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THE DERIVE - NEWSLETTER #83 USER GROUP

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1. we finally And after a translation x x u and y y v with u and v a T yield equation 3 U Example 2 a parabola The terms of the parabola are called t4 t5 and t6 and the appropriate graphs are f4 x f5 x and f6 x New variables have been introduced to avoid side effects in the actual problem The entry line in the G amp G windows has been hidden by Ctrl G the original term and its graph 4 x7 4 x yt4 y 2 x 6 y 7 x74x 4 y 2 44 y7 6 y 7 fAx zeroslx2 x 4y 2 4 y2 6 y 7 y ii rotation first 5 rotpoll t4 tan 2 5 x2 2 5 x 2 5 y 7 fslx zeros s x2 2 5 x 2 f5 y 7y Done i f l then the translation t6 trapol 5 E 5 x 2 f5 y Done va f lx ZerosS 5 x72 Js Yy ra 9 99 Conclusion We can constrain the investigation of equation 1 to two cases It is sufficient that either b d e Oorthatb c d f 0 ora b e Pf The first case yields ellipses and hyperbolas and their degenerations straight lines or point In the other case the conic section is a parabola 3 Classification using a matrix M CS 3 1 3 By an operation H M H where matrix H is either a rotation matrix cos sin 0 0 10 u R sin 0 cos 0 ora translation matrix T 0O 1 v matrix M 0 0 1 0014 QTD ooo gt DOD Q pe Wolfgang Propper On Quadratics and Conic Sections a 0 0 a 0 O0 can be transformed to O
2. 5 10 10 5 1 5 b 0 Inclination lin degrees 0 10 10 Center M 0 0 e 1 10 10 d 0 ant E e 0 ra 2 4 10 10 ind 10 10 gt gt This is the screen after moving some sliders i e changing some coefficients Inclination in degre res 35 2 enter M 2 97 0 867 i a Tr b 6 u En En 0 ian inclination lin bg 2 fp Center tif S5 a b a 4 2 The Focus Feature CS 5 For ellipses the sum of the focus rays i e the line segments connecting one point on the ellipse with the two focus points is constant That means d F X d F2X k const For a hyperbola this condition is d F X d F gt X k Both expressions on the left side are containing roots One can eliminate these roots by quite elaborate al gebraic transformations see CS 5 2 in order to obtain a rather complicated expres sion in x and y which depends on the focus point coordinates f 1 f 2 fo1 foo and the constant k The value of k can be varied by a slider For the focus point coordinates we store the coordinates of two arbitrary points in f11 f12 and f21 f22 respectively Now the graph Wolfgang Propper On Quadratics and Conic Sections can be plotted using again the zeros function By dragging the foci and changing k we can create ellipses and hyperbolas see CS 5 3 In general it would also be possibl
3. 23 B7 COSCt J37 SINCt 70 2 37 COST 2 37 SINCt 39 Circle Klaus Korner Airplane and Conformal Mapping I generalized the mapping and introduced a slider for k It was interesting to observe that only for k 25 a wing profile appeared Cire le Next step was dealing with Mr Korner s problem 1 e solving the system of nonlinear equations for x and y I assumed that DERIVE with Groebner bases implemented should be able to do the job 25 25 SOMUTIONS u ull e el 5 2er 2 2902 Y 6 This did not work because the Groebner algorithm can be applied on systems of polynomials only So I rewrote both equations free of fractions and then I tried again I kept the system as general as possible and replaced 25 by a variable k k 2 2 2 2 2 2 u x 1 lixs y uk y 2 eee y kD Hr 2 2 x y k 2 2 2 2 2 2 v y ll lI amp 0 Oe y Dove y kij 5 2 2 xX 2 2 2 2 2 2 2 2 9 SOLUTIONSCUeR y jz y kd Ama y dave y kd Ex 3 And I received a solution not only one but five including the trivial solution x y 0 But the oth ers were bulky enough with two of them containing complex expressions u a 2 2 2 4 22 4 2 2 2 2 2 4 22 4m feeCOfie k amp k cudeu Dumm v 9 4k tu viele Ekv ultu Bev Im Seley as 2 2 2 2 4 2 4 2 2 2 2 2 4 m um GEk Sky udeu Dumm tyd ektu yO k kl ud eu 2 Seley a 2 2 2 4 2 2 4
4. aloha david s Von Michel Beaudin etsmtl ca Gesendet Mittwoch 18 Mai 2011 02 22 An nojo boehm pov at Betreff RE RE Textbooks and the arctanh Dear Josef this email is sent only to you because some months ago sent my new integral table to David Jeffrey and Albert Rich but not to you David Stoutemyer also received it In fact this high level discussion started last November when was teaching single variable calculus see the abstract of my Derive session and when started to be fascinated by Rubi So here is the table of integrals that would like to use and also here is the table actually used at ETS in the single variable calculus course If you can t make it to Houston this table of integrals will give you an idea of what will be talking about in my lecture Best regards Michel Do you know this solid Find more on page 38 p2 Textbooks and the arctanh This is Michel Beaudin s very individual table of integrals which he is finding to be sufficient n l n l L x dx n l 0 fx dx arctanh a a r feoi x dx ln sin x Il Jesex dx arctanh cos x a lt 2 4 Jese x dx cotx 15 Jese xcotx dx csc Ri dx arcsin a 1 a 2 Lyf Be dv aresinh 9 7 ae x ay 2 2 f X dx arctanh Klaus Korner Airplane and Conformal Mapping p 25 An email came in which led to an interesting applic
5. Let s try finding a hyperbola 2 2 con3 18 x 34 x y 13 x 6 y 1 7 y 12 ans Type Center Real axis 2a Imaginary axis 2b Hyperbola 1 013812154 1 455801104 4 532953869 2 030942852 Real Vertices Asymptotes 2 105449700 t 1 013812154 1 455801104 2 322866802 3 306015585 0 4 2 7 7084 74 t 1 013812154 2 39959 7608 t 0 2952424933 0 3944133754 1 455801104 1 244452697 t p 18 Josef Bohm A Conics Explorer and a Conics Trainer The next task is not so easy Model the arc as a conic using the slider bars 2 2 1 aex bD x y c y d x e y 1 Ze 5 P x 5 00 2 eee 5 00 4 3 a f 4 IN E e NS gt u N You can find conics everywhere I used the sliders to model the first three ellipses Three more are remaining waiting to be plotted Textbooks and the arctanh p 19 Textbooks and the arctanh An email discussion From Michel Beaudin To Albert Rich David Jeffrey David Stoutemyer Josef Boehm Subject Textbooks and the arctanh Hello Albert and my friends it is always exciting for me when I am writing an email to the fathers of Derive to my best ROC friend ROC stands for Rest of Canada Quebecers love this word and to my best friend in Europe My question is do you know why textbooks at least calculus ones are so quiet about the arctanh function I understand that arctanh x has the op
6. All r n20 lf ALL ofther n are positive or zero r n 2 0 the common situation inequality holds be cause the left side s terms are smaller than the right side s factors For an example let N 3 and see statements 5 6 7 and 8 Note that in 7 a rearrangement of 2 with N 3 the left side is 1 and ALL of the variables on the right side are positive Therefore a SUFFICIENT condition for the usual condition inequality is simply that all all rjn gt 0 1 CaseMode Sensitive N N 2 2 ra m lee see n l n hel n N N 3 O MAL frf n 1 n n l f H H 4 I I ae ee ne n 1 n Bel n For example if N 3 3 3 5 a a Ne ra De Re N Er no n n l n Statement 5 can be written as Ser See esr Sl a eed Pec ee e pe Ss ee a 6 al 2 3 3 2 1 pa Roger Folsom Growth Rates and Comparative Statics In statement 6 multiplying out the r n 1 factors would eliminate the 1 r 3 r 2 r 1 expression Doing that by using Derive s Simplify Factor command gives statement 7 Irene ee eet Near 3 2 al 1 lt r r r r r r r rer 8 1 2 3 2 3 iS 1 2 Note I have done the same calculations above using N 5 Given the sufficient condition that all r n 2 0 statements 7 and 8 are obviously true be cause the left side is negative and every term on the right side is positive And since statements 7 and 8 are merely a rearrangement of 5 which is an example of sta
7. Consequently we are doing more and not less mathematics if we avoid absolute values in the integral of 1 x Best regards Michel p22 Textbooks and the arctanh On 11 05 17 03 00 PM Josef B hm wrote Dear friends I am very fascinated by your high level discussion Unfortunately I can not contrib ute because I never had to do with hyperbolic functions and their inverse BTW in German the inverse hyperbolic functions are called Area functions eg Area sinus hyperbolicus etc I never taught on a technical school where the hyperbolic func tions might be part of the curriculum What I can contribute is the fact that in Austria secans and cosecans are widely unknown and unused I am quite sure if l d ask some math teachers about secans and cosecans they would have never heard about them They don t appear in our textbooks I never heard about sec and cosec during my math study on the Techni cal University Many thanks and I will follow your discussion with high interest and I will not hesitate including it into the next DERIVE newsletter It is a nice coincidence that I am just revising DNL 26 with David s paper In x or In x Best regards to you all Josef From David Jeffrey Subject Re AW RE Textbooks and the arctanh Dear All My Russian wife has been teaching North American calculus for the first time and also com mented that she was not taught secant and cosecant guess in Europe you ar
8. ly E aea Se ay SS The first steps confirmed my conjecture about treating g h as a Boolean expression Some steps are following until coming to an end which is given in 16 If n and m are integers n a m bitwise logical and of n and m 16 5 x 12 y 5v n 5 x 12 y a 5 x 12 y 45 v 5 x 12 y 5 x 2 y 5v2 A 5 x 12 y 5 v 3 Repeating This Would be the correct form of the Boolean expression 17 gah which gives the correct truth value for the contradiction 18 false My explanation for 4 from above 19 Sex 12 y 4 5 A Sex 12Psy 4 3 The expression g h or better g lt his true if Sx 14y 5 AND bx 12y 4 3 Le itis false for 5x 12 5 OR 5x 12y 3 Hence the whole plane is shaded true with exeption of fanky some the points on the lines g and h Why only same points This could be a deficiency of the shading algorithm I tried with two other equations of parallel lines and found out that the non plotting of the forbidden points seems in fact to depend on the resolution of the screen and connected with this on the scaling of the plane See the next example Walter Klinger An Equation of Equations pag 1 Mae See Gey ee aa a eee a Say 5 2 SOLYECH gj 3 jig Sey l A See Sey 5 Many white spots within the shaded plane I change the scaling and there are no spots at all p 50 Walter Klinger An Equation of Equations There are the spots again but not
