Grade 11 / Reference functions and derivatives 94 exercises (100% corrected)

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ChingQuizz : 4 exercises available for Quizz assessment : 1. Introduction to derivative functions (Example 1) E.598 Consider the function f defined on R by the relation: f ( x ) = 1 4 · x 2 . Below is the curve C f representing the function f in an or-thonormal coordinate system O ; I ; J : We assume that the curve C f has a tangent at each of its points. 1 Let c ( x ) denote the slope of the tangent to the curve C f at the point with abscissa x : that is, the slope of the tangent to C f passing through the point with coordinate ( x ; f ( x )) . Complete the following table : x − 3 − 1 0 2 4 c ( x ) 2 Consider the function g defined on R by: g ( x ) = 0 ; 5 · x . Complete the following table : x − 3 − 1 0 2 4 g ( x ) E.592 Consider the function f defined on R by the relation: f ( x ) = 1 10 · x 3 . Below is the curve C f representing the function f in an or-thonormal coordinate system O ; I ; J : Assume that the curve C f has a tangent at each of its points. Hint: Calculations should be rounded to the nearest hun-dredth. 1 Let c ( x ) be the slope of the tangent to the curve C f at the point with abscissa x : that is, the slope of the tan-gent to C f passing through the point with coordinates ( x ; f ( x )) . Complete the following table : https://chingmath.fr chapExoCorrec/598 sacados/598 -4-3-2-1234I234JO chapExoCorrec/592 sacados/592 -4-3-2-1234I-6-5-4-3-2-123456JO
x − 3 ; 5 − 1 0 2 3 c ( x ) 2 Consider the function g defined on R by: g ( x ) = 3 10 × x 2 . Complete the following table : x − 3 ; 5 − 1 0 2 3 g ( x ) 2. Introduction to derivative functions (Example 2) E.4655 The C curve, representative of the square function, is shown below in orthogonal reference frames. In each of these representations, a tangent to the curve C is drawn : 1 By graphical reading, complete the following table : x − 3 − 1 2 0 1 2 1 f ( x ) Coeff. dir. tangente 2 Make a conjecture as to the expression of a function as-sociating with the real number x the directing coefficient of the tangent at the point of abscissa x . E.4656 The curve C , representative of the in-verse function, is shown below in orthogonal reference frames. In each of these representations, a tangent to the curve C is drawn : 1 By graphical reading, complete the following table : x − 2 − 1 2 3 f ( x ) Coeff. dir. tangente 2 Make a conjecture as to the expression of a function as-sociating with the real number x the slope of the tangent at the point of abscissa x . https://chingmath.fr chapExoCorrec/4655 sacados/4655 0120123CT1 -4-3-246810121416CT2 -10101CT3 -10101CT4 chapExoCorrec/4656 sacados/4656 -3-2-10123-3-2-10123CT1 23456701CT2 -4-3-2-10-3-2-10CT3 012340123CT4
E.4657 The curve C , representative of the square root function, is shown below in orthogonal reference frames. In each of these representations, a tangent to the curve C is drawn : 1 By graphical reading, complete the following table : x 1 4 1 4 9 f ( x ) Coeff. dir. tangente 2 Make a conjecture as to the expression of a function as-sociating with the real number x the slope of the tangent at the point of abscissa x . 3. Variations and derivative numbers E.4730 Consider the function f defined and derivable on the interval − 4 ; 4 whose representative curve C f is given in the frame O ; I ; J orthonormal below : All the questions in this exercise will be answered with refer-ence to the graph above. 1 Draw up the table of variations of the function f on − 4 ; 4 . 2 a Consider the tangent ( T 1 ) the tangent to the curve C f at the point of abscissa − 3 . Give the sign of the slope of the tangent ( T 1 ) . b Consider the tangent ( T 2 ) the tangent to the curve C f at the point of abscissa 0 . Give the sign of the slope of the tangent ( T 2 ) . c Consider the tangent ( T 3 ) the tangent to the curve C f at the point of abscissa − 2 . Give the sign of the slope of the tangent ( T 3 ) . 3 a What is the sign of the number derived from the function f in x = − 1 ? b What is the sign of the number derivative of the func-tion f in x =2 ? c What is the sign of the number derived from the func- tion f in x =2.5 ? 4 Note f the derivative function of the function f . Draw up the sign table for the function f . E.10621 Consider a function f defined and derivable on − 3 ; 3 . We denote f its derivative function. The graphical representation of the function f is given in the frame below 1 Justify that the function f is increasing on the interval − 3 ; 0 . 2 Determine the direction of variation of the function f on 0 ; 3 . https://chingmath.fr chapExoCorrec/4657 sacados/4657 -10123012CT1 00.5100.51CT2 0246811.522.53CT3 4567891001234CT4 chapExoCorrec/4730 sacados/4730 -4-3-2-1234I-2-12JOCf chapExoCorrec/10621 sacados/10621 -4-3-2-1234I2JOCf
E.10622 Consider a function f defined and derivable on − 3 ; 3 . We denote f the derivative function of the function f . The graphical representation of the function f is given in the frame below Determine the directions of variation of the function f on − 3 ;3 . E.6061 Let f be a function f defined on − 4 ; 4 whose table of variation is given below : Determine the sign of the derivative number of the function f in 1 . E.6062 Consider a function f for which the sign table of its derivative function is given below : x − 5 − 2 1 4 f ( x ) − 0 + 0 − Consider the tangent ( T ) to the curve C f at the point of abscissa 2 . What is the direction of variation of the tangent ( T ) ? E.4858 Consider a function f defined on − 3 ; 5 whose derivative admits the following sign table : We have the following values and relations : f ( − 1) = 3 f (5) = − 2 · f ( − 1) f ( − 3) = f (5) + 5 f (2) = f ( − 1) · f (5) In the previous table, complete the line of variations of the function f . E.4856 1 Consider the function f defined on R admitting a strictly positive derivative on R . In addition, we have the infor-mation : f (2)=0 . Draw up the sign table for the function f on R . 2 Consider the function g defined on R admitting a deriva-tive g verifying: For any real x , we have : g ( x ) < 0 Furthermore, we know that : g ( − 3)=0 . Draw up the sign table for the function g at R . E.3004 The table below shows the table of variations of a function f defined on R : Complete the rows for the sign of the function f and the sign of the function f . 4. Polynomials: derivative functions E.9702 Proposition: the tables below give the derivatives of the monomials : https://chingmath.fr chapExoCorrec/10622 sacados/10622 -4-3-2-1234I-1JOCf chapExoCorrec/6061 sacados/6061 −4−2−14−24−3−1Variationdefx chapExoCorrec/6062 sacados/6062 chapExoCorrec/4858 sacados/4858 +-0+0−3−125VariationdefSignedefx chapExoCorrec/4856 sacados/4856 chapExoCorrec/3004 sacados/3004 -∞−4−2−12∞150−2−12−3SignesdefVariationdefSignedefx chapExoCorrec/9702 sacados/9702 Pour toutaRf(xaf(x0f(x1f(x0f(x5f(x0 Pour toutnN∗f(xxnf(xn·xn−1g(xxg(x1j(xx3j(x3x2h(xx2h(x2xk(xx4k(x4x3 Pour toutaR; nN∗f(xa·xnf(xa×n·xn−1g(x2xg(x2j(x7x3j(x21x2h(x−2x2h(x−4xk(x−x4k(x−4x3
