ANIMATION OF LAGRANGE MULTIPLIER METHOD AND MAPLE Pavel Prazak University of Hradec Kralove, Czech Republic

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1 ANIMATION OF LAGRANGE MULTIPLIER METHOD AND MAPLE Pavel Prazak University of Hradec Kralove, Czech Republic Experience shows that IT is a powerful tool for visualization of hidden mathematical concepts. This article presents how to use animation produced with the help of CAS for demonstration of constrained maximum and minimum of function of two variables with subject to one implicit given constraint. The rule of Lagrange multipliers is demonstrated in this case, too. In this particular - the easiest - case there exist two vectors that have the same direction and Maple allow us to show it. INTRODUCTION Motion belongs to the basic philosophical conceptions. This phenomenon draws human attention since time immemorial. There is no doubt that motion can reveal hidden phenomena. An animal hidden in a forest can be found if it moves. A comet hidden in the night sky can be found because it changes the known patterns of constellations. Mathematical concepts are also hidden. To understand mathematics often means to see mathematical ideas. However, we understand mathematics rather by insight instead of eyesight, cf. (Stewart, 1997). This makes mathematics difficult to learn and to teach as well. Fortunately, with the help of IT there are ways how to transfer the inner images of mathematics onto screens of our computers. Keeping in mind that motion can very effectively attract human attention it would be desirable to present mathematical objects like animations. In some fortunate cases it is possible to make it and in this article we describe how to animate a basic idea hidden under the term Lagrange multipliers method. Our students of informatics and management get acquainted with the method of Lagrange multipliers during the second term of their study of mathematics. It is well known that this method represents a necessary condition for the existence of the constrained maxims and minima of functions of several variables. The plan of our lectures doesn t allow us to discuss these problems in more details. Due to this lack of time our students have some problems with understanding the problem of constrained extremes. The idea of Lagrange multipliers causes them some difficulties too. The basic optimisation problems and methods of their solutions belong to basic courses of mathematics. The aim of this paper is to describe a Maple s worksheet we have prepared for our students to help them understand the concept of constrained extremes of functions of several variables and the method of Lagrange multipliers. Particularly, since Maple can visualise 3-D objects we deal with the problem maximize (minimize) f ( x, subject to g ( x, = 0. In this paper we omit the general problem for functions of more than two variables and for the constraint given by more than one equation, see also (Pražák, 003). THE INTUITION CONCERNING THE CONCEPT OF CONSTRAINED EXTREMES The worksheet begins by a simple example taken from management real-life. This problem should show the necessity of the new concept and give us the motivation for the new mathematical definition. Particularly we use this problem (Marsden, 000): 1

2 A company s production function is Q(x, =xy. The cost of production is C(x, =x+y. If this company can spend C(x, =10, what is the maximum quantity that can be produced? After the discussion on how to solve this problem we use the method of substitution and change it to the problem of function with one real variable. Then we introduce the visualization of this problem. THE FORMULATION OF THE CONCEPT We continue by the exact formulation of the problem. Definition: Let f : R R and g : R R be real functions defined on the open set U. Let us denote the zero level set of function g by M = { x U g( x). If there is a point a M U such that f ( x) f ( a) for all x M U we say that f has a local maximum at the point a subject to the constraint g ( x) = 0. The equation g( x) = 0 is called the constraint equation. Further we introduce two other examples and carefully illustrate the above concept. We do it in a few steps. First we introduce the graph of the function f ( x,. Then we introduce the graphical representation of the constraint g ( x, = 0. We continue by construction of the graph of the function f on the set D( f ) M, where M = {( x, D( f ) g( x,, finally we show this graph. In particularly, we demonstrate here only this problem: maximize (minimize) f ( x, = x xy + y subject to g ( x, = x + y 1 = 0. The illustration of the given problem can be watched in the figures 1-4. Fig.1: The graph of the function f. Fig.: The constraint g ( x, = 0.

