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Gino, Mario and Joseph-Louis
A family trip Gino’s lecture Tasks Gino, Mario and Joseph-Louis ———————————– Il teorema di Lagrange presentato in modalità CLIL Prof. James Silipo docente di Matematica e Fisica presso l’IIS di Sersale (CZ) Lezione destinata agli alunni delle classi VA e VB del Liceo Scientifico anno scolastico 2015-2016 Glossary A family trip Gino’s lecture Summary 1 A family trip 2 Gino’s lecture 3 Tasks 4 Glossary Tasks Glossary A family trip Gino’s lecture Tasks The story Exercise 1: Write down all the words and constructions you don’t know. Glossary A family trip Gino’s lecture Tasks The story Exercise 1: Write down all the words and constructions you don’t know. Gino Glossary A family trip Gino’s lecture Tasks The story Exercise 1: Write down all the words and constructions you don’t know. Gino is a 17 years old student equally gifted in Mathematics and in squandering his father’s money! Glossary A family trip Gino’s lecture The story The story Gino lives in Sersale, a lovely small town Tasks Glossary A family trip Gino’s lecture The story The story near the town of Catanzaro Tasks Glossary A family trip Gino’s lecture Tasks The story The story His family planned to visit the universal exhibition hosted in Milan from May 1 to October 31, 2015. Glossary A family trip Gino’s lecture The story The story Although Sersale is quite far from Milan Tasks Glossary A family trip The story The story Mario, Gino’s lecture Tasks Glossary A family trip The story The story Mario, Gino’s father, Gino’s lecture Tasks Glossary A family trip Gino’s lecture Tasks The story The story Mario, Gino’s father, thought that traveling by car could give them the chance to visit some of the nice places along the way. Glossary A family trip Gino’s lecture Tasks The story The story Mario’s travel-planning was very well designed and the whole family enjoyed both the trip and the Expo! Glossary A family trip Gino’s lecture Tasks The story The story Mario’s travel-planning was very well designed and the whole family enjoyed both the trip and the Expo! The experience turned out to be a real success and once back home they were really happy! Glossary A family trip Gino’s lecture Tasks The story The story Mario’s travel-planning was very well designed and the whole family enjoyed both the trip and the Expo! The experience turned out to be a real success and once back home they were really happy! Glossary A family trip Gino’s lecture The story The story However, bad news were yet to come! Tasks Glossary A family trip Gino’s lecture The story The story However, bad news were yet to come! Tasks Glossary A family trip Gino’s lecture Tasks Glossary The story The bad news Ten days later Mario received a notice by which the police fined him 500 e for driving too fast somewhere between Bari and Pescara. A family trip Gino’s lecture Tasks Glossary The story The bad news Ten days later Mario received a notice by which the police fined him 500 e for driving too fast somewhere between Bari and Pescara. The notice also indicated that the infraction had been detected by a system called safety tutor between 5:00 pm and 7:10 pm on September 27, 2015. A family trip Gino’s lecture Tasks Glossary The story The bad news Ten days later Mario received a notice by which the police fined him 500 e for driving too fast somewhere between Bari and Pescara. The notice also indicated that the infraction had been detected by a system called safety tutor between 5:00 pm and 7:10 pm on September 27, 2015. The first reaction Mario was rather upset by the rough estimation of the place and the time in which the infringement would have been committed! A family trip Gino’s lecture Tasks Glossary The story The bad news Ten days later Mario received a notice by which the police fined him 500 e for driving too fast somewhere between Bari and Pescara. The notice also indicated that the infraction had been detected by a system called safety tutor between 5:00 pm and 7:10 pm on September 27, 2015. The first reaction Mario was rather upset by the rough estimation of the place and the time in which the infringement would have been committed! In fact, his trip of 306 km form Bari to Pescara lasted 2 hours and 10 minutes, so he wondered how the police could infer the existence of a single traffic infraction in such a long distance and time. A family trip Gino’s lecture Tasks The story Foreword to the second reaction You have to know that Mario is a nice person but he has a terrible flaw: Glossary A family trip Gino’s lecture Tasks The story Foreword to the second reaction You have to know that Mario is a nice person but he has a terrible flaw: he is disgustingly stingy. Glossary A family trip Gino’s lecture Tasks The story Foreword to the second reaction You have to know that Mario is a nice person but he has a terrible flaw: he is disgustingly stingy. That’s why he payed great attention to each speed trap all the way long. Glossary A family trip Gino’s lecture Tasks Glossary The story Foreword to the second reaction You have to know that Mario is a nice person but he has a terrible flaw: he is disgustingly