Determine the expression of the derivative function of each of the functions below : 1 f ( x ) = 5 x 2 + 2 x + 3 2 g ( x ) = 3 x 4 − 5 x + 2 3 h ( x ) = 5 − 3 x 2 4 j ( x ) = 3 x 2 − x + 1 E.7735 Determine the expression of the derivative functions of each of the functions below : 1 f ( x )= x 5 +3 · x 2 − x +10 2 f ( x )=2 · x 7 − x 2 − 2 · x +1 E.4670 Determine the numbers derived in 1 for each of the following functions : 1 f : x ↦−→ 2 x + 4 2 g : x ↦−→ 5 − 3 x 3 k : x ↦−→ x 4 + x 2 + 1 4 ‘ : x ↦−→ 2 x 4 − 2 x 3 − 8 x E.8392 Consider the function f defined by: f ( x )= 5 3 · x 3 − 2 3 · x 2 +3 · x − 4 Determine the expression of the function f derived from the function f . E.104 Determine the expression of the derivative functions of the following polynomial functions : 1 f : x ↦−→ − 3 · x + 2 2 g : x ↦−→ 4 · x 2 − 4 3 h : x ↦−→ 2 · x 2 + 3 · x 4 j : x ↦−→ 5 · x 3 − 2 · x 2 E.118 Determine the expression of the derivative functions of the following polynomial functions : 1 f : ↦−→ 3 · x + 2 2 g : ↦−→ x 2 + 4 3 h : ↦−→ x 2 + x 4 j : ↦−→ x 3 + 2 · x 2 E.5219 Determine the expression of the derivative of each of the following functions : E.7534 The function f is defined for any real x element of the interval 1 ; 7 by: f ( x ) = 1.5 · x 3 − 9 · x 2 + 24 · x + 48 Let f be the derivative function of the function f and f  its second derivative on 1 ; 7 . For any real x of the interval 1 ; 7 : 1 Calculate f ( x ) 2 Calculate f  ( x ) . E.10375 Determine the numbers derived in 1 for each of the following functions : 1 h : x ↦−→ 2 x 2 + 3 2 j : x ↦−→ 5 x − 3 x 2 − 1 3 k : x ↦−→ − 2 · x 2 + 2 · x 4 k : x ↦−→ 3 x 2 − 2 · x E.10376 Determine the expression of the derivative functions of the following polynomial functions : 1 f : x ↦−→ (3 · x + 11)(4 − x ) 2 g : x ↦−→ ( x + 1)(2 · x − 4) E.11397 Determine the expression of the derivative of each of the following functions : 5. Polynomials: link between derivative function and tangent E.7649 Consider the second-degree function f whose representative curve is given in the graph below : 1 a The straight line ( d 1 ) is the tangent to the curve C f at the point with coordinates (1 ; − 0.5) . Determine the slope of the line ( d 1 ) . b The straight line ( d 2 ) is the tangent to the curve C f at the point with coordinates ( − 2 ; − 2) . Determine the slope of the line ( d 2 ) . 2 The expression of the function is defined by: f ( x ) = 0.5 · x 2 + x − 2 a Determine the expression of the function f derived from the function f . b Calculate the following images by the function f : f (1) f ( − 2) 6. Polynomials: tangents E.4682 Proposition: let a be a real number and f a function deriv-able at a . The tangent ( T ) at the point of abscissa a to the curve C f of the function f has the slope-intercept form y = f ( a ) · x − a + f ( a ) Consider the function f defined on R by the relation: f ( x ) = 1 2 · x 3 − 3 2 · x 2 + x + 1 In a reference frame O ; I ; J , note C f the representative curve of the function f . 1 a Determine the expression of the derivative function https://chingmath.fr chapExoCorrec/7735 sacados/7735 chapExoCorrec/4670 sacados/4670 chapExoCorrec/8392 sacados/8392 chapExoCorrec/104 sacados/104 chapExoCorrec/118 sacados/118 chapExoCorrec/5219 sacados/5219 chapExoCorrec/7534 sacados/7534 chapExoCorrec/10375 sacados/10375 chapExoCorrec/10376 sacados/10376 chapExoCorrec/11397 sacados/11397 chapExoCorrec/7649 sacados/7649 xxyy-5-4-3-2-10123-4-3-2-11Cf(d1(d2 chapExoCorrec/4682 sacados/4682
f of the function f . b Give the value of f (2) . 2 a Give the coordinates of point A of C f having ab-scissa 2 . b Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 2 . 3 Using the calculator, check that the straight line obtained is indeed the tangent ( T ) . E.4683 Consider the function f defined on R by the relation: f ( x ) = − 2 3 · x 3 − 3 · x 2 + x + 10 In a reference frame O ; I ; J , note C f the representative curve of the function f . 1 a Determine the expression of the derivative function f of the function f . b Give the value of f ( − 3) . 2 a Give the coordinates of point A of C f having ab-scissa − 3 . b Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa − 3 . 3 Using the calculator, check that the straight line obtained is indeed the tangent ( T ) . E.7782 Consider the function f defined on the interval − 3.5 ; 0.5 by the relation: f ( x ) = 0.25 · x 3 + x 2 + x + 0.5 Note C f the representative curve of the function f in the reference frame below : 1 Determine the expression of the function f derived from the function f . 2 Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 0 . 3 Draw the tangent ( T ) in the above reference frame. E.4684 Consider the function f defined on R by the relation: f ( x ) = − 1 2 · x 3 − x 2 + 5 2 · x + 2 In a reference frame O ; I ; J orthonormal, the curve C f representative of the function f : 1 a Determine the expression of the derivative function f of the function f . b Give the value of f ( − 2) . 2 a Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa − 2 . b Plot the tangent ( T ) in the above reference frame. https://chingmath.fr chapExoCorrec/4683 sacados/4683 chapExoCorrec/7782 sacados/7782 x-3-2-10y-2-11Cf chapExoCorrec/4684 sacados/4684 -4-3-2-123I-4-3-2-1234JOCf
E.4706 Consider the function f defined on R by the relation: f ( x ) = 1 2 · x 3 − 3 4 · x 2 − 5 2 · x + 3 2 In the plane provided with a 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 Consider the linear function g defined by: g ( x ) = 1 2 · x + 13 4 a Draw the straight line ( d ) representative of the func-tion g . b Determine the derivative number of the function f in − 1 c Show that the straight line ( d ) is the tangent to the curve C f at the point of abscissa − 1 . 3 a Solve the equation : f ( x )= 1 2 b Deduce the slope-intercept formof a straight line (Δ) parallel to ( d ) and tangent to the curve C f at another point. E.4731 Consider the function f defined by the relation: f ( x ) = 1 2 · x 4 − x 3 − 3 2 · x 2 + x + 2 The plane is provided with a reference frame O ; I ; J or-thonormal in which is represented the curve C f representative of the function f : 1 a Draw the straight line ( d ) of equation y = − x . b What conjecture can be made about the line ( d ) rela-tive to the curve C f . 2 Establish the previous conjecture. https://chingmath.fr chapExoCorrec/4706 sacados/4706 -4-3-2-1234I-4-3-2-1234JOCf chapExoCorrec/4731 sacados/4731 -3-2-123I-3-2-123JOCf
