Grade 11 / Derivative functions 94 exercises (100% corrected)

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x-4-2024y-4-224C1 x-4-2024y-4-224C2 x-4-2024y-4-224C3 x-4-2024y-4-224C4 -8-6-4-22468I-4-22JOCf -6-4-2246I-4-22JOC1 -6-4-2246I-4-22JOC2 -6-4-2246I24JOC3 -6-4-2246I24JOC4 ChingQuizz : 5 exercises available for Quizz assessment : 1. Derivative function: graphical relationship E.7748 Consider a function f defined and derivable on the interval − 4 ; 4 and whose following proper-ties are known : function f admits a maximum in − 2 on the interval − 3 ; 0 . the function f admits a minimum in 3 on the interval 1 ; 4 . Of the four representative curves below, only one is the rep-resentative curve of the function f derived from the function f . E.5020 Consider the function f defined and differentiable on R , which is represented by the curve C f in the coordinate system O ; I ; J : Among the four curves C 1 , C 2 , C 3 , and C 4 shown below, de-termine the curve representing the function f , derived from the function f . Justify your choice : https://chingmath.fr chapExoCorrec/7748 sacados/7748 x-4-2024y-4-224C1 x-4-2024y-4-224C2 x-4-2024y-4-224C3 x-4-2024y-4-224C4 chapExoCorrec/5020 sacados/5020 -8-6-4-22468I-4-22JOCf -6-4-2246I-4-22JOC1 -6-4-2246I-4-22JOC2 -6-4-2246I24JOC3 -6-4-2246I24JOC4
xxyy-12-10-8-6-4-202468-4-224CfACBD -4-3-2-1234I-2-123JOCf -4-3-2-1234I-2-123JOCg -4-3-2-1234I-2-123JOCh -4-3-2-1234I-2-123JOCj -4-3-2-1234I-2-123JOCk -4-3-2-1234I-2-123JOC‘ E.5731 Consider the function f defined on R whose representative curve C is given below in an orthonor-mal plot : The curve C f admits two horizontal tangents at points A and C with respective abscissas − 7 and 1 2 ; The curve C f intercepts the abscissa axis at points B and D of coordinates ( − 1 ; 0) et (3 ; 0) . 1 Consider the function g , whose derivative is the function f ( g = f ) . Draw up the table of variations of the function g . 2 Consider the function h which is the derivative of the function f ( f = h ) . Draw up the sign table for the func-tion h . E.2926 Consider the six functions f , g , h , j , k , l defined on [ − 4 ; 4] whose representative curves are given below : Associate three of these functions with their three respective derivatives. 2. Product derivatives: polynomial function E.7511 Consider the two functions u and v defined on R by the relations : u ( x ) = 3 · x − 2 ; v ( x ) = 2 − x We define the function f defined by the relation f = u · v . Determine the images below by the function f : a f (1) b f (3) c f − 1 3 E.105 Proposition: For two functions u and v defined on an in-terval I , the product function u · v has as its derivative the function denoted by u · v defined for all x ∈ I by: u · v ( x ) = u ( x ) · v ( x ) + u ( x ) · v ( x ) 1 Complete the table below, where the function u (resp. v ) is the derivative of the function u (resp. v ) : u ( x ) v ( x ) u ( x ) v ( x ) 3 · x 2 +3 · x 2 · x + 2 x 8 3 − x 2 For each row of the table, show that the function f has the function f as its derivative : f ( x ) f ( x ) (3 · x 2 + 3 · x )(2 · x + 2) 18 · x 2 + 24 · x + 6 x 8 · 3 − x x 7 · 24 − 9 · x https://chingmath.fr chapExoCorrec/5731 sacados/5731 xxyy-12-10-8-6-4-202468-4-224CfACBD chapExoCorrec/2926 sacados/2926 -4-3-2-1234I-2-123JOCf -4-3-2-1234I-2-123JOCg -4-3-2-1234I-2-123JOCh -4-3-2-1234I-2-123JOCj -4-3-2-1234I-2-123JOCk -4-3-2-1234I-2-123JOC‘ chapExoCorrec/7511 sacados/7511 chapExoCorrec/105 sacados/105
E.4686 1 For each of the functions u (resp. v ) , give the expression of its derivative function u (resp v ) : u ( x ) v ( x ) u ( x ) v ( x ) 3 · x 2 − 2 8 − x 2 Determine the expression of the derivative function f of the function f defined by: f : x ↦−→ 3 · x 2 − 2 8 − x E.4689 Determine, for each function, the expression of its derivative function : 1 f : x ↦−→ ( x 2 − 3 · x + 1)(1 − 2 · x ) 2 g : x ↦−→ ( − x 3 + 2 · x + 3) · ( x 2 + 1) E.7085 Determine the expression of the derivatives of the following functions : 1 f : x ↦−→ x 5 · x 2 − 1 2 g : x ↦−→ 2 · x 2 − 5 x +1 1 − x 2 3. Product derivatives: inverse function E.10466 Proposition: For two functions u and v defined on an in-terval I , the product function u · v has as its derivative the function denoted by u · v , defined for all x ∈ I by: u · v ( x ) = u ( x ) · v ( x ) + u ( x ) · v ( x ) 1 Complete the table below, where the function u (resp. v ) is the derivative of the function u (resp. v ) : u ( x ) v ( x ) u ( x ) v ( x ) 1 x x 5 + 1 1 x 3 − x 2 2 For each row of the table, show that the function f has the function f as its derivative : f ( x ) f ( x ) 1 x · ( x 5 + 1) 4 x 5 − 1 x 2 1 x · (3 − x 2 ) − x 2 − 3 x 2 E.7736 Consider the two functions f and g defined on R ∗ whose expressions are given in the table below. Establish the expressions of their derivative functions given in the table : f ( x ) = x 2 − 1 · 1 x f ( x ) = x 2 + 1 x 2 g ( x ) = 2 − 3 · x 2 · 1 x g ( x ) = − 3 · x 2 − 2 x 2 E.10463 1 For each of the functions u (resp. v ) , give the expression of its derivative function u (resp v ) : u ( x ) v ( x ) u ( x ) v ( x ) 1 x x 2 − 1 5 · x + 2 x 3 − 2 · x 3 2 For each of the functions below, determine the expression of its derivative function : a f : x ↦−→ 1 x · x 2 − 1 c g : x ↦−→ 5 · x + 2 x 3 − 2 · x 3 E.5226 Determine the expression of the derivatives of the following functions : 1 f : x ↦−→ 3 − x · 1 x 2 g : x ↦−→ x · x + 1 x The expression of the derivative functions will be given as a simplified quotient . 4. Product derivatives: square root function E.10465 Proposition: For two functions u and v defined on an in-terval I , the product function u · v has as its derivative the function denoted by u · v defined for all x ∈ I by: u · v ( x ) = u ( x ) · v ( x ) + u ( x ) · v ( x ) 1 Complete the table below, where the function u (resp. v ) is the derivative of the function u (resp. v ) : u ( x ) v ( x ) u ( x ) v ( x ) 2 · x 2 +1 x 2 x x 2 For each row of the table, show that the function f has the function f as its derivative : https://chingmath.fr chapExoCorrec/4686 sacados/4686 chapExoCorrec/4689 sacados/4689 chapExoCorrec/7085 sacados/7085 chapExoCorrec/10466 sacados/10466 chapExoCorrec/7736 sacados/7736 chapExoCorrec/10463 sacados/10463 chapExoCorrec/5226 sacados/5226 chapExoCorrec/10465 sacados/10465
