITOR pl Dear DUG Members Producing the first issues of the DNL I thought that it would be difficult or impossible to write a Letter of the Editor for each DNL On the contrary I ve problems to bring all the additional facts I want to tell you into one column In this DNL I d like to draw your attention to Sergey Biryukov s utility You can find the numerous files together with two demos in the folder lt LABEL gt It must have been an enormous work to design and edit the font sets Sergey and his team in front of the curtain It is a great pleasure for me to see that many of you do not only browse the DNL to find special rai sins for their own purpose but are dealing seri ously with the offered items You can find one result in the User Forum two others are so volumi nous that they are worth an own contribution I must admit that I have to make good something I don t know why but I have forgotten to place G nter Scheu s 2 DERIVE book on our Book Shelf until now I want to point especially to the fact that a couple of physics applications can be found in this book Satisfied Gunter Please take notice of the information enclosed Concerning the DERIVE Journal I would like to remind you that as a DUG member you benefit from a considerable reduction Now I have one good and one bad news for you Let me take the bad one first We will raise the DUG Membership fee up to AS 340 for Austria DM 52 or B
3. 3 41 3 412 3 414 3 416 3 418 3 12 10 30679813 10 30679185 10 30678803 10 30678668 10 3067878 3 122 10 30679055 10 30678457 10 30678106 10 30678002 10 30678146 3 124 10 3067845 10 30677882 10 30677562 10 3067749 10 30678146 3 126 10 30677998 10 30677461 10 30677173 10 30677131 10 30677338 3 128 10 306777 10 30677194 10 30676936 10 30676927 10 30677165 Then proceed as shown above D N L 16 Heinz Rainer Geyer Der Fermat Punkt p21 II Differentiation der Abstandssummenfunktion Differentiation of the distance sum function F r die Sch ler der Klasse 11 lag es nahe den Extrempunkt der Abstandssummenfunktion L x y ber die Ableitung zu suchen Als Funktion zweier Ver nderlicher entzieht sie sich allerdings dem f r Sch ler bekannten L sungsverfahren Einer der Sch ler hatte hier die Idee die partiellen Ableitungen von L x y als Funktion der einen Variablen bei konstantem Wert der zweiten Variablen zu plotten Hierzu definierte er einen Vektor dessen Komponenten jeweils die so bestimmten partiellen Ableitun gen waren Die dargestellten Graphen f hren zwar nicht direkt zu den Koordinaten des gesuchten Punktes zeigen aber auff lliges Verhalten in der N he der Koordinaten des Fermat Punktes und insbesondere an den Grenzen der durch die Dreieckspunkte bestimmten Definitionsbereiche The pupils of form 11 tried to find the extremal pont of the distance sum function using the derivative Being a function of two variables it exceeds
4. MOD x 0 2 0 8 0 2 FA 1 6 1 4 1 2 1 0 8 0 6 0 4 0 2 Doe ee We Ue dee ee dee ie ae How to automate the process In lectures on Fourier analysis students construct a function f t by summing harmonics When the original function 1s T periodic they can match it with their own constructed func tion To avoid unnecessary arithmetic one could automate the process by defining the next function PERIODIC f t a b 11m f t MOD t a b a a From now on it s rather easy to use PERIODIC SIN t t L 2 1 p36 J C M Verhoosel T periodic Functions D N L 16 PERIODIC t t 0 5 1 5 4 3 2 l rp A q4 5 PERIODIC EXP t t 0 3 rn We have now the possibility to compare a Fourier approximation of f x with f x itself as you can see below 5 PEE i yl SE Dee a yl dt 225 2 PERIODIC F t t L 2 5 FOURIER F t t L 2 5 5 Conclusion DERIVE can be a useful tool for plotting T periodic functions Although it is possible to define and use T periodic functions with CHI and SUM they can t be plotted The use of MOD x T is simpler and more elegant Nevertheless the process of defining true T periodic functions with CHI and SUM and the use of transformations is very learningful The use of PERIODIC appears to be very useful in the classroom D N L 16 J M C Lopes Ill Conditioned Problems p37 DERIVE Newton Method and Ill Conditioned Problems
5. 2 0 0 eis 17 seit ASh 0 3 0 4 0 3 0 12 8 0 81 ta 10 3 IT JT 18 ser 2 5 2 0 2 Oe tee Cie Oo 2 31 aS eS 5 19 SEIETZEABELLEABELT L5 aS Miara S 28 0 a 10s Os 20 SHIFT_LABELCLABEL 1 5 2 5 1 2 1 1 5 8 a 0 0 0 05 07 21 BOXC L 5 3 5 5 5 3 511 As you can see in the following figures Sergey s tools are working with DERIVE 6 in the same way ar VAII i IG Di Ad p16 Biryukov 2D Plots Labeling D N L 16 Some Additional info about 2D Plot Labeling Utility 1 The Utility needs DERIVE 2 04 or later 2 Its main programming ideas FLOOR function emulation When using modern DERIVE versions you have to exclude FLOOR amp MOD definitions and replace TRUNC by FLOOR amp A_V by APPEND Strings comparison and search in a vector Complex structured arguments that simplifies using defined functions and carries the idea of structuring not only functions but data also 3 There are 3 additional useful functions in LABEL MTH not described in the paper APPROW p1 p2 1 draws an arrow from point p to point p2 with arrow sides length BOX p1 p2 draws a rectangle with lower left corner p and upper right corner p2 SHIFT_LABEL v sxy shifts a vector of broken lines v letters or curves by sxy x y 4 In LABEL_FD MTH the functions INSERT_ELEMENT v i a REPLACE_EL v i a and REPETITIONS a v improves the VECTOR MTH utility file The first is a complementa
6. Cost en B RAD AUTO REAL p28 Thomas Weth A Lexicon of Curves 5 D N L 16 Verwendung der Konchoide zur W rfelverdoppelung Doubling a die Zu konstruieren ist die Kantenlange x eines W rfels der den doppelten Inhalt besitzt wie ein W rfel mit der Kantenl nge a also xX 2a oderx a V2 Dazu zeichnet man eine Konchoide mit K a und zeichnet durch U unter 60 eine Gerade zur x Achse die den rechten Kurvenzweig in S schnei det Die Gerade OS schneidet die Leitkurve C dann im Punkt T x OT ist dann die gesuchte W rfel i kante mitx 2a 4 OT x is the edge of the cube withx 2a AE 3 00 0 N Nach dem Strahlenlsatz ist n mlich a x a 2a y bzw x y 2a D Mit dem Lot von O auf die Gerade US ist das Dreieck OSL rechtwinkelig und es gilt zunachst LU a cos 60 a 2 Damit folgt fiir das Dreieck AOSL x a b z OL und f r das Dreieck AOLU a T OL 2 Subtraktion liefert x a 7 y bzw x 2ax y ay I und II vereinfacht man durch Elimination von y zu x 2a You can simplify I and II by eliminating y and you obtain x 2a Verwendung der Konchoide zur Drittelung des Winkels How to use the conchoid to trisect an angle Ist a der zu drittelnde Winkel so tr gt man auf dem einen Schenkel eine beliebige Strecke OU mit der L nge a ab Das Lot in U auf die Gerade OU schneidet den anderen Schenkel im Punkt N Mit ON c zeic
7. bw by ex cy Lx yh 10 as Solving y G and y H for ix gives the co ordinates of the Fermat point Now let s try with our special values D N L 16 Heinz Rainer Geyer Der Fermat Punkt p23 SFL 2675 3 3229 43 1973 il eR 2 2 a a er re oo 1226 36 78 36 78 1225 12 f_ptto DO 7 2 3 5 3 41483525 3 129905399 We repeat the calculation for the Nspire Triangle from page 19 14 2 844703633 2 550873696 15 triangle 0 7 2 3 5 cr 8 00 5 00 7 The Cabri plot with the rotated triangle ATB which is used for the proof Die Darstellungen zeigen deutlich da das Problem sicher nicht abschlie end behandelt wurde Es gibt noch viele M glichkeiten zur Verfeinerung zB beim Problem der Ableitungen oder bei der Schnittpunktsberechnung Ich hoffe aber da deutlich wurde wie ergiebig dieses Fermat Problem f r den Mathematikunterricht in vielen Klassenstufen ist und wie hervorragend sich dabei ein Programm wie DERIVE einsetzen l t Der experimentelle Charakter der mathematischen Arbeit in unserem Projekt bewirkte jedenfalls einen erkennbaren Motivationsschub der dem Mathematikunterricht ins gesamt bestimmt zu gute kommt The presentations are showing very clear that we couldn t deal with the problem suffi ciently There are many possibilities for refinement But hope that could show how produc tive this problem could be for many forms and how excellent you can use DERIVE for thi
8. MOD a b Because MOD x a already is a periodic function it offers more perspectives Moreover it is not necessary for a to be an integer for instance MOD 5 2 1 will yield 0 8 For students this function often gives surprises It s not automatically clear for them why the plot of x MOD x or MOD x 2 looks so strange These plots invite the students to discuss simplifications in DERIVE involving MOD x Can we understand what is happening here What is the connection between FLOOR x and MOD x After some experimenting in DERIVE they find it rather easy to understand that MOD x x FLOOR x and x MOD x simplifies tox 2 x FLOOR x p32 J C M Verhoosel T periodic Functions D N L 16 f En F u EA D plot 1 1 mm 2D plot 1 2 R 2D plot 1 3 tC BE 2p plot 1 5 MODX x FLOOR x EX FE 2p plot 1 4 BEIE 1 M DtX 2 x FLOORGx 3 x MOOCx x 2 FLOOR Cx 4 x FLOOR x 5 FLOOR x MOD x 23 5 Figure 2 The usage of MOD x in plotting periodic functions To understand how MOD x y can be used to plot T periodic functions we must understand how MOD x y works see window 4 of figure 2 for a plot of MOD x MOD x can be used to make sure that for instance f 3 2 will be reduced to f 0 2 This means that f x can be periodic the part on the domain 0 1 is repeated MOD x can be used to draw other random composed periodic functions MOD x has period 1 every x is
