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THE DERIVE - NEWSLETTER #21 USER GROUP

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1. 2 j 2 2 11 2 2 121 1 2 y il 4 l6 11 y 177 18 2 I y 4 8 19 fertig ready Repeating once more delivers the list of all intermediate results as a whole which is stored in the global variable steps It looks a little bit better than the step by step procedure from above page 6 because can influence the factorization of the square 2 ll y 2 y f 2 2 11 y 11 2 11 y ly 2 4 2 2 steps 11 y 2 121 ly il 11 2 177 ly 8 fertig ready Please not the use of the quote Operator which is used to suppress evaluating and simplifying expressions You can use any variables This program could be extended to transform randomly generated quadratics Random poly variable 2 integer has the disadvantage that the leading coeffi cient is always 1 See an example P 8 33 34 35 DERIVE USER FORUM D N L 21 qu RANDOM POLY Cu 2 20 Z qu u 19 u 18 transform qu Vereinfache step auf step Simplify step by step 36 3 38 39 40 41 E 19 y 19 2 u 19 u I 18 I s d 19 2 361 u 18 fertig ready However will include this tool into my Training Skills Library Josef You can influence the exponential expressions using Manage Logarithm Collect amp Expan
2. l l 2 2 8 a a 3 b r t 2e 3 ar Cai C ee CO Si en u 3 E 3 r t 2 omit the rest of this expression because it is too large for printing here Example 3 3 4 2 2 3 2 19 RED x y 3 X y 3x y Y X Y2 2Z 4 2 3 20 y y 2 needs 0 015 sec Example 4 4 4 4 2 2 2 21 x a b c u a b c 9 22 za b c 24 Ws a Lb LC p54 Johann Wiesenbauer s RED function D N L 21 23 RED x w 4 3 2 2 3 2 24 2 b b 4 c 12 b 6 c 36 c 54 b 4 c 36 c 108 c 4 3 2 108 2 c 12 c 54 c 108 c 81 2 2 25 RED u w 2 b b 2 c 6 2 c 6 c 2 2 2 26 RED v w 3 b GB c 3 b c 6 c 9 3 3 c 9 c 1 4 3 2 2 3 2 27 x new 2 b b 4 c 12 b 6 c 36 c 54 b 4 c 36 c 4 3 2 108 c 108 2 c 12 c 54 c 108 c 81 2 2 28 u new 2 b b 2 c 6 2 c 6 c 2 2 2 29 v new 3 b 3 c 3 b c 6 c 9 3 3c 9 c 1 Next step 30 RED x new u new 0 12 c 36 c 81 3 2 31 RED v_new u_new 0 3 c 9 c 3 3 2 3 2 32 x next 12 c 36 c 81 v next 3 c 9 c 3 33 RED x_next v next 69 No try applying the implemented GROEBNER BASIS function 2 2 2 3 3 3 34 GROEBNER BASIS Ia b c 9 a b c 24 a b c 3 la 3 2 2 2 b c Lc 3 c 1 b b c 3 c 3 a b rc 3 Solving vor c and considering the symmetry of all solutions try a co
3. 104 BP 23 5517 SPF 23 5517 105 m_MT SPF_1 23 5517 106 m_MT 0 38792 107 MTanF x m_MT x MTanF_ x 108 If 0 x 23 5517 MTanF x Length of segment O BP 2 2 109 L MTan 23 5517 SPF 23 5517 110 L MTan 25 2616 Hellmut Scheuermann Skispringen im Blickpunkt Arc length of the front part from PT to Normalpoint P 79 0009 2 J 1 SPF 169 dx 23 5517 111 LRI 112 LRI 65 01 Calculation of width W Rech from the ramp to the K point 113 W Rech L MTan LR1 30 114 W Rech 120 271 Error in percent is W Rech 120 115 100 0 226441 120 The error is 0 23 Solution of problem 4 116 Rek 132 LSK 120 271 Distance to point K on the arc 117 LKRek Rek LSK 118 LKRek 11 729 Calculation of the upper boundary needs some calculation time Solution of problem 4 116 Rek 132 LSK 120 271 Distance to point K on the arc 117 LKRek Rek LSK 118 LKRek 11 729 Calculation of the upper boundary needs some calculation time x 2 119 SOLVE J 1 R2F_1 z dz LKRek x x 112 644 goM Landing point for the world record 120 112 644 R2F 112 644 112 644 67 3489 x 2 119 SOLVE JG R2F_1 z dz LKRek x x 112 644 goM Landing point for the world record 120 112 644 R2F 112 644 112 644 67 3489 Hellmut Scheuermann Skispringen
4. bers with Prof Douros files I called Karl Heinz and he told that he had contacted Prof Douros because of some problems and he will submit the outcome of this ODE discussion for the next DNL By the way Karl Heinz suggested that the authors of DNL articles should add their e mail address in the future if they have one Hellmut Scheuermann Hofheim Taunus Germany Hellmut Scheuermann wrote a long letter which could be of interest for many of you So I try to translate and summarize his ideas Josef I sent my last letter in March 95 So it is obvious that some DERIVE questions have arised since then e Is it intentional that the SIGN function is not defined at location x 0 e Is there a possibility to transform the quadratic using DERIVE step by step f x ax 2 bx ce gt f x a x n 2 h e Is it possible to perform a x b x amp a b x similar to a x y S a x a y e have learned in the DERIVE manual that it is able to create more special characters using the keys Ctrl P followed by Alt ASCII Code e g Alt 127 for A I liked this idea because I often use Ax Ay etc But unfortunately even in DERIVE s Word Mode Ax looks like A x But even more interesting is the following If you enter in a DERIVE comment the character gt Ctrl P Alt 026 e g 26 function f leads for x gt towards 1 save the file and reload it then the file appears until the x All the following part of the file seems to be aban
5. x 1 It would be fine if RED and SOLVE could be combined This could be a challenge for Albert Rich to improve DERIVE Now we can solve the system because of the implemented Groeber Bases 2 2 0 cal X Y y 113 y NN LN en Do yl 61 16 x m 15 Ay mE A XS Ss hye e 5 5 Example 2 See Johann s Example together with his comments We start with the Van de Waals equation 8 3 dm Vos p a v b r t 9 V We solve for p and then introduce p as a function of v alone 10 SOLVE p a v 2 v b zr t p 11 pz ax b v serst v 2 v 2 v b 2 a b v retev 12 p v 2 v v b The critical values of v t are obtained by solving the following system of nonlinear equations d d 2 13 p v 20 p v 0 dv dv 2 2 3 2 a b 2 b v v r t vV 3 2 v b v D N L 21 Johann Wiesenbauer s RED function 3 2 2 3 4 3 a b 3 b v 3 b v v r t v 0 4 3 v b v The resulting system 14 looks rather frightening doesn t it This is where our RED function from DNL 20 comes into play d d 2 r t 3 b v eof p v E vol lt ae 15 dv dv 3 3 v b r t 3 b v SOLVE v v o vv 3 bvr t 0 16 3 3 v b eurprisingly enough the last two steps cannot be combined d d 2 317 aud eo p v a 000 j dv dv 18 2 2 2 2 8 a 36 a b r t 27 b r t ASIN
6. 2 9 5 2 45 6 15 SOLVE x 7 OX X 8 8 5 5 5 20 16 45 4 16 x UMP dpt 109 545 5 Only the first solution is valid for our problem Then back substitution and calculation of r and s 20 16 45 2 17 SIN r 109 545 20 16 45 18 r ASIN 109 545 4 45 2 19 r ATAN a 19 19 20 r 0 3504054128 J0 2 45 21 s ASIN SINCr 4 J5 1 6 6 23 s 0 2031689460 p40 Physikalisch Chemisches Bei diesen Berechnungen des Fu balls wird nur der mathematische Aspekt angesprochen Die chemisch physikalischen Untersuchungen und Anwendungen etwa Erzeugung von Supraleitern bei hohen Temperaturen oder Verwendung bei neuen Arzneimitteln bleiben hier au er Betracht Eine gute bersicht ber diese neue Stoffklasse der Fullerene ist in einem Artikel von R F Curl und R E Smalley im Scientific American 10 91 enthalten Die Bezeichnung Fulleren oder auch der saloppe Name Bucky ball ist dem Architekten Buckminster Fuller zu Ehren gew hlt worden weil dieser schon Kuppeln geodesic domes nach dem Muster der Ceo Modifikation entwickelt hatte Aktuelle Informa tionen und interessante Bilder zu diesem Themenbereich kann man an vielen Stellen im Internet finden Hier ist eine kleine Auswahl http www chemie im alltag de articles 0079 http www godunov com bucky fullerene html htto
7. 3 Erl uterungen FIS 1992 S 34 zur Abb 1 Schanzenprofilbest tigung p26 Hellmut Scheuermann Skispringen im Blickpunkt D N L 21 Br hlwiesenschule Hofheim Subject Mathematics Work sheet Ski Jump 1 Object Calculus Class 12 1 The Big M hlenkopf Ski Jump Construction and building of ski jumps has to follow precise rules of the International Ski Federation FIS The permission for execution of national and international competitions will be given only if FIS specialists have inspected the jump and given a jump profile certificate This so called Homolo gation Certificate contains all specific data of the respective jumping hill see fig 1 and fig 3 Tasks 1 Considering the data from fig 1 approximate The front structure of the jump starts at the base of the the inrun and the landing run by piecewise de Jump jumping hill platform with a slope bo approxi fined functions use the equations from fig 2 mate value by b 6 and ends at point P with a slope Plot the profile of the complete ski jump facil p The profile of the front building should ensure good ity cut the run out landing conditions for short jumps and minimum flight heights for long jumps as well The profile can be defined by a cubic Using the coordinates from fig 2 Use Calculus to find a cubic general for approximating the front structure of the jump Compare your results with the FIS rules see fig 3 y px qx tanb
8. 8 95817 yM 81 7640 p28 Hellmut Scheuermann Skispringen im Blickpunkt M p x Yu T S P Xs Ys T p x y Figure 4 Ideas and illustration for calculating the center of the circle Finding point S xS yS 22 c 35 x SM R1 SIN c y SM RI COS c 23 xS xM x SM yS yM y SM 24 xS 8 95817 x SM yS 81 764 y SM 25 xS 38 075 yS 14 5935 26 38 075 14 5935 End of this alternative 111 Equation for the line EG length E c1 35 We have to find the parameters m E and b E 27 EGF x m_E x b E 28 cl c m E TAN c1 EF ist tangent of the circle thus 29 SOLVE RIF 1 x m E x 30 x 55 9913 v x 38 0751 Begin of the arc 31 guR1 38 0751 32 RIF guR1 14 5934 Point R1E 38 0751 14 5934 is osculation point of EGF with KR 33 RIE 38 0751 14 5934 2nd equation to find b_e 34 SOLVE CEGF 38 0751 R1F 38 0751 b e b E 12 0669 35 b_E 12 0669 36 EGF x 0 700207 x 12 0669 D N L 21 Hellmut Scheuermann Skispringen im Blickpunkt Figure 5 Illustration for calculation of the arc length Calculation of the arc length 6 3368 2 37 LR1 J 1 RIF 169 dx 38 0751 38 LRI 34 7058 Alternative way to find the arc length 180 180 c a 39 iD im S M 1 40 0 24 25 4
9. E l 4 M the TI 92 qf C37 85 FUME 12730 HIH Both the pv and qf user defined functions can be placed in a special folder and on a custom menu as well Also user defined functions can involve multi line statements The following is an ex ample of such a user defined function It is typed directly in the entry line Define examp a b Func lf a lt 0 or b lt 0 Then Return undefined Else Retum V a 2 b 2 Endlf EndFunc There is a lovely contents function F6 found in the VAR LINK menu that shows exactly what you defined i Fer Fir Fir FE FR Se T TETE R EG Tf ace ar eto ThentReturn Tun E ara 7 B SR E HE d RE TRS E x _ Kisten a ge asec Bafta b c FUHC 29 Here is an exercise Eigenvalue is not a built in command on the TI 92 u Define exampia by Func Create a definition for the TI 92 that solves this problem Bexampl 3 7 undefined n exampt5 7 r4 See page 59 EEA 7 8 TI 92 Tipp of the day Leave your calculator in the radian mode When you want degrees paste in the degree symbol by pressing 2 D after the number Hint A keyboard map of the 2 func tion keys can be found by pressing K SE Buesbeaksis pisse ersntoelea ace a Define exampia bi Func Done 5 undefined undefined undefined r 3 11 12 4 p58 THE TI 92 CORNER D N L 21 TI 92 TIPPS AND TRICKS BY BERNHARD KUTZLER In this article we will provide you with
10. Newsletter refer to Maple Surgery Q amp A How can I get Maple to substitute a mathematical expression For example have the following gt f x 3 y 4At3 x 2 yt3 x y 2ty 3 y 2 I want to substitute all occurrences of x y with z If use subs nothing changes The answer use the siderels option of the simplify command For ex ample gt simplify f xty z y 4 y 2 tz 3 In example 3 will show how to tackle this problem with Johann s RED function And finally found a suitable fourth example In the last edition of Spektrum der Wissen schaft German edition of Scientific American contains some contributions focussing on various computeralgebra systems Among other read one sentence about Groeb ner bases Damit kann man sogar a b c berechnen wenn a b und c nur implizit n mlich ber die Bedingungen a b c 3 a b c 9und a b c 24 bekannt sind Das Ergebnis ist 69 In 1996 Groebner Bases were not implemented in DERIVE In the meanwhile they are as you will see later In the following you can find the above described applications of the RED function p52 Johann Wiesenbauer s RED function D N L 21 1 RED u v ITERATECRHS v QUOTIENT u_ LHS v REMAINDER u_ LHS v u_ U Example 1 2 2 2 R X NN y 113 v y x y x A1 126 y 13 RED u v 70 3 2 y 1 16 4 y vy 7 5 2 x 113 5 SOLVE RED u v y y
11. One problem of the last Bavarian end examination was to investigate the function 4x 4 l f x a In one part of this problem the students had to show that f x 1s reversible in 2x 00 0 and after verifying f N2 V2 give reasons why S N2 V2 is the intersection point of fand its inverse f The students were not asked to find the inverse function But the TI 92 freak tries to let the machine do this work Let s go on solve f y x y ysO should help We receive the following result Ell m x 2 m ee x42 x x X x Each other restriction behind the with operator will lead to the same results The mathematician in the teacher will see immediately that only the second solution contains the required solution Using Copy and Paste we could define g x 2 solution Now our little machine should evaluate the func tion value for x V2 so ans 1 x 2 What has happened now Has the TI 92 really lost its mind It presents two remarkable expressions N 2042 2 4242 5 4 202 2 4242 iw ft After some attempts it maybe that you try solve f y x y x lt O You will find again the two solutions but now ans 1 x v 2 gives y 2 2 or y 2 Could anybody give an explanation why the TI 92 is calculating the same problem in different ways Pe Aigebralcaiclother Prantolcieara z H gt Slarsebralcarejotner erantolciear Up R golvet s y TIE I AR e xti or y n cr be BU lt or y 3