23456I-4-3-2-1234JO -4-3-2-1234I234JOCf f ( x ) f ( x ) (2 · x 2 + 1) x 10 · x 2 + 1 2 x 2 x · x − 1 x · x E.10464 1 For each of the functions u (resp. v ) , give the expression of its derivative function u (resp v ) : u ( x ) v ( x ) u ( x ) v ( x ) x x x 2 + 1 x 2 For each of the functions below, determine the expression of its derivative function : a f : x ↦−→ x · √ x b g : x ↦−→ x 2 + 1 · √ x E.10462 Determine the expression of the derivative function of the function g defined below : g : x ↦−→ x 2 − 3 · x The expression of the derivative function g will be given in the form of a simplified quotient . 5. Products: derivative functions to tangents E.4715 Consider the function f defined by the relation is : f ( x ) = x · 1 2 · x 2 − 4 x + 6 In the plane provided with an orthonormal reference frame O ; I ; J , we note C f the representative curve of the func-tion f : 1 a Draw the line ( d ) whose equation is : y = − 1 2 · x − 2 b Draw the line (Δ) whose equation is : y = − 7 4 · x + 17 4 2 a Determine the expression of the derivative function of the function f . b Give the values of the numbers derived from the func-tion f in 1 and 4 . 6. Products: tangents E.7719 Consider the function f defined on R ∗ by: f ( x ) = 1 x · x + 1 x 1 Establish that the function f derived from the function f admits as expression : f ( x )= − 2 x 3 2 We give the curve C f representative of the function f in a reference frame O ; I ; J : https://chingmath.fr chapExoCorrec/10464 sacados/10464 chapExoCorrec/10462 sacados/10462 chapExoCorrec/4715 sacados/4715 23456I-4-3-2-1234JO chapExoCorrec/7719 sacados/7719 -4-3-2-1234I234JOCf
234I-2-12JOCf 23456I-3-2-12JOCf a Give the value of the numbers f (2) and f (2) . b Deduce the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 2 . c Draw the tangent ( T ) in the above reference frame. E.8193 Consider the function f defined on R + by the relation: f ( x ) = (2 · x − 3) · x The curve C f representing the function f is given in the or-thonormal coordinate system O ; I ; J : 1 Establish that the function f , derived from the function f , can be expressed as : f ( x ) = 6 x − 3 2 x 2 a Determine the reduced equation of the tangent ( T ) to the curve C f at the point with abscissa 1 . b Draw the tangent ( T ) in the coordinate system above. E.10643 Consider the function f defined on R + by: f ( x ) = 3 − x · x 1 Establish that : f (9) = − 4 2 Let C f be the representative curve of the function f in a coordinate system. Deduce the reduced equation of the tangent to the curve C f at the point with abscissa 9 . E.10642 Consider the function f defined on R + by: f ( x ) = 2 x + 2 · x 1 Establish that : f (4) = 13 2 2 Let C f be the representative curve of the function f in a coordinate system. Deduce the reduced equation of the tangent to the curve C f at the point with abscissa 4 . E.5227 Consider the function f defined on R + by the relation: f ( x ) = ( x − 4) · x The curve C f representing the function f is given in the or-thonormal coordinate system O ; I ; J : 1 Determine the reduced equation of the tangent ( T 1 ) to the curve C f at the point with abscissa 4 . Plot the tan-gent ( T 1 ) in the coordinate system. 2 Determine the reduced equation of the tangent ( T 2 ) to the curve C f at the point with abscissa 1 . Draw the tangent ( T 2 ) in the coordinate system. E.10644 Consider the function f defined on R + by: f ( x ) = 4 − x · x 1 Establish that : f (8) = − 5 · 2 2 2 Let C f be the representative curve of the function f in a coordinate system. Deduce the reduced equation of the tangent to the curve C f at the point with abscissa 8 . 7. Products: variations E.2668 Consider the function f defined on R + by the expression : f ( x ) = 5 x 2 + 5 x − 4 · x 1 Establish that the function f , derivative of the function f , admits as expression : f ( x ) = 25 · x 2 + 15 · x − 4 2 · x 2 Draw up the sign table for the function f on R ∗ + . 3 Assuming the following two limits: lim x ↦→ 0 + f ( x ) = 0 ; lim x ↦→ + ∞ f ( x ) = + ∞ Draw up the table of variations of the function f . https://chingmath.fr chapExoCorrec/8193 sacados/8193 234I-2-12JOCf chapExoCorrec/10643 sacados/10643 chapExoCorrec/10642 sacados/10642 chapExoCorrec/5227 sacados/5227 23456I-3-2-12JOCf chapExoCorrec/10644 sacados/10644 chapExoCorrec/2668 sacados/2668
E.2842 Consider the function f defined on R + by the expression : f : x ↦−→ x · − 5 x 2 − 5 x − 1 + 1 2 1 Determine the expression of the derivative function f of f . 2 Draw up the complete table of variations of the function f . The following limit is accepted : lim x ↦→ + ∞ f ( x ) = −∞ 3 a Justify that the function f is zero only once over its domain. Note ¸ this value. b Determine the images of 1 10 and 15 100 by the function f , rounded to the nearest ten-thousandths. Deduce a range for ¸ at 8. Products: tangents and variations E.10520 Consider the function f defined on R ∗ + by: f ( x ) = x 2 − x · x 1 Establish that the function f , derived from the function f , can be expressed as : f ( x ) = 5 · x 2 − 3 x 2 · x 2 In the coordinate system O ; I ; J , we denote C f as the curve representing the function f . Determine the reduced equation of the tangent ( T ) to the curve C f at the point with abscissa 1 . 