Outside the high school program / Functions in general 37 exercises (including 32 corrected)

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-8-7-6-5-4-3-2-1234567I-4-3-2-12345JOCf 0123456789105101520253035404550556065707580859095100105110 1. Graphic reading: image, antecedents E.4494 The graphical representation of the function f is given in the reference frame O ; I ; J below : 1 Justify each of your observations : a What is the image of the number 2 by f ? b What are the antecedents by f of the number 2 ? 2 Draw up the table of variations of the function f . 3 a What are the coordinates of the highest point on the curve C f ? b Deduce the maximum value taken by the function f . 4 Give the minimum value taken by the function f and the value of x for which it is reached. E.4493 A website specializes in video dis-tribution on the Internet. The site manager has noticed that video loading times vary according to the number of users connected simultaneously. The aim is to estimate the loading time as a function of the number of people connected simultaneously. Two functions are proposed to model this situation. In the orthogonal reference frame below, we have drawn the representative curve of a function f that models the above situation. We note x the number, expressed in thousands, of Internet users connected simultaneously and f ( x ) the loading time ex-pressed in seconds. 1 By graphical reading, estimate the loading time, in sec-onds, for 8 000 connected people. 2 a Graphically determine an antecedent of 15 by f . b Give an interpretation of this result. https://chingmath.fr chapExoCorrec/4494 sacados/4494 -8-7-6-5-4-3-2-1234567I-4-3-2-12345JOCf chapExoCorrec/4493 sacados/4493 0123456789105101520253035404550556065707580859095100105110
0,5I0,5JOCf -5-4-3-2-12I-224JOCfCg Temps (en heure)012345678910111213141516Concentration (g/‘)0,20,40,60,811,21,41,61,822,2 E.4892 In one country over a two-year period, food prices have risen uncontrollably. The state decides to impose a reduction on each of the food-stuffs in order to fix inflation at 5 % over this two-year period. 1 Assuming that the inflation percentage was 40 % , give the characteristics of the reduction that the state must impose in order to bring this inflation down to 5 % . We note x the rate of inflation and y the rate of the reduction imposed by the state to bring inflation down to 5 % . 2 Give a relationship linking the values x et y . It is assumed that the value of y is given as a function of x by the relationship : y = x − 0.05 1+ x 3 Consider the function f defined by the relation: f ( x ) = x − 0.05 − x 1 + x In an orthonormal coordinate system O ; I ; J , we con-sider the curve C f representing the function f : We will answer the following questions by reading the graph and give the results rounded to the nearest tenth : a When inflation is 30 % , what is the inflation rate set by the government? b When the government imposes a price reduction of 40 % , what is the inflation rate? E.4539 Consider the two functions f and g de-fined on R whose presentations, C f and C g , are given in the orthogonal frame O ; I ; J below : 1 Graphically, determine the set of solutions to the equa-tion : f ( x ) = g ( x ) 2 Graphically, determine the relative position of the curves C f and C g . 2. Graphical reading: inequalities E.7250 A patient is injected with a drug and, for 15 hours, the concentration, in grams in liters, of this drug in the blood is measured regularly. The curve shown below is obtained : With the precision permitted by the graph, indicate : https://chingmath.fr chapExoCorrec/4892 sacados/4892 0,5I0,5JOCf chapExoCorrec/4539 sacados/4539 -5-4-3-2-12I-224JOCfCg chapExoCorrec/7250 sacados/7250 Temps (en heure)012345678910111213141516Concentration (g/‘)0,20,40,60,811,21,41,61,822,2