E.98 A company wants to manufacture slides for young children with a profile like the curve shown oppo-site. The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j ) . We’ll take 3 cm as the graphic unit. The object of the exercise is to model this profile using the representative curve C of a function defined on the interval [0 ; 3] verifying the following conditions : The curve C passes through the points A (0 ; 2) and B (3 ; 0) ; The curve C admits at each of the points A and B a tangent parallel to the x-axis. Part 1 1 a Let f be the function defined on the interval R by: f ( x ) = − 2 3 · x 2 + 2 Study the variations of the function f (the study of limits is not required) . b Let g be the function defined on the interval R by: g ( x ) = 1 3 · x 2 − 2 · x + 3 Study the variations of the function g (the study of limits is not required) . 2 We note C f and C g respectively, the representative curves of the functions f and g . a Show that C f and C g pass through through the point K 1 ; 4 3 and have the same tangent T at this point. b Plot on the same graph, the line T , the part of C f corresponding to the points with abscissas between 0 and 1, and the part of C g corresponding to points with abscissas between 1 and 3. The curve obtained by joining the two parts of the curves is a representation of the problem posed. Part 2 The design office determined that the slide profile could also be modeled using part of the representative curve C h of the function h defined on R by: h ( x ) = 4 27 · x 3 − 2 3 · x 2 + 2 1 Demonstrate that the function h verifies conditions (1) and (2) . 2 Determine the coordinates of the point at C h abscissa 1 and the directing coefficient of the tangent at this point. E.2844 Consider the function f defined on R by the relation: f : x ↦−→ 3 2 · x 4 + 3 · x 3 − 9 2 · x 2 − 5 · x + 6 Below, we give the representative curve of the function f in a reference frame ( O ; I ; J ) : The curve C f representative of this function admits a straight line ( d ) of directrix 1 as tangent at two points. Determine the equation of this line and the coordinates of these two points. Any trace of research or initiative, however incom-plete, will be taken into account in the assessment. 7. Polynomials: tangents, points of intersection, relative positions E.4707 Consider the function defined by the relation: f ( x ) = x 2 − 6 · x + 5 In a plane with an orthonormal coordinate system, note C f the representative curve of the function f . We denote ( d ) and (Δ) the two tangents to the curve C f respectively at the abscissa points 2 and 5 . 1 Determine the expression of the function f derived from the function f . 2 Determine the equation of the tangent ( d ) . 3 Determine the equation of the tangent (Δ) . 4 Determine the coordinates of the point of intersection of the straight lines ( d ) and (Δ) . E.7738 Consider the function f defined on R by the relation: f ( x ) = x 3 − 2 · x 2 + 3 · x − 2 Note C f the representative curve of the function f in a refer-ence frame O ; I J 1 a Determine the expression of the function f derived from the function f . b Deduce the expression of the tangent ( T ) to the curve C f at the point of abscissa 1 . 2 a Study the sign of the polynomial x · x 2 − 2 · x +1 . b Deduce the relative position of the curves C f and ( T ) . https://chingmath.fr chapExoCorrec/98 sacados/98 France - Septembre 2002 - 8 points 012312 chapExoCorrec/2844 sacados/2844 -4-3-2-1234I-22468JO chapExoCorrec/4707 sacados/4707 chapExoCorrec/7738 sacados/7738
E.7739 Consider the function f defined on R by the relation: f ( x ) = 2 · x 3 − 4 · x 2 + 1 Note C f the representative curve of the function f in a refer-ence frame O ; I J 1 a Determine the expression of the function f derived from the function f . b Deduce the expression of the tangent ( T ) to the curve C f at the point of abscissa 1 . 2 a Study the sign of the polynomial 2 · x · x 2 − 2 · x +1 . b Deduce the relative position of the curves C f and ( T ) . E.2347 Consider the function f whose image of x is defined by the relation: f ( x ) = 1 8 · x 3 − 1 2 · x 2 − 1 2 · x + 3 Note C f the representative curve of the function f in an or-thonormal reference frame. 1 Give the expression of the function f derived from the function f . 2 Consider the tangent ( T ) to the curve C f at the point of abscissa 2 . a Give the value of the directing coefficient of ( T ) . b Determine the reduced equation of the tangent ( T ) . c In the reference frame below, draw the tangent ( T ) . 3 Consider the line ( d ) admitting the reduced equation : ( d ) : y = − x + 3 Determine the coordinates of the intersection points of the line ( d ) and the curve C f . E.7741 Consider the function f defined on R by the relation: f ( x ) = − x 3 + 2 · x 2 − 2 x + 1 We note C f the representative curve of the function f in a reference frame O ; I J Consider the tangent ( T ) to the curve C f at the point of abscissa 1 . Determine the relative position of the curves C f and ( T ) . 8. Polynomials: introduction to variations E.5230 Consider the function f defined on R by the relation: f ( x ) = 1 3 · x 3 − x 2 − 3 x + 1 The curve C f representative of the function f is given in the reference frame O ; I ; J orthogonal below : 1 Graphically, draw up the table of variations of the func-tion f on the interval − 3 ; 6 . (image values will not be shown) 2 a Determine the expression of the function f . b Draw up the sign table for the function f at R . 3 What do we notice? E.4840 Consider the function f defined on R by the relation: f ( x ) = x 3 − 3 2 · x 2 − 6 · x + 2 In a reference frame O ; I ; J orthonormal, we give the curve C f representative of the function f : 1 Graphically and on the interval − 5 2 ; 11 4 , draw up the table of variations of the function f . 2 a Determine the expression of the function f derived from the function f . b Study the sign table of the function f on R . https://chingmath.fr chapExoCorrec/7739 sacados/7739 chapExoCorrec/2347 sacados/2347 fichierPlus/2347/diapo-correction.pdf -3-2-123456I234JOCf chapExoCorrec/7741 sacados/7741 chapExoCorrec/5230 sacados/5230 -3-2-123456I-8-6-4-224JOCf chapExoCorrec/4840 sacados/4840 -4-3-2-1234I-8-6-4-2246JOCf
