Grade 12 / Integral calculation 56 exercises (including 55 corrected)

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IJOfx−x2CfIJOfx−x2CfCf15R1Cf25R2Cf35R3Cf45R4 IJOfx↦−x2CfIJOfx↦−x2CfCfCfCfCfCfCfCfCfCfCfCfCfCfCf1n ijijCfS(x0x0 ijijCfx0x0h A1A3ijA2 1. Introduction E.3921 Consider the square function, de-noted f and its representative curve C f in the reference frame O ; I ; J . Rectangles are constructed to ˇ fill ı the area between the curve C f and the x-axis. The two representations below divide the interval 0 ; 1 into n equal parts forming rectangles of equal width : for the figure on the left n =5 ; the figure on the right represents any figure. Part A : n =5 Note A 5 the shaded area under the curve. 1 Justify equality: A 5 = 4 k =0 1 5 · k 5 2 2 Establish equality: A 5 = 6 25 Part B: in the general case n is a natural number greater than or equal to 2 . We denote A n the hatched area under the curve C f when the segment [0 ; 1] is divided into n equal parts. This area is as-sumed to have the value : A n = n − 1 k =1 1 n · k n 2 1 Using reasoning by recurrence, establish the following equality for any natural number n such that n 2 : n − 1 k =1 k 2 = ( n − 1) · n · (2 n − 1) 6 2 Deduce the following equality: A n = ( n − 1) · n · (2 n − 1) 6 · n 3 3 a Determine the value of the limit: lim n ↦→ + ∞ 2 n 3 − 3 n 2 + n 6 n 3 b Deduce the measure of the area included : vertically: between the two straight lines with equa-tions x =0 and x =1 . horizontally: between the curve C f and the straight line with equation y =0 . E.3922 Let f be a continuous, strictly increas-ing function on R + . Note C f its representative curve in an orthonormal reference frame. For a ∈ R + , note S ( a ) the area comprised : vertically: between the straight lines of equations : x =0 and x = a ; horizontally: between the curve C f and the straight line of equation y =0 . The figure on the left shows the image of x 0 by the function S : We wish to determine the derivative function of the function S . To do this, consider the real number h où h> 0 and the figure on the right above. 1 This representation highlights the following three areas : Which of these areas represents the area defined by: S ( x 0 + h ) − S ( x 0 ) 2 a Compare the three areas A 1 , A 2 and A 3 . b Give a frame for the difference below, using the func-tion f , x 0 and h : S ( x 0 + h ) − S ( x 0 ) 3 Deduce the following limit: lim h ↦→ 0 + S ( x 0 + h ) − S ( x 0 ) h = f x 0 https://chingmath.fr chapExoCorrec/3921 sacados/3921 IJOfx−x2CfIJOfx−x2CfCf15R1Cf25R2Cf35R3Cf45R4 IJOfx↦−x2CfIJOfx↦−x2CfCfCfCfCfCfCfCfCfCfCfCfCfCfCf1n chapExoCorrec/3922 sacados/3922 ijijCfS(x0x0 ijijCfx0x0h A1A3ijA2
IJOC ijC1d1 ijC2d2 ijC3d3 IJOFigure1CfIJOFigure1CfCf15R1Cf25R2Cf35R3Cf45R4 IJOFigure2CfIJOFigure2CfCfCfCfCfCfCfCfCfCfCfCfCfCfCf1n 123456ABCDEFGn=6012345A= E.6010 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 . 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 function verifying the following condi-tion : F = f (i.e. ∀ x ∈D f ; F ( x )= f ( x ) ) 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. 2. First manipulations of primitives E.5231 Let f be a strictly positive function on the interval a ; b . Consider the function F defined on a ; b by the relation: F ( x ) = x a f ( t ) d t 1 Draw up the table of variations of the function F on the interval a ; b . 2 Justify the existence of a single real x 0 verifying: F ( x 0 ) = 1 2 · F ( b ) E.6756 Consider the square function, denoted f and its representative curve C f in the reference frame O ; I ; J . Rectangles are constructed to ˇ fill ı the area between the curve C f and the x-axis. The two representations below divide the interval 0 ; 1 into n equal parts forming rectangles of equal width : for the figure on the left n =5 ; the figure on the right represents any figure. Part A : n =5 In figure 1 , we note A 5 the area of the shaded part. 1 Justify equality: A 5 = 4 k =0 1 5 · k 5 2 2 Establish equality: A 5 = 6 25 Part B: with OpenCal 1 a In a new worksheet, enter the following values : b Enter in cell A4 the formula : =1/$B$1*(A3/$B$1)^2 Extend this formula over the range A4:F4 . c Write a formula in cell B6 giving the sum of the values present in row 4 . d Justify that this value is the approximate value of A 6 . 2 Modify your spreadsheet to calculate A 7 . 3 Similarly to obtain an approximate value of A 50 . 3. Area calculation https://chingmath.fr chapExoCorrec/6010 sacados/6010 IJOC ijC1d1 ijC2d2 ijC3d3 chapExoCorrec/5231 sacados/5231 chapExoCorrec/6756 sacados/6756 IJOFigure1CfIJOFigure1CfCf15R1Cf25R2Cf35R3Cf45R4 IJOFigure2CfIJOFigure2CfCfCfCfCfCfCfCfCfCfCfCfCfCfCf1n 123456ABCDEFGn=6012345A=