12. Puig Adam with permission As a qualified tennis coach this co author is familiar with the Tennis Rules and regulations and hence the tennis analogy eroanes eucmos sim ucm es rOanes eucmos sim ucm es ABSTRACT Computer Algebra Systems are sometimes unable to completely solve some problems In some of these cases it is desirable to merge Symbolic and Numerical computing A simple example is given here To calculate the area covered by a tennis net in the position which it is usually to be found in tennis clubs and to compare this with the theoretical position as recommended by the Tennis Federa tion Surprisingly the difference is not small as will be shown using elementary mathematics and DERIVE 3 INTRODUCTION On a tennis court the posts of the net should be positioned differently for singles 2 players and doubles 4 players Official tennis rules state that to play singles the posts should be in the singles position 1 e closer to each other The wire from which the net hangs should be very taut and it s height controlled at the center by a vertical tape the other end of the tape is fixed to the surface of the court This is how the net is positioned for championships Therefore we shall call it championship position In this position the shape of the upper border of the net is that of two equal and consecutive segments forming a wide open V But the more common net position for singles is with the n
13. VN gru OE and Schanzenvorbaus Vergleichen Sie Ihr Er NC NZ gebnis mit den Vorgaben der FIS siehe Abb 2 u mH S N lgb V igb tgb ist 3 berpr fen Sie inwieweit die L nge W De Abb 2 Auszug aus FIS 1992 S 37 finition aus der Schanzenprofilbest tigung mit der L nge Ihrer Schanzennachbildung bereinstimmt prozentualer Fehler 4 Der Schanzenrekord wurde w hrend des Weltcup Springens 1995 aufgestellt Der aiaa Funaki sprang 132m weit Markieren Sie den Rekord im Diagramm aus 1 Geometrische Elemente einer Sprungschanze P Beginn der Landestrecke K Kritischer Punkt am Ende der Landestrecke M Lange der Landestrecke W Entfernung von Schanzentischkante bis zum kriti schen Punkt K H hendifferenz zwischen Schanzentischkante und K Neigung der Landebahn Neigung des Vorbauprofils am Schanzentischfu Neigung des geradlinigen Teils der Anlaufbahn Radius des Bogens von der Anlaufbahn zum Schanzentisch Radius des Bogens vom K zum Auslauf L nge des Schanzentisches H he des Schanzentisches L nge der Anlaufbahn vom obersten Startplatz bis H hendifferenz zwischen Schanzentischkante und zum Beginn des Schanzentisches P Bereich der Startpl tze Horizontaldifferenz zwischen Schanzentischkante L nge des Auslaufes und P Projektierte Geschwindigkeit auf dem Schanzen Neigung des Schanzentisches tisch Em CT P Fa Horizontaldifferenz zwischen Schanzentischkante und K m GB D A n Abb
14. a DERIVE function PF n that returns a list of all prime factors of the natural number together with their multiplicities eg PF 360 2 3 3 2 5 1 360 2 37 5 PF n can be used as an auxiliary function in a lot of other important functions of number theory Give as an example the definitions of the DERIVE routines for t n 1 e the num ber of positive divisors of n and Q n which is the number of prime divisors of n each one counted with its multiplicity e g 1 360 24 and 0 360 6 3 And here a really tough one for those readers who fear neither death nor devil when it comes to programming in DERIVE Define a DERIVE routine DIVS n that returns a list of all positive divisors of n in ascending order Furthermore write a DERIVE routine for o n 1 e the sum of all positive divisors of n Try to avoid all auxiliary functions if pos sible To exclude trivial solutions I should also mention that all your routines in 2 and 3 are supposed to take only a few seconds for numbers with up to 20 digits p48 Johann Wiesenbauer Titbits 7 D N L 21 4 Write an efficient DERIVE routine LUCAS n for calculating the n th Lucas number L based on the recursion L A by SEL DT 4 2 1 n Furthermore use the formula F an ZL L 5 to derive from it a DERIVE routine FIB n for the n th Fibonacci number and compare the calculation times with the corresponding function in the utility file NUMBER MTH 5 You say yo
15. are S P Np Hp lt b lt bo lt b 6 3 2 87 F x a3 x a2 x al x a0 2 88 F_1 x 3 a3 x 2 a2 x al m Np TAN b and m 0 TAN bO 89 S Hp z Np 90 SOLVEC FCO S FCNp Hp F_1 Np m Np F 1000 m 0 a0 al a2 a3 3 Hp Np 2 m O0 m Np 3 5 a0 S a al mO A a2 A 43 91 2 Np 2 Hp Np m_0 m Np 2 5 3 Np Equation according to the FIS rules 3 2 92 F_FIS x p x qx m_0 x S 93 u Hp S Np m 0 v m_Np m 0 2 u v Np 3 u v Np p q 94 3 2 Np Np 2 Hp Np m O0 m Np 2 S p oio m q E 95 3 Np 3 Hp Np 2m 0 m Np 3 5 2 Np Comparing the coefficients we see the identity Hellmut Scheuermann Skispringen im Blickpunkt Solution of Problem 3 See the definition for measuring the width of a jump We need a tangent from the edge of the ramp to the cubic of the landing section see Figure 6 Figure 6 Illustration for calculation of width W Osculation point BP xT SPF xT b 96 Np guM Hp MF guM S 3 15 bO p 97 m Np TANCb m O TANCbO Equation of the measurement tangent 98 MTan x m MT x d 99 SPF_1 x SPF x dx 2 100 SPF 1 x 0 0000669531 x 0 0134602 x 0 108046 101 MTan x SPF_1 xT x 102 SOLVE SPF xT MTan xT xT 103 xT 147 538 v xT 20 3096 v xT 23 5517
16. d d 3 s_ k_ d_ s_ k_ d_ 0 2 5 JI t tL 1 nil And now a second try applying brute force P n ITERATE IF n lt DIMENSION p p v m_ APPEND p v p APPEND v_ IF 3 m 2 m y2 DIMENSION p_ 1 OR3 m 2 m_ 2 DIMENSION p_ 1 1 Gm_ 1 0 IF 3 m 2 m 2 DIMENSION p 1 m_ 1 m_ p_ v_ m_ 1 1 1 n 1 1 PC 100 190569292 0 0 sec F 100 190569292 0 047 sec D N L 21 Johann Wiesenbauer s RED function p5l I don t know whether the performance of these implementations lives up to Albert s expecta tions but one thing is for sure They are both considerably faster than the one above PARTS n Check it Any comments or improvements This is my email address wiesenbauer a tuwien ac at Applications of Johann s RED function from DNL 20 Please remember Johann s RED function from the last DNL It is indeed a remarkable inven tion want to illustrate this using four examples Johann and talked about the next DNL and by the way he mentioned that his RED function would be able to solve nonlinear systems of equations and that he had found a fine example with the van der Waals equation He prom ised to fax the example In the meanwhile checked his words with a problem from one of my textbooks finding the intersection points of two conics which is example 1 in the following Example 2 will be Johann s contribution By chance found one problem in a Maple
17. in dem 1 gilt 1 sins sin36 sin r 2 Verbindet man die Mittelpunkte dreier benachbarter F nfecke erh lt man ein gleichseitiges Dreieck mit der Seite 27 2s und den Winkeln 72 Nach dem Winkel Kosinussatz gilt 2 cos 72 1 cos 72 2 cos 2r 2s 2 sin 72 D N L 21 Richard Schorn C4 The Buckyball p37 1 Connecting the center of a pentagon with two adjacent vertices gives an isosceles trian gle which leads to equation 1 1 sins sin 36 sin r 2 Connecting the centers of three adjacent pentagons results in n equilateral triangle with side length 2r 2s and angles 72 which leads to equation 2 angle cosine rule cos 72 1 cos 72 sin 72 As there is a closed form for 36 and 72 both equations 1 and 2 can be simplified to 1 and 2 The next steps are obvious one has to apply the the trigonometric sum rules in 2 in order to substitute for sin s from 1 For avoiding uncomfortable roots we will try to have only one variable x sin r The boring simplifications can easily be done using DERIVE and they finally lead to a quadratic 3 Only one of its two solution is valid for our problem 4 This solution gives the result which is presented in the box 5 2 cos 2r 2s Da die trigonometrischen Funktionen f r den Winkel 36 und damit nat rlich auch f r 72 n ge schlossener Form angegeben werden k nnen lassen sich beide Beziehungen vereinfachen
18. k 1 070000 0 5 0 0016532702217014 3 65857 79550064 1 86 k 1 0752688172043 604 81 k sete 07 0 006824 1 i gt fx cosh 0 006824 x 0 93 one 0 006824 0 006824 ts 87 Ix 5 035 or lx gt 6 485 undef else dus se gt 6 485 undef else 3 035 gar ga xi dx Pi gg ki x 5 035 5 0352 HAIN RAD EnncT FUNC 29750 ffx Hellmut Scheuermann Skispringen im Blickpunkt Skispringen im Blickpunkt der Mathematik oki Jumping in the Focus of Mathematics Hellmut Scheuermann Hofheim Taunus Germany Wer kennt sie nicht Spektakul re Fernsehbilder von Skispringern die sich ganz von oben aus der Anlaufluke in die Tiefe st rzen Dennoch und trotz zunehmender Variationen der Bildeinstellungen durch immer aufsehenerregendere Kamerapositionen k nnen reale Ein dr cke nur eingeschr nkt ber und vermittelt werden Der Fernsehzuschauer kann sich kaum vorstellen wie steil es auf der Anlaufstrecke heruntergeht oder wie weit der Skisprin ger nach unten fliegt Die Fernseh bertragung des Weltcup Skispringens in Willingen im Januar 1995 waren f r mich der Ansto u a diesen Fragen nachzugehen und speziell diese Schanzenanlage auf ihre spezifischen Kenndaten L nge H he und insbesondere die Steigungen an verschie denen markanten Stellen zu analysieren Dar ber hinaus wollte ich auch erkunden wie viel Mathematik zur Planung und Konstruktion abgesehen von d
19. my head in my hat think you should keep your head just where it is Pat will have no use with a head less Carl And where would you fix your diving mask and snorkel We are looking forward to meeting you both in Bonn You will also note another expansion of the DNL Four additional pages are dedicated to the TI 92 Of course we will deal with the other features of this magic machine in the future too My special thank goes to Bert Waits and Bernhard Kutzler who helped to produce the first TI 92 section I will encourage you to sub mit papers and requests concerning the TI 92 One of my personal goals for the DUG in 1996 is to integrate TI DERIVE into the DNL This is closely associated with the second goal to increase the number of our members from more than 500 up to approximately 600 think we may master this ambitious task Please take notice of our Materials Ex change on the Information page For 1996 can promise a couple of very inter esting contributions and together with you am looking forward to having a thrilling DERIVE for WINDOWS amp TI 92 year 1996 hope to meet many of you in July in Bonn Sincerely yours P2 E DI TORIA HL The DERIVE NEWSLETTER 1s the Bulle tn of the DERIVE User Group It 1s 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
20. n monthly payments of R at the end of each month each at an annual per KU LLI l centage rate APR of 12 The basic present value annuity formula is given by PV The following TI 92 screens show the results of using the TI 92 Define command along with illustrating other TI 92 CSA commands 1 For Fir Fur FE Fh 1 Fer Fir Fur FE Fh a RAR RER PTO oo S ifiti FI Dane E solve pur i n5 a r pp Se 07125 C ae 2 pu 1485 56 Iz 180 leoni S nEo luet put 1495 56 1 180 164000 i 07125 no solution found solve pur 12 07 180 164000 r S nEoluer put 1485 56 i 180 164000 13 ip MEE BL HU e 07125 12 180 gt 164000 1 180 gt 164000 1 gt 11 gt 0 and 141 RAD AUTO FUNC 330 RAD AUTO FUNC Bret u Define puir i ni Note in the last line of the second panel the use of the vertical bar with 2 K to apply bounds i gt 0 and i lt 1 for the nSolve algorithm to find a numerical solution for i there is no explicit closed form solution for 7 We are using the default display digits setting of 6 However when selected and pasted in the entry line all digits available are shown and used in the computation of 12 i It is very useful to note that any expression or answer in the history screen can be selected and pasted in the entry line by repeated pressing of the up cursor key on the large wheel and then pressing ENTER in the desired line 1 For For Fu FE FR a
21. the straight line of equation y 2x 7 solve the equation f1 x 7 2 e ls there a tangent which passes point 1 2 2 Use Substitution to replace x and y by 1 2 and 2 solve for xO and assign the solution you have found to x0 Of course one draws on the screen the straight lines obtained fig 1 and this graphic capa bility of Derive is here most interesting because the pupil can thus illustrate and check his result by himself and do it again in case of mistake with his teacher s support Figure 1 Some tangents to a curve Here Derive is used in semi automatic mode the student unburdens himself onto the com puter of the details of calculation developing simplifying solving which are not the aim of the lesson but he keeps the leading role of the mathematical operations by commanding the program and by making it finding the answers On the other hand the tangent function will be very useful if my pedagogical objective is dif ferent for example in the class of Terminale calculating the approximated value of an inte gral with the method of tangents and visualizing the trapezia calculating an area fig 2 solv ing an equation with the Newton Method and illustrating graphically the procedure fig 3 and making a curve appearing as the envelope of its tangents fig 4 Figure 2 Approximation of an integral using tangents f x x 2 1 a 1 b 2 10 tangents area 4 48875 D N L 21 D Lymer DERIVE Automa