3 What conjecture can be made between the sign of the derivative function f and the direction of variation of the function f . 9. Polynomials: variations E.4851 Proposition: let f be a function derivable on I . If f ( x ) > 0 for all x ∈ I then f is strictly increasing on I . If f ( x ) < 0 for any x ∈ I then f is strictly decreasing on I . If f ( x )=0 for any x ∈ I then f is constant over I . Example: consider the function f defined on R by: f ( x )= x 2 − 2 x +3 The derivative function of the function f admits for expres-sion : f ( x ) = 2 x − 2 The function f admits the sign table : x −∞ 1 + ∞ 2 x − 2 − 0 + We deduce the table of variation of the function f : Consider the function f defined on R by the relation: f ( x ) = − 2 · x 3 + 3 · x 2 + 12 · x − 2 1 Determine the expression of the function f derived from the function f . 2 Establish the sign of the function f on R . 3 Draw up the table of variations of the function f . E.11400 Consider the function f defined by: f ( x ) = x 3 − 3 x 2 − 9 x + 1 Let f be the derivative of the function f . 1 Establish that f is expressed by: f ( x ) = 3 x − 9 x + 1 2 a Establish the sign table for the function f . b Draw up the variation table for the function f . E.4850 Consider the function f defined on the intervalee R whose image of a real number x is given by the formula : f ( x ) = x 3 − 6 · x 2 + 9 · x + 3 1 Determine the expression of the function f derived from the function f . 2 Establish the sign table of the function f . 3 Draw up the table of variations of the function f . E.1756 Each of the functions below is de-fined on R . Study the variations of each of these functions : 1 f ( x ) = x 3 − 9 · x 2 + 15 · x − 7 2 g ( x ) = − x 3 − 3 · x 2 − 3 · x + 3 3 h ( x ) = − 1 3 · x 3 + 1 2 · x 2 − 1 2 · x − 1 (we will indicate in the table of variations the values of the local extremums) E.10449 Consider the function f defined on R by: f ( x ) = − 2 · x 3 − 4 · x 2 + 8 · x + 1 1 Determine the expression of the function f derived from the function f . 2 Draw up the table of variations of the function f . Indication : we will not indicate the values in the variation table E.10632 Consider the function f defined on R by: f ( x ) = 8 · x 3 + x 2 − x + 5 1 Determine the expression of the function f on R . 2 a Draw up the sign table for the function f on R . b Draw up the table of variations of the function f on R . E.11398 Consider the function f defined by: f ( x ) = 4 x 4 + 4 x 3 − x Let f be the derivative of the function f . 1 Establish that f is expressed by: f ( x ) = 2 x + 1 2 4 x − 1 2 a Establish the sign table for the function f . b Draw up the variation table for the function f . E.11399 Consider the function f defined by: f ( x ) = 4 x 4 + 4 x 3 − 2 x 2 − 3 x Let f be the derivative of the function f . 1 Establish that f is expressed by: f ( x ) = 2 x − 1 2 x + 1 4 x + 3 2 a Establish the sign table for the function f . b Draw up the variation table for the function f . https://chingmath.fr chapExoCorrec/4851 sacados/4851 −∞1∞∞2∞Variationdefx chapExoCorrec/11400 sacados/11400 chapExoCorrec/4850 sacados/4850 chapExoCorrec/1756 sacados/1756 chapExoCorrec/10449 sacados/10449 chapExoCorrec/10632 sacados/10632 chapExoCorrec/11398 sacados/11398 chapExoCorrec/11399 sacados/11399
E.4843 Consider the function f defined by the relation: f ( x ) = x 3 + 3 · x 2 − 9 · x + 5 1 a Determine the value of the reals a , b and c realizing the equality: f ( x )=( x +5)( a · x 2 + b · x + c ) b Draw up the sign table for the function f . 2 a Determine the expression of the function f derived from the function f . b Draw up the sign table for the function f . c Draw up the table of variations of the function f . 10. Polynomials: tangents and variations E.10404 Consider the function f defined and derivable on R by: f ( x ) = x 3 − 5 x 2 + 7 x − 2 Note C f the curve representing the function f in the plane with an orthonormal coordinate system. 1 Determine the expression of the function f derived from the function f . 2 Determine the equation reduce the tangent ( T ) to the curve C f at the point of abscissa 2 . 3 Determine the variations of the function f on R . E.10405 Consider the function f defined and derivable on R by: f ( x ) = x 3 + 6 x 2 + 9 x + 9 Note C f the curve representing the function f in the plane with an orthonormal coordinate system. 1 Determine the expression of the function f derived from the function f . 2 Determine the equation reduce the tangent ( T ) to the curve C f at the point of abscissa − 2 . 3 Determine the variations of the function f on R . E.10406 Consider the function f defined and derivable on R by: f ( x ) = x 3 + 4 x 2 + 5 x + 2 Note C f the curve representing the function f in the plane with an orthonormal coordinate system. 1 Determine the expression of the function f derived from the function f . 2 Determine the equation reduce the tangent ( T ) to the curve C f at the point of abscissa − 2 . 3 Determine the variations of the function f on R . E.10407 Consider the function f defined and derivable on R by: f ( x ) = − x 3 − 3 x 2 − 2 x + 4 Note C f the curve representing the function f in the plane with an orthonormal coordinate system. 1 Determine the expression of the function f derived from the function f . 2 Determine the equation reduce the tangent ( T ) to the curve C f at the point of abscissa − 1 . 3 Determine the variations of the function f on R . 11. Polynomials: sign tables and variations E.4844 Consider the function f defined by the relation: f ( x ) = x 3 + 3 · x 2 − 2 1 a Establish equality: f ( x )= x +1 x 2 +2 · x − 2 b Draw up the sign table for the function f . 2 a Determine the expression of the function f derived from the function f . b Draw up the sign table for the function f . c Draw up the table of variations of the function f (the exact values of the local extremums should be given) . E.4842 Consider the function f defined by the relation: f ( x ) = − x 3 + 3 · x 2 + 9 · x − 2 1 a Establish equality: f ( x )=( x +2)( − x 2 +5 · x − 1) b Draw up the sign table for the function f . 2 a Determine the expression of the function f derived from the function f . b Draw up the sign table for the function f . c Draw up the table of variations of the function f . 12. Polynomials: extremum E.7783 A company manufactures metal parts for the automotive industry every day. Daily production varies between 0 and 25 parts. The amount of expense corresponding to the manufacture of x pieces, expressed in euros, is modeled by the function C defined on the interval 0 ; 25 by: C ( x ) = x 3 − 30 · x 2 + 400 · x + 100 https://chingmath.fr chapExoCorrec/4843 sacados/4843 chapExoCorrec/10404 sacados/10404 chapExoCorrec/10405 sacados/10405 chapExoCorrec/10406 sacados/10406 chapExoCorrec/10407 sacados/10407 chapExoCorrec/4844 sacados/4844 chapExoCorrec/4842 sacados/4842 chapExoCorrec/7783 sacados/7783
