Grade 12 / Annals on integration 29 exercises (100% corrected)

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Temps (en jours)020406080100120140160180200220Hauteur (en mètres)0.20.40.60.811.21.41.61.822.2 IJOCf ijC1d1 ijC2d2 ijC3d3 1. Areas and function studies E.5429 Part 1 We are interested in the evolution of the height of a corn plant as a function of time. The graph below represents this evolution. Height is in meters and time in days. We decide to model this growth by a logistic function of the type : h ( t ) = a 1 + b · e − 0.04 t où a and b are positive real constants, t is the time variable expressed in days and h ( t ) denotes the plant height, expressed in meters. We know that initially, for t =0 , the plane measures 0.1 m and that its height tends towards a limiting height of 2 m . Determine the constants a and b so that the function h corre-sponds to the growth of the corn plant under study. Part 2 The growth of the corn plant is now considered to be given by the function f defined on 0 ; 250 by: f ( t ) = 2 1 + 19 · e − 0.04 t 1 Determine f ( t ) as a function of t ( f denoting the deriva-tive function of the function f ) . Deduce the variations of the function f on the interval 0 ; 250 . 2 Calculate the time required for the corn plant to reach a height greater than 1.5 m . 3 a Verify that for any real t belonging to the interval 0 ; 250 , we have : f ( t ) = 2e 0.04 t e 0.04 t + 19 . Show that the function F defined on the interval 0 ; 250 by F ( t )=50 · ln e 0.04 t +19 is a primitive of the function f . b Determine the mean value of f on the interval 50 ; 100 . Give an approximate value to the nearest 10 − 2 and interpret this result. 4 We are interested in the growth rate of the corn plant ; this is given by the derivative function of f . The growth rate is maximum for a value of t . Using the graph given in the appendix, determine an ap-proximate value of this. Then estimate the height of the plant. E.6009 Let f be the function defined and derivable on R . Note C its representative curve in the plane provided with a reference O ; −→ i ; −→ j . Part A The graphs below show the C curve and three other curves C 1 , C 2 , C 3 with the tangent at their point of abscissa 0 . 1 By graphical reading, give the sign of f ( x ) according to the values of x . 2 We denote by F a primitive of the function f on R . a Using the curve C , determine F (0) and F ( − 2) . b One of the curves C 1 , C 2 , C 3 is the representative curve of the function F . Determine which one, justifying the elimination of the other two. Part B In this part, it is assumed that the function f discussed in part A is the function defined on R by: f ( x ) = ( x + 2) · e 1 2 x 1 Observing the curve C allows us to conjecture that the function f admits a minimum. a Demonstrate that for any real x : f ( x ) = 1 2 · x + 4 · e 1 2 x b Deduce a validation of the previous conjecture. 2 We pose : I = 1 0 f ( x ) d x . a Geometrically interpret the real I . b Let u and v be the functions defined on R by: u ( x ) = x ; v ( x ) = e 1 2 · x Vérifier que : f = 2 · u · v + u · v . c Deduce the exact value of the integral I . 3 The function f of the algorithm below is given : Function f(n) s ← 0 For k from 0 to n − 1 https://chingmath.fr chapExoCorrec/5429 sacados/5429 Temps (en jours)020406080100120140160180200220Hauteur (en mètres)0.20.40.60.811.21.41.61.822.2 chapExoCorrec/6009 sacados/6009 IJOCf ijC1d1 ijC2d2 ijC3d3
IJOC ijABC s ← s+ 1 n · f k n End of loop Return s Note s n the number returned by the function f when called with the strictly positive integer n . a Justify that s 3 represents the area, expressed in area units, of the hatched area on the graph below où the three rectangles have the same width. b What about the value returned by the function f when the value passed as argument n , when called, becomes large? E.6409 On the graph below, in the plane provided with an orthonormal reference frame O ; −→ i ; −→ j , the representative curve C of a function f defined and deriv-able on the interval 0 ; + ∞ . The following information is available: points A , B , C have respective coordinates (1 ; 0) , (1 ; 2) , (0 ; 2) ; curve C passes through point B and line ( BC ) is tangent to C at B ; there are two positive reals a and b such that for any strictly positive real x : f ( x )= a + b · ln x x 1 a Using the graph, give the values of f (1) and f (1) . b Verify that for any strictly positive real x : f ( x ) = ( b − a ) − b · ln x x 2 c Deduce the reals a and b . 2 a Justify that for any real x belonging to the interval 0 ; + ∞ , f ( x ) has the same sign as − ln x . b Determine the limits of f in 0 and in + ∞ . Note that for any strictly positive real x : f ( x ) = 2 x + 2 · ln x x c Deduce the table of variations of the function f . 3 a Show that the equation f ( x )=1 admits a single so-lution ¸ on the interval 0 ; 1 . b By analogous reasoning, we show that there exists a single real ˛ of the interval 1 ; + ∞ such that : f ( ˛ )=1 . Determine the integer n such that : n<˛ <n +1 4 The algorithm below is given : a ← 0 b ← 1 As long as b − a>0.1 m ← 1 2 · a + b If f(m)<1 Then a ← m Otherwise b ← m End of If End of As long as a Run this algorithm step by step and complete in the table below the values successively taken by the vari-ables a , b and m . Etape 1 Etape 2 Etape 3 Etape 4 Etape 5 a 0 b 1 b − a m b What do the values of the variables a and b represent at the end of the execution of this algorithm? c Modify the algorithm below so that the valuesbelow so that the values a and b at the end of execution of the algorithm are the two bounds of a frame of ˛ of amplitude 10 − 1 . 5 The aim of this question is to demonstrate that the curve C divides the rectangle OABC into two domains of equal areas. a Justify that this is equivalent to showing that : 1 1 e f ( x ) d x = 1 b Noting that the expression of f ( x ) can be written 2 x +2 × 1 x × ln x , complete the demonstration. 2. Areas, study of functions and sequences https://chingmath.fr IJOC chapExoCorrec/6409 sacados/6409 ijABC