-6-5-4-3-2-123I-2-1234JOCf -2-1234I-224JOCf -3-2-1234I-123JOCfCg the concentration at the initial instant ; the time interval during which the concentration is greater than or equal to 0.4 gram per liter. The necessary construction lines will be shown on the graph. E.4495 Consider the function f whose represen-tation is given below in the reference frame ( O ; I ; J ) : 1 Give the definition set of the function f . 2 Give the images, by the function f , of 0 and 1 . 3 Give the antecedents of the numbers 0 and 1 by the func-tion f . 4 Draw up the table of variations of the function f . E.4503 Consider the function f whose represen-tation is given below in the orthonormal frame O ; I ; J ) : We are interested in the affine function g defined by the rela-tion : g : x ↦−→ x + 1 1 Draw the representative curve of the function g in the above reference frame. 2 Graphically, solve the equation : f ( x )= g ( x ) . 3 Graphically solve the inequation : f ( x ) g ( x ) E.4516 Consider the function f whose image of a real number x is defined by the relation: f ( x ) = x 2 + x + 1 x 2 + 1 1 Answer the following questions using the calculator: a Determine the minimums and maximums of the func-tion f . b Draw up the table of variations f at R . 2 a Solve the equation : f ( x )= − 3 · x +1 . b Check your result with a calculator. E.7330 Consider the two functions f and g de-fined on R by the relations : f ( x ) = x x 2 + 1 ; g ( x ) = − 1 2 · x + 1 Using the calculator, give the coordinates of the intersection point of the two curves C f and C g representative of the func-tions f and g respectively. E.4540 Consider the two functions f and g de-fined on R whose representations C f and C g are given in the frame O ; I ; J orthonormal below : 1 Graphically solve the equation : f ( x )= g ( x ) . 2 Give the relative positions of the curves C f and C g at R . https://chingmath.fr chapExoCorrec/4495 sacados/4495 -6-5-4-3-2-123I-2-1234JOCf chapExoCorrec/4503 sacados/4503 -2-1234I-224JOCf chapExoCorrec/4516 sacados/4516 chapExoCorrec/7330 sacados/7330 chapExoCorrec/4540 sacados/4540 -3-2-1234I-123JOCfCg
Nombre de pièces par jour024681012141618202224Montant en euros1000200030004000500060007000CCCA x-6-5-4-3-2-10123456y-1123456(d1(d2(d3 xxyy-6-5-4-3-2-10123456-2-112345(d1(d2 E.7467 A company manufactures metal parts for the automotive industry every day. Daily production varies between 0 and 25 pieces. 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 It is assumed that the company sells its daily production ev-ery day. Each piece is sold at a price of 247 euros. Sales are modeled by the function A defined on the interval 0 ; 25 by: A ( x ) = 247 · x In the reference frame below are represented the curves C C and C A respectively of the functions C and A : 1 Graphically, conjecture the relative position of these two curves. 2 Using the calculator, give the interval for which the company is profitable. (Boundaries of the interval are rounded to the nearest hundredth) . 3. Affine functions E.4497 In an ( O ; I ; J ) orthogonal reference frame, we represent the five straight lines below. Determine the reduced equations of the lines ( d 1 ) , ( d 2 ) , ( d 3 ) . E.7737 Consider the plane provided with a ref-erence frame O ; I ; J . 1 Consider the points A ( − 2 ; − 1.5) and B (0.5 ; 2.25) . Determine the reduced equation of the line ( AB ) . 2 Consider the points C ( − 1 ; 1) and D (1 ; − 0.5) . Determine the reduced equation of the line ( CD ) . E.4496 The graph below gives the representation of two straight lines in a reference frame O ; I ; J orthonor-mal: 1 Consider the two points A ( − 2 ; 3) and B (4 ; 0) belonging to the line ( d 1 ) : a Show that the directing coefficient of the line ( d 1 ) has the value − 1 2 . b Determine the reduced equation of the line ( d 1 ) . 