Grade 12 - Comp. / Integration 46 exercises (including 45 corrected)

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ABCDx(en mètres)012345678910y(en mètres)12345 x0123456789y1234567891011Ch 1. Around the areas E.7031 An advertiser is considering the installation of a rectangular billboard under part of a skateboard ramp. The profile of this ramp is modeled by the representative curve of the function f defined on the interval 0 ; 10 by: f ( x ) = 4 · e − 0.4 x This curve C f is plotted below in a frame of reference with origin O : The rectangle ABCD represents the billboard and satisfies the following constraints : the point A is located at the origin of the reference frame, the point B is on the x-axis, the point D is on the y-axis and the point C is on the curve C f . It is assumed that the point B has abscissa x =2 . Show that the area of the billboard has a value of 3.6 m 2 , rounded to the nearest tenth of a square meter. 2. Area framing E.7010 Consider the function h defined on 0 ; 7 and represented by the curve below : Among the propositions below, only one is correct. Which one? a 5 0 h ( x ) d x = h (5) − h (0) b 20 < 5 0 h ( x ) d x< 30 c 15 < 5 0 h ( x ) d x < 20 d 5 0 h ( x ) d x = 20 E.7030 Instruction : For each of the following five statements, in-dicate whether it is true or false and justify your answer. One point is awarded for each correct answer with a valid justification. Answers without justification will not be con-sidered. No penalty will be applied for unanswered questions. Note that R refers to the set of real numbers. Below is the representative curve C g of a function g defined on R . We assume that g is differentiable on R and recall that g denotes the derivative of the function g . The tangent T to the curve C g at point A of this curve, with abscissa 1 and ordinate 2 , is shown as a dotted line. This tangent intersects the x-axis at the point with x-coordinate 2 . https://chingmath.fr chapExoCorrec/7031 sacados/7031 ABCDx(en mètres)012345678910y(en mètres)12345 chapExoCorrec/7010 sacados/7010 Extrait Liban Mai 2016 x0123456789y1234567891011Ch chapExoCorrec/7030 sacados/7030
-123I-2-12345678JOCgA -10123450,511,522,53CfBT1 −10−5310724−6xVariationdeg Statement 1: g (1) = − 2 Statement 2: 1 0 g ( x ) d x < 3 E.7038 In an orthonormal plane, we give the representative curve C f of a function f defined and deriv-able on the interval − 1 ; 5 . Let f be the function derived from f . The curve C f passes through the point A (0 ; 1) and through the point B of abscissa 1 . The tangent T 1 to the curve at the point B is parallel to the x-axis. Which of the following four statements is correct? A frame of 2 0 f ( x ) d x by successive natural numbers is : a 3 2 0 f ( x ) d x 4 b 2 2 0 f ( x ) d x 3 c 1 2 0 f ( x ) d x 2 d autre réponse E.7769 Of the four propositions, only one is correct. Which one? Justify your answer. Consider the function g defined on the interval − 10 ; 10 whose table of variations is given below : Note I = 3 − 5 g ( x ) d x . It can be stated that : a − 5 I 3 b 2 I 4 c 16 I 32 d 4 I 8 3. Calculating integrals and finding primitives E.7015 Let f be the function defined on the interval 3 ; 13 by: f ( x ) = − 2 x + 20 − e − 2 x +10 Calculate the integral 13 3 f ( x ) d x . Give the exact value and then the value rounded to the near-est 10 − 3 . 4. Calculating integrals: using primitives E.7535 The population change of a sea-side resort for the summer 2015 was modeled by a function f , defined on the interval 0 ; 70 , whose representative curve is given below. Where x is the number of days that have elapsed since July 1 er , f ( x ) denotes the population in thousands of inhabitants. Thus, x =30 corresponds to July 31 and f (30) represents the population expected to be present on July 31 . It is estimated that each inhabitant will use between 45 and https://chingmath.fr -123I-2-12345678JOCgA chapExoCorrec/7038 sacados/7038 Extrait Asie Juin 2016 -10123450,511,522,53CfBT1 chapExoCorrec/7769 sacados/7769 −10−5310724−6xVariationdeg chapExoCorrec/7015 sacados/7015 Extrait Liban Mai 2016 chapExoCorrec/7535 sacados/7535 Extrait Antilles-Guyane Septembre 2015