9. At the first view it seems to be that I am not facing Klaus Korner s sign problem But see what is happening when plotting the circles in steps 6 sols deze sol3 m z t lt 27 P a 4 F aA meta 27 4 2 2 4 There are strange arcs belonging to the other circle which I don t know how to explain I must admit that I am not an expert in conformal mappings Now I ll deal with the second circle It s centre seems to lay on y x and it passes 5 0 and 0 5 This is not enough to find its parameters I didn t know better than to use sliders for estimating the coordi nates of the centre and the radius Centre 0 73 0 73 Radius 4 36 What is the result when applying the con formal mapping from above to this sec ond circle Let s see p 34 Klaus Korner Airplane and Conformal Mapping 25 25 SUBST x 1 y 1 x y _ 4 36 COS t 0 73 4 36 SIN t 2 2 2 2 X y X y u 4 5 A 0 01 436 C0S t 73 3 1828 10 COS t 3 1828 10 SIN t 2 25377 10 4 4 5 3 1828 10 COS t 3 1828 10 SIN t 1 00377 10 4 4 4 0 01 436 SIN t 73 3 1828 10 COS t 3 1828 10 SIN t 2 4623 10 4 4 5 3 1828 10 COS t 3 1828 10 SIN t 1 00377 10 Obviously we receive the same profile as result of the mapping applied to two different circles which are surely related in any form but which one Who knows It is also possi
10. David You are absolutely right and I will add more After their calculus course my students will take the ODE course They will learn how to solve by hand easy ODEs and also they will learn how to use the desolve command of their CAS calculator Voyage 200 and starting in September Nspire CX CAS Those who have set the complex format to Real will sometimes get a Non real result message on their screen when trying to solve some ODE With complex format set to Rectangular we get a solution involving the number i but choosing the appropriate domain and this is a nice opportunity to recall the existence and uniqueness of solution the i will disappear and students will obtain a real solution In my talk at ACA 2007 Using In abs x as an Antiderivative for 1 x is a Bad Choice I used the separable ODE y 4 y x 2 9 y 0 5 In order to remove the imaginary unit i from the solution found by the device and also Derive Maple students have to use the fact that we are looking for a solution defined in the strip 3 lt x lt 3 because the initial condition is at the point 0 5 according to the existence and uniqueness theorem this ODE has a unique solution defined for 3 lt x lt 3 because both f and dif f y are continuous in some rectangle centred at the point 0 5 with 3 lt x lt 3 here f stands for 4 y x 2 9 And restricting the domain gives this unique and real solution
11. Fi 2 2 2 2 4 nu U Jek Sky udtu Rum 944k u y IEHLEk Sketvy UF HU u 2 Bulle nu 2 2 2 4 2 2 4 2 2 2 2 2 4 ma U feieGhe k Bk C udtu Rum v 344k u I dEk Bkv UF HU Bee 2 Sally oe p28 Klaus K rner Airplane and Conformal Mapping I sent my results to Mr K rner and he he does not use DERIVE transferred the expressions to MS Excel and then had some satisfying results He wrote the most important parts of his letter will follow in a translation Sehr geehrter Herr B hm nun habe ich endlich Zeit gefunden um Ihnen nochmals Dank zu sagen f r die bermittelten Formeln als Ergebnis der DERIVE Rechnungen Zun chst zu Ihrer Frage woher ich Ihre Adresse bekommen habe Nachdem ich vor mehreren Jahren von einem Geschaftsfreund in S dafrika auf DERIVE hingewiesen wurde er hatte eine h here Position in einer Raffinerie und war nebenbei Frei zeitmathematiker ist mir die Idee zur Nutzung dieses Programms f r mein anstehendes Problem in den Sinn gekommen Im Internet habe ich dann bei der DERIVE USER GROUP Ihren Namen gefunden Im Internet habe ich dann auch irgendwo Ihren Artikel ber die moderne Unterrichtung des Faches Mathematik gelesen und finde ihn echt zukunftsweisend leider finde ich ihn nicht mehr Bitte um Hilfe Zu meiner Zeit und wahrscheinlich auch noch zu Ihrer Zeit wurde Mathematik sehr trocken gelehrt und dadurch hatte dieses Fach auch wenig Freude f r die Sch ler
12. Ich erinnere mich noch an meine Oberstufe wo Kurvendiskussion besprochen doch niemals ein Beispiel aus dem t glichen Leben angef gt wurde Daf r wurden wir mit sph rischer Trigonometrie Himmelskunde gequ lt Beim Studium allgem Maschinenbau an der TH M nchen waren die Mathe Vorlesungen auch sehr schwer und oft undurchsichtig Ein ganzes Semester besch ftigte sich mit Funk tionentheorie konformen Abbildungen auch 3 dimensional und auch mit geschlossenen und offenen Torusformen Im Nachhinein halte ich das f r ein Maschinenbaustudium f r v l lig bertrieben Doch nun zu unserem Rechenbeispiel Ich habe mir die Beschreibung des DERIVES Programms von Frau Maria Koth Universitat Wien angesehen und festgestellt welche traumhaften M glichkeiten dieses Programm bie tet Aus Ihren Formeln habe ich die 3 L sung und die 4 L sung programmiert und ganz eigen artigerweise festgestellt dass teilweise die Ergebnisse der 3 L sung richtig sind und teilwei se die der 4 L sung Damit Sie eventuell mein Beispiel mit den Zeichenmethoden von DERIVE nachempfinden k nnen m chte ich folgende Daten nennen Der Kreis hat den Mittelpunkt M 1 1 und einen Radius von Wurzel aus 37 Damit kann man schrittweise ich habe immer 5 genommen die Umfangskoordinaten berechnen Mit den Ihnen bekannten Formeln u und v ergibt sich dann eine Form die einem Tragfl gel hnelt Klaus Korner Airplane and Conformal Mapping
13. Textbooks and the arctanh Hello Michel Not being an educator I am probably least qualified of your addressees to know why textbooks are so quiet about the arctanh function Or more generally why the inverse trigonometric functions get so much more ink than the inverse hyperbolic ones This is despite the undeniable symmetry be tween these two sets of functions This symmetry is evidenced by the identical parallel treatments they receive in Chapter 4 of the recently released NIST Handbook of Mathematical Functions which is the authoritative successor to Abramowitz and Stegun p20 Textbooks and the arctanh So good luck on your quest to make the inverse hyperbolic functions and hyperbolic functions first class objects of study The father of Rubi Albert De David Jeffrey Objet Re Textbooks and the arctanh My guess is that trigonometric functions are introduced early in high school because of the geometric application Hyperbolic functions on the other hand have to wait until the exponential function has been covered and thereafter they never catch up to trig functions in the mathematical mind One advantage of arctanh sin x or invtanh sin x as prefer to write it is that it is always a real value unlike In sec x tan x For the same reason arctanh x 2 1 x 2 1 is a better integral of 1 x than In x but don t think it will catch on DJJ From Michel Beaudin Subject RE Textbooks and
14. amp CAS TI User Group It is published at least four times a year with a content of 40 pages minimum The goals of the DNL are to enable the ex change of experiences made with DERIVE TI CAS and other CAS as well to create a group to discuss the possibilities of new methodical and didactical manners in teaching mathematics Editor Mag Josef B hm D Lust 1 A 3042 W rmla Austria Phone 43 0 660 3136365 e mail nojo boehm pgv at Contributions Please send all contributions to the Editor Non English speakers are encouraged to write their contributions in English to rein force the international touch of the DNL It must be said though that non English articles will be warmly welcomed nonethe less Your contributions will be edited but not assessed By submitting articles the author gives his consent for reprinting it in the DNL The more contributions you will send the more lively and richer in contents the DERIVE amp CAS TI Newsletter will be Next issue December 2011 Preview Contributions waiting to be published Some simulations of Random Experiments J B hm AUT Lorenz Kopp GER Wonderful World of Pedal Curves J B hm AUT Tools for 3D Problems P L ke Rosendahl GER Hill Encription J B hm AUT Simulating a Graphing Calculator in DERIVE J B hm AUT Do you know this Cabri amp CAS on PC and Handheld W Wegscheider AUT An Interesting Problem with a Triangle Steiner Point P Luke Rosen
15. c 0 and in case ac b 0 to 0 0 e or respec 0o 0 f 0 e 0 00 d tively to O c O As the kind of a conic section is neither changed by a translation d 0 nor by a rotation we can use matrices for the classifications provided in paragraph 2 For doing this we introduce new quantities A det M d det 4 and S atc They are invariant with respect to the rotations and translations from above So we get Empty set Hyperbola Intersecting pair of straight lines fe Patoa Pair of parallel straight lines Table 3 The general case first m b c e sinl cosla of 72 0 1 a b sin 6 0 rot def 0 0 1 the determinantes det m det roz det tra the translation with u and v from 2 3 m tra m tralu and v b e c d a e b d t a c b a c b det m 1 det m the rotation of ml with 8 from 2 4 1 2 b m2 rof m1 rop anf this matrix must be simplified using tCollect and specifications for signla c and la c ac 0 4 m2 tCollectlm 2 1 a c 2 2 2 2 2 a 2 a ct 4 b c Again a lot of algebra Representation of Conic Sections In this section some constructional ideas for conic sections are presented All these ideas use the zeros command from TI Nspire CAS So they can not be used in the nonCAS version of TI Nspire If p x y is a 2 degree polynomial in x and y then zeros
16. e und nochmals Danke Klaus K rner Ein Teil der Excel Datei und die zugeh rigen Grafiken folgen der englischen ber setzung der wichtigen Teile des Briefes finally found time to thank once more for the sent formulae which resulted from the DE RIVE calculations Using your expressions took the real solutions 3 and 4 and found out that partially re sults based on 3 and partially based on 4 are correct If you would like to reproduce my example with DERIVE I ll give the following data The circle is given by its centre M 1 1 and its radius v37 I calculated the points of the cir cumference in 5 steps Then using the formulae u and v one obtains the figure which is similar to an airplane wing These u and v values were used applying the extended expressions of solutions 3 and 4 for achieving a back transformation to get the circle Curiously this does not work without problems found out that the problems are connected with the sign of the respective y values of the initial circle For y gt 0 one has to take the 4 solution otherwise the 3 one When y changes its sign then the results of 3 and 4 show a jump p 30 Klaus K rner Airplane and Conformal Mapping As I want to start with a certain given wing profile in order to investigate the form resulting from the back transformation cannot guess in advance whether y changes its sign or not hope and am pretty sure t
