maths course exercises Liceo Scientifico Isaac Newton - Roma continuity

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1 maths course exercises Liceo Scientifico Isaac Newton - Roma in accordo con il Ministero dell Istruzione, Università, Ricerca e sulla base delle Politiche Linguistiche della Commissione Europea percorso formativo a carattere tematico-linguistico-didattico-metodologico scuola secondaria di secondo grado teacher Serenella Iacino

2 Indice Modulo Strategies Before Prerequisites Linking to Previous Knowledge and Predicting con questionari basati su stimoli relativi alle conoscenze pregresse e alle ipotesi riguardanti i contenuti da affrontare Italian/English Glossary Strategies During Video con scheda grafica Keywords riferite al video attraverso esercitazioni mirate Conceptual Map Strategies - After Esercizi: Multiple Choice Matching True or False Completion Flow Chart Think and Discuss Summary per abstract e/o esercizi orali o scritti basati su un questionario e per esercizi quali traduzione e/o dettato Web References di approfondimento come input interattivi per test orali e scritti e per esercitazioni basate sul Problem Solving Answer Sheets 2

3 1 Strategies Before Prerequisites Maths the prerequisites are Domain of a function Limit of a function Right - limit Left - limit Indeterminate forms Fundamental limits Inverse function CONTINUITY 3

4 2 Strategies Before Linking to Previous Knowledge and Predicting 1. How do you define a real function? 2. Are you able to calculate the domain of a function? 3. Do you know the concept of neighborhood of a point c? 4. What is the definition of limit of a function as x goes to c? 5. When is a function strictly growing? 6. When is a function invertible? 7. What is the definition of Maximum and minimum of a function? 8. Are you able to removable the indeterminate forms? 9. Do you know the fundamental limits? 4

5 3 Strategies Before Italian / English Glossary Asse Continua a destra Continua a sinistra Continuità Continua Discontinuità Dominio Eliminabile Equazione Estremo Finito Funzione Funzione esponenziale Funzione inversa Funzione logaritmica Funzione polinomiale Funzione trigonometrica Grafico Illimitata Infinito Insieme dei numeri reali Intervallo Intervallo aperto Intervallo chiuso Intorno Invertibile Limitato Massimo Minimo Potenza Prodotto Quoziente Raggio dell intorno Retta Salto Somma Axis Right - continuous Left - continuous Continuity Continuous Dis Domain Removable Equation Extreme Finite Function Exponential function Inverse function Logarithmic function Polynomial function Trigonometric function Graph Unlimited Infinite/infinity Set of real number Interval Open interval Closed interval Neighborhood Invertible Limited Maximum Minimum Power Product Quotient Radius of neighborhood Straight-line Jump Sum 5

6 4 Strategies During Keywords Circle the odd one out: Real numbers jump dis limit right neighborhood closed interval domain sum - invertible function straight line intersection minimum extreme left infinite finite dis centre radius equation distance variable dis equation real point - power quotient system of equations tangent axis. 6

7 5 Strategies During Conceptual Map Complete the conceptual map using the following words: dis image domain of a function left continuous limit first kind second kind third kind right 7

8 6 Strategies After Multiple Choice 1) Let f(x) be a function so defined: x² -2 x < 0 2x - x² 2x x 2 2 < x 3 a. the Weierstrass Theorem is checked and f(x) has only an absolute Maximum at x = 4 b. f(x) has an absolute Maximum at x = 4 and an absolute minimum at x = 0 c. f(x) doesn t have neither Maximum nor minimum d. f(x) has an absolute minimum at x = 0 2) Let f(x) be a function so defined: 2x x < 0 -x²+ 4x x 4 a. the Weierstrass Theorem is checked in the interval [ -2, 2 ] b. the function f(x) is continuous at the point x = 0 c. the function f(x) has at least one point of intersection with the x axis in the interval [ 0, 2 ] d. the function f(x) doesn t have points of intersection with the x axis in the interval [ 0, 4 ] 8

9 3) Let f(x) be a function so defined: 4x³ - 4x²+ x 2x² + x - 1 a. f(x) has a removable dis at the point x = -1 b. f(x) has a jump dis at the point x = 1 2 c. f(x) has an infinite dis at the point x = -1 d. f(x) is always continuous 4) Let f(x) be a function so defined: x + 1 x 1 3 k x² x > 1 the value of k so that f(x) is always continuous is: a. k = -1 b. k = -2 c. k = +1 d. k doesn t exist 9

10 5) Let f(x) a function so defined: x² - 9 x -3 ax² + x < x < 4 b log (x 1) 3 x 4 the values of a and b, so that f(x) is always continuous, are: a. a = 91 b = 9 b. a = 4 b = 9 c. a = 9 b = d. a and b don t exist 10

11 7 Strategies After Matching 1) Match the graphs of the functions with the definitions: Y 1 Y 2 Y 3 0 c X 0 X 0 c X Y 4 Y 5 Y 6 0 c X 0 c c X 0 X a b c d e f c is a point of dis of the third kind c is a point of dis of the second kind the function is right-continuous at the point c the function is always continuous the function is left-continuous at the point c c is a point of dis of the first kind 11