2 Determine the reduced equation of the line ( d 2 ) . 4. Relative position of curves E.4542 Let f and g be two functions defined on −∞ ; 3 by the relations : f ( x ) = − x + 3 ; g ( x ) = x − 1 1 a Solve the equation : − x +3=( x − 1) 2 b Check whether the two solutions found in question a are solutions of the equation : f ( x ) = g ( x ) 2 a On −∞ ; 3 , establish that the function f is decreas-ing. b Justify that the function g is increasing. https://chingmath.fr chapExoCorrec/7467 sacados/7467 Nombre de pièces par jour024681012141618202224Montant en euros1000200030004000500060007000CCCA chapExoCorrec/4497 sacados/4497 x-6-5-4-3-2-10123456y-1123456(d1(d2(d3 chapExoCorrec/7737 sacados/7737 chapExoCorrec/4496 sacados/4496 xxyy-6-5-4-3-2-10123456-2-112345(d1(d2 chapExoCorrec/4542 sacados/4542
-2-123456I-123JO(dCf -∞−325∞-2030-1Variationdefx -4-3-2-123456I234JOCf 3 Deduce the relative position of the curves C f and C g . E.7419 Consider the function f defined on R by the relation: f ( x ) = 0.5 · x 2 − x + 1 whose representative curve is given in the reference frame O ; I ; J below : The straight line ( d ) shown below passes through the points A (1 ; 1) and B (3 ; 2) . 1 a Determine the reduced equation of the line ( d ) . b Justify that the inequation f ( x ) g ( x ) is equivalent to the inequation 0.5 · x 2 − 1.5 · x +0.5 0 . 2 a Using a calculator, solve the inequation : 0.5 · x 2 − 1.5 · x +0.5 0 b Conjecture the relative positions of the curves C f and ( d ) on R . 5. Signs of a function E.4855 Consider the function f defined on R whose table of variations is given below : 1 On which of these intervals, the function f is either pos-itive or negative. Specify its sign : a − 3 ; 5 b −∞ ; 2 c − 3 ; + ∞ 2 On which of these intervals, the function f is either pos-itive or negative. Then specify its sign : a − 3 ; 5 b −∞ ; 2 c − 3 ; + ∞ E.7468 Consider the function f defined on R by the relation: f ( x ) = 4 3 · x − 3 2 + 1 In the reference frame O ; I ; J below, we give the curve C f representative of the function f : 1 Graphically, graphically solve the inequation f ( x ) 2 . 2 a Establish the following identity: f ( x ) − 2 = − 18 · x 2 + 36 · x − 16 3 · x − 3 2 + 1 b Establish the sign table of the polynomial: − 18 · x 2 +36 · x − 16 . c Solve f ( x ) 2 , justifying your approach. 6. Study of variations E.4505 Consider the function f admitting the sign table below : x −∞ − 3 5 + ∞ f ( x ) + 0 − 0 + Answer the following statements with ˇ vrai ı, ˇ faux ı ou ˇ we can’t know ı : 1 f (2)=6 . 2 The equation f ( x )=0 admits exactly two solutions. 3 The function f is an affine function. 4 The inequation f ( x ) < 0 has the solution set ] − 3 ; 5[ . 5 The point A (0 ; 5) belongs to the representative curve of the function f . 6 Si f (1)= − 4 , then the minimum of the function f sur R est − 4 . https://chingmath.fr chapExoCorrec/7419 sacados/7419 -2-123456I-123JO(dCf chapExoCorrec/4855 sacados/4855 -∞−325∞-2030-1Variationdefx chapExoCorrec/7468 sacados/7468 -4-3-2-123456I234JOCf chapExoCorrec/4505 sacados/4505
-4-3-2-1234I-2-12JOCf xVariationdef−∞-201∞537-13 E.4857 Consider the function f defined on − 4 ; 4 whose representative curve C f is given below : To answer the following questions, rounded values obtained by graphical reading will be used where appropriate. 1 Draw up the table of variations of the function f . 