nombre de jours024681012141618202224consommation(m350100150200250300350400450500550Cg 55 liters of water per day. We note g as the function defined on the interval 0 ; 70 by: g ( x ) = 110 + 11 · x · e − 0.025 · x +1 When x is the number of days that have passed since July 1 er , g ( x ) represents the maximum water consumption expected on that day, expressed in m 3 . Let the function G be defined on the interval 0 ; 70 by: G ( x ) = 110 · x − 440 · x + 17 600 · e − 0.025 · x +1 We assume that the function G is a primitive of the function g . The sum S = g (10)+ g (11)+ g (12)+ ··· + g (20) represents the maximum water consumption from day 10 e to day 20 e ex-pressed in m 3 . 1 Illustrate this on the curve C g below and give a graphical interpretation in terms of areas of the sum S . 2 Deduce an approximate value for this amount of water consumed from day 10 e to day 20 e . E.7021 Consider the function f defined on 0 ; 15 by: f ( x ) = 9 · x 2 · 1 − 2 · ln x + 10 We give the function F defined on the interval 0 ; 1.5 by: F ( x ) = 10 · x + 5 · x 3 − 6 x 3 · ln x 1 Show that F is a primitive of the function f on 0 ; 1.5 . 2 Calculate 1.5 1 f ( x ) d x . We will give the result rounded to the hundredth. E.7727 Consider the function f defined and derivable on the interval 0 ; 7 of expression : f ( x ) = 2 · x · e − x +3 Consider the function F defined on the interval 0 ; 7 by: F ( x ) = − 2 · x − 2 · e − x +3 1 Justify that F is a primitive of f on the interval 0 ; 7 . 2 Calculate the exact value of the area, in area units, of the plane domain bounded by the straight lines of equation x =1 , x =3 , the x-axis and the curve C . E.7731 Consider the function f defined on the interval − 20 ; 20 by: f ( x ) = − 2 · x + 30 · e 0.2 · x − 3 Formal calculation software gives the following results : 1 Dériver − 10 · x + 200 · e 0 , 2 · x − 3 − 2 · x + 30 · e 0 , 2 · x − 3 2 Dériver 2 · x + 30 · e 0 , 2 · x − 3 − 0.4 · x + 4 · e 0 , 2 · x − 3 3 Dériver − 0 ; 4 · x + 4 · e 0.2 · x − 3 − 0 ; 08 · x + 0.4 · e 0.2 · x − 3 Calculate the exact value of 15 10 f ( x ) d x . E.7732 The preceding function f is de-fined on the interval − 4 ; 10 by: f ( x )= x +4 · e − 0.5 · x Consider the function F defined by: F ( x )= − 2 · x − 12 · e − 0.5 · x . 1 How can we show that F is a primitive of f on the inter-val − 4 ; 10 ? This check is not requested. 2 Calculate: S = 4 2 f ( x ) d x Give the exact value and then the value rounded to the hundredth. E.7728 We assume that the function f is defined by: f ( x ) = x 2 − 2 · x + 1 · e − 2 x +6 Using computer algebra software, we obtain the following re-sults, which can be used without proof : L1 f(x):=(x^2-2x+1)*e^(-2x+6) → f ( x ) = x 2 − 2 · x + 1 · e − 2 x +6 L2 f’(x):=Derivative f(x) → f ( x ) = − 2 · x 2 + 6 · x − 4 · e − 2 x +6 L3 g(x):=Derivative f’(x) → g ( x ) = − 16 · x · e − 2 x +6 + 4 · x 2 · e − 2 x +6 + 14 · e − 2 x +6 L4 Factorize g(x) → 2 · e − 2 x +6 · 2 · x 2 − 8 · x + 7 L5 Solve g(x)=0 → x = − 2 + 4 2 ; x = 2 + 4 2 L6 F(x):=Primitive f(x) → F ( x ) = 1 4 · − 2 · x 2 + 2 · x − 1 · e − 2 · x +6 We set I = 5 3 f ( x ) d x . Calculate the exact value of I then the value rounded to 10 − 1 . 5. Area between two curves https://chingmath.fr nombre de jours024681012141618202224consommation(m350100150200250300350400450500550Cg chapExoCorrec/7021 sacados/7021 chapExoCorrec/7727 sacados/7727 chapExoCorrec/7731 sacados/7731 chapExoCorrec/7732 sacados/7732 Extrait Antilles-Guyane Juin 2017 chapExoCorrec/7728 sacados/7728
x-2-1012y12CfT 02468101214161820-40-30-20-101020304050Cf -10123450,511,522,53Cf E.7770 Consider f the function defined on R by: f ( x ) = x · e − x + 1 Note C f the representative curve of the function f in an or-thonormal plane and f the derivative function of f . 1 a Show that, for any real x : f ( x )=e − x · 1 − x b Show that the reduced equation of the tangent T to C f at the point of abscissa 0 is : y = x +1 . The tangent T is assumed to lie above the curve C f . 2 The curve C f and the tangent T have been plotted below in an orthonormal reference frame. a Consider the function F defined on R by: F ( x ) = e − x · − 1 − x + x Show that F is a primitive of the function f on R . b Calculate, in area units, the area of the hatched do-main between the curve C f , the tangent T and the straight lines with equations x =0 and x =1 then give the result rounded to the nearest 10 − 3 . 