9. PC using Microsoft Windows for memory management With 16 megabytes of RAM it can compute n to 524 200 decimal places With 32 megabytes of RAM it can compute x to 1 048 500 decimal places PiW uses a Fast Fourier Transform FFT to speed up multiplications It is written in Borland C for Windows All source code is included Also includes a copy of rn to 500 000 places that was computed in 1 55 days using a 33 MHz 1486 It is a good example of object oriented programming OOP re to 500 000 places is in file PiW500K Zip Every thing else is in file PIW 100 Z p Harry Message 3288 From CHRIS LAMOUSIN to JERRY GLYNN about PLOTTING I m having trouble plotting a function with Derive I m still figuring the program out The function is y x 3 2 x 2 3 I can get f x for x gt 0 but not for x lt 0 The program is set for all real numbers Any Ideas Message 3289 From JERRY GLYNN to CHRIS LAMOUSIN about 3288 PLOTTING Yes I know what you need Do Manage Branch Real and replot Derive thinks correctly that 8 2 3 is a complex number However this is an uncomfortable viewpoint in Calculus or before so Derive has this flexibil ity You ll probably want to leave the setting this way so do Transfer Save State In commonly asked questions at the back of your manual you ll see this discussed Message 3303 From JERRY GLYNN to PUBLIC about INTELLIGENCE In all symbolic algebra programs I ve tried 2 3001 2 3000 has produced a correct a
10. Wurmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef B hm Herstellung Selbstverlag DERIVE Alfonso J Poblaci n Valladolid Spain I send you a parametric plot with DERIVE It seems to be a tridimensional net that was found by chance by Stanley S Miller Concord Massachusetts I found it in Scientific American Ist equations are SEN LOO Oat COSTS OE COS LO as DEE SENSE It is not known if it is closed cyclic or infinite This plot is made with t e 300 300 but you can increase these values without finishing I tried up to 1000 1000 Making little variations in the coef i Sas i et ar ren Tg Me ia i heat tare Me ee 1 LSINCO 99 t 0 7 005 3 01 t COS l Ol t 0 1 SIN15 03 t 2 LSINGCL a t 6b C05 23 01 t COSCCL a t ce SIN 15 03 3 Glyn D Williams Gwynedd Wales I recently upgraded to DERIVE 3 0XM to make better use of the extended memory on my computer and to take advantage of the extra facilities on DERIVE 3 0 In the pre release notes a copy of which was given to me at the conference at Plymouth the facility of changing the menu to suit one s own needs is mentioned Unfortunately the manual does not explain how to do this and I would like to do this because I feel that the default menu is in parts badly thought out with commands like Save which are likely to be needed frequently on a lower menu and approX having an aw
11. eu mth English Upper case letters E Graeme lO shies O 46 68 4 wal gO go hy Oe Oe a gms gamma_low fi_low omega_low delta_up D N L 16 Biryukov 2D Plots Labeling p17 See first the font sets from 1994 and then the changed sets adapted for DERIVE 6 mer mth oo y o y g y HA y er j B mee y j gt j y en y a mie ea m Eee eS Se a i at ee a ee eS we ee a Oe eg f nO mth 3 g l a N myn WATT VUE g lI lI i as ee 1 LOADED DFD DNL DNL94 MTHLS LABEL LABEL MTH 2 LOADED DFD DNL DNL944 MTHLS LABEL F_MEF MTH 3 LABEL 4 051 7 005 el release nt die pe 4 LABELI 2 L 9 3 0 5 04 0 05 er ave 8 0 amp gt 53 3 af antl don appr 3 LOADED DFDYDNL DNLSAMTHLSYLABELYAF_MF2 MTH 6 LABEL 2 0 005 0 25 G4 1000 3 0 bose a aie nie 227 boxe Bf LOADED DTD DNL DNL94 MTHL64 LABEL F_NO MTH 5 LABEL Aa 9d el ec aa a Seems rae a Dam re a oe Pt ee The next sets are still valid f s mth pi double Sitz ge eg pe r Se e 2 MO LE a ae og ae ee mO IN non Bon W e a u i ai a nen we ee a Ar PB ee 2 Eg were ee ae see A re IM N 5 a Oar we we wei Sn use ie Ww ar ae a 4 f t mth i double quotes Mae N wey 7 LAJ Gare a re ae a ae mg ae 2 Ma MPN ri ry oar DIN a NEON on a a non en ae ue Ma Pha T mar na nn a It is important that the two lists vectors forming the fon
12. needs some knowledge experience skills and phantasy in manipulating trig expressions Josef What is necessary to know sin x cos y sin x y sin y x 1 sin Z x COS x 2 ine lt and cys 3 2i 2 a b a b a a b ab b 4 cos 2x 2cos x 1 5 cos 3x 4cos x 3cos x 6 2sin cos sin sin 1 Ie DE ne OTE 37 7 3 4 2 sin 5 sin 7 r sin 3 a os 77 2 Ari Ari 22 42 e4 e 4 Ft E4 E E E E E E E E cos sin ee ee 14 14 2i 2i 2i 3ri 3ri ri ri E E E EF el4 e 4 en el eS Se 0 M E E Mum m _ e 4 e 4 3 4 2i 2i 2i 2i K i G4 sin 72 sin Z 2e0s Z 14 14 14 37 2 cos 4cos 3 cos I u sin 2 sin Z SB agg ue 2 cos 2cos 42 5 6 14 14 200 Z 1 2 c0 sZ cos E qued We continue with Hadud s mail More generally consider oe oe with n k both odd Letting a and proceeding as above we get 2n 2n 2n cos ka sin n k 1 a sin n k 3 a sina oe p10 DERIVE Bulletin Board Service If k is fixed at 3 we have the identity l _ T _ _ z _ et Be sin n 4 a sin n 6 a sin n 8 a sina cos 2a with a T By interchanging the roles of sines and cosines corresponding formulas may be found for cosine terms on the LHS Message 3265 From HARALD LANG to SWH about 3256 MYSTERIOUS TRIG IDENTITY Perhaps the
13. not using DERIVE XM posting the last 40 of 500 000 digits of on this BBS so I can confirm that the DERIVE result is correct Please include your source Even on a 33mHz 486 with 8Mb of memory I was unable to get Maple V Release 2 to compute more than about 100 000 digits of n before it exhausted memory I would be interested to know if anyone can get Maple or Mathematica to compute 500 000 digits of n and how much time and memory it takes Aloha Al Rich Soft Warehouse Inc Message 3250 From MICHAELWALSH to SOFT WAREHOUSE about 3247 500 000 DIGITS OF PI Hi Al I ve been involved in a discussion about computing r on Prodigy I know that 7 4 1 1 3 1 5 1 7 1 9 this is a Taylor expansion of arcsin 1 But this alternating se ries doesn t converge very rapidly My question is what algorithm did you use to compute 0 5M digits of 1 Thanks for your help I ve always been interested in number theory Mike Detroit Message 3255 From SOFT WAREHOUSE to MICHAELWALSH about 3250 500 000 DIGITS OF PI The algorithm DERIVE uses to compute 7 is based on Ramanujan s formula published in the Quarterly Journal of Pure and Applied Math Volume 45 page 350 1914 The formula is that 4 1 equals the sum of the terms Him sb UA Stee den BETZ Ce Br as m goes from 0 to infinity DERIVE then uses some tricks to compute the series efficiently I am still waiting for someone to post on this BBS the last 40 of 500 000 digits of n
14. so I can verify DERIVE XM s result and sleep better at night Aloha Al Rich Soft Warehouse Inc Message 3256 From SOFT WAREHOUSE to PUBLIC about MYSTERIOUS TRIG IDENTITY In response to a user inquiry I stumbled across the following trig identity 2 SN 1 14 COS ry 7 COS T 1 2 or equivalently SIN 317114 SIN 7n 14 COSTI 172 It looks like it should be easy to derive using the various rules for simplifying trig products half angles etc However I have been stumped at finding a derivation If there is a generalization of this identity it could be incorporated into DERIVE s trig simplifier Then expres sions on the LHS of the identity could be transformed to the obviously simpler equivalent expressions on the RHS However the first step to finding such a generalization is to derive the above identity I would greatly appreci ate if someone could provide a derivation and or generalization of this identity Aloha Al Rich SWH D N L 16 DERIVE Bulletin Board Service p 9 Message 3263 From HADUD to SOFT WAREHOUSE about TRIG IDENTITY Your formula can be derived from the trivial equality sin 7 3 cos 22 l ur DE E E E For brevity let E e 4 Then the LHS may be written ae een 2i eo 37 T 14 m 3 2x 1l Th sin 2cos E D us sins um A cos 14 3 cos 14 7 Q Ta cannot believe that this is so clear for the average student In my opinion it
15. structure For a polynomial equation we can substitute the evaluation of the multi ple root of Kx 9 by the evaluation of the single root of u x 0 where fo t x see for example Chapra e Canale 1990 This is another situation in which DERIVE can be a very good pedagogical tool We can graphically show how the structural change of the problem improves dramatically the path to the root We can see some examples Example 1 f x x 3 2x 5 Derive has no difficulty to solve this equation but the graphical representation suggests that the New ton method converges on the multiple root x 3 at a very slow rate notice that the function does not cross the x axis at x 3 The situation is very different with u x p38 J M C Lopes Ill Conditioned Problems D N L 16 Example 1 Ul ul 5 1 fixjis fx 3 2 x 5 fix u x single root 2 d a 2 5 3 3 5 4 multiple root Example 2 f x 5x 14 x 3 UK Example 2 i 2 3 fix 5 x 14 ix I fix ux amp 2 8007 2 8004 i ae double root dx lI rJ m Wo wo oo If the lecture is designed for beginners there will probably be some interest in a more detailed approach like the one prposed by Dyer 1993 to show the results of some interations In addition the second example points out the importance of the manipulation of the mathemati cal expressions in order to minimize the number of arithmetic operations