22. x t 2 does not return x 2 but t x And solve f x 0 x t 2 returns t instead of 2 And if I want to plot graph f x t 2 then I receive an error message Yes I know there are existing other ways e solve x t 2 0 x t 2 or graph x t 2 t 2 But why did I define the function earlier e tis nice to have a list for the values 1 0 2 t and then be able to produce all solutions and all the graphs in one step but having done the job I have to delete the variable with DelVar t e Or I define f x t 2 and go on But doing so I will miss the functional point of view e Or I define f x t x t 2 and use the with operator I regret that the function variable x lies in the same logical level as the parameter t I would like the idea that D Stoutemyer could teach the chip to make all the variables of the function term local to an actual folder in order that all the variables could be substituted applying the with operator This is possible Look at the following 1 Fer Fr Fur FE Fh stw c LIZ A FEK s EFI L lt 2 T hyt s o L lt 2 Male RAD AUTO FUNC zn These problems bugs have been fixed in later Operations Systems See the respective screens of the TI 92 or Voyage 200 There are no problems with TI NspireCAS id Fe Y F4 a For a Zoom Tr ace Kedi ah dath R zolvee f xi 20 x3 t Graph f00 t 2 do uM E a EN L T L m A J t E p60 THE TI 92 CORNER D N L 21 Another thing
23. x S with 1 we define 3 Check if length W definition from the Cer tificate of Jumping Hill matches with your Jump Hill Model percentage error 2u v N 3u vN Tr ge and P D u H S N tanb v tanb tanb D Figure 2 Excerpt from FIS 1992 page 37 4 The jump hill record of 132m was reached at the world cup competition 1995 by the Japanese ath lete Kozuyashi Funaki Mark this record jump on your plot from task 1 Geometric elements of a ski jump begin of the landig section slope of the landing run critical point at the end of the landing section slope of the profile at the platform base length of the landing section slope of the linear part of the inrun distance between platform edge and radius of arc between inrun and platform critical point K height difference between platform edge radius of arc between K and landing run and K horizontal difference between platform length of the jumping platform edge and K heigth of the jumping platform height difference between platform edge length of the inrun from the topmost star and P ting position to the begin of the starting position horizontal difference between platform zone of starting positions edge and P length of the landing run slope of the landing hill assumed velocity on the jumping platform Figure 3 Explanations FIS 1992 page 34 to Fig 1 Certificate of Jumping Hill Hellmut Scheuermann Skispringen im Blickpunkt 1
24. 1 0 24 25 Arc length 42 lg 82 0 43 Ig Length EG 85 85m 34 7058m 50 1442m lt c 35 34 7058 44 EG 50 1442 45 guE EG COS c guRl 46 guE 79 1508 EGF_ x 47 If 79 1508 lt x lt 38 075 EGF x RIF_ x 48 If 38 075 lt x lt 6 3368 RIF x 49 Start guE EGFCguE SP 0 01 50 Start R1E TR1 SP TF GO RIF OO EGF_ x Plot the complete hill Hellmut Scheuermann Skispringen im Blickpunkt 50 7 0 60 50 40 30 20 10 10 Part 2 Equation of the landing 1 Equation for the line segment M length M 30m b 37 point K 102 96m 60 74m 51 K 102 96 60 74 M 30 b 37 52 mM TANCb Equation for MF 53 MFOO m_M x bM 54 MF 102 96 b_M 77 5859 55 SOLVE MF 102 96 60 74 b_M 56 b_M 16 8459 57 b_M 16 8459 58 goM 102 96 59 guM goM M COS b 60 guM 79 0009 MF_ x 61 If 79 0009 lt x lt 102 96 MF x 11 Equation for the arc KR2 radius R2 125m Point K 62 R2 125 The lower half of the circle and its 1st derivative 2 2 63 R2F x R2 x x02 y02 d 64 R2F_1 x R2F x dx x x02 R2F 160 65 2 2 JC x 2 x02 x x02 15625 Now similar to Part 1 15 18 K is osculation point qtih tangent M 66 R2FCgoM MF goM R2F 1C goM m MJ 67 APPROX SOLVE R2F_1 goM m M x02 68 x02 27 7331 v x02 178 186 He
25. 12 TTS 2 T EJ EU 2 2 EQUES a or l or y de 241 or u 2 uy 2 585786 or y 1 41421 apron Caneci oo approxtansti MAA RAD AUTO FUNC 77 Z0 HAIN RAD AUTO UHE Brs You can see two screens from the TI 92 left and one screen from the Voyage 200 right Later Operating Systems behaved better The values given by the first TI 92 generation were correct see the approximated values Evaluating nested roots form a serious problem for computeralgebra systems The DERIVE expressions below confirm the identity of the ex pressions Josef JC 2 G 2 2 G 2 2 JC 2 0 2 2 G 2 2 EU sem d d J 1 2 2 2 J2 42 SWHH Albert Rich Concerning the Tl 92 tech support question from W Propper TI wants TI 92 tech support questions to go through their designated tech support so that they can keep records of re quests for future products They in turn occasionally refer new questions to Dave Stoute myer that only he can answer and then he replies to TI I D Jeffrey and Albert Rich Simplifying Square Roots of Square Roots by Denesting in Computer Algebra Systems ed by Michael J Wester Wiley 1999
26. 2915 1222119123 52219129 29 024 29 9129520 91264271 1260 44 7127429 15127726 151282917 120 30 1 30 31 30 47 31 32 31 42 32 33 33 34 34 35 34 50 35 36 I 021416 a2 E 129 99 49955914129 12014132392 99 T 41 A 152 43 42 56 43 44 44 45 45 46 45 57 46 47 47 48 48 49 48 58 r 49 530 50 51 51 52 51 52 52 53 93 54 54 55 54 60 56 57 5 6 60 37 58 58 52 59 60 food VECTOR Cl pote SUB CL SUBI SUB 1 pre SUB SUB SUB 2 71 BIN D N L 21 Richard Schorn C4 The Buckyball p43 used the respective ACD file which is plain ASCII text and imported the data to DERIVE 6 to reproduce the red green plot of the bucky ball something similar to an analglyph On the next page you will find the short DERIVE 6 version of the calculation followed by a screen shot for our TI Nspire users Information about ACROSPIN can be found at http www nordwest net doering math lkl acrospin index htm p44 Richard Schorn C4 The Buckyball D N L 21 COS 72 01 COS 27 JCIU0 2 45 415 SIN aar RR IE zZ d 5 SIN 72 2 SIN s SIN 36 SIN r dL 2 453 E35 SIN s SINCE 4 45 Hd COSL Z P 2 8 5 5 Tr igonometry Expand Trigpower Sines Z 2 2 45 6 4 SIN r COS r SINGS COSES SIM r 4 5IM E 2 z SIN 5 1 5 2 z 2 2 Z 45 Er 4 SINO
27. 58 pi ATAN 2 SORT 5 1 2 pi 59 71 5 ALAND PSORE HEIZ ZIP 60 9 pi 5 ATAN 2 SQRT 5 1 2 pi The 60 points given in world coordinates in a text file CGOR1000 TXT Pon u 343 Uy 99 pA 264 202 172 PAL t sO Soy 202 315 B 2 10 67 LO 1959 P22 d 099 40472172 BAZO Do dose PLS X 599y9090y 315 Par 899 404 172 Poo 1 100 5205799 39 EL ME oL o 15 P40 9064 202 172 POU 36 218 20 2 939 p42 Richard Schorn C4 The Buckyball D N L 21 My short utility function converts the 60 points from C60LAPHI MTH into the coordinates given in C60R1000 TXT So simply add HOLS OTS Fl AD FS wo 60 62 H x FLOOR x 0 5 63 pts VECTOR H 1000 COS p SUB i SUB 2 COS p SUB i SUB 1 COD Tp SUB 4 SUB 2 9S LNCD SUB x SUB 1 H P000 SILINTD SUB T SUBE 2 1 1 DIMENSION p HCLOUDA 7644 pusscpps422 0 939 44 106742260 959 273 202 7939 Ai 95217904 2024 93997 ET 2201909 Mo ede Ty 220549I 1999909955 1212 0994 2 T re ao do A Y A 790 e Y o ee A A o dar o o 9999592024045 4 5 9397 404 7 PA The next two expressions are from 2009 in order to produce the 3D plot in DERIVE 6 65 1 1 21 1 51 1 61 12 31 12 9 3 41 3 121 4 51 4 151 5 18 6 7 6 20 7 8 7 21 8 9 8 24 19 101 10 11 10 25 11 12 11 2 9 12 13 13 14 13 29 14 15 14 32 15 16 16 17 16 33 17 18 o ES 59 1129 21 729 377 232 2 22 121 20 1224
28. ACTORS function so the solutions from 1996 don t work in DERIVE 5 or DERIVE 6 3 Comment in 2009 DIVISORS n and DIVISOR_SIGMA n are implemented now ka DIVISOURS nm sor VECTOR Tu U Neal el E NE De M wenns 1 2 DIVISORS 360 1 2 3 4 5 6 8 8 10 12 15 18 20 24 30 36 40 45 60 72 90 120 180 360 v L2 lee J DIVISOR SIGMA k mics I Bowe ovo FACTORS j 1 DIVISOR SIGMA 1 360 1170 4 Comment in 2009 LUCAS n and FIB n are Johann S routines from 1996 while LUCAS n and FIBONACCI n are the respective functions of DERIVE 6 LUCAS n ITERATE IF c n a b c d e IF e gt d a 2 2 1e a b 1 c 2c d e 2 a b l1 c b 2 2 1y e 1 2e 1 d e e 2 a b c d e L 2 1 0 nITERATE IF2x n x 2x 5x DD 1 LUCAS 10000 LUCAS 10000 0 WECTOR LUCAS nj n O 1000 needs 1 56 sec VECTORCLUCAS n n 0 1000 needs 1 53 sec FIB n ITERATE IF c n a b c d 2b a y5 IF e gt d a 2 2 1 c a b 1 c 2c d e 2 a b 1p c b 2 2 1y e 1 2e 1 d e e 2 a b c d e L 2 1 0 n ITERATE IF 2x gt n x_ 2 x_ x_ 1 5 FIB 10000 and FIBONACCI 10000 give in an instant a 2090 digit number 5 Comment in 2009 MISC MTH from the earlier DERIVE versions has been replaced by MiscellaneousFunctions mth Johann s POL DEG from original DNL 21 does not work because of FACTORS but we can compare his POL COEFF with the implemented POLY COEFF
29. Gilligan GILMAR Cincinnati OH 1996 208 pages 5 Symbolrechner TI 92 Bernhard Kutzler Addison Wesley ISBN 3 89319 952 7 192 pages 6 Atlas mathematischer Bilder Leo Klingen Addison Wesley ISBN 3 89319 947 0 224 pages diskette Exchange for DERIVE Teaching B rse f r DERIVE Unterrichts materials in the DNL materialien im DNL The wheel has not to be invented twice Das Rad muss nicht zweimal erfunden werden I can offer Binomial Theorem GCD amp LCM System of Coordinates in English and in Ger man as well Modeling Word Problems with DERIVE SET EXE MENGE EXE International DERIVE and TI 92 Conference Computeralgebra and Matheducation Schlo Birlinghoven Bonn 2 6 Juli 1996 7 Keynote Lectures M Artigue J Berry B Buchberger W Herget W Koepf J Palmiter B Waits 57 Lectures from Carmen Arriero to Nurit Zehavi 14 DERIVE Workshops from Josef B hm to Helen Surovyatkina 12 TI 92 Workshops from Roger Brown to Thomas Weth Banquet Choice of 9 coach tours on Thursday Family programme for Wednesday and Friday Information B rbel Barzel Heinrich K nn Str 225 D 40625 D sseldorf D N L 21 Liebe DUG Mitglieder Ungebrochen sind der Ideenreichtum und die Schaffenskraft vieler DERIVE Freunde aus aller Welt auch im 6 Jahr des Bestehens der DUG zu dem ich Sie alle recht herzlich begr e Der Um fang der diesmal angebotenen Sport Artikel machte es n tig die versprochene Konstruk
30. Precision Mixed Notation Decimal 2 InputMode Word CaseMode Sensitive 3 NotationDigits 6 Solution of problem 1 Part 1 Equation ofthe ramp 1 Equation for segment T length T 6 45m a 10 75 4 a 10 75 1 T 6 45 TF x equation for T guT lower bound for T then plot in the interval gu T O 5 m T TANCa TFOO m_T x guT T COS a 6 m T 0 189855 TF x 0 189855 x guT 6 3368 7 TFCx 0 189855 x m T 0 189855 guT 6 3368 TF_ x 8 If 6 3368 lt x lt 0 TFOO 11 Equation for the arc KR1 radius R1 82m 9 R1 82 Starting point of KR1 is endpoint of segment T 10 TRI guT TF guT 11 TR1 6 3368 1 20307 The lower half of the circle and its first derivative 2 2 12 RIFOO R1 x x0 yO d 13 RIF_100 RIF x dx x x0 RIF 10x 14 2 2 ex 2 x0 x x0 6724 We know the osculation point of KR1 with tangent T and we know that TF is tangent of the circle this leads to two equations 15 LRIF CguT TFCguT R1F_1 guT mT 16 APPROXCSOLVEC RIFCguT TFCguT R1F_1 guT 2m T x0 y0 17 x0 8 95809 yO 81 7640 x0 21 6316 yO 81 7640 The center of the circle 18 x0 8 95809 y0 81 764 This is another way to find the center M xM yM 19 x TM R1 SINCa y TM R1 COS a 20 xM 6 3368 x TM yM 1 20307 y TM 21 xM
31. THE DERIVE NEWSLETTER 21 ISSN 1990 7079 THE BULLETIN OT THE USER GROUP Contents Letter of the Editor Editorial Preview DERIVE User Forum Dominique Lymer DERIVE Automatic or Semi automatic mode E Roanes Lozano amp E Roanes Macias About the Tennis Net with DERIVE Hellmut Scheuermann Ski Jumping in the Focus of Mathematics Richard Schorn Ceo The Buckyball Carl Leinbach amp Marvin Brubaker Carl and Marvin s Laboratory 1 Johann Wiesenbauer Titbits 7 The Art of Programming Applications of Johann s RED function Bert Waits Bernhard Kutzler amp Frank Demana The TI 92 Corner revised Version 2009 January 1996 D N L 21 INFORMATION Book Shelf D N L 21 1 Analyse Bekijken en Begrijpen met DERIVE Jan Vermeylen amp Marcel Dams Uitgeverij Rhombus Kapellen Belgium 1996 82 pages diskette Although this is a Dutch book it an easily be understood by non Dutch readers It contains a lot of ideas to forward the concepts of calculus The figure shows one example Josef SS 2 Matematica con il PC Introduzione a DERIVE Bernhard Kutzler Media Direct Bassano del Grappa VI 1995 159 pages 3 Neue Medien im Mathematikuntericht DERIVE mehr als ein Assistent H J Kayser Verlag f r Schule und Weiterbildung Druck Verlag Kettler 1995 ISBN 3 8165 1782 X 187 pages 4 Mastering the TI 92 Explorations from Algebra through Calculus N Rich J Rose L