E.3257 Part A - Preliminary study of a function f defined on R by ’ ( x )=(2 − x )e x − 1 1 Determine the limits of the function ’ in −∞ and + ∞ . 2 Show that the function ’ is continuous and derivable on R and study the sign of its derivative. Deduce the variations of the function ’ and specify the values of ’ ( − 2) , ’ (0) , ’ (1) and ’ (2) . 3 Prove that the function ’ cancels only at two values, which we will name ¸ and ˛ . We’ll take ¸<˛ . Then study the sign of the function ’ on the set of real num-bers and summarize this study in a table. 4 Using the calculator, provide an amplitude frame 10 − 2 of the values ¸ and ˛ . 5 Show that : e α = 1 2 − ¸ Part B - Study of a function f defined by f ( x )= e x − 1 e x − x and integral calculus. 1 Show that e x − x does not cancel at R . Deduce that f is defined on R . 2 Determine the limits of the function f at −∞ and + ∞ . 3 Calculate the derivative f of the function f and then, using the results from part A , construct the table of variations of f . 4 Show that f ( ¸ )= 1 ¸ − 1 , the number ¸ being the smaller of the two values for which the function ’ of the part A cancels. 5 Determine a primitive of the function f on R . Give an exact value and then an approximate decimal value to the nearest 0.01 of the integral: 1 0 e x − 1 e x − x dx Part C - Study of two suites 1 Specify the set of definition D g of the function g defined on this set by g ( x )=ln 1 2 − x où ln denotes the neperian logarithm function. Prove that the function g is increasing on its defining set and that the image by g of the interval I =[ − 2 ; 0] is included in this interval. 2 a Let be the sequence ( u n ) defined for any natural number n by: u 0 = − 2 u n +1 = g ( u n ) Show that u 1 belongs to the interval I =[ − 2 ; 0] . Prove by recurrence, using the variations of the function g , that the sequence ( u n ) has all its terms in the interval I and is increasing. b Consider the sequence ( v n ) defined for any natural number n by: v 0 = 0 v n +1 = g ( v n ) Calculate the term v 1 and show that : − 2 u 1 v 1 v 0 0 . Establish by recurrence, using the growth of the func-tion g over the interval [ − 2 ; 0] , that for any strictly positive natural number n , we have : − 2 u n v n v n − 1 0 Specify the direction of variation of the sequence ( v n ) . 3 a Let m be the function defined on 0 ; + ∞ by: m ( x ) = x − ln(1+ x ) Show that m is increasing and calculate m (0) . Deduce that, for any positive x , we have : ln(1+ x ) x b Verify that, for any integer n : v n +1 − u n +1 = ln 1+ v n − u n 2 − v n . Deduce that : v n +1 − u n +1 v n − u n 2 − v n Knowing that, for any integer n , the terms of the se-quence ( v n ) belong to the interval − 2 ; 0 , give a frame for 1 2 − v n and establish that : v n +1 − u n +1 1 2 v n − u n Prove then that, for any natural number n , v n − u n 1 2 n v 0 − u 0 What can we deduce for the general term sequence v n − u n and for the sequences ( u n ) and ( v n ) ? 4 Using the calculator, give an amplitude frame 10 − 4 of u 10 and v 10 . 3. Area between two curves E.6046 Part A f is a function defined and derivable on R . f is the derivative function of the function f . In the plane with an orthogonal reference frame, we name C 1 the representative curve of the function f and C 2 the repre-sentative curve of the function f . The point A with coordinates (0 ; 2) belongs to the curve C 1 . The point B with coordinates (0 ; 1) belongs to the curve C 2 . 1 In the three situations below, we have drawn the repre-sentative curve C 1 of the function f . On one of them, the curve C 2 of the derivative function f is drawn ap-propriately. Which is it? Explain your choice. https://chingmath.fr chapExoCorrec/3257 sacados/3257 Antilles chapExoCorrec/6046 sacados/6046
xxyy-2024-2246810C1C2Situation 1 xxyy-2024-2246810C1C2Situation 2 xxyy-2024-2246810C1C2Situation 3 ijDEGFC1 2 Determine the slope-intercept formof the line Δ tangent to the curve C 1 in A . 3 We know that for any real x , f ( x )=e − x + ax + b où a and b are two real numbers. a Determine the value of b using the information given in the statement. b Prove that a =2 . 4 Study the variations of the function f on R . 5 Determine the limit of the function f in + ∞ . Part B Let g be the function defined on R by: g ( x )= f ( x ) − ( x +2) . 1 a Show that the function g admits 0 as a minimum on R . b Deduce the position of the curve C 1 with respect to the straight line Δ The figure 2 below represents a company logo. To design this logo, its creator used the curve C 1 and the straight line Δ , as shown in figure 3 below. In order to estimate painting costs, he wishes to determine the area of the part colored gray. The logo outline is represented by the trapezoid DEFG où : D is the point with coordinates ( − 2 ; 0) , E is the point with coordinates (2 ; 0) , F is the point of abscissa 2 of the curve C 1 , G is the point of abscissa − 2 of the curve C 2 . The part of the logo colored gray corresponds to the surface between the straight line Δ , the curve C 1 , the straight line with equation x = − 2 and the straight line with equation x =2 . 2 Calculate, in units of area, the area of the part of the logo coloured grey (we will give the exact value then the value rounded to 10 − 2 of the result) . E.6257 Let f and g be the functions de-fined on R by: f ( x ) = e x ; g ( x ) = 2 · e x 2 − 1 Note C f and C g the representative curves of the functions f and g in an orthogonal reference frame. 1 Show that the curves C f and C g have a common point of abscissa 0 and that at this point, they have the same tangent Δ of which we will determine an equation. 2 Study the relative position of the curve C g and the straight line Δ . Let h be the function defined on R by: h ( x )=2 · e x 2 − x − 2 a Determine the limit of the function h in −∞ . b Justify that, for any non-zero real x : h ( x ) = x · e x 2 x 2 − 1 − 2 x . Deduce the limit of the function h in + ∞ . c Let h be the derivative function of the function h on R . For any real x , calculate h ( x ) and study the sign of h ( x ) depending on the values of x . d Draw up the table of variations of the function h on R . e Deduce that, for any real x : 2 · e x 2 − 1 x − 1 f What can we deduce about the relative position of the curve C g and the straight line Δ ? 3 Study of the relative position of the curves C f and C g . a For any real x , expand the expression e x 2 − 1 2 . b Determine the relative position of the curves C f and C g . 4 Calculate, in units of area, the area of the domain be-tween the curves C f and C g and the straight lines with equations x =0 and x =1 respectively. https://chingmath.fr xxyy-2024-2246810C1C2Situation 1 xxyy-2024-2246810C1C2Situation 2 xxyy-2024-2246810C1C2Situation 3 ijDEGFC1 chapExoCorrec/6257 sacados/6257
-123456I-1-0.50.51.5JO E.3268 Part A : study of an auxiliary function Let ’ be the function defined on R by: ’ ( x ) = x 2 + x + 1 · e − x − 1 1 a Determine the limits of ’ in −∞ and in + ∞ . b Study the direction of variations of ’ and then draw up its table of variations on R . 2 Show that the equation ’ ( x )=0 has two solutions in R , one of which lies in the interval 1 ; + ∞ , which will be noted ¸ . Determine a frame of amplitude 10 − 2 of ¸ . 3 Deduce the sign of ’ ( x ) at R and present it in a table. Part B: study of the relative position of two curves and calculation of area On the attached sheet are plotted the representative curves of two functions f and g . The functions f and g are defined on R by: f ( x ) = (2 x + 1) · e − x ; g ( x ) = 2 x + 1 x 2 + x + 1 Their representative curves in an orthogonal reference frame O ; −→ i ; −→ j are noted C f and C g . 1 a Show that the two curves pass through the point A of coordinates (0 ; 1) and admit at this point the same tangent. b Show that, for any real number x : f ( x ) − g ( x ) = (2 x + 1) · ’ ( x ) x 2 + x + 1 où ’ is the function studied in part A . c Using a table, study the sign of f ( x ) − g ( x ) on R . d Deduce the relative position of the curves C f and C g . 