Cf234I234JO 00,20,40,60,811,20,20,40,60,8Cf 1x0x15 0120,10,20,3AC E.3929 Consider the function f , de-fined on 0 ; 4 , whose graph-ical representation is given be-low : Determine a frame for theıinteger: 4 0 f ( x ) d x E.6922 The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . Consider a function f defined on R and whose curve C f is shown below : Using the graph and explaining the procedure, propose a frame with an amplitude of 0.05 for the area of the domain bounded by: the straight lines with equations x =0 and x =1 ; the curve C f and the x-axis E.5224 Consider the function f defined by the relation: f ( x ) = 1 2 · x 2 − 3 x + 4 1 Determine the value of the integral: 5 1 f ( x ) d x . 2 Below is given the curve C f representative of the function f in a O ; I ; J orthonormal reference frame : We wish to determine the value of the area of the part shown in grey. a Determine the zeros of the function f which will be denoted x 0 and x 1 such that x 0 <x 1 . b Determine the value of : x 0 1 f ( x ) d x + 5 x 1 f ( x ) d x c Determine the value of : x 1 x 0 f ( x ) d x d Deduct the area, in area units, of the shaded part. E.3975 Let f and g be two functions defined and continuous on the interval 0 ; 1 . Say whether the following proposition is correct or not. Jus-tify your answer. If 1 0 f ( x ) d x = 1 0 g ( x ) d x then f = g on the interval [0 ; 1] . E.3993 The plane is provided with an orthogonal reference frame O ; −→ i ; −→ j . 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 the 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 https://chingmath.fr chapExoCorrec/3929 sacados/3929 Cf234I234JO chapExoCorrec/6922 sacados/6922 Extrait d'Antilles-Guyane Septembre 2016 00,20,40,60,811,20,20,40,60,8Cf chapExoCorrec/5224 sacados/5224 1x0x15 chapExoCorrec/3975 sacados/3975 chapExoCorrec/3993 sacados/3993 0120,10,20,3AC
Cf-123456789I-4-3-2-12JO E.3974 The plane is given the reference frame O ; −→ i ; −→ j orthonormal with graphic unit 3 cm . Consider the function f defined on 1 ; e by: f ( x ) = ln x x Note C the representative curve of the function f . 1 Show that : e 1 f ( x ) d x = 1 2 . 2 Deduce the area of the region of the plane bounded by the straight lines of equation x =1 and x = e , the x-axis and the curve C . We’ll express this area in cm 2 . E.5238 Consider the function f defined on R by the relation: f ( x ) = 1 2 x − 2 − 1 4 · ⏐ ⏐ x − 4 ⏐ ⏐ Below is given the curve C f representative of the function f in the O ; I ; J orthonormal reference frame : We wish to determine the area of the hatched part. 1 Simplify the expression of the function f on each of the intervals 0 ; 4 and 4 ; 8 . 2 Determine the area of the shaded surface. 4. Calculating integrals E.3951 Calculate the following integrals : a 2 − 3 x + 1 d x b 5 0 2 x − 5 2 d x c 1 − 3 1 − x 3 d x d 4 1 x 2 · x 2 + 1 2 d x e 6 4 2 · x x 2 − 3 d x f 3 1 1 x 2 − 1 x d x E.3952 Calculate the following integrals : a 5 4 ln x x d x b 4 − 2 1 x + 3 d x c 3 − 1 1 x + 4 d x d 1 − 1 x 2 · 2 · x 3 + 2 2 d x E.3995 Consider the function g defined on the interval 0 ; + ∞ by: g ( x ) = x 2 · e − x 1 Let H be the function defined on the interval 0 ; + ∞ by: H ( x ) = − x 2 + 2 · x · e − x Calculate the derivative H of the function H . 2 Deduce a primitive on the interval 0 ; + ∞ of the func-tion g . 3 Deduce the value of the integral: 1 0 g ( x ) d x E.4321 Let h be the function defined on 0 ; + ∞ by: h ( x ) = x · ln x − x 1 Show that the function h is a primitive of the natural logarithm function on 0 ; + ∞ . 2 Calculate the exact value, then an approximate value to 10 − 2 near, of the area bounded by: the x-axis and the curve C representing the natural logarithm function ; the lines with equations : x =1 ; x =2 https://chingmath.fr chapExoCorrec/3974 sacados/3974 chapExoCorrec/5238 sacados/5238 Cf-123456789I-4-3-2-12JO chapExoCorrec/3951 sacados/3951 chapExoCorrec/3952 sacados/3952 chapExoCorrec/3995 sacados/3995 chapExoCorrec/4321 sacados/4321 Extrait des Antilles Juin 2011