6. Properties of primitives E.7018 The representative curve of a func-tion f defined and continuous on the interval 0 ; 18 is given below : Which of the following is correct? a All primitives of the function f on the interval 0 ; 18 are negative on the interval 0 ; 2 . b All primitives of the function f on the interval 0 ; 18 are negative on the interval 8 ; 12 . c All primitives of the function f on the interval 0 ; 18 are increasing on the interval 0 ; 2 . d All primitives of the function f on the interval 0 ; 18 are increasing on the interval 8 ; 12 . 7. Function studies E.7039 In an orthonormal plane, we give the representative curve C f of a function f defined and deriv-able on the interval − 1 ; 5 . Let f be the derivative function of f . https://chingmath.fr chapExoCorrec/7770 sacados/7770 x-2-1012y12CfT chapExoCorrec/7018 sacados/7018 02468101214161820-40-30-20-101020304050Cf chapExoCorrec/7039 sacados/7039 Extrait Asie Juin 2016 -10123450,511,522,53Cf
-3-2-12I-2-12JOCf 1 We admit that the function F defined on − 1 ; 5 by: F ( x ) = − x 2 + 4 · x + 5 · e − x is a primitive of the function f . a Deduce the expression of f ( x ) on − 1 ; 5 . b Calculate, in units of area, the exact value of the area of the domain of the plane bounded by the curve C f , the x-axis and the straight lines with equations x =0 and x =1 . 2 Show that on the interval 1 ; 5 , the equation f ( x )=1 admits at least one solution. 8. Average E.7768 In this exercise, we study the evo-lution of French household spending on audiovisual programs (audiovisual license fees, movie tickets, videos,. . . ) . We denote D n as household spending on audiovisual pro-grams, expressed in billions of euros, during the year 1995+ n . year 1995 1996 1997 1998 1999 2000 2001 2002 n 0 1 2 3 4 5 6 7 D n 4.95 5.15 5.25 5.4 5.7 6.3 6.55 6.9 year 2003 2004 2005 2006 2007 2008 2009 2010 n 8 9 10 11 12 13 14 15 D n 7.3 7.75 7.65 7.79 7.64 7.82 7.89 8.08 Let f be the function defined, for any real number x , by: f ( x ) = − 0.0032 · x 3 + 0.06 · x 2 + 5 For any integer n satisfying 0 n 20 , we decide to model French household spending on audiovisual programs, ex-pressed in billions of euros, during the year 1995+ n by the number f ( n ) . 1 Calculate f (5) . 2 Determine the percentage p of the error made by replac-ing D 5 with f (5) . (The percentage error is obtained by calculating p = valeur réelle-valeur estimée valeur réelle and the result will be given to the nearest 0.1 % . ) . 3 Using function f , what estimate can be made of the to-tal expenditure for the year 2013 ? (The result will be rounded to the nearest hundredth of a billion euros) . 4 We want to use function f to estimate the average house-hold expenditure between January 1 er 1995 and January 1 er 2015 . To do this, we calculate: M = 1 20 · 20 0 f ( x ) d x a Determine a primitive of F of the function f over the interval 0 ; 20 . b Calculate M . E.7792 Consider the function f de-fined, for any real x in the interval − 2 ; 4 by: f ( x ) = x + 2 · e − x +1 Formal calculation software gives the following results : 1 factoriser dériver − ( x +1) ∗ exp( − x +1) x ∗ exp − x + 1 2 intégrer x +2 ∗ exp( − x +1) − ( x + 3) ∗ exp − x + 1 Using these results, answer the following questions : 1 Show that : 1 − 2 f ( x ) d x = − 4+e 3 2 Deduce the average value, rounded to the thousandth, of the function f on the interval − 2 ; 1 . 9. Area calculations E.7243 Consider the part of the plane shown in grey oppo-site : Determine the area of the shaded part. https://chingmath.fr chapExoCorrec/7768 sacados/7768 chapExoCorrec/7792 sacados/7792 chapExoCorrec/7243 sacados/7243 -3-2-12I-2-12JOCf
CfIJOABC ABCDx(en mètres)012345678910y(en mètres)12345 DI2JOC A1A2aI2JOC E.7241 Consider the function f defined piece-wise by the relation below : f ( x ) = 3 4 · x − 3 sur l’intervalle −∞ ; 4 f ( x ) = 1 4 · x − 1 on the interval 4 ; + ∞ Below is given the curve C f representative of the function f in the O ; I ; J orthonormal coordinate system : The points A , B and C are points on the curve C f where the point C has ordinate 1 . 1 Determine the coordinates of the points A , B and C . 