MODELLING OF GRAPHS OF FUNCTIONS IN INTEGRAL CALCULUS TAUGHT IN FIRST TERM AT FAI OF THE TBU IN ZLÍN FIALKA Miloslav URBANČOK Lukáš CHARVÁTOVÁ Hana, ČR Abstract The article focuses on the use of means for more effective teaching of single variable integral calculus, which the latest software Wolfram Mathematica offers. This concerns in particular the graphic resources in education. Integral calculus is lectured in the first term at the Faculty of Applied Informatics and the other three faculties of the Tomas Bata University in Zlín. It is also already very often taught at many secondary schools. Three faculties of our university obtained the Mathematica software licence, which is now being used by students of these faculties. Here we are presenting figures of animations that are available to students in the Moodle environment in the course Matematika1(Fialka) at the Faculty of Applied Informatics server. Animations show the areas of surfaces computed in Mathematica software, which are defined by elementary functions of one variable. These are examples from the basic study literature. Some of the examples calculated by Mathematica the students can verify by classical calculation. In case of interest these animations can be also used by students or teachers at secondary schools or universities. Key words: Wolfram Mathematica, computer graphics, animations, integral calculus, function of one real variable, Czech technical norm. MODELOVÁNÍ GRAFŮ FUNKCÍ V INTEGRÁLNÍM POČTU VYUČOVANÉM V PRVNÍM SEMESTRU NA FAI UTB VE ZLÍNĚ Resumé V článku se zaměřujeme na využití některých možností k zefektivnění výuky integrálního počtu funkcí jedné reálné proměnné, jež nabízí aktuální software Wolfram Mathematica. Týká se to zejména grafických prostředků ve výuce. Integrální počet je přednášen na Fakultě aplikované informatiky Univerzity Tomáše Bati ve Zlíně a na dalších třech fakultách v prvním semestru. Je také velmi často vyučován už i na mnoha středních školách. Tři fakulty naší univerzity získaly na software Mathematica licenci, kterou využívají také studenti těchto fakult. Uvádíme obrázky animací, které jsou studentům k dispozici v prostředí Moodle v kurzu Matematika1(Fialka) na serveru Fakulty aplikované informatiky. Animace zobrazují počítané obsahy ploch pomocí softwaru Mathematica, které jsou definovány funkcemi jedné proměnné. Jde o příklady ze základní studijní literatury. Studenti si softwarem vyčíslené výsledky mohou v některých příkladech ověřit klasickým výpočtem. V případě zájmu mohou uvedené animace využít také studenti nebo učitelé středních škol nebo univerzit. Klíčová slova: Wolfram Mathematica, počítačová grafika, animace, integrální počet, funkce jedné reálné proměnné, Česká technická norma. Introduction Department of Mathematics of Faculty of Applied Informatics of the Tomas Bata University in Zlín focuses on the use of the latest trends in education of mathematical 423
subjects. Integral calculus, from which the presented graphic samples of 2D animations come, is taught by our department for a total of 2000 students at four faculties of our university. Mathematics I subject content of the Faculty of Applied Informatics at the Tomas Bata University in Zlín, which we deal with in the article, is also supported by the textbook of the author M. Fialka. (M. Fialka, 2009) This textbook contains mathematical signs and symbols that meet the valid Czech technical norm. (ČSN ISO 31-11, 1999) Mathematics I includes the integral calculus of functions of one real variable with applications. This calculus contains the following sections and topics: Antiderivative of a function, indefinite integral, decompositional method. Integration by parts, integration by substitution Integration of rational functions, integration of goniometrical functions Definite integral, applications of definite integrals Improper integral 1 Main results In the presented HTML sample each file represents the specific phase of chosen animations that are available to students of four faculties of our university. Figures and animations of Mathematica can be exported to the simple JPEG or GIF format. Figure 1: Animations sample of numerical quadrature of chosen elementary functions 424