32. The results given above can be obtained as follows The appendix will show a shorter version performed with DERIVE 6 4 5 JQ0 2 45 J5 Simplification of expression 3 followed by rewriting in order to prepare for squaring 2 2 2 J5 6 4 SIN r COS r SIN s COS s SIN r 4 SIN s 2 2 SIN s 1 5 2 2 2 7 4 SIN r COS r SIN Cs COS s SIN r 4 SIN s 2 2 SIN OS 1 5 E 2 2 2 J5 8 4 SIN r COS r SIN Cs COS s SINCr 2 4 SIN s 2 SIN s 1 5 2 2 2 J5 9 4 SIN r COS r SIN s COS s SIN r 2 4 SIN s 2 SIN s 5 2 4 4 2 2 2 4 10 SINCr 16 SIN s 16 SIN s SINCr 16 SIN s 16 SIN s A 4 2 2 4 8 5 SINCr 16 SIN s 16 SIN s 4 SINCr sse l 5 2 4 45 4 4 45 2 2 45 6 16 SIN s 4 4 SIN s 4 SIN s irs 5 5 5 5 D N L 21 Richard Schorn C4 The Buckyball p39 Expression 8 squared all terms collected on the left side and then simplified once more 2 4 2 8 5 SIN s 4 5 4 12 4 SIN r SINCr A 4 SIN s E 5 5 4 45 2 2 45 6 J snc Ey E 5 5 5 Substitution for sin s according 2 and then simplified 13 45 55 4 9 5 2 2 45 6 13 l sme gt J sio Ep 8 8 5 5 5 Substitution sin r x 13 45 55 2 9 5 2 45 6 14 x 7 X 8 8 5 5 5 13 45 55
33. Thema Analysis Klasse 12 1 Die Gro e M hlenkopf Schanze Die Konstruktion und der Bau von Skisprungschanzen sind genauen Vorgaben des Internatio nalen Skiverbandes FIS unterworfen Eine Freigabe zur Durchf hrung nationaler und interna tionaler Wettbewerbe erhalten Skisprungschanzen nur dann wenn die Schanze von Sachver st ndigen der FIS abgenommen wurde und eine Schanzenprofilbest tigung erteilt werden konnte Dieses sog Homologationszertifikat f hrt alle spezifischen Daten der jeweiligen Schanze auf vgl Abb 1 Erl uterungen in Abb 3 Aufgaben 1 N amp hern Sie unter Ber cksichtigung der Da Der Schanzenvorbau beginnt am Schanzentischfuss ten aus Abb 1 die Anlauf und Aufsprung mit einer Tangentenneigung von b Richtwert bahn durch abschnittweise definierte Funkti b b 6 und endet bei P mit der Tangentenneigung b onen Gleichungen aus Abb 2 anwenden Das Profil des Vorbaues soll bei kurzen Sprungweiten Nehmen Sie das Diagramm der kompletten gute Landebedingungen und gleichzeitig bei langen g X p Fl gen m glichst geringe Flugh hen gew hrleisten Sprunganlage auf Auslauf k rzen Das Profil kann mit einer kubischen Parabel definiert s N s werden Mit den Koordinaten gem ss Abb 1 ist 2 Ermitteln Sie mit Hilfe der Differentialrech 2 l nung eine ganzrationale Funktionsgleichung P PE tgb x 5 3 Grades allgemein zur N herung des _ 2u
34. Web page has been getting a lot of activity since it came on line A number of web page visitors have suggested that we include more information on the DUG If we have enough room perhaps we could also put MTH files from the recent Newsletters on line so they could easily be downloaded The MTH files would be useful to someone only if he or she was a member of the DUG and had a copy of the corresponding Newsletter DNL Your challenge was faced by J Wiesenbauer Look at Johann s Titbits in this issue Your idea concer ning the MTH files is great So e mail users have not to wait until the dikette of the year I ll forward you a diskette with all MTH files immediately after having mailed the DNLs So you can put them on your Web page Torbj rn Alm Eker Sweden A happy New Year to DUG The ODE file was the Gem of the Year A real Herculean effort by Professor Douros Karl Heinz Keunecke Kiel Germany Hallo Josef ich habe k rzlich die Arbeit von Douros in DNL 20 gelesen Die darin besprochene Funktion ODE aus dem Utility file ODE MTH hat mich begeistert und ich wollte sie gerne haben Der Hinweis auf den Listserver von mailbase war wenig ergiebig Da ich annahm dass es sich um eine Utility von DERIVE 3 handelte habe ich bei Bernhard Kutzler angefragt Er verwies mich auch auf mailbase weil Douros die Utili ty dort abgelegt hat WeiBt Du Doch eine andere M glichkeit an die Utility heranzukommen Lieber Josef danke f r Deinen An
35. aces are used only for lists Use only parentheses to delimit function argu ments E Why doesn t atan or arctan simplify to These do not designate the inverse of the tangent function Use 27 TAN for the arc tangent function m tanii tan ix E Why doesn t lim simplify to Although it is abbreviated to lim on the home screen the name of the function is limit see line editor limittsin Cx gt x x 0 gt rp ita AUTO FU e is an ordinary variable just like x To denote the exact base of the natural loga rithm italic e 2 7182 press 2 LN H i A O AUTO Func 5730 then press the lt key twice to delete the AP and enter a closing parenthesis R Ln e mI TRT E Why does solve a x 0 x return the fol lowing result Bsolyve a x B x solve Carx 0 0 DE HAN RA AUTO Os a 0 means that when a 0 any value of x satisfies the equation D N L 21 THE TI 92 CORNER p59 eee AlgebralCalc Other PramI0iClean Up Eigenvectors and Eigenvalues are now built in functions in the Tl handheld devices and in TI NspireCAS as well 3 2 1 1 8 4 Wolfgang Pr pper N rnberg German The TI 92 seems not to like families of curves I define a family of parabolas Define f x x t 92 If I want to substitute t by 2 using the wonderful with operator then I am failing f
36. ally this simple problem with the help of DERIVE The conclusion is that the area covered by the tennis net for singles in its recommended position 1s bigger than that usually found in tennis clubs by more tan 0 4 square meters Therefore those who practise this sport at a certain level must check more than just the mere height at the center of the net RELATED BIBLIOGRAPH Y 1 J L Llorens Introducci n al uso de DERIVE Pub Univ Polit cnica de Valencia 1993 2 P Puig Adam Curso te retico pr tico de Ecuaciones Diferenciales aplicado a la Fisica y T c nica Biblioteca Rey Pastor Puig Adam 1950 3 A Rich J Rich D Stoutemyer DERIVE User Manual Soft Warehouse 1994 4 Reglamento de la Real Federaci n Espanola de Tenis RFET 1993 Appendix from 2009 TI CAS implemented in Voyage 200 and TI NspireCAS as well is able to solve the differen tial equation in one single step Using the shade function we can visualize the difference in areas on the V200 See the respective Tl Nspire and V200 screen shots P 1 y and y 0 0 93 and lajny Ed o ta nl x so po 93 k 100 100 4 e so e2 x 93 x 100 e 50 i Y k 6 405 and y 1 07 100 k 1281 k 1281 k 1281 k jy 2 9 u 100 4 93 k 100 e 2 50 100 100 k 1281 k 1281 k 1281 k 107 e 0 bo 100 93 x 100 e 200 s 100 100 k 0 500000 0 001653 365857 795501 1 860000 k 1 075269 604 861799 1 000
37. ands Navigate through directory trees using standard Windows dialog boxes to load save and print files Context sensitive help Press the F1 key to instantly get detailed help on any command dialog box Extensive on line help is available on DERIVE s built in mathematical functions and operators The utility files distributed with DERIVE are also fully documented on line Expression Entry and Editing Greek toolbar Entering Greek variable names and mathematical symbols on the author line is as easy as clicking on the desired name or symbol in the Greek toolbar Or if you prefer enter them using Ctrl key combinations 2D matrix input After entering the dimensions of the matrix DfW displays a two dimensional expres sion entry array ready for your input You can move around the array using the mouse or direction keys Multiple expression highlighting Click and drag with the mouse to highlight multiple expressions Then you can easily remove them or drag and drop them at a new location Subexpression highlighting Click on an expression to highlight the entire expression Then click on a subexpression to highlight it Just keep clicking to highlight more deeply nested subexpressions Printing Print page setup Customize the header and footer lines of DERIVE expressions and plot printouts Using amp macros it is easy to include the file name page number date and time in headers and footers You can also specify the page size margins and or
38. d and Approximate Mode If you are still encountering problems please let me know exactly how to reproduce the problem Dave Theresa and I are overloaded rigth now working on DERIVE for Windows Dave is also con tinuing to work with TI on improving TI 92 As soon as we release DfW we should have more time to write some interesting articles for inclusion in the DUG Newsletter Thanks for all your help and enthusiasm These are good news concerning set theory indeed Using SET MTH from the 92 diskette lt MTH07 gt and extending this file I helped myself to work with DERIVE in set theory I hope to in clude this in one of the next DNLs I offer SET EXE MENGE EXE is the German version men tioned by Albert among other Teaching Materials See the information page Josef produced an extended set theory teaching and training program based on DERIVE 6 which was presented in the frame of my ACAO09 presentation It will be published in DNL 75 or DNL 76 The DOS programs SET EXE MENGE EXE are still working and are still avail able Josef Recent versions of DERIVE provide set operations UNION VU INTERSECTION and DIFFERENCE miss the complement of a finite set with respect to a given universal set See the help file since the complement of a finite set is not represent able We have the function MEMBER u v but we don t have a function testing if a set u is a subset of set v So let s add these two functions In the
39. d but you have to declare the bases as Positive See the following 1 2 3 FA 5 6 7 8 X X a b a Real 0 b e Real 0 0 Logarithm Collect Now simplify 1 x a b a e Real b Real simplify 1 again X X a b D N L 21 DERIVE USER FORUM p 9 Albert Rich Soft Warehouse Hawaii Hello Josef thank you for sending your ACD EXE program for transforming MTH into ACD files Dave looks forward to seeing your demonstration of t at the Bonn conference in July You also men tioned that R Schorn had produced another tool for creating analglyphs of space curves Will that also be demonstrated Y es of course Josef I enjoyed using your set theory teaching program SET EXE I am happy to report that the next version of DERIVE will include finite sets as a data type and have operators for all the usual set operations DERIVE will use a syntax for enter sets similar to that used by the popular ISETL programming lan guage For example 248 4 Ado INTER orerar AU will simplify to 3 5 7 11 13 In response to H Scheuermann Unfortunately the character ASCII 26 that displays as a right arrow is also the character used to indicate the logical end of a file Therefore DERIVE quits reading the MTH file when it encounters this logical EOF character Using several different versions of DERIVE I had no trouble solving systems of simultaneous linear equations in Exact Mixe
40. doned but if you will view it in a text editor you will find the complete file Maybe ASCII 026 is inter preted as a loading stop character e The last point During my last end examinations I found myself in big troubles and I sweated a lot I had installed DERIVE INI in the net with Precision Mixed In this case DERIVE was unable to solve the system of simultaneous linear equations Why that 4 4 2 1 Fix a x b x e cx e dex e 3 2 2 F l x 4 a x 3 b x 2 cex d 2 3 Fc 0 FO 2 Fi 5 F 1 2 7 J FO dx 2 4 Precision Mixed 5 Notation Mixed gt 6 vra soLve Eco 0 FC 2 F10 5 F 1 2 7 FG dx 2 a b c d a 7 FE Precision Exact 9 Notation Rational 2 10 oulle 0 F2 2 F 1 00 5 F 1 2 7 FO dx 2 a b c d a O 15 11 E ES Ab 164c 154 d 5 4e 0 4 This is Hellmut s problem executed with DERIVE 6 In Precision Mixed the SOLVE Button automati cally switches to approximative solving which results in presenting no solution Josef P6 DERIVE USER FORUM D N L 21 There is an extended discussion on sign 0 in DNL 55 which can be downloaded from our website Transforming the quadratic is a nice challenge In original DNL 21 didn t provide any answer Now many years later tried and can offer a solution using DERIVE s capability simplifying and factoring subexpressions and finally programming the procedure 2 1 d y 1l
41. e TI 92 Section Bert Waits Bernhard Kutzler Frank Demana and Setif FRA Vermeylen BEL Leinbach USA Aue GER Halprin AUS Weth GER Wiesenbauer A Keunecke GER Weller GER Zehavi ISR Impressum Medieninhaber DERIVE User Group A 3042 W rmla D Lust 1 AUSTRIA Richtung Fachzeitschrift Herausgeber Mag Josef B hm Herstellung Selbstverlag D N L 21 DERIVE USER FORUM p 3 G P Speck Wanganui New Zealand I have received DERIVE NEWSLETTER 20 which draws attention to several matters of interest to me Many thanks for the diskette which you forwarded with DNL 20 It provides some very useful additions to the DERIVE programs Last year I was in some doubt as to whether or not joining the DERIVE User Group would prove worthwhile This year I can say without hesitation that the DERIVE Newsletters alone provide excellent value for subscrip tion money Keep up your good work I have a number of mathematics articles at various stages of completion and several of these employ DERIVE in useful ways which may be of interest to DERIVE users I will forward some of these uses of DERIVE from time to time for your consideration for publication in the DNL Lorenz Kopp Neumarkt Germany Ich unterrichte an einer bayerischen Fachoberschule Mathematik Physik und Informatik Seit einigen Monaten bereite ich an Hand von B chern den Einsatz von DERIVE f r den Unterricht vor Als Fachbetreuer muss ich auch meine Kollege
