Derivative of a function at a point. Ján Gunčaga, Štefan Tkačik

Transcription

1 XII th Czech-Polish-Slovak Mathematical School Derivative of a fnction at a point Ján Gnčaga, Štefan Tkačik Abstract: We present a concept of differentiable fnctions and derivatives. The notion of a differentiable fnction f at a point x is based on the existence of a fnction φ sch that f(x+ ) f(x) φ() for all from some neighborhood of 0 and φ is continos at 0. Key words and phrases. Derivation of linear and polynomial fnctions, definition and basic properties of a differentiable fnction, existence of the derivative of a fnction at a point Introdction The notion of a derivative of a fnction at a point is often introdced via the limit of a fnction at a point. This is sally too difficlt for nderstanding by stdents at secondary grammar schools (see (), (3), (7)), since the definition of a limit is difficlt for them, too. We se the continity of a fnction at a point as a prime notion (see (), (4), (6)). The difference of the vale of a fnction f at a point x and at another point in a neighborhood of the point x is given by the formla f(x+) f(x), where is distance from the point to x. To qantify the velocity of the change, we se the fnction f ( x + ) ϕ ( ). This fnction is defined for all 0 for which is x + from a set, on which f is defined. Is it possible to define the fnction ϕ() at the point 0, so that it will be continos? Fnction φ() and its properties Linear fnction f(x) If f(x) is a linear fnction, then f(x) ax+b, where a,b are real nmbers. This fnction is defined for all real nmbers. For 0 we have: f ( x + ) ( a( x + ) + b) ( ax + b) a ϕ ( ) a. This means, that the fnction ϕ() does not depend on the variable, that is, ϕ() is a constant fnction. The graph of a linear fnction f(x) is a line. The vale of the fnction ϕ() is the slope of this line. If the fnction f(x) is interpreted as the law of motion of a particle on a straight-line, then x and x+ represent instants of time and the vales f(x) and f(x+) the corresponding positions of the particle. The difference f(x+) f(x) is the displacement of the particle dring the time--interval between the instants x and x+. The particle moves at a constant velocity given by the fnction ϕ(). The velocity is the rate of displacement. Let f(x) be the costs of prodcing x nits of the given commodity, f is the costs fnction of this commodity and ϕ() is the marginal costs. Let f(x) be the amont of heat needed to raise the temperatre of a nit mass of the sbstance from 0 to x (measred in degrees). Then ϕ() is the amont of heat needed to raise

2 the temperatre of a nit mass of the sbstance by one degree; ϕ() is the specific heat of the sbstance. Temperatre extensibility can be approximated by linear fnction ll 0 (+αδt). The vale of the fnction φ()l 0 α describes the change of longitde of a solid according to the nit change of temperatre. Non-linear fnction f(x) If the fnction f(x) is non-linear, then ϕ() is not constant. For instance, consider the fnction yx. Fig. If 0 < <, then f ( x + ) < f ( x + ) and hence ϕ ) < ϕ ). ( ( Theorem (Principle of continos extension) Let f and g be fnctions continos (left-continos, right- continos) at a point a. If in every neighborhood (left- neighborhood, right- neighborhood, respectively) of a there is a point x a sch that f(x) g(x), then f(a) g(a). Theorem implies that the vale at a of a fnction continos at the point a is niqely determined by the vales at points x a in a neighborhood of a. If the vales of a fnction at points x a are given and if it is possible to choose the vale at a so that the fnction becomes continos at a, then it can be done in only one way. The next example illstrates the theorem.