16. the pupils known algorithms One of the students had the idea to plot the partial derivatives of L x y wrt one variable keeping the other one constant The graphs dont lead directly to the point we are looking for but they are showing a very strange behav iour in the neighbourhood of the point and especially at the borders ofthe domain defined by the an gles of the given triangle Partial derivative of Locyi wt x which willbe evaluated for 2 y 4 with an increment of 0 5 d vecron Lx yJ y 2 4 5 dx Partial derivative of Loc yi wit y which will be evaluated for 3 lt x 3 5 with an Increment of 0 1 d vecron Lx vo x 3 3 5 dy Caution The earlier Derive versions were not able to plot implicit functions and they accepted all variables as the independent variable Now we can do implicit plots and the graph of the 2 family has another ap pearance p22 Heinz Rainer Geyer Der Fermat Punkt D N L 16 II Nachbildung des geometrischen Beweises als Schnittproblem zweier Geraden Reproducing the geometric proof as intersection point of two lines Hierzu mu ten die Koordinaten der gedrehten Punkte C und C bestimmt werden Die Sch ler expe rimentierten dazu zun chst mit Tangens und Pythagoras Zur Vereinfachung wurden dann die erfor derlichen Drehungen nur auf Drehungen um den Ursprung Punkt A beschr nkt Au erdem stellte ich den Sch lern die Matrix einer Drehung um den Ursprun
17. 1 2 etc The next step is to define this function in terms of SUM Apparently it s not a problem when the number of terms is finite On 0 1 we get 4 CHI 0 x 1 x 0 5 2 The quadratic function defined on 1 2 would be f x 4 CHI 1 x 2 x 1 5 2 the one on k A 1 4 CHI k x k 1 x 0 5 k 2 So when we take the first 5 terms we get 5 f x 2 4 CHI k x k 1 x 0 5 k k 0 This function isn t a true periodic function To produce a true periodic function f x must become symmetric and extended to infinity and negative infinity So let us define g x g x gt 4CHI k x k 1 x 0 5 k k 0 Although DERIVE is capable of plotting f x it accepts g x but seems not capable of plotting it Probably DERIVE tries to evaluate the function and can t handle infinity Using IF What about the IF function We can use this function to make a piecewise defined function for in stance f x IF x lt 0 OR x gt 1 x x A short explanation When x lt 0 or x gt 1 then f x x everywhere else f x x Another example g x IF x lt 1 AND x lt 2 x When we plot g x we can see that outside 1 2 g x is not forced to 0 because the function is not defined there Nevertheless we have the same problems with the IF command as those we encountered with the CHI functions it is not possible to plot true piecewise defined functions although it is possible to de fine them Using
18. 10 30679185 10 30678803 10 30678668 10 3067878 10 30679055 10 30678457 10 30678106 10 30678002 1 30678146 9 10 3067845 10 30677882 10 30677562 10 3067749 10 30677665 10 30677998 10 30677461 10 30677173 10 30677131 10 30677338 10 306777 10 30677194 10 30676936 10 30676927 10 30677165 10 FERM 3 415 3 127 0 001 10 30676992 10 3067701 10 3067709 10 30677232 10 30677436 10 30676901 10 30676927 10 30677015 10 30677165 10 30677377 11 10 30676848 10 30676882 10 30676978 10 30677136 10 30677356 10 30676834 10 30676876 10 3067698 10 30677146 10 30677373 10 30676858 10 30676908 10 3067702 10 30677194 10 3067743 Value of the distance sum function 12 L 3 415 3 13 10 30676834 Die beiden Beispiele aus dem DERIVE Programm zeigen da die Auswertung der Matrizen zwar schwierig weil un bersichtlich ist aber die Lage des Fermat Punktes l t sich so doch gut einen gen Both examples show that the evaluation of the matrices is difficult because not easy to sur vey but we can find the position of the Fermat point approximatively The numerical approach would be easier to follow if we could include the x and y values as well in the table This is possible with little manipulating the matrix and the respective function could be pro vided as a tool by the teacher Producing a table including the x and y values 13 values s t h APPENDCLAPPEND VECTOR y y t t 4h h FERM s t h 14 values 3 41 3 12 0 002
19. 3x 14 c G x 4x 6x 18x 20 d G x 10 18x 12x 16x e G x 2x 9x 3x f G x 16x 12x 18x 10 4 Draw the graph of F x sin x V3 1 a G x 2 sin x cos x b G x gt Sin x 5 COSC c G x cos d G x V2 sin x V2 cos x e G x sin x cos x f G x 1 3 sin x 4 sin x All the trig functions in the above are in radians not degrees Hint Use the well known fractional multiples of rn 5 Is it always possible to map a quadratic function to another quadratic function by a series of transformations Experiment with your own quadratic equations Can you prove any of your findings Give some examples What is the maximum number of different single transformations that 1s required p40 Keith Eames Functions Transformations D N L 16 Answers 1 Use a trial and error method to find possible answers Simplify F x 1 Compare the given answers with G x a F x b F x c F x d F x 3 e 3 F x f F 2x 2 If Derive does not give the answer shown Expand the expressions and compare a F x 9 b F x c F x 3 d 2F x e F 3x f F x 2 5 3 a F x 10 b F x 1 c 2 F x d F 2x e F x 2 4 f F 2x 20 4 a F 2x b Fx n 6 F 2x d 2 F x 7 4 e V2F x n 4 f F 3x 1 5 F x ax bx c can be transformed to G x rx sx t by at most 4 transformations this depends on the values of a and r Ifr and a are both posit
20. 4 On a LAN it is usually preferable to start DERIVE using a DOS batch file In addition to starting DERIVE the batch file can automatically switch to the desired working directory and or redirect printer output to the desired network printer The batch file should be named DERIVE BAT and saved on the file server in the standard batch file directory or any directory to which a command file search path has been established For example if the file server 1s drive F the batch file ECHO OFF F DERIVE DERIVE EXE 1 starts DERIVE using the current DOS drive and directory as the working directory If installing DERIVE XM change the last line of the batch file to read F DERIVEXM DERIVE EXE 1 Alternatively the batch file ECHO OFF C CD C DERIVE F DERIVE DERIVE EXE 1 makes the DERIVE directory on drive C the working directory Finally the batch file D N L 16 DERIVE Bulletin Board Service p 7 ECHO OFF F CD F WORKING F DERIVE DERIVE EXE 1 makes the WORKING directory on the file server the working directory 5 If you want to redirect DERIVE printer output to a network printer you can include the appropriate printer redirection command before the DERIVE EXE line in DERIVE BAT see your LAN manuals for details For example if you are running a Novell LAN and the Netware program CAPTURE EXE is in the PUBLIC directory on drive F the command line F PUBLIC CAPTURE L 1 Q name NT NFF TI 5 redirects DERIVE pri
21. H 2 LOADED OF0 0NL 0NLS4 MTH16 LABEL LABEL MTH 3 numb x LL ei EF 6 4 Ee 2 0 2 4 6 4 numb y LL 6 4 el bale 2 4 5 5 6 2 8 5 6 2 0 3 0 4 numb_x o 5 0 8 X 5 aa ALL_AXES s 0 0 0 3 0 4 numby 0 5 0 8 Y fi TT 5 las LABEL 1 22 dee a ana I oO T Ea a Sl 4 7 aa la y x 0 75 Don t forget to enter all the characters under quotes eg numb_x n Q s ma ors Nee 2 ae a o wa A 6 D N L 16 Biryukov 2D Plots Labeling p15 6 COMMAND Center Delete Help Move Options Plot Quit Range Scale Transfer Window aXes Zoom n Enter optio ross x 1 y 2 Derive ZD plot Screenshots from original DNL 16 December 1994 DUG for YOU and the arrows are added by me then tried to create a New Year s screen Josef 8 LOADED DFDYDNL DNLSA MTHLS LABELYFLE MTH 5 LABELC 8 3 0 5 1 0 6 0 1 D U 6 fro rr Y 0 U 10 SHIFT_LABELCLABELCL 8 3 0 5 1 0 6 0 1 D US f o r 0 UD 0 1 0 17 11 ARROW CCO 0 7 2 8 0 53 ARROWC O 0 4 7 3 2 0 53 ARROWC O O 2 3 3 7 0 51 12 LOADCD DFDONL DNLS4 MTH16 LABEL F_E MTH 13 LABEL 3 5 2 5 0 6 2 0 8 UO 14 7 fy ay 088 14 LABELE 3 0 5 0 8 1 1 0 N e w 2 a ri 15 LOADCD DFD ONL DNLS4 MTH16 LABEL F_NUMB3 MTH 18 LABELC 1 5 2 5 a2 2 2255 0
22. P 22 for Europe and US 40 for other destinations But here is the hopely good news the contents of the DNL will be extended overpro portionally to at least 40 pages instead of 30 Please take this fact as a call for further contribu tions I must confirm that you have been very active and reliable until now No contribution large or small 1s getting lost If one or another will not be published for a longer time then this is caused only by technical reasons amount of space subject not suited for the issue translation For 1995 I ve in mind to dedicate one issue to geometry and another one shall deal mainly with activities in Austrian classrooms Please pay attention to the enclosed invoice for the renewal of your membership You will automati cally receive a receipt together with DNL 18 My wife Noor and I wish you all personally known or not a Merry Christmas and a successful New Year 1995 Sincerely yours p 2 E D I T OR IAL The DERIVE NEWSLETTER is the Bulle tin of the DERIVE User Group It is pub lished at least four times a year with a con tents 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 possibilities of new methodical and didactical manners in teaching mathematics Editor Mag Josef Bohm A 3042 W rmla D Lust 1 Austria Phone 43 0 2275 8207 D N L 16 Contributions Plea
