y - TIME 2010
Contents
1. LoL rE tant iva Derivatives k Sf Tat d Une EE kei l12ztLance anaent Hec w mm a Mor eee UU D Dir wO Kraft keng A d De d Su r Let s see this using Voyage 200 Theme C Theory and CAS Since OS 1 7 Nspire CAS has a Generalized Series built in function ft 1 Actions Lk 2 Number y o Algebra d 4 Calculus Je 5 Probability 4 Limit x Statistics Sum Matrix BAD Product T Taylor Polynomial hum 4 Generalized Series damum 3 Dominant Term 4 Remainder of Polynomial 2 Quotient of Polynomial 5 A 1 Derivative 4 Derivative ata Paint 3 Integral Greatest Common Dirvisor Ceefficients of Polynomial 3 Degree of Poly H quation Solver nomial e Question from complex analysis Compute the following line integral il f z dz C the boundary of the square C with vertices 43 43i cos z sinh z ea and f z We usually compute this kind of integral using the residue integration method Iff has an isolated singularity at z and Laurent expansion f for 0 z z R n then the coefficient c 1s called the residue of f at Zp By the residue theorem we have I 277i Bj B where B residue at the triple pole z 1 and B residue at the triple pole z 1 cos z sinh z 2 2 I e Te Nspire CAS will show tha2. Technology and its Integration into Mathematics Education July 6 10 2010 Tim E E T S l Telecomunicaciones Malaga Spain Using the Real Power of Computer Algebra Michel Beaudin Who I Am What I Do Mathematics professor at ETS engineering school since 1991 single and multiple variable calculus differential equations linear algebra complex analysis I started to use Derive 1n 1991 and I continue to use it today Who I Am What I Do With the venue of the TI 92 the real possibilities of using computer algebra IN THE CLASSROOM were finally unified Who I Am What I Do In an engineering school it was natural to use an affordable CAS since 1999 each undergraduate student at ETS has to buy a TI 92 Plus now Voyage 200 Who I Am What I Do In my engineering school we have a ereat privilege being able to use Computer Algebra in the classroom when we want and during exams all semester long We don t have to block internet access Voyage 200 remains a calculator Who I Am What I Do Organizing TIME 2004 and ACA 2009 conferences would have been impossible without them Gilles Picard l E Pineau Who I Am What I Do My same and only wife since 3 2 years J il f A d A o E akata de e ep i T v ER ke INN o v M de E um i r t kW i A Rob Ze af iy f i KH Who I Am What I Do B
3. e And no built in Dirac Delta function How can you deal with this Let us show that these absent functions can be turned into a better understanding of some mathematical concepts Let s find all real solutions and one complex solution for the equation n Note some systems Maple for example can solve this equation because they have defined a special function Lambert W which is the inverse function of f x 2 x exp x Now let s solve the differential equation y 4y 250 t z y 0 10 y 0 5 Note the Dirac delta function or T can be seen as a limit of indicator function over a very small interval of time with area 1 Le CHI z t T amp in Derive Lis rt T Tg dsolve2_iv p q r t to yo vo 1s the command that solves the second order linear differential equation in Derive y p y q y zs PU y tg 9 yo y to 9 vo y 4y 506 t z y 0 210 y 0 5 Derive 1s using the method of variation of parameters which involves the computation of integrals And Derive has no problem to integrate piecewise functions Example 5 Using Voyage 200 how can I find the coordinates of the point of intersection of two parametric 2D defined curves The F5 Math menu shows the item Intersection 1n Function graphic mode only r x zoon rrace Resraen neu TE TYFE OF USE cat EEHTERJ DE AMD EE C12 CHHCEL a E SCH T Ai tat
4. ed here Esos ieeehes 0 Cl SE m i EM ul t u2 xcI3 uci 2 03675 CHECK RHEZ AND IMITIRL CHNDITIDNZ HAIN RAD AUTO D I So the value y 3 2 04 1s correct Why not use the power of the CAS Especially for Derive its ability to integrate piecewise continuous functions Find the steady state solution of the following problem y 2y y z f t 0 0 y 0 20 sin t O lt t lt az 0 VU IN SB ro f 2z2 f t One can show using the undetermined coefficients method and Fourier series that the particular steady state solution looks like this l l z COS t T d d P SSINCZ t COSCZ t bam But according to Laplace transform theory the whole solution is given by a convolution integral If y 42y y f t yO 0 y 0 0 then y t A t f t hc f t r dc where h t te 0 This is because the inverse Laplace transform of I s 2s 1 is te No one will try to compute the convolution integral by hand and how can we define easily a non trivial periodic input We can use Derive in order to compute the convolution of f and h Let us recall that le SO D sin f O t z ro fJ t 22 2 f t h t te Theme B My CAS can t do this What can I do e Example 4 There is no LambertW function implemented into Deriv