It is assumed that the company sells its daily production ev-ery day. Each piece is sold for 247 euros. 1 Note B the profit function, expressed in euros. Justify that the expression of B ( x ) on the interval 0 ; 25 is : B ( x ) = − x 3 + 30 · x 2 − 153 · x − 100 2 Let B be the derivative function of the function B . Calculate B ( x ) , for any real number x belonging to the interval 0 ; 25 . 3 Justify the following table : x 0 3 17 + ∞ Signe de B ( x ) − 0 + 0 − 4 Deduce the complete table of variations of the function B on the interval 0 ; 25 . 5 Determine the number of pieces that the company must produce each day for the maximum profit to be made. What is this maximum profit worth? E.7784 A company produces and sells a rectangular cotton fabric 1 meter wide ; we note x its length expressed in kilometers, x being a number between 0 and 10 . The total production cost in euros of this fabric is given, as a function of x , by: C ( x ) = 15 · x 3 − 120 · x 2 + 350 · x + 1 000 The market price offers a price of 530 e per kilometer of fab-ric manufactured by the company. For any x ∈ 0 ; 10 , we note R ( x ) the revenue and B ( x ) the profit generated by the company’s production and sale of x kilometers of fabric. 1 Express R ( x ) as a function of x . 2 Show that for any x ∈ 0 ; 10 : B ( x ) = − 15 · x 3 + 120 · x 2 + 180 · x − 1 000 3 Determine B ( x ) for x ∈ 0 ; 10 où B denotes the func-tion derived from B . 4 Study the sign of B ( x ) and deduce the variations of the function B on 0 ; 10 . 5 a For what length of fabric produced and sold does the company make maximum profit? b Then give the value of this maximum profit? 13. Polynomials - modeling: maximizing the area of a box E.112 In the pattern below, we want to make a rectangular box without a lid. The lengths are expressed in cm . 1 What are the possible values of x ? 2 Verify that the volume V of this box is expressed, as a function of x , by: V ( x ) = 4 · x 3 − 52 · x 2 + 160 · x . 3 a Check that : V ( x )=12 · x 2 − 104 · x +160 Study its sign on the interval [0 ; 5] . b Construct the table of variations of the function V on the interval [0 ; 5] . c Deduce the dimensions of the final box so that the maximum volume. E.73 A manufacturer of cardboard boxes uses rolls to produce a 32 cm wide strip of cardboard, from which he traces and cuts out box patterns before gluing them on. He arranges his patterns as shown in the drawing below : The boxes, in the shape of straight blocks, have two x cm -square faces, fitted with two 1 cm -wide tabs for gluing, and four other sides whose cm dimensions are x and y , as well as a flap for closure. 1 The manufacturer uses the full width of the cardboard strip, so we have : y =30 − 2 · x . a Explain why we necessarily have : 0 <x< 15 . b Demonstrate that the volume V , in cm 3 and as a func-tion of x , of the box admits as expression : V ( x ) = 30 · x 2 − 2 · x 3 2 Let f be the function defined on the interval [0 ; 15] by: f ( x ) = 30 · x 2 − 2 · x 3 a Determine the derivative function f of the function f https://chingmath.fr chapExoCorrec/7784 sacados/7784 chapExoCorrec/112 sacados/112 16cmxx10cm chapExoCorrec/73 sacados/73 1xyx132
and study the sign of f ( x ) on the interval [0 ; 15] . b Deduce the table of variations of the function f . (val-ues will not be shown) 3 For what value of x , is the volume V maximum? What is the value of this volume? What special feature does the box have in this case? E.9703 Consider the cube shown here whose edges measure 5 cm . From this cube, we cut out a right paving block shown in grey, some of whose measurements are indicated on the figure. We note V the volume of this right paving block in cm 3 . 1 a What are the possible values for x ? b Establish the expression for the volume V as a function of x : V = x 3 − 10 · x 2 + 25 · x 2 a Determine the expression of the function V derived from the function V . b Draw up the sign table for the function V . c Draw up the table of variation of the function V . Hint: it is not required to complete the values in the table of variations. 3 Deduce the value of x for which the volume V of the right block is maximum. E.10450 Consider the right block below with dimensions : 8 cm × 5 cm × 5 cm . In this figure, using a number x , the dimensions of another right paving stone shown in gray are indicated and the volume is noted V . 1 a What are the possible values for x ? b Establish the expression for volume V as a function of x : V = x 3 − 13 · x 2 + 40 · x 2 a Determine the expression of the function V derived from the function V . b Draw up the table of variation of the function V . Hint: it is not required to complete the values in the table of variations. 3 Deduce the maximum value of the volume V of the greyed-out right paving stone. E.5245 A rectangular box without a lid is to be made in the pattern below. The lengths are expressed in cm . 1 a What values can the variable x take in this problem? b Give the expression for the volume V as a function of the value of x . 2 a Determine the expression of the function V derived from the function V . b Draw up the table of variations of the function V . c Justify that the function V admits a maximum value on the interval 0 ; 11 2 . 3 What is the maximum volume you can get with this type of box? https://chingmath.fr chapExoCorrec/9703 sacados/9703 xxx5cm chapExoCorrec/10450 sacados/10450 xxx5cm5cm8cm chapExoCorrec/5245 sacados/5245 35cmxx11cm