42. e trick with in Opwinfunctie page 14 and in Integraal van Leibniz page 73 to produce a car toon like simultaneous plotting of two figures which gives a special effect Maybe this trick was al ready published before I don t know D N L 21 D Lymer DERIVE Automatic or Semi automatic Mode DERIVE Automatic or Semi automatic Mode Dominique Lymer Vieux Cond France teach mathematics in France to Lycee students 16 22 years old If use Derive to solve an equation to factorize an algebraic expression build a tangent to a curve calculate an integral or what else get the correct result almost at once This is already great But if wish to explain and to make the pupils understand how one goes about it and this is an important part of a teachers job the raw and even brutal use of the program does not bring me anything since Derive gives the result too fast and chiefly since the result flashing on the sceen explains nothing about the method used Starting from this consideration have tried to conceive for pupils pedagogical sequences targetting a precise mathematical objective where they are asked to use Derive in semi automatic mode speculate thus if the student manages to make the hardware doing the job then he has somehow understood the mathematical notions and he has made headway What a computeralgebra system can bring in a classroom remains to be studied more accu rate This is o
43. en blichen Berechnun gen zur Baustatik einer Skisprungschanze ben tigt wird Das Organisationskomitee Weltcup Willingen bersandte mir auf Anfrage detaillierte Un terlagen zur Sprungschanze die die angef hrten Fernsehbilder durch beeindruckende Da ten der Schanze erg nzen und somit zu einer realistischeren Einsch tzung der Wahrneh mung des Skispringens beitragen k nnen Die zus tzlich beigelegte Internationale Skiwett kampfordnung f r Skisprung und Skifliegen f hrt in einem Kapitel die Normen zum Bau von Sprungschanzen aus Diese Richtlinien ordnen u a jedem Teilst ck der Schanze einen speziellen Funktionstyp zu und beschreiben Vorgaben und Kriterien zur Bestimmung der Funktionsparameter Nat rlich interessierte bei Bearbeitung dieser Fragestellung auch inwieweit sich der Prob lemkontext zur Formulierung einer anwendungsorientierten Aufgabe f r den Mathematikun terricht eignet Als Adressatenkreis stellte ich mir Sch ler aus den Klassen 11 ll oder 12 I des Gymnasiums oder vergleichbarer Schulformen z B Fachoberschule vor Die mathe matischen Anforderungen werden durch die traditionellen Oberstufeninhalte abgedeckt Aufgrund der Komplexit t der Aufgabe erscheint mir der Einsatz des Computers mit ent sprechender Software z B DERIVE sinnvoll ja pr destiniert Damit kann sich der Sch ler im Wesentlichen auf den Modellbildungsprozess konzentrieren Zur Nachbildung der Willinger Skisprungschanze an einem Com
44. er Vienna This time I am going to put special emphasis on examples which are very illustrative of what one might call the art of programming in DERIVE Yes it 1s true programming 1s certainly an art both in general and particularly in DERIVE in view of the few tools which are avail able for this purpose On the other hand it s always a great feeling f a self made DERIVE program works perfectly at last and usually very fast into the bargain In the following I have put together some nice programming examples and I strongly recom mend that you have a try at some of them on your own before looking at the solutions given afterwards Whether you are an inexperienced programmer or a dab hand at programming at any rate you will profit from it In the first case it will help you memorize some common tricks of the trade whereas in the second case you might be given a booster by coming up with a solution that is equivalent or even better Wow than mine Well let s go down to brass tacks Here are the tasks you are expected to do if you feel like it l Letubea list of nonnegative inegers Write a DERIVE function DIFF u that returns the Boolean value true 1f all elements of u are different and false otherwise as well as a DERIVE function SORT u that sorts u in ascending order in the first cas e g DIFF VECTOR MOD 51 k 101 k 100 true SORT VECTOR MOD 51 k 101 k 100 1 2 3 4 5 6 7 98 99 100 2 Define
45. er gezogen dass es nicht durchh ngen kann Considerations for measuring of the jump widthd see W on the certifica te Bestimmung der Bogenl nge f r einen Teil des Schanzenvorbaus Calculation of the arc length for a part of the front part of the ski jump R ckinterpretation Back Interpretation Vergleich der zum einen theoretisch ermittelten und zum anderen der real gemessenen L nge W Comparison of the theoretical and the real measured width W Interpretation der Koordinaten des Schanzenrekords verdeutlichen die Leistungen des Springers aus einem anderen Blickwinkel der Rekordsprung impliziert eine ca 67m tie fe vertikale Fallstrecke ganz abgesehen von der wirklichen Flugkurve die ja am An fang des nach oben gerichteten Absprungs sogar noch ansteigt Die Markierung des Rekords weist eine Aufsprungstelle mit deutlich geringerem Gef lle als vor dem kriti schen Punkt K auf Dies ist ein Indiz f r die zunehmende Belastung des Springers bei der Landung Interpretation of the coordinates of the record jump shows the athlete s perfor mance which is a fall of 67m The landing position has an incline which is significantly less than close to the critical point K which indicates growing physical stress for the jumper The next two pages show the work sheets which were given to the students This is a picture of the ski jump which was Gro e M hlenkopfschanzel renovated in 2000 Anlauf The technical details o
46. et posts in the doubles position which is further apart the net slack rather than taut and without a vertical tape at the center The net height at the center can be controlled by the players by adjusting the tension of the net In this position the shape of the upper border of the net is a catenary curve This position is what is found in most tennis clubs so we shall refer to it as the club position 1 DIMENSIONS Height of the posts singles and doubles 1 07m Height of the net at it s center singles and doubles 0 93m Distance from the center of the net to the posts in singles 4 115m 0 93m 5 035m Distance from the center of the net to the posts in doubles 4 115m 1 37m 0 93m 6 405m p16 Roanes amp Roanes About the Tennis Net with DERIVE D N L 21 1 EQUATIONS Let us introduce a coordinate system in the vertical plane that contains the net The x axis is the ground line and the y axis 1s the symmetry axis of the net 2 1 CLUB POSITION For the sake of simplicity the weight of the nylon net will be incorporated with that of the iron wire which supports it Let us denote by T the horizontal tension at the midpoint M of the wire If P is an arbitrary point of the wire s 1s the length of the arc of wire whose extreme points are M and P and m 1s the mass per unit of length of wire then the weight of the arc of wire whose extreme points are M and P will be mgs Therefore the tension 7 of the wire
47. f the recent jump right Picture and table from http de wikipedia org wiki M hlenkopfschanze Anlaufl nge Neigung des Anlaufz v Anlaufgeschwindigkeit Schanzentisch Tischh he Tischl ange Neigung des Schanzertisches a Aufsprung Hillsize Kanstruktionspunkt H hendifferenz Tischkante bis K Punkt h L ngendifferenz Tischkante bis K Punkt n Yerh ltnis H hen zu L ngendifferenz hin R Punkt Heigungswinkel Auslauf Lange des Auslaufs Gr e 107 m 35 26 m s 83 6 km h 3 25 m 6 m 195 m 130 m 65 73 m 111 24 m 0 590 35 118 m Hellmut Scheuermann Skispringen im Blickpunkt Br hlwiesenschule Hofheim Fach Mathematik Aufgabenblatt Sprungschanze 2 Thema Analysis Klasse 12 1 FEDERATION INTERNATIONALE DE SKI INTERNATIONAL SKI FEDERATION INTERNATIONALER SKI VERBAND ES 1o CERTIFICATE OF JUMPING HILL CERTIFICAT DE CONFORMIT SCHANZENPROFILBEST TIGUNG Date of issue Valid till Etabli le 1 10 9 Valable jusq au 51 12 99 Ausgestellt am G ltig bis Place Lit Ge N o2 S N P Ri amp 2 e 10 HENK 0 59 R2 425 Vo 25 m s HIN4By gt gt gt B D 59 5 M So B 1 220 250 Q 3o os B 2 22 980 A 180 w Abb 1 Homologationszertifikat der Grossen M hlkopf Schanze in Willingen D N L 21 Hellmut Scheuermann Skispringen im Blickpunkt p25 Br hlwiesenschule Hofheim Fach Mathematik Aufgabenblatt Sprungschanze 1
48. function K 1 k POL COEFF Qu x kj QUOTIENTCREMAINDER CuU x J x 0 1 el I Giese oe 1s 64 1 1 needs 0 031 sec O 1 E NT E 64 1 1 needs 0 188 sec O 1 paLy_pecree M x J 710 Tel p50 Johann Wiesenbauer Titbits 7 D N L 21 The last examples remind me of Albert Rich s appeal to the DERIVE community in DNL 20 to improve his implementation PARTS Alx n mJ If n lt mm nE 1 CPARTS Allin k_ k k_ m FLOOR n 21 PARTS n Ifae U PARTS AllX n 1 for the number p n of partitions of a natural number n Well there is an obvious improve ment namely to remove the IF part in PARTS n since the correct value of p 0 is definitely 11 Ah those pesky initial values I remember with a smile that at one time mod n 0 was simplified to 0 by DERIVE DERIVE 6 contains a really improved function PARTS n in its utility file CombinatoricsFunc tions mth which works very fast PARTS 50 204226 needs 8 38 sec try PARTS 100 PARTS 50 204226 PARTS 100 190559292 in an instant A more serious approach could be based on Euler s formula for the partition function p n namely po Y cp dR MR n21 p n 0 for n 0 First a less memory consuming implementation of it PO n ITERATE APPEND ITERATE IF k x DIMENSIONt t s 1 d_ 1 t Ik t Ik d 253 k d d 3 IF k d 2 3 lt DIMENSION t s 1 4 d_ lnr Ik d 2 3 k
49. hten linkes Auge rechtes Auge erstellt Am einfachsten geht das indem man aus vorhandenen Koordinaten durch gering f gige nderung der Werte einen zweiten Daten satz erzeugt Auch hier hat ein Pascal Programm geholfen Richard Schorn Can The Buckyball p41 For the Red Green Representation it is necessary to produce two views left eye right eye which can be achieved by slight change of the given coordinates This was also done applying a Pascal program You will find the ACD files and the PASCAL program among the files collected in the com pressed file MTH21 ZIP ACROSPIN is still running in the DOS Window You might try to execute acrospin c60mono and acrospin c60raumk It should work if you have still ACROSPIN In one of the next DNLs I will publish a contribution how to produce ACD files for polyhedrons solids given in parameter form and others ACROSPIN was distributed to gether with the DERIVE DOS versions in order to enable 3D plots As we do have 3D plotting in the recent versions of DERIVE we can do without ACROSPIN but it has its own charm to have a look back to the nineties The 3D plot of the bucky ball is presented at the end of this contribution Josef Here are the first and the last lines of C6OLAPHI MTH longitude A latitude q of the 60 verti ces of the bucky ball 1 O ATAN 2 SORT 5 1 2 2 2 p1 5D ATAN C 2 SORT 5 1 723 3 FARrBL SFATAEN M2 ESOR ECO FLZ 2
50. i i 1 i 1i 1 R nSolue puarid55 56 i 158903 2 164000 15 ho solution found pSolve p 1485 56 i 1850 164000 i ib gzolue purr i1 n 7a rJ r HSA 05937471941 0432 12 Hrliz3 0057374717410452 12 Male RAD AUTO FUNC 7720 tvmPV 180 7 125 1485 56 0 12 12 0 164000 nSolve tvmPV 180 i 1485 56 0 12 12 0 164000 7 12497 THE TI 92 CORNER As you can see in the screen shot above TI NspireCAS and the Voyage 200 TI 92 PLUS as well have the finance mathematics functions implemented Time Value Money Solver Lists and the Graph command can be used to illustrate interesting relationships between the vari ables in this case between the present value pv and the interest rate 7 We also are illustrating a use of the split screen mode as well Page 2 of the MODE menu EE c oon race TO EHET s 0059374719410452 12 07125 1 B pucr Bi mi rz 1000 and n 360 LEFT RIGHT az7218 3 1 Window Editor pucr 2 01 011 812 8132 m r 21b IS 197215 3 89135 2 52196 1 76187 35 Efir S Ratio ii Graph putr i m r 1000 and n 260 i SE Ras AUTO a SS eee ene pu r 1 n Ir 1000 and n 360 i XL a FUNC 10730 Fo lAlgebralcatc other Pra Note the use of the STOP key SS re eg T 2 uda SEED ce rather than the Define com Done 2 cSoluela x 4 bx ec B xl 5 afta b ch ge TA np x mand to define functions Ei Done NE afta b c ther method may be used on T
51. ientation of DERIVE printouts p 4 DERIVE USER FORUM D N L 21 Print preview Preview the print image of DERIVE expressions and plots before actually sending them to the printer Zoom the preview in and out and view one or two pages at a time Printer support If your printer is set up for Windows it is set up for DERIVE Support is provided for all types of black amp white and color printers Plotting fast and accurate plotting Plotting is as easy as highlighting an expression and clicking on the plot command Almost instantly your 2D or 3D plot appears Cross positioning With just a click of the mouse move the cross to any point in a 2D plot window and view the point s coordinates displayed on the status bar Plot range Use a drag box to specify the boundaries of a new 2D plot range or enter boundary co ordinates in a dialog box Command toolbar The 2D plot window toolbar makes it easy to annotate and print plots position the plot region and scale the plot Plot annotation Annotations can be placed at any location in a 2D or 3D plot window They can in clude Greek letters or mathematical symbols as well as normal text Each annotation can be dis played in any color for example to match the color of a particular plot line Albert Rich Soft Warehouse Hawaii Dear Josef Got DUG Newsletter 20 Looks great I wonder if anyone will respond to my mathematical challenges that you have published The Soft Warehouse DERIVE