17. of tools that let me state restrictions on my inequality s variables then state the inequality itself and returns either true or false or or the same object if the restrictions are too weak to determine whether the inequality is true or false The r n variables below are annual or some other time period rates of growth such as in terest rates inflation and deflation rates real output growth or decline rates etc Rates of growth r n can be either positive zero or negative The number N is the number of time periods being considered Sometimes total rates of growth over N time periods are calculated by summing the N periodic rates of change r n n 1 2 N which is less accurate than multiplying the N corresponding 1 r n factors Summing the r n can give results that are either larger or smaller than the more accurate multiplication of the 1 r n However a common situation is that over N time periods more rates of growth r n are posi tive than negative If that difference is sufficiently large summing the N periodic rates of change r n gives a lower total rate of growth over N periods than does multiplyng the corre sponding 1 r n numbers so that summation understates the true total rate of change over N periods Mathematically that common situation inequality can be written in three basic ways See statements 2 3 and 4 below SATISFYING THE COMMON SITUATION INEQUALITY A SUFFICIENT CONDITION
18. p x y y yields a list of root expressions in x The G amp G application interprets this list as a family of functions in x with two elements maximum So a definition f1 x zeros p x y y in G amp G plots a graph with two colors and labels f1_1 and f1_2 Many thanks to Philippe Fortin France for providing most of the ideas realized in the following paragraphs 4 1 The Conic Section Generator CS 4 The classification of conic sections from paragraph 2 and 3 can easily be studied with the ConicSectionGenerator CSG The workspace is divided into two G amp G windows a smaller Geometry and a wider Graphs window In the left window we insert 6 sliders for variables a through f having a range from 10 to 10 which can easily be changed The step size is 0 1 for a b c and 0 5 for d e fand may also be changed as needed The right window has the definition f1 x zeros a x 2bxy cy 2dx 2e y f 0 y in function mode which plots the maximum two components of the conic section a bd In addition the determinants A al C det 2 e and the sum S a tc def are shown in the upper left corner In the upper right corner we see angle of inclination of the main axis and the coordi nates of the centre point By modifying parameters a through f ellipses hyperbolas parabolas and straight lines can be created and can be compared with the summaries given in tables 1 2 and 3 isi er u a
19. p29 Die berechneten u und v Werte habe ich nun verwendet um mit den langen Gleichungen der 3 und 4 L sung eine R ck Transformation in Richtung Kreis zu erreichen Eigenartigerweise geht das dann nicht ganz ohne Probleme Ich habe n mlich festgestellt dass es am Vorzeichen des jeweiligen y Werts des Ausgangskreises liegt Im Bereich positi ver y Werte stimmt das Ergebnis der 4 L sung sonst das Ergebnis der 3 L sung Beim Vorzeichenwechsel von y machen die Ergebnisse der 3 und der 4 L sung einen Sprung Da ich aber eine bestimmte Profilform vorgeben m chte um dann mit der R cktransformation zu sehen welche Form herauskommt kann ich ja nicht vorausahnen ob y einen Vorzei chenwechsel macht oder nicht Ich hoffe doch und glaube auch dass ich bei der Umsetzung dieser komplizierten Formeln nach EXCEL keinen Fehler gemacht habe Bitte probieren Sie es vielleicht einmal mit DERIVE Es gibt ja mehrere Versionen von DERIVE Ich habe auch geh rt dass die Version DERI VES besser sein soll als DERIVE6 Mit welcher Fassung arbeiten Sie Sollten Sie mit den Derive Plot Funktionen auch die eigenartigen Ergebnisse wie ich mit meinen EXCEL Methoden erhalten dann bitte ich Sie doch mit Mathematica nochmals einen Versuch zu starten Vielleicht kommt dann irgendeine andere Losung heraus Als Anhang schicke ich Ihnen meine EXCEL Ergebnisse Darin k nnen Sie ndern oder er weitern wie Sie m chten F r heute viele Gr
20. suggestions After downloading all of the zipped MTH files from the DUG website one at a time I then saw and fol lowed the recommendation to look at the back of the PDF membership file which contained the statement Disk of the year containing all MTH files is included It s been a long day almost 2 00 a m California time so I likely am operating in stupid mode but In what is that Disk included With or without a disk it might be useful if the DUG website had three large zip files that included each of the three groups of num bered MTH zip files Similarly since the newsletters bulletins apparently include arti cles that explain the use of those zipped mth files it might be useful if the DUG website had zipped packages of dnl pdf files so each pdf file wouldn t have to be downloaded individually I ll close with a very minor request Mathematicians and also Derive like to write polynomial functions downhill meaning with the largest exponent term first followed by successively smaller exponent terms and then a constant I do understand that that se quence has mathematical advantages for example by facilitating di viding one polynomial by another But economists often like to write polynomials uphill constant term first followed by terms with successively larger exponents because we often don t know how big the polynomial needs to be until we use data to estimate its co efficients And it is more convenient t
21. the imaginary part is mirrored wrt to the origin Real Part From the eDUG Forum Tensors and DERIVE for DOS From the eDUG Forum Tensor Algebra and DERIVE DOS Thomas Fowler I posted two new versions of the Tensor Algebra file updating the fine work done by Hans Dudler The version for Derive 5 amp 6 is TensorAlgebraExtd mth The version for Derive 3 last DOS release is TENSORNE MTH I also posted an updated and expanded explanation of the operations Ten sorAlgebra doc The extensions are primarily for functions in use n the study of general relativity and include the Riemann Curvature function K Tom See pages 39 41 Rick Nungester I have Derive for DOS v3 04 working on Windows XP but it always starts full screen instead of in a small terminal like text window Right clicking on the shortcut and seeing its Properties shows tabs General Program Font Memory Screen Misc Compatibility and Summary I ve played with vari ous settings but it always starts full screen Can Derive for DOS be made to open n a small window instead of full screen I also have Derive for Windows v5 06 but use Derive for DOS v3 04 on my circa 1994 HP 200LX palmtop and want the same functionality on Windows Rick Thomas Fowler Until TI decides to issue a version of DERIVE for Android devices which would be a cash cow for them you can still get most of the functionality if you have your old DOS release The last DOS re le
22. the same as above My next question What will happen in case of a solution Take equations h and J 4 SOLYECh j 5 aix Sey 12 v Zn Gey 5S A Din 4 Gey Gov See Sey E 12 The h system has a unique solution but 7 cannot be plotted 6 SOLYECh j x 12 Sey 5 1 y 1 Sey Sea yw oy a S nr Wr HE o a O OOO 3 2 3 2 8 SOLVEfh hd 9 true Any comments on this interesting behavior are welcome I tried other Computer Algebra Systems Most of them do not accept the equation of equations I tried with changed resolution of the screen 1024 x 768 instead of 1280 x 1024 and received other patterns in the shaded plane
23. 71 1708292528 0 91027781213277 x 6 5542685029883 37 4498 0 026702 y 1 x2410 4995 x434 71 17 0 910278 x 6 55427 0 026702412295927 y 1 0000002420262 x2 10 499493646781 x 34 711708292528 0 91027781213277 x 6 5542685029883 0 910278 x 6 55427 0 910278 x 0 026702 y 5 96621 1 x2 10 4995 x 34 71 17 3 24439 8 x 0 000001 2 lo 910278 x 0 026702412295927 5 96620778706 1 0000002420262 x 10 499493646781 x 34 711708292528 0 828606 x 0 029334 y 223 433 7 1 2410 4995 x 34 71 17 2 2 1 0000004840525 x 10 499493646781 x 34 71 1708292528 rightlo 82860603 2 expandlienl0 828606037284 x 0 029334348732944 y 223 43328838384 0 171394 x 0 048613 x y 0 362317 x 0 000713 y2 0 318624 y 0 88391 0 The next contribution will do this job for you p 14 Josef Bohm A Conics Explorer and a Conics Trainer A Conics Explorer and a Conics Trainer Josef Bohm W rmla Austria As promised on page 13 here is the analysis of the bridge conic 2 2 task 0 171394 x 0 048613 x y 0 362317 x 0 000713 y 0 318624 y 0 88391 0 ans Type Center Real axis 2a Imaginary axis 2b Real Vertices 10 99451200 85 94367265 Hyperbola 5 249788684 44 47126414 83 73679034 12 79439536 0 4949346356 2 998855624 Asymptotes 0 09253588869 t 5 249788684 6 620111512 t 44 47126414 1 888548332 t 5 249788684 44 47126414 6 345693384 t It seems to be a hyperbola I say it see
24. 94 9 y w4y24dk4 416k v2 4 u yt Ak F BYW amp uf ives ad 1 i pres ed a Infiz ParametricPlot sol4 a b 25 t 0 2 Pi AspectRatio gt Automatic p 36 Klaus K rner Airplane and Conformal Mapping At first I performed the MATHEMATICA plots using u_ and v_ from page 32 Then I asked MATHEMATICA to do the substitution and received quite other expressions c and d in In 16 Infisj x 14 25 x2 y 2 we 1 25 x 24 y 2 Z x gt Sqrt 37 Cos t 1 y gt Sqrt _ 25 25 Duff S ay a O E Se eal E ea LI E T E ae C37 eet ra a ra nie e 37 cos 1 1 0 F 437 cos t 1 437 sin 1 d 37 sin t 1 1 S 437 cos 1 437 sin 1 But the resulting plot is pretty the same It might be a good exercise for students proofing the identity of c and u_ d and v_ For us it is easier let s ask DERIVE ce 37 COS t 1 1 i 37 COS t D 37 SIN t 1 Precision Exact Notation Rational uU_ c 0 Solutions 1 and 2 are complex back transformation applying them gives also complex expressions and we cannot obtain any plot I wondered what the plot of the real part and the imaginary part too of course would look like and I was presented a big surprise RECsoll u_ v_ 25 IM soll u_ v_ 25 Real Part Imaginary Part EIS The real part was again the basic wing profile Solution 2 gave also the wing