23. See the plots of the factor ized and the expanded function There is no difference now in higher versions of DERIVE Josef subtractive cancelation 2 5x 14 x 3 3 ya 140 x 271x 420 x SBA Scale x 5 104 y 5 10 5 Accuracy 9 Precision Exact Screen Shot from the 1994 DERIVE for DOS version References Bohm J 1993 Newton Raphson s Chaos The DERIVE Newsletter 12 p 10 13 Chapra S C amp Canale R P 1990 Numerical Methods for Engineeers 2 ed Mc Graw Hill Dyer D 1993 The Bisection Method and DERIVE The DERIVE Newsletter 11 p 11 13 D N L 16 Keith Eames Functions Transformations p39 Find here a worksheet of Keith Eames Exercise Use the graphical capability of Derive to answer these questions Check your answers age braically using Derive Explain how you can get form F x by a single transformation whenever possible to each G x n all of the following questions 1 Draw the graph of F x x 2x 3 a G x x 2x 3 b G x x 2x 3 c G x x 2x 3 d G x x 4x 6 e G x 3x 6x 9 c G x 4x 4x 3 2 Draw the graph of F x 2x 1 3x 5 a G x 6x2 7x 4 b G x 6x 7x 5 c G x 6x 29x 28 d G x 10 14x 12x e G x 54x 21x 5 f G x 6x 31x 28 3 Draw the graph of F x 2x 3x 9x 10 a G x 2x 3x 9x b G x 2x 9x7
24. THE DERIVE NEWSLETTER 16 ISSN 1990 7079 THE BULLETIN OF THE USER GROUP Contents Letter of the Editor Editorial Preview DERIVE User Forum Bulletin Board Service Sergey Biryukov 2D Plots Labeling Heinz Rainer Geyer The Fermat Point in a Triangle Thomas Weth A Lexicon of Curves 5 The Conchoid J C M Verhoosel DERIVE and Plotting T periodic Functions J M Cardia Lopes Il Conditioned Problems Keith Eames Functions Transformations A Worksheet revised Version 2008 September 1994 D N L 16 INFORMATION Book Shelf D N L 16 1 DERIVE im Mathematik und Physikunterricht Ginter Scheu F Dummler Verlag Bonn D mmlerbuch 4592 1994 2 Mathematik am PC Einf hrung in DERIVE Bernhard Kutzler Soft Warehouse Hagenberg 1994 3 Mathematics on the PC Bernhard Kutzler Soft Warehouse Hagenberg 1994 First North American DERIVE User Group Meeting Sunday November 20 1994 9 00 13 00 Room Europe 10 Walt Disney World Dolphin Lake Buena Vista Florida USA Report by Bernhard Kutzler The first North American DERIVE User Group Meeting was organized in the frame of the International Conference on Teaching Collegiate Mathematics ICTCM Nov 17 20 1994 Despite the early Sunday morning time at the end of the conference and despite of the temp tation of being in the middle of Disney World with its many fun parks approximately 30 peo ple attended the meeting among them Bert Waits o
25. ant and characters step in a text can be defined Axes with arbitrary point of their cross numbers near ticks and labels near arrows can be plotted One character plotting CHAR_ xy wh ch rot is the base function of the utility It prepares a set of points for character ch plotting in Options gt Display gt Points gt Connected and Small mode xy x_coordinate y_coordinate lower left point of the character rectangle and wh width height defines rectangle width and height of this rectangle Optional scalar parameter rot defines character rotation counterclockwise in radians It is zero by default It is necessary to preload a vector font see next paragraph to plot characters CHAR_ 1 2 3 4 A 0 for example simplifies to 4 2 3453355 15 25 61 2 5 3 515 15215 1 55 355 15135 5 31 and gives a 3 units width and 4 units height upper case letter A The lower left corner of the letter is at the point 1 2 Sergey Biryukov 2D Plots Labeling Vector Fonts Characters in the utility are divided into several groups and combined in different combinations in 5 font files Tiny t mth digits arithmetic operations and letters x y amp z Small f s mth Tiny English upper case letters Small Russian sr mth Small Russian upper case letters Full mth all characters except Russian letters and Russian Full Font fr mth Full upper amp lower case Russian letters All characters available
26. che und Transzendente Kurven 5 Thomas Weth W rzburg Germany Die Hundekurve Die Konchoide des Nikomedes Die n dieser Folge behandelten Kurven s nd Erweiterungen der sog Hundekurve oder Tractrices Tractrices entste hen wenn ein Herrchen seinen Hund auf einem geraden Weg f hrt w hrend der Hund bei konstanter Leinenl nge sei nem Lieblingsbaum zustrebt Mathematisch formuliert handelt es sich bei dem Weg des Hundes um eine spezielle Form von Konchoiden The curves dealt with in in this sequel are extension of the so called Dog s curve or Tractrices They are created when a man walks along a line while his dog kept on a lead with constant length wants to visit his favourite tree From the mathematical point of view the dog s way is a special form of a conchoid Erzeugung von Konchoiden How to create a conchoid Konchoiden im allgemeinen ergeben sich nach folgender Konstruktionsvorschrift Gegeben ist eine ebene Kurve C und ein Punkt O in ihrer Ebene Von einem Punkt Q der Kurve aus tr gt man auf der Geraden OQ in die beiden m glichen Richtungen jeweils eine Strecke konstanter Lange K ab die Endpunkte P und P dieser Strecken sind dann Konchoidenpunkte zur Kurve C This is the instruction how to obtain a conchoid given is a plane curve and a point O in its plane Have any point QeC The two points P and P on the line OQ with QP QP K const are points of a con
27. choid to C Ifthe curve C is a line the conchoids are especially simple they are tractrices Niko medes 200 BC invented this curve for doubling a die and for trisecting an angle Its name is derived from the greek word for shell x yxn Die Konchoide des Nikomedes Herleitung der algebraischen Kurvengleichung Besonders einfache Konchoiden ergeben sich wenn die Leitkurve C eine Gerade ist wie im Beispiel der Tractrix Diese Kurven wurden von Nikomedes ca 200 v Chr zur W rfelverdoppelung und zur Dreiteilung des Winkels erfunden Der Name ist wegen der Kurvenform abgeleitet vom griechischen k yxn Muschel F r die Kurvenpunkte P ergibt sich in Polarform vgl obige Zeichnung p26 Thomas Weth A Lexicon of Curves 5 D N L 16 a P r 1 2 K oder o K 0 und daraus f r die gesamte Kurve cos o cos o 2 r K r _ K 0 oder kurz r _ K COS o COS o COS Polargleichung der Konchoide des Nikomedes this is the polar form of Nikomedes con choid Now you easily can find the Cartesian form Den bergang zu kartesischen Koordinaten erh lt man mit x r cos und y r sin 2 r K Ersetzt man nun noch r x y DERIVE und faktorisiert die Summe so x erhalt man die algebraische Kurvengleichung 2 2 2 _ 2 2 x y x a Kx Aus der Kurvengleichung l t sich nun entnehmen dass die Kurven f r verschiedene K e symmetrisch zur x Achse s nd e
28. congratulations are in order D N L 16 DERIVE Bulletin Board Service p11 Message 3284 From HARRY SMITH to SOFT WAREHOUSE about 500 000 DIGITS OF PI Al I know your 500000 digits of m has been confirmed but I thought you might be interested in my program for computing 71 computed accurate to 500 000 decimal digits has the following as its last 60 digits 96959399375495362232222 19746596193325290740424876025 13819524 The next 26 digits are 2697391017563719753430045 This was computed by the program PiW Compute PI to a million or so decimal places in Windows Version 1 00 last revised 1992 12 16 0600 hours Copyright c 1981 1992 by author Harry J Smith 19628 Via Monte Dr Saratoga CA 95070 All rights reserved I used PiW to compute r to 500 000 decimal digits This was done on an IBM AT compatible 33 MHz 486 DX computer using Windows 3 1 with 16 megabytes of RAM and 22 mega bytes of virtual memory It took 37 3 hours for algorithm a and 31 0 hours for algorithm b The results were the same The divide and square root routines have been improved since these runs so the program is a little faster now 10 to 20 If I upload the program it will be in the two files PiW100 Zip and PiWS500K Zip Its online description is PiW v1 00 By Harry J Smith Computes Pi to a million decimal places Key words Pi Math Windows C Precision FFT PiW is a program to compute rn to a million or so decimal places on an IBM compatible
29. die Leitgerade C mit der Gleichung x a zur Asymptote haben und e im Ursprung einen evtl isolierter Kurvenpunkten haben From the curve s equation you can see that for different values for K the curves e are symmetric in respect to the x axis e have line C with x a as an asymptote and e the origin is a eventually isolated point of the curve F r die F lle K lt a K a und K gt a ergeben sich die drei Kurvenformen DERIVE 6 has two remarkable features which we can use to reproduce the construction of this curve the slider bars and the possibility to switch between rectangle coordinates and polar coordinates in one session This one more occasion to narrow the gap between com puter algebra and dynamic geometry VIl try to explain the procedure very briefly guided by the following screen shots Josef D N L 16 Thomas Weth A Lexicon of Curves 5 p27 start in Rectangular coordinates Flot 1 and 2 and introduce slider bars for a 0 a s 10 and p_I mM2 p_ lt m2 1 xo a 2 x TAN Cp Moving the slider bars you can change a and see the rotating line Change now to polar coordinates Introduce a third slider bar fork 0 ks 10 Plotting 3 you can move the points on the wo branches of the curve exchanging p_ by p gives the polar form of the curve and so does the implicit farm in 4 You can alter a k and p_ as you like da ee E COS po da f Be COS p_ a fa fu 4 _ ik