3 Example: What is f(3), if yo know that the fnction f is continos at 3 and f (x) Soltion: x x 9 x 3 x x x + 3 Becase x 9 x 3 x 9 for every x -3 and x 3? ( ) x 3 x 9, for every x -3 and x 3, x + 3 the fnction defined by x, x 3, is continos at 3, hence the vale f(3) is given x + 3 by f () The fnction x x 3, 3 x 9 x 3 is continos at the point 3. x 3 6 Definition: A fnction f is said to be differentiable at a point x if there exists a fnction ϕ, continos at 0, sch that f(x+) f(x) ϕ(). for every in some neighborhood of 0. Hence, the problem of differentiability of a fnction f at a point x redces to the problem of a sitable choice of ϕ (0). The principle of continos extension says that if ϕ (0) can be chosen so that ϕ becomes continos at 0, then sch a choice niqe. The nmber ϕ (0) is then called the derivative of the fnction f at the point x and is denoted by Df(x). Example: Let f(x) x 3 for every x (-, ). Let s show that the fnction f is differentiable at the point and the derivative of fnction f at is eqal to. Soltion: Note that f(+) f() (+) for every. Hence.ϕ() If ϕ() for every then the fnction is continos at 0 and f(+) f() ϕ() for every. By definition this means that the fnction f is differentiable at the point. The derivative of fnction f at is eqal to becase in this case ϕ() + 6 +, for every (, ) and hence ϕ (0). Existence and non-existence of derivation of a fnction at a point Or approach to the introdction of derivation of a fnction at a point was tested for mistakes among stdents of the forth class at the secondary grammar school. The fnction f ( x + ) ϕ ( ) was replaced by the fnction of the slope of chord given by f ( a) formla s f, a ( x). We illstrate or procedre in next example. x a Teacher: Calclate the derivation of the fnction y x at the point from the definition! Robert: y x, a.

4 x ( ) x, x ( x )( x + ) s f, x x x + k x. x x x + x, s f,( x) k x. Teacher: Do yo know to describe the graph of the fnction y x +? Robert: The line. Teacher: More precisely. Robert: The straight line. Fig. Teacher: What is it possible to add so that the previos fnction becomes continos? Miroslava: We have to fill the circle. Teacher: How? Ivan: By nmber. Teacher: What does it mean for the vale of derivation of the fnction y x at the point? Robert: It is eqal to. Teacher: We considered fnctions with derivation at every point of the domain. Now, we are going to deal with fnctions having no derivation at least at one point. x f '()? Pal: s f, ( x x) x k x, x. x x x (; ) : x x x ( x ) x ( ;) : x x Fig. 3 Teacher: Is it possible to extend the fnction (to define its vale at ) so that it becomes continos? Lke, Lcy: No, it isn t.

5 Teacher: What does it mean for the derivation at the point? Pal: It doesn t exist. Conclsion At the end we borrow few lines from (5): If the reader does not vale mathematics and mathematical analysis more than a comfortable feeling that the way calcls is taght at his and other famos niversities is essentially all right, then for him the present paper does not have mch to say. We feel that the qality and the amont of intellectal activities needed to transform the mathematics nderstood (derivation of a fnction at a point) into the mathematics sitable for teaching shold never be ndervaled. The effort needed to nderstand mathematical knowledge matches the effort to invent it. If one wants to write a good calcls book, he has to carry ot a mathematical research in the sal sense of the word. In or paper we wanted to follow the idea cited above. We believe that mch more effort is needed to help stdents to nderstand and to master the process of derivation. We have a positive experience with teaching derivation based on continity. This way the process of derivation is visalised. Stdents practise graphs of fnctions and it is easier for them to decide whether the derivative of a fnction exists. Literatre () Flier J.: Fnkcie a fnkčné myslenie vo vyčovaní matematickej analýzy. Nitra, UKF, 00. () Gnčaga J.: Limitné procesy v školskej matematike. Dizertačná práca. Nitra, UKF, 004. (3) Hischer H., Scheid H.: Grndbegriffe der Analysis: Genese nd Beispiele as didaktischer Sicht. Heidelberg; Berlin; Oxford, Spektrm Akademischer Verlag, 995. (4) Klvánek I.: Prípravka na diferenciálny a integrálny počet. Žilina, VŠDS, 99. (5) Klvánek I.: Čo nie je dobré vo vyčovaní matematickej analýzy?. In: Matematické obzory, 99, č. 36, s. 3-49/č. 37, s (6) Tkačik Š.: Spojitosť a limity troch inak. In: Zborník konferencie Setkání kateder matematiky České a Slovenské repbliky připravjící bdocí čitele, Ústí nad Labem, 004, s (7) Eisenmann P.: Propedetika infinitezimálního počt. Ústí nad Labem, UJEP, 00. Athors PaedDr. Ján Gnčaga, PhD., RNDr. Štefan Tkačik, PhD. Katedra matematiky a fyziky PF KU Námestie A. Hlink Ržomberok Slovakia gncaga@fed.k.sk, tkacik@fed.k.sk