IJOCf ijC -123I-123JOCf E.6011 Consider the function f defined on R by: f ( x ) = ( x + 2) · e 1 2 x Note C its representative curve in the plane provided with a reference frame O ; −→ i ; −→ j . We pose : I = 1 0 f ( x ) d x . 1 Geometrically interpret the real I . 2 Let u and v be the functions defined on R by: u ( x ) = x ; v ( x ) = e 1 2 x Check that : f = 2 · u · v + u · v . 3 Deduce the exact value of the integral I . E.6012 Consider the function f defined on 0 ; + ∞ by the relation: f ( x ) = 2 x + 3 · e x x + 4 2 Consider the number I defined by: I = 2 0 f ( x ) d x . The curve C representative of the function f is given in the reference frame below : 1 Hatch on the above represen-tation a domain of the plane with an area of I . Justify your approach. 2 a Consider the two functions u and v defined by: u ( x ) = 2 · e x ; v ( x ) = x + 4 Prove that we have the following relationship on 0 ; + ∞ : f = u · v − u · v v 2 b Deduce the exact value of the number I . E.3979 Let n be a non-zero natural num-ber, we define the function f n by: f n ( x ) = 4 · e n · x e n · x + 7 1 For n a non-zero natural number, determine a primitive of the function f n . 2 Let u n be the sequence defined for any non-zero natu-ral number n by: u n = n ln 7 · ln 7 n 0 f n ( x ) d x . Show that the sequence u n is constant. E.3967 1 Let f be the function f defined on D = − 1 ; + ∞ by the relation: f ( x ) = x 2 x + 1 The representative curve C f of the function f is given in the reference frame O ; I ; J , we have : a Determine the value of the real numbers a , b and c verifying the following equality for any x ∈D : f ( x ) = a · x + b + c x + 1 b Determine the expression of a primitive of the function f . c Calculate, in area units, the area A of the part of the plane bounded by the curve ( C f ) and the lines of equa-tions : x = − 3 4 ; x = 5 2 ; y = 0 d Knowing that this representation is made with the scale : 1 unit = 1.5 cm Give the area A in cm 2 rounded to the unit. 2 Calculate the following integrals : a 3 − 1 x · e x 2 +1 d x b 2 0 x · e x 2 e x 2 + 1 2 d x E.5530 Consider the function f defined on R by: f ( x )= 1 1+e − x 1 Show that, for any real x : f ( x )= e x 1+e x 2 We define the number: I = 1 0 f ( x ) d x . Show that : I =ln 1+ e 2 Give a graphical interpretation of I . https://chingmath.fr chapExoCorrec/6011 sacados/6011 IJOCf chapExoCorrec/6012 sacados/6012 ijC chapExoCorrec/3979 sacados/3979 chapExoCorrec/3967 sacados/3967 -123I-123JOCf chapExoCorrec/5530 sacados/5530 Extrait du Liban Mai 2013
234I-2-1234JOeC -1234I-1234JO -2-123I2JOCfCg 5. Linearity of the integral E.5233 Consider the following two integrals : I = 1 0 e x e x + 1 d x ; J = 1 0 1 e x + 1 d x 1 Justify the equality: I + J =1 . 2 a Determine the value of the integral I . b Deduce the value of the integral J . E.3980 Consider the sequence u n de-fined for any natural number n by: u n = 1 0 e − n · x 1 + e − x d x 1 Show that : u 0 + u 1 =1 2 Show that, for any non-zero natural number n : u n +1 + u n = 1 − e − n n E.4293 Consider the plane provided with an orthonormal reference frame O ; I ; J whose unit mea-sures 2 cm . Let f be the function defined on R ∗ + by the rela- tion : f ( x ) = x + ln x x Below is given the curve C representative of the function f . 1 Show that : e 1 ln x x d x = 1 2 2 Deduct the area of the region of the plane bounded by: the straight lines of equation x =1 and x = e ; the x-axis and the curve C . We’ll express this area in cm 2 . Hatch this region on the graph. 6. Chasles relationship E.5234 Consider the integer function E , which returns the integer part of any real number. 1 In the reference frame below, plot the representative curve of the function E on the interval − 1 ; 4 . 