5. n the same course we should try to link different subjects and show the Students how to use their handheld For example let s solve using power series the following problem 2x 11 y y 3xy 0 y 0 210 y 0 4 Then we will use it to estimate y 3 series solution y x kk C C x CX Lee n 0 with c 210 and c 4 By hand the students will find the following recurrence formula 2n d fis 5 NND C SS I In n 1 Then c 2 11 c 2 167 363 c 651 5324 If we use only the first 5 terms we will conclude that y 3 1 96 But the correct answer is y 3 2 04 Students can obtain this answer with the help of their device Here 1s how Voyage 200 has a SEQ graphic mode Let s use it aFLOTs wl uil i z 11 4 ess ulz 11 n n 1 105 MAIN RAD AUTO DUNNO Fey Yq FR vi lAlgebrs P e S uice 3 scp n O mst 4 s10 Ezz SR SC ZE e zi 30 s430 gt HAIN 2 n2 n Sluttn 1 3 ulin RAC AUTO SEG Bs Far Done 1 96225 Ze 2B C z D35r15 2 03721 2 03721 Voyage 200 also has a DIFF EQUATIONS eraphic mode We have to convert by hand the second order ODE into a first order system o z nn X0 10 20 4 2x 11 We type the system into the Y Editor plot the eraph of yl the solution and observe the value of y 3 RK method is being us
6. t B 5 ON RES ptm EE IC 16 But we can also use the definition of a line integral Of course it would be long to integrate over each side of the square Using the principle of deformation of paths we can impose a continuous deformation of the square into the circle of radius 2 located at the origin cos z sinh z ea f z This means that we can compute the value of the integral by using the definition b ff Codz f r dt where r ab a is a smooth curve 1n the plane Let s use r t 2exp it for the circle of radius 2 centered at 0 Now take a look at the power of Nspire CAS The importance to see it In Derive ROTATE X simplifies to a matrix A such that A v rotates the 3D coordinate vector v through an angle of 0 radians about the x axis counterclockwise when viewed from the positive x axis toward the origin Can we see an example Conclusion What are the main benefits of using technology in the classroom Some personal answers Computer algebra allows me to continue to teach almost the same courses with a taste of new make new from old Josef Bohm This situation has many advantages namely Retirement can wait can revisit some mathematical results When I have to prepare new material CAS explorations are helping me But using technology has also some disadvantages Weaker studen
7. ts can succeed thanks to the CAS Is this really a problem In some cases less time CAN be spent on proving results because more time MUST be spent on learning how to use the CAS This 1s why we need a good mathematical assistant As far as I am concerned Derive Voyage 200 and Nspire CAS are in this category And don t forget this adapted from the Derive user manual Making mathematics more exciting and enjoyable should be the driving force behind the development of a CAS program It gives you the freedom to explore different approaches to problems Something you probably would never consider if you had to do the calculations by hand But I have to tell you something why do I like so much to attend TIME conferences Or why Computer Algebra is so powerful according to me It gives me the chance to meet nice people or at least think of them 47 I have the opportunity to travel very far and have beer with friends I can see almost yearly very SERIOUS colleagues 53
8. ut my main occupation for the past 2 years 1 Ideas and Themes of the Presentation e Theme A We can teach mathematics with a CAS Instead of removing items from the curriculum we should revisit it with the CAS example 1 V200 Nspire CAS With a CAS you can sometimes solve the same problem using 2 different approaches examples 2 and 3 V200 Derive Ideas and Themes of the Presentation Theme B CAS are powerful but the teacher remains important If the CAS you are using does not have some built in function sometimes you can overcome this by using an appropriate approach examples 4 and 5 Derive V200 Ideas and Themes of the Presentation Theme C When computer algebra and theory become partners Some years ago it was not easy to keep students interest for concepts in analysis or linear algebra To much theory not enough practice no graphs examples 6 and 7 Nspire CAS Derive Comment about the examples to come These selected examples are coming from my daily teaching exchanges with colleagues at ETS and David Jeffrey from UWO This means that the level of mathematics involved is the one at university level for engineering studies But one can easily apply it to any level of teaching Comment about the examples to come Some examples will be now have been performed live if someone would like to obtain a file sho
9. wing the details please email michel beaudin etsmtl ca Theme A We can teach with CAS ES expressions oT sin x 3x 4 5x 25x 45x 18 15 Sh See R5 xD 25 x 45 x 18 X 2ox o nox 25 x45 x 48 M A afd Zox o Set t 25 x d5 u 18 x 29 1nlx 9 3 1nt3 x IDN 2 Intx th Tttt 4 25 424 45 x 18 2132 i 574 7 31 Xx drb5bxx 42425xx 2 45xx 18 2fT8 JE XJ e bk Saadler ue x a 1 zx 04 cnx 25 x7 45 x 18 3 dc 9 geti d gt t patenam i 1866 x 9 USM Gx 1 30 x 2 S che 5 et 4 25 e2445 e 18 a Pactorl 3 X 45 xo 25 x 45 y E x lim Loo Ax AEX 0000 Sa EPR 2 liia EE propFracCans i 0 00 0 HESE AE E zeg KE AS CC KEE A RAC AUTO FAR Ed zu RAD AUTO 16 E ES ei eeler ern ro cieSr ue a Gggx cilg 5 y sinco 8 tsini 18 sinoo T 48 Sint x d uisi PU 4 cg 2 5 STE O5 Ge 8 14 ec RE Ae EIS ree jaraebra cs kuch ue 5 sinis E Graine x a E ul eis dE els d FM z pen ZS _ S ine _ 5 An Cinco ease SEENEN anz amp i bkzin RAC AUTO FAR Hd zu MHAIN RAD AUTO FAR Hd cp 1 4 Buc B sinlx 8 sine 4 10 sinbe e15 10 nel 15 ERR l n an I PRAD AUTO RECT 1 2 ax 5 sinlx 5 sin x 7 5 OUS 778 ae glo b E I
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