E.4864 Consider the rectangular paral-lelepiped shown below : The number x is used to define the measurements of this solid as shown in the figure. 1 What values can the variable x take? 2 Note V ( x ) the volume of this solid as a function of x . Give the expanded and reduced form of V . 3 a Draw up the table of variations of the function V . b Determine the value of x for which the volume of the parallelepiped is maximum. E.4632 We wish to construct a parallelepiped-shaped box from a cardboard sheet of dimen-sions 10 cm by 16 cm . To do this, we cut four squares from the corners of this sheet whose sides measure x cm . It is assumed that the value of x must lie within the interval 0 ; 6 . Determine the value of x for which the volume of the box is maximum. 14. Polynomials - modeling: using a curve E.4863 Under a shed, whose roof is ˇ parabolic ı, we wish to install a parallelepiped-shaped dwelling. The drawing below illustrates the problem : The dwelling is assumed to extend the full length of the shed. The aim of this exercise is to determine the dimensions of the façade of this habitat in order to maximize its volume. We model this problem on the figure below : The rectangle DEFG admits the straight line ( CO ) as its axis of symmetry. We note x the measure of length AG . In the reference frame A ; I ; J , the curve C f is the repre-sentative curve of the function f defined on 0 ; 6 by the relation: f ( x ) = − 1 4 · x 2 + 3 2 · x Note A ( x ) the area of rectangle DEFG as a function of x . 1 The point G belonging to the segment [ AO ] , what are the possible values for the variable x ? 2 Show that for x ∈ 0 ; 3 : A ( x ) = 1 2 · x 3 − 9 2 · x 2 + 9 · x 3 a Determine the table of variations of the function A on the interval 0 ; 3 . b Deduce the value of x for which the area of the rectan-gle DEFG is maximum. https://chingmath.fr chapExoCorrec/4864 sacados/4864 5−xx1x1 chapExoCorrec/4632 sacados/4632 12cm24cmx chapExoCorrec/4863 sacados/4863 Cfx6mABCODEFGIJ
E.6618 Consider the function f de-fined by the relation: f ( x ) = 1 20 · x 2 + 1 10 · x + 7 16 Note C f the representative curve of the function f in a refer-ence frame O ; I ; J We wish to frame the area of the plane domain between the curve C f and the x-axis and between the two straight lines with equations x = 1 2 and x = 5 2 . To do this, we’ll measure two trapezoidal surfaces : The first surface is formed from the chord [ MN ] to the curve C f : its area majors the desired area; The second surface will be constructed by considering the tangent (Δ) to the curve C f at the point of abscissa 3 2 : its area undermines the desired area. Note: A and B the points on the x-axis with respective abscis-sas 1 2 and 5 2 . M and N the points on the curve C f admitting 1 2 and 5 2 as abscissae respectively. P and Q the points on the line (Δ) with abscissas 1 2 and 5 2 respectively The figure below illustrates, in an orthogonal reference frame to emphasize the difference between these two surfaces, the two areas to be calculated : Recall the formula for calculating the area of a trapezoid : B + b × h 2 1 a Determine the coordinates of points M and N . b Deduce the area of the trapezoid ABNM . 2 a Determine the slope-intercept formof the tangent (Δ) . b Determine the coordinates of points P and Q . c Deduce the area of the trapezoid ABQP 3 a Deduce a framework for the area A . b What is the magnitude of this frame. E.5279 Consider the function f de-fined on R by the relation: f ( x ) = 4 − x 2 Below is given the curve C f representative of the function f in the plane provided with a reference O ; I J : The point M is a point on the x-axis with coordinates ( x ; 0) où x ∈ 0 ; 2 . From the point M , we construct the rectangle MNPQ whose sides are parallel to the axes. Determine the position of point M so that the area of rectan-gle MNPQ is maximum. In this exercise, any trace of research or initiative, however incomplete, will be taken into account in the assessment. E.5246 Consider the function f defined by the relation: f ( x ) = x 2 + x The graphical representa-tion is shown opposite : Consider the point J with coordinates (0 ; 1) and M , a point on the curve C f . Determine the position of point M for which the length JM is minimal. Hint: we can use the factorization : 4 x 3 + 6 x 2 − 2 = 2( x + 1)(2 x 2 + x − 1) https://chingmath.fr chapExoCorrec/6618 sacados/6618 fichierPlus/6618/ Cf1252IJOMN CfBA32IJOMNPQ bBh chapExoCorrec/5279 sacados/5279 MNPQ2−24Cf chapExoCorrec/5246 sacados/5246 -2-12I-1234JOMCf
E.7573 Joe the stuntman and his ˇ 2CV ı have to perform a jump pictured below for a shoot. He can choose his speed and the angle of inclination of the starting springboard, but to optimize his landing, he wants to land on the finishing springboard with the same inclination as it. To prepare for his jump, he scouts the production sites (with a landmark O ; −→ i ; −→ j orthonomized shown on the represen-tation) and obtains the following data : The point A is at a height of 1 m from the ground. The two stepping stones are 9 m apart. The point B is at a height of 4 m . The slope of the landing board has a length of 5 m . Using his knowledge of 1 o S , he models his trajectory by the curve C f of a function f of the second degree. Let’s note this function : f ( x ) = a · x 2 + b · x + c where a; b; c ∈ R Determine the coefficients of this polynomial. Any trace of research or initiative, however incomplete, will be taken into account in the assessment. 15. Reference functions E.5215 Proposition: below are the derivatives of the inverse func-tion and the square root function. Determine the expression of the derivatives of the following functions : 1 f ( x ) = 3 x 2 2 g ( x ) = 1 12 · x 6 3 h ( x ) = 4 x 4 j ( x ) = x 2 5 k ( x ) = 1 2 x 6 l ( x ) = − 2 x E.7679 For each question, a function f is proposed as well as the expression of the function f derived from the function f . Establish the expression of the proposed function f : f ( x ) f ( x ) 1 1 x − x 2 − 2 · x 3 − 1 x 2 2 x + x 2 + 1 x 2 · x 3 + x 2 − 1 x 2 3 x 2 + x 4 · x · x + 1 2 · x E.4685 Determine the expression of the derivative functions of each of the functions below : 1 f : x ↦−→ x + 1 x 2 g : x ↦−→ 2 3 · x 3 − x 3 h : x ↦−→ 3 · x − 2 x 4 4 j : x ↦−→ 3 x − 1 x E.93 Determine the derivative functions as-sociated with the following functions : 1 f : x ↦−→ x − 2 x 2 g : x ↦−→ 2 × 1 x 3 h : x ↦−→ − 5 x + x 4 k : x ↦−→ x 2 − 1 x E.1945 Determine the expression of the derivative of each of the functions below : 1 f : x ↦−→ x − 1 x 2 g : x ↦−→ 2 · x 3 h : x ↦−→ 3 x − 2 x 4 j : x ↦−→ 2 · x 3 + 2 x We’ll present the derivative expression in quotient form. E.4829 Determine the expression of the derivative functions associated with each of the following func-tions : 1 f : x ↦−→ 5 + x + 1 x 2 g : x ↦−→ 3 x + 2 √ x 3 h : x ↦−→ 5 · x 3 − 3 x The derivatives of the functions g and h will be presented in quotient form. 16. Reference functions and tangents https://chingmath.fr chapExoCorrec/7573 sacados/7573 ¸OABCD−i−j chapExoCorrec/5215 sacados/5215 Formule générale:f(x1xf(x−1x2g(x5xg(x−5x2h(x−73xh(x73x2 Formule générale:f(xxf(x12xg(x3xg(x32xh(x2x3h(x13x chapExoCorrec/7679 sacados/7679 chapExoCorrec/4685 sacados/4685 chapExoCorrec/93 sacados/93 chapExoCorrec/1945 sacados/1945 chapExoCorrec/4829 sacados/4829