stingy. That’s why he payed great attention to each speed trap all the way long. To that purpose, before leaving he endowed his smartphone with a fully functioning app providing him the precise location of the speed traps on the whole Italian road network. A family trip Gino’s lecture Tasks Glossary The story Foreword to the second reaction You have to know that Mario is a nice person but he has a terrible flaw: he is disgustingly stingy. That’s why he payed great attention to each speed trap all the way long. To that purpose, before leaving he endowed his smartphone with a fully functioning app providing him the precise location of the speed traps on the whole Italian road network. Of course he had been slavishly obeying each of the app warnings commanding him to slow down his car! A family trip Gino’s lecture Tasks The story The second reaction Mario realized that, because of the traffic offense, his driving license would no longer be clean and consequently this would (slightly) affect his insurance premium! Glossary A family trip Gino’s lecture Tasks The story The second reaction Mario realized that, because of the traffic offense, his driving license would no longer be clean and consequently this would (slightly) affect his insurance premium! Glossary A family trip Gino’s lecture The story The question What might have gone amiss? Tasks Glossary A family trip Gino’s lecture Tasks The story The question What might have gone amiss? The answer Mario is still regretting to have trusted his smartphone app! Although the app was up to date, it probably did not notice those diabolic equipments called safety tutors. Glossary A family trip Gino’s lecture Tasks Glossary The story The question What might have gone amiss? The answer Mario is still regretting to have trusted his smartphone app! Although the app was up to date, it probably did not notice those diabolic equipments called safety tutors. The lesson Mario decided to learn more about safety tutors in order to establish their reliability and the extent to which he could file an appeal against the fine. A family trip Gino’s lecture Tasks Glossary The story The question What might have gone amiss? The answer Mario is still regretting to have trusted his smartphone app! Although the app was up to date, it probably did not notice those diabolic equipments called safety tutors. The lesson Mario decided to learn more about safety tutors in order to establish their reliability and the extent to which he could file an appeal against the fine. With some surprise he discovered that they are based on Lagrange’s theorem: a mathematical statement concerning differentiable functions. A family trip Gino’s lecture The story The third reaction Tasks Glossary A family trip Gino’s lecture Tasks The story The third reaction Mario wanted to thoroughly understand the reason of his misfortune and decided to ask Gino for a short talk about Lagrange’s theorem. Glossary A family trip Gino’s lecture Tasks The story The third reaction Mario wanted to thoroughly understand the reason of his misfortune and decided to ask Gino for a short talk about Lagrange’s theorem. Gino’s cue Glossary A family trip Gino’s lecture Tasks The story The third reaction Mario wanted to thoroughly understand the reason of his misfortune and decided to ask Gino for a short talk about Lagrange’s theorem. Gino’s cue Gino accepted his father’s request against a small reward: the money for dining out with Diana, his girlfriend! Glossary A family trip Gino’s lecture Tasks The story The third reaction Mario wanted to thoroughly understand the reason of his misfortune and decided to ask Gino for a short talk about Lagrange’s theorem. Gino’s cue Gino accepted his father’s request against a small reward: the money for dining out with Diana, his girlfriend! Mario damned his idea of visiting the Expo, but he was determined to go all the way and then he accepted his son’s extortion. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Mario’s idea To avoid incurring further extortions by Gino, Mario decided to study by himself all the concepts which were necessary to understand Lagrange’s theorem, so he learned about limits of real functions of a real variable and related theorems Glossary A family trip Gino’s lecture Tasks Gino’s lecture Mario’s idea To avoid incurring further extortions by Gino, Mario decided to study by himself all the concepts which were necessary to understand Lagrange’s theorem, so he learned about limits of real functions of a real variable and related theorems continuous real functions of a real variable and related theorems Glossary A family trip Gino’s lecture Tasks Gino’s lecture Mario’s idea To avoid incurring further extortions by Gino, Mario decided to study by himself all the concepts which were necessary to understand Lagrange’s theorem, so he learned about limits of real functions of a real variable and related theorems continuous real functions of a real variable and related theorems differentiable real functions of a real variable and Rolle’s theorem Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture The car trip from Bari to Pescara can be understood as a