30. e figure s Josef 2 99 2 592 R d4 15 EG AUTO FUNC AT 2 85682 u PA AT 3 82416 u BT 5 77972 1 Pa BT 4 73466 u CT 3 62119 u gt OAN CT 3 40821 u 7 33991 1 08423 7 33991 1 08423 FP 2 8472 2 55365 T 1 69019 2 30318 Totsum 12 2577 Totsum 11 967 Zur analytischen Bestimmung von F wurde vereinbart das Dreieck in einem kartesischen Koordina tensystem mit A 0 0 zu beschreiben Die Sch ler kamen nun selbst ndig je nach Kenntnisstand auf folgende L sungsans tze I Berechnung des numerischen Abstandswertes als Funktion des Punktes F x y und Minimieren durch sukzessive Einengung der Definitionsbereiche f r x und y Calculation of the numerical value as a function of the point F x y and minimization by a step by step restriction of the domains for x and y Definition of the TOTAL sum of distances 1 b1 b2 c1 c2 2 2 2 da x y J x y 2 2 3 db x y b1 b2 x b1 y b2 2 2 4 dc x y cl c2 x c1 y c2 5 TOTAL x y b1 b2 c1 c2 da x y db x y b1 b2 dc x y cl1 c2 p20 Heinz Rainer Geyer Der Fermat Punkt D N L 16 Substitution of the co ordinates of the given vertices of the triangle 6 L x y TOTAL x y 7 2 3 5 It follows an evaluation of function L in a 5 x 5 matrix with increment h starting with x sand y t 7 FERM s t h VECTOR VECTOR L x y x S S 4 h h y t t 4 h h 8 FERM 3 41 3 12 0 002 10 30679813
31. einen Kurz vortrag zu halten uns etwas vorzuzeigen W nsche oder Anregungen darzulegen w re ich f r eine Vorinformation sehr dankbar Ich hoffe sehr mit Ihrer Hilfe ein kurzes Programm zusammenstellen zu k nnen Ich danke jetzt schon Herrn Wolfgang Propper f r die Organisation des Treffens N heres erfahren Sie im n chsten DNL M rz 1995 D N L 16 Liebe DUG Mitglieder Anfangs dachte ich es w rde schwierig werden f r jeden DNL einen Letter of the Editor zu schreiben Ganz m Gegenteil ich habe immer Probleme all das was ich Ihnen so nebenbei noch mitteilen will n einer Spalte unterzubringen In diesem DNL m chte ich Sie auf die bereits ange k ndigte Utility von Sergey Biryukow hinweisen Die zahlreichen Files finden Sie ebenso wie die beiden Demos im Unterverzeichnis lt LABEL gt Es mu eine ungeheure Arbeit gewesen sein die Zei chens tze zu entwerfen Sergey und sein Team bitte vor den Vorhang F r mich ist es besonders erfreulich feststellen zu k nnen da der DNL vielfach nicht nur rasch durchgebl ttert wird um spezielle Rosinen f r seinen Bedarf zu finden sondern da sich viele Leser ernsthaft mit den angebotenen Themen aus einandersetzen Ein Ergebnis finden Sie m User Forum zwei weitere werden erw hnt Sie sind so umfangreich da sie einen eigenen Artikel wert sind Nun mu ich ein Vers umnis nachholen Leider habe ich bisher vergessen G Scheu s zweites DERIVE B
32. except Russian letters are on Fig 1 Each font file has two vectors the vector of character names chars names and the main vector chars of the form character_namel pointl point2 char_name2 Points are in a standard for this font pattern rectangle The size of this rectangle is defined in each font file char_rect_width 6 char_rect_height 8 for all our fonts A lot of memory space was saved by defining character points as pairs of small integers but not rationals or decimals Font designer Label fd mth supports new font files design by combining data from existing font files add ing deleting new characters to from font files checking fonts amp character points input in the form of long integer that reduces the number of keys to be pressed 3 times For example N 60533813001353 simplifies to L6 0 5 31 3 81 1 31 0 01 1 31 5 3 and gives an upper case letter A after plotting Only nonnegative integer coordinates less than 10 can be input with NC function and the first point can t be 0 0 Text Plotting Labeling LABEL xy wh dxy t rot function prepares a vector of vectors of points for plotting text t given as vector of characters t t e x t xy x y starting point wh char width char_height dxy char_step_x char_step_y rot characters rotation counterclockwise in radians optional default 0 Axes drawing numbering and labeling DRAW_AXES xyl1 xy2 axes_cr
33. following simple identity can help ie m 2 IT 142 3 108 005 106 COS 5 T f AT h _oo 77 azco 1 42 H 77 3 si 14 3 Message 3266 From HARALD LANG to SWH 3256 MYSTERIOUS TRIG IDENTITY an a oO LY IT Be 3 More seriously simplifying trig expressions enters when DERIVE solves 3 and 4 degree equations For exam ple try solving aa E 4 In my version of DERIVE I get three roots one of which is the other two look similar 5 ATAN 7 iT iT S n serro cos 0 5009585579 4 al f APPROA 0 90096 GrG 3 But this is actually equal to COS pi 7 a much simpler expression This follows easily from the SWH Hans Dudler formula I just figured out that I should use Std Char set and print to a File in derive then I can download the formula just as s Harald Lang DERIVE 6 produces another but equivalent output for the solutions of the equation given above 2 1 SOLUTIONS 2 2 z Zz 2 4 E f os ATAN ATAN AOT g g iT 7 SIN esis Fre er 3 3 3 3 0 2225209339 0 6234898018 0 9009688579 Message 3269 From JERRY GLYNN to PUBLIC about 500000 DIGITS OF PI Al Rich recently calculated 500000 digits of pi on Derivexm He asked for confirmation on this BBS We are happy to announce his last 40 digits match up with two different sources so it look like
34. g und ihre Anwendung auf Ortsvektoren sozusagen als Black Box Funktion zur Verf gung Entscheidend war dass die Koordinaten der ge drehten Punkte C_ und B_ zur Verf gung standen Damit konnten die Geradengleichungen aufgestellt werden und das Schnittproblem auf dem Niveau der Klasse 9 gel st werden The co ordinates of the roteted points C and C were necessary At first the students experi mented with Tangens and Pythagoras As a simplification made the rotation matrix avail able to the students as a Black Box Having the points C and B the problem was easy to solve for pupils even of level form 9 A ABC given by A0 0 Blox byl Ciox oy we yl 1 Jinelxl yl x2 y2 x ia x x1 yl x2 xl ax ay Br by 2 triangletax ay be by cx cy iz cK CY ax ay Rotation matrix around the origin AlQ D COSte SING 3 rotla SINCa COS Calculation of rotated points Cand Bic and b_i 4 e_lex cy ts of ten cy 5 EIER cy i l CK 3 cy J3 CX cy Z u a a 5 b bx by t o im by E be Sy by J3 bax 7 b lbw by i TOO 2 2 2 2 Lines connecting B with C line Gi and B with iine HY G x bx by cx cy is Time c_tex cyll Ce bex cyl bx by 1 2 FR Hix bx by cx cy i linellb_ bx rl b bx byl cx cy 9 1 2 f_ptiax ay bx by cx cy SOLUTIONS Cy Glx her by ex cy Ay Hix
35. hnet man zur Leitgeraden NU eine Konchoide mit O als Pol und K 2c Durch N zeichnet man die Parallele zu OU welche die Konchoide im Punkt P trifft Dann ist ZPOU z NOU alpha 21 8709 MOU beta 7 2905 NOU alpha 38 0766 MOU beta 12 6948 Thomas Weth A Lexicon of Curves If you want to trisect a then take any arbitrary point U with OU a on one side The perpen dicular intersects the other side of a in N With ON c draw a conchoid with respect to the line NU with O as ist pole and K 2c Let NP OU then POU is the third of NOU Begrundung Ist M die Mitte von QP so liegen P Q und N auf dem Thaleskreis tiber der Strecke PQ Also ist PM MQ MN c ON Damit gilt IZNPM ZMOU als Z Winkel Da das Dreieck MPN gleichschenkelig ist gilt f r den Au enwinkel ZOMN ZOMN 2p Da auBerdem das Dreieck OMN gleichschenkelig ist gilt wegen der Kongruenz der Basiswinkel M is in the middle of PQ then P Q and N are lying on a Thales circle so PM MQ MN c ON Then you will find two isosceles triangles AMPN and OMNA giving simple relations between the angles ZMON ZOMN 22 also ita 2 oder 6 a 3 Es sei hier ausdr cklich darauf hingewiesen da es sich bei den Konstruktionen zur W rfelverdoppe lung und zur Winkeldreiteilung nicht um elementare Zirkel und Linealkonstruktionen im Pla ton schen Sinne handelt Wie man an den
36. ive then ist 3 transformations Jr b s s b de er Der How does this formula change if a is positive and r is negative When you have finished this assignment and you are happy if your results and conclusions ask for the next worksheet Answer the questions on it without using Derive A correction In the last DNL published H Scheuermann s phone number One digit was wrong this is the right number 06192 27568 Please don t forget to renew your subscription for 1995 in time Administration would be much easier for us If you don t cancel then we assume that you will stay a member Many thanks for your cooperation Bitte denken Sie daran rechtzeitig Ihre Mitgliedschaft zu erneuern Falls Sie nicht k n digen d rfen wir annehmen dass Sie Mitglied bleiben Herzlichen Dank f r Ihr Ver st ndnis
37. kward letter X I know that the menu is kept in a file called DERIVE MEN but my attempts to adjust the menu have usually led to the computer locking up and needing to be rebooted I should be grateful for any information on this because I am finding it very frustrating that I cannot adjust the menu to suit my own way of working and my left handedness pa DERIVE USER FORUM D N L 16 DNL have sent Mr Williams the information he wants and announce for the next DNL to publish the way to tune one s own menu Dr Klaus Kuenzer Bruneck Italy I ve read Eugenio Roanes contribution about the Operations on Polynomials in DNL 15 very keen I enjoyed it very much because he used recursive algorithms I tried the functions and had a lot of trou bles with calculation times so it needed for MCD 32456 28588 28 8 sec on a 486DX or MCD POL 1021 3 sec instead of the 4 8 sec in DNL 15 So I tried another way Dr Kuenzer explained his ideas by one example he simulates a kind of WHILE DO loop See here his functions checked Eugenio s times and can confirm them ed a bestela B a ron b ggtla bj CITERATECIF x 0 x k o rest ix A Jh x a 6 Zz a 1 1 100 100 garen sels J 1 2 9 sec In 1994 0 05 sec in 2005 u lim u u lim u l w O 1 l w O 1 Infeptp vj is ITERATE IFlu DO u 1 u u 1 u u ii Eee 1 2 ali 2 1 Aut pgrad p vw infopi