25. C con randem choice from above Zufallsaufgabe aus den obigen DISPLAYC DISPLAYC ans gives the analysis ant liefert die Analyse DISPLAYC 2 start start_ 3 task z coniclfa b c rd Frog rd RANDOMCS Loop a iz 20 RANDOM C413 b 20 RANDOM C413 4 rt 30 RANDOM C613 If f a b U Frog task a x b y 2 f 0 a x 2 b y 0 a y b x 0 task IF rd lt 6 task 1 IF rd e 7 task task 13 RETURN task Josef B hm A Conics Explorer and a Conics Trainer p 17 Eanes D C d e ro Prog rd RANDOM 8 a 20 RANDOM 41 b 20 RANDOM 41 5 c z 20 RANDOM 41 d 20 RANDOM 41 e 30 RANDOM 61 ask gt a3 2 b 2 zer 4 oa yre 0 arg rdy te U by 2 4 ce fy ee task IF rd lt 6 taskil IF rd lt 7 task 2 task 3 RETURN task rl gt crd e f Prog a 20 RANDOM 41 b x 20 RANDOM 41 c 20 RANDOM 41 6 d 20 RANDOM 41 e x 20 RANDOM 61 f 30 RANDOM 61 task ZEN Pc by ver rdyvrexyars o RETURN EXPANDCtask 7 Corl cornielt 8 conz coniczt 9 con3 conic3 con cor cond con3 10 RANDOM 3 1 Expression 11 not printed here is the main function ch_conic It investigates the random conic con from 10 which is also stored as global variable task and stores the results in ans and ant for the German users 12 ans ch_conic task 13 ant ch_conic task
26. DOS diskettes were for Derive 4 0 7 I have a Derive 3 manual in addition to the Derive 6 manual by Bernhard Kutzler and Vlasta Kokol Voljc so apparently my use of Derive goes back at least to Derive 3 x The Derive 3 manual is quite useful in providing more detail about various Derive features and capabili ties gt Applications in Economics would be highly appreciated lt Roger Folsom Growth Rates and Comparative Statics You may already have some I have on paper a message posted by RaN Eolsshm on the Derave BBS 2 April 1993 about Folsom s Comparative Statics Technique I was responding to a message 2896 19 April 1993 on Derive Mathware apparently a different BBS by Harald Lang to Roger Folsom with the title Harald Lang s Use of Taylor s Series to Specify the Chain Rule And that was preceded again on Derive Mathware by message 2893 by Hadud to Roger Folsom about Response To Questions which appar ently was a response to a set of 7 questions posted by me Unfortu nately I don t have the seven questions but Hadud s answers give pretty good clues about what my questions were However they ap parently were about how Derive worked and did not have anything to do specifically with economics But Comparative Statics is something that economists do fre quently If you havea Systen of Tunclaons Is a model describing a set of markets aS in microeconomics or describing in b
27. THE DERIVE NEWSLETTER 83 ISSN 1990 7079 THE BULLETIN OF THE DLI tv LLZ Ieou Ogg Contents Letter of the Editor Editorial Preview Wolfgang Pr pper On Quadratics and Conics Josef B hm A Conics Explorer and a Conics Trainer Textbooks and the arctanh Klaus K rner Airplane and Conformal Mapping Tensor Algebra and DERIVE for DOS Tom Fowler Tensor Algebra Roger Folsom Growth Rates and Comparative Statics Walter Klinger An Equation of Equations September 2011 Sam information sans Proceedings of former DERIVE amp CAS TI and ACDCA Conferences You are invited to browse and to download lectures and workshops all in df format of almost all earlier Conferences starting with Krems 1992 and ending with Malaga 2010 http rfdz ph noe ac at acdca konferenzen html Krems 1992 Krems 1993 Derive Days Dusseldorf 1995 Bonn 1996 Gettysburg 1998 Portoroz 2000 Liverpool 2000 Vienna 2002 Montreal 2004 Dresden 2006 Buffels poort South Africa 2008 Malaga 2010 Some proceedings have been published originally in printed form only I scanned them all and interlinked them with the respective contents page Many thanks to Peter Hof bauer who set all files on the web Happy Browsing and Happy Remembering for many of you and don t forget Attend TIME2012 in Tartu Estonia July 11 14 2012 http time2012 ut ee Browse a library of over 2 400 educational videos at http www kha
28. and c have equal signs the equation from above is only fulfilled by the origin In all other cases there will be no more genuine intersection figure i e the solution set is empty Except if a c f 0 Then equation 2 is fulfilled by the whole xy plane The following table shows this in a clear form Wolfgang Propper On Quadratics and Conic Sections leere Pair of parallel lines with respect to x axis EICHE Pair of parallel lines with respect to y axis 0 Double line the x axis lel eve ole fs ere os erevee o ofo moms Table 2 The impact of coefficients b d and e CS 2 3 and 2 4 If just b O and further on d e 0 then a rotation where x is replaced by x cos y sin 6 and y is replaced by x sin 6 y cos 0 can bring the coefficient of the mixed term to zero without introducing linear terms in x or y This means that we have again one of the previous cases System Matrix syst translation to remove linear terms koe ftermlx u ytv _ Yo 2 s Coefficient Matrix koef X XY y x y consi 9 a a2bec 2 a utb v d 2 b u c v e au 2wlbv d cv 2ev f a b a u b v d System Matrix syst b c butec vie 2 autbyv d bute vte a u 2 u b vid te v 2 e vtf Done b e c d a e koef termlx u y v u and yv a c b yy x const Coeffici I 1 9 2 Coefficient Matrix koef le fe2 b2 2 b d e c d z a c b 0 0 gt a c b Sys
29. ase was DERIVE 3 I have it running on an Android tablet under aDOSBox It works well but you really need an external keyboard You can speed it up by going into the menu option from the F1 key selecting DOSBox and increasing the CPU cycle variable I am running mine at 87350 about 20x faster than the default which naturally emulates and 8088 Tom Aleksey Tetyorko IMHO The last DOS release was DERIVE 4 11 with the extended syntax Sorry one question can you switch to a windowed mode by pressing Alt Enter I cannot My DOS Derives are 3 11 and 4 11 But when I ve changed the Derive Display Mode to Text Option Display Mode Text I made my Derive into a window by Alt Enter Do your Derive and XP behave in this manner Aleksey p 38 From the eDUG Forum Tensors and DERIVE for DOS Rick Nungester Thank you Aleksey that is just what I needed Now I can start Derive for DOS 3 04 from a Desktop icon directly into a window without using Alt Enter at all Here is what I did In Derive make Text mode the start up setting Start Derive no other changes before the next steps Option Display Mode Text Transfer Save State DERIVE INI Y overwrite Quit In Windows XP make window the start up choice Desktop icon right click Properties Screen tab Usage Window OK Now whenever I double click the Desktop icon Derive for DOS 3 04 starts in a text window Plotting still works drawing lines as a series of
30. ation translation of the most important parts of the mail next page Sehr geehrter Herr Bohm durch Zufall bin ich auf Ihre e mail Adresse gesto en und m chte Sie wenn es fur Sie keine allzu gro e M he macht um eine mathematische Hilfe bitten Ich habe geh rt dass die Lo sung meines Problems mit DERIVE m glich w re doch kann ich selbst damit nicht umge hen Einige S tze zu meiner Person Ich bin 1939 geboren habe in M nchen an der TU Maschi nenbau studiert war dann mein ganzes Berufsleben in einer Turbinenfabrik in N rnberg be sch ftigt In relativ jungen Jahren wurde ich zum Leiter der technischen Berechnung und der technischen EDV Abteilung gemacht und hatte mit Thermodynamik Str mungsmechanik Festigkeitslehre Schwingungslehre Materialfragen Lagerauslegung Wellenschwingungen und Wuchttechnik zu tun Mein ganzes Berufsleben hat mir viel Spal gemacht Als techni sches Gewissen unserer Vertriebsleute hatte ich die M glichkeit fast die ganze Welt zu be reisen doch gesehen hatte ich dabei relativ wenig weil nie Zeit war Nun im Ruhestand denke ich manchmal ber Probleme nach die mich w hrend meines Be rufslebens irgendwie begleitet haben doch deren L sung aus Zeitmangel immer verschoben wurde Es geht um das Wissensgebiet konforme Abbildungen Wenn Sie sich auch damit ausken nen dann berlesen Sie bitte meine nachfolgenden kurzen einf hrenden Worte Mittels konformer Abbildung kann man einen Kreis i
31. ays conic sections have nearly disappeared from textbooks Nevertheless it is an in teresting field containing a lot of nontrivial algebra 1 General 2 Variable Quadratic Equations The intersection of a circular cone with the equation z x y in R with a plane ax y yz e yields aso called quadratic which only has terms in x y x y x y and a constant term It looks like 1 ax 2bxy cy 2dx 2ey f 0 Due to its geometrical genesis its graph is called conic section or just conic The left side of this equation is elegantly expressed by W M W with M asymmetric ma a bd X trix M E C and Wa vector W This construction leads to the factors 2 in def 1 the coefficients of the mixed and the linear terms CS 2 2 the declarations oh 1 termlx y w x y m w xy 1 1 eg nlx y term x y 0 for the examples we must declare a specific matrix M r 5 a x 2 b xyt2 d xte y 2 e y f 0 d Josette 2nd degree polynomial in x and ce e f i 2 2 a 0 d PRS LIE NE PERTINENT ax 2 d x e y f o term x y a general expression for a parabola zi e f w we become specific 0 eqn x y 225 ja an ellipsi rnd factor egn x y a pair of straight line seqn x y H x7 1 a parabola 11 99 1 With this contribution is connected an extended TI Nspire document OnConicSections_NN tns NN is 20 or 30 depending on w
32. ble to generate trochoids in two different ways I am much more surprised by the two little pieces of the circles Do you have any explanation for this Who is an expert Josef Just now finalizing this DNL I performed the calculations with MATHEMATICA and received the same results some in another form I found that the DERIVE results are much more compact You can find parts of the MATHEMATICA treatment after Mr Korner s next mail And I had one more question which I tried to answer for myself and for you too of course But see first Klaus K rner s response Klaus Korner Airplane and Conformal Mapping p 35 Hello Mr Bohm would like to inform you about an interesting link http www excelformeln de uberuns html You can find numerous meaningful and also less meaningful EXCEL applications free of charge Mr Klaus K hnlein is the main organizer of this publication Finally l d like to thank for your efforts Your results agree completely with my Excel solutions will try to back transform the profile of a real turbine blade in order to find out whether it results as a more or less regular figure Best regards Klaus Korner These are the MATHEMATICA calculations and plots Infid ParametricPlot sol3 a b 25 ft 0 2 Pi AspectRatio gt Automatic Outf10 Graphics Infiij sol4 u_ v_ Kk 1 7 2v3 u v2 dk 16k v4 u v 4k r UET E a T Bu