2 Draw up the table of signs of the function f derivative of the function f . E.7416 Consider the function defined on R by the relation: f ( x ) = x 2 + 2 x 2 + 1 1 For any real number a and b , establish the identity: f ( a ) − f ( b ) = b − a a + b a 2 + 1 b 2 + 1 2 Deduce that the function f is increasing on −∞ ; 0 ( R − ) . E.4504 Consider a function f defined on R whose table of variations has been given below : Say whether the statements below are true or false, justifying the answer. a 3 is an antecedent of the number − 2 b f (1) >f ( − 1) c f (1) is a number positif d Pour x ∈ ]0 ; 1[ , we have : f ( x ) 0 e Le minimum of function f is − 1 . E.4545 Consider the function f defined by the relation: f ( x ) = 2 · (5 − x ) 3 + 1 Establish that the function f is decreasing on R . E.4537 Let f be the function whose image of a number x is defined by the relation: f ( x ) = − 2 · x + 1 + 3 1 Justify that the definition set of the function f is : D f = − 1 ; + ∞ 2 Establish that the function f is decreasing on its defining set. E.4570 1 Let f be the function defined by the relation: f ( x ) = − 2 · x 3 + 2 Justify that the function is decreasing on −∞ ; 1 . 2 Let g be the function defined by the relation: g ( x ) = − 2 x + 1 3 Justify that the function g is strictly increasing on the interval − 1 ; + ∞ . E.7334 Consider the function f defined on R by the relation: f ( x ) = x 3 + x + 2 1 Let a and b be any two real numbers. Déterminer l’identité: f ( a ) − f ( b ) = a − b · b 2 + a · b + a 2 + 1 2 Establish that the function f is strictly increasing on −∞ ; 0 E.7485 Consider the function f defined on R by the relation: f ( x ) = x 3 − x 1 a Let a and b be any two real numbers. Establish the identity: f ( a ) − f ( b ) = a − b a 2 + a · b + b 2 − 1 b Deduce that the function f is increasing on 1 ; + ∞ . 2 a Establish the sign table for the polynomial: − 3 · x 2 − x + 2 b Consider the function g defined on R by the relation: g ( x ) = x 3 + 3 · x 2 − 2 In an orthonormal coordinate system O ; I ; J , we de-note C f and C g as the representative curves of the functions f and g . Determine the relative position of the representative curves of the functions f and g . E.7466 Consider the function f defined on R by the relation: f ( x ) = x 2 + x − 2 1 For any real number a and b , establish the equality: f ( a ) − f ( b ) = a − b b + a + 1 2 Establish that the function f is increasing on 0 ; + ∞ . 7. Points of intersection E.4669 Consider the two functions defined on R by the relations : f ( x ) = 1 2 · x 2 + x − 2 ; g ( x ) = x + 1 The plane is provided with a reference frame O ; I ; J or-thonormé. The graph below shows the curve C f representa-tive of the function f . https://chingmath.fr chapExoCorrec/4857 sacados/4857 -4-3-2-1234I-2-12JOCf chapExoCorrec/7416 sacados/7416 chapExoCorrec/4504 sacados/4504 xVariationdef−∞-201∞537-13 chapExoCorrec/4545 sacados/4545 chapExoCorrec/4537 sacados/4537 chapExoCorrec/4570 sacados/4570 chapExoCorrec/7334 sacados/7334 chapExoCorrec/7485 sacados/7485 chapExoCorrec/7466 sacados/7466 chapExoCorrec/4669 sacados/4669
-5-4-3-2-123I-3-2-12JOCf D1D2ABCDE2345I23JO largeur(‘Longueur(LRectangleALבpetitebase(bgrandeBase(Bhauteur(hTrapèzeABb×h2côté(cCarrédAc2ABCTrianglerectangleAAB×AC2base(bhauteur(hTriangleAb×h2rayon(rDiamètre(DCercleP2×ı×rAı×r2 1 Complete the table of values : x − 2 0 2 g ( x ) 2 We consider the straight line ( d 1 ) passing through the point with coordinates (0 ; − 2) et having as its directing coefficient g (0) . a Determine the reduced equation of the line ( d 1 ) . b Draw the representation of the line ( d 1 ) in the above reference frame. 3 Consider the straight line ( d 2 ) passing through the point of abscissa − 2 de the curve C f et having g ( − 2) pour directing coefficient. Draw the representation of the line ( d 2 ) in the above reference frame. 4 a What conjecture can be made about the relationship of the curve C f et of the function g ? b Use this conjecture to plot the tangent ( d 3 ) to the curve C f au point of abscissa 2 . 