38. ne of the two founders of the ICTCM conference series several delegates from Europe and one from Australia had the great honor of chairing the meeting and started by reading Josef s welcome note It was only half as charming as if Josef would have spoken himself but people told me that just by hearing his words they almost could see him there What a marvellous compliment from those who know Josef personally David Stoutemyer one of the two fathers of DERIVE was the first soeaker He gave some insight in DERIVE version 3 His lecture was followed by presentations of Jeanette Palmiter USA David Sjostrand Sweden Lisa Townsley Kulich USA Terence Etchells UK Robert Mayes USA and Bernhard Kutzler Austria We had many nice discussions following the lectures met with new DERIVE friends and once again enjoyed sampling Hawaiian macadamia nuts which were sponsored by Soft Warehouse Inc It was a very nice start for more North American DERIVE activities Bernhard many thanks for your nice report I am jealous not because of Disney World but because of the macadamia nuts There is a challenge for us in Europe Dear German members and members from the countries in the neighbourhood I m glad to announce the 2 Deutsches DERIVE User Group Treffen im Rahmen der MNU 1995 Nurnberg Ostern 1995 Mittwoch 12 4 1995 14 00 16 00 Uhr G ste und Interessierte sind nat rlich herzlich willkommen Falls Sie den Wunsch haben
39. nswer of 2 but has obviously done this by calculating both terms and dividing We would not do it this way All of these programs produce 2 as an answer to 2 x 1 2 x so it s not that they can t do algebra but rather they can t do algebra when no vari ables are present Could this ability pass for an example of human intelligence Are there other examples out there If we knew what humans were good at and what algebra systems were good at maybe we could see how to work together and then maybe train people for these jobs Do symbolic algebra systems simulate intelligence or are they just wonderfully fast calculators of symbolic expressions Message 3304 From GREG SMITH to JERRY GLYNN about 3303 INTELLIGENCE The fact that you had to ask the question at all hints strongly at the answer pl2 DERIVE Bulletin Board Service D N L 16 Message 3308 From HARALD LANG to JERRY GLYNN about INTELLIGENCE I have a similar experience It seems like DERIVE has a difficulty to handle redundant information I have pointed out earlier the curious fact that she correctly simplifies 1 3 1 3 1 3 Gee amp x 16 2 ex 54 to 0 but if you replace x by SQRT 5 then she fails I don t know if this is typical for computer algebra pro grams in general but no doubt we humans are not fooled by x being replaced by a number Harald Message 3309 From JERRY GLYNN to HARALD LANG about 3308 INTELLIGENCE Excellent example again algeb
40. nter output from the LPT1 port to the network printer named name Also tabs are not expanded extra form feed characters are not sent at the end of print jobs and data is sent to the printer after A 5 second time out 6 If the DERIVE user does not specify a drive and or directory when attempting to load a utility file DERIVE searches for the file in a the DOS drive and directory current when DERIVE was started b the DERIVE directory on the current drive and finally c the directory from which DERIVE was invoked Thus to provide access for all users to a utility file copy the file to the DERIVE directory on the file server Then the file will be found by search path c above 7 The directories listed above are also searched to find the DERIVE INI initialization file when DERIVE is started and to find the DERIVE HLP help file when a request is made for help see the DERIVE User Man ual for details 8 You may want the state of DERIVE to be the same for all users when it starts If this is other than the default initial state save the desired state in a DERIVE INI initialization file in the DERIVE directory on the file server see Section 2 12 of the DERIVE User Manual for details Then if individual users wish to modify the initial state they can save their own DERIVE INI files in their own working directory Message 3226 From SOFT WAREHOUSE to PUBLIC about COMPUTING LARGE POWERS Normally DERIVE raises numbers to integer powers Fo
41. ong with DERIVE when it begins execution In Version 3 this can be done by following DERIVE on the DOS command line with the name of the INI file For example the command DERIVE REVERSE INI ARITH DMO starts DERIVE initializes DERIVE using the settings in REVERSE INI and then runs the ARITH DMO demonstration file You might want to put a note to this effect in the DNL since it is not yet in the DERIVE User Manual Aloha DNL Thank you for the answer to the question how to save the background and work color of the plot screen using Transfer Save State and then calling these settings Dr H J Kayser Dusseldorf Germany Dr Kayser wrote a short comment on Terence Etchell s INVNORM from DNL 15 In the next DNL I will publish his An alternative to the function INVNORM from DNL 15 EI DERIVE Bulletin Board Service D N L 16 Peter Baum Kassel Germany Peter Baum dealt with Th Weths Lexicon of Curves and sent an extensive completion together with a couple of TIF files of nice pictures Mr Baum uses only the polar form and he yields interesting results The paper is too large to be part of the User Forum so it will be a contribution in 1995 But I would like to include one of his plots Message 3225 From SOFT WAREHOUSE to PUBLIC about INSTALLING DERIVE ON A LAN The compact DERIVE and DERIVE XM executable files are extremely small by today s standards Also neither program uses overlay files while running i e once loaded
42. ose in figure 1 4 4 5 3 2 5 2 15 1 0 5 eave ae Oo Ses De Figure 1 difficult periodic function We will discuss the possibilities and problems connected with plotting this kind of functions The tools Derive knows the following functions a CHI a x b a function which returns 1 for x a b and 0 outside a b b The IF function We can use this function for piecewise definded expressions using conditions For instance IF x lt 0 x x isa function which for x lt 0 returns a parabola and for x gt 0 the straight line y x c MOD a b which approximates to a modulo b When MOD x 1 is plotted you get a saw shaped function MOD x rn also returns a saw shaped function but now with period 7 d FLOOR x is narrowly connected to MOD x Simplifying MOD x 1 yields x FLOOR x In other words MOD x 1 x FLOOR x Plotting using CHI IF and MOD The usage of CHI Let s look at the following piecewise defined function fix x ifx e 0 1 and 0 ifx 0 1 We could Author this with f x CHI 0 x 1 x A short explanation The fact that the product is forced to 0 outside 0 1 makes it rather difficult for us to plot functions only defined on an interval DERIVE plots 0 outside the interval while the function is not defined over there J C M Verhoosel T periodic Functions Let s suppose we want a function like the one in figure 1 We can define f x on 0 1 on
43. oss scale_xy DRAW_AXES x1 y1 x2 y2 ac_x ac_y sc_x sc_y J prepares points for drawing axes arrows points up amp right amp axes ticks in the window with the lower left corner coordinates xy x1 y1 amp upper right corner xy2 x2 y2 axes crosses at the point ac_x ac_y scale xy defines distance between ticks on x amp y axes ac_x sc_x amp ac_y Sc_y must be integers wh x label x 1 LABEL AXES xy2 axes_cross labels LABEL_AXES xy2 axes cross wh_y label y prepares points for labels near arrows wh_x amp wh_y are width amp height of characters in x amp y axes text labels label x amp label y NUMBER AXES axs numbers NUMBER AXES axes_cross scale xy p14 Biryukov 2D Plots Labeling D N L 16 prepares points for plotting num_x amp num_y vectors of text under ticks If the tick is in the point of axes cross appropriate text is shifted left or down num x text_for_left_tick text_for_2nd_tick Three previous functions are combined in ALL_AXESC one that has more clear input parameters and prepares points for plotting axes axes labels and axes numbering wh_x amp wh_y in the second and third arguments are different and define characters width and height for the right neighbour text ALL AXES axs numbers labels xyl xy2 wh x num x wh_x label x ALL AXES axes cross wh y num y wh_ y label y scale xy Example 1 LOADED N DFON DNLYDNLSA MTHLSYLABELY F_S MT
