Gino, Mario and Joseph-Louis - Sersale (CZ) · A family trip Gino’s lecture Tasks Glossary Gino,...

A family trip Gino’s lecture Tasks Glossary

Gino, Mario and Joseph-Louis———————————–

Il teorema di Lagrange presentato in modalita 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 Scientificoanno scolastico 2015-2016


Summary

1 A family trip

2 Gino’s lecture

3 Tasks

4 Glossary


The story

Exercise 1: Write down all the words and constructionsyou don’t know.

Gino

is a 17 years old student equally gifted in Mathematics and insquandering his father’s money!


The story


Gino



The story


Gino



The story

The storyGino lives in Sersale, a lovely small town


The story

The storynear the town of Catanzaro


The story

The storyHis family planned to visit

the universal exhibition hosted in Milan from May 1 to October31, 2015.


The story

The storyAlthough Sersale is quite far from Milan


The story

The storyMario,

Gino’s father, thought that traveling by car could give them thechance to visit some of the nice places along the way.


The story

The storyMario,

Gino’s father,

thought that traveling by car could give them thechance to visit some of the nice places along the way.


The story

The storyMario,

Gino’s father, thought that traveling by car could give them thechance to visit some of the nice places along the way.


The story

The storyMario’s travel-planning was very well designed and the wholefamily enjoyed both the trip and the Expo!

The experience turned out to be a real success and once backhome they were really happy!


The story

The storyMario’s travel-planning was very well designed and the wholefamily enjoyed both the trip and the Expo!The experience turned out to be a real success and once backhome they were really happy!


The story

The storyMario’s travel-planning was very well designed and the wholefamily enjoyed both the trip and the Expo!The experience turned out to be a real success and once backhome they were really happy!


The story

The storyHowever, bad news were yet to come!


The story

The storyHowever, bad news were yet to come!


The story

The bad newsTen days later Mario received a notice by which the police finedhim 500e for driving too fast somewhere between Bari andPescara.

The notice also indicated that the infraction had been detectedby a system called safety tutor between 5:00 pm and 7:10 pmon September 27, 2015.

The first reactionMario was rather upset by the rough estimation of the placeand the time in which the infringement would have beencommitted! In fact, his trip of 306 km form Bari to Pescaralasted 2 hours and 10 minutes, so he wondered how the policecould infer the existence of a single traffic infraction in such along distance and time.


The story

The bad newsTen days later Mario received a notice by which the police finedhim 500e for driving too fast somewhere between Bari andPescara.The notice also indicated that the infraction had been detectedby a system called safety tutor between 5:00 pm and 7:10 pmon September 27, 2015.



The story


The first reactionMario was rather upset by the rough estimation of the placeand the time in which the infringement would have beencommitted!

In fact, his trip of 306 km form Bari to Pescaralasted 2 hours and 10 minutes, so he wondered how the policecould infer the existence of a single traffic infraction in such along distance and time.


The story




The story

Foreword to the second reactionYou have to know that Mario is a nice person but he has aterrible flaw:

he is disgustingly stingy.

That’s why he payed great attention to each speed trap all theway long.

To that purpose, before leaving he endowed his smartphonewith a fully functioning app providing him the precise location ofthe speed traps on the whole Italian road network.

Of course he had been slavishly obeying each of the appwarnings commanding him to slow down his car!


The story

Foreword to the second reactionYou have to know that Mario is a nice person but he has aterrible flaw: he is disgustingly stingy.





The story






The story






The story






The story

The second reactionMario realized that, because of the traffic offense, his drivinglicense would no longer be clean and consequently this would(slightly) affect his insurance premium!


The story

The second reactionMario realized that, because of the traffic offense, his drivinglicense would no longer be clean and consequently this would(slightly) affect his insurance premium!


The story

The questionWhat might have gone amiss?

The answerMario is still regretting to have trusted his smartphone app!Although the app was up to date, it probably did not noticethose diabolic equipments called safety tutors.

The lessonMario decided to learn more about safety tutors in order toestablish their reliability and the extent to which he could file anappeal against the fine.With some surprise he discovered that they are based onLagrange’s theorem: a mathematical statement concerningdifferentiable functions.


The story





The story



The lessonMario decided to learn more about safety tutors in order toestablish their reliability and the extent to which he could file anappeal against the fine.

With some surprise he discovered that they are based onLagrange’s theorem: a mathematical statement concerningdifferentiable functions.


The story





The story

The third reaction

Mario wanted to thoroughly understand the reason of hismisfortune and decided to ask Gino for a short talk aboutLagrange’s theorem.