33. ctivates the case sensitive and word input modes As an introduction to the concepts described in this document the demonstration files Tensor1 dmo and Tensor2 dmo are recommended They should be loa ded using the File gt Load gt Demo File command 1 Tensor Representation in DERIVE From the algebraic point of view tensors are an extension of the vector matrix concept to higher order arrays Thus a tensor of rank order 1 is a vector A rank 2 tensor is a matrix which DERIVE treats as a vector of vectors A rank 3 tensor is a vector of matrices a rank 4 tensor is a matrix of matrices and so on The obvious way to handle tensors in DERIVE is therefore to extend the vector concept to higher ranks orders The equivalent of the ELEMENT function to extract an element from a tensor is the function see function descriptions below EL T A iv where A is the name of the tensor and iv is the index vector quotes used to distinguish a DERIVE vector from a rank 1 tensor As an example consider the tensor of rank order 3 and dimension 3 NS So x Then EL T A 2 3 2 returns q EL T A 3 1 3 returns u As is clear from the example the first index element 1 of the index vector selects the ma trix the second index selects the row vector in the selected matrix and the third index finally selects the element in the row vector Note that under DERIVE the indices must range from 1 to n the space dimension zero often used in r
34. d y we get five equations An equation like a 2b c 1 avoids that a band c are zero at a time and we Call it normalization condition This system yields a unique solution for a through f For the implementation we define a function fivepoint P1 P2 P3 P4 P5 where P1 P5 are 2 dimensional column vectors with components p1 p5 and q1 q5 This function solves the system mentioned above and assigns the solution to ax 2bxy cy 2dx 2ey f We only have to draw five different points in a G amp G window assign their coordinates to p1 p5 and q1 q5 and define f1 x zeros fivepoint P1 P2 P3 P4 P5 y A conic section through those five points appears Now we can grab one or another point and drag it around observing what is happening have provided a stand alone document to run on handhelds It is called FivePoint_NN tns and consists of only two pages a short introduction and the con struction in G amp G The function fivepoint is contained in the document but no longer di rectly visible Two screenshots of fivepoint are given on the next page PY 3 50 7 PY 1 2 3 3 Ps 3 18 Pa 1 5 0 3 rs oa Additional Comments of the Editor It might be a good question for students how to explain the fact that there are 6 unknown coefficients in the quadratic but only five points are sufficient to define a conic 2 Dividing 1 by fyields a x b xy c y d x e y 1 0 wit
35. dahl GER Overcoming Branch amp Bound by Simulation J Bohm AUT Graphics World Currency Change P Charland CAN Cubics Quartics Interesting features T Koller amp J Bohm AUT Logos of Companies as an Inspiration for Math Teaching Exciting Surfaces in the FAZ Pierre Charland s Graphics Gallery BooleanPlots mth P Schofield UK Old traditional examples for a CAS what s new J Bohm AUT Truth Tables on the TI M R Phillips USA Where oh Where is It GPS with CAS C amp P Leinbach USA Embroidery Patterns H Ludwig GER Mandelbrot and Newton with DERIVE Roman HaSek CZ amp Rob Gough UK Tutorials for the NSpireCAS G Herweyers BEL Some Projects with Students R Schroder GER Dirac Algebra Clifford Algebra D R Lunsford USA Treating Differential Equations M Beaudin G Piccard Ch Trottier CAN A New Approach to Taylor Series D Oertel GER Statistics with Tl Nspire G Herweyers BEL Cesar Multiplication G Sch dl AUT Henon amp Co Find your very own Strange Attractor J Bohm AUT Rational Hooks J Lechner AUT Cubus Simus H Ludwig GER Simulation of Dynamic Systems with various Tools J Bohm AUT and others Impressum Medieninhaber DERIVE User Group A 3042 W rmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef Bohm On Quadratics and Conic Sections Wolfgang Propper Nurnberg Germany w proepper franken online de Nowad
36. e taught to write 1 tan 2 1 cos 2 instead of our 1 tan 2 sec 2 We always boast that mathematics is a universal language but maybe not quite as universal as we hope Some years ago James Davenport was working on the open math project and reminded us of even more obscure functions such as versin x 1 cos x See http en wikipedia org wiki File Circle trigb svg for an attractive diagram David J Textbooks and the arctanh p 23 Von David R Stoutemyer Betreff Re AW RE Textbooks and the arctanh Tables of csc sec and cot usefully replaced a division with a multiplication in the days of manual computation However those days are long gone and most students quickly forget even whether csc is the reciprocal of cos or sin Therefore we should stop wasting valuable curriculum time torturing students with these secondary trigonometric functions and their hy perbolic counterparts just as we no longer torture students with other trivially related trigo nometric functions such the haversine which was convenient for manual celestial navigation computations We don t have a special name for the reciprocal of In abs etc Also think that hyperbolic functions are way less important than trigonometric because exponentials are often a reasonable alternative to hyperbolics whereas complex exponen tials are a less attractive alternative to sinusoids Yes even prefer results absent i when offers no strong advantage
37. e to insert the focus point condition directly in G amp G that means to define f1 x zeros abs norm x y f1 1 f12 norm x y f21 f22 K y Wolfgang Pr pper On Quadratics and Conic Sections p 11 If previously the coordinates and k have been created like mentioned above we should get the graph of a conic section However it takes quite a long time depending on the performance of the computer to receive a result And if the foci are dragged around or k is changed Nspire is blocked for several minutes and the dynamics of the application gets lost A final remark As for this example there are not so many objects on the screen it can also run on a TI Nspire CAS Handheld device with reasonable visibility The compli cated calculations from CS 5 2 are omitted and just the result is inserted The file Focus_NN tns where again NN 20 indicates a document for Nspire version 2 x and NN 30 is for version 3 x is among the files connected with this DNL 5 3 OnConicSe _30 s afl x 5 3 OnConicSe _30 s afl gt lt k 96 Art des kbsHiligiseit2 Hyperbe FO i i Art desakl Hiliiseit2 Hyperbe Oe F ee r 1 8 1 5 20 LS ee 20 4 5 5 2 9 e NE eA 012 52 gt gt Two handheld screenshots of FOCUS 4 3 The Five Point Feature CS 6 The quadratic a x 2b xy cy 2d x 2e y f 0 in x and y contains 6 coeffi cients With the coordinates of five different points inserted for x an
38. echnische Universit t Graz http www math tugraz at ganster lv_vektoranalysis ss 10 17 tensor definition pdf GERMAN pa Roger Folsom Growth Rates and Comparative Statics Von R N Folsom rnfolsom redshift com Gesendet Donnerstag 16 Dezember 2010 09 24 Ans Bohm Joser Betreff A Constrained Inequality s Truth Josef First thank you very much for your message of 13 December I very much appreciate your work supporting the Derive User Group and me I have attached a zipped file GrowthRates Summed vs Multi plied N3 Derive that contains a dfw file of the same name which describes a problem I am having I hope you can pass the file on to someone who might know the answer or that you decide it is worth including in the next newsletter bulletin But I would be happy to revise this document if you think I Should do S0 Sending the Comparative Statics Technique information is defi nitely on my short term todo List Thanks again for your help As you will see below am trying to formally prove the circumstances under which a particu lar inequality is true and Derive s Simplify or Solve command actually says that it is true For now all need is sufficient conditions But have not been able to figure out the Derive6 1 command or commands that will do that My sources are Bernhard Kutzler and Vlasta Kolol Voljc s Introduction to Derive 6 Ad vanced Mathematics for Your PC hereafter Der
39. elativity is not allowed From the above ex ample we deduce that EL T A i j k lt gt ELEMENT ELEMENT ELEMENT A j j k in this case and the extension to higher ranks is obvious Note that EL T A i lt gt ELEMENT A I First page of the documentation one example follows pa Tom Fowler Tensor Algebra CURVE T T T s vector components needed for curvature and geodesics in terms of arc length parameter s Calculates components of b as d x gt Or ax dx dx dt b E eS x t E E 1 ds FO ds dt ds And arc length parameter s s t given by t dx dx s t EQ du 2 Si du du Easiest to understand with an example First define G metric tensor T curve whose curva ture is desired and coordinate vector x 1 0 HUR ee ee a 2 2 T a 5ECt t 0 xl Next compute Christoffel Symbols of 2 Kind 1 102 G_ CHRIS G G x 0 0 1 Sa 103 G_ z 1 i x1 0 1 x 1 0 Now convert G to parameterized coordinates by Simplify Variable Substitution and calculate arc length parameter s Replace u with t then solve for t in terms of s 0 0 1 0 eee 103 G_ z af x1 0 x1 x1 0 1 0 2 104 G z a 0 2 COS t 105 ARC_LENGTH_P T G t a TAN u 106 SOLVE s a TAN t t Real s g g 107 t araf E P ey Bs aral j FEREYE ava lal a a am paea oo 2 2 5 109 T a 5 SIGN a ee ad Tom Fowler Tensor Algebra p 41 In