2 a Show that the function h defined on R by: h ( x ) = ( − 2 x − 3) · e − x − ln x 2 + x + 1 is a primitive on R of the function x ↦→ f ( x ) − g ( x ) . b Deduct the area A , expressed in units of area, of the part of the plane bounded by the two curves C f and C g and the straight lines of equations x = − 1 2 and x =0 . Give the exact value and then the value rounded to 10 − 4 of this area. E.3168 Part A Consider the functions f and g defined on R by: f ( x ) = e − x 2 ; g ( x ) = x 2 e − x 2 Note respectively C f and C g the representative curves of f and g in an orthogonal frame O ; −→ i ; −→ j , the plots of which can be found on the attached sheet. The figure is to be com-pleted and returned with the copy. 1 Identify C f and C g on the figure provided. (Justify the answer given) . 2 Study the parity of the functions f and g ( off programme ) . 3 Study the direction of variation of f and g . Investigate the possible limits of f and g in + ∞ . 4 Study the relative position of C f and C g . Part B Consider the function G defined on R by: G ( x ) = x 0 t 2 e − t 2 d t 1 What does G represent for the function g ? 2 Give, for x> 0 , an interpretation of G ( x ) in terms of ar-eas. 3 Study the direction of variations of G on R . We define the function F on R by: F ( x ) = x 0 e − t 2 d t for any real x 4 Demonstrate that : G ( x ) = 1 2 F ( x ) − x e − x 2 for any real x (we can start by comparing the derivative functions of G and x ↦−→ 1 2 F ( x ) − x e − x 2 ) . We admit that the function F admits a finite limit ‘ in + ∞ , and that this limit ‘ is equal to the area, in units of area, of the domain A bounded by the curve C f and the half-lines O ; −→ i et O ; −→ j . 5 a Show that the function G admits a limit in + ∞ that we will specify. b Interpret in terms of areas the real: N = 1 0 1 − t 2 e − t 2 d t . c Assuming that the limit of G in + ∞ represents the area P in area units of the domain D bounded by the semidroite O ; −→ i and the curve C g , justify graphi-cally that : 1 0 1 − t 2 e − t 2 d t ‘ 2 (we can illustrate the reasoning on the figure provided) https://chingmath.fr chapExoCorrec/3268 sacados/3268 -123456I-1-0.50.51.5JO chapExoCorrec/3168 sacados/3168
-3-2-123IJO AABB00.20.40.60.80.511.522.53Cf E.4017 Part A Consider the function f defined on 0 ; + ∞ by: f ( x ) = x − 1 x + 1 − e − x and denote by ( C ) its representative curve in an orthonormal reference frame O ; −→ i ; −→ j d’unité 3 cm . 1 Calculate the limit of f ( x ) when x tends to + ∞ . What can we deduce for the curve ( C ) ? 2 Calculate f ( x ) , deduce the variations of f for x belong-ing to 0 ; + ∞ . 3 Determine an equation of the tangent ( T ) at ( C ) at its point of abscissa 0 . 4 Show that the equation f ( x )=0 admits a unique solu-tion u . Show that u belongs to 1 ; 2 and determine an amplitude frame 10 − 1 of u . 5 Draw ( T ) and ( C ) on the same figure. 6 a Determine the real numbers a and b such that, for any x =1 : x − 1 x + 1 = a + b x + 1 b Deduce the area in cm 2 of the plane domain bounded by ( T ) , ( C ) and the straight line of equation x =1 (we’ll admit that T is above ( C ) . Part B n denotes a non-zero natural number. Consider the function f n defined on 0 ; + ∞ by: f n ( x ) = x − n x + n − e − x 1 Calculate f n ( x ) and give its sign on 0 ; + ∞ . Specify f n (0) and lim n ↦→ + ∞ f n ( x ) . Draw up the table of variations of f n . 2 a Calculate f n ( n ) ; what is its sign? b Demonstrate by recurrence that, for any n of N : e n +1 > 2 · n + 1 Deduce the sign of f n ( n +1) . c Show that the equation f n ( x )=0 admits a unique so-lution on n ; n +1 ; this solution is denoted u n . 3 Calculate lim n ↦→ + ∞ u n , then lim n ↦→ + ∞ u n n . 4 a Noting that, for any x from 0 ; + ∞ : x − n x + n = 1 − 2 · n x + n Show that the mean value M n of f n over 0 ; u n is equal to : 1 − 1 u n + e − n u n − 2 · n u n · ln u n n +1 b Deduct : lim n ↦→ + ∞ M n . E.5241 Let f be the function defined on 0 ; 1 by: f ( x ) = x · e x We denote by C the representative curve of f in the plane provided with an orthogonal reference frame O ; −→ i ; −→ j . Let a be a real number belonging to the interval 0 ; 1 . On the curve C , plotted above, we have placed the points A and B of abscissas a and 1 respectively. The segments [ OA ] and [ AB ] were drawn. We hatched the part of the plane bounded by the segments [ OA ] and [ AB ] and the curve C . We placed the points : A ( a ; 0) ; B (1 ; 0) . The aim of the exercise is to determine the value of the real number a for which the area of the part of the plane hatched in the appendix is minimal. Part A : 1 a Show that the function f admits as primitive the function F defined by: F ( x ) = ( x − 1) · e x b Establish that : 1 0 x · e x d x = 1 2 a Give the area of the triangle OAA and show that the area of the trapezoid ABB A is equal to : 1 2 · − a 2 · e a + a · e a − a · e + e b Deduce that the area of the hatched part of the plane is equal to : 1 2 · a · e a − a · e + e − 2 Part B: Let g be the function defined on 0 ; + ∞ by: g ( x ) = x · e x − e + e − 2 1 Let g be the derivative function of the function g . Cal-culate g ( x ) for any real x of 0 ; + ∞ . https://chingmath.fr -3-2-123IJO chapExoCorrec/4017 sacados/4017 chapExoCorrec/5241 sacados/5241 AABB00.20.40.60.80.511.522.53Cf
−∞02∞∞04e−20xVariationdef -3-2-12345I246810121416JO(C Verify that the second derivative function g  is defined on 0 ; + ∞ by: g  ( x ) = (2 + x ) · e x 2 Deduce the variations of the function g on 0 ; + ∞ . 3 Establish that the equation g ( x )=0 admits a unique so-lution ¸ in the interval 0 ; + ∞ . Determine an approximate value of ¸ to the nearest 10 − 1 . 4 Deduce the variations of the function g on 0 ; + ∞ . 5 Using the answers to the questions in parts A and B , show that there is a value of a for which the area of the hatched part of the plane is minimal. Give this value of a . E.3209 Part A The table of variations of a function f derivable on R is given : We define the function F on R by: F ( x )= x 2 f ( t ) d t 1 Determine the variations of the function F on R . 2 Show that : 0 F (3) 4 · e − 2 Part B The function f considered in part A is the function defined on R by: f ( x )= x 2 e − x We call g the function defined on R by: g ( x )=e − x We denote by ( C ) and (Γ) the curves representing respectively the functions f and g in an orthogonal frame O ; −→ i ; −→ j The curves are plotted in the appendix. 1 a Show that the variations of the function f are indeed those given in part A . No justification of the limits is required. b Study the relative positions of the curves ( C ) and (Γ) . 2 Let h be the function defined on R by: h ( x )= x 2 − 1 · e − x a Show that the function H defined on R by: H ( x ) = − x 2 − 2 x − 1 · e − x is a primitive of the function h on R . b Let ¸ be a real greater than or equal to 1 . Consider the part of the plane bounded by the curves ( C ) and (Γ) and the straight lines with equations x =1 and x = ¸ . Determine the area A ( ¸ ) , expressed in units of area, of this part of the plane. c Determine the limit of A ( ¸ ) when ¸ tends to + ∞ . 