differentiable real function s of a real variable which is nothing but the time t. In particular: the trip from Bari to Pescara lasted 2 hours and 10 minutes, (i.e. 7800 seconds), and consisted of 306 km, (i.e. 306000 m) Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture The car trip from Bari to Pescara can be understood as a differentiable real function s of a real variable which is nothing but the time t. In particular: the trip from Bari to Pescara lasted 2 hours and 10 minutes, (i.e. 7800 seconds), and consisted of 306 km, (i.e. 306000 m) the initial spacial position (Bari) can be set to 0 m, so that the value of the function s at the time t = 0 s is s(0) = 0 Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture The car trip from Bari to Pescara can be understood as a differentiable real function s of a real variable which is nothing but the time t. In particular: the trip from Bari to Pescara lasted 2 hours and 10 minutes, (i.e. 7800 seconds), and consisted of 306 km, (i.e. 306000 m) the initial spacial position (Bari) can be set to 0 m, so that the value of the function s at the time t = 0 s is s(0) = 0 the final spacial position (Pescara) has to be set to 306000 m, so that the value of the function s at the time t = 7800 s is s(7800) = 306000. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture A good idea is to plot a cartesian graph of the situation Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture A good idea is to plot a cartesian graph of the situation s 306000 t 7800 A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Any smooth line joining the starting point with the ending one would represent a way of driving form Bari to Pescara in 2 hours and 10 minutes, so there may be several possibilities: s 306000 t 7800 A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture By analyzing the different plots it turns out that all of them but one show variable speed. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture By analyzing the different plots it turns out that all of them but one show variable speed. The red plot is the only one with a constant speed. It is quite a simple plot, however it is not realistic because such a speed would be equal to v= m km 306000 − 0 = 36, 42 = 141, 23 , 7800 − 0 s h Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture By analyzing the different plots it turns out that all of them but one show variable speed. The red plot is the only one with a constant speed. It is quite a simple plot, however it is not realistic because such a speed would be equal to v= m km 306000 − 0 = 36, 42 = 141, 23 , 7800 − 0 s h too high a speed to drive instantly from the beginning till the end of the trip. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Glossary Gino’s lecture Gino’s lecture Nevertheless, the red plot is still useful because it seems that all the other plots admit at least a point where the tangent line is parallel to the red one. s 306000 t 7800 A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture In each of those points the speed is equal to 141, 23 km/h whereas the limit is set to 130 km/h plus a 3% of tolerance, so no more than 133, 9 km/h. It follows that each marked point represents an infringement. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture In each of those points the speed is equal to 141, 23 km/h whereas the limit is set to 130 km/h plus a 3% of tolerance, so no more than 133, 9 km/h. It follows that each marked point represents an infringement. The examples we have analyzed suggest that the such an infringement point should exist for every sufficiently regular curve joining the starting point and the ending one. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Gino’s lecture In each of those points the speed is equal to 141, 23 km/h whereas the limit is set to 130 km/h plus a 3% of tolerance, so no more than 133, 9 km/h. It follows that each marked point represents an infringement. The examples we have analyzed suggest that the such an infringement point should exist for every sufficiently regular curve joining the starting point and the ending one. In order to tackle the problem efficiently, let us look at the problem in a more general mathematical setting. Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Lagrange point Definition: Let f : [a, b] → R be a function. A Lagrange point of f is a point c ∈ (a, b) at which f is differentiable and f 0 (c) = f (b) − f (a) . b−a A Lagrange point c is just the abscissa of a point of the graph of f where the tangent line is parallel to the line joining (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Gino’s lecture Lagrange point Definition: Let f : [a, b] → R be a function. A Lagrange point of f is a point c ∈ (a, b) at which f is differentiable and f 0 (c) = f (b) − f (a) . b−a A Lagrange point c is just the abscissa of a point of the graph of f where the tangent line is parallel to the line joining (a, f (a)) and (b, f (b)). Yet, we don’t know if such a point does exist for any given f ! A family trip Gino’s lecture Tasks Glossary Gino’s lecture Lagrange point Definition: Let f : [a, b] → R be a function. A Lagrange point of f is a point c ∈ (a, b) at which f is differentiable and f 0 (c) = f (b) − f (a) . b−a A Lagrange point c is just the abscissa of a point of the graph of f where the tangent line is parallel to the line joining (a, f (a)) and (b, f (b)). Yet, we don’t know if such a point does exist for any given f ! Lagrange’s theorem answers this question in the affirmative under suitable hypotheses on f ! A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 1: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 1: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 1: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x Work in pairs and discuss your answers with your deskmate. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 2: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 2: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 2: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x Work in pairs and discuss your answers with your deskmate. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 3: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 3: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 3: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x Work in pairs and discuss your answers with your deskmate. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 4: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 4: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 4: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x Work in pairs and discuss your answers with your deskmate. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 5: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 5: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for the good hypotheses on f Exercise 5: Is there a Lagrange point in the following graph? Is this function continuous in [a, b]? y f(b) f(a) a b x Work in pairs and discuss your answers with your deskmate. A family trip Gino’s lecture Tasks Gino’s lecture Searching for the good hypotheses on f Exercises 1, 2, 3 show that without continuity Lagrange points may fail to exist. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Searching for the good hypotheses on f Exercises 1, 2, 3 show that without continuity Lagrange points may fail to exist. However, exercises 4 and 5 prove that without differentiability Lagrange points may fail to exist too. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Searching for the good hypotheses on f Exercises 1, 2, 3 show that without continuity Lagrange points may fail to exist. However, exercises 4 and 5 prove that without differentiability Lagrange points may fail to exist too. So we can ask ourselves if differentiability is enough for a Lagrange point to exist! Glossary A family trip Gino’s lecture Tasks Gino’s lecture Searching for the good hypotheses on f Exercises 1, 2, 3 show that without continuity Lagrange points may fail to exist. However, exercises 4 and 5 prove that without differentiability Lagrange points may fail to exist too. So we can ask ourselves if differentiability is enough for a Lagrange point to exist! The answer is YES, but let us build a proof together! Glossary A family trip Gino’s lecture Tasks Gino’s lecture Building a proof together Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Glossary A family trip Gino’s lecture Tasks Gino’s lecture Building a proof together Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Let us write down the equation of the line r joining the points (a, f (a)) and (b, f (b)). Glossary A family trip Gino’s lecture Tasks Gino’s lecture Building a proof together Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Let us write down the equation of the line r joining the points (a, f (a)) and (b, f (b)). y − f (a) = f (b) − f (a) b−a (x − a) , Glossary A family trip Gino’s lecture Tasks Gino’s lecture Building a proof together Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Let us write down the equation of the line r joining the points (a, f (a)) and (b, f (b)). y − f (a) = f (b) − f (a) b−a (x − a) , or equivalently y = g(x), where f (b) − f (a) f (a)b − f (b)a g(x) = x+ . b−a b−a Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Building a proof together Let h : [a, b] → R be the function defined by f (b) − f (a) f (a)b − f (b)a h(x) = f (x)−g(x) = f (x)− x− . b−a b−a A family trip Gino’s lecture Tasks Glossary Gino’s lecture Building a proof together Let h : [a, b] → R be the function defined by f (b) − f (a) f (a)b − f (b)a h(x) = f (x)−g(x) = f (x)− x− . b−a b−a h(x) is just the difference between the value of f (x) and the ordinate of the point of the line r whose abscissa is x. y f(b) f(a) a b x A family trip Gino’s lecture Gino’s lecture Exercise 6: Prove the following statement: Tasks Glossary A family trip Gino’s lecture Tasks Gino’s lecture Exercise 6: Prove the following statement: h is continuous on [a, b] and differentiable on (a, b). Glossary A family trip Gino’s lecture Tasks Gino’s lecture Exercise 6: Prove the following statement: h is continuous on [a, b] and differentiable on (a, b). Exercise 7: Prove the following statement: Glossary A family trip Gino’s lecture Tasks Gino’s lecture Exercise 6: Prove the following statement: h is continuous on [a, b] and differentiable on (a, b). Exercise 7: Prove the following statement: h(a) = h(b) = 0. Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Exercise 6: Prove the following statement: h is continuous on [a, b] and differentiable on (a, b). Exercise 7: Prove the following statement: h(a) = h(b) = 0. Exercise 8: Prove the following statement: Let c ∈ (a, b), then c is a critical point of h if and only if c is a Lagrange point of f. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Exercise 6: Prove the following statement: h is continuous on [a, b] and differentiable on (a, b). Exercise 7: Prove the following statement: h(a) = h(b) = 0. Exercise 8: Prove the following statement: Let c ∈ (a, b), then c is a critical point of h if and only if c is a Lagrange point of f. We are now ready for the proof! A family trip Gino’s lecture Tasks Gino’s lecture Lagrange’s theorem Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Then f admits at least a Lagrange point c ∈ (a, b). Glossary A family trip Gino’s lecture Tasks Gino’s lecture Lagrange’s theorem Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Then f admits at least a Lagrange point c ∈ (a, b). Proof By virtue of exercises 6 and 7, the function h satisfies the hypotheses of Rolle’s theorem. Then, there exist a point c ∈ (a, b) such that h0 (c) = 0. Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Lagrange’s theorem Let f : [a, b] → R be continuous on [a, b] and differentiable on (a, b). Then f admits at least a Lagrange point c ∈ (a, b). Proof By virtue of exercises 6 and 7, the function h satisfies the hypotheses of Rolle’s theorem. Then, there exist a point c ∈ (a, b) such that h0 (c) = 0. Thanks to exercise 8, the point c is a Lagrange point of f . A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for Lagrange points In order to find the Lagrange points of a function f satisfying the hypotheses of Lagrange’s theorem, one has to solve for x in the following equation A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for Lagrange points In order to find the Lagrange points of a function f satisfying the hypotheses of Lagrange’s theorem, one has to solve for x in the following equation f (b) − f (a) f 0 (x) = . b−a A family trip Gino’s lecture Tasks Glossary Gino’s lecture Searching for Lagrange points In order to find the Lagrange points of a function f satisfying the hypotheses of Lagrange’s theorem, one has to solve for x in the following equation f (b) − f (a) f 0 (x) = . b−a Example Let f : [1, 4] → R be given by f (x) = −3x 2 + 1. The difference quotient of f is −48 + 1 + 3 − 1 −45 f (4) − f (1) = = = −15 . 4−1 3 3 the derivative of f is f 0 (x) = −6x, so the Lagrange points are the solutions to the equation −6x = −15, i.e. 5/2. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Important remark Lagrange’s theorem cannot be reversed. However it is possible to prove the following more general version. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Important remark Lagrange’s theorem cannot be reversed. However it is possible to prove the following more general version. Generalized Lagrange Theorem: Let f : [a, b] → R be a continuous function that is differentiable in (a, b) \ S, where S ⊂ (a, b) is a finite set of inflection points with vertical tangent. Then f admits at least a Lagrange point c ∈ (a, b) \ S. A family trip Gino’s lecture Tasks Glossary Gino’s lecture Important remark Lagrange’s theorem cannot be reversed. However it is possible to prove the following more general version. Generalized Lagrange Theorem: Let f : [a, b] → R be a continuous function that is differentiable in (a, b) \ S, where S ⊂ (a, b) is a finite set of inflection points with vertical tangent. Then f admits at least a Lagrange point c ∈ (a, b) \ S. The proof is based on a corresponding generalization of Rolle’s theorem. A family trip Gino’s lecture Tasks Gino’s lecture Back to Mario’s problem After the lecture, Mario fully understood how the safety tutor could infer the existence of a Lagrange point in his trip from Bari to Pescara! Glossary A family trip Gino’s lecture Tasks Gino’s lecture Back to Mario’s problem After the lecture, Mario fully understood how the safety tutor could infer the existence of a Lagrange point in his trip from Bari to Pescara! Although Mario is quite miserly, he had to admit that Gino’s lecture was worthy of the small reward required. Glossary A family trip Gino’s lecture Tasks Gino’s lecture Back to Mario’s problem After the lecture, Mario fully understood how the safety tutor could infer the existence of a Lagrange point in his trip from Bari to Pescara! Although Mario is quite miserly, he had to admit that Gino’s lecture was worthy of the small reward required. At the end of the story Mario realized his love for mathematics and he felt grateful to the police for the fine because without it he would never discover his passion. Glossary A family trip Gino’s lecture Tasks Glossary Gino’s lecture Back to Mario’s problem After the lecture, Mario fully understood how the safety tutor could infer the existence of a Lagrange point in his trip from Bari to Pescara! Although Mario is quite