40. en interval 1 lt x lt 1 as domain and this can cause a problem for students when they will be integrating 1 1 x 2 if the bounds of their definite integral are for example 3 to 5 But it is quite easier for them to use the antiderivative arctanh x instead of 2 logs or one log with quotients because of the symmetry with the antiderivative arctan x for 1 1 x 2 Moreover when textbooks will start to use arctanh x for the antiderivative of 1 1 x 2 it will be natural to perform the integral of sec x sec x 1 cos x cos x cos 2 x cos x 1 sin 2 x so the sub stitution u sin x leads to 1 1 u 2 and that will be a good reason to use arctanh sin x instead of log abs sec x tan x what textbooks are still using as an antiderivative of sec x Looking back to an old copy of the DNL journal David Stoutemyer wrote Calculus students are introduced to complex numbers in high school algebra and they are taught that antiderivatives can differ by arbitrary constants Why not exploit logarithms of negative numbers to reinforce both ideas This is not to say that we should generalize all elementary calculus to complex numbers only that if a complex number is most correct and natural in a few places exploit it rather than fight it DNL 26 page 5 Things do not change as fast as I would like Regards Michel Von Albert Rich An Michel Beaudin David Jeffrey David Stoutemyer Josef Boehm Betreff Re
41. h a Fb Now it is easy to see that 5 points are sufficient One could also divide by a TI Nspire has implemented a long desired DERIVE feature Background pictures load a picture of a bridge jpg bmp not far from my home and would like to analyze the form of the arc Ask students and teachers and there first answer will be This is a parabola Now drag the five points as accurate as possible on one of the arcs and you will find a wonderful curve Go back to the Calculator App and ask for f1 x As Wolfgang mentioned above you will find two branches take one of them do a little algebra and get the quadratic Then follow the recipe given in CS for classifying and analyzing the curve Or you write a program which does the job for you You could also use the CSG page 10 to model the bridge or other pictures This is a little bit more sophisticated but the students might find out the impact of the various parameters Finally there is a German version of this contribution at http www ti unterrichtsmaterialien net imgserv php id Propper pdf Wolfgang Propper On Quadratics and Conic Sections 25 ee lalla Let s do a little algebra to find the quadratic in its implicit form ia fix 37 4498 x2410 4995 x 34 7117 0 910278 x 6 55427 37 4498 x2 10 4995 x 34 71 17 0 910278 e 6 55427 y ftly 1 y 37 4498 x2410 4995 x 34 71 17 0 910278 x 6 55427 y 37 449809063834 x2 10 499493646781 x 34
42. hat I didn t make a mistake by transferring the bulky expres sions to EXCEL Please try with DERIVE There are several versions of DERIVE I ve heard that DERIVE 5 should be better than version 6 Which version do you use attach my Excel results You are free to change or to extend the file as you like Many thanks again Klaus Korner Parts of Klaus Korner s Excel worksheet Calculations of conformal mapping and back transformation Centre of the circle 1 Entry Yin 1 Entry Squared radius of the circle 37 Entry Factor k 25 Entry Backtransformation Backtransformation Combination of 4th Deg Pts on Circumference Conformal Mapping 4th DERIVE Solution 3rd DERIVE Solution and 3rd solution x y U Y NeW new NEw Ynew J 3 083 1 000 98618 0068 4 735 0 932 S gt 060 1 530 goo 0 169 Ade 1 269 10 4 990 056 3273 0292 4 253 1 765 15 4 875 org 0 009 0457 4 0710 le 20 4 7716 3 050 0 432 0653 3 716 At Figure 1 Base form is the circle Mi 1 1 andr 3705 The profile results farm the conformal mapping according ta the formulae u and Y Klaus Korner Airplane and Conformal Mapping p 31 Figure 2 This the back transformation applying the 4th DERIVE solution The upper circle is the right one Figure 3 Back transformation using DERIVE s 3rd solution Now the lower circle is the correct one For O to 185 the results of the 4th solution are fitting excactly to the given circle The g
43. hich release of TI Nspire CAS it should run 20 is for release 2 x and 30 for the latest release 3 x When referring to this document in this contribution we denote this by CS Pr Pg where Pr means problem and Pg the page in this problem pa Wolfgang Propper On Quadratics and Conic Sections 2 Classification of conic sections At first we will investigate the case b d e 0 Equation 1 is reduced to 2 ax cy f 0 In case a c f 0 the classification is done easily by analyzing the signs of these coeffi cients If a c and f have equal signs then the solution set is empty Otherwise we have ellipses or hyperbolas so called central curves with a centre point at the origin of the coordinate system and their half axes on the coordinate axes pate ff we _ Table 1 If however a cf 0 i e at least one of the coefficients disappears then we have 3 in stances 1 Pair of straight lines exactly one coefficient is zero lf f 0 and a and c have different signs the left side of equation 2 decomposes into two linear factors This results in two symmetric straight lines through the origin If a 0 or c 0 we get a pair of parallels with regard to the x or the y axis if the signs of a or c and f are different 2 Double line exactly two coefficients are zero In case a and for c and f disappear then the x axis or the y axis is the only one solu tion of the corresponding equation 3 Point lf f 0 and a
44. ive6 and Albert and Joan Rich Theresa Shelby and David Stoutemeyer s Derive User Manual Version 6 hereafter Derive3 Derive6 s equation solver page 57 when used to determine whether a relationship is True or False doesn t apply to what am trying to do because it doesn t allow conditions to be imposed on the equation s or in my case the inequality s variables Derive6 s discussion of the IF tool on pages 166 and 227 didn t help because didn t un derstand page 166 I don t know any Analytic Geometry and although think that did un derstand page 227 couldn t figure out how to apply it to my problem Derive3 s discussion of the IF tool pages 309 310 made sense but again couldn t figure out how to apply it to my problem My understanding is that the IF tool has four fields a test condition a then clause of re sults if the test condition is satisfied an else clause of results if the test condition is NOT satisfied and an unknown clause Derive3 s opening IF example page 309 is a payroll problem IF h lt 40 10h 400 15 h 40 where h is hours worked and 10h and 15 h 40 are worker payment amounts Unfortu nately the problem does not include the unknown clause which if not included but is needed turns out to be the entire IF tool An example that does include an unknown clause would have been helpful Roger Folsom Growth Rates and Comparative Statics p 43 What want is a tool or set
45. knowledge conformal mappings Applying a conformal mapping it is possible to transform a circle into a figure similar to a wing of an airplane This figure is also looking similar to a coop profile My idea is to transform a given coop profile back in order to check if the back transformation results in a beautiful re gular figure or something like a potato in the latter case would conclude that the given profile is not optimal in fluidic regard it could be a bluff body Given is a circle by its radius and the coordinates of its centre Applying the formulae u x 1 kx y and v y 1 k x y we obtain as trans formed figure the profile of a wing in a uv coordinate system My problem is to express the variables x and y as functions of u and v have calculated my fingers awry but I could not get a solution I was told that DERIVE should be able to solve the problem It would be great if you could provide any advice or the solution of the problem My Josef s answer was First of all I d like to present the airplane wing profile as result of the conformal mapping given in Mr Korner s mail 1 eas tia i sae SLN ty a l 2 x Ss COSCt 1 y i 437 SINCt 1 25 25 SUBST z 1a mH ya s ae 2 gt 2 2 x F x F 2 26C37 Cos ta J37 COSCt CIS7 SINCt 333 J37 SINCt 323 Lal 4 2 7 eCOS ty 2 37 SINet 39 CZF SINCE
46. last years and finally came back now to his first love DERIVE He explained the problem on the mobile phone and then sent a short DERIVE file to explain his problem more detailed An Equation of Equations Walter Klinger Stockerau Austria Textbook 5th form 15 years Example 899 1 eos se 2 sa NS Sea Dee He Gein What do we expect when solving this equation Input produced by students In fact there are two equations with tow unknowns 3 SoLYEig h x y 4 Dig Lily 4 5 A Dig L2ey 3 Strange answer How to interpret this answer Lets plot 1t What 1S an equation of two equations Plot of expression 2 So ORS ROR RI RI IIIS SS EEES BESETE SERMON EEEE EEEE E O E AA AEE A teen R RERRERERRERR EEE ERITERETTEEEER CAAA Plot of expression 4 p 48 Walter Klinger An Equation of Equations My first answer via phone was that I cold imagine that DERIVE is interpreting h and g as statements and then treats expression h g as a Boolean expression But I was not really sure and asked myself for an explanation for 4 What to do Simply ask DERIVE applying its really unique Stepwise simplification 7 g h 8 Ses a ley 2 Hs Gan Ley 5 fp and g are of logical type P oe Pe y 9 Sex L vy 3 e Dex liry 5 peq p qia q pP 10 Sas l2iy gt 5ix dee 5J A 5 X l2 y 5 o 5ix ley 3 e Dix ley 5 P 4q gt na PY y ell mha lay See Saya SA a lye na Dk
47. ms to be because unlike to translation and rotation the conics can change their type after applying a perspective projection which is given by taking a picture of the object with a camera So it could be indeed a parabola In fact the program would not give the answer parabola even if it would be one because of numerical reasons As Wolfgang explained on page 6 the discriminant must become zero and I am quite sure that the pixel coordinates would not allow this This example was the introduction The Conics Explorer is the core part of my Conics Trainer Load conics mth asa Utility file and simply Simplify start 1 LOADCD DOKUS DNLS DNLIT MTHS3 con1cs mth He start Conics Trainer Trainer f r Kegelschnitte conl Center 1n origin unrotated Kegelschnitt in Ursprungs lage con2 shifted origin unrotated verschobener Kegelschnitt con3 conic shifted and rotated Kegelschnitt 1n allgemeiner Lage con random choice from above Zufallsaufgabe aus den obigen You can enter your own conic as task ans gives the analysis ant liefert die Analyse Josef Bohm A Conics Explorer and a Conics Trainer Let s assume that your students should train the conics of type 2 shifted origin and not rotated con2 delivers a randomly generated quadratic The students do their investigation with or without technology and then they can check their solution 2 4 con 9 x 16 y 7