3 We admit that, for any real m strictly greater than 4e − 2 , the straight line with equation y = m intersects the curve ( C ) at the point P ( x P ; m ) and the curve (Γ) at the point Q ( x Q ; m ) . The objective of this question is to show that there is a single value of x P , belonging to the interval −∞ ; − 1 such that the distance PQ is equal to 1. a Show approximately on the graph (proposed in ap-pendix) the points P and Q such that : x P ∈ −∞ ; − 1 et PQ =1 . b Express the distance PQ as a function of x p and x Q . Justify the equality: f ( x P )= g ( x Q ) . c Determine the value of x P such that : PQ =1 . 4. With trigonometric functions E.6936 Parts A and B can be treated independently Part A Recall that the real part of a complex number z is denoted Re ( z ) . 1 Determine the exponential writing of the complex num-ber: u =1 − i . 2 Determine, for any real „ , the algebraic form and expo-nential writing of the complex number e i θ · 1 − i . 3 Deduce from the previous questions that, for any real „ : cos „ + sin „ = 2 · cos „ − ı 4 Part B In this part, we admit that, for any real „ : cos( „ ) + sin( „ ) = 2 · cos „ − ı 4 Consider the functions f and g defined on the interval 0 ; + ∞ by: f ( x )=e − x · cos x et g ( x ) = e − x We define the function h on 0 ; + ∞ by: h ( x )= g ( x ) − f ( x ) https://chingmath.fr chapExoCorrec/3209 sacados/3209 −∞02∞∞04e−20xVariationdef -3-2-12345I246810121416JO(C chapExoCorrec/6936 sacados/6936
234567I0,20,40,60,8JOCfCgCh ex0x0h2I2345JO Graphical representations of C f , C g and C h of the functions f , g and h are given, below, in an orthogonal reference frame. 1 Conjecture : a the limits of functions f and g in + ∞ . b the relative position of C f to C g ; c the value of abscissa x for which the deviation between the two curves C f and C g is maximum. 2 Justify that C g lies above C f over the interval 0 ; + ∞ 3 Show that the straight line with equation y =0 is hori-zontal asymptote to the curves C f and C g . 4 a Let h be the derivative function of the function h on the interval 0 ; + ∞ . Show that, for any x of the interval 0 ; + ∞ : h ( x ) = e − x · 2 · cos x − ı 4 − 1 b Justify that, on the interval 0 ; ı 2 : 2 · cos x − ı 4 − 1 0 and that, on the interval ı 2 ; 2 ı : 2 · cos x − ı 4 − 1 0 c Deduce the table of variations of the function h on the interval 0 ; 2 ı . 5 We admit that, on the interval 0 ; + ∞ , the function H defined by: H ( x ) = 1 2 · e − x · − 2 + cos( x ) − sin( x ) is a primitive of the function h . Note D the plane domain bounded by the curves C f and C g and the straight lines with equations x =0 and x =2 ı . Calculate the area A of the domain D , expressed in area units. 5. Framing integrals E.3232 Consider the function f , defined on 1 ; + ∞ by: f ( t ) = e t t 1 a Justify the continuity of f over 1 ; + ∞ . b Show that f is increasing on 1 ; + ∞ 2 Organized knowledge transfer Reasoning may be based on the graph provided. For any real x 0 of [1 ; + ∞ [ , we note A ( x 0 ) the area of the domain bounded by the curve representing f in an orthogonal reference frame, the x-axis and the straight lines with equations x =1 and x = x 0 . We propose to show that the function thus defined on 1 ; + ∞ is a primitive of f . a What is A (1) worth? b Let x 0 be any real from [1 ; + ∞ [ and h be a strictly positive real. Justify the following framing : f ( x 0 ) A ( x 0 + h ) − A ( x 0 ) h f ( x 0 + h ) c When x 0 > 1 , what framing can be obtained for h< 0 such that x 0 + h 1 ? d Deduce the derivability in x 0 of the function A as well as the derivative number in x 0 of the function A . e Conclude. 6. Sequences and integrals of families of functions https://chingmath.fr 234567I0,20,40,60,8JOCfCgCh chapExoCorrec/3232 sacados/3232 ex0x0h2I2345JO
011ijDC1C2C3C4C5C15C60 011ijC0C1C2C3 E.6259 Part A In the plane provided with an orthonormal reference frame, we denote by C 1 the representative curve of the function f 1 defined on R by: f 1 ( x ) = x + e − x 1 Justify that C 1 passes through the point A of coordinates (0 ; 1) . 2 Determine the table of variations of the function f 1 . We’ll specify the limits of f 1 in + ∞ and in −∞ . Part B The purpose of this part is to study the sequence I n defined on N by: I n = 1 0 x + e − n · x d x 1 In the plane provided with an orthonormal reference frame O ; −→ i ; −→ j , for any natural number n , note C n the representative curve of the function f n defined on R by: f n ( x ) = x + e − n · x On the graph below, we have drawn the curve C n for several values of the integer n and the straight line D of equation x =1 . a Geometrically interpret the integral I n . b Using this interpretation, formulate a conjecture about the direction of variation of the sequence I n and its possible limit. We will specify the elements on which we rely to conjecture. 2 Show that for any natural number n greater than or equal to 1 . I n +1 − I n = 1 0 e − ( n +1) · x · 1 − e x d x Deduce the sign of I n +1 − I n then demonstrate that the sequence I n is convergent. 3 Determine the expression of I n as a function of n and determine the limit of the sequence I n . E.3838 Let n be a natural number. We nobte f n , the function defined on the set R of real numbers by: f n ( x ) = e − n · x 1 + e − x We denote C n the representative curve of f n in an orthogonal reference frame O ; −→ i ; −→ j . The curves C 0 , C 1 , C 2 and C 3 are shown below : Part A : Some properties of functions f n and curves C n . 1 Show that for any natural number n the curves C n have a point A in common. We will specify its coordinates. 2 Study the function f 0 . a Study the direction of variation of f 0 . b Specify the limits of the function f 0 in −∞ and + ∞ . Interpret these limits graphically. c Draw up the table of variations of the function f 0 at R . 3 Study of function f 1 . a Demonstrate that f 0 ( x )= f 1 ( − x ) for any real number x . b Deduce the limits of the function f 1 in −∞ and + ∞ , as well as its direction of variation. c Give a geometric interpretation of question 3 a for the curves C 0 and C 1 . 4 Study the f n function for n 2 . a Verify that for any natural number n 2 and for any real number x , we have : f n ( x ) = 1 e nx + e ( n − 1) x b Study the limits of the function f n in −∞ and in + ∞ . c Calculate the derivative f n ( x ) and draw up the table of variations of the function f n on R . Part B: Study of a sequence related to functions f n We pose, for any natural number n : u n = 1 0 f n d x 1 Calculate u 1 then show that u 0 + u 1 =1 . Deduce u 0 . 2 Show that, for any integer n : https://chingmath.fr chapExoCorrec/6259 sacados/6259 011ijDC1C2C3C4C5C15C60 chapExoCorrec/3838 sacados/3838 011ijC0C1C2C3