52. im Blickpunkt This is the original plot from 1996 EDU E 3 J 8 WR Dea u Fi 7 L Hormalpunkt an Kritis cher M Schanzenrekord m COMMAND TIE ETE Center Delete Help Move Options Plot Quit Range Scale Transfer Window aXes Zoom Enter aptian Cross x 8H u 7TH Scale x ZB u zH a pam j In the meanwhile the record was improved by Janne Ahonen He jumped 152 m in January 2005 on a renovated jump hill It was last Sunday when watched a ski jump competition which was held in Hakuba Japan in the frame of the Summer Grand Prix 2009 and saw Kazuyoshi Funaki jump ing He is now 13 years later among the sen iors of the jumpers And he performed very well Josef p36 Richard Schorn C4 The Buckyball D N L 21 Zur Can Modifikation des Kohlenstoffs Richard Schorn Kaufbeuren Germany Historisches Historical Im Jahre 1985 entdeckten Harold W Kroto Universit t von Sussex und Richard E Smalley Rice Universit t eine neue Modifikation des Kohlenstoffs F nf Jahre sp ter konnte Wolfgang Kr tschmer Max Planck Institut f r Kernphysik Heidelberg den neuen Stoff in gr eren Mengen herstellen Die Modifikation besteht aus 60 Kohlenstoffatomen die im molekularen Bereich einen Fu ball auf spannen Harold W Kroto University of Sussex and Richard E Smalley Rice University discovered a
53. images google at images client firefox a amp rls org mozilla de official amp channel s amp hl de amp source Richard Schorn Can The Buckyball D N L 21 Physical Chemical At these computations of the soccer ball we only refer to the mathematical aspect Chemical physical investigations and appli cations e g production of superconductors at high temperatures or its use in connec tion with new drugs cannot be considered at this place You can find a survey about the Fullerenes in an article written by R F Curl and R E Smalley in Scientific American 10 91 The name Fulleren or Buckyball was chosen to honour the architect Buck minster Fuller who developed geodesic domes modelled after the Cso modification There are a lot of Internet resources for in formation together with interesting pictures about this topic In the following you can find a selection hp amp ga zBuckyball amp umz1 amp ie UTF 8 amp eizOTOiSri1 G9aMsAbn N3SBA amp sazX amp oiz image result group amp ct title amp resnum 4 http en wikipedia org wiki Fullerene http www physik uni oldenburg de bucky htmis buckintro html Geographisches Mit den Ergebnissen kann man nun die Koordi naten der Eckpunkte als geographische Koordi naten Lange A Breite y im Gradmaf oder wie in C60LAPHI MTH im Bogenma angeben F r Plotterausgaben Raytracing Verfahren oder auch ACROSPIN Dateien empfehlen sich Berechnun gen von karte
54. in P will be obtained by adding 7 horizontal with the reaction mgs vertical against the weight of the arc of wire When balanced the direction of the wire in P will be the direction of 7 Therefore denoting by y the function that gives the position of the wire when balanced and denoting by o the angle of T with T results in dy mgs ang 9 dx E d Differentiating with respect to x mg ds m N SIT T dx So denoting by K the quotient ze the differential equation of the wire of the tennis net is obtained as y sk ly With the change of variable y z it is transformed into the equation of separable variables that will be integrated with the help of DERIVE The previous equation can be integrated by issuing the expression 2 102 SEPARABLE Ck 1 z 3 Pee ecol Sh as z y 0 when x 0 horizontal incline D N L 21 Roanes amp Roanes About the Tennis Net with DERIVE p17 2 1 2 1 SEPARABLECK 1 z eau xn m 2 2 LNGCz 1 4 2 kex 2 3 SOLVE LN Oz 1 z kx 23 k x k x e e 4 Em 2 2 k x k x e e 5 SEPARABLE RHS vxo ay 0 0Sa 2 2 kx k x e e 1 a3 5 y T ek eek k 100 Simplifying 1 gives equation 2 which is solved wrt variable z By undoing the change of variable y z another equation of separable variables is obtained As y 0 93 when x 0 height of the net at its center then this can be integrated by
55. is wonderful International DERIVE Newsletter We recognize that the CSA is not a complete DERIVE system but the TI 92 is sufficiently powerful to be of great practical and pedagogical value to both students and teachers in many levels of mathematics The CORNER is for YOU we need your contributions We hope to have enough reader contribu tions to make the TI 92 CORNER a regular feature of the International DERIVE Newsletter We solicit TI 92 contributions in the following areas Mathematical examples Pedagogical examples Interactive scripts using the TI 92 text editor Programs Any combination of any above Letters to the editor including questions about the TI 92 CSA vm cU qox p56 THE TI 92 CORNER D N L 21 Please send the contributions to the TI 92 Corner Editors c o Bert Waits The Ohio State University Department of Mathematics 231 W 18 Ave Columbus Ojio 43210 USA Contributions may also be sent electronically to waitsb math ohio state edu Bert Waits Berhnard Kutzler and Frank Demana March 1996 You can teach the TI 92 mathematics it does not know by Bert Waits and Frank Demana The ability to easily great user defined functions on the TI 92 is a very useful feature For exam ple the present value value today of a series of future payments is not a built in function on the TI 92 However such a mathematics of finance function can become a permanent feature of the TI 92 Consider a series of
56. l tool that we have at our disposal The following is an illustration of such a problem First we begin with a description of another of DERIVE s plotting features DERIVE will do a 2 dimensional plot of any expression that requires a single numerical input and returns a unique numerical value This it can plot a function defined as an IF statement This is an obvious way to do a piecewise defined function Another way is to use DERIVE s built in CHI characteristic function This function is authored in the following way CHI lt left end gt lt variable gt lt right end We will consider a situation based on a farm irrigation device that consists of several nozzles mounted on a pipe that rolls along a field Each nozzle is a foot long and sprays water in a circle of radius 20 feet with uniform wetting of the area covered The sprayers are to be positioned on the pipe to give the most uniform coverage possible as the pipe 1s rolled along the field The criterion we use for uniform is the time under the spray multiplied by the number of nozzles reaching a point The fig ure below is a picture of a stationary position of part of the sprayer We are assuming that at most two sprayers overlap Our problem gives rise to the following function to describe within a constant multiple the amount of water sprinkled on a point V400 x 41 x 0 xxl F x d 244400 x 1 lt x lt d 20 4400 x 4 400 x d d 20
57. lieder sollte von mehr als 500 auf ann hernd 600 angehoben werden k nnen Wenn wir alle ein wenig mithelfen k nnte das gelingen Bitte beachten Siebitte die Materialienb rse auf der Infoseite Ich kann Ihnen f r 1996 einige sehr interessante Beitr ge im DNL versprechen und freue mich mit Ihnen auf ein spannendes DERIVE for WINDOWS amp TI 92 Jahr 1996 Au erdem hoffe ich viele von Ihnen im Juli in Bonn zu treffen Mit den besten Gr en Ihr LETTER OF THE EDITOR p 1 Dear DUG Members Unbroken is the ingenuity and the creative power of so many DERIVE friends from all over the world even in the 6th year of the DUG s existence would like to welcome you all old and new members from all continents The volume of the sportive contributions pub lished in this issue makes it necessary to show the construction of the 17 edge in the next DNL In addition J Wiesenbauer has sent me another instruction for drawing this figure will compare both constructions next time Unfor tunately the next part of Th Weth s Lexicon of Curves has to wait for DNL 22 but then with the TI 92 Cabri You will find two innovations in this Newsletter Carl and Marvin s Labor atory is a collection of some out of the ordinary DERIVE activities and am glad that Cart agreed the labs to be published in the DNL What an honor to have a section entitled Cart and Marvin s Laboratory don t know if will be able to keep
58. llmut Scheuermann Skispringen im Blickpunkt 69 x02 178 186 70 APPROX SOLVE R2F goM MF goM y02 71 y02 39 0900 72 y02 39 09 73 R2F 178 186 85 91 Center of circle R2F 178 186 39 0900 Finding the endpoint R2A of the arc We find the x coordinate by shifting the circle x02 178 186m and the y coordinate 85 91 74 R2A 178 186 85 91 R2F x 75 If 102 96 lt x lt 178 186 R2F x 111 Equation for line segment A length 180m point R2A A is a horizontal tangent ot arc KR2 AF Cx in 76 If 178 186 lt x lt 178 186 180 R2F x02 IV Equation for the part beween the ramp and point P First finding point P 77 P guM MF guM 78 P 79 0009 42 6855 Approximation by a cubic according to the FIS rules b 79 Np guM Hp MF guM S 3 15 b0 6 80 u Hp S Np TANCbO v TANCb TAN bO 2 u V Np 3 u v Np D qQ S 81 3 2 Np Np 3 2 82 SPF x pex qx TAN bO x S 3 2 83 SPF x 0 0000223177 x 0 00673013 x 0 108046 x 3 15 SPF x 84 If 0 lt x lt 79 0009 SPF x 85 AS 0 S 86 K R2A P AS SPF_ x ME GO R2F OO AF 60 Plot the landing zone Hellmut Scheuermann Skispringen im Blickpunkt TRl 10 1 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 20 40 Point P Point K Solution of problem 2 Given
59. lt x lt S p46 Carl s and Marvin s Laboratory 1 D N L 21 This can be entered n DERIVE n the following way 1 F x d CHI 0 x 1 N 400 x 2 N 1 x 2 CHI 1 x d 20 N 400 x 2 CHI d 20 x d 2 N 400 x 2 4N 400 x d 2 The question is now what value of d i e for what spacing of the sprayers will we get the best coverage of the area This 1s a very fuzzy question We will graph the function for several values of the variable d to try and get a handle on it Beneath is the graph of F x d for several values of d 24 26 28 30 32 34 and 36 together with the lines y 19 3 We chose 19 since that is the value for each function at x 0 the position under the nozzle To save time we use the VECTOR function Author and Plot 2 VECTOR FE x YV d 24 36 2 DERIVE will plot this array of functions one function at a time Unless we change the defaults it plots each function in a different color 30 25 It appears that when d 34 the spraying best fits our arbitrary criterion like this laboratory because of some aspects 1 it is an unusual but daily life problem 2 it emphasizes the modelling process 3 itis really a fuzzy problem we call it an open ended question and 4 the use of the computer supports in a high degree to find a solution Josef D N L 21 Johann Wiesenbauer Titbits 7 p47 T tbits from Algebra and Number Theory 7 by Johann Wiesenbau
60. mbination of the solutions 3 2 2 T TT 35 SOLVECC 3 c 1 c c 1 2 COS v c 2 COS 1vc 9 9 TT 1 2 SIN 18 2 T 4 7 4 T 4 36 1 2 COS 2 COS 1 1 2 SIN 9 9 18 2 T TT TT 2 7 37 2 COS COS 16 SIN 20 22 43 SIN 9 9 18 9 TT 16 SIN 65 18 38 69 D N L 21 THE TI 92 CORNER p55 We try solving the system with DERIVE 6 in one single step 2 2 2 3 3 3 39 SOLVE a b c 9 0 Aa b c 24 0 r1a b c 3 0 la b c faco cos es 40 9 9 18 TT a C SIN 1 2 18 This is again a huge expression which gives approximated 41 a2 2 879385241 A b 0 5320888862 v a 0 5320888862 A b 2 879385241 c 0 6527036446 v a 0 6527036446 b 0 5320888862 A c 2 879385241 v b 2 879385241 c 0 5320888862 v a 0 5320888862 c 2 879385241 v a 2 879385241 c 0 5320888862 b 0 6527036446 One solution combination for a b and c leads to the final result 69 THE TI 92 CORNER edited by Bert Waits Bernhard Kutzler and Frank Demana Introduction We have waited many years for a powerful easy to use inexpensive hand held DERIVE like com puter symbolic algebra CSA system like the TI 92 The algorithms of DERIVE are the foundation of the TI 92 CSA system We are very pleased that Josef has allowed us to offer this new TI 92 CORNER as an addition to h
61. n found satisfies the differential equation as well as the initial conditions 2 SOLVECT5 x 24 x 3660 D x TA L 4 Y X 5 5 4 PB faz Zi os d SHI psi 5 5 5 d 4 t 5 i d a feeb feet ea d h feet O z COS SIN dt 5 5 5 5 5 5 4 t 5 d a 76 b FE fava 4 h l TA T lt e COS SIN 5 5 5 5 5 5 3 soLve d z coy 50 7 10 z1 0 all La BJ a 0 0507 a b D 0025 4 78 7B zit d sll man ad 0 0006 s l 5 5 5 4 t 5 7B t 793 8 SIN 5 1000 ZUCH 2 507 2 0 Z1 0 10000 d 42 Ze DE E ZE d 15 224 24 214 3660 7 t O Draw the Curve representing the solution fig 5 Thus the student unburdens himself onto the computer of the laborious and boring part of the calculations that his knowledge or abilites in maths do not allow him to deal with but keeping the lead in the calculations and making the machine work he can concentrate more on the mathematical method of solving as well as the interpretation of the results attained Figure 5 The hit of a hammer D N L 21 Roanes amp Roanes About the Tennis Net with DERIVE p15 About the Tennis Net with DERIVE Eugenio Roanes Lozano amp Eugenio Roanes Macias Dep Algebra Fac Educacion Univ Complutense de Madrid Translation of an article published in the Boletin de la Soc