2 Determine the area of the shaded surface. E.7358 An advertiser is considering plac-ing a rectangular billboard under part of a skateboard ramp. The profile of this ramp is modeled by the representative curve of the function f defined on the interval 0 ; 10 by: f ( x ) = 5 x + 1 This curve C f is plotted below in a frame of reference with origin O : The rectangle ABCD represents the billboard and satisfies the following constraints : the point A is located at the origin of the reference frame, the point B is on the x-axis, the point D is on the y-axis and the point C is on the curve C f . It is assumed that the point B has abscissa x =2 . Show that the area of the billboard is 3.3 m 2 , rounded to the nearest square metre. E.7242 Consider the function f defined by: f ( x ) = 2 − x for all real numbers x in the interval 0 ; 1 . We assume that : f ( x ) > 0 , for all real numbers x in the interval 0 ; 1 . Let C be the curve representing the function f in an orthonor-mal coordinate system, and D the plane domain bounded on one side by the x-axis and the curve C , and on the other side by the lines with equations x =0 and x =1 . The curve C and the domain D are shown opposite. The aim of this exercise is to divide the domain D into two domains of equal area, first by a line parallel to the y-axis (part A ) , then by a line parallel to the x-axis (part B ) . Part A Let a be a real number such that 0 a 1 . Let A 1 be the area of the do-main between the curve C , the axis Ox , the lines with equa-tions x =0 and x = a , then A 2 be the area of the domain be-tween the curve C , Ox and the lines with equations x = a and x =1 . A 1 and A 2 are expressed in units of area. Determine the value of a so that the areas A 1 and A 2 are equal Part B Let b be a positive real number. In this part, we propose to divide the domain D into two domains of equal area by the line with equation y = b . We assume that there is a unique positive real number b that is the solution. Determine the value of b . 10. Area framing with rectangles E.7036 The curve ( C ) below represents, in an orthonormal frame, a function f defined and derivable on 0.5 ; 6 . https://chingmath.fr chapExoCorrec/7241 sacados/7241 CfIJOABC chapExoCorrec/7358 sacados/7358 ABCDx(en mètres)012345678910y(en mètres)12345 chapExoCorrec/7242 sacados/7242 DI2JOC A1A2aI2JOC chapExoCorrec/7036 sacados/7036
0123456-2-112345(C x048121620y-448C −10−5310724−6xVariationdeg 234567I2345678JOCfAB Give a square of the area, in units of area and to the nearest unit, of the domain between the curve ( C ) , the x-axis and the straight lines with equations x =1 and x =2 . E.7494 The representative curve C of a function f defined on the interval 0 hasbeenplottedbelowintheplaneprovidedwithanorthonormalcoordinatesystem ; 20 . Determine a frame, of amplitude 4 , by two integers of : I = 8 4 f ( x ) d x E.7520 Among the four propositions below, only one is correct. Which one? Consider the function g defined on the interval − 10 ; 10 whose table of variations is given below : Note I = 3 − 5 g ( x ) d x . It can be stated that : a − 5 I 3 b 2 I 4 c 16 I 32 d 4 I 8 E.7440 On the graph below is plotted the curve C f of a function f defined and continuous on the interval 0 ; 7 . The points A and B have coordinates A (2 ; 5) and B (4 ; 6.8) . The straight line ( AB ) is tangent to the curve C f at the point A . a The tangent to the curve C f at the point A has the equa-tion : Assertion 1: y = − 0.9 · x + 6.8 Assertion 2: y = 0.9 · x + 3.5 Assertion 3: y = 0.9 · x + 3.2 Assertion 4: y = 1.8 · x + 1.4 b Assertion 1: f (0) 5 0 f ( x ) d x f (5) Assertion 2: 2 7 2 f ( x ) d x 7 Assertion 3: 18 5 0 f ( x ) d x 19 Assertion 4: 25 7 2 f ( x ) d x 31 https://chingmath.fr 0123456-2-112345(C chapExoCorrec/7494 sacados/7494 Extrait Antilles-Guyane Juin 2013 x048121620y-448C chapExoCorrec/7520 sacados/7520 −10−5310724−6xVariationdeg chapExoCorrec/7440 sacados/7440 234567I2345678JOCfAB