xy00,20,40,60,810,20,40,60,81f1f2f3f50f200 0 u n 1 0 e − nx d x 3 Calculate integral: 1 0 e − nx d x . Deduce that the sequence u n is convergent and specify its limit. E.6258 Let n be a natural number greater than or equal to 1 . Let f n be the function defined for any real x in the interval 0 ; 1 by: f n ( x ) = 1 1 + x n For any integer n 1 , we define the number I n by: I n = 1 0 f n ( x ) d x = 1 0 1 1 + x n d x 1 The graphical representations of some functions f n ob-tained using software are plotted below. Carefully explaining your approach, conjecture, for the sequence I n the existence and possible value of the limit, when n tends to + ∞ . 2 Calculate the exact value of I 1 . 3 a Demonstrate that, for any real x in the interval 0 ; 1 and for any natural number n 1 , we have : 1 1 + x n 1 b Deduce that, for any natural number n 1 , we have I n 1 . 4 Show that, for any real x in the interval 0 ; 1 and for any natural number n 1 , we have : 1 − x n 1 1 + x n 5 Calculate the integral: 1 0 1 − x n d x . 6 Using the previous questions, show that the sequence I n is convergent and determine its limit. 7 Consider the function f below taken from an algorithm: Function f(n,p) I ← 0 . For k ranging from 0 to p − 1 x ← k p I ← I+ 1 1+x n × 1 p End To Return I a What value, rounded to the hundredth, does this function return when called with argument : n =2 ; p =5 ? Justify the answer by reproducing and completing the following table with the different values taken by the variables, when calling the function f . Values of I will be rounded to the thousandth. k x I 0 4 b Explain why the function f approximates the integral I n . E.6007 In all that follows, m denotes any real number. Part A Let f be the function defined and derivable on the set of real numbers R such that : f ( x ) = x + 1 · e x 1 Calculate the limit of f in + ∞ and −∞ . 2 Let f be the derivative function of the function f on R . Show that for any real x : f ( x ) = x + 2 · e x 3 Draw up the table of variations of f at R . Part B We define the function g m on R by: g m ( x ) = x + 1 − m · e − x and we note C m the curve of the function g m in a reference frame O ; −→ i ; −→ j du plan. 1 a Demonstrate that g m ( x )=0 if, and only if, f ( x )= m . b Deduce from A , without justification, the number of points of intersection of the curve C m with the x-axis as a function of the real m . 2 Shown below are the curves C 0 ; C e and C − e (obtained by taking for m the values of 0 , e and − e respectively) . https://chingmath.fr chapExoCorrec/6258 sacados/6258 Asie Juin 2014 xy00,20,40,60,810,20,40,60,81f1f2f3f50f200 chapExoCorrec/6007 sacados/6007 Antilles-Guyane Juin 2013
xxyy-2-12345I-2-12345JOCourbe2Courbe3Courbe1 0120,10,20,3AC Identify each of these curves on the figure, giving reasons. 3 Study the position of the curve C m with respect to the straight line D of equation y = x +1 according to the val-ues of the real m . 4 a We call D 2 the part of the plane between the curves C e , C − e , the axis ( Oy ) and the straight line x =2 . Hatch D 2 on the figure above. b In this question, a denotes a positive real, D a the part of the plane between C e , C − e , the axis ( Oy ) and the straight line Δ a of equation x = a . We denote by A ( a ) the area of this part of the plane, expressed in units of area. Show that for any positive real a : A ( a ) = 2 e − 2 · e 1 − a Deduce the limit of A ( a ) when a tends to + ∞ . E.3981 Part A Organized restitution of knowledge. The following results will be assumed to be known : e 0 = 1 ; for all real x and y : e x × e y = e x + y 1 Show that for any real x : e − x = 1 e x 2 Demonstrate that for any real x and for any natural num-ber n : e x n =e n · x Part B Consider the sequence u n defined by: u n = 1 0 e − n · x 1 + e − x d x , for any natural number n 1 a Show that : u 0 + u 1 =1 . b Calculate u 1 . Deduce u 0 . 2 Show that for any natural number n : u n 0 . 3 a Show that for any non-zero natural number n : u n +1 + u n = 1 − e − n n . b Deduce that for any non-zero natural number n : u n 1 − e − n n 4 Determine the limit of the sequence u n . 7. Suites and integrals of a function E.3994 The plane is provided with an orthogonal reference frame O ; −→ i ; −→ j . Part A The curve ( C ) , given in the appendix, is the representative curve of a function f derivable on 0 ; + ∞ , of derivative func-tion f continuous on 0 ; + ∞ The curve ( C ) passes through the points O and A 1 ; 1 2 e and, on 0 ; 1 , it is above segment [ OA ] . 1 Show that : 1 0 f ( x ) d x = 1 2 · e . 2 Show that : 1 0 f ( x ) d x 1 4 · e Part B We now know that the function f considered in part A is defined on 0 ; + ∞ by: f ( x ) = x · e − x x 2 + 1 1 Determine the limit of f in + ∞ . Interpret graphically the result obtained. 2 Consider the function g defined on 0 ; + ∞ by: g ( x ) = x 3 + x 2 + x − 1 Establish that the equation g ( x )=0 admits a unique so-lution ¸ in the interval 0 ; + ∞ . 3 a Show that for any x from 0 ; + ∞ , f ( x ) and g ( x ) are of opposite signs. b Deduce the variations f on 0 ; + ∞ . 4 Consider the sequence u n defined for any natural num-ber n by: u n = 2 n n f ( x ) d x a Show that for any x of 0 ; + ∞ : 0 x x 2 + 1 1 2 b Show that for any natural number n : 0 u n 1 2 · e − n − e − 2 · n . c Deduce the limit u n when n tends to + ∞ . https://chingmath.fr xxyy-2-12345I-2-12345JOCourbe2Courbe3Courbe1 chapExoCorrec/3981 sacados/3981 Liban Juin 2010 5 points chapExoCorrec/3994 sacados/3994 0120,10,20,3AC
23456I-0.2-0.10.10.20.30.40.50.60.70.80.91.1JO 8. Chasles sequences and relations E.3263 Consider the function f defined on R by: f ( x ) = 1 e x + e − x and we denote by Γ its representative curve in an orthogonal reference frame O ; −→ i ; −→ j Part A 1 Study the parity of f . What can be deduced for the curve Γ ? 2 Demonstrate that, for any real x positive or zero: e − x e x . 3 a Determine the limit of f in + ∞ . b Study the variations of f on 0 ; + ∞ . 4 Consider the functions g and h defined on 0 ; + ∞ by: g ( x ) = 1 e x ; h ( x ) = 1 2 · e x Below are plotted, in the O ; −→ i ; −→ j the curves repre-sentative of g and h , noted Γ 1 and Γ 2 respectively. a Demonstrate that, for any real x positive or zero: h ( x ) f ( x ) g ( x ) b What can we deduce from this for the curves Γ , Γ 1 and Γ 2 ? Plot Γ in the graph above, specifying its tangent at the point of abscissa 0. Part B Let ( I n ) be the sequence defined on N by: I n = n +1 n f ( x ) d x 1 Justify the existence of ( I n ) , and give a geometric inter-pretation of ( I n ) . 2 a Demonstrate, that for any natural number n : f ( n +1) I n f ( n ) b Deduce that the sequence ( I n ) is decreasing. c Show that the sequence ( I n ) is convergent and deter-mine its limit. Part C Let ( J n ) be the sequence defined on N by: J n = n 0 f ( x ) d x 1 Using the framing obtained in question A 4 a , show that, for any natural number n : 1 2 · 1 − e − n J n 1 − e − n 1 2 Show that the sequence ( J n ) is increasing. Deduce that it converges. 3 Let L be the limit of the sequence ( J n ) and admit the following theorem : ˇ If u n , v n and w n are three convergent sequences of lim-its a , b and c respectively and if, from a certain rank we have for all n , u n v n w n then b c ı. Give a frame for L . 