E.2310 1 Give the reduced equation of the tangent to the curve of the square function at the point of abscissa − 2 . 2 Give the reduced equation of the tangent to the repre-sentative curve of the inverse function at the point of abscissa 3. E.10451 Consider the function f defined on R ∗ by: f ( x ) = 1 x − x + 2 In the plane provided with an orthonormal reference frame O ; I ; J , consider the curve C representative of the func-tion f given below : Note ( d ) the tangent to the curve C at the point of abscissa 1 . 1 Determine the slope-intercept formof the tangent ( d ) . y 2 Draw in the reference frame below the tangent ( d ) . Indication : we will indicate the coordinates of the points used to draw the tangent. E.5216 Consider the function f defined on 0 ; + ∞ by the relation: f ( x ) = x + 2 x − 2 The curve C f representative of the function f is given below in an O ; I ; J orthonormal coordinate system : 1 Show that the function f admits as derivative the func-tion f whose expression is given by: f ( x ) = x 2 − 2 x 2 2 We wish to determine the slope-intercept formof the tan-gent ( T ) to the curve C f at the point of abscissa 2 . a Give the slope of the tangent ( T ) . Justify your ap-proach. b Determine the slope-intercept formof the tangent ( T ) . c Draw the straight line ( T ) in the above reference frame. 3 Consider the straight line ( d ) of slope-intercept form ( d ) : y = 1 2 · x a Sur 0 ; + ∞ , study the sign of the expression : f ( x ) − 1 2 · x b Deduce the relative position of the curve C f and the straight line ( d ) . 17. Reference functions: variations E.10533 Consider the function f defined on R ∗ by: f ( x ) = 4 x + 1 x − 5 1 Establish that the function f admits as deriva-tive function, the function f defined by: f ( x ) = 2 · x + 1 2 · x − 1 x 2 2 In a reference frame O ; I ; J , consider the curve C of the function f . Determine the slope-intercept formof the tangent ( T ) to the curve C at the point of abscissa 1 . 3 a Draw up the sign table for the function f at R ∗ . b Deduce the variations of the function f on 0 ; + ∞ . E.10631 Consider the function f defined on 0 ; + ∞ by: f ( x ) = 3 x + 4 x − 7 We equip the plane with an orthonormal coordinate system O ; I ; J and we give the curve C f representative of the func-tion f . 1 Establish that the function f , derived from the function f , can be expressed as : f ( x ) = 3 · x 2 − 4 x 2 2 Determine the reduced equa-tion of the tangent to the curve C f at the point of abscissa 2 . 3 a Draw up the sign table for the function f . b Draw up the table of vari-ations of the function f on 0 ; + ∞ . https://chingmath.fr chapExoCorrec/2310 sacados/2310 chapExoCorrec/10451 sacados/10451 2I2345JOAB chapExoCorrec/5216 sacados/5216 2345I23JOCf chapExoCorrec/10533 sacados/10533 chapExoCorrec/10631 sacados/10631 23I-12345JOCf
18. Reference functions: modeling E.1990 Part A Consider the function f defined on the interval I =[20 ; 150] by: f ( x ) = 2 · x + 13 122 x 1 Show that on the interval I : f ( x )= 2 x 2 · ( x − 81)( x +81) Deduce that on the interval I : f ( x ) is of the sign of ( x − 81) . 2 Draw up the table of variations of the function f on the interval I . 3 The graphical representation of the function f is given below : Determine with the precision allowed by the graph, an ap-proximate value of the solutions of the equation : f ( x )= 350 (the graph is not to be returned with the copy) Part B A club manager needs to organize a trip. The total journey is 600 km and the club has a bus whose fuel consumption, expressed in liters per hour, is given by 5+ v 2 300 where v represents the vehicle’s average speed in kilometers per hour. The price per liter of fuel is 1 e et the driver is paid 16.87 e par hour. 1 We denote by t the total duration of the trip, expressed in hours. a Express t as a function of v . b Demonstrate that the cost of fuel, expressed in euros, for the total trip is equal to : 3 000 v + 2 · v c Show that the cost of transport, expressed in euros, is equal to f ( v ) . 2 Using part A: a Give the average speed at which the bus must travel for the cost of transport to be minimal. What is this cost? b The club manager has at most 350 e pour the trans-port. For safety reasons, the average speed of the bus may not exceed 90 kilometers per hour. Determine the range within which the average speed of the bus must lie, so that the cost of transport does not exceed 350 e . E.96 A rectangular play area of 450 m 2 is to be constructed alongside a building. Further-more, it is desired that the dimensions of this rectangle be greater than or equal to 10 m . This playing area is surrounded on three sides by a 3 m -wide driveway as shown in the sketch below. The set is fenced on three sides [ AB ] , [ BC ] and [ CD ] . We are interested in the length L of the fence : L = AB + BC + CD . We note x and y the dimensions in meters of the playing area. 1 a Demonstrate that y = 450 x , then justify that x be-longs to the interval 10 ; 45 . b Express the length L as a function of x . 2 Let f be the function defined on the interval 10 ; 45 by: f ( x ) = 2 · x + 12 + 450 x a Determine the derivative function f of the function f . b Show that, for any x belonging to 10 ; 45 , f ( x ) has the same sign as ( x 2 − 225) . Deduce the sign of f ( x ) depending on the values of x . c Draw up the table of variations of f . 3 Deduce from the previous study the dimensions to be given to the play area so that the length of the fence is as small as possible. What is this length? 19. Square root function https://chingmath.fr chapExoCorrec/1990 sacados/1990 France - Septembre 2005 - 7 points 010203040506070809010011012013014015050100150200250300350400450500550600650700 chapExoCorrec/96 sacados/96 3yxABCD