2 Determine the measure of the integral: 4 0 E ( x ) d x E.125 Consider the two functions f and g de-fined by: f ( x ) = e x − 1 ; g ( x ) = e − x In the reference frame O ; I ; J orthonormal, note C f and C g the respective representative curves of the functions f and g . The shaded area above is defined by: it lies between the straight lines of equations x = − 1 and x =2 . it lies above the x-axis. it is located below the two curves C f and C g . Determine the area of this domain. 7. Positivity of the integral E.3978 Let f be a function defined on R and admitting the following table of variations : https://chingmath.fr chapExoCorrec/5233 sacados/5233 chapExoCorrec/3980 sacados/3980 Extrait de Liban Juin 2010 chapExoCorrec/4293 sacados/4293 Extrait d'Antilles-guyane Septembre 2010 234I-2-1234JOeC chapExoCorrec/5234 sacados/5234 -1234I-1234JO chapExoCorrec/125 sacados/125 -2-123I2JOCfCg chapExoCorrec/3978 sacados/3978
−∞0e∞−∞0ln50xVariationdef Determine, if possible, the sign of the following integrals : a 3 1 f ( x ) d x b 0 − 2 f ( x ) d x c 1 − 1 f ( x ) d x d e 3 f ( x ) d x e ln 2 ln 1 2 f ( x ) d x f e 1 2 1 f ( x ) d x E.6018 For any natural number n , consider the function f n defined on 0 ; 1 by the relation: f n ( x ) = x n 1 + x We define the sequence u n of real numbers by: u n = 1 0 f n ( x ) d x for all n ∈ N 1 Establish, for any natural number n , the comparison : u n 1 0 x n d x 2 Deduce that the sequence u n is convergent and con-verges to 0 . E.3991 Let n be a natural number. Let f n be the function defined on the set R of real numbers by: f n ( x ) = e − n · x 1 + e − x We pose, for any natural number n : u n = 1 0 f n ( x ) d x 1 Calculate u 1 then show that u 0 + u 1 =1 . Deduce u 0 . 2 Show that, for any integer n : 0 u n 1 0 e − n · x d x 3 Calculate the integral: 1 0 e − n · x d x Deduce that the sequence u n is convergent and specify its limit. E.3977 Consider the function f defined on the interval 0 ; 1 by the relation: f ( x ) = x n · x pour x ∈ 0 ; 1 Consider the sequence u n defined, for n ∈ N , by: u n = 1 0 x n · x d x 1 Justify that all terms in the sequence are positive. 2 By studying the sign of the function : x ↦−→ x n +1 · x − x n · x , Show that the sequence u n is decreasing. 3 Deduce that the sequence u n is convergent. E.5389 Consider the sequence I n de-fined for n non-zero natural number by: I n = 1 0 x n · e x 2 d x 1 a Let g be the function defined on R by: g ( x ) = x · e x 2 . Show that the function G defined on R by: G ( x ) = 1 2 · e x 2 is a primitive on R of the function g . b Deduce the value of I 1 . 2 We admit the following relationship for any integer n greater than or equal to 1 : I n +2 = 1 2 e − n + 1 2 · I n The following algorithm is considered : n ← 1 u ← 1 2 e − 1 2 Tant que n<21 u ← 1 2 e − n+1 2 · u n ← n+2 End As long as At the end of the algorithm, which term of the sequence I n is assigned to the variable u ? 3 a Show that, for any non-zero natural number n : I n 0 . b Show that ( I n ) is decreasing. c Deduce that the sequence I n is convergent. We note I its limit. 4 In this question, any trace of research, however incom-plete, or initiative, however unsuccessful, will be taken into account in the assessment. Determine I . 8. Positivity of the integral and variable bounds E.3973 Consider the sequence u n n ∈ N defined, for any natural number n , by the relation: u n = n 0 x · e − x d x Justify that the sequence u n is increasing. https://chingmath.fr −∞0e∞−∞0ln50xVariationdef chapExoCorrec/6018 sacados/6018 chapExoCorrec/3991 sacados/3991 Extrait de Centres Etrangers Juin 2009 chapExoCorrec/3977 sacados/3977 chapExoCorrec/5389 sacados/5389 chapExoCorrec/3973 sacados/3973