48. n eine tragfl gelf rmige Figur verwan deln Diese Figur sieht dann auch so hnlich aus wie ein Schaufelprofil Meine Idee ist es nun ein gegebenes Schaufelprofil zur ck zu transformieren und zu sehen ob die R cktrans formation eine sch ne ebenm ige Figur ergibt oder so etwas wie eine Kartoffel In die sem Fall wurde ich daraus schlie en dass dieses vorgegebene Profil str mungstechnisch nicht optimal ist Gegeben sind die Koordinatenwerte eines Kreises mit vielen x und y als Funktion von Radius und den Mittelpunktskoordinaten Mit den Formeln u x 1 k x y und v y 1 k x y ergibt sich als transfor mierte Figur eine Schaufelprofil in einem Koordinatensystem u v Mein Problem liegt nun darin aus diesen beiden vorgenannten Formeln u und v die Werte x und y als Funktion von u und v darzustellen Ich habe mir schon die Finger krumm gerechnet und bin zu keiner L sung gekommen Man sagte mir allerdings dass eine solche Formelumkehr mit DERIVE m glich w re Da Sie sich offensichtlich mit diesem System auskennen wage ich es Sie hier um eine fach liche Hilfe zu bitten Ich w rde mich sehr ber eine Antwort freuen gegebenenfalls auch auf einen Hinweis wie ich hier weiterkommen k nnte Viele Gr e aus N rnberg Klaus K rner p26 Klaus Korner Airplane and Conformal Mapping Klaus Korner s problem Inverse Conformal mappings My problem deals with the field of
49. nacademy org Nur in deutscher Sprache in German only Franz Embachers Mathe Online wurde erweitert und verbessert http www mathe online at neu html DUG Mitglied Johannes Zerbs ist sehr aktiv Gerne ver ffentliche ich den Link zu seinen Publikationen und w nsche weiterhin gutes Gelingen und viel Erfolg MATHE CHILLED ONLINE ein Buch eine Sammlung ein R tselblock von bungschilis kapiteln nach den neuen Standards und Kompetenzen der Unterstufe http www johanneszerbs at visionen index php option com docman amp task cat view amp s d 41 amp Itemid 12 Alle Infos mit der Buchbeschreibung unter Klick auf www johanneszerbs at gt linker Men punkt Online Mathe bungscenter UE klasse center gt UE klasse Worddokus Von der Homepage gelangt man auch zu einer mathematischen Millionenshow und zu einem Mathe Quiz Josef D N 1 83 Letter of the Editor pt Liebe DUG Mitglieder eigentlich wollte ich in diesem Newsletter vor allem die Beitr ge von Dietmar Oertel Piotr Trebisz und Duncan McDougall wei ter bzw zu Ende f hren Ich war auch schon ziemlich weit mit den bersetzun gen der deutschen Beitr ge aber dann erreichten mich einige Anfragen und mails die schlie lich ber weitere Kommu nikation eine so heftige Eigendynamik entwickelten dass ich sie nicht l nger liegen lassen konnte Nur Robert Setifs Anfrage musste auf den n chsten DNL verschoben werden Ich m chte Sie be
50. o add or delete a high order term if it is at the end of the polynomial rather than at its begin ning because if you add or delete a term at the beginning of the polynomial the coefficients of the previous version if identified by subscripts have to be renumbered I wouldn t want to change the way Derive works But if you know of a Derive operation probably created by some user since I m pretty sure it s not in Derive itself that rearranges a downhill polyno mial and any polynomial subcomponents and rewrites it and its subcomponents to go uphill rather than downhill please let me know That would be a lot faster than doing reordering the polyno mial manually particularly if it is in a Lactored form Thanks again for welcoming me into the Derive Users Group Cordially Roger Folsom BTW it 1s not easy to find a meaningful translation of Comparative Statics into German Josef Interesting readings on this issue can be found among others at http en wikipedia org wiki Comparative_ statics http www applet magic com compstat htm http www economics utoronto ca osborne Math Tutorial DIFF HTM http www duke edu dgraham handouts ComparativeStatics pdf Walter Klinger An Equation of Equations Some days ago I was phoned by Walter Klinger a DERIVIAN from the first days about an interesting behavior of DERIVE It must be said that Walter has used several technologies in math teaching Lower and Upper Secondary during the
51. raph shows the combination of both series of data One could be happy but why do not fit the results for 355 and 350 Values for 355 would be fine if we would take the 4th solution Klaus Korner Airplane and Conformal Mapping I define four functions for the nontrivial solutions soll and sol2 are the complex ones sol3 and sol4 the real ones Sols v k 2 2 2 4 a 4 2 2 2 2 2 4 J2 Q01G k Bk u Iru 4 2 0 Sy YS LE FU v ICE EEE u yeu Fj 8 U V ie 4 2 2 2 ov FU IH Ak U aw he 2 2 av 2 2 2 4 202 4 2 2 AEk Beck SU Jo FA N ee ae Si a V mn nn on 4 2 sol4 u v k 2 2 2 2 4 2 2 4 2 2 2 2 JoCa lack FR ei EU FZ u av a Sa eae ev ICI bk Bike el eee ae 8 U V 2 4 252 4 2 2 u u 2 U v V 4k u v Vv 2 2 2 2 4 2 2 4 2 2 J2itlouwk Fak op eu 20 Er ltak U Ev 4 I try the functions with u and v being the parameter form of the wing and k 25 and hope to come back to the circle 2 2 37 C0S t 37 COS t 37 SIN t 33 37 SIN t 32 t Se eee 2 37 COS t 2 37 SIN t 39 2 37 SIN t 1 37 COS t 37 SIN t 7 V ee 2 37 COS t 2 37 SIN t 39 sol3 u_ v_ 25 sol4 u_ v_ 25 sola with U lt t lt 27 sold with D z t lt 27 Klaus Korner Airplane and Conformal Mapping p 33 This is interesting and surprising as well I don t receive the expected circle but I receive two circles one of them being the expected one
52. rive wouldn t use the chain rule Harald Lang came up with one solution and I used his notation to come up with another solution namely not substituting any function into any other function before taking derivatives and calculating Jacobian matrices for the system If you do not already have the messages I have described above let me know and I will send them I d do so right now except that they all are paper documents and the ink is not as dark as I wish it were so I would need some time to fuddle around with my scanner and optical character recognition software to send them electronically I don t have time to do that right now because I am facing deadlines for two entirely different projects but I think I could get to it in January My apologies for taking your time in trying to explain the terms comparative statics and endogenous and exogenous I did so because I don t know if those terms are used by non economist mathe maticians Of Iin Countries o tside the WS and Canadas Erz Roger Folsom Growth Rates and Comparative Statics I briefly visited your Didactics of Computer Algebra http www acdca ac at and tentatively concluded that it focuses on teaching computer algebra But the site s index frequently uses the CAS acronym where presumably C is for Computer and A is for Al gebra but I m wondering what the S is for Studies Or a non English word At your convenience please satisfy my curiosity A couple of
53. road terms the entire economy macroeconomics the variables fall into two categories endogenous whose values are determined by the model s functions and exogenous whose values are determined out side of the model e g Russia s awful wheat destroying fires this past summer And you want the model to tell you what the exogenous event will do to markets not merely the market for wheat but also to Markets for Grain Substitutes such as corn Other things thart grow such as cotton markets for beef since cattle often are fed COLTS Sera In short you want to know more accurately to esti mate the effects of an exogenous variable or possibly multiple ex ogenous variables on endogenous variables such as the prices and quantities of corn cotton beef etc Comparative statics usually maybe always involves the chain rule derivatives or finite difference quotients of y Y x and x X a where x and y are endogenous and a is exogenous and where I have used capital letters instead of f to indicate a function because I ll probably want to use f for something else such as fire severity In short you want to know not merely dy dx and dx da but also dy da dy x 28 dx a dal which in the U S and maybe elsewhere is commonly known as the chain rule B t Tor a mu ulti runction System To calculate a Tull seu of tos tal derivatives of all of the endogenous variables with respect to each of the exogenous variables De
54. s 4 0 a c f e b 92 b d e c d B 7 a c b now we have to calculate 6 that the mixed term disappears solvelsysi 1 2 0 8 2 2b result The condition for 6 is tan 26 a c because m was not changed we can now take the statement from above and substitute eth terme cosld ysinl xsin y cos o Wo tan 24 a c There is a lot of algebra in the Calculator Window isn t it As one easily can see these conditions are only valid if a c b 0 This case has not been discussed up to now Here we have a parabola as we will see in the next section The parabola case CS 2 5 If acc b 0 and a 0 we can omit c in equation 1 and we get b 1 a x 2b x y y 2d x 2e y f 0 a This equation can be brought to the form 3 a X 2e y 0 with a and e only depending on the coefficients of equation 1 If in 1 the term with x should be omitted then 3 takes the form c y 2d x 0 The calculations are principally the same Wolfgang Propper On Quadratics and Conic Sections In both cases we have parabolas so called non central curves They are symmetric with respect to the y or x axis and they are passing the origin Applying a rotation by angle 8 with tan g 2 we can remove the mixed term and the a term which is quadratic in y in equation 1 as well So it takes the form a x 2d x 2e y f 0 a d a f