E.5220 Consider the function f defined on R + by the relation: f ( x ) = − x + 2 x In the reference frame O ; I ; J below, is given the curve C f representative of the function f . 1 a Show that the function f admits as derivative, on R ∗ + , the function f whose expression is given by: f ( x ) = 1 − x x b Determine the value of the numbers derived from the function f in 1 4 and 4 . 2 Note ( d ) and (Δ) the tangents to the curve C f at points of abscissa 1 4 and 4 respectively. a Determine the slope-intercept forms of the tangents ( d ) and (Δ) . b Show that the two straight lines ( d ) and (Δ) intercept at the coordinate point 1 ; 3 2 . c Draw on the graph the straight lines ( d ) and (Δ) . 20. Unclassified financial years E.7051 Shown below is the representative curve of a function g defined and derivable on 0 ; 5 and its horizontal tangent at point A abscissa 3 . Which of the following four statements is correct? The sign of the derivative function of g is : a négatif sur 0 ; 1 b positif sur 3 ; 4 c négatif sur 1 ; 4 d change en x =4 E.7008 The graphical representation of a function f defined and derivable on R is plotted belowbelow, along with the respective tangents at abscissa points − 3 and 0 . Which of the following four answers is correct: a f (0) = − 1 b f ( − 1) = 0 c f ( − 3) = − 1 d f ( − 3) = 3 https://chingmath.fr chapExoCorrec/5220 sacados/5220 -12345678I-3-2-12JOCf chapExoCorrec/7051 sacados/7051 01234-112A chapExoCorrec/7008 sacados/7008 Extrait Liban Mai 2016 -7-6-5-4-3-2-12345I-4-3-2-12345JOCf
E.7037 Note: In this multiple-choice questionnaire, no justifica-tion is required. For each question, only one of the answers provided is correct. A correct answer is worth 0 ; 75 points. An incorrect answer or no answer does not result in any points being deducted or awarded. Write the question number and your answer on your answer sheet. In an orthonormal coordinate system, we are given the repre-sentative curve C f of a function f that is defined and differ-entiable on the interval − 1 ; 5 . Let f be the derivative of f . The curve C f passes through the point A (0 ; 1) and the point B with abscissa 1 . The tangent T 0 to the curve at point A passes through point C (2 ; 3) , and the tangent T 1 at point B is parallel to the x-axis. 1 The exact value of f (1) is : a 0 b 1 c 1 ; 6 d other answer 2 The exact value of f (0) is : a 0 b 1 c 1 ; 6 d other answer 3 The exact value of f (1) is : a 0 b 1 c 1 ; 6 d other answer E.7048 Consider a function f defined and derivable on the interval − 2 ; 5 , increasing on − 2 ; 2 and decreasing on 2 ; 5 . Note f the derivative function of the function f . The curve ( C ) plotted below represents the function f in the plane provided with an orthonormal reference frame ; it passes through the points A ( − 2 ; 0) , B 2 ; 4 3 and C (4 ; 0) . It admits at each of the points A and B a tangent parallel to the abscissa axis and its tangent ( T ) at the point C passes through the point D (2 ; 3) . For each of the following four propositions, indicate whether it is true or false and justify the answer chosen. Justification may be based on the graph or on a calculation. Proposition 1: f (4)= − 2 3 Proposition 2: f (2) = 0 https://chingmath.fr chapExoCorrec/7037 sacados/7037 Extrait Asie Juin 2016 -10123450,511,522,53CfACBT0T1 chapExoCorrec/7048 sacados/7048 Extrait Asie Juin 2014 -2-1012345-2-1123(C(TABCD
E.7035 The curve ( C ) below represents, in an orthonormal reference frame, a function f defined and derivable on 0.5 ; 6 . The points A (1 ; 3) and B with abscissa 1.5 are on the curve ( C ) . The tangents to the ( C ) curve at points A and B are also shown dotted on this graph, the tangent at point B is hori-zontal. Note f the derivative function of f . 1 Determine f (1.5) . 2 The tangent to curve ( C ) passing through A passes through the point with coordinates (0 ; 2) . Determine an equation of this tangent. E.7022 Let the function f be defined for any strictly positive real x by: f ( x ) = 5 − x + 2 · ln x Shown below is the representative curve C of the function f , as well as T , the tangent to the curve C at the point A of abscissa 4 . Which of the following four statements is correct? On the interval 0 ; 10 , the equation f ( x )=0 admits : a Aucune solution b Une only solution c Deux solutions d More than two solutions E.7062 The function f is defined for any real x element of the interval 1 ; 7 by: f ( x ) = 1.5 · x 3 − 9 · x 2 + 24 · x + 48 We denote f the derivative function of the function f and f  its second derivative on 1 ; 7 1 Calculate f ( x ) 2 Calculate f  ( x ) E.7533 Consider the function f defined on the interval − 3.5 ; 0.5 by the relation: f ( x ) = 0.25 · x 3 + x 2 + x + 0.5 Note C f the representative curve of the function f in the reference frame below : 1 Determine the expression of the function f derived from the function f . 2 Determine the values of f (0) and f (0) by the function f . 3 a Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 0 . b Draw the tangent ( T ) in the above reference frame. E.7090 Consider the function defined by the relation: f ( x ) = x 2 − 6 x + 5 In the plane with an orthonormal reference frame, note C f the representative curve of the function f . We denote ( d ) and (Δ) the two tangents to the curve C f respectively at the abscissa points 2 and 5 . 1 Determine the expression of the function f derived from the function f . 2 Determine the equation of the tangent ( d ) . 3 Determine the tangent equation (Δ) . https://chingmath.fr chapExoCorrec/7035 sacados/7035 0123456-2-112345(CAB chapExoCorrec/7022 sacados/7022 012345678910-1123456Cf chapExoCorrec/7062 sacados/7062 chapExoCorrec/7533 sacados/7533 x-3-2-10y-2-11Cf chapExoCorrec/7090 sacados/7090
E.7251 A company makes dog food. Ev-ery day, it makes between 0 and 80 tonnes of them. The manufacturing cost, in euros, of x tonnes is modelled by the function C defined by: C ( x ) = x 3 − 105 · x 2 + 3700 · x + 4000 One ton of croquettes is sold 1 900 e . The revenue, for x tons sold, is therefore given by a function R defined on the interval 0 ; 80 . 1 a For x belonging to the interval 0 ; 80 , give the ex-pression for R ( x ) . b Deduce that the profit made by selling x tons of kib-ble is given by the function B defined on the interval 0 ; 80 by: B ( x ) = − x 3 + 105 · x 2 − 1800 · x − 4000 2 Calculate B ( x ) où B denotes the derivative of the func-tion B . 3 Justify that the sign of B ( x ) is given by the following table : x 0 10 60 80 Signe de B ( x ) − 0 + 0 − 4 Deduce the table of variations of the function B on the interval 0 ; 80 . 5 How much kibble must the company sell to make the maximum profit? What is this profit worth? E.7295 In the reference frame O ; I ; J given below, is given the curve C f representative of a function f defined on 0.5 ; 7 : Without justification, draw up the sign table of the function f derived from the function f https://chingmath.fr chapExoCorrec/7251 sacados/7251 chapExoCorrec/7295 sacados/7295 2345678I234JOCf