miserly, he had to admit that Gino’s lecture was worthy of the small reward required. At the end of the story Mario realized his love for mathematics and he felt grateful to the police for the fine because without it he would never discover his passion. Anyway, he did not regret his avarice: after all, had he not be so cheap he would never be so motivated to learn about Lagrange’s theorem! A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. functions admit Lagrange points, other do not . Lagrange’s theorem ensures their existence for the admit functions which satisfy its hypotheses, however, Lagrange’s theorem implies thing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: times it will have Lagrange points, times it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, other do not . Lagrange’s theorem ensures their existence for the admit functions which satisfy its hypotheses, however, Lagrange’s theorem implies thing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: times it will have Lagrange points, times it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not . Lagrange’s theorem ensures their existence for the admit functions which satisfy its hypotheses, however, Lagrange’s theorem implies thing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: times it will have Lagrange points, times it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies thing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: times it will have Lagrange points, times it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies nothing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: times it will have Lagrange points, times it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies nothing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: sometimes it will have Lagrange points, times it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies nothing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: sometimes it will have Lagrange points, sometimes it will not. way, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies nothing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: sometimes it will have Lagrange points, sometimes it will not. Anyway, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms of the points in (a, b) solves the figuring in it do exist! If preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies nothing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: sometimes it will have Lagrange points, sometimes it will not. Anyway, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms figuring in it do exist! If none of the points in (a, b) solves the preceding equation, then it means that f does not satisfies of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary First task Fill in the blanks using the words some, any, no, none. Some functions admit Lagrange points, some other do not admit any. Lagrange’s theorem ensures their existence for the functions which satisfy its hypotheses, however, Lagrange’s theorem implies nothing about the functions which do not satisfy its hypotheses. What really happens to the latter class of functions depends on the shape of the particular function: sometimes it will have Lagrange points, sometimes it will not. Anyway, if you are looking for the Lagrange points of a function f : [a, b] → R you can always try to solve for x in the equation f 0 (x) = (f (b) − f (a))/(b − a) , provided that all the terms figuring in it do exist! If none of the points in (a, b) solves the preceding equation, then it means that f does not satisfies some of Lagrange’s theorem hypotheses. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange point; however, Lagrange’s theorem or may have no , provided that f is on [a, b] ensures their on (a, b). and Let Lf denote the of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points point; however, Lagrange’s theorem or may have no , provided that f is on [a, b] ensures their on (a, b). and Let Lf denote the of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem , provided that f is on [a, b] ensures their on (a, b). and Let Lf denote the of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem on [a, b] ensures their existence, provided that f is on (a, b). and Let Lf denote the of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] on (a, b). and Let Lf denote the of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] and differentiable on (a, b). Let Lf denote the of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] and differentiable on (a, b). Let Lf denote the number of Lagrange points of a function f . If f the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] and differentiable on (a, b). Let Lf denote the number of Lagrange points of a function f . If f satisfies the hypotheses of Lagrange’s theorem, then Lf > 1. number Oscillating functions prove that, for any positive a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] and differentiable on (a, b). Let Lf denote the number of Lagrange points of a function f . If f satisfies the hypotheses of Lagrange’s theorem, then Lf > 1. Oscillating functions prove that, for any positive integer number a function f such that