55. sonders darauf hinwei sen dass nun die Tagungsb nde aller DE RIVE und ACDCA Konferenzen im pdf Format verf gbar sind Beachten Sie bit te den Hinweis auf der Info Seite Ich freue mich wieder einen TI Nspire Beitrag vorstellen zu k nnen Mit Version 3 kann man sehr ordentlich arbeiten Nun sind wirklich schon einige von uns lange gew nschten DERIVE M glichkeiten imp lementiert worden Besonders spannend f r mich war die Be sch ftigung mit der konformen Abbildung bzw deren Umkehrung zu der mich ein Brief von Klaus K rner gebracht hat Ich habe auch sehr wenig Ahnung ber Tensore gehabt Dank Tom Fowler wei ich nun ein wenig mehr dar ber Roger Folsom schreibt einiges ber Comparative Sta tics wof r ich keine passende deutsche bersetzung finden konnte Ich habe aber ein paar sehr sch ne Unterlagen im Web gefunden die zu einer weniger bekannten Anwendung der Differentialrechung im wirtschaftlichen Bereich f hren Walter Klinger ist wieder zum Einsatz von DERIVE im Unterricht zur ckgekehrt und erlebt mit der Version 6 immer wieder berraschungen die er gerne mit mir bespricht Walter vielen Dank f r die anregenden Gespr che Dear DUG Members Actually T intended to continue the con tributions of Dietmar Oertel Piotr Tre bisz and Duncan McDougall in this news letter I was very busy translating the German papers but then I received some requests and emails which together with the resul
56. tem Matrix syst l b e c d b d a e Result A translation with u and v lets the coefficients of x and y disappean gt a c b a c b Erz Wolfgang Pr pper On Quadratics and Conic Sections The condition for removing the mixed term is tan 20 aD In case a c the conic sec a c tion is a circle which is invariant with respect to rotations around its centre Likewise at an arbitrary b and d e 0 the translation x x u and y y v may re move the coefficients of the linear terms in x and y The conditions are u Aes and ees The coefficients of x x y and y remain unchanged a c Here we determine a rotation that the coefficient of x y disappear first we assign syst from page 3 to m m syst ab 0 be 0 a c f e2 b2 42 b d e c d 00 gt a c b Statement to calculate the angle of rotation koeff term x cos 4 y sin x sin y cos 4 oefficient Matrix koef 2 2 x xy y x y const 2 _ 2 a c f e b r2 b a 1 lcosl8 2 5 cos 9 c sin 8 sinl8 2 a sin 6 cos 6 0 2 cos 2 1 c sin cos 6 a sin 2 b sin c cos 8 cos 0 0 2 4 j gt cos a 1 c sin cos 0 b c os c sin sin a sinl cosl 2 b ystem Matrix syst a sin 6 Yr 2 co s a 6 2 1 c sin sin a cos a sin a 7 2 2 b sin a ea cos 4 co
57. tements 4 3 and 2 statement 7 demonstrates that if all r n 2 0 the usual situation inequality holds so that summing N periodic rates of change r n instead of multiplying cor responding 1 r n numbers understates the true total rate of change over N periods HOWEVER need to learn how to persuade Derive6 1 to do a formal proof results either true false or unknown of the preceding paragraph s results because my next step is to experiment with less restrictive conditions such as all r n gt 1 TO DO Figure out a formal proof for any statement 5 through 8 that when 10 be low holds comes up with true rather than the wrong answer of false For example Simplify Basic of either IF 10 7 or IF 7 10 returns the same written out IF statement presumably meaning that Simplify could not determine whether the inequality was true or false despite the all r n gt 0 variable conditions But Solve Expression of either IF 10 7 or IF 7 10 concludes that the constrained ine quality is false when know that it is true from the analysis that follows 8 clearly am not using IF properly It may not be intended for what am trying to do In that case need to know what else to do VECTOR E ae 9 n i gt OAP gt Oar gt a 10 3 Pa 1 Roger Folsom Thanks for welcoming me to the Derive Users Group Actually it Turns out prior to the Derive 6 1 CD that als though my latest MS
58. text characters instead of using graphics But then switching to Option Display Mode Graphics switches to full screen and good looking graphics If desired then switching back to text mode and doing Alt Enter will go from nice full screen graphics back to a smaller text only window Great Rick Aleksey Tetyorko For your service You may try using DosBox DOS emulator for XP and you can do it without switching to Derive text mode I ve test this method some hours before now My DosBox version is 0 73 maybe not the last I cannot recommend it because of slow operation Aleksey When informing about tensors in E Weisstein s Concise Encyclopedia of Mathematics I came across this pretty quartic surface a TANGLECUBE Josef x 5x y S5y z 5z 11 8 15 0 Tom Fowler Tensor Algebra p 39 KRERERRERERRRERERERRERERRRRRRRRRRRRRERERERERERRERERERER The Derive Tensor Algebra and Analysis Package Documentation for the utility file Tensor mth kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk Written by Hans A Dudler 619 420 1787 25 November 1994 Updated for Derive 5 10 November 2001 New functions added documentation expanded by Tom Fowler September 2011 This file describes the DERIVE utility file Tensor mth which provides definitions and functions for tensor algebra and analysis Tensor mth should normally be loaded as a utility file using the File gt Load gt Utility File command Note that Tensor mth a
59. the arctanh David if the first goal when choosing an antiderivative is to have a real value then I prefer In abs x instead of arctanh x 2 1 x 2 1 for the integral of 1 x these 2 expressions are identical In the same idea In abs sec x tan x is always real so those who are writ ings calculus textbooks will continue to use it instead of arctanh sin x for the integral of SEC X Michel Von David R Stoutemyer Betreff Re RE Textbooks and the arctanh Hi Michel can remain quiet no longer The absolute value 1 unnecessarily makes the antiderivative incorrect for non real endpoints and some people will eventually use non real endpoints not realizing the incorrectness Textbooks and the arctanh p 21 2 completely hides the non integrable singularity if someone integrates from a negative to a positive value With In x you get a some warning from the i pi term 3 makes it very difficult to simplify subsequent expressions that use the antiderivative for examples mixtures of In x and In abs x As another example iterated integrals become very difficult if an inner integral generates a In abs x With all of its extra closure the complex domain is so much easier than the real domain don t know why we put these extra hurdle constraints on students who have been exposed to complex numbers in earlier courses aloha david s Von Michel Beaudin Betreff RE RE Textbooks and the arctanh Hi
60. this case we have chosen the last of the three possibilities for t and assumed that a gt 0 giving t ATAN s a and converting T to s variable by substituting this and invoking Simplify Basic This allows us to get rid of the SIGN function so we have 2 2 S 110 T a 5 nl d Use these values for xi s in terms of s to substitute into G_ 0 0 1 0 er EEE nee eT 1 2 2 111 G_ ee PER ee Ka 5 2 2 a s 2 2 late 0 Invoking CURVATURE on this will yield 0 which indicates that the original curve was a geo desic 112 CURVE_T T G_ s 0 0 There are problems with the I in the DERIVE file because I is reserved for the Gamma function I took G_ instead Josef About Tensors from CRC Concise Encyclopedia of Mathematics An nth rank tensor is a mathematical object in m dimensional space which has n indices and m components and obeys certain transformation rules Each index of a tensor ranges over the number of dimensions of space If the components of any tensor of any rank vanish in one particular coordinate system they vanish in all coordinate systems Tensors of rank 0 are calles scalars tensors of rank one are vectors and tensors of rank two are matrices Two recommended links to introductory papers on Tensors An Introduction to Tensors for Students of Physics and Engineering NASA http www grc nasa gov WWW k 12 Numbers Math documents Tensors TM2002211716 pdf A paper in GERMAN T
61. ting communication got a strong self dynamics and I could not leave them on my to do list any longer Only Robert Setif s request must be left for the next issue I d like to inform you that the proceed ings of all earlier DERIVE and ACDCA Conferences are available in pdf format and can be downloaded More info on the information page I am very pleased about being able to present an TI Nspire related article again Version 3 allows comfortable and powerful working Finally some of long desired DERIVE features have been im plemented Extra exciting for me was dealing with the conformal mapping and its inverse provoked by Klaus Korner s mail I must admit that I knew nothing whatso ever about tensors Thanks Tom Fowler I now have at least a weak idea about these mathematical objects Roger Folsom writes about Comparative Statics I couldnt find an appropriate German translation What I could find are some fine resources on the web They are in forming about a not so well known applica tion of calculus in economics Walter Klinger returned to use DERIVE in secondary level math teaching He is com ing across some surprises with DERIVE 6 which he likes to discuss with me Many thanks for the inspiring talks With my best regards Mit herzlichen Gr en Download all DNL DERIVE and TI files from heine wen austronati at augs ive E D I T OR IAL D N L 83 The DERIVE NEWSLETTER is the Bulle tin of the DERIVE
62. y 30 Type Vertex Parameter Focus Axis amp Directrix 9 t 1969 Fi 5 ans 1969 7 9 59 i 64 576 32 Parabola ee 576 32 32 18 32 1025 9 t 7 288 64 32 Finally they can plot the curve together with all the important details directrix focus axis con3 gives the most complicated cases shifted and rotated 6 cons oes G x y 8 xK or I y 16 7 ans Typ Mitte Ipkt Hauptachse Nebenachse Scheite 8 Type Center Major ax s Minor axis Vertices Ellipse 0 6969696969 0 9696969696 Imaginary imaginary imaginary vertices 9 cons s 5 x y 13 x cone y 21 10 Gigs Typ Mitte Ipkt Hauptachse Nebenachse Scheitel Type Center Major ax s Minor ax s Vertices 2 671401567 0 6926204161 11 0 9790938750 0 4772358007 Ellipse 0 8461538461 0 1076923076 3 833364128 2 587418828 0 4513429306 1 339685972 1 240964761 1 124301356 p 16 Josef Bohm A Conics Explorer and a Conics Trainer The quadratic in expression 6 turns out as an imaginary conic but in expression 9 we find a rotated and shifted ellipse The program code start_Cdummy Prag dummy RANDOMCO x Complex y e Complex DISPLAYC Conics Trainer Trainer f r Kegelschnitte DISPLAYC conl Center in origin unrotated Kegelschnitt in Ursprungs lage 1 DISPLAYC con2 shifted origin unrotated verschobener Kegelschnitt DISPLAYC con3 conic shifted and rotated Kegelschnitt in allgemeiner Lage DISPLAY

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