Gino’s cueGino accepted his father’s request against a small reward: themoney for dining out with Diana, his girlfriend!

Mario damned his idea of visiting the Expo, but he wasdetermined to go all the way and then he accepted his son’sextortion.


The story

The third reactionMario wanted to thoroughly understand the reason of hismisfortune and decided to ask Gino for a short talk aboutLagrange’s theorem.




The story


Gino’s cue

Gino accepted his father’s request against a small reward: themoney for dining out with Diana, his girlfriend!



The story





The story





Gino’s lecture

Mario’s ideaTo avoid incurring further extortions by Gino, Mario decided tostudy by himself all the concepts which were necessary tounderstand Lagrange’s theorem, so he learned about

limits of real functions of a real variable and relatedtheorems

continuous real functions of a real variable and relatedtheoremsdifferentiable real functions of a real variable and Rolle’stheorem


Gino’s lecture


limits of real functions of a real variable and relatedtheoremscontinuous real functions of a real variable and relatedtheorems

differentiable real functions of a real variable and Rolle’stheorem


Gino’s lecture


limits of real functions of a real variable and relatedtheoremscontinuous real functions of a real variable and relatedtheoremsdifferentiable real functions of a real variable and Rolle’stheorem


Gino’s lecture

Gino’s lectureThe car trip from Bari to Pescara can be understood as adifferentiable real function s of a real variable which is nothingbut the time t . In particular:

the trip from Bari to Pescara lasted 2 hours and 10minutes, (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 thatthe value of the function s at the time t = 0 s is s(0) = 0the final spacial position (Pescara) has to be set to306000 m, so that the value of the function s at the timet = 7800 s is s(7800) = 306000.


Gino’s lecture


the trip from Bari to Pescara lasted 2 hours and 10minutes, (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 thatthe value of the function s at the time t = 0 s is s(0) = 0

the final spacial position (Pescara) has to be set to306000 m, so that the value of the function s at the timet = 7800 s is s(7800) = 306000.


Gino’s lecture


the trip from Bari to Pescara lasted 2 hours and 10minutes, (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 thatthe value of the function s at the time t = 0 s is s(0) = 0the final spacial position (Pescara) has to be set to306000 m, so that the value of the function s at the timet = 7800 s is s(7800) = 306000.


Gino’s lecture

Gino’s lectureA good idea is to plot a cartesian graph of the situation

t

s

7800

306000


Gino’s lecture

Gino’s lectureA good idea is to plot a cartesian graph of the situation

t

s

7800

306000


Gino’s lecture

Gino’s lectureAny smooth line joining the starting point with the ending onewould represent a way of driving form Bari to Pescara in 2hours and 10 minutes, so there may be several possibilities:

t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture

Gino’s lectureBy analyzing the different plots it turns out that all of them butone show variable speed.

The red plot is the only one with a constant speed. It is quite asimple plot, however it is not realistic because such a speedwould be equal to

v =306000− 07800− 0

= 36,42ms

= 141,23kmh,

too high a speed to drive instantly from the beginning till theend of the trip.


Gino’s lecture

Gino’s lectureBy analyzing the different plots it turns out that all of them butone show variable speed.The red plot is the only one with a constant speed. It is quite asimple plot, however it is not realistic because such a speedwould be equal to

v =306000− 0

7800− 0= 36,42

ms

= 141,23kmh,



Gino’s lecture

Gino’s lectureBy analyzing the different plots it turns out that all of them butone show variable speed.The red plot is the only one with a constant speed. It is quite asimple plot, however it is not realistic because such a speedwould be equal to

v =306000− 0

7800− 0= 36,42

ms

= 141,23kmh,



Gino’s lecture

Gino’s lectureNevertheless, the red plot is still useful because it seems thatall the other plots admit at least a point where the tangent lineis parallel to the red one.

t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture


t

s

7800

306000


Gino’s lecture

Gino’s lectureIn each of those points the speed is equal to 141,23 km/hwhereas the limit is set to 130 km/h plus a 3% of tolerance, sono more than 133,9 km/h. It follows that each marked pointrepresents an infringement.

The examples we have analyzed suggest that the such aninfringement point should exist for every sufficiently regularcurve joining the starting point and the ending one.

In order to tackle the problem efficiently, let us look at theproblem in a more general mathematical setting.


Gino’s lecture





Gino’s lecture





Gino’s lecture

Lagrange point

Definition: Let f : [a,b]→ R be a function. A Lagrange point off is a point c ∈ (a,b) at which f is differentiable and

f ′(c) =f (b)− f (a)

b − a.

