On the examining of functions

Transcription

1 On the eamining of functions Djurdjica Takaµci Deartment of Mathematics and Informatics Facult of Science, Universit of Novi Sad Radivoje Stojković High School "Jovan Jovanović Zmaj" Novi Sad The grah of the function f() =, is given on the Figure. 6 6 Figure. The domain of the function is: R n f ; g: The students have no roblems with determining the domain of f, even the oints are irrational numbers. The function is odd.

2 f ( ) = ( ) = f () is true, f ( ) = f () is true central smmetr of the grah. The function has three zeroes = ; = ; =. Some students change the oints where the function is not de ned and zeros. Looking at the grah the avoid this roblem. The function has two vertical asmtotes = ; = ; because lim! f () = ; lim! + f () = ; lim! f () = ; lim! + f () = : The asmtotic behaviour of function f; in the neighborhood of ; and can be checked on the grah. The slanted asmtote = ; k = lim! f() = ; n = lim! (f () + ) = : The rst derivative obtained with the comuter is: f () = ( ) 8 + The zeroes of the rst derivatives obtained b comuter are: f () =, Solution is: ; i.e., + ; + ; ; = + = : 77 8; = + = : 77 8; = + = :68 ; = + = :68 :

3 Figure, f black, f red The sign of the rst derivative also di cult to be obtained usuall, b determining the sign of 8 + : B using the comuter we get the following eressions. f () <, Solution is: ; [ + ; f () >, Solution is: + ; [ ; + [ + [ + ; + + ; ; + [ The students often make mistakes in determining the solution of the revious inequalit. The almost forget that the oints do not belong to the domain. The second derivative obtained b comuter is: f () = ( ) + 9 :

4 The zeros of the second derivative = is ver eas to be determined. The sign of the second derivative can be b using comuter. f () >, Solution is: ; f () <, Solution is: [ ; ; [ ; : Figure. f blue, f black The questionnaire The method eosed in revious section was eosed to one fouth ear class in high school Isidora Sekulić in Novi Sad in Serbia. After a week these stedents got their written task in which the had to eamine classicall three functions. The functions were given together with their grahs and with the rst and the second derivatives. The same questionnaire was given to the the other class of the same high school who was taught classicall, without the comuter. The questionnaire was given in the following

5 . Eamine the function f() = 6 : Instructions: Show that the rst derivative of f can be written in the f () = + 6 ( 6) ; the second derivative of f can be written in the f () = ( 6) : the grah of the given function f is given on the Figure... Figure

6 . Eamine the function f() = ( ) e + : Instructions: Show that the rst derivative of f can be written in the f () = e + ; the second derivative of f can be written in the f () = e the grah of the given function f is given on the Figure.... Fifure. 6

7 . Eamine the function f() = Instructions: Show that the rst derivative of f can be written in the f () = ( 9) ( ) ; the second derivative of f can be written in the f () = 6 ( + 9) ( ) the grah of the given function f is given on the Figure 6. Figure 6 7

8 The results were the following comuter no comuter :eamle 8% 9% :eamle 9% % :eamle 7% 9% (%) students of 6, did all three eercises. References [] J.Schmeelk, Dj.Takaµci, A.Takaµci, Elementar Analsis through Eamles and Eercises, Kluwer Academic Publishers, 99. [] D. Tall, A. Vinner, Concet Image and Concet De nitionin Mathematics with articular reference to Limits and Continuit, Education Studies in Mathematics,, 9-69, 98. [] D. Tall, The Transition to Advanced Mathematical Thinking: Functions, Limits, In nit, and Proof, in Grouws D. A., Handbook of Reseach on Mathematcs Teaching and Learning, Macmillan, New York, 9-, 99. [] D. Tall, Resent Develoements in the Use of Comuter to Visualize and Smbolize Calculus Concets, The Laborator Aroach to Teaching Calculus, M.A.A. Notes Vol.,-, 99. 8