a01123456C -5-4-3-2-101234567891011-1123456C 2345678I2345678910111213141516JOC 234I2JO(d E.7240 Consider the function f defined on the interval 0 ; 1 by: f ( x )=4+e − 5 x The curve C representative of the function f has been plotted in a planar coordinate system. The hatched area D on the figure is the area bounded by the curve C , by the x-axis, the y-axis and the straight line of equation x =1 . We want to divide the hatched domain into two domains of equal area by a straight line of equation y = a , parallel to the x-axis, according to the example given below. Justify that the value a =3 is not suitable. 11. Area framing with triangles E.7239 The curve C below is the represen-tative curve in the plane provided with an orthonormal frame of reference of a function f defined on the interval − 4 ; 10 . The shaded area S on the figure is the area between the curve C , the x-axis, the straight line with equation x =2 and the straight line with equation x =4 . Determine, by graphical reading, a frame by two consecutive integers of the area of the domain S shaded on the figure. E.7238 The answer will be given without justifica-tion, with the precision allowed by the graph opposite : The graph opposite shows, in a reference frame with origin O , the representative curve C of a function f defined and differentiable on the interval 0 ; 7 . The measurement of the shaded area belongs to only one of the following intervals. Which one? a 9 ; 17 b 18 ; 26 c 27 ; 35 12. Introduction to integral calculus E.7344 Below is represented, in a reference frame O ; I ; J , the straight line ( d ) admitting as reduced equation : ( d ) : y = − 0.25 · x + 1.5 1 a Place the points A (3 ; 0) , C (0 ; 1.5) and the point B having abscissa 3 and belonging to the line ( d ) . https://chingmath.fr chapExoCorrec/7240 sacados/7240 Extrait Antilles Juin 2017 a01123456C chapExoCorrec/7239 sacados/7239 Extrait Antilles Juin 2017 -5-4-3-2-101234567891011-1123456C chapExoCorrec/7238 sacados/7238 2345678I2345678910111213141516JOC chapExoCorrec/7344 sacados/7344 234I2JO(d
aABC0,51,5I0,51,522,5JODf 2I234JO(d 2I234JO(d -2-12IJO b Determine the area of the trapezoid OABC . 2 For any strictly positive real x , we admit that the area of the trapezoid OMNC where the points M and N have abscissa x and belong respectively to to the abscissa axis and to the line ( d ) is determined by the image of x by the function F defined by: F ( x ) = − 0.125 · x 2 + 1.5 · x a Check the result of question 1 b . b Consider the domain D bounded by: the straight line ( d ) and the x-axis; the straight lines with equations x =1 and x =3 . Using the function f , determine the area of the domain D . E.7490 Let f be the function defined on 0 ; 1 by: f ( x ) = 2 − 2 · x Below is the line D f , the graphical representation of the func-tion f in an orthonormal frame O ; I ; J of the plane. The point C has coordinates (0 ; 2) . Δ is the part of the plane inside the triangle OIC . Let a be a real number between 0 and 1 ; note A the point with coordinates ( a ; 0) and B the point with coordinates D f ( a ; f ( a )) . The aim of this exercise is to find the value of a , such that the segment AB divides Δ into two parts of equal area. Determine the exact value of a , then its value rounded to the nearest hundredth. 13. Towards the use of primitives E.7345 Consider the square function denoted f : f ( x ) = x 2 The integral of the function f between 0 and a is given by the formula : a 0 f ( x ) d x = 1 3 · a 3 1 Determine the area of the hatched part : 2 Determine the area of the hatched part : E.7355 Consider the function f defined on R by the relation: f ( x ) = 1 − x 2 1 + x 2 2 Below is the graphical representation C of the function f in an orthonormal coordinate system : The shaded area is bounded by: the curve C and the x-axis; the lines with equations x = − 0.5 and x =0.5 . For any number a that is a number in the interval − 1 ; 1 , we denote I a as the value of the integral of the function f between − 1 and a . Here is a table of values rounded to 10 − 3 : a − 1 − 0.75 − 0.5 − 0.25 0 0.25 0.5 0.75 1 I a 0 0.02 0.1 0.265 0.5 0.735 0.9 0.98 1 Determine the area of the shaded region. https://chingmath.fr chapExoCorrec/7490 sacados/7490 aABC0,51,5I0,51,522,5JODf chapExoCorrec/7345 sacados/7345 2I234JO(d 2I234JO(d chapExoCorrec/7355 sacados/7355 -2-12IJO