Mathematica allows us to convert mathematical expressions into TeX format, which is freely available mathematical typesetting editor. TeX can be also used in graphical presentations, which are written in HTML. The Figure 1 shows formulas for definite integrals of elementary functions at exactly defined limits on its left side. On the right side the area of plane surface is drawn, which continuously widens from lower to upper limits, i. e. from the left to the right. Below the calculation of definite integral is compared, which can be calculated by numerical or classical symbolic method. Geometrically it is the area P of surface bounded by the closed interval of integration and half-periodic sine curve. The end points of interval are actually lower and upper limits of the interval [0, π]. 1.4 1.2 1.0 y 0.8 0.6 P 0.4 0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x Figure 2: Area of surface under half-periodic sine curve Classical calculation follows: Figure 3: Dynamic function Manipulate for an interactive demonstration of definite integral of square root function use 425
Figure 4: Dynamic function Manipulate for an interactive demonstration of the definite integral of exponential function use Since the release of Wolfram Mathematica in version 6 the command Dynamic has been used. The function Manipulate represents one of its parts. This contribution includes a notebook with examples of figures on Figure 3 that are the outputs of elementary functions from the above mentioned course Moodle Matematika1(Fialka). Wolfram notebook module contains a dynamic calculation of the area of surface. In that sample the Dynamic module is used. In this example there are several functions, which can use the slider to set up the upper and lower limit. The area is then recalculated and displayed above the graph. Another strong feature of the function Manipulate is the ability to start up the animation course over the whole defined interval of the chosen function. One can regulate the speed of the transition with arrows, as shown on Figure 4. Conclusion The detailed quantitative research, based on standard of ECTS grading, conducted in recent time period, has confirmed that students at the Tomas Bata University in Zlín have been evaluating usage of the animations in teaching very positively. Article demonstrates how useful it is to engage capable students in the publication activities of the departments and utilize the results of this activity for teaching or research process. We believe that mutual cooperation is inspiring for both teachers and also for talented students. The presented animations of integral calculus include the subject matter that is common to many secondary schools and universities. The reader of our article, who would like to create or obtain analogous application source files in Wolfram Mathematica can obtain necessary information or files directly in the above mentioned course in Moodle environment. We will appreciate if the general public is interested in the results of our work. 426
Bibliography 1. ČSN ISO 31-11. Veličiny a jednotky - část 11: Matematické znaky a značky používané ve fyzikálních vědách a v technice. Praha: Český normalizační institut, 1999, 27 s. 2. FIALKA, M., CHARVÁTOVÁ, H. Matematika I. Dotisk 2. vyd. Zlín: Univerzita Tomáše Bati ve Zlíně, 2009, 107 s. ISBN 978-807318-584-8 3. FIALKA, Miloslav. Diferenciální počet funkcí více proměnných s aplikacemi. 3. vyd. Zlín: Univerzita Tomáše Bati ve Zlíně, 2008a. 145 s. ISBN 978-80-7318-665-4. 4. FIALKA, Miloslav. Integrální počet funkcí více proměnných s aplikacemi. 3. vyd. Zlín: Univerzita Tomáše Bati ve Zlíně, 2008b. 103 s. ISBN 978-80-7318-668-5. 5. KLUČKA, Dalibor. Inovace výuky předmětu Matematika I na FAI UTB ve Zlíně obsahující ukázky řešení úloh v prostředí Mathematica. Příručka k bakalářské práci. Zlín: Univerzita Tomáše Bati ve Zlíně, 2011. Assessed by: Ing. Bc. Bronislav Chramcov, Ph.D. Contact address: Miloslav Fialka, RNDr. CSc., Ústav matematiky, Fakulta aplikované informatiky UTB ve Zlíně, Nad Stráněmi 4511, 760 05 Zlín, ČR, tel. 00420 576 035 002, e-mail: fialka@fai.utb.cz Hana Charvátová, Ing. Ph.D., Ústav automatizace a řídicí techniky, Fakulta aplikované informatiky UTB ve Zlíně, Nad Stráněmi 4511, 760 05 Zlín, ČR, tel. 00420 576 035 274, e-mail: charvatova@fai.utb.cz Lukáš Urbančok, student, Obor Bezpečnostní technologie, Fakulta aplikované informatiky UTB ve Zlíně, Nad Stráněmi 4511, 760 05 Zlín, ČR, e-mail: l_urbancok@fai.utb.cz 427