A Lagrange point c is just the abscissa of a point of the graph off 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 affirmativeunder suitable hypotheses on f !


Gino’s lecture

Lagrange point


f ′(c) =f (b)− f (a)

b − a.


Yet, we don’t know if such a point does exist for any given f !

Lagrange’s theorem answers this question in the affirmativeunder suitable hypotheses on f !


Gino’s lecture

Lagrange point


f ′(c) =f (b)− f (a)

b − a.


Yet, we don’t know if such a point does exist for any given f !Lagrange’s theorem answers this question in the affirmativeunder suitable hypotheses on f !


Gino’s lecture

Searching for the good hypotheses on fExercise 1: Is there a Lagrange point in the following graph? Isthis function continuous in [a,b]?

x

y

a b

f(a)

f(b)

Work in pairs and discuss your answers with your deskmate.


Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture


x

y

a b

f(a)

f(b)



Gino’s lecture

Searching for the good hypotheses on fExercises 1, 2, 3 show that without continuity Lagrange pointsmay fail to exist.

However, exercises 4 and 5 prove that without differentiabilityLagrange points may fail to exist too.

So we can ask ourselves if differentiability is enough for aLagrange point to exist!

The answer is YES, but let us build a proof together!


Gino’s lecture






Gino’s lecture






Gino’s lecture






Gino’s lecture

Building a proof togetherLet 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

g(x) =(

f (b)− f (a)b − a

)x +

(f (a)b − f (b)a

b − a

).


Gino’s lecture



y − f (a) =(

f (b)− f (a)b − a

)(x − a) ,


g(x) =(

f (b)− f (a)b − a

)x +

(f (a)b − f (b)a

b − a

).


Gino’s lecture



y − f (a) =(

f (b)− f (a)b − a

)(x − a) ,


g(x) =(

f (b)− f (a)b − a

)x +

(f (a)b − f (b)a

b − a

).


Gino’s lecture



y − f (a) =(

f (b)− f (a)b − a

)(x − a) ,


g(x) =(

f (b)− f (a)b − a

)x +

(f (a)b − f (b)a

b − a

).


Gino’s lecture

Building a proof togetherLet h : [a,b]→ R be the function defined by

h(x) = f (x)−g(x) = f (x)−(

f (b)− f (a)b − a

)x−

(f (a)b − f (b)a

b − a

).

h(x) is just the difference between the value of f (x) and theordinate of the point of the line r whose abscissa is x .

x

y

a b

f(a)

f(b)


Gino’s lecture

Building a proof togetherLet h : [a,b]→ R be the function defined by

h(x) = f (x)−g(x) = f (x)−(

f (b)− f (a)b − a

)x−

(f (a)b − f (b)a

b − a

).

h(x) is just the difference between the value of f (x) and theordinate of the point of the line r whose abscissa is x .

x

y

a b

f(a)

f(b)


Gino’s lecture

Exercise 6: Prove the following statement:

h is continuous on [a,b] and differentiable on (a,b).


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 off .

We are now ready for the proof!


Gino’s lecture

Exercise 6: Prove the following statement:h is continuous on [a,b] and differentiable on (a,b).


h(a) = h(b) = 0.





Gino’s lecture



h(a) = h(b) = 0.





Gino’s lecture


Exercise 7: Prove the following statement:h(a) = h(b) = 0.





Gino’s lecture







Gino’s lecture







Gino’s lecture

Lagrange’s theoremLet 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).

ProofBy virtue of exercises 6 and 7, the function h satisfies thehypotheses of Rolle’s theorem. Then, there exist a pointc ∈ (a,b) such that h′(c) = 0.Thanks to exercise 8, the point c is a Lagrange point of f . �


Gino’s lecture


ProofBy virtue of exercises 6 and 7, the function h satisfies thehypotheses of Rolle’s theorem. Then, there exist a pointc ∈ (a,b) such that h′(c) = 0.

Thanks to exercise 8, the point c is a Lagrange point of f . �


Gino’s lecture


ProofBy virtue of exercises 6 and 7, the function h satisfies thehypotheses of Rolle’s theorem. Then, there exist a pointc ∈ (a,b) such that h′(c) = 0.Thanks to exercise 8, the point c is a Lagrange point of f . �


Gino’s lecture

Searching for Lagrange pointsIn order to find the Lagrange points of a function f satisfying thehypotheses of Lagrange’s theorem, one has to solve for x in thefollowing equation

f ′(x) =f (b)− f (a)

b − a.