44. p E 5 E 3 4 x rgrad 3 x 2 x 1 3 x 3 gt x 2 x x 8 2 I sec in 1994 0 02 sec in 2008 lkoeff p v Cinfop p v 3 5 3 3 4 X 3 lkoeff 3 x 2 x 1 3 x 3 x 2 X ee 2 2 0 9 sec in 1994 0 03 sec in 2008 rest_pol p q v ITERATE IF pgrad u v lt pgrad u v u u L 2 1 u_ lkoeff u v pgrad u v pgrad u v 2 1 1 2 ena a aea a U a N lkoeff u v 2 2 3 2 3 2 rest_pol 8 x 26 x 15 x 63 12 x 49 x 25 x 42 x 2 5 4 x 19 x 21 a 1 3 sec in 1994 0 03 sec in 2008 D N L 16 DERIVE Bulletin Board Service p 5 p rest_pol p q v quo_pol p q v Z M q 3 2 3 2 quo_pol 8 amp x 26 x 15 x 63 12 x 49 x 25 x 42 x 2 3 p limp Vva ruffini p a v limp V a Voa 7 5 4 2 r ffini 3 x X 25 X Ss X 6 5 4 3 2 3 x 9 x 26 x 80 x 240 x 719 x 2157 6471 gat pollo g v3 gt CITERATECIFCU 0 u i rest_pol u u 2 ke 2 2 1 2 2 rest_pol u U eal u p q1 1 2 1 g t polpa vJ CITERATECIFCU 0 i rest_pol u u me je 2 2 1 2 2 rest polgu u 2 Us ip gi 1 2 1 4 3 2 3 2 ggt_pol x 7 x 18 x 22 x 12 3 x 13 x 8 x 12 x 2 34 x 5 x 6 9 9 4 sec in 1994 0 14 sec in 2008 Al Rich SWH Hawaii In order to get the background color saved in an INI file to be used by DERIVE the INI file must be loaded al
45. point To explain the problem I showed the following proof by construction Zur Erkl rung f hrte ich den folgenden geometrischen Konstruktionsbeweis vor Sei T zun chst ein beliebiger Punkt im 3 Dreieck AABC Dreht man das Teildrei F eck ACAT um 60 in positiver Drehrich i tung um den Punkt A so ergibt sich ein Polygonzug C T TB dessen L nge die gesuchte Abstandssumme ist Die L nge des Polygonzuges ist minimal wenn er auf der Geraden C B liegt Analog dreht man das Teildreieck ATBC um B um 60 Der gesuchte Fermat Punkt F ergibt s ch also als Schnittpunkt der beiden Geraden C B und C A This is the original screen shot from 1994 The labels were produced using Sergey Biryukov s labeling tool T is an arbitrary point in AABC Having rotated ACAT about 60 in positive direction you ob tain the sum of the distances with the polygon CT TB Ist length is minmal if it coincides with the line CB In the same way you can rotate ATBC round B by 60 negative The Fermat point is the intersection point of CB with CA It was interesting to observe how the pupils found different ways to solve the problem ac cording to their mathematics knowledge D N L 16 Heinz Rainer Geyer Der Fermat Punkt p19 We can use dynamic geometry to illustrate the problem and to verify that interection point FP is indeed the solution point show the Cabri figure PC and TI 92 Voyage 200 and the Tl Nspir
46. r example 2 3 simplifies to 8 However before com puting powers DERIVE determines if the answer is going to be too big to store in memory For DERIVE the largest power it will compute is roughly 2450000 1612500 3 16 10415051 For DERIVE XM the largest power it will compute is roughly 2 400000 16100000 9 96 104120411 Although it does not raise numbers to huge powers directly DERIVE can be used to convert such expressions to scientific notation using common logarithms like we learned our high trig class For example to convert 31000000 set the precision level to 12 digits set the notation to decimal and ap proximate the expression LOG 31000000 10 to give 477121 254719 Thus 341000000 is approximately 1040 254719 10477121 Finally highlight and approximate the first factor to give 1 7977 104477121 which is 31000000 in scientific notation Aloha Al Rich SWH pe DERIVE Bulletin Board Service D N L 16 Message 3247 From SOFT WAREHOUSE to PUBLIC about 500 000 DIGITS OF PI In response to a challenge by a user I had DERIVE XM compute z accurate to 500 000 digits on a 25mHz 486 with 4Mb of memory It took about a week to compute and another week to save the result in a file It requires almost 50 pages to print using a very small type font No I am not going to upload 500 000 digits on this BBS The last 20 of the 500 000 digits are 40424876025 13819524 I would greatly appreciate someone independently i e
47. ra is something with variable s and arithmetic is something without variables and they are treated differently When I wrote my comment I expected you to respond How nice to be able to predict such a response at such a great distance I m inspired to think of more Any more responses out there Derive 6 works as expected 1 3 1 3 1 3 5 2 8 5 16 27 5 54 0 In the next DNL you will find a fine example how an idea is growing on the Bulletin Board 2D Plots Labeling Dr Sergey V Biryukov Moscow Abstract Derive utility files for 2D plots labeling and axes drawing numbering and labeling are described Vector scaleable and rotatable fonts with 250 characters are avail able User fonts design is supported Introduction Derive makes excellent 2D plots It is for bright pupils students and scientists But ordinary pupils have some difficulties in keeping in mind plots and axes names and in axes numbering calculation from scale values and cross position Furthermore axes cross is restricted to 0 0 point Our aim was to make 2D plots more clear and easy to percept Utility overview Utility files set consists of the main file label mth 5 font files f mth font designer label fd mth and appropriate document files doc It supports plotting an arbitrary ASCII char acter except graphic and control ones or sequence of characters text in the 2D Plot window Char acters position width height rotation sl
48. reduced to a value between 0 and 1 Replacing x with MOD x gives us a 1 periodic func tion A nice way to create a T periodic functions is to transform the function defined on 0 1 by multiplica tion translation etc An example We want a periodic parabola through the points 0 1 and 1 1 which is symmetrical on the axes x 0 5 see figure 3 1 5 1 0 5 ee Sd Se eae ee Figure 3 the quadratic curve J C M Verhoosel T periodic Functions First we must find the formula of the quadratic curve f x 4 x 0 5 This function surely is de fined on 0 1 Now we replace x by MOD x which gives us the function fix 4 MOD x 0 5 Plot the formula and you get figure 4 re ere Figure 4 2 periodic quadratic curve Suppose we want to translate this function so that the minimum of the quadratic curve occurs for x 0 then we must replace x by x 0 5 In the same way we can get a 2 periodic function by replac ing x by x 2 etc Another example We want a series of half circles At first you can define a half circle on 0 1 see figure 5 Solving for y gives us 2 possibilities we choose the positive one The only thing we must do now is to replace x by MOD x In the same way we get a 2 periodic series by replacing x by x 2 The period is twice as big but not the amplitude So we ll have to multiply our function as well by a factor 2 Now we have a true 2 periodic series of half circle
49. ry function to DELETE ELEMENT v i It inserts expression a in a vector v at position i The oe replaces the i th element of v with axpression a The 3 returns the number of elements of vector v identical to expression a Conclusion Utility described can be easily used on every IBM PC compatible computer Only 2 files are to be loaded as utilities Label mth amp one of the font files mth Only 2 functions are needed by the user LABEL for writing graph labels amp ALL_AXES for axes drawing labeling and numbering Full Fonts are rather slow on IBM PC XT so Tiny amp Small Fonts for these computers are highly rec ommended Acknowledgments My thanks to Dr B Kutzler for the idea of writing this paper V B Biryukov for draft fonts design amp Prof N V Soina for friendly advices and support Comments of the editor do neither list LABEL MTH nor LABEL_FD MTH you can download the files from the subdi rectory lt LABEL gt Included are many files with font sets am adding here the most important names which are packed in separated mth files which must be preloaded before using had to change some 1994 files because the DOS characters are not compatible with the Unicode characters of DERIVE 6 and it might be too complicated to include them now This is the reason that copy some screens with the respective characters f e mth English Upper case and Lower case letters f el mth English Lower case letters f
50. s fey Algebra 1 EA 5D plot 1 1 Sele 2 a a 1 x 0 5 y 05 2 2 7 2 SOLWVEC x 0 5 y 05 y 3 y ll x vy 4x 1 x 4 Jx C1 x 5 JCMODCx fl MiblkT yVCFLOOR 24 2 YeNE 4 FLOORG2 2 6 j vaD 1 HOD e fe a 3 2 5 2 1 5 1 0 5 Figure 5 a series of half circles It is interesting that DERIVE simplifies the functions when we have them annotated in the plot window Josef Noting that MOD x 2 already has a period 2 it must be possible to do things more easy Suppose we plot y Vx VI x J C M Verhoosel T periodic Functions Suppose we plot y Vx V1 x To make sure that the next period is produced we must be sure that x 2 1 for instance is reduced to x 0 1 This can be done by MOD x 2 So we replace x by MOD x 2 and get y 4 MOD x 2 41 MOD x 2 When we plot this function we get 2 periodic function in the next figure 2 1 5 1 0 5 at de Ve ae Let s try this on another function Suppose f x is defined as fix x when x e 0 2 0 4 and 0 6 periodic We are tempted to replace x by MOD x 0 6 Why does this not work 5 l fies MODx 0 6 1 8 6 0 4 2 Wee Uga Vie Use 1 bee 0 2 Looking at x 0 7 we see that MOD x 0 6 reduces it to 0 1 while it should be reduced to 0 1 We have to transform MOD x The period is all right 0 6 The starting point is not correct I