3 a Draw up the sign table for the function f . b Study the variations of the function f on R ∗ + . E.10521 Consider the function f defined on R ∗ + by: f ( x ) = 3 · x 2 − 5 · x · x 1 Establish that the function f , derivative of the function f , has the expression : f ( x ) = 15 x · x − 1 2 · x 2 In the reference frame O ; I ; J , note C f the representa-tive curve of the function f . Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 1 . 3 Study the variations of the function f on R ∗ + . 9. Quotients: derivatives of rational functions E.10461 Consider the two functions u and v defined on R by the relations : u ( x ) = 3 · x − 2 ; v ( x ) = 2 − x We define the function g defined by the relation g = u v . Determine, if possible, the images below by the function g : a g (0) b g (2) c g − 1 4 E.4688 Proposition: Let u and v be two functions defined on an interval I such that v does not cancel on v . Consider the function f defined on I by: f ( x ) = u ( x ) v ( x ) The function f admits as derivative function the function f defined by: f ( x ) = u ( x ) · v ( x ) − u ( x ) · v ( x ) v ( x ) 2 1 For each line, give the expression of the function u (resp. v ) derived from the function u (resp. u ) : u ( x ) v ( x ) u ( x ) v ( x ) 5 · x + 2 3 · x − 2 x 2 − 3 x + 1 2 For each of the functions f below, establish the proposed expression of its derived function f : https://chingmath.fr chapExoCorrec/2842 sacados/2842 chapExoCorrec/10520 sacados/10520 chapExoCorrec/10521 sacados/10521 chapExoCorrec/10461 sacados/10461 chapExoCorrec/4688 sacados/4688
f ( x ) f ( x ) 5 · x + 2 3 · x − 2 − 16 (3 · x − 2) 2 x 2 − 3 x + 1 x 2 + 2 · x + 3 ( x + 1) 2 E.5225 1 For each line, give the expression of the function u (resp. v ) derived from the function u (resp. u ) : u ( x ) v ( x ) u ( x ) v ( x ) 3 − 2 x x + 1 x 2 2 x + 1 2 For each of the lines below, establish the expression of the function f derived from the function f : f ( x ) f ( x ) 3 − 2 · x x + 1 − 5 ( x + 1) 2 x 2 2 · x + 1 2 · x 2 + 2 · x 2 · x + 1 2 E.4691 Consider the function f defined by: f ( x )= 3 · x +1 2 · x − 1 Establish the following equality: f ( x )= − 5 (2 · x − 1) 2 E.8398 Consider the function f defined by: f ( x ) = 3 2 − x Determine the expression of the function f , derivative of the function f . E.8393 Determine the expression of the derivative functions associated with each of the following func-tions : 1 f ( x ) = 1 x 5 + 1 2 g ( x ) = 5 · x − 2 3 · x + 1 E.10468 Consider the function f defined by: f ( x ) = 4 x 2 − 2 x + 3 Establish that the function f , derived from the function f , has the expression : f ( x ) = − 8 x + 8 ( x 2 − 2 x + 3) 2 E.10491 Consider the function f defined by: f ( x ) = x 2 − 3 x + 1 2 · x + 1 Determine the expression of the function f , derivative of the function f . E.10490 Consider the function f defined by: f ( x ) = x 2 + 4 · x − 1 2 · x − 1 Show that the function f , derived from the function f , has the expression : f ( x ) = 2 · x 2 − 2 · x − 2 (2 · x − 1) 2 E.10493 Consider the function f defined by: f ( x ) = 2 · x − 1 x 2 + x Show that the function f , derived from the function f , can be expressed as : f ( x ) = − 2 · x 2 − 2 · x − 1 x 2 · ( x + 1) 2 E.7106 Consider the function f defined by: f : x ↦−→ x 2 − 3 x 2 x − 4 Determine the derivative of the function f . E.7108 Consider the function f defined by the relation: f ( x ) = 5 · x − 2 · x 2 + x − 2 4 · x − 3 Let f be the derivative of the function f . Determine the so-lutions of the equation : f ( x ) = 0 E.10467 Consider the function f defined by: f ( x ) = x 2 + x + 1 2 · x 2 − 1 Establish that the function f , derived from the function f , has the expression : f ( x ) = − 2 · x 2 − 6 · x − 1 (2 · x 2 − 1) 2 E.10489 Consider the function g defined by: g ( x )= 5 · x − x 2 3 − x 2 Establish the following equality: g ( x )= 5 · x 2 − 6 · x +15 3 − x 2 2 E.10494 Consider the function f defined by: f ( x )= 4 · x 2 + x − 3 3 · x 2 +2 · x − 1 Show that the function f , derivative of the function f admits for expression : f ( x ) = 5 3 · x − 1 2 E.8397 Consider the function f defined by: f ( x )= 2 x 2 − 2 2 x 2 − 3 x +1 Establish that the function f derived from the function f has the expression : f ( x )= − 6 2 x − 1 2 https://chingmath.fr chapExoCorrec/5225 sacados/5225 chapExoCorrec/4691 sacados/4691 chapExoCorrec/8398 sacados/8398 chapExoCorrec/8393 sacados/8393 chapExoCorrec/10468 sacados/10468 chapExoCorrec/10491 sacados/10491 chapExoCorrec/10490 sacados/10490 chapExoCorrec/10493 sacados/10493 chapExoCorrec/7106 sacados/7106 chapExoCorrec/7108 sacados/7108 chapExoCorrec/10467 sacados/10467 chapExoCorrec/10489 sacados/10489 chapExoCorrec/10494 sacados/10494 chapExoCorrec/8397 sacados/8397
-4-3-2-1234I-2-1234JO E.10492 Consider the function f defined by: f ( x ) = 4 · x 2 + x 3 · x 2 − x Show that the function f , derived from the function f , ad-mits as expression : f ( x ) = 7 3 · x − 1 2 E.11404 Consider the function g defined by: g : x ↦−→ − 2 · x 2 + x − 3 4 · x 2 + 3 · x Determine the expression of the derivative of the function g . E.4828 Consider the function h whose im-age of x is defined by the relation: h ( x ) = x 2 − 2 · x + 1 x 2 − 5 · x + 6 1 Determine the definition set of the function h . 2 Show that the derivative number of h in x is expressed as : h ( x ) = − 3 · x 2 + 10 · x − 7 ( x 2 − 5 · x + 6) 2 10. Quotients: derivative functions and square roots E.2320 The table below shows you, for each line, the expression of the image of x by a function and the expression of the number derived in x of this function. Check the accuracy of the expression of the derivative number in x : Fonction Image de x Nombre dérivé en x f x x + 1 − x + 1 2 x · ( x + 1) 2 g ( x 2 − 3) · x 5 · x 2 − 3 2 · x E.5349 Consider the two functions f and g defined by the relations : f ( x ) = x 2 − 3 · x · √ x ; g ( x ) = x + 1 x Determine the expressions of the derivative functions f and g as simplified quotients. 11. Quotients: derivative functions and tangents E.4717 Consider the function f defined by the relation is : f ( x ) = 4 · x + 2 2 · x 2 + x + 1 In the plane provided with an orthonormal reference frame O ; I ; J , note C f the representative curve of the function f : 1 a Draw the line ( d 1 ) whose equation is : y = 1 2 · x − 1 2 b Draw the line ( d 2 ) whose equation is : y = 2 · x + 2 c Draw the line ( d 3 ) whose equation is : y = − x + 5 2 2 a Determine the expression of the derivative function of the function f . b Give the values of the numbers derived from the func-tion f in − 1 , 0 and 1 2 . 12. Quotients: tangents https://chingmath.fr chapExoCorrec/10492 sacados/10492 chapExoCorrec/11404 sacados/11404 chapExoCorrec/4828 sacados/4828 chapExoCorrec/2320 sacados/2320 chapExoCorrec/5349 sacados/5349 chapExoCorrec/4717 sacados/4717 -4-3-2-1234I-2-1234JO