Example

Let f : [1,4]→ R be given by f (x) = −3x2 + 1. The differencequotient of f is

f (4)− f (1)4− 1

=−48 + 1 + 3− 1

3=−45

3= −15 .

the derivative of f is f ′(x) = −6x , so the Lagrange points arethe solutions to the equation −6x = −15, i.e. 5/2.


Gino’s lecture


f ′(x) =f (b)− f (a)

b − a.

Example


f (4)− f (1)4− 1

=−48 + 1 + 3− 1

3=−45

3= −15 .



Gino’s lecture


f ′(x) =f (b)− f (a)

b − a.

Example


f (4)− f (1)4− 1

=−48 + 1 + 3− 1

3=−45

3= −15 .



Gino’s lecture

Important remarkLagrange’s theorem cannot be reversed. However it is possibleto prove the following more general version.

Generalized Lagrange Theorem: Let f : [a,b]→ R be acontinuous function that is differentiable in (a,b) \ S, whereS ⊂ (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’stheorem. �


Gino’s lecture





Gino’s lecture





Gino’s lecture

Back to Mario’s problemAfter the lecture, Mario fully understood how the safety tutorcould infer the existence of a Lagrange point in his trip fromBari to Pescara!

Although Mario is quite miserly, he had to admit that Gino’slecture was worthy of the small reward required.

At the end of the story Mario realized his love for mathematicsand he felt grateful to the police for the fine because without ithe would never discover his passion.

Anyway, he did not regret his avarice: after all, had he not be socheap he would never be so motivated to learn aboutLagrange’s theorem!


Gino’s lecture






Gino’s lecture






Gino’s lecture






First task

Fill in the blanks using the words some, any, no, none.

Some

functions admit Lagrange points,

some

other do notadmit

any

. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies

no

thing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions depends on the shape of the particular function:

some

times it will have Lagrange points,

some

times it will not.

Any

way, if you are looking for the Lagrange points of a functionf : [a,b]→ R you can always try to solve for x in the equationf ′(x) = (f (b)− f (a))/(b − a) , provided that all the termsfiguring in it do exist! If

none

of the points in (a,b) solves thepreceding equation, then it means that f does not satisfies

some

of Lagrange’s theorem hypotheses.


First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points,

some

other do notadmit

any


no


some


some

times it will not.

Any


none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit

any


no


some


some

times it will not.

Any


none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies

no


some


some

times it will not.

Any


none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies nothing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions depends on the shape of the particular function:

some


some

times it will not.

Any


none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies nothing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions depends on the shape of the particular function:sometimes it will have Lagrange points,

some

times it will not.

Any


none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies nothing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions depends on the shape of the particular function:sometimes it will have Lagrange points, sometimes it will not.

Any


none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies nothing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions 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 functionf : [a,b]→ R you can always try to solve for x in the equationf ′(x) = (f (b)− f (a))/(b − a) , provided that all the termsfiguring in it do exist! If

none


some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies nothing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions 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 functionf : [a,b]→ R you can always try to solve for x in the equationf ′(x) = (f (b)− f (a))/(b − a) , provided that all the termsfiguring in it do exist! If none of the points in (a,b) solves thepreceding equation, then it means that f does not satisfies

some



First task

Fill in the blanks using the words some, any, no, none.Some functions admit Lagrange points, some other do notadmit any. Lagrange’s theorem ensures their existence for thefunctions which satisfy its hypotheses, however, Lagrange’stheorem implies nothing about the functions which do notsatisfy its hypotheses. What really happens to the latter class offunctions 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 functionf : [a,b]→ R you can always try to solve for x in the equationf ′(x) = (f (b)− f (a))/(b − a) , provided that all the termsfiguring in it do exist! If none of the points in (a,b) solves thepreceding equation, then it means that f does not satisfiessome of Lagrange’s theorem hypotheses.


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 theoremensures 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

numbern, 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 pointis a

Lagrange

point, so that Lf becomes infinite in this case.

Word list: points, least, Lagrange, differentiable, satisfies, such,number, existence, integer, continuous.


Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no

such

point; however, Lagrange’s theoremensures their

existence


continuous

on [a,b]and

differentiable


number


satisfies


integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures their

existence


continuous

on [a,b]and

differentiable


number


satisfies


integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures their existence, provided that f is

continuous

on [a,b]and

differentiable


number


satisfies


integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures their existence, provided that f is continuous on [a,b]and

differentiable


number


satisfies


integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures their existence, provided that f is continuous on [a,b]and differentiable on (a,b).Let Lf denote the

number


satisfies


integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures 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


integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures 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 fsatisfies the hypotheses of Lagrange’s theorem, then Lf > 1.Oscillating functions prove that, for any positive

integer


least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures 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 fsatisfies the hypotheses of Lagrange’s theorem, then Lf > 1.Oscillating functions prove that, for any positive integer numbern, there exists at

least


Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures 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 fsatisfies the hypotheses of Lagrange’s theorem, then Lf > 1.Oscillating functions prove that, for any positive integer numbern, 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 pointis a

Lagrange




Second task

Fill in the blanks using the words listed below the text.A continuous function f : [a,b]→ R may have Lagrange pointsor may have no such point; however, Lagrange’s theoremensures 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 fsatisfies the hypotheses of Lagrange’s theorem, then Lf > 1.Oscillating functions prove that, for any positive integer numbern, 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 pointis a Lagrange point, so that Lf becomes infinite in this case.



Third task

Crossword puzzle

1 2 3 4 5

6 7

8

9 10 11

12 13

14 15 16 17

18 19 20

21

22 23

24 25

T U R I N L I N EO B L I V I O N

N W EN L I N E H A M

O D YPG O T L A

G H T Y Y YO H O E S

J U D G E A OB R O H E R

THEOREM

JL

ROLLE

NOXIOUS

Across: 1 Where was Lagrange born? 3 It can be tan-

gent or secant. 6 Forgotten state in which you usually fall

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 Posses-

sive 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 Differ-

ent. 19 Any divine being. 20 Yes vote. 22 Joseph-Louis.

23 Connects alternatives.


Third task

Crossword puzzle

1 2 3 4 5

6 7

8

9 10 11

12 13

14 15 16 17

18 19 20

21

22 23

24 25

T U R I N L I N EO B L I V I O N

N W EN L I N E H A M

O D YPG O T L A

G H T Y Y YO H O E S

J U D G E A OB R O H E R

THEOREM

JL

ROLLE

NOXIOUS

Across: 1 Where was Lagrange born? 3 It can be tan-

gent or secant. 6 Forgotten state in which you usually fall

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 Posses-

sive 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 Differ-

ent. 19 Any divine being. 20 Yes vote. 22 Joseph-Louis.

23 Connects alternatives.


Fourth task

Sequence orderPut the steps in the correct order to get our proof of Lagrange’sTheorem.

6

Proving that critical points of h are exactlythe Lagrange points of f .

2

Defining the auxiliary function h(x) = f (x)− g(x).

5

Applying Rolle’s theorem to the auxiliary function h.

3

Showing that the auxiliary function h is continuousin [a,b] and differentiable in (a,b).

4

Proving that h(a) = h(b) = 0.

1

Defining the function g whose graph is the linejoining the points (a, f (a)) and (b, f (b)).


Fourth task


6


2

Defining the auxiliary function h(x) = f (x)− g(x).

5


3


4


1Defining the function g whose graph is the linejoining the points (a, f (a)) and (b, f (b)).


Fourth task


6


2 Defining the auxiliary function h(x) = f (x)− g(x).

5


3


4




Fourth task


6



5


3Showing that the auxiliary function h is continuousin [a,b] and differentiable in (a,b).

4




Fourth task


6



5



4 Proving that h(a) = h(b) = 0.



Fourth task


6


2 Defining the auxiliary function h(x) = f (x)− g(x).5 Applying Rolle’s theorem to the auxiliary function h.





Fourth task


6Proving that critical points of h are exactlythe Lagrange points of f .

2 Defining the auxiliary function h(x) = f (x)− g(x).5 Applying Rolle’s theorem to the auxiliary function h.





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) = −3x3 + 2x2 + 5x − 6.5 f : [0,3]→ R, f (x) = ex .6 f : [−2,2]→ R, f (x) = |x |.


Glossary

English ItalianoDifference quotient Rapporto incrementaleDifferentiable function Funzione derivabileDerivative of f at the point c Derivata di f nel punto cSpeed Velocita scalareVelocity Velocita vettorialeBounded and closed interval Intervallo chiuso e limitatoBounded and open interval Intervallo aperto e limitatoSolve for x in an equation Risolvere un’equazione rispetto ad xLagrange point Punto di LagrangeCorner point Punto angolosoCusp CuspideInflection point with vertical tangent Punto di flesso a tangente verticale


Thank you.

Date post:	23-Feb-2019
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