12 Strategies After Matching 2) Match the functions with the domains: log( x² - 1 ) 1 e 1 x 2 3 x x³ - 1 x² + 2x a IR b IR - {1,-3} c ( -,-1 ) U ( 1,+ ) d IR - {0} 12

13 Strategies After Matching 3) Match the graphs of the functions with the features: Y Y L 0 3 X 0 3 X 1 3 Y a This function has a jump dis at x=0 but here it is also left-continuous; besides at x=3 it has a point of dis of the second kind. 0 3 X b This function has a jump dis at x=0 but here it is also right-continuous; besides at x=3 it has a point of infinite dis. 2 c This function has a jump dis at the point (3;0); besides it has a removable dis at x=0. 13

14 Y Y 0 3 X 0 3 X 4 5 Y d This function has a jump dis at x=3; besides at x=0 it has a point of dis of the second kind. 0 3 X 6 e This function is continuous for x=0; besides at x=3 it has a point of infinite dis. f This function has a dis of the third kind at x=0; besides it has an infinite dis at x=3. 14

15 8 Strategies After True or False State if the sentences are true or false. 1) If a function is continuous at the point x = 0, then lim f(x) isn t equal to x 0 + lim f(x). x 0 - T F 2) If a straight-line, having equation x = k, is an asymptote for the function f(x), then the point k is a point of dis of the second kind for f(x). T F 3) If f(0) = 0 and lim 1, we can say the point x = 0 is a point of removable x 0 - dis for f(x). T F 4) If lim -1 and if lim 1, we can say the point x = 0 is a point of x 0 - x 0 + jump dis for f(x). T F 5) Let f(x) be a continuous function in the interval [ a, b ]; if f(a) > 0 and f(b) > 0 then f(x) can t have points of intersection with the x axis belonging to the interval [ a, b ]. T F 15

16 6) Let f(x) be a continuous strictly growing function in the interval [ a, b ]; if f(a) < 0 and f(b) > 0, then f(x) has only one point of intersection with the x axis belonging to the interval [ a, b ]. T F 7) Let f(x) be a continuous strictly growing function in the interval [ a, b ]; if f(a) < 0 and f(b) > 0, then f(x) has at least a point of intersection with the x axis belonging to the interval [ a, b ]. T F 16

17 9 Strategies After Completion Complete the following definitions. 1) A function is continuous at the point c if ) A function defined in a closed interval [ a, b ] is here continuous if ) The Bolzano Theorem states that if f(x) is.. 4) The dis of the third kind is called also.. because. 5) We have a dis of the first kind at the point c if.... 6) The sum of two continuous functions is.. so also their product. 7) Given a continuous and invertible function f(x), defined in an interval [a,b], its inverse function.. 8) The Weierstrass Theorem states that if f(x) is... 9) A function can be continuous at the point c in only one direction

18 10 Strategies After Flow Chart Consider the parabola of equation y = ax² + bx + c and the straight- line y = mx + q; A and B are the points of intersection between the parabola and the straight-line. If P is a point of the AB arc of the parabola, calculate: lim P A PK PH where PK is the distance between P and the tangent in A to parabola and PH is the ordinate of P. Complete the flow chart using the terms listed below: I calculate the equation of the tangent in A to the parabola I calculate A and B by a system between the equations of the parabola and the straight-line I calculate the limit I calculate the distance PK I consider a point P belonging to the parabola 18

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20 11 Strategies After Think and Discuss The following activity can be performed in a written or oral form. The teacher will choose the modality, depending on the ability (writing or speaking) that needs to be developed. The contexts in which the task will be presented to the students are: A)The student is writing an article about the of a function. B)The student is preparing for an interview on a local TV about the parabolic movement. The student should: 1) Write an article or prepare an interview. 2) Prepare the article or the debate, outlining the main points of the argument, on the basis of what has been studied. 3) If the written activity is the modality chosen by the teacher, the student should provide a written article, indicating the target of readers to whom the article is addressed and the type of magazine / newspaper / school magazine where the article would be published. 4) If the oral activity is the modality chosen by the teacher, the student should present his point of view on the topics to the whole class and a debate could start at the end of his presentation. 20

21 12 Strategies After Summary Given a function f(x) defined in a closed interval [ a, b ], if c is a point belonging to this open interval, f(x) is said to be continuous at the point c if: 1) f(c) exists; 2) lim f(x) exists, is finite and is equal to l, so that lim lim f(x) x c x c + x c - 3) f(c) = l ; this means that lim f(c). x c It s possible that a function is continuous in only one direction, either from the left or from the right; in fact f(x) is right continuous if the lim f(c), likewise f(x) x c + is left continuous if the lim f(c). x c - If also only one of these conditions isn t satisfied, the function f(x) isn t continuous at the point c, and we say that f(x) has a dis at that point. There are three types of dis: we have a dis of the first kind at the point c, that is also called jump dis, if the limits from the right and from the left, as x goes to c, exist, are finite but aren t equal. We have a dis of the second kind at the point c, that is also called infinite dis, if at least one of the limits from the right or from the left, doesn t exist or is equal to infinite. Besides we have a dis of the third kind at the point c, that is also called removable dis if: 1) lim l exists and is finite but the function isn t defined at the point c x c 2) lim l exists and is finite but the value of the limit isn t equal to f(c). x c In addition a function f(x) is said to be continuous in the interval [ a, b ], if it s continuous at every point of this interval. All elementary functions are continuous functions like for example the polynomial functions, the exponential function, the logarithmic function when its argument is greater than zero, and the trigonometric functions. 21