-123456I-2-123JOCf -3-2-123I-3-2-1JOCf -3-2-123I-3-2-1JO -2-12IJOCf E.4830 Consider the function f defined on − 1 ; + ∞ and whose image of a number x is defined by the relation: f ( x ) = 3 x − 2 x + 1 In the plane provided with a reference frame O ; I ; J , con-sider the curve C f representative of the function f : Note ( d ) the tangent to the curve C f at the point of abscissa 1 . 1 Determine the slope-intercept formof the line ( d ) . 2 Draw tangent ( d ) in frame O ; I ; J . E.4699 Consider the function f defined on R whose image of x is defined by the relation: f ( x ) = − 16 4 · x 2 + 7 1 Establish that the function f admits as derivative the function f whose expression is : f ( x ) = 128 · x 4 · x 2 + 7 2 2 Determine the equation of the tangent ( T ) to the curve C f at the point of abscissa − 1 2 . 3 In, the orthonormal reference frame O ; I ; J , is repre-sented the representative curve C f of the function f . Plot the graphical representation of ( T ) . E.7218 Consider the function f defined on R whose image of x is defined by the relation: f ( x ) = − 16 4 · x 2 + 7 1 Determine the equation of the tangent to C f at the point of abscissa − 1 2 . 2 The orthonormal reference frame O ; I ; J below repre-sents the representative curve C f of the function f . Plot the graphical representation of ( T ) . E.6665 Consider the function f defined on R by the relation: f ( x ) = 2 · x + 3 x 2 + 4 Note C f the representative curve of the function f in a refer-ence frame O ; I ; J . Below is a portion of the C f curve. 1 Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 0 and plot it in the above reference frame. (the coordinates of the two points used to draw the tangent will be given) . 2 Determine the abscissa(s) of the points on the curve C f où it admits a horizontal tangent. E.4719 The plane is given a reference frame O ; I ; J and consider the function h defined by: h ( x ) = 3 · x 2 − x + 2 x + 1 Note C h its representative curve in the plane. 1 Determine the equation of the tangent (Δ) to the curve C h at the point of abscissa 2 . 2 Check your results with the calculator https://chingmath.fr chapExoCorrec/4830 sacados/4830 -123456I-2-123JOCf chapExoCorrec/4699 sacados/4699 -3-2-123I-3-2-1JOCf chapExoCorrec/7218 sacados/7218 -3-2-123I-3-2-1JO chapExoCorrec/6665 sacados/6665 -2-12IJOCf chapExoCorrec/4719 sacados/4719
-3-2-1234I234JOCf -4-3-2-1234I-2-1JOCf E.4883 Consider the function f defined on − 1 ; + ∞ whose expression is given by the relation: f ( x ) = x 2 + x + 1 x + 1 In the plane provided with a reference frame O ; I ; J or-thonormal, consider the curve C f representative of the func-tion f : 1 Establish that the function f derived from the function f has the expression : f ( x ) = x 2 + 2 · x ( x + 1) 2 2 Consider the straight lines ( d ) and (Δ) tangent to the curve C f at the points of abscissas − 1 2 and 1 respectively. a Determine the slope-intercept forms of the tangents ( d ) and (Δ) . b Draw the straight lines ( d ) and (Δ) . E.7734 Consider the function f defined on R by the relation: f ( x ) = − 4 · x − 2 x 2 + 2 Note C f the representative curve of the function f in a refer-ence frame O ; I ; J . Below is a portion of the C f curve. 1 Give the coordinates of the point A on the curve C f hav-ing abscissa 0 . 2 a Establish that the function f , derived from the func-tion f , has the expression : f ( x ) = 4 · x 2 + 4 · x − 8 x 2 + 2 2 b Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 0 . c Plot the tangent ( T ) in the above reference frame. (the two points used to draw the tangent will be indicated) . 3 Determine the abscissa(s) of the points on the curve C f où it admits a horizontal tangent. E.2827 Consider the two functions f and g defined by: f : x ↦−→ 2 x 2 − 5 x + 2 ; g : x ↦−→ 3 x − 2 1 − 2 x In a reference frame O ; I ; J , note C f and C g the represen-tative curves of the functions f and g , respectively, and the straight line ( T ) of slope-intercept form : y = − x Show that the straight line ( T ) is a tangent for the curve C f and the curve C f . (the abscissas of the points of contact of ( T ) with each of these two curves will be given) E.2839 Consider the function f whose im-age of x is defined by the relation: f ( x ) = − 2 · x 2 + x + 1 4 · x − 1 1 Establish that the function f derived from the function f has the expression : f : x ↦−→ − 8 · x 2 − 4 · x + 5 (4 · x − 1) 2 2 a Does the function f admit tangents whose directing coefficient is − 1 ? b If so, determine their reduced equations. 13. Quotients: tangents and points of intersection E.2395 Consider the function f de-fined on R whose image of x is defined by the relation: f ( x ) = − 16 4 x 2 + 7 1 a Determine the equation of the tangent to C f at the point of abscissa − 1 2 . b The orthonormal reference frame ( O ; I ; J ) below rep-resents the representative curve C f of the function f . Plot the graphical representation of ( T ) . 2 a Establish the following factorization : 8 x 3 + 20 x 2 + 14 x + 3 = (2 x + 1) 2 · (2 x + 3) b Study the relative position of the straight line ( T ) rel-ative to the curve C f . https://chingmath.fr chapExoCorrec/4883 sacados/4883 -3-2-1234I234JOCf chapExoCorrec/7734 sacados/7734 -4-3-2-1234I-2-1JOCf chapExoCorrec/2827 sacados/2827 chapExoCorrec/2839 sacados/2839 chapExoCorrec/2395 sacados/2395 fichierPlus/2395/