x-4-3-2-101234y1234C x-2-1234Iy-1JOC x-3-2-1234Iy-12JOC -1234I2JO E.7359 Consider the function f defined on R by the relation: f ( x ) = 0.25 · x 3 + 0.375 · x 2 − 1.5 · x + 1 Consider the domain D bounded by: the curve C and the x-axis. the two lines with equations x = − 2 and x =2 . For any number a belonging to the interval − 3 ; 3 , we de-note I a as the value of the integral between − 3 and a . Here is a table of values rounded to 10 − 3 : a − 3 − 2 − 1 0 1 2 3 I a 0 3.0625 6.25 8.0625 8.5 9.0625 12.75 Determine the area of the domain D . 14. Calculator and integrals E.7360 Consider the function f defined on R by the relation: f ( x ) = 2 · x x 2 + 1 Below is given the curve C representative of the function f : The shaded area above is D . 1 Describe the domain D . 2 Using the calculator, determine the area of the domain D , rounded to the nearest 10 − 4 . E.7361 Consider the function f defined on R by the relation: f ( x ) = 0.5 · x 2 − 0.5 · x − 1 Below is given the curve C representative of the function f : The shaded area above is D . Using the calculator, determine the area of the domain D to the nearest 10 − 4 . 15. Junction of two curves E.7362 Consider the two reference functions : the square function noted f and the inverse function noted g . We give the representation of the curves C f and C g respec-tively of the functions f and g in the reference frame O ; I ; J below : Consider the domain D shaded above and : bounded by the straight lines of equations x =0 and x =1 ; and between the curves C f and C g and the y-axis. 1 a For any positive or zero real a , we assume that the integral of the function f between 0 and a has the ex-pression : a 0 x 2 d x = 1 3 · a 3 Determine the domain under the curve C f between 0 and 1 . b Using a calculator, determine the measure of the do-main under the curve C g between 1 and 3 , rounded to the nearest thousandth. 2 Deduce the measure of the greyed-out D domain, rounded to the nearest thousandth. https://chingmath.fr chapExoCorrec/7359 sacados/7359 x-4-3-2-101234y1234C chapExoCorrec/7360 sacados/7360 x-2-1234Iy-1JOC chapExoCorrec/7361 sacados/7361 x-3-2-1234Iy-12JOC chapExoCorrec/7362 sacados/7362 -1234I2JO
-2-1234I2JOCfCg2;5 x-3-2-1234Iy23JOC(d 23I2JO(d1(d2 E.7493 Consider the two functions f and g de-fined on R by the relation: f ( x ) = − 0.24 · x 2 + 0.6 · x + 1.4 ; g ( x ) = 0.4 · x + 0.4 In a coordinate system O ; I ; J , we have the representa-tions of the representative curves C f and C g respectively of the functions f and g : For a a real number between − 1 ; 4 , we have the following additional information : For a , a real number in the interval 0 ; 4 , we denote I a as the area of the domain under the curve C f between the lines x =0 and x = a . Here is a table of values of I a rounded to 10 − 3 : a 0 0.5 1 1.5 2 2.5 3 3.5 4 I a 0 0.765 1.62 2.505 3.36 4.125 4.74 5.145 5.28 To calculate the area of the region under the curve C g between the lines x = − 1 and x = a , we use the following integral calculation: a 0 g ( x ) d x = 0.2 · a 2 + 0.4 · a Determine the shaded area, showing your steps. 16. Domain between two curves E.7363 Consider the function f defined on R by: f ( x ) = − 0.6 · x 2 + 0.4 · x + 1.9 Below is the curve C representing the function f in the coor-dinate system O ; I ; J : Consider the line ( d ) , representative curve of the function g defined by: g ( x ) = − 0.2 · x + 0.7 We assume that the curve C lies above the line ( d ) on the interval − 1 ; 2 . The domain D is the domain of the plane with the following characteristics : delimited by the lines x =0 and x =2 ; located between the line ( d ) and the curve C . The following data will be used : For any number a belonging to the interval − 1 ; 2 , we denote I a as the area of the domain bounded by the curve C , the x-axis, and the lines with equations x = − 1 and x = a . Below is a table of values rounded to 10 − 3 : a − 1 − 0.5 0 0.5 1 1.5 2 I a 0 0.625 1.5 2.475 3.4 4.125 4.5 For any number a ∈ 0 ; 2 , we assume that the integral of the function g from 0 to a has the value : a 0 g ( x ) d x = − 0.1 · a 2 + 0.7 · a Determine the area of the domain D . E.7444 Consider the two functions f and g de-fined on R by: f ( x ) = 0.6 · x + 0.2 ; g ( x ) = − 0.4 · x + 1.2 whose graphical representations are respectively the lines ( d 1 ) and ( d 2 ) shown below : For any number a belonging to the interval 0 ; 3 , we de-note I a the domain bounded by the line ( d 1 ) , the x-axis, and the lines with equations x =0 and x = a . Here is a table of values : a 0 0.5 1 1.5 2 2.5 3 I a 0 0.175 0.5 0.975 1.6 2.375 3.3 For a real number a belonging to 1 ; 3 , we assume that the area of the domain located under their respective curves is determined by the integral calculation: a 1 g ( x ) d x = − 0.2 · a 2 + 1.2 · a − 1 Deduce the area of the shaded region. 17. Area between two curves and calculators https://chingmath.fr chapExoCorrec/7493 sacados/7493 -2-1234I2JOCfCg2;5 chapExoCorrec/7363 sacados/7363 x-3-2-1234Iy23JOC(d chapExoCorrec/7444 sacados/7444 23I2JO(d1(d2
