Grade 12 - Comp. / Continuity, derivability, limits 23 exercises (100% corrected)

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−515104-6-1-13xVariationdef 1. Study of functions E.7006 Consider the function g defined on the interval 1 ; 15 by: g ( x ) = − 0.6 · x + 4 + e − x +5 We admit that the function g is derivable on the interval 1 ; 15 and we note g its derivative function : 1 a Calculate g ( x ) for any real x in the interval 1 ; 15 . b Deduce that the function g is decreasing on the interval 1 ; 15 . 2 a Draw up the table of variations of the function g on the interval 1 ; 15 , specifying the values g (1) and g (15) rounded to the unit. b The table of variations shows that the equation g ( x )= 0 admits a single solution ¸ on the interval 1 ; 15 . Give an approximate value of ¸ to the nearest 0.1 . c Deduce from the previous questions the table of signs of g ( x ) on the interval 1 ; 15 . 2. Intermediate value theorem E.7521 Consider the function f defined on the interval − 20 ; 20 by: f ( x ) = − 2 · x + 30 · e 0.2 · x − 3 1 a Show that f ( x )= − 0.4 · x +4 · e 0 , 2 · x − 3 for any real x from the interval − 20 ; 20 . b Draw up the table of variations of the function f on the interval − 20 ; 20 . The exact value of the maximum of f will be specified. 2 a Show that, on the interval − 20 ; 20 , the equation f ( x )= − 2 admits a single solution ¸ . b Give a frame for ¸ of amplitude 0.1 . E.7430 We admit that the function f is defined for any real x of − 3 ; 2 by: f ( x ) = − x 2 + 2.5 · x − 2 · e x + 5 1 Verify that for any real x in the interval − 3 ; 2 : f ( x ) = − x 2 + 0.5 · x + 0.5 · e x 2 Study the sign of f then draw up the table of variations of f on − 3 ; 2 . 3 a Justify that the equation f ( x )=0 admits a unique solution ¸ on 1 ; 2 . b Give the value of ¸ rounded to the hundredth. 3. Intermediate value theorem and quotient E.7435 Consider the function f de-fined and differentiable on the interval 1 ; 25 by: f ( x ) = 10 − e 0.2 x +1 x A computer algebra system provides the following results that can be used : f(x) :10 - e^(0.2 x+1)/x x ↦−→ 10 − exp(0.2 x + 1) x factorize(derive(f(x))) exp 0.2 · x + 1 · 1 − 0.2 · x x 2 factorize(derive(derive(f(x)))) exp 0.2 · x + 1 · − x 2 + 10 · x − 50 25 · x 3 1 Find the factorized expression of f ( x ) where f is the derivative of f . 2 Study the sign of f on the interval 1 ; 25 and draw up a table of variations of f on the interval 1 ; 25 . Round the values to three decimal places. 3 Consider the equation f ( x )=0 . a Show that the equation f ( x )=0 has no solution on the interval 1 ; 5 . b Show that equation f ( x )=0 has a unique solution ¸ on the interval 5 ; 25 . c Determine an amplitude range 10 − 2 for the solution ¸ 4. Introduction to intermediate values E.7286 Consider a function f which admits the following table of variations : https://chingmath.fr chapExoCorrec/7006 sacados/7006 chapExoCorrec/7521 sacados/7521 Extrait Centres Etrangers Juin 2017 chapExoCorrec/7430 sacados/7430 Extrait Asie Juin 2017 chapExoCorrec/7435 sacados/7435 Extrait Antilles-Guyane Septembre 2017 chapExoCorrec/7286 sacados/7286 −515104-6-1-13xVariationdef
−4243−92-1xVariationdef −534107-5-112xVariationdef 01∞−121xf(xa 01∞−32−2xf(xb −∞01∞0-231xf(xc 025∞025−2xf(xd 025∞0-21xf(xe 025∞0-5−2xf(xf -4-3-2-1234I-2-123JO 1 Justify that the equation f ( x )=7 admits no solution. 2 Without justification, give the number of solutions to the equation f ( x )=0 on the interval − 5 ; 10 . E.7287 Consider a function f defined on the interval − 4 ; 4 admitting the following table of variations : No justification for the answers is expected. 1 How many solutions has the equation f ( x )=0 ? 2 Discuss the number of solutions to the equation f ( x )= m as a function of the following values of m : a f ( x ) = − 6 b f ( x ) = − 4 c f ( x ) = 2 E.7863 Consider a function f defined on − 5 ; 3 ∪ 3 ; 10 and whose table of variations is given below : Without justification, give the number of solutions to the equation : f ( x )=0 . E.7288 Consider the function f defined on R by the relation: f ( x ) = 2 · x 3 − 3 · x 2 − 12 · x + 1 1 a Determine the expression of the function f derived from the function f . b Draw up the table of variations of the function f on the interval − 2 ; 3 . 2 Without justification, give the number of solutions of each of the following equations on the interval − 2 ; 3 : a f ( x )= − 10 b f ( x )=15 c f ( x )= − 6 E.7367 We consider a function f defined on R + of which we have partial results of its study via a formal calculus software: L1 f(x) : =. . . . . . f ( x ) = : : : : : : L2 g(x): =Dérivée f(x) g ( x ) = − 5 · x 2 − 3 · x + 2 2 · x L3 Résoudre f(x)=0 x =0 ; x =1 L4 Résoudre g(x)=0 x = 2 5 Of the tables of variations below, only one is the table of variations of the function f . Which is it? 5. Introduction to continuity E.7316 Below is the representative curve C f of the function f 1 Give the intervals on which the function f is continuous. 2 Give the intervals on which the function f is monotonic. E.7311 Consider the function f defined by the relation: f ( x ) = x − 1 2 x 2 − 4 · x + 3 1 a Determine the images of 0 and 2 by the function f . b For the equation f ( x )=0 , conjecture the existence or non-existence of solution for this equation on the inter-val 0 ; 2 . 2 a Using the calculator, draw the representative curve of the function f . b Does the function f admit antecedents of 0 in the in-terval 0 ; 2 . https://chingmath.fr chapExoCorrec/7287 sacados/7287 −4243−92-1xVariationdef chapExoCorrec/7863 sacados/7863 −534107-5-112xVariationdef chapExoCorrec/7288 sacados/7288 chapExoCorrec/7367 sacados/7367 01∞−121xf(xa 01∞−32−2xf(xb −∞01∞0-231xf(xc 025∞025−2xf(xd 025∞0-21xf(xe 025∞0-5−2xf(xf chapExoCorrec/7316 sacados/7316 -4-3-2-1234I-2-123JO chapExoCorrec/7311 sacados/7311
−3−101−6−1−24xVariationdef −1123−22−1−0;5xVariationdef 6. Intermediate value theorem E.7056 The table of variations of a func-tion f defined on the interval − 3 ; 1 is given below : Determine whether the following proposition is true or false, justifying the answer: Proposition: the equation f ( x )=0 admits a single so-lution in the interval − 3 ; 1 . E.7041 We are given the table of varia-tions of a function f defined on the interval − 1 ; 3 : Which of the following four statements is correct? In the interval − 1 ; 3 , the equation f ( x ) = 0 has : a exactly 3 solutions b exactly 2 solutions c exactly 1 solutions d no solutions E.7328 Consider the function f defined by: f ( x ) = 2 · x 3 + 3 · x 2 − 12 · x + 4 1 Draw up the table of variations of the function f . 2 a Justify that the equation f ( x )=0 admits a single solution, denoted ¸ , on the interval − 2 ; 1 . b Using a calculator, give a value approximated to one hundredth of the solution ¸