-3-2-123I-3-2-1JO IJOCf -5-4-3-2-123I-12JO -4-3-2-1234I-2-12JO E.6617 Consider the function f defined on R by the relation: f ( x ) = 1 x 2 + 2 · x + 2 The curve C f representative of the function f is given in the O ; I ; J orthonormal frame below : Let (Δ) be the tangent to the curve C f at the point of abscissa 0 . 1 a Establish that the derivative function of the function f admits for expression : f ( x ) = − 2 · x − 2 x 2 + 2 · x + 2 2 b Establish that the slope-intercept formof the line (Δ) admits as expression : y = − 1 2 · x + 1 2 . 2 Study the relative position of the curve C f and the straight line (Δ) . E.4709 Consider the function f defined by the relation: f ( x ) = 3 · x + 1 x 2 + 3 In the plane provided with an orthonormal reference frame O ; I ; J , we give the curve C f representative of the function f : 1 Determine the expression of the function f derived from the function f . 2 a Determine the derivative number of the function f in 1 . b Deduce the equation of the tangent ( d ) to the curve C f at the point of abscissa 1 . c Plot the line ( d ) . 3 a Determine the value of the reals a , b and c realizing the following identity: x 3 + 3 · x 2 − 9 · x + 5 = ( x − 1)( a · x 2 + b · x + c ) b Deduce the factorized form of the polynomial: x 3 + 3 · x 2 − 9 · x + 5 . c Deduce the set of solutions of the equation : f ( x ) = 1 4 · x + 3 4 4 Give the set of coordinates of the intersection points of the curve C f and the tangent ( d ) . E.4708 Consider the function f defined by the relation: f ( x ) = x − 1 x 2 + 1 In the plane provided with an orthonormal reference frame O ; I ; J , we give the curve C f representative of the function f : 1 Show that the function f derived from the function f admits as expression : f ( x ) = − x 2 + 2 x + 1 x 2 + 1 2 . 2 a Determine the derivative number of the function f in 1 . b Deduce the equation of the tangent ( d ) to the curve C f at the point of abscissa 1 . c Plot the line ( d ) . 3 a Determine the value of the reals a , b and c realizing the following identity: x 3 − x 2 − x + 1 = ( x − 1)( a · x 2 + b · x + c ) b Determine the factorized form of the polynomial: x 3 − x 2 − x + 1 c Deduce the set of solutions to the equation : f ( x ) = 1 2 · x − 1 2 4 Give the set of coordinates of the intersection points of the curve C f and the tangent ( d ) . 14. Quotients: variations https://chingmath.fr -3-2-123I-3-2-1JO chapExoCorrec/6617 sacados/6617 IJOCf chapExoCorrec/4709 sacados/4709 -5-4-3-2-123I-12JO chapExoCorrec/4708 sacados/4708 -4-3-2-1234I-2-12JO
-∞1−√31√3∞Variationdefx E.10522 Consider the function f defined for any x ∈ R by: f ( x ) = 2 − 3 · x 4 · x 2 + 1 1 Determine the expression of the function f , derivative of the function f . 2 a Draw up the table of variations of the function f . b Deduce the variations of the function f on R . E.7296 Consider the function f defined on R by the relation: f ( x ) = 4 − 3 · x x 2 + 1 The following results can be obtained with the help of a cal-culus program and can be used without being demonstrated : L1 f(x):=(4-3*x)/(x^2+1) f ( x ) = 4 − 3 x x 2 + 1 L2 g(x):=Dérivée f(x) g ( x ) = 3 x 2 − 8 x − 3 x 2 + 1 2 L3 Résoudre f(x)=0 x = 4 3 L4 Résoudre g(x)=0 x = − 1 3 ; x =3 1 Draw up the sign table for the function f . 2 Give the set of solutions to the inequation f ( x ) > 0 . E.5278 Consider the function f defined on R by the relation: f ( x ) = 3 x − 4 x 2 + 1 1 a Determine the expression of the derivative function f . b Draw up the sign table for the function f . 2 Draw up the table of variations of the function f . The following two limits are assumed : lim x ↦→−∞ f ( x ) = 0 ; lim x ↦→ + ∞ f ( x ) = 0 3 Does the function f admit extremums? If so, specify their characteristics. E.2964 Consider the function f defined by the relation: f ( x ) = 3 x 2 − 2 x − 2 2 x 2 + x + 1 1 Justify that the function f is defined on R . 2 a Establish that the derivative function f admits the following expression : f ( x ) = 7 x 2 + 14 x 2 x 2 + x + 1 2 b Draw up the table of variations of the function f . We admit the following two limits: lim x ↦→−∞ f ( x ) = 3 2 ; lim x ↦→ + ∞ f ( x ) = 3 2 3 Deduce that the function f admits as minorant the num-ber − 2 and as majorant the number 2 . E.10646 Consider the function f , defined on R \ − 1 3 , by: f ( x ) = 3 3 x + 1 + x + 1 . Let f be the derivative of the function f . 1 Establish that for all x ∈ R \ − 1 3 : f ( x ) = 3 x − 2 3 x + 4 3 x + 1 2 2 a Establish the sign table for the function f . b Draw up the table of variations of the function f . (Note: the values of its extrema will not be indicated, nor will the boundaries of its domain) . E.2326 Consider the function f whose image of x is defined by the relation: f ( x ) = x 2 − x + 3 2 x 2 − 2 x + 3 1 Show that the denominator never cancels. Thus, the function f is defined on the set of real numbers R . 2 Establish that the derivative function f of the function f admits as expression : f ( x ) = − 6 x + 3 2 x 2 − 2 x + 3 2 3 a Draw up the sign table for f at R . b Draw up the table of variations of the function f . The following two limits are assumed : lim x ↦→−∞ f ( x ) = 1 2 ; lim x ↦→ + ∞ f ( x ) = 1 2 4 Deduce the extremums of the function f . E.6666 Consider the function f whose image of a number x is defined by the relation: f ( x ) = − x 2 − 3 · x + 1 x − 1 1 Justify that the function f admits the table of variations : (we won’t indicate the images in the table) 2 In a O ; I ; J , note C f the representative curve of the function f and ( T ) the tangent to the curve C f at the point of abscissa 0 . Determine the relative position of the curves C and ( T ) on R \{ 1 } . https://chingmath.fr chapExoCorrec/10522 sacados/10522 chapExoCorrec/7296 sacados/7296 chapExoCorrec/5278 sacados/5278 chapExoCorrec/2964 sacados/2964 chapExoCorrec/10646 sacados/10646 chapExoCorrec/2326 sacados/2326 chapExoCorrec/6666 sacados/6666 -∞1−√31√3∞Variationdefx