8. Modeling problems: equation E.7418 Consider the inverse function f defined on R ∗ by the relation: f ( x ) = 2 x whose representative curve is given below in the coordinate system O ; I ; J : For x ∈ 0 ; 5 , the points shown above are defined by: A ( x ; 0) and E is the point on the curve C f with abscissa x . D (5 ; 0) and C is the point on the curve C f with abscissa 5 . Points B and D are placed so that the quadrilateral ABCD is a rectangle Consider the domain 1 Justify that for all x ∈ 0 ; 5 , we have : A 1 ( x ) = 1 ; A 2 ( x ) = 2 − 2 5 · x 2 Using a calculator, determine the value of x for which the two domains D 1 and D 2 have the same value. Reminders: https://chingmath.fr -5-4-3-2-123I-3-2-12JOCf chapExoCorrec/7418 sacados/7418 D1D2ABCDE2345I23JO largeur(‘Longueur(LRectangleALבpetitebase(bgrandeBase(Bhauteur(hTrapèzeABb×h2côté(cCarrédAc2ABCTrianglerectangleAAB×AC2base(bhauteur(hTriangleAb×h2rayon(rDiamètre(DCercleP2×ı×rAı×r2
x012345678y102030405060708090100 01234567820406080100120140160180200CRCC E.4889 In an economic context, a satis-faction function is defined as a function f that is defined and differentiable on a subset of R and has values in the interval 0 ; 100 . We say that there is ˇ saturation ı when the satisfaction func-tion takes the value 100 . The function v , derived from the function f , is called the ˇ envy ı function. We therefore have : v = f . We say that there is ˇ envy ı when v is positive, otherwise we say that there is ˇ rejection ı. Charlotte has to write a research paper. She wants to know the daily working hours that suit her best, knowing that she can devote between 0 and 8 hours per day to it. At the beginning of the day, she is increasingly efficient, but after a certain amount of time, she is no longer satisfied with her productivity. She models her satisfaction rate based on the number of hours x she spends working each day. The curve representing his satisfaction f is given below. The tangent to this curve at the point with abscissa 4 is par-allel to the abscissa axis. The curve passes through the origin of the coordinate system, and the tangent at this point passes through the point with coordinates (1 ; 50) . 1 Using the graph, answer the following questions : a For what daily working time is there ˇ saturation ı? b Over what interval is there ˇ desire ı? c Over what interval is there ˇ rejection ı? d Give v (4) . We will assume that the function v is here an affine function defined on the interval 0 ; 8 . 2 Justify that its expression is : v ( x )= − 25 2 · x +50 3 a Justify that the function f has the expression : f ( x ) = − 25 4 · x 2 + 50 · x + c where c ∈ R b Determine the value of the real number c . 4 Deduce the values of x for which satisfaction takes the value 75 . 9. Annales 1L E.7484 An entrepreneur launches new high-end shells for cell phones onto the market. We model : revenues by the function R defined on 0 ; 6.5 by: R ( x ) = − 2 · x 3 + 4.5 · x 2 + 62 · x costs by the function C defined on 0 ; 6.5 by: C ( x ) = 20 · x + 10 On the graph below are the curves representing revenues C R and costs C C as a function of the number of products manu-factured, expressed in hundreds of units. Revenues and costs are expressed in thousands of euros. 1 Graphically, conjecture the relative position of these two curves. 2 Using the calculator, give the interval for which the com-pany is profitable. (the limits of the interval are rounded to the nearest hundredth) . https://chingmath.fr chapExoCorrec/4889 sacados/4889 x012345678y102030405060708090100 chapExoCorrec/7484 sacados/7484 01234567820406080100120140160180200CRCC