Lf = n. Moreover, n, there exists at if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] and differentiable on (a, b). Let Lf denote the number of Lagrange points of a function f . If f satisfies the hypotheses of Lagrange’s theorem, then Lf > 1. Oscillating functions prove that, for any positive integer number n, there exists at least a function f such that Lf = n. Moreover, if one takes a linear function f , it is easy to show that any point point, so that Lf becomes infinite in this case. is a Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Second task Fill in the blanks using the words listed below the text. A continuous function f : [a, b] → R may have Lagrange points or may have no such point; however, Lagrange’s theorem ensures their existence, provided that f is continuous on [a, b] and differentiable on (a, b). Let Lf denote the number of Lagrange points of a function f . If f satisfies the hypotheses of Lagrange’s theorem, then Lf > 1. Oscillating functions prove that, for any positive integer number n, there exists at least a function f such that Lf = n. Moreover, if one takes a linear function f , it is easy to show that any point is a Lagrange point, so that Lf becomes infinite in this case. Word list: points, least, Lagrange, differentiable, satisfies, such, number, existence, integer, continuous. A family trip Gino’s lecture Tasks Glossary Third task Crossword puzzle Across: 1 Where was Lagrange born? 3 It can be tangent or secant. 6 Forgotten state in which you usually fall 1 3 2 4 5 during the math oral test. 8 Plural nominative of I. 9 On 7 6 the internet. 11 In a sandwich. 12 Fish eggs. 13 The first 8 9 10 get. 17 Los Angeles. 18 Past tense of may. 20 The first 13 12 15 14 18 three letters of Odyssey. 14 Former. 15 Simple past of 11 16 19 letter of Yesterday thrice. 21 Garden tool. 22 Preside at 17 trial. 24 Informal abbreviation of a male sibling. 25 Posses- 20 sive of she. 21 23 22 24 Down: 1 We have just studied Lagrange’s one. 2 His theorem is used in the proof of Lagrange’s one. 25 3 Louis Vuitton. 4 Currently. 5 Not a friend. 7 Incompetent. 10 Bad for health. 11 Sacred. 15 Grand Hotel. 16 Different. 19 Any divine being. 20 Yes vote. 22 Joseph-Louis. 23 Connects alternatives. A family trip Gino’s lecture Tasks Glossary Third task Crossword puzzle Across: 1 Where was Lagrange born? 3 It can be tangent or secant. 6 Forgotten state in which you usually fall 1 2 U R 6 O L 10 N L O E X 19 I G O O 22 J U D L S T H E 9 O 12 R 14 E 18 M I N 7 B L I N I N E P 15 16 G O T H T 21 H O G E 24 B R O 3 4 5 L I N E V I O N 8 W E 11 H A M 13 O D Y 17 L A 20 Y Y Y S E 23 O A 25 H E R during the math oral test. 8 Plural nominative of I. 9 On the internet. 11 In a sandwich. 12 Fish eggs. 13 The first three letters of Odyssey. 14 Former. 15 Simple past of get. 17 Los Angeles. 18 Past tense of may. 20 The first letter of Yesterday thrice. 21 Garden tool. 22 Preside at trial. 24 Informal abbreviation of a male sibling. 25 Possessive of she. Down: 1 We have just studied Lagrange’s one. 2 His theorem is used in the proof of Lagrange’s one. 3 Louis Vuitton. 4 Currently. 5 Not a friend. 7 Incompetent. 10 Bad for health. 11 Sacred. 15 Grand Hotel. 16 Different. 19 Any divine being. 20 Yes vote. 22 Joseph-Louis. 23 Connects alternatives. A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. 1 Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. 2 1 Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. 2 3 1 Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. 2 3 4 1 Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. 2 5 3 4 1 Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Glossary Fourth task Sequence order Put the steps in the correct order to get our proof of Lagrange’s Theorem. 6 2 5 3 4 1 Proving that critical points of h are exactly the Lagrange points of f . Defining the auxiliary function h(x) = f (x) − g(x). Applying Rolle’s theorem to the auxiliary function h. Showing that the auxiliary function h is continuous in [a, b] and differentiable in (a, b). Proving that h(a) = h(b) = 0. Defining the function g whose graph is the line joining the points (a, f (a)) and (b, f (b)). A family trip Gino’s lecture Tasks Fifth task Find the Lagrange points of the following functions, if any. Explain your answers. 1 f : [−π, 3π] → R, f (x) = x + cos x. 2 f : [−2, −1] → R, f (x) = 1/x. 3 f : [e, 2e] → R, f (x) = −x + ln x. 4 f : [−3, 4] → R, f (x) = −3x 3 + 2x 2 + 5x − 6. 5 f : [0, 3] → R, f (x) = ex . 6 f : [−2, 2] → R, f (x) = |x|. Glossary A family trip Gino’s lecture Tasks Glossary English Difference quotient Differentiable function Derivative of f at the point c Speed Velocity Bounded and closed interval Bounded and open interval Solve for x in an equation Lagrange point Corner point Cusp Inflection point with vertical tangent Italiano Rapporto incrementale Funzione derivabile Derivata di f nel punto c Velocità scalare Velocità vettoriale Intervallo chiuso e limitato Intervallo aperto e limitato Risolvere un’equazione rispetto ad x Punto di Lagrange Punto angoloso Cuspide Punto di flesso a tangente verticale Glossary A family trip Gino’s lecture Thank you. Tasks Glossary