-8-6-4-22468I-4-224JO 012345678910110,511,52 E.5347 Consider the function f whose image of a number x is defined by the relation: f ( x ) = 5 · x + 2 x 2 + x + 1 In the plane provided with a reference frame O ; I ; J , con-sider the curve C representative of the function f given below : 1 Justify that the function f is defined for any real number. 2 Draw up the table of variations of the function f on R . Approximate values, to the nearest tenth, of the ex-tremums will be given only. 3 Note ( T ) the tangent to the curve C f at the point of ab-scissa 0 . Study the relative position of the curve C f and the straight line ( T ) . E.10523 For a ∈ R fixed, consider the func-tion f a defined on R \{− 1 } by: f a ( x ) = a · x 2 + 1 x + 1 Determine the set of values of a such that the function f a is strictly decreasing on R \{− 1 } . Indication : all traces of research, even incomplete, will be taken into account in the evaluation. 15. Quotients: modeling economic problems E.379 A supermarket wants to buy fruit from a supplier. This supplier offers prices per kilogram, decreasing according to the weight of fruit ordered. For an order of x kilograms of fruit, the price P ( x ) in euros per kilogram of fruit is given by the formula : P ( x ) = x + 300 x + 100 for x ∈ 100 ; + ∞ . For example if the supermarket buys 300 kilograms of fruit, this fruit is sold to it: P (300) = 600 400 = 1.50 euros per kilogram. In this case, the supermarket will have to pay 300 × 1.5=450 euros to the supplier for this order. Part A : Study the price P proposed by the supplier. 1 Show that : P ( x )= − 200 ( x +100) 2 on 100 ; + ∞ . 2 Give the direction of variations of the function P on 100 ; + ∞ . Part B: Study of the sum S to be spent by the supermarket. We call S ( x ) the sum in euros to be spent by the supermarket for an order of x kilograms of fruit fruit (this fruit sold by the supplier at P ( x ) euros per kilogram) . This sum is therefore equal to : S ( x )= x · P ( x ) for x ∈ 100 ; + ∞ . 1 Show that for any x belonging to 100 ; + ∞ : S ( x ) = x 2 + 200 · x + 30 000 ( x + 100) 2 2 Show that for any x belonging to 100 ; + ∞ : S ( x ) = x + 200 − 20 000 × 1 x + 100 E.7044 Antibiotics are molecules with the property of killing bacteria or limiting their spread. The table below gives the concentration in blood as a function of time of an antibiotic injected in a single dose to a patient. Temps en heure 0.5 1 1.5 2 3 4 5 6 7 8 9 10 Concentration en mg = 1.6 2 1.9 1.6 1.2 0.9 0.8 0.7 0.6 0.5 0.4 0.4 These data lead to the modelling of concentration as a func-tion of time by the function g defined on the interval 0 ; 10 by: g ( t ) = 4 · t t 2 + 1 Where t represents the time elapsed, in hours, since the an-tibiotic was injected, g ( t ) represents the concentration in mg = of the antibiotic. The graph below represents the data in the table and the representative curve of the function g . 1 By graphical reading, give without justification : a the variations of the function g on 0 ; 10 ; https://chingmath.fr chapExoCorrec/5347 sacados/5347 -8-6-4-22468I-4-224JO chapExoCorrec/10523 sacados/10523 chapExoCorrec/379 sacados/379 chapExoCorrec/7044 sacados/7044 012345678910110,511,52
012345678910100020003000400050006000700080009000 b the maximum antibiotic concentration during the first 10 hours ; c the time interval during which the antibiotic concen-tration in the blood is greater than 1.2 mg = . 2 a The function g is derivable on the interval 0 ; 10 and its derivative is g . Show that : g ( t ) = 4 · 1 − t 2 t 2 + 1 2 b Using the expression of g ( t ) , show that the maximum concentration would, with this modeling, be reached exactly 1 hour after injection. E.4885 The CoTon company produces cotton fabric. This fabric is manufactured in 1 meter widths and in lengths of x expressed in kilometers, with x ranging from 0 to 10 . The total production cost in euros for the CoTon company is given as a function of length x by the formula : C ( x ) = 15 x 3 − 120 x 2 + 500 x + 750 . The graph below gives a graphical representation of the func-tion C . Parts A and B of this exercise are independent. Part A : Profit analysis If the market offers a price p in euros for one kilometer of this fabric, then the revenue of the CoTon company for the sale of a quantity x is equal to R ( x ) = p · x . 1 Plot the line D 1 with equation : y = 400 · x . Explain, based on this graph, why the CoTon company cannot make a profit if the market price p is equal to 400 euros. 2 In this question, we assume that the market price is equal to 680 euros. a Plot the line D 2 with equation : y = 680 · x . Determine graphically, with the accuracy allowed by the graph, for which quantities produced and sold, the CoTon company makes a profit if the market price p is 680 euros b Consider the function B defined on the interval 0 ; 10 by: B ( x ) = 680 · x − C ( x ) Show that for any x belonging to the interval 0 ; 10 , we have : B ( x ) = − 45 · x 2 + 240 · x + 180 c Study the variations of the function B on 0 ; 10 . Deduce for which quantity produced and sold the profit made by the CoTon company is maximum. Give the value of this profit. Part B: Study of average cost Recall that the average cost of production C M measures the cost per unit produced. Consider the function C M defined on the interval 0 ; 10 by: C M ( x ) = C ( x ) x 1 Show that for any x belonging to the interval 0 ; 10 , we have : C M ( x ) = 30 · ( x − 5)( x 2 + x + 5) x 2 2 a Prove that for any x belonging to the interval 0 ; 10 , C M ( x ) is of the sign of ( x − 5) . Deduce the variations of the function C M on the inter-val 0 ; 10 . b For what quantity of fabric produced is the average production cost minimum? What are the average production cost and total cost in this case? E.4773 Part A 1 The price of an item is 120 euros. This price undergoes a first evolution at the rate of 25 % , then a second evolu-tion that brings it back to its initial value. What is the rate of the second evolution? 