xxyy-0,200,20,40,60,811,2-0,20,20,40,60,811,21,41,61,822,22,42,6CgCf 024681012141618100200300400500600700800900CfAB E.7436 A company wants to use a decorative motif for its communication. To make this pattern, we model its shape using two functions f and g defined for any real x from 0 ; 1 by: f ( x ) = 1 − x · e 3 x ; g ( x ) = x 2 − 2 · x + 1 Their representative curves will be noted C f and C g . Determine the area of the shaded part, on the graph, between the two curves C f and C g on the interval 0 ; 1 , rounded to the nearest thousandth. (The steps in its reasoning will be justified) 18. Unclassified financial years E.8422 Part A The curve C below, associated with a function f defined on the interval 0 ; 19 , represents the daily audience of a televi-sion channel between January 1 er 2000 (year number 0 ) and January 1 er , 2019 (year number 19 ) , i.e., daily viewers, in thousands. Thus, on January 1 er 2000 , the channel was watched by ap-proximately 460 000 viewers. 1 Describe the evolution of the daily audience for this tele-vision channel between January 1 er and January 1 er . 2 Give an approximate value for the number of viewers on January 1 er 2014 . 3 The line ( AB ) , where the points A and B have coor-dinates A (0 ; 460) and B (3 ; 82) , is the tangent to the curve C at point A . Determine the value of f (0) where f denotes the deriva-tive of the function f represented by C ? Part B We now want to predict the evolution of this television chan-nel’s audience over the next ten years. We consider that the daily number (expressed in thousands) of viewers of the channel is modeled by the function f defined on the interval 0 ; 29 by f ( x ) = 20 x 2 − 80 x + 460 · e − 0 , 1 x where x represents the number of years since 2000 (for exam-ple x =19 for the year 2019) . 1 Give an approximate value to the nearest thousand of the number of viewers of the channel on January 1 er 2014 . 2 We denote f the derivative of f over the interval 0 ; 29 . a Show that f is defined by: f ( x ) = − 2 x 2 + 48 x − 126 · e − 0 , 1 x b Consider the equation : − 2 x 2 +48 x − 126=0 A computer algebra system gives : Instruction : Result Solve( − 2 x 2 + 48 x − 126 = 0 ) 3 and 21 Find this result by calculation. c Deduce the sign of f ( x ) on the interval 0 ; 29 and construct the table of variations of f on the interval 0 ; 29 . Round the elements of the table to the nearest whole number. d Will the daily number of viewers of this television chan-nel exceed one million before the year 2029 ? Justify your answer. 3 Show that equation f ( x )=800 has a unique solution ¸ in the interval 3 ; 21 . Determine an amplitude bound 1 for ¸ . In which year will the daily number of viewers of the television channel exceed 800 000 ? 4 We assume that the function F defined on the interval 0 ; 29 by: F ( x ) = − 200 x 2 − 3 2000 x − 36 600 · e − 0 , 1 x is a primitive of the function f . Determine, to the nearest thousand, the average daily au-dience of viewers of the television channel between Jan-uary 1 er 2018 and January 1 er 2019 . https://chingmath.fr chapExoCorrec/7436 sacados/7436 xxyy-0,200,20,40,60,811,2-0,20,20,40,60,811,21,41,61,822,22,42,6CgCf sacados/8422 024681012141618100200300400500600700800900CfAB
En année, après2000012345678910111213141516En10000passagers12345678910CACP -8-7-6-5-4-3-2-1234I-4-3-2-123JOCf E.7057 Starting January 1, 2000 , an air-line is offering a new ticket purchase option, the Advantage option, which is in addition to the existing Privilege option. A study has modeled the evolution of the number of passen-gers carried since year 2000 , and the airline acknowledges that this model is valid for the period from year 2000 to year 2016 . The number of passengers choosing the Privilege option is modeled by the function P defined on the interval 0 ; 16 , and the number of passengers choosing the Advantage package is modeled by the function A defined on the interval 0 ; 16 . The graph below shows the representative curves C p and C A of these two functions. When x represents the time in years from year 2000 , P ( x ) rep-resents the number of passengers, expressed in tens of thou-sands, choosing the Privilege option, and A ( x ) represents the number of passengers, expressed in tens of thousands, choos-ing the Advantage option. The estimates will be obtained by graph reading. 