3 Fig.3: The construction of the graph of f on the set D( f ) M. Fig. 4: The graph of f on the set D( f ) M. THE INTUITION CONCERNING THE METHOD OF LAGRANGE MULTIPLIERS. It is necessary to emphasize that a constraint is often given from which it isn t possible to gain one variable as a function of the second one or which we are not able to parameterise. Therefore the method of substitution can t be used and we need another method. It is the reason why in the above example we also introduce an animation with two vectors. The first one is the tangent vector to the curve of the constraint g ( x, = 0. The second one is the tangent vector to the graph of the function f on the set D( f ) M. We try to show to our students that at the point where is the constrained maximum or minimum, the two above mentioned tangent vectors are parallel. Two sequences of this animation can be seen on the figures 5,6. Fig.5: Moving tangent vectors. Fig.6: Parallel tangent vectors at the point of constrained minimum. 3

4 Our derivation continues with the idea that it is easier to find vectors that are perpendicular to the tangent vectors. Such vectors that both lie in the xy plane are namely gradients of functions f respectively g. Now we can formulate the following theorem. Theorem: (the necessary condition for constrained relative extremes - the method of Lagrange multiplier) Let f : R R and g : R R are continuously differentiable real functions on the open set U. Let a U, g ( a) = 0 and M = { x U g( x) is the level set for g. Assume that g ( a) 0. If f has a local maximum or minimum on the set M U at the point a - it means a constrained maximum or minimum, then there is a real number λ such that f ( a) = λ g( a). SKETCH OF THE PROOF In the thorough proof the implicit function theorem is usually applied but this theorem can be accepted only on the second thoughts and our students are not very confident with it. Instead of the full proof we provide only some essential steps of the proof and show an animation of these ideas. The sketch of the proof is based on the presumption that the constraint g ( x, = 0 can be parameterised by a curve c (t), where t is a real parameter. Then could be used the first derivative test for the function of one variable. The idea that the gradients of functions f and g at the point of constrained extreme are parallel is given there, too. This idea is animated and two out of the sequences can be watched in the following figures 7,8. Fig.7: The start of the animation that illustrate the essential steps of the proof. Fig.8: The animation continues: f ( a) g( a). CONCLUSION Students live very quickly at the beginning of the 1st century. They have a lot of activities. Some of them complain that they are not able to understand hidden ideas of mathematics, because they cannot see them. They are not ready to spend time with the traditional learning aids such as books, pencil and paper, but they spend a lot of time with computers or mobile phones. Thus teachers of mathematics have to change their classic teaching aids like chalk and blackboard as well. Experience shows that IT is a powerful tool for visualization of hidden mathematical concepts. 4

5 Management and informatics students at our university study mathematics only for the first two semesters of their studies. During the whole curriculum they learn a lot about management and marketing. Models that they meet are rather descriptive and usually they do not use mathematics. The concept of constrained extremes and the idea of the method of multipliers are quite old. According to (Alexejev, 1991) the first mention of the method of multipliers was included in Leonhard Euler s works from In 1788 Joseph-Louis Lagrange implemented this method for a broad class of the variational problems in his Mécanique analytique. Later in 1797 in the book Théorie des fonctions analytique he used it for finite dimensional problems, too. Our approach to the considered problem is not analytical and instead of the logical proof without gaps we demonstrate a geometrical interpretations of the method of multipliers with the help of modern IT. It means that we replace the old and commonly used way of evidence of mathematical knowledge by a geometrical operation. We choose this way since we only want to demonstrate the method of multipliers to students that need this rule in their applications. The described Maple s worksheet has been used third time this summer term. The discussions with our students imply that with the help of the above-mentioned visualizations they gained a better insight into the problem and its solution. That is very important for applications in optimisation problems they meet in macroeconomics and microeconomics. So we plan to spread a similar approach to the other concepts we teach in the basic courses of mathematics. REFERENCES Alexejev, V.M., Tichomirov, V.M., Fomin, S.V. (1991): Matematická teorie optimálních procesů, (in Czech), Prague, Academia. Marsden, J.E., Tromba, A.J. (000): Vector Calculus, New York, W.H. Freeman and Comp. Monagan, M.B. et. al. (00): Maple 8, Introductory Programming Guide, Waterloo, Maple Inc. Ostaszewski, A. (1993): Mathematics in Economics, Oxford, Blackwell Publishers. Pražák, P. (003): Constrained Extremes, Lagrange Multipliers and Maple, ICTMT6, Volos Stewart, I. (1997): Does God Play Dice?, Penguin, London 5