2 The price of an item is 120 euros. This price undergoes a first evolution at a rate of − 20 % , then a second evolu-tion that brings it back to its initial value. What is the rate of the second evolution? Part B Generally speaking, a price P undergoes two successive evo-lutions, the first at a rate of x , and the second at a rate of y . It then returns to its initial value P . 1 Show that x and y verify: (1+ x )(1+ y )=1 . We then assume that : y = − x 1+ x 2 We want to study on the interval − 0.5 ; 2 the function f such that : f ( x ) = − x 1 + x . Let C be the representative curve of f in the orthonormal plane. a Let f be the derivative function of the function f . Cal-culate f ( x ) . b Determine the variations of the function f on the in-terval − 0.5 ; 2 and draw up the table of variations of f on this interval. 3 Using the graphical representation of the curve C given in the appendix, or using a calculation, answer the fol-lowing questions : a What change must a price increased by 50 % undergo to return to the initial price? b What evolution must a price decreased by 50 % un-dergo to recover the initial price? https://chingmath.fr chapExoCorrec/4885 sacados/4885 Bacalaureat ES Nouvelle-Caledonie Mars 2011 7 points 012345678910100020003000400050006000700080009000 chapExoCorrec/4773 sacados/4773
IJO 024681012 234IJOCfM E.6113 Antibiotics are molecules with the ability to kill bacteria or limit their spread. The table below shows the concentration of an antibiotic in-jected into a patient in a single dose, as a function of time. Temps en heure 0.5 1 1.5 2 3 4 5 6 7 8 9 10 Concen tration en mg = 1.6 2 1.9 1.6 1.2 0.9 0.8 0.7 0.6 0.5 0.4 0.4 These data lead to the modeling of the concentration as a func-tion of time by the function g de-fined on the interval 0 ; 10 by: g ( t ) = 4 · t t 2 + 1 Where t represents the time elapsed, in hours, since the an-tibiotic was injected, g ( t ) repre-sents the concentration in mg = of the antibiotic. The graph opposite represents the data in the table and the rep-resentative curve of the function g . 1 The function g is derivable on the interval 0 ; 10 and its derivative is g . Show that : g ( t ) = 4 1 − t 2 t 2 + 1 2 2 Using the expression of g ( t ) , show that the maximum concentration would, with this modeling, be reached ex-actly 1 hour after injection. E.7063 A company manufactures and markets an item whose production is between 1 000 and 7 000 items per week. We model the manufacturing cost, expressed in thousands of euros, by the function f defined by: f ( x ) = 1.5 · x 3 − 9 · x 2 + 24 · x + 48 où x denotes the number of thousands of items manufactured. Let c be the function defined on 1 ; 7 representing the aver-age cost per item manufactured, expressed in euros. We have, therefore, for any x from 1 ; 7 : c ( x ) = f ( x ) x = 1.5 · x 2 − 9 · x + 24 + 48 x We admit that the function c is derivable on 1 ; 7 . We note c its derivative function. 1 Show that, for any x in the interval 1 ; 7 , we have : c ( x ) = 3 x − 4 x 2 + x + 4 x 2 2 a Study the variations of the function c on the interval 1 ; 7 . b Determine, in thousands, the number of items to be manufactured so that the average cost per item is min-imal. 16. Quotients: modeling and functions E.5244 Consider the function f defined on R + by the relation: f ( x ) = 1 x 2 − x + 1 In the plane provided with a reference frame O ; I ; J , con-sider the curve C f representative of the function f : Consider a point M of the curve C f of abscissa x and the rectangle represented above où : points O and M are two opposite vertices. https://chingmath.fr IJO chapExoCorrec/6113 sacados/6113 024681012 chapExoCorrec/7063 sacados/7063 chapExoCorrec/5244 sacados/5244 234IJOCfM
Mx234IJOCf 234I2JOCfMNP CfMPQO63 2IJOAMNP its sides are parallel to the axes of the frame of reference. Note A ( x ) the area of this rectangle as a function of the value of x . 1 Give the expression of the function A . 2 a Show that the function A derived from the function A has the expression : A ( x ) = 1 + x 1 − x x 2 − x + 1 2 b Draw up the sign table for the function A . c Draw up the table of variations of the function A . 3 Justify that the area of the rectangle is maximum when the point M has abscissa 1 . E.10650 Consider the function f defined on R + by: f ( x ) = 5 − x 3 x + 5 Let C f be the curve representing the function f in the coor-dinate system provided below : Let M be the point on the curve C f with abscissa x and x 0 . Let A be the function that associates, with an integer x be-longing to R ∗ + , the area of the rectangle with vertices O and M and sides parallel to the axes. 1 Denoting A as the derivative of the function A , establish the identity: A = x + 5 5 − 3 x 3 x + 5 2 2 a Establish the sign table for the function A . b Deduce the table of variations for the function A . Note: only indicate the directions of variation. 3 Deduce the value of x so that the area A is maximized. E.2828 Consider the function f defined on the interval 2 3 ; + ∞ by the relation: f ( x ) = x + 1 3 x − 2 The C f representation is given below : Consider a point M belonging to the curve C f and the rectan-gle MNOP constructed from the point O and M and whose sides are parallel to the axes. Note A ( x ) the area of rectangle MNOP where x is the ab-scissa of point M . The aim of the exercise is to determine for which values of x , the area A ( x ) is minimal. 