62. n informieren Ihre Newsletter sind mir dabei eine wertvolle Hilfe Wilfried Wieland W stenrot Germany Vielleicht k nnen Sie in der n chsten Ausgabe etwas ber die versprochene Windows Version yon DERIVE sagen DNL Fortunately I received yesterday an e mail from AI Rich which might be of interest for you and not only for you I m sure A Windows version of DERIVE DfW will soon be available DfW will carry on the tradition by combining the point and click ease of Windows with the reliability users have come to expect from DERIVE SWHH has targeted Summer 1996 as the release date for DfW Its suggested retail price will be 250 Registered users of DERIVE and DERIVE XM will soon be receiving their discount offer The following are a few of the many innovations incorporated into Derive for Windows Command toolbar Issuing frequently used DERIVE commands is as easy and effortless as a mouse click Just highlight an expression and click on the desired operation on the command toolbar The algebra window toolbar includes buttons for authoring simplifying solving differentiating and inte grating expressions Helpful tools indicate the purpose of each button Substitution dialogs Simply click on the variables from the list that DfW presents to you and then enter the desired value for that variable If you want to change a value for a variable already entered simply click again on that variable and edit its value File menu comm
63. ne of the aims of the study currently conducted in France with the project of utilisation of Derive led by the Ministry of National education Here are two examples to illustrate uses of Derive in automatic and in semi automatic mode Example 1 The notion of tangent to a curve The DIF APPS MTH utility file includes a tangent function which automates the calculation of equations of tangents but does not teach how to get them In class of Premiere one year before the Baccalaureat where pupils discover differentia tion think that having explained the formula y yo f1 xo x xo one can suggest the fol lowing activity with Derive Define the function f by f x 2x 3x 1 for example Calculate the derivative function f1 Calculus Derive Simliplify then f1 x State a real variable xo then assign yo yO f x0 After solving for y the formula gives the general equation of a tangent to the curve Then questions in French or in English can be asked which the students must first under stand then rephrase to make Derive finding the answer e What is the tangent to the abscissa point 2 xo 2 ordinate 4 First solve the equa tion f x 4 for x then assign the result to x0 e Find the tangents at the points of intersection with the axes solve the equation y 0 then assign the solutions to x0 p12 D Lymer DERIVE Automatic or Semi automatic Mode D N L 21 e sthere a tangent parallel to
64. new modification of carbon in 1985 Five years later W Kr tschmer Max Planck Institute for Nuclear Physics Heidelberg was able to produce larger amounts of this new stuff Mathematisches Mathematical Als Modell verwendet man einen blichen Fu Use a regular soccer ball as a model Its ball in Spielzeuggesch ften gibt es auch kleine Surface consists of 12 mostly black col Plastikb lle die beim Tischfu ball verwendet Oured pentagons and 20 white hexagons werden Die Oberfl che besteht aus 12 meist In the vertices are all the 60 carbon atoms schwarz gef rbten F nfecken und 20 wei en located Sechsecken An den Ecken sitzen dann die insge samt 60 Kohlenstoffatome Im Folgenden sei der Radius der Kugel 1 Der Radius des sph rischen F nfecks ist ein Gro kreisbogen der L nge r und die Seitenl ngen der regelm igen Vielecke seien 2s nat rlich auch wie alle weiteren Angaben als Gro kreis b gen gedacht Die Gleichungen zur Berechnung der beiden Gr en s und r erh lt man aus sph rischen Dreiecken Let the radius of the sphere 1 The radius of the spherical pentagon is an arc of a great circle with length r and the edges of the regular polygons have length 2s also arcs of great circles opheric triangles lead to the equations to find r and s l Verbindet man den Mittelpunkt eines F nfecks mit zwei benachbarten Ecken des F nfecks ergibt sich ein gleichschenkliges Dreieck
65. next file define a universal set and some subsets co subset universal set gives the complementary set and subset u v gives true if u is subset of v and false otherwise p10 1 He 3 4 5 6 7 8 9 10 11 12 13 DERIVE USER FORUM u I0 1 7 3 4 5 6 7 8 9 10 Ll 12 s1 fi 7 3 5 6 7 9 10 s 12 3 4 6 10 3 1D 1 3 7 amp Y 54 12 5 10 55 1 2 3 4 9 10 11 15 cols g lt QV S co sl u 10 4 11 12 subset u v i Ifunv u true False False subset s4 u true subset 55 u false subset u true subset s 52 true Some examples 14 15 16 17 18 19 20 21 22 23 A verification of De Morgan s Rule co sl n s2 u 10 1 4 5 7 8 9 11 12 co sl u u co s u 10 1 4 5 7 8 9 11 12 co sl u s2 u 10 11 12 co sl u n co s2 u 10 11 12 SZ us3 u 55 1 2 0 1 2 3 4 6 7 8 9 10 11 15 52 n s3 n s55 3 De Morgan s Rule generalized anb a ub true aub a nob true Jan Vermeylen Kapellen Belgium Of course you can reprint any material from ANALYSE for the DNL As my favorites I am a bit proud of the 3D projection I used in Volume Omweltelingslichaam Volume of a solid of revolution page 80 See information page I modified a function from GRAPHICS MTH And also th
66. nnis Net with DERIVE p19 As a consequence the equation of the upper border of the whole net in the championship position is G x 0 93 m ABS x 2 2 DIFFERENCE OF AREAS The difference between both areas can be calculated by integrating the difference of functions F and Gl The lower and upper limits of integration are 0 and the distance from the center of the net to one of the single posts respectively As the net is symmetric the area to be calculated will be twice the value of this definite integral Calculating it with the help of DERIVE i e integrating between the theoretical positions of the single posts we obtain 5 035 a Fix GLS 11 dx 0 414 70585548 which 1s as expected a negative value 3 GRAPHIC REPRESENTATION OF THE NET The shape of the net can be obtained in DERIVE with the following functions defined using function F and G from above FFix 22 If ABS x gt 6 405 Fix 7 Gx t If ABS x lt 5 035 G x If 4 CABS lt 5 035 v ABS x gt 6 405 1 07 z S x e E ee a SE 21 FFix s y s Ga x a a lt 5 035 6 405 U 6 405 U 5 035 U 5 035 U 622 i 1 i 6 405 1 07 6 405 1 07 5 035 1 07 5 035 1 07 21 gives the shading of the difference area and 22 gives the plot of the posts p20 Roanes amp Roanes About the Tennis Net with DERIVE D N L 21 4 CONCLUSION The correct combination of Computer Algebra and numerical techniques enabled us to solve automatic
67. possibilities of new methodical and didactical manners in teaching mathematics We include a section dealing with the use of the TI 92 Editor Mag Josef B hm A 3042 W rmla D Lust 1 Austria Phone 43 0 2275 8207 Preview Contributions for the next issues D N L 21 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 D N L It must be said though that non English articles are very welcome nonetheless Your contributions will be edited but not assessed By submitting articles the author gives his consent for reprinting it in the D N L The more contributions you will send the more lively and richer in contents the DERIVE Newsletter will be Graphic Integration Probability Theory Linear Programming Bohm A LOGO in DERIVE Lechner A DREIECK MTH Wadsack A IMP Logo and Misguided Missiles Sawada HAWAII 3D Geometry Reichel A Parallel and Central Projection Bohm A Vector and Vector Indices Sorting Biryukov RUS Algebra at A Level Goldstein UK Tilgung fremderregter Schwingungen Klingen GER Utility for Complex Dynamic Systems Lechner A Some notes on DERIVE 2 6 functions and limits Speck NZL Linear Mappings and Computer Graphics Kummel GER Julia Sets K mmel GER Solving Word Problems with DERIVE Bohm A DERIVE and ACROSPIN Schorn amp B hm A GER Th
68. puter werden vom Sch ler Modellbildungsqualit ten gefordert wie sie z B bei Kaiser Blum Schober 1982 beschrieben werden Das Homologierungszertifikat vgl Abb 1 der Aufgabenbl tter stellt bereits eine Idealisierung des realen Modells dar Die L sungen zu verschiedenen Funktionsfindungs aufgaben Mathematisierung er ffnen den bergang zum mathematischen Modell Der Einsatz des Computers zum einen als Graphikwerkzeug er zeichnet die mathematische Nachbildung der Schanzenprofillinie und zum anderen als Rechenhilfe zur Bearbeitung der weiteren Fragen aus dem Aufgabenblatt mathematische berlegungen f hren zu mathe matischen Resultaten Die R ckinterpretation der erzielten Ergebnisse auf das reale Modell aber vor allem auf die reale Situation sollte u a zur Verbesserung der eingangs erw hnten Vorstellungen vom Skispringen beitragen Hellmut Scheuermann Skispringen im Blickpunkt Who does not know them spectacular TV pictures of ski jumpers who plunge into the depth AI though we can see more and more thrilling views the real facts yet cannot be transferred unrestricted The person sitting in front of a TV set can neither imagine the steepness of the approach nor how far the jumpers are gliding down through the air The telecast of the World Cup ski jumping from Willingen 1995 gave the idea to investigate these questions and to analyse the parameters of this special ski jump I wanted to explore how much mathematics is necessa
69. ri SINS ffl SINS VL SINEr I SIM r 4 SIN S 2 z SIN S 1 5 e e Z JCR 45 SIN r E1 SIM r 0 4005 5 SINCr 8 5 5 d l 45 13 q5 8 SN SIN eS n Z 4 Z Z Z x eaea x 5 5 8 4 5 45 n 45 13 5 3 SF ee L X plc Z Z x 5x5 411 x DJS 45 5 8 4 l 5 45 Z 45 E JS 10 _ SQ LYE X 24xz 1 IX Z d d 5 24013625 zl80 45 24013625 2180 57 A lA 222222 T _ _ 22 545 545 24013625 2180 45 2 4013625 2180 45 12 ASTM ASIN 1 545 545 4 5 2 4 5 2 13 ATAN ATAN ma 19 19 19 19 amp 14 0 3504054128 0 3504054128 SIN 36 2 4J 13525 2180 45 215 ASI q a e e En 545 45 l 16 ATAN I 6 6 17 0 203168946 cos 4 sin z cos r sin s co 2 A E pata and s sin y x 0 343278613032 or x 0 343278513032 sin 0 343278613032 0 350405412847 sin 36 0 35040541284736 0 205963133995 Carl s and Marvin s Laboratory 1 An Optimization Problem A Non Calculus Example Carl Leinbach amp Marvin Brubaker USA Every optimization problem requires a careful understanding of underlying method and definition of the objective function Whenever possible it is wise to plot this objective function in the region defined by the constraints Sometimes this 1s the only mathematica
70. ruf Ich habe nicht erwartet dass die von mir gesuchten Files bereits auf der Diskette 95 sind Ich habe sie bereits gefunden und werde damit loslegen Aus meiner vielleicht einseitigen Sicht ist das eine der besten Arbeiten die im DNL ver ffentlicht wurden Vielleicht k nnen die Autoren in Zukunft ihre e mail Adresse angeben Man kommt dann leichter mit Ihnen ins Gespr ch Douros hat z B eine ich habe sie erfahren und ihn schon angeschrieben Es gr t aus eingeschnei tem Haus Karl Heinz Hallo Josef inzwischen habe ich die Arbeit von Douros mit seinem Utility file nachvollzogen besser es nur mit m gem Erfolg versucht H ufig erhalte ich nicht seine L sung sondern die Fehlermeldung No ordinary differential equation Einiges in der Nomenklatur erscheint mir unklar und wenigstens in einem Falle erhalte ich mit den Douros Funktionen eine falsche L sung einer DGL Wie weit hast Du Dich damit befasst Hast Du Reaktionen von anderen Lesern dieser Arbeit erhalten D N L 21 DERIVE USER FORUM p gt DNL Karl Heinz had some problems to find Prof Douros Utility file containing ODE from DNL 20 He was surprised that it was already on the diskette of the year 95 in subdirectory lt DOUROS gt Karl Heinz esti mates this contribution as one of the best articles published in the DNL so far In the meanwhile he has worked with this file but the results are not as good as he had expected Are there any experiencies of other DUG mem
71. ry for planning and constructing such a facility The organizing committee sent detailed materials of the jump and enclosed the nternational Competi tion Rules for Ski Jumping and Ski Flying These rules define each single part of the ski jump by a special function and describe the criteria to determine the parameters of the functions I was interested to find out if the problem would be able to give an application oriented problem for the maths class The mathematical requirements are covered by the traditional SII level The complex ity of the task seemed to make the use of the PC not only useful but predestined So the students can concentrate on the modelling process They are forced to show modelling abilities Kaiser Blum Schober 1982 The Certificate of Jumping Hill is already an idealisation of the real model The construction of several functions mathematiza tion process opens the change to the mathematical model The use of the PC to plot the profile and to work on the other problems given on the work sheet leads to mathematical results Back interpretation of these results for the real model but especially for the real situation should contribute to a much more better imagination of ski jumping M Meinen Dank mochte ich insbesondere dem Sekret r des Organisationskomitees Weltcup Willin gen Herrn Manfred Stede aussprechen der mir spontan alle Unterlagen zuschickte und der im wei teren durch zahlreiche fachliche Erl uterungen z