4 Let u be the function defined on R by: u ( x ) = 1 1 + x 2 . Let v be the primitive of u on R such that : v (1)= ı 4 . We admit that the representative curve of v admits in + ∞ an asymptote of equation y = ı 2 . a Show that, for any real x : f ( x )= e x e x 2 +1 . b Demonstrate that, for any real x , f is the derivative of the function x ↦→ v (e x ) . c Deduce the exact value of L . E.3998 Consider the sequences u n and v n defined, for any non-zero natural number n , by: u 1 = 1 u n = u n − 1 + 1 n pour n 2 ; v n = u n − ln n for n 1 1 a Calculate u 2 , u 3 and u 4 . b Show that, for any non-zero natural number n : u n = n k =1 1 k 2 a Show that, for any non-zero natural number k : 1 k + 1 k +1 k 1 x d x 1 k b Deduce that, for any integer n greater than or equal to 2 , we have the following inequalities: u n − 1 ln n u n − 1 n ; 0 v n 1 3 a Show that, for any non-zero natural number n : v n +1 − v n = 1 n + 1 − n +1 n 1 x d x b Deduce the direction of variations of the sequence v n . 4 Show that the sequence v n converges. We note ‚ the limit of the sequence v n (no attempt will be made to calculate ‚ ) . What is the limit of the sequence u n ? https://chingmath.fr chapExoCorrec/3263 sacados/3263 23456I-0.2-0.10.10.20.30.40.50.60.70.80.91.1JO chapExoCorrec/3998 sacados/3998
9. Calculating integrals, areas and volumes E.3193 We denote by f the function defined on the set R of real numbers by: f ( x ) = 1 1 + e − x Note C the representative curve of f in an orthonormal frame O ; −→ i ; −→ j , (unité graphique: 5 cm ) Part A. Study of the function f 1 Verify that for any real number x : f ( x ) = e x 1 + e x . 2 a Determine the limits of f in −∞ and in + ∞ . Inter-pret graphically the results obtained. b Calculate f ( x ) for any real number x . Deduce the variations of f at R . c Draw up the table of variations of f . 3 Draw the curve C and any asymptotes in the reference frame O ; −→ i ; −→ j Part B. Some graphical properties. 1 Consider the points M and M of the curve C of abscis-sas x and − x respectively. Determine the coordinates of the midpoint A of the segment [ MM ] . What does the point A represent for the curve C ? 2 Let n be a natural number. We denote by D n the domain of the plane bounded by the line of equation y =1 , the curve C and the straight lines with equations x =0 and x = n , A n denotes the area of the domain D n expressed in area units. a Calculate A n . b Investigate the possible limit of A n when n tends to + ∞ . Part C. Calculation of a volume. Let – be a positive real. We denote V ( – ) the integral: V ( – ) = 0 − λ ı f ( x ) 2 d x It is accepted that V ( – ) is a measure, expressed in units of vol-ume, of the volume generated by rotation about the abscissa axis, of the portion of the curve C obtained for − – x 0 . 1 Determine the real numbers a and b such that for any real number x : e 2 x (e x + 1) 2 = a e x e x + 1 + b e x (e x + 1) 2 2 Express V ( – ) as a function of – . 3 Determine the limit of V ( – ) when – to + ∞ . 10. Modeling E.6901 A helicopter hovers over a plain. A passenger drops a parachute-equipped package vertically. Part 1 Let v 1 be the function defined on 0 ; + ∞ by: v 1 ( t ) = 5 × e 0.3 · t − 1 e 0.3 · t + 1 1 Determine the direction of variation of the function v 1 . 2 In this question, it is assumed that the parachute is oper-ating correctly. It is assumed that t seconds after it was released, the speed of the package (expressed in m · s − 1 ) is equal, before reaching the ground, to v 1 ( t ) . The package is considered to arrive safely on the ground if the arrival velocity does not exceed 6 m · s − 1 . Is there a risk of damage to the package when the parachute opens correctly? Justify. Part 2 It is assumed in this part that the parachute does not open. It is assumed that, in this case, before the parcel reaches the ground, its velocity (expressed in m · s − 1 ) , t seconds after be-ing dropped by the passenger, is given by: v 2 ( t ) = 32.7 · 1 − e − 0.3 · t 1 What is the speed, expressed in m · s − 1 , reached by the package after 10 seconds? Arrondir à 0.1 m · s − 1 . 2 Solve the equation v 2 ( t )=30 m · s − 1 . Give a concrete in-terpretation of the solution of this equation in the context of this exercise. 3 It is known that the fall of the package lasts 20 seconds. We assume that the distance, in meters, from the heli-copter to the package, T seconds after being dropped by the passenger, is given by: d ( T ) = T 0 v 2 ( t ) d t a Show that, for any real T in the interval 0 ; 20 : d ( T ) = 109 · e − 0.3 · T + 0.3 · T − 1 b Determine an approximate value to the nearest 1 m of the distance the package travels when it reaches the ground. 4 Determine a frame of magnitude 0.1 s of the time taken for the package to reach the ground if it had been dropped from a height of 700 meters. https://chingmath.fr chapExoCorrec/3193 sacados/3193 chapExoCorrec/6901 sacados/6901
vantaildedroitevantaildegauchepilierdroitpiliergauche 00.511.522.50.511.5La distance entre le bas du portail et le sol est de0;05m E.6266 We want to build a gate as shown below. Each leaf measures 2 meters wide. Part A : modeling the upper part of the gate The upper edge of the right gate leaf is modeled with a func-tion f defined on the interval 0 ; 2] by: f ( x ) = x + 1 4 · e − 4 x + b où b is a real number. Let f be the derivative function of the function f on the interval 0 ; 2 . 1 a Calculate f ( x ) , for any real x belonging to the in-terval 0 ; 2 . b Deduce the direction of variation of the function f on the interval 0 ; 2 . 2 Determine the number b so that the maximum gate height equals 1.5 m . In the following, the function f is defined on the interval 0 ; 2 by: f ( x ) = x + 1 4 · e − 4 x + 5 4 Part B: determining an area Each leaf is made from a metal plate. We want to calculate the area of each of the plates, knowing that the bottom edge of the leaf is 0.05 m height from the floor. 1 Show that the function F defined on the interval 0 ; 2 by: F ( x ) = − x 4 − 1 8 · e − 4 x + 5 4 · x is a primitive of the function f . 2 Deduce the area in m 2 of each leaf. The exact value and then an approximate value to the nearest 10 − 2 of this area will be given. (The focus here is on the object ˇ vantail ı without reference to its surroundings) . Part C : using an algorithm We wish to make a gate of the same shape but from disjointed rectangular boards of width 0.12 m , spaced 0.05 m . For the right-hand leaf, the top left corner of each board is located on the top edge of the leaf and the bottom of each board at 0.05 m in height. The boards are numbered from 0 : thus the first board on the left carries the number 0 . 1 Give the area of board number k . 2 Copy and complete the algorithm, below, so that at the end of execution, the value of the variable S corresponds to the sum of the areas of the boards in the right-hand leaf. S ← 0 X ← 0 As long as X+0.17<... S ← S+... X ← X+0.17 End As long as https://chingmath.fr chapExoCorrec/6266 sacados/6266 vantaildedroitevantaildegauchepilierdroitpiliergauche 00.511.522.50.511.5La distance entre le bas du portail et le sol est de0;05m
DI2JOC A1A2aI2JOC rstABCDEG rstABCDEG E.6008 Consider the function g defined by: g ( x ) = 1 + e − x for any real x in the interval 0 ; 1 . We assume that : g ( x ) > 0 , for any real x in the interval 0 ; 1 . Note C the representative curve of the function g in an orthonor-mal reference frame, and D the plane area between the x-axis and the curve C , on the other hand, between the equa-tion lines x =0 and x =1 . The C curve and D domain are shown opposite. The aim of this exercise is to divide the domain D into two domains of equal area, first by a straight line parallel to the ordinate axis (part A ) , then by a straight line parallel to the abscissa axis (part B ) . Part A Let a be a real such that 0 a 1 . We note A 1 the area of the domain between the curve C , the axis Ox , the straight lines with equations x =0 and x = a , then A 2 that of the domain be-tween the curve C , Ox and the equation lines x = a and x = 1 . A 1 and A 2 are expressed in units of area. 1 a Demonstrate that : A 1 = a − e − a + 1 . b Express A 2 as a function of a . 2 Let f be the function defined for any real x in the inter-val 0 ; 1 by: f ( x ) = 2 x − 2 · e − x + 1 e a Draw up the table of variations of the function f on the interval 0 ; 1 . The exact values of f (0) and f (1) will be specified. b Demonstrate that the function f cancels once and only once on the interval 0 ; 1 in a real ¸ . Give the value of ¸ rounded to the hundredth. 3 Using the previous questions, determine an approximate value of the real a for which the areas A 1 and A 2 are equal. Part B Let b be a positive real. In this part, we propose to divide the domain D into two do-mains of equal area by the straight line of equation y = b . We admit that there exists a single positive real b solution. 1 Justify the inequality b< 1+ 1 e . A graphical argument may be used. 2 Determine the exact value of the real b . E.6932 Parts A and B are independent The manufacturer of padlocks under the brand name ˇ K ‘’ wants to print a logo for its company. This logo is in the form of a stylized capital letter K , inscribed in a square ABCD with sides of length one, and satisfying the following conditions C 1 and C 2 : Condition C1 : the letter K must ficonsist of three lines : one of the lines is the segment [ AD ] ; a second line has as its endpoints the point A and a point E on the segment [ DC ] ; the third line has as its endpoints the point B and the point G located on the second line. Condition C2 : the area of each of the three surfaces de-limited by the three lines drawn in the square must fi be between 0.3 and 0.4 , the unit of area being that of the square. These areas are denoted r , s , t in the figures below. A design workshop proposes two possible designs, shown be-low : To carry out the following studies, we use the orthonormal coordinate system A ; −−→ AB ; −−→ AD . Proposal A Proposal B Part A : study of proposal A In this proposal, the three lines are segments and the three areas are equal: r = s = t = 1 3 Determine the coordinates of points E and G . Part B: Study of proposal B This proposal is characterized by the following two condi-tions : the line segment A and E is a portion of the graph of the function f defined for all real numbers x 0 by: f ( x )=ln 2 x +1 ; the line of endpoints B and G is a portion of the graph-ical representation of the function g defined for all real numbers x> 0 by: g ( x ) = k 1 − x x où k is a positive real number that will be determined. 1 a Determine the x-coordinate of point E . b Determine the value of the real number k , given that the x-coordinate of point G is equal to 0.5 . 2 a Show that the function f has as its primitive the function F defined for all real numbers x 0 by: F ( x ) = x + 0.5 · ln 2 · x +1 − x https://chingmath.fr chapExoCorrec/6008 sacados/6008 DI2JOC A1A2aI2JOC chapExoCorrec/6932 sacados/6932 rstABCDEG rstABCDEG
BCDABCDIJO BCDIJOCf BCDABCDB1B1B2B2BkBkBkBkIJO b Prove that : r = e 2 − 1 3 Determine a primitive G of the function g on the interval 0 ; + ∞ 4 We assume that the previous results allow us to establish that : s = ln(2) 2 + ln(2) − 1 2 Does the proposition B meet the conditions imposed by the manufacturer? E.6935 A municipality has decided to in-stall a skateboard module in a local park. The drawing opposite provides a cavalier perspective. The quadrilaterals OAD D , DD C C and OAB B are rectangles. The face plane ( OBD ) is provided with an orthonormal ref-erence frame O ; I ; J . The unit is the meter. The module’s width is 10 meters, in other words, DD =10 , its length OD is 20 meters. The aim of the problem is to determine the area of the various surfaces to be painted. The profile of the skateboard module was modeled from a photo by the function f defined on the interval 0 ; 20 by: f ( x ) = ( x + 1) · ln x +1 − 3 x + 7 We denote f the derivative function of the function f and C the representative curve of the function f in the reference frame O ; I ; J . Part 1 1 Show that for any real x belonging to the interval 0 ; 20 , we have : f ( x ) = ln x +1 − 2 2 Deduce the variations of f on the interval 0 ; 20 and draw up its table of variation. 3 Calculate the slope of the tangent to the curve C at the point of abscissa 0 . The absolute value of this coefficient is called the inclination of the skateboard module at point B . 4 We admit that the function g defined on the interval 0 ; 20 by: g ( x ) = 1 2 · x + 1 2 · ln x +1 − 1 4 · x 2 − 1 2 · x has as derivative the function g d efined on the interval 0 ; 20 by: g ( x )=( x +1) · ln x +1 . Determine a primitive of the function f on the interval 0 ; 20 . Part 2 The three questions in this part are independent 1 Are the following statements correct? Justify the an-swers. P 1 : the difference in height between the highest and lowest points of the runway is at least 8 meters. P 2 : The inclination of the runway is almost twice as great at B as at C . 2 We want to cover the four sides of this module with a coat of red paint. The paint used covers an area of 5 m 2 per liter. Determine, to the nearest 1 liter, the minimum number of liters of paint required. 3 The rolling track, i.e. the upper surface of the module, is to be painted black. In order to determine an approximate value for the area of the part to be painted, consider in the reference frame O ; I ; J of the face plane, the points B k ( k ; f ( k )) for k varying from 0 to 20 . Thus : B 0 = B . We decide to approximate the arc of the curve C from B k to B k +1 by the segment B k B k +1 . Thus, the area of the surface to be painted will be ap-proximated by the sum of the areas of rectangles of the type B k B k +1 B k +1 B k (see figure) a Show that for any integer k ranging from 0 to 19 : B k B k +1 = 1 + f ( k +1) − f ( k ) 2 b Complete the algorithm so that we can recover from the values of its variables an estimate of the area of the rolling part. Function f(x) Renvoyer (x + 1) ln x+1 − 3x + 7 S ← 0 For K varying from ... to ... S ← ... End For Which variable will contain an estimate of the area of https://chingmath.fr chapExoCorrec/6935 sacados/6935 BCDABCDIJO BCDIJOCf BCDABCDB1B1B2B2BkBkBkBkIJO
0;600;800;400;20Image A0;360;640;160;04Image B0;770;890;630;45Image C 00.510.51Cf1 the rolling part? E.6267 A black-and-white digital image is made up of small squares (pixels) ranging in color from white through all shades of gray to black. Each shade is coded by a real x as follows : x =0 for white ; x =1 for black; x =0.01 ; x =0.02 and so on up to x =0.99 in steps of 0.01 for all intermediate shades (from light to dark) . The image A , below, is composed of four pixels and gives a sample of these shades with their codes. Image retouching software uses digital functions known as ˇ re-touching functions ı. A function f defined on the interval 0 ; 1 is said to be ˇ func-tion of retouche ı if it has the following four properties : f (0)=0 ; f (1)=1 ; f is continuous on the interval 0 ; 1 ; f is increasing on the interval 0 ; 1 . A shade coded x is said to be darkened by the function f if f ( x ) >x , and lightened, if f ( x ) <x . if f ( x )= x 2 , a pixel of coded shade 0.2 will take on the coded shade 0.2 2 =0.04 . Image A will be transformed into image B below. If f ( x )= x , the coded shade 0.2 will take on the coded shade 0.2 ≈ 0.45 . The image A will be transformed into the image C below. Part A 1 Consider the function f 1 defined on the interval 0 ; 1 by: f 1 ( x ) = 4 x 3 − 6 x 2 + 3 x a Demonstrate that the function f 1 is a touch-up func-tion. b Graphically solve the inequation f 1 ( x ) x , using the graph given below, showing the useful dotted lines. Interpret this result in terms of lightening or darken-ing. 2 Consider the function f 2 defined on the interval 0 ; 1 by: f 2 ( x ) = ln 1 + ( e − 1) · x We admit that f 2 is a touch-up function. We define on the interval 0 ; 1 the function g by: g ( x ) = f 2 ( x ) − x . a Establish that, for any x in the interval 0 ; 1 : g ( x ) = ( e − 2) − ( e − 1) · x 1 + ( e − 1) · x b Determine the variations of the function g on the in-terval 0 ; 1 . Show that the function g admits a maxi-mum at e − 2 e − 1 , a maximum whose value rounded to the hundredth is 0.12 . c Establish that the equation g ( x )=0.05 admits on the interval 0 ; 1 two solutions ¸ and ˛ , with ¸<˛ . We’ll assume that : 0.08 <¸ < 0.09 ; 0.85 <˛ < 0.86 Part B Note that a grade change is only visually noticeable if the ab-solute value of the difference between the original grade code and the changed grade code is greater than or equal to 0.05 . 1 In the algorithm described below, the function g takes as arguments the variables x (initial shade) , y (retouched shade) , E (deviation) and uses the function f which des-ignates a retouch function. In the function code g , the variable c serves as a counter. Function g(x,y,E) c ← 0 For k from 0 to 100 x ← k 100 y ← f(x) E ← | y − x | If E 0.05 Then c ← c+1 End if End for https://chingmath.fr chapExoCorrec/6267 sacados/6267 0;600;800;400;20Image A0;360;640;160;04Image B0;770;890;630;45Image C 00.510.51Cf1
00.510.51 2cm5cm CfTerrainCuveTerrainDBAT012345612I Return c When calling the function g , this algorithm returns the value of the variable c . What does this algorithm do? 2 What value will be returned by the function g if applied to the function f 2 defined in the second question of the A part? Part C In this section, we’re interested in retouching functions f whose ef-fect is to brighten the image as a whole, i.e. such that, for any real x in the interval 0 ; 1 , f ( x ) x . We decide to measure the overall brightening of the image by calcu-lating the area A f of the portion of the plane between the x-axis, the curve representing the function f , and the straight lines with respective equations x =0 and x =1 . Between two functions, which will have the effect of brighten-ing the image the most the one corresponds to the smallest area. We’d like to compare the effect of the following two functions, which we assume to be retouching functions : f 3 ( x ) = x · e ( x 2 − 1) ; f 4 ( x ) = 4 x − 15 + 60 x + 4 1 a Calculer A f 3 . b Calculer A f 4 . 2 Which of these two functions has the effect of brightening the image the most? E.6939 A private individual wants to have a water recuperator manufactured. This water recuperator is a tank that must comply with the following specifications : it must be located two meters from his house ; the maximum depth must be two meters ; it must be five meters long; it must follow the natural slope of the land. This tank is shown in the diagram opposite. The curved part is modeled by the curve C f of the function f on the interval 2 ; 2 e defined by: f ( x ) = x · ln x 2 − x + 2 The curve C f is shown below in an orthonormal reference frame of unit 1 m and is a profile view of the tank. Consider the points A (2 ; 2) , I (2 ; 0) and B (2 e ; 2) . Part A The aim of this part is to evaluate the volume of the tank. 1 Justify that the points B and I belong to the curve C f and that the x-axis is tangent to the curve C f at point I . 2 Note T the tangent to the curve C f at point B , and D the point of intersection of the line T with the x-axis. a Determine an equation of the line T and deduce the coordinates of D . b The area of the domain bounded by the curve C f is called S , the straight lines with equations y =2 , x =2 and x =2 e . S can be framed by the area of the triangle ABI and that of the trapezoid AIDB . What framework for the volume of the tank can be deduced from this? 3 a Show that, on the interval 2 ; 2 e , the function G defined by: https://chingmath.fr 00.510.51 chapExoCorrec/6939 sacados/6939 2cm5cm CfTerrainCuveTerrainDBAT012345612I
xf(x01234561234 G ( x ) = x 2 2 · ln x 2 − x 2 4 is a primitive of the function g defined by: g ( x )= x · ln x 2 b Deduce a primitive F of the function f on the interval 2 ; 2 e . c Determine the exact value of the area S and deduce an approximate value of the volume V of the tank to the nearest m 3 . Part B For any real x between 2 and 2 e , let v ( x ) denote the volume of water, expressed as m 3 , being in the tank when the height of water in the tank is equal to f ( x ) . We admit that, for any real x of the interval 2 ; 2 e . v ( x ) = 5 · x 2 2 · ln x 2 − 2 x · ln x 2 − x 2 4 + 2 x − 3 1 What volume of water, to the nearest m 3 , is there in the tank when the height of water in the tank is one meter? 2 Recall that V is the total volume of the tank, f is the function defined at the beginning of the exercise and v is the function defined in part B . Consider the algorithm shown opposite and be interested in the value of the variable d at the end of its execution. Interpret the result that this algorithm assigns, at the end of execution, to the variable d a ← 2 b ← 2e As long as v(b) − v(a)>10 − 3 c ← a+b 2 Si v(c)< V 2 Then a ← c Otherwise b ← c End If End as long as d ← f(c) https://chingmath.fr xf(x01234561234