23IJO E.3989 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = x · e − x 2 The representative curve of the function f is given below : We admit that the function f is strictly decreasing on 2 2 ; + ∞ . Consider the sequence u n defined for any nat-ural number n by: u n = n +1 n f ( x ) d x No attempt will be made to explain u n 1 Demonstrate that, for any natural number n other than 0 and 1 , we have : f ( n +1) u n f ( n ) 2 What is the direction of variation of the u n n 2 ? 3 Show that the sequence u n converges. What is its limit? E.6915 Let f be the function defined and derivative on the interval 0 ; + ∞ such that : f ( x )= x e x − x Let the sequence I n be defined for any natural number n by: I n = n 0 f ( x ) d x No attempt will be made to calculate the exact value of I n as a function of n . 1 Show that the sequence I n is increasing. 2 We admit that for any real x of the interval 0 ; + ∞ : e x − x e x 2 a Show that, for any natural number n : I n n 0 2 · x · e − x d x b Let H be the function defined and derivable on the interval 0 ; + ∞ such that : H ( x )= − x − 1 · e − x . Determine the derivative function H of the function H . c Deduce that, for any natural number n : I n 2 . 3 Show that the sequence I n is convergent. The value of its limit is not asked. 9. Integrals and study of functions E.6920 Let a be a real number between 0 and 1 . Let f a be the function defined on R by: f a ( x ) = a · e a · x + a Let I ( a ) be the integral of the function f a between 0 and 1 : I ( a ) = 1 0 f a ( x ) d x Is there a value of a for which I ( a ) is equal to 2 ? If so, give a frame of amplitude 10 − 2 . E.3976 We want to bound the integral: I = 1 0 e x 1+ x d x We define the function f on the interval 0 ; 1 by: f ( x ) = e x 1 + x 1 Study the variations of f on 0 ; 1 . 2 For any integer n between 0 and 5 , we set : S n = n k =0 f k 5 . a Justify that for any integer k between 0 and 4 , we have : 1 5 · f k 5 k +1 5 k 5 e x 1 + x d x 1 5 · f k +1 5 Interpret the previous inequalities graphically using rectangles. b Deduce that : 1 5 · S 4 1 0 e x 1 + x d x · 1 5 · S 5 − 1 c Give approximate values to 10 − 4 near S 4 and S 5 re-spectively. Deduce the range : 1 ; 092 1 0 e x 1 + x d x 1 ; 164 https://chingmath.fr chapExoCorrec/3989 sacados/3989 23IJO chapExoCorrec/6915 sacados/6915 chapExoCorrec/6920 sacados/6920 Extrait d'Asie Juin 2016 chapExoCorrec/3976 sacados/3976
x01y0.20.4C1C2C3C10C20C30 0.20.40.60.81.2I0.20.40.60.8JOCf0Cf1Cf2Cf3 E.5237 Part A Denote by f the function defined on the interval 1 ; + ∞ by: f ( x ) = 1 x + 1 + ln x x + 1 1 Determine the limit of the function f in + ∞ . 2 Show that for any real x in the interval: f ( x ) = 1 x ( x + 1) 2 Draw up the table of variations of the function f . 3 Deduce the sign of the function f on the interval 1 ; + ∞ . Part B Consider the sequence u n defined for n ∈ N ∗ defined by: u n = 1 + 1 2 + 1 3 + · · · + 1 n − ln n 1 Show that for any strictly positive integer n : u n +1 − u n = f ( n ) où f is the function defined in part A . Deduce the direction of variation of the sequence u n . 2 a Let k be a strictly positive integer. Justify the in-equality: k +1 k 1 k − 1 x d x 0 Deduce that : k +1 k 1 x d x 1 k Prove the inequality: ln( k +1) − ln k 1 k b Write the previous inequality by successively replacing k by 1 , 2 , . . . , n and show that for any strictly positive integer n : ln( n +1) 1 + 1 2 + 1 3 + · · · + 1 n c Deduce that for any strictly positive integer n : u n 0 3 Prove that the sequence u n is convergent. We do not ask to calculate its limit. 10. Integral and function families E.5242 We denote by I n the sequence defined for any integer n greater than or equal to 1 by: I n = 1 0 x n · e − x d x 1 a Consider the two functions f and g defined by: f ( x ) = x · e − x ; g ( x ) = ( − x − 1) · e − x Show that the function g is a primitive of the function f . b Calculate I 1 . 2 In this question, any trace of research or initiative, how-ever incomplete, will be taken into account in the assess-ment. On the graph below, we have plotted the portions of the curves C 1 , C 2 , C 3 , C 10 , C 20 , C 30 within the band defined by 0 x 1 . a Formulate a conjecture about the direction of variation of the sequence I n , describing your approach. b Demonstrate this conjecture. c Deduce that the sequence I n is convergent. d Determine : lim n ↦→ + ∞ I n . E.5236 Consider the sequences I n and J n defined for any natural number n by: I n = 1 0 e − nx 1 + x d x ; J n = 1 0 e − nx (1 + x ) 2 d x 1 Shown below are the functions f n defined on the interval 0 ; 1 by: f n ( x ) = e − nx 1 + x for different values of n . a Formulate a conjecture about the direction of variation of the sequence I n , explaining the procedure. b Demonstrate this conjecture. 2 a Show that for any integer n 0 and for any real num-ber x in the interval 0 ; 1 : 0 e − nx (1 + x ) 2 e − nx 1 + x e − nx b Show that the sequences I n and J n are convergent https://chingmath.fr chapExoCorrec/5237 sacados/5237 chapExoCorrec/5242 sacados/5242 x01y0.20.4C1C2C3C10C20C30 chapExoCorrec/5236 sacados/5236 0.20.40.60.81.2I0.20.40.60.8JOCf0Cf1Cf2Cf3