12345678910ABCxf(xb(x0200000300000204060801001201401606900-2100 x020406080100120140160y-10001000200030004000500060007000 E.199 A craftsman sells jars of honey and jars of home-made jam to supermarkets and stores specializing in local produce. Part A During the month of January the artisan sold 900 jars. We know that : 2 3 are honey pots, of which 55 % are sold to specialist stores ; 20 % jars of jam are sold to supermarkets. Complete Table 1 in Appendix 2, to be returned with the copy. Part B 1 Manufacturing and packaging of confiture Consider the function f defined on the interval [0 ; 160] by: f ( x ) = 0.25 x 2 + 500 The complete manufacture of the jam and its packaging in cartons represents a cost for the artisan. For x cartons, ready for sale, this cost (in euros) is given by f ( x ) . a What formula can bebe entered in cell B2 of table 2 ap-pendix (obtained using a table) to obtain by automatic copy to the base the numbers f ( x ) ? Complete column B . b The graphical representation, noted F , of the function f is one of the two curves on the graph in the appendix. Identify the curve F on the graph. 2 Sale of the confiture A carton of jam is sold for 30 euros. Consider the function g which, for the integer x of car-tons sold, associates the selling price g ( x ) , in euros, of these x cartons ( for x belonging to the interval [0 ; 160] ) a Express g ( x ) as a function of x . b Plot on the graph in the appendix the representative curve G of the function g . c By graphical reading indicate for which values of x we have g ( x ) f ( x ) . 3 Study of bénéfice Consider the benefit function b defined on the interval [0 ; 160] by: b ( x )=30 x − f ( x ) a What formula can be entered in cell C2 of the table to obtain by automatic copy down the numbers b ( x ) ? Then complete column C . b On the graph in the appendix, identify the representa-tive curve of the function b and note this curve E . With the help of the graph and Table 2, give the table of variations of the function b . c Deduce from the previous question the number of car-tons to be sold for the profit made to be maximum. What is this maximum profit? Annexe Pots de miel Pot de confiture artisanale Total Supermarkets Magasins Spécialisés Total 600 900 E.201 new Caledonia ffl March 2006 ffl 10 points In this exercise, we are interested in the profile of a skateboard track in a leisure park. A design office wishes to give this eight-meter-wide track a parabolic shape with a maximum difference in height (differ-ence in height is the difference in altitude between two points.) . Given the constraints of the terrain, to find a runway model, this office uses functions f defined on the interval [0 ; 8] by: f ( x ) = ax 2 + bx + c where a , b and c are three given real coefficients. The curves representative of these functions will be possible profiles for this track. To do this, we are interested in two particular functions f 1 and f 2 . We provide, in appendix (to be returned with the copy) , a ta-ble obtained from a spreadsheet. This table gives the coefficients a , b and c for each of these functions. Thus, we have : f 1 ( x ) = 4 x 2 − 32 x + 28 ; f 2 ( x ) = 4 x 2 − 28 x + 28 . On this appendix, we also provide the portions of parabolas corresponding to these two functions. Part A Profile recognition 1 What formula should be entered in cell B6 so that, by copying to the right, we obtain the values taken by the function f 1 when x varies? Complete line 6, using a calculator if necessary. 2 What formula should be entered in cell B7 so that, by copying to the right, we obtain the values taken by the function f 2 when x varies? Complete line 7 using a calculator if necessary. 3 Indicate on the graph which of the two curves represents the function f 1 . It will be denoted P 1 . Justify this choice. 4 If we copy down the formula entered in cell B7 in question 1 , will we obtain the formula entered in B7 in question 2 ? Justify. Part B Search for the profile with the highest gradient https://chingmath.fr sacados/199 Guadeloupe, Guyane, Martinique - Juin 2005 - 12 points 12345678910ABCxf(xb(x0200000300000204060801001201401606900-2100 x020406080100120140160y-10001000200030004000500060007000 sacados/201