1 Provide an estimate of the number of passengers who, during the year 2002 , chose the Privilege . 2 package. Estimate the difference that the company can expect in 2015 between the number of passengers who chose the Advantage package and those who chose the Privilege package. 3 How can the coordinates of the intersection point of the two curves be interpreted in relation to the proposed sit-uation? 4 Justify that, according to this model, the airline can esti-mate that the total number of passengers who chose the Privilege package during the period between 2007 and 2015 will be between 240 000 and 320 000 E.5556 We denote by f the function de-fined on the interval 0 ; 6 by: f ( x ) = 1 − ( x + 1) · e − x 1 Show that f ( x ) = x · e − x where f denotes the derivative function of the function f . 2 Demonstrate that the equation f ( x )=0.6 admits a unique solution ¸ on the interval 0 ; 6 . Determine a rounded value of ¸ at 0.01 . 3 We admit that the function F defined on 0 ; 6 by: F ( x ) = x + ( x + 2) · e − x is a primitive of f on 0 ; 6 . Give the exact value and then a value rounded to 10 − 3 of : I = 6 0 f ( x ) d x E.5561 Consider the function f defined on R whose representative curve C f is plotted below in an orthonormal frame : 1 By graphical reading, indicate the values of f (2) and f (0) . 2 Freehand, reproduce the reference frame and the repre-sentative curve of the function f , then cross-hatch a do-main of the plane having the area value A where : A = 2 0 f ( x ) d x . (no attempt will be made to not determine the value of A .) 3 a Draw, freehand, the tangent ( T ) to the curve C f at the point of abscissa 2 . b By graphical reading, give the value of the derivative number f (2) , rounded to the nearest tenth. https://chingmath.fr chapExoCorrec/7057 sacados/7057 En année, après2000012345678910111213141516En10000passagers12345678910CACP chapExoCorrec/5556 sacados/5556 chapExoCorrec/5561 sacados/5561 -8-7-6-5-4-3-2-1234I-4-3-2-123JOCf
-10-9-8-7-6-5-4-3-2-12345678910I-4-3-2-12345678JOC1C2C3 123456festdé∏niesur:−∞0−∞00∞RLafonctionfadmetpourdérivéelafonctionfdontl’expressionest2x·e−2x−2x·e−2x−2·e−2x2·e−2xSursonensemblededé∏nition,lafonctionfestcroissantedécroissanteconstanteNicroissante,nidécroissanteAupointd’abscisse0,lacourbeCfadmetpourtangenteladroited’équation−2x−2x1−2x−12x−1Uneprimitivedelafonctionfadmetpourexpression−2·e−2x2·e−2xe−2x2−e−2x2LedomaineDdé∏nieparl’axedesabscissesetlacourbeCf,lesdroitesd’équationsxetxapourmesured’aire12·1e−212·1−e−212·−1e−212·−1−e−2 E.5562 Consider the function f defined on R by the relation: f ( x ) = ( − x + 2) · e 0.5 x Note C f the representative curve of the function f in an or-thonormal frame. 1 a Consider the function F defined on R by the rela-tion : F ( x ) = ( − 2 x + 8) · e 0.5 x Show that the function F is a primitive of the function f on R . b Calculate the exact value of 2 0 f ( x ) d x and give a value rounded to the nearest 10 − 2 . 2 Consider G another primitive of f on R . Of the three curves C 1 , C 2 and C 3 below, only one is the graphical representation of G . Determine the appropriate curve and justify the answer: E.5554 Consider the function f defined by the relation: f ( x ) = e − 2 x We denote C f its representative curve in an orthonormal co-ordinate system. For each statement, only one answer is correct. Indicate the correct answer and justify your reasoning : https://chingmath.fr chapExoCorrec/5562 sacados/5562 -10-9-8-7-6-5-4-3-2-12345678910I-4-3-2-12345678JOC1C2C3 chapExoCorrec/5554 sacados/5554 123456festdé∏niesur:−∞0−∞00∞RLafonctionfadmetpourdérivéelafonctionfdontl’expressionest2x·e−2x−2x·e−2x−2·e−2x2·e−2xSursonensemblededé∏nition,lafonctionfestcroissantedécroissanteconstanteNicroissante,nidécroissanteAupointd’abscisse0,lacourbeCfadmetpourtangenteladroited’équation−2x−2x1−2x−12x−1Uneprimitivedelafonctionfadmetpourexpression−2·e−2x2·e−2xe−2x2−e−2x2LedomaineDdé∏nieparl’axedesabscissesetlacourbeCf,lesdroitesd’équationsxetxapourmesured’aire12·1e−212·1−e−212·−1e−212·−1−e−2