72. simplifying expression 5 which can be entered as SEPARABLE RHS 4 1 2 y 0 0 93 where RHS is the right hand side of an equation which is new in DERIVE 3 Simplifying 5 re sults in the following k zd Q uot y 0 93 i e the catenary of equation 0 93 PL T y 0 S Now the constant k has to be obtained from the condition y 1 07 when x 6 405 which is the height of the net at the posts k 6 405 k 6 405 e l 1 07 0 93 2 k 2k k Let us observe that unlike other CAS DERIVE transforms all numerical inputs even those writ ten in floating point into fractions Unfortunately DERIVE cannot solve the previous transcendental equation with respect to vari able k So a numerical approximation will be computed To obtain this numerical approximation the bisection method will be used This method is based on Bolzano s Theorem and due to its recursive nature can easily be implemented in DERIVE BOLZANOCF x pl pz e is If ABS pl pz lt e w LIMCf x pl If 0 pl p2 2 15 If LIMA x plJ LIM T x pl elf gt 0 BOLZAMOCf x pl p2l 2 pz el BOLZANOCF x pl pl pzZJ 2 el p18 Roanes amp Roanes About the Tennis Net with DERIVE D N L 21 The inputs to the BOLZANO function are f function x variable pl lower extreme p2 upper extreme and e upper boundary for the error It is supposed that F p1 and F p2 are of different sign In this case f
73. sischen Koordinaten Mit dem Radius 1 erh lt man die Koordinaten x cos Q COS A y cos g sin A z sin Die Berechnungen sind nicht nur mit DERIVE sondern auch mit Hilfe von Turbo Pascal Programmen durchgef hrt worden Das Netz soll eine bersicht zu den 60 Koordinatenpaaren und tripeln geben Au er diesen Angaben ben tigt man nat rlich auch Informationen ber die Verbindungslinien der einzelnen Punkte Geographical All these results obtained so far allow to pre sent the coordinates of the vertices in geo graphic coordinates longitude 4 latitude The computations were performed not only using DERIVE but also using Turbo Pascal programs The net gives a survey of the 60 pairs and triples of coordinates In addition to the points you will need information how to connect the points by segments produced two ACROSPIN files C60MONO ACD and C60RAUMB ACD which contain the respec tive data Using a similar program obtained the Car tesian coordinates for the world radius 1000 given in C60R1000 TXT D N L 21 Die ACROSPIN Dateien C60MONO ACD and C60RAUMB ACD enthalten entsprechende An gaben die durch Pascal Programme erzeugt wor den sind C60ACD PAS berechnet die Koordina ten f r das Mono Bild Mit einem hnlichen Pro gramm wurden die kart Koordinaten f r den Erdradius 1000 ermittelt C601000 TXT F r die Rot Gr n Darstellung ist es notwendig dass man zwei Ansic
74. tic or Semi automatic Mode Figure 3 Visualizing Newton s method for solving xg 20 ee Figure 4 The curve of f x exp x as the envelope of its tangents Example 2 Solving a differential equation The study of the hit of a hammer onto an anvil leads to the particular solution of the differen tial equation 15 Z 24 Z 3660 z 0 with z 0 50 7 10 3 and z 0 0 The calculations end up becoming simpler but are difficult for the students and many do not know how to solve it in manual mode no use of a CAS If my goal is to visualize the oscillations caused by the hit on the screen author with DERIVE automatic mode DPOLVE2 EViZ47 15 S000 Loy Uy ty Dy rer 0 and represent the solution graphically If my goal is to teach the pupils to solve a differential equation can use this example but the automatic mode does not suit me because it pulls the solution out of too dark a box as if by miracle The following activity semi automatic mode can help me to convey the technique for solving p14 D Lymer DERIVE Automatic or Semi automatic Mode D N L 21 Write the characteristic equation and solve it Using the lesson s formulas write the general solution of the differential equation Calculate the first derivative z t then define z1 t Then set the systeme of equations which fits the initial conditions and solve this system Replace the constants by the values found Check whether the functio
75. tion des 17 Ecks zu den letzten Titbits auf den DNL 22 zu verschieben Au erdem lie mir J Wiesenbauer noch eine zweite Konstruktionsvorschrift zukom men die ich gerne vergleichsweise darstellen mochte Leider muss auch die n chste Folge des Kurvenlexikons auf DNL 22 warten daf r auch mit dem TI 92 Cabri In diesem Newsletter finden Sie zwei Neuerungen Carl und Marvin s La boratory ist eine Sammlung von nicht allt glichen DERIVE Aktivit ten und es freut mich dass Carl sofort seine Zustimmung zum Abdruck gegeben hat What an honor to have a section entitled Carl and Marvin s Laboratory I don t know if I will be able to keep my head in my hat Ich denke Du solltest Deinen Kopf dort behalten wo er ist Pat braucht sicher keinen kopflosen Carl Und wo w rdest Du Deine Tauchermaske und den Schnorchel fest machen Wir freuen uns auf ein Wiedersehen in Bonn Zweitens werden Sie eine neuerliche Erweiterung des DNL bemerken k nnen Vier zus tzliche Seiten sind vor allem dem CAS im TI 92 gewidmet Nat r lich sollen auch die weiteren TI 92 M glichkeiten in Zukunft behandelt werden Dank gilt Bert Waits und Bernhard Kutzler die mitgeholfen haben die erste TI 92 Sektion zu gestalten Ich erwarte gerne Beitr ge und Anfragen zum TI 92 Eine meiner pers nlichen Zielsetzungen f r die DUG im Jahr 1996 ist es das TI 92 DERIVE in den DNL zu integrieren Damit habe ich ein zweites Ziel verkn pft die Anzahl unserer Mitg
76. u have had enough Let us conclude then with two easy routines namely POL COEFF u x n which yields the coefficient of the term x of the polynomial expres sion u in x and POL DEG u x which computes the degree of the polynomial expression u in x By a polynomial expression u in x we mean here that expand u 1s a polynomial in x in the strict sense of the word You say those functions are already contained in the utility file MISC MTH That s handy Thus they could serve us well as a benchmark for our own routines Now it s high time we turned to my solutions Here you are 1 Comment in 2009 The SORT routine is now a built in command DERIVE DIFFEw z If DIMCv DIMCTERMSCv VECTOR Cx k k_ v2322 true false 2 Comment in 2009 Now we have the FACTORS command which does the job Z 3 FACTORS 360 3 2 5 1 Comment in 2009 We do now have DIVISOR_TAU n which can easily be defined by using FACTORS n This can be found in NumberTheoryFunctions mth which is one of the files in the Math directory and is replacing the old NUMBER MTH Q n is not con tained but it is no problem to define it DIVISOR _OMEGA n DIVISOR_TAUCH Mv 1 v_ FACTORS 2 DIVISOR TAU C360 24 DIVISOR OMEGA n CCFACTORSCn22112 DIVISOR OMEGA 360 6 D N L 21 Johann Wiesenbauer Titbits 7 p49 Johann s original solutions for tasks 2 3 and partially 5 are using FACTORS from 1996 Recent versions of DERIVE have another F
77. um Entstehen dieser Anwendungsaufgabe beitrug Die einzelnen Phasen des Modellbildungsprozesses lassen sich wie folgt pr zisieren The phases of the modelling process can be defined as follows Mathematisierung Mathematization Mathematische Beschreibung des realen Modells durch ganzrationale Funktionen bis 3 Grades und Kreisfunktionen Mathematical description of the real model by polynomial functions up to cubis and trig functions L sung von Funktionsfindungsaufgaben zur Bestimmung der Parameter Solution of problems finding the functions in order to determine the parameters Mathematische berlegungen und Rechnungen mit DERIVE Mathematical considera tions and calculations supported by DERIVE Graphische Darstellung der abschnittsweise definierten Funktionen L ngenbestim mung von Geraden und Kreisst cken Graphic representation of the piecewise defined functions including finding the lengths of line segments and arcs of circles Verifikation der Funktions Gleichungen zur Beschreibung des Schanzenvorbaus laut Wettkampordnung Verification of the functions equations desribing the jump s front part according to FIS rules Hellmut Scheuermann Skispringen im Blickpunkt A berlegungen zur Messung von Sprungweiten vgl Definition der Weite W Modell vorstellung Ein Ma band wird mit dem Anfang an der Schanzentischkante befestigt und zur Messung der Weite so stramm den Berg herunt
78. useful hints on the use of the TI 92 Some of the examples are taken from our work at the bk teachware product hotline The other examples are taken from a FAQ frequently asked questions document produced by David Stoutemyer E How can I interrupt the plotting of a graph Simply press ON nC aesaet art a A li E How can one plot the family of curves sinx sin 2x sin 3x Use one of the following commands Graph sin 1 2 3 x Graph seq sin k x k 1 3 E Can the TI 92 solve a generic quadratic equation I get the following result p R sclue cat pN s x x 44 px and 4 px B or x 2 q qOUIIC M AE LARES IRIN RAD AUTI Obviouslx you have entered px instead of p x px designates a 2 character variable different There is no character mode for entering variables like in DERIVE so you have to use the multiplication symbol or a space fesse es Sar x 1 2 ms G 4 JF 4 a p 4 P q Py solyet x 2t qt p x 0 E 0l 2 2Z HAIK L m H FUNC 350 E How can one see all the second functions of the QWERTY keyboard Simply press 0 K to obtain the respec tive keyboard map see page 57 E What do cos inf cos inf or cos undef mean in an answer These designate the interval 1 through 1 Why can t one use curly braces to delimit function arguments Unlike elementary arithmetic and algebra curly br
79. will be the function given in 79 which has different signs in 0 001 and 0 02 as can be checked with DERIVE s command SIGN al 1 H3 wik COSH 6 405 k 0 93 1 07 k k 10 BOLZAMO QwCk k 0 001 0 02 0 00001 11 0 005871533203 Therefore the recursice process can start from p1 0 001 and p2 0 02 If e 0 00001 is chosen as upper boundary for the error when DERIVE is asked to calculate 10 the system returns 0 00682153 So we shall consider as an approximative value for k ko 0 00682 Comment of the editor It is no problem to solve the equation from above numerically k 6 405 k 5 405 e e 1 ga 12 HSOLVE 1 07 zeke zeke k 100 poder UL 13 k 0 0068241182346 We consider the function of the net in club position as l l F x cosh k x 0 93 x A ko x A 0 0 DERIVE returns approximations for F 0 and F 6 405 14 kO D 00682 1 al 15 Fix ci COSH KO x 0 93 k k 16 FCO F G6 405 0 93 1 069914171 2 2 CHAMPIONSHIP POSITION According to the introduction the upper border of the right half of the net is a segment whose ex treme points are 0 0 93 and 5 035 1 07 Therefore the equation of the straight line that contains this segment is GI x 0 93 mx where according to paragraph 1 the inclination of the straight line m is 1 07 0 93 5 035 D N L 21 Roanes amp Roanes About the Te
80. y 7 2 11 2 Jy y 7 2 2 11 11 y 11 y 3 Jy cy Pd 2 4 4 2 A y 11 11 Y Ep x ENE E E l6 4 2 4 y 11 121 5 N E 16 8 2 4 y 11 121 88 2 I F l6 2 4 y 11 177 Er Oo 2 16 8 4 y 11 2 8 so II Fr S 08 l6 8 The last expressions shows the identity Now let me collect all steps into one program The tool consists of two programs and one assignment transform q v c2 c1 cO c4 c5 Prog v VARIABLES g 1 C2 O q v 2 2 o q V SUBST c1 v 0 SUBST q v 0 c1 2 c2 c4N2 steps l C2 v 2 c1 v c2 c0 c2 v42 c1 c2 v c4 2 cO c2 c1 2 c2 2 c2 v cA 2 cO c2 c5 c2 v c4 2 cO c2 c1 2 c2 2 fertig ready stn 0 DISPLAY Vereinfache step auf step Simplify step by step DISPLAY step dummy Prog stn stn 1 steps stn 1 step step How to run this tool D N L 21 DERIVE USER FORUM p 7 2 13 transform 2 y 11 y 7 Vereinfache step auf step Simplify step by step enter transform 2y 2 11y 7 and simplify this expression receive the instruction to simplify step by step follow this instruction and enter step followed by pressing ENTER button repeat and repeat until reach the final result 11 y 15 2 y f 2 ll y 11 y 11 y 16 2 y
81. zu 1 und 2 17 sins sin 10 283 2 cos 2r 2s E Der weitere Gang ist klar man wendet auf 2 die Additionstheoreme der Trigonometrie so weit an dass man schlie lich f r die Gr e sin s aus 1 substituieren kann Damit keine unerfreulichen Wur zelterme auftreten wird man die Gleichung so umformen dass neben den Konstanten nur noch x sin r vorkommt Diese Umformungen sind zu Fu u erst m hsam k nnen aber mit DERIVE in k rzester Zeit bewerkstelligt werden Sie f hren schlie lich zu einer quadratischen Gleichung 3 3 655 275 x 280 7245 x 1645 48 0 Die Gleichung besitzt zwei L sungen 4 100 2164 6 4 j gt 545 von denen die erste ausscheidet Sie wird durch das Quadrieren Vermeidung von Wurzeln produ ziert Die zweite L sung f hrt zu folgendem Ergebnis r arcsin pude arctan 45 2 x 0 3504054 545 19 5 S arcsin Ba aretan 0 2031689 p38 Derivisches Die oben angef hrten Ergebnisse k nnen mit den folgenden DERIVE Anweisungen ermittelt wer den Im Anhang gibt es eine k rzere Version mit DERIVE 6 C0S 72 1 C0S 72 SING6 1 2 SIN 72 10 2 45 2 SIN s SINCr J5 43 COS 2 r 2 5 5 Forcing to express the equation in sines 4 Trigonometry Expand 5 Trigpower Sines Richard Schorn C4 The Buckyball D N L 21 Derivical

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