1234567ABCDEFGHIJKCoe∑.abcpourf14-3228pourf24-2828x01233,545678f1(x280-20-32-36f2(x284 largeur(enm)-10123456789altitude(enm)-40-30-20-10102030405060 Remember that a vertical drop is the difference in altitude between two points. 1 Using the table and graph, give the tables of variation of f 1 , then of f 2 over the interval [0 ; 8] . Give the maximum and minimum for both functions on this interval. 2 Calculate the maximum vertical drop, in meters, for each of the two profiles. 3 Which profile offers the greater vertical drop? E.200 In this exercise we wish to study a market law relating to a magazine entitled Mots as a function of the annual subscription price. Consider the function f defined on the interval 0 ; 200 by: f ( p ) = − 50 · p + 12 500 We admit that this function gives the number of subscribers as a function of the price p , in euros, of the annual subscrip-tion to this magazine Mots . Part A Number of subscribers 1 When the subscription is set to 50 e , what is the number of subscribers? 2 What is the image of 52 by f ? What does this image represent? 3 Justify that any increase of 2 e in the annual subscription price decreases the number of subscribers to this maga-zine by 100 Mots . 4 The number of subscribers to Mots is 5 000 . What then is the annual subscription price? 5 Using the function f , justify that for this product : ˇthe more expensive a product, the lower the demandı. Part B Recipe study We call revenue the total amount of annual subscriptions to the magazine Mots collected by the magazine publisher. 1 The subscription price is equal to 50 e . Calculate the corresponding revenue. 2 The subscription price is set at 40 e . Calculate the corresponding revenue. 3 The number of subscribers is equal to 5 000 . Calculate revenue. 4 The subscription price is equal to p euros. Express the revenue as a function of p and f ( p ) . 5 We define the function R on the interval 0 ; 200 by: R ( p ) = − 50 · p 2 + 12 500 · p Verify that R ( p ) is equal to the revenue corresponding to a subscription price equal to p euros. 6 The graph of the function R is given in Appendix (to be returned with the copy) . Using this graph and allow-ing all necessary plots to appear, answer the following questions : a What is the price of the annual subscription to this magazine March which makes the revenue maximum? What is the revenue amount then? b Give the set of solutions of the inequation : R ( p ) 500 000 7 Calculate the number of subscribers that corresponds to the maximum revenue. https://chingmath.fr 1234567ABCDEFGHIJKCoe∑.abcpourf14-3228pourf24-2828x01233,545678f1(x280-20-32-36f2(x284 largeur(enm)-10123456789altitude(enm)-40-30-20-10102030405060 sacados/200 Guadeloupe - septembre 2004 - 8 points
Prix de l’abonnement(ene)050100150200Recette(ene)0100 000200 000300 000400 000500 000600 000700 000800 00090 E.206 In this exercise, we’re interested in the pro-file of a skateboard track in a leisure park. A design office wishes to give this eight-meter-wide track a parabolic shape with a maximum gradient (gradient is the difference in altitude between two points) . Given the constraints, linked to the terrain, this office uses to find a runway model, functions f defined on the interval [0 ; 8] by: f ( x ) = ax 2 + bx + c where a , b and c are three given real coefficients. The curves representative of these functions will be possible profiles for this track. To do this, we are interested in two particular functions f 1 and f 2 . We provide, in appendix (to be returned with the copy) , a table obtained from a table. Thus, we have : f 1 ( x )=4 x 2 − 32 x +28 and f 2 ( x )=4 x 2 − 28 x +28 . On this appendix, we also provide the portions of parabolas corresponding to these two functions. Part A Profile recognition 1 What formula should be entered in cell B6 so that, by copying to the right, we obtain the values taken by the function f 1 when x varies? Complete line 6 using a calculator if necessary. 