00,20,40,60,810,20,40,60,8C0C1C2C3C4C10C50 ij−22C and determine their limit. E.6921 The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . For any natural number n , consider the function f n defined and derivable on the set of real numbers R by: f n ( x ) = e − ( n − 1) x 1 + e x We denote by C n the representative curve of f n in the refer-ence frame O ; −→ i ; −→ j . The curves C n for different values of n are shown below. Let u n be the sequence defined for any natural number n by: u n = 1 0 f n ( x ) d x 1 What conjectures can be made about the variations and convergence of the sequence u n ? 2 We admit that the sequence u n is convergent and we note ‘ its limit. a Show that, for any integer n greater than or equal to 1 , we have : u n + u n +1 = 1 − e − n n b Deduct the value of ‘ . 11. Area of a domain between two curves E.3984 Let f be the function defined on R by: f ( x )= x +2 − 4 · e x e x +3 We denote by C its representative curve in the plane referred to an orthonormal reference O ; −→ i ; −→ j of graphic unit 2 cm . 1 a Determine the limit of f in −∞ . b Consider the straight line D 1 of equation y = x +2 . Study the position of C relative to D 1 . 2 a Determine a primitive of the function g defined on R by: g ( x ) = e x e x + 3 b Let – be a strictly negative real. Let A ( – ) be the area, in area units, of the domain bounded by D 1 , C and the straight lines of equations : x = – ; x = 0 Show that : A ( – )=4 · ln4 − 4 · ln e λ +3 c Calculate: lim λ ↦→−∞ A ( – ) . E.126 In a coordinate system O ; I ; J , consider the two curves C and Δ , where : Curve C is the graph of the function f defined by: f ( x )=e − x +2 x +1 The line Δ has the reduced equation : y = x +2 Consider the shaded region shown above and defined by: located between the lines with equations x = − 2 and x =2 ; located between the curve C and the line Δ . 1 a Examine the variations of the function f defined by: g ( x )= f ( x ) − ( x +2) . b Prove that the line Δ lies below the curve C 2 Find the area of the shaded region. https://chingmath.fr chapExoCorrec/6921 sacados/6921 00,20,40,60,810,20,40,60,8C0C1C2C3C4C10C50 chapExoCorrec/3984 sacados/3984 Extrait d'Antilles Guyane Septembre 2008 chapExoCorrec/126 sacados/126 ij−22C
-3-2-123I-12JOCfCg -3-2-1012345-11234ijCf E.6918 Consider the function f defined for any real x by: f ( x ) = x · e 1 − x 2 and the function g defined for any real x by: g ( x ) = e 1 − x On the graph below, the representative curves C f and C g of the functions f and g respectively have been plotted on a reference frame. It is assumed that, at R , the C g curve lies above the C f curve. 1 Find a primitive F of the function f on R . 2 Deduce the value of : 1 0 e 1 − x − x · e 1 − x 2 d x . 3 Interpret this result graphically. E.6913 Let f be the function defined on R by the relation: f ( x ) = 3 1 + e − 2 x On the graph below, we have plotted, in an orthogonal ref-erence frame O ; −→ i ; −→ j , the representative curve C of the function f and the straight line Δ of equation y =3 . Let h be the function defined on R by: h ( x )=3 − f ( x ) 1 Justify that the function h is positive. 2 We denote by H the function defined on R by: H ( x ) = − 3 2 · ln 1 + e − 2 x Show that H is a primitive of h on R . 3 Let a be a strictly positive real. a Give a graphical interpretation of the integral: a 0 h ( x ) d x b Demonstrate that : a 0 h ( x ) d x = 3 2 · ln 2 1+e − 2 a . c Note D the set of points M ( x ; y ) of the plane defined by: x 0 f ( x ) y 3 Determine the area, in area units, of the domain D E.6914 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = 1 − 1 x · ln( x ) − 2 + 2 The representative curve of the function f in an orthogonal reference frame is called C . Let C be the curve with equation : y =ln( x ) 1 Show that, for any real x in the interval 0 ; + ∞ : f ( x ) − ln( x ) = 2 − ln( x ) x Deduce that the curves C and C have a single common point whose coordinates we will determine. 