22 In addition, if f(x) and g(x) are two continuous functions at the point c, then the f sum ( f + g ), the product ( f g ), the quotient, the power f g, the g composition f [g (x) ] are still continuous functions at the point c. 1. Answer the following questions. The questions could be a answered in a written or oral form, depending on the teacher s objectives. a) When do you say that a function is continuous at the point c? b) Can a function be continuous only from the left or only from the right? c) When does a function have a jump at the point c? d) How many types of dis do you know? e) When do you say that a function is continuous in the interval [ a, b ]? f) How many continuous functions do you know? g) How do you determine the points of dis of a function? 2. Write a short abstract of the summary (max 150 words) highlighting the main points of the video. 22

23 Web References Website designed to provide parents and classroom teachers with the means to better employ visual imagery. This website is intended to help students from secondary school through college. Teachers, other tutors, and parents will also find this site to be very useful. These websites offers students the opportunity to expand their knowledge on the study of a function

24 13 Activities Based on Problem Solving Solve the following problems: 1) Let f(x) a function so defined: ax + b x < 0 x² +bx + 3a x + c x x < 3 x 3 Determine the values of a, b, and c so that f(x) is always continuous and passes through the point P ( -1 ; 2 ). 2) Determine the type of dis of the following functions: x² - x - 2 x² - 3x - 4 x x² - 9 x² - 1 x² - x 24

25 3) Let f(x) be a function so defined: x³ - 3x² - x + 3 check it satisfies the Bolzano Theorem in the interval [ 0, 2 ] and determine the points of this interval in which 0. 4) Using the Intermediate value Theorem, explain why the following function 1 x attains at least once the value for some x belonging to the interval [ 0, 1 ] 5 5) Check if the Weierstrass Theorem is valid for the following functions in the interval indicated, and if so, determine the Maximum and the minimum of them: 2 x [ -1, 2 ] 2 x - 1 [ -3, -2 ] ln ( x + 4 ) [ 0, 2 ] ln sin x [ 1, 4 ] 25

26 6) Determine and classify the points of dis of the following functions: x² - 1 x + 1 x < -1 cos x x < 0 x² -1 2 x 0 2x e - 1 x 1 sin 2x x x < 0 x = 0 x > 0 2 x 1 x - 1 x - 1 x >1 x x⁴ - x²

27 Answer Sheets Keywords: Finite dis, centre, system of equations, tangent, variable dis. Conceptual Map: domain of a function limit dis image first kind second kind third kind left right 27

28 Multiple Choice: 1a, 2c, 3c, 4c, 5b Matching: 1) 1b, 2d, 3e, 4f, 5a, 6c 2) 1c, 2d, 3a, 4b 3) 1b, 2a, 3c, 4e, 5d, 6f True or False: 1 F, 2 T, 3 F, 4 T, 5 T, 6 T, 7 F Completion: 1) f(x) is defined in c so that f(c) exists; lim L, where L is finite ; f(c) = L x c 2) It s continuous at every point of this interval 3) A function defined and continuous in a closed interval [a,b] and if f(a) f(b) < 0, then there s at least a point c belonging to the open interval (a,b) such that f(c) = 0 4) removable dis because this dis can be removed to make f(x) continuous at c 5) the limits from the right and from the left, as x goes to c, exist, are finite, but aren t equal 6) still a continuous function 7) is still continuous 8) defined and continuous in a closed interval [a,b], then it attains its Maximum and its minimum in this interval 9) either from the left or from the right if lim f(c) or lim f(c) x c - x c + Activities Based on Problem Solving: 1) a = 1, b = 3, c = 18 2) x = -1 dis of the third kind; x = 3 dis of the third kind; x= 0 dis of the second kind and x = 1 dis of the first kind 28

29 3) x = ) f(x) is continuous in [0,1]; f(0)=1=m and f(1)= =m, while < < 1 ; so there s x = ) m = M = 4; m = M = ; m = ln 4 M = ln 6; here the Weierstrass Theorem isn t valid 6) x = -1 dis of the first kind and x= 0 dis of the third kind; x = 0 dis of the third kind; x = 1 dis of the third kind; x = 2 and x = - 2 dis of the second kind Flow Chart: start I calculate A and B by a system between the equations of the parabola and the straight-line I consider a point P belonging to the parabola I calculate the equation of the tangent in A to the parabola I calculate the distance PK I calculate the limit. end Materiale sviluppato da eniscuola nell ambito del protocollo d intesa con il MIUR 29