2 What formula should be entered in cell B7 so that, by copying to the right, we obtain the values taken by the function f 2 when x varies? Complete line 7, using a calculator if necessary. 3 Indicate on the graph which of the two curves represents the function f 1 . We will note it P 1 . Justify this choice. 4 If we copy down the formula entered in cell B6 in question 1 will we obtain the formula entered in B7 in question 2 ? Justify. Part B Search for the profile with the greatest drop Remember that a vertical drop is the difference in altitude between two points. 1 Using the table and graph, give the tables of variation of f 1 , then of f 2 over the interval 0 ; 8 . Give the maximum and minimum for the two functions on this interval. 2 Calculate the maximum vertical drop, in meters , for each of the two profiles. 3 Which profile offers the greater vertical drop? https://chingmath.fr Prix de l’abonnement(ene)050100150200Recette(ene)0100 000200 000300 000400 000500 000600 000700 000800 00090 sacados/206
largeur (enm)-10123456789Altitude (en dm)-40-30-20-10102030405060 1234567ABCDEFGHIJKCoe`cientabcPourf1Pourf2x01233;545678f1(x280-20-32-36f2(x284 Temps en heures01234567Taux d’alcoolémie eng=l0.20.40.60.811.21.4 E.2026 Part A At a given moment, the blood alcohol level corresponds to the quantity of pure alcohol contained in one liter of blood. It is expressed in grams (of pure alcohol) per liter (of blood) : g=‘ . After ingesting alcohol, the alcohol level in the blood increases and very quickly reaches its maximum. This maximum blood alcohol level can be estimated by the following formula (Wid-mark formula) : T = A P × K where T is the maximum blood alcohol level, P is the mass of the person, in kilograms, K is the diffusion coefficient : it is of 0.7 for men and 0.6 for women A is the mass of pure alcohol ingested, in grams. It is estimated that a glass of alcoholic beverage (a glass of wine, 25 c‘ of beer, a glass of aperitif. . . ) contains approxi- mately 10 g of pure alcohol. For example, a 60 kg man who has consumed 4 glasses of alcoholic beverage reaches a maxi-mum blood alcohol level of : 40 60 × 0.7 ≈ 0.95 . 1 Estimate the maximum blood alcohol level of a 70 kg man who has drunk an aperitif and four glasses of wine. Round the result to the hundredth. 2 Estimate the mass of alcohol ingested by a 50 kg woman with a maximum BAC of 1.02 g=‘ . Part B A person’s blood alcohol content also varies with time. The graph below represents the evolution of the blood alcohol level, as a function of time, of a 80 kg man who has consumed several alcoholic beverages in a short period of time. The time origin (hour 0) is the moment of ingestion, i.e. alcohol intake. 1 a How long after ingestion is the maximum BAC reached? b What is this man’s maximum BAC? 2 a What is this man’s blood alcohol level 3 hours after alcohol ingestion? b What is the percentage decrease in BAC 3 hours after alcohol ingestion from its maximum value? Round the result to 1 % . 3 In France, according to current legislation, the permitted blood alcohol level for driving a vehicle must not exceed 0.5 g=‘ . a Two hours after alcohol ingestion, why can’t the ob-served person drive? b How long after alcohol ingestion can this person drive? https://chingmath.fr largeur (enm)-10123456789Altitude (en dm)-40-30-20-10102030405060 1234567ABCDEFGHIJKCoe`cientabcPourf1Pourf2x01233;545678f1(x280-20-32-36f2(x284 sacados/2026 Temps en heures01234567Taux d’alcoolémie eng=l0.20.40.60.811.21.4