2 We admit that the function H defined on the interval 0 ; + ∞ by: H ( x ) = 1 2 · ln( x ) 2 is a primitive of the function h defined on the interval 0 ; + ∞ by: h ( x ) = ln( x ) x Calculate: I = e 2 1 2 − ln( x ) x d x Interpret this result graphically. 12. Average of a function E.4013 Let f be the function defined on R by the relation: f ( x ) = 4 · e x e x + 7 1 Determine a primitive of the function f on R . 2 Calculate the mean value of f on the interval 0 ; ln7 . E.4015 For the question below, three an-swers are proposed, only one of which is correct. The choice of answer must be justified : The mean value of the function f is defined on the interval 0 ; 1 by f ( x )= 1 1+ x 2 is equal to : a − ı 2 b ı 4 c ı 2 https://chingmath.fr chapExoCorrec/6918 sacados/6918 Extrait d'Antilles-Guyane Juin 2016 -3-2-123I-12JOCfCg chapExoCorrec/6913 sacados/6913 -3-2-1012345-11234ijCf chapExoCorrec/6914 sacados/6914 chapExoCorrec/4013 sacados/4013 chapExoCorrec/4015 sacados/4015 Extrait d'Asie Juin 2009
E.5529 Consider the function f defined on the interval 0 ; 250 by the relation: f ( t ) = 2 1 + 19 · e − 0.04 t 1 Verify that for any real t belonging to the interval 0 ; 250 , we have : f ( t ) = 2 · e 0.04 t e 0.04 t + 19 2 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 . 3 Determine the mean value of f over the interval 50 ; 100 . Give an approximate value to the nearest 10 − 2 . 13. A little further on E.4019 Consider the function f defined on R by: f ( x ) = 1 e x + e − x 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 . 1 Show that, for any real x : f ( x )= e x e x 2 +1 2 Demonstrate that, for any real x , f is the derivative of the function : x ↦−→ v e x . 3 Consider the sequence J n defined on N by: J n = n 0 f ( x ) d x We admit that the sequence J n is convergent and ad-mits for limit a real L . Determine the exact value of L . E.3987 Consider the function f defined on R by: f ( x ) = ln e x +2 · e − x Note C the representative curve of the function f in an or-thonormal reference frame. 1 Show that, for any real x : f ( x ) = x + ln 1+2 · e − 2 · x We admit that, for any real x : f ( x )= − x +ln 2+e 2 · x 2 Calculate lim x ↦→ + ∞ f ( x ) and show that the line ( d ) of equa-tion y = x is asymptotic at C . Study the relative position of C and ( d ) . 3 We pose : I = 3 2 f ( x ) − x d x a Give a geometric interpretation of I . b Show that, for any X ∈ 0 ; + ∞ : ln 1+ X X c Deduce that : 0 I 3 2 2 · e − 2 · x d x and give an amplitude frame of I of amplitude 0.02 . 14. Lessons - Integrals E.5501 If f is a continuous and positive function on the interval a ; b , the function F defined on a ; b by: F : x ↦−→ x a f ( t ) d t is derivable on a ; b and admits as derivative the function f . E.5502 Let a and b be two real numbers such that a<b . We admit that a positive function f defined on a ; b admits as primitive the function x ↦−→ x a f ( t ) d t 1 Show that any continuous function on the interval a ; b admits primitives on R . (It will be admitted that any continuous function on an interval a ; b closed admits a minimum) . 2 Let f be a continuous function on the interval a ; b ad-mitting the function F as a primitive. Show that for any other primitive G of the function f , there exists a real k such that : G ( x ) = F ( x ) + k . https://chingmath.fr chapExoCorrec/5529 sacados/5529 chapExoCorrec/4019 sacados/4019 Extrait d'Amerique du Sud Novembre 2003 chapExoCorrec/3987 sacados/3987 chapExoCorrec/5501 sacados/5501 chapExoCorrec/5502 sacados/5502