51. s purpose The experimental character of the mathematical work in our project caused a Sig nificant motivation which could be very useful for maths teaching in total Additional comments for the revised version of DNL 16 will follow on the next page p24 Heinz Rainer Geyer Der Fermat Punkt D N L 16 The students had the idea of applying calculus So one could provide the suggestion to equate both partial derivatives to zero and then try to solve the resulting system of equations Inspecting the partial derivaties might discourage the students from solving the system But they might also have heard or even learned to apply the Newton Raphson algorithm for approximatively solving equations Why not extend for more variables Using the DERIVE Help they might find NEWTONS u x x0 32 Precisteanbigits 6 d d reverse neurons Lix aae Ltx Bra 33 33 dx dy 3 2 1 3 41483 3 12990 34 3 41483 3 12990 3 41483 3 12990 Here we see the last three iterations of the algorithm And finally it might be a welcome surprise to show the solution in a 3D environment L x y from expression 6 is a function of two variables which can be plotted immediately in the 3D Plot window The students have to find appropriate settings for the box Using the Trace tool they are asked to find the Minimum Point and then they can add the point Xrp Vep L xrp Vep Thomas Weth A Lexicon of Curves 5 Ebene Algebrais
52. se 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 D N L It must be said though that non English articles are very welcome nonethe less Your contributions will be edited but not assessed By submitting articles the author gives his consent for reprinting it in D N L The more contributions you will send the more lively and richer in contents the DERIVE Newsletter will be Preview Contributions for the next issues Stability of systems of ODEs Kozubik SLO Prime Iterating Number Generators Wild UK Graphic Integration Probability Theory Linear Programming Bohm AUS DERIVE in Austrian Schools some examples Lechner Voigt Eisler a o AUS Tilgung fremderregter Schwingungen Klingen GER Continued Fractions and the Bessel Functions Cordoba a o ESP Turtle Commands in DERIVE Lechner AUS DREIECK MTH Wadsack AUS IMP Logo and Misguided Missiles Sawada HAWAII Reverse Discussion of Curves Reichel AUS Reichel Klingen Bohm Splines A thriathlon AUS amp GER 3D Geometry Reichel AUS Parallel and Central Projection Bohm AUS Conic Sections Fuchs AUS A Trick for Plotting 2D Plots Roanes amp Roanes ESP Setif France Vermeylen Belgium Leinbach USA Lymer FRA Baum GER Kayser GER and others Impressum Medieninhaber DERIVE User Group A 3042
53. t set mth file are called char_names the short one and chars the huge one Have a look at this vector and admire the enor mous work lying in them Maybe you will create another font set for us Editor p18 Heinz Rainer Geyer Der Fermat Punkt D N L 16 Der Fermat Punkt im Dreieck The Fermat Point in the Triangle H R Geyer Wiesbaden GER W hrend einer Projektwoche an unserer Schule Gymnasium im Fr hjahr 1993 besch ftigte sich eine Gruppe von 7 Sch lern der Jahrg nge 9 bis 13 mit den Einsatzm glichkeiten von DERIVE im Mathematikunterricht Bis auf einen Sch ler hatten sie keine Erfahrungen mit dem Programm Nach dem Einf hrungstag 4 Schulstunden hatten sich alle soweit in die Bedienung eingearbeitet dass sie in der Lage waren sich an aktuellen Problemen des Mathematikunterichts der jewei ligen Jahrgangsstufe zu versuchen Am 3 Tag stellte ich allen gemeinsam eine Aufgabe Es ist die Lage desjenigen Punktes in einem spitzwinkeligen Dreieck zu suchen dessen Abstandssumme zu den drei Eckpunkten minimal ist der Fermat Punkt During a project week at our school grammar school in spring 1993 a group of 7 students age 15 19 dealt with DERIVE After a short introduction 4 hours they were able to use DERIVE in connection with actual problems of their curriculum On the 3rd day I set them all the same task Find the point in an acute triangle ABC with a minimal sum of distances to the points ABC the Fermat
54. t should be 0 2 so will MOD x 0 2 0 6 work Not yet Suppose we look again at x 0 7 Now it is re duced to MOD 0 7 0 3 0 6 0 3 still not correct We must realize that we should subtract 0 2 so it should be MOD x 0 2 0 6 0 2 We can generalize this to the procedure shown in the next figure ya 2 MODx 0 2 0 6 0 2 ee u Ge a 1 1 2 1 0 8 0 6 0 4 0 2 _O 2 ie Combining There is no restriction on the form of the function D Z which is to be made periodic We can join x on l 115 0 2 0 4 with 0 56 x on 0 4 0 6 for instance ea In this case we make use of the IF or CHI 0 05 function 0 2 0 1 zer a J C M Verhoosel T periodic Functions We can generalize this to the procedure shown in the next figure 2 x x 0 2 x 0 4 x 0 4 x 0 6 x 0 56 Substitute x by MOD x 0 2 0 8 0 2 and plot the resulting expression MOD x 0 2 0 8 0 2 x 0 2 MOD x 0 2 0 8 0 2 0 4 x 0 4 MOD x 0 2 0 8 0 2 0 6 MOD x 0 2 0 8 0 2 0 56 O 2 te 1 8 1 6 1 4 1 2 1 0 amp 5 0 6 0 4 0 2 ee u4 0G 0S 1 12 14 16 1 8 This works also with an IF construction instead of the CHI function for the piecewise defined function Josef M IF 0 2 lt x lt 0 4 x IF x lt 0 6 0 56 x lA IF 0 2 lt MOD x 0 2 0 8 0 2 lt 0 4 MOD x 0 2 0 8 0 2 IF MOD x 0 2 0 8 0 2 lt 0 6 0 56
55. they make no further access to the hard or floppy disk Both these facts make installing and running DERIVE and DERIVE XM on a Local Area Network LAN simple and straight forward The following describes how to install DERIVE on a LAN The procedure for installing DERIVE XM is the same except where noted 1 Login with system supervisor privileges and create a directory named DERIVE on the LAN file server If installing DERIVE XM name the directory DERIVEXM to avoid a possible name conflict with an existing DERIVE directory Give DERIVE users read only rights to this directory and give the system supervisor all rights see your LAN manual for details Note that it is NOT necessary to establish a command file se arch path to this directory 2 Using the MS DOS COPY or XCOPY commands transfer all the files on the DERIVE or DERIVE XM distribution diskette to the new directory 3 DERIVE users often need to save mathematical expressions and or graphics screen images in files Nor mally these files should NOT be stored in the DERIVE directory on the file server Instead they should be saved in a working directory on a diskette in a directory on the user s own computer or in the user s own di rectory on the file server The DOS drive and directory current at the time DERIVE is started is the default working directory Therefore DERIVE should be started from the desired working directory and NOT from the DERIVE directory on the file server
56. uch vorzustellen Das wird jetzt gerne mit einer Bitte um Entschuldigung nachgeholt Ich m chte besonders darauf hinweisen da hier zahl reiche physikalische Anwendungen behandelt wer den Zufrieden G nter Beachten Sie bitte die beiliegenden Informationen Beim DERIVE Journal m chte ich nochmals daran erinnern da Sie als DUG Mitglied einen betr cht lichen Preisvorteil genie en Nun habe ich eine gute und eine schlechte Nach richt f r Sie Zuerst die schlechte wir werden den DUG Mitgliedsbeitrag erhohen und zwar auf S 340 fiir Osterreich DM 52 bzw BP 22 fiir Europa und US 40 fiir Ubersee Aber jetzt die hoffentlich gute der Umfang des DNL wird berproportional auf mindestens 40 Seiten erweitert Fassen Sie das auch als Bitte um weitere Beitrage auf Sie sind aber recht fleiBig Kein Beitrag geht verloren und wenn der eine oder andere langere Zeit nicht erscheint dann hat dies ausschlie lich technische Ursachen Platzbedarf Themenzugeh rigkeit bersetzung F r 1995 habe ich eine Ausgabe mit Schwerpunkt Geometrie vorgesehen eine andere soll sich mit dem Geschehen in ster reichischen Schulklassen besch ftigen Beachten S e bitte die beiliegende Rechnung f r den Mitgliedsbeitrag 1995 S e erhalten automa tisch ein Quittung mit dem DNL 18 Ich w nsche Ihnen ein frohes Fest und ein erfolg reiches Neues Jahr 1995 Mit den besten Gr en bis zum n chsten Mal LETTER OF THE
57. vorhergehenden Beispielen sieht ist n mlich die Wurfelver doppelung und die Winkeldreiteilung sehr wohl geometrisch m glich nur eben nicht wie zB in der Galois Theorie bewiesen wird mit Zirkel und Lineal alleine In den obigen Konstruktionen wurde n mlich eine Gerade mit einer Kurve der Konchoide geschnitten also eine Operation durchgef hrt die mit Zirkel und Lineal alleine nicht m glich ist Abschlie end sei bemerkt dass sich f r einen Kreis als Leitlinie andere bekannte Konchoiden u a die Pascalschen Schnecken und die Kardioide ergeben Wir werden in einer sp teren Folge darauf zu sprechen kommen I want to emphasize that these constructions are not elementary constructions using ruler and compasses only in Platon s sense because you need the intersection point between a conchoide and a line and this operation can t be done with the tools mentioned The Galois Theory proofs this fact Finally would like to mention that using a circle as leading line we will find other well known conchoids as Pascal s snail and the cardiode In a later sequel we will talk about p30 J C M Verhoosel T periodic Functions D N L 16 DERIVE and plotting T periodic functions Drs J C M Verhoosel Eindhoven Netherland Introduction For most simple periodic functions it is rather easy to plot them Think of y sin x for instance Just Author sin x press twice on Plot and voila It s more difficult to plot functions like th
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