1 Give the expression of A ( x ) as a function of x . 2 Determine the expression of the derivative function of A . 3 Establish the direction of variation of the function A . 4 Deduce the position of point M so that the area of rect-angle MNOP is minimal. E.5348 Consider the function f defined on 0 ; 6 by: f ( x ) = 12 − 2 x x + 4 In the reference frame O ; I ; J orthonormal below, is given the curve C f representative of the function f : Let M be a point on the curve C f . Consider the points P and Q belonging to the x-axis and y-axis respectively, so that the quadrilateral OPMQ is a rectangle. Determine the position of point M so that the area of rectan-gle OPMQ is maximum. 17. Modelling: going further E.6667 Consider the plane provided with an O ; I ; J orthonormal coordinate system shown below : https://chingmath.fr chapExoCorrec/10650 sacados/10650 Mx234IJOCf chapExoCorrec/2828 sacados/2828 234I2JOCfMNP chapExoCorrec/5348 sacados/5348 CfMPQO63 chapExoCorrec/6667 sacados/6667 2IJOAMNP
The point A has coordinates A (1 ; 1) . For any real number x belonging to the interval 0 ; 1] , con-sider the two points M and N defined by: M ∈ [ OJ ] ; JM = x N ∈ [ OI ) ; N ∈ [ OI ] ; IN = x The point P is defined by the intersection of the straight lines ( MN ) and ( AI ) . Determine the value of x so that the ordinate of point P is maximum. 18. Compounded by an affine function E.2826 Proposition: Let f be a differentiable function on I , a and b be any two real numbers. The function defined by: x ↦−→ f ( a · x + b ) is a differentiable function on any interval J such that : x ∈ J = ⇒ ax + b ∈ J and its derivative function is expressed as : x ↦−→ a · f ( a · x + b ) Determine, for each function, the expression of the derivative function : a f : x ↦−→ 4 x − 2 7 b g : x ↦−→ 1 5 − 3 x E.2841 Determine the expression, in simpli-fied quotient form, of the function f (resp. g ) derived from the function f (resp. g ) : a f ( x ) = 1 3 x − 2 b g ( x ) = 3 x − 1 E.5065 Consider the function f defined by the relation: f ( x ) = − x 2 + 4 x − 3 x 2 − 4 x + 4 1 Determine the definition set of the function f . 2 Show that the function f admits as derivative the func-tion f whose expression is : f ( x ) = − 2 ( x − 2) 3 3 Consider the function g defined by the relation: g ( x ) = f 2 x − 1 a Determine the simplified expression of the number g ( x ) as a function of x . b Determine, by the method of your choice, the expres-sion of the function g derived from the function g . 19. Compounded by an affine function: tangent and variation E.3510 Let f be the function defined on R by the relation: f ( x ) = 1 2 · x + 1 4 Determine the equation of the tangent to the curve C f repre-sentative of the function f at the point of abscissa 2. E.3305 Let g be the function whose image of a number x is defined by: g ( x ) = 2 − 3 · x In the plane provided with a reference frame, note C g the representative curve of the function g . 1 Give the definition set of the function f . 2 Determine the equation of the tangent to the curve C g at the point of abscissa − 2 . E.2325 Consider two functions f and g de-fined respectively on R and on 5 3 ; + ∞ by: f ( x ) = 2 3 − 2 x 5 ; g ( x ) = 3 x − 5 1 Determine the expression of the derivative functions f and g associated with the functions f and g . 2 a Determine the sign of the functions f and g on their derivation set. b Draw up the table of variations for each of these func-tions. 20. Compounded by an affine function: product and quotient E.119 Determine the expression of the derivative function for each of the functions below : a f : x ↦−→ 1 1 − 2 x b g : x ↦−→ (2 x + 1) · 3 x − 1 E.2667 Consider the function f defined on 1 3 ; + ∞ by the relation: f ( x ) = x · 3 x − 1 Determine the expression of the function f derived from the function f (The expression of f will be given as a simplified https://chingmath.fr chapExoCorrec/2826 sacados/2826 chapExoCorrec/2841 sacados/2841 chapExoCorrec/5065 sacados/5065 chapExoCorrec/3510 sacados/3510 chapExoCorrec/3305 sacados/3305 chapExoCorrec/2325 sacados/2325 chapExoCorrec/119 sacados/119 chapExoCorrec/2667 sacados/2667
quotient) . E.8401 Consider the function f defined on − 1 4 ; + ∞ by: f : x ↦−→ 3 − 2 x 3 · 4 x + 1 Hint: we will give the expression of f in the form P ( x ) 4 x + 1 where P ( x ) is a polynomial with 3 − 2 x 2 as a factor 21. Compound by an affine function: product, quotient, tangent and variations E.2521 Consider the function f whose image of x , for x ∈ [1 ; + ∞ [ , is defined by the relation: f ( x ) = ( x 2 − 4 x + 3) 2 x − 2 Note C f the representative curve of the function f in an or-thonormal frame. 1 Determine the value of the derivative number of the func-tion f in 3 . 2 Determine the expression of the tangent to the curve C f at the point of abscissa 3. E.2348 Let f be the function defined by the relation: f : x ↦−→ ( x + 5) 1 − 2 x 1 Give the definition set of the function f . 2 Determine the expression of the derivative function of f . 3 Draw up the table of signs of f . 4 Deduce the table of variations of the function f 5 Justify that f admits a global extremum in − 4 3 E.2338 Consider the function f defined on − 3 ; + ∞ by: f : x ↦−→ x 2 + 2 x + 1 x + 3 1 Show that the number derived from f into x is written : f ( x ) = 3 · x 2 + 14 · x + 11 2( x + 3) x + 3 2 Draw up the sign table for the function f . 3 a Deduce the variations of the function f on − 3 ; + ∞ . b Give the minimum of the function f on its defining set. https://chingmath.fr chapExoCorrec/8401 sacados/8401 chapExoCorrec/2521 sacados/2521 chapExoCorrec/2348 sacados/2348 fichierPlus/2348/diapo-correction.pdf chapExoCorrec/2338 sacados/2338