x0x0h2I2345JO ABCDEFGij E.3298 Reasoning can be based on the graph provided. For any real x 0 of [1 ; + ∞ [ , note A ( x 0 ) the area of the do-main bounded by the curve representing f in an orthogonal datum, 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 . 1 What is A (1) worth? 2 Let x 0 be any real from [1 ; + ∞ [ and h be a strictly pos-itive real. Justify the following framing : f ( x 0 ) A ( x 0 + h ) − A ( x 0 ) h f ( x 0 + h ) 3 When x 0 > 1 , what framing can be obtained for h< 0 such that x 0 + h 1 ? 4 Deduce the derivability in x 0 of the function A as well as the derivative number in x 0 of the function A . 5 Conclude. E.3962 Prerequisite: Let a and b be two real numbers such that a<b , f , and g are two continuous functions on the interval [ a ; b ] . We assume that the following results are known : b a f ( t ) + g ( t ) d t = b a f ( t ) d t + b a g ( t ) d t If for all t ∈ [ a ; b ] , f ( t ) 0 then b a f ( t ) d t 0 . Show that : if for all t ∈ a ; b then : b a f ( t ) d t b a g ( t ) d t . 15. Unclassified financial years E.8143 In this exercise, we’re interested in the volume of a low-energy light bulb. Part A - Modeling the shape of the bulb The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . Consider the points A ( − 1 ; 1) , B (0 ; 1) , C (4 ; 3) , D (7 ; 0) , E (4 ; − 3) , F (0 ; − 1) and G ( − 1 ; − 1) . The cross-section of the bulb is modeled by a plane through its axis of revolution using the figure below : The part of the curve above the abscissa axis is decomposed as follows : the portion between the points A and B is the graphical representation of the constant function h defined on the interval − 1 ; 0 by h ( x )=1 ; the portion between the points B and C is the graphi-cal representation of a function f defined on the interval 0 ; 4 by: f ( x ) = a + b · sin c + ı 4 where a , b and c are fixed non-zero reals and where the real c belongs to the interval 0 ; ı 2 ; the portion between points C and D is a quarter circle of diameter [ CE ] . The part of the curve below the abscissa axis is obtained by symmetry with respect to the abscissa axis. 1 a We call f the derivative function of the function f . For any real x in the interval 0 ; 4 , determine f ( x ) . b It is imposed that the tangents at points B and C to the graphical representation of the function f are parallel to the x-axis. Determine the value of the real c . 2 Determine the reals a and b . Part B - Approximation of ampoule volume By rotating the previous figure around the x-axis, we obtain a model of the light bulb. In order to calculate its volume, we break it down into three parts, as illustrated below : https://chingmath.fr chapExoCorrec/3298 sacados/3298 x0x0h2I2345JO chapExoCorrec/3962 sacados/3962 Nouvelles Caledonie Novembre 2010 sacados/8143 ABCDEFGij
ABCDEFGij ABCDEFGijVue dans le plan(BCE ABCDEFGijVue dans l’espace Recall that : the volume of a cylinder is given by the formula ı · r 2 · h where r is the radius of the base disk and h is the height ; The volume of a ball of radius r is given by the formula · ı · r 3 . We also admit that, for any real x in the interval 0 ; 4 , f ( x ) = 2 − cos ı 4 x . 1 Calculate the volume of the cylinder of section the rect-angle ABFG . 2 Calculate the volume of the half-sphere of section the half-disc of diameter CE . 3 To approximate the volume of the solid of section the shaded area BCEF , we divide the segment OO into n segments of equal length 4 n and then construct n cylin-ders of equal height 4 n . a Special case: in this question only, we choose n =5 . Calculate the volume of the third cylinder, shaded in the figures below, then give the value rounded to 10 − 2 . b General case: in this question, n denotes any non-zero natural number. The volume of the solid of section BCEF is approxi-mated by the sum of the volumes of the first n cylinders thus created by choosing a sufficiently large value of n . Copy and complete the following algorithm so that, at the end of its execution, the variable V contains the sum of the volumes of the n cylinders created when b is entered. E.5567 Consider the function f defined on R by the relation: f ( x ) = 1 1 + e − x 1 a Determine the expression of the function f derived from the function f . b Deduce the variations of f on R 2 a Demonstrate that, for any real x , we have : f ( x )= e x 1+e x b Establish equality: 1 0 f ( x ) d x =ln 1+ e 2 https://chingmath.fr ABCDEFGij ABCDEFGijVue dans le plan(BCE ABCDEFGijVue dans l’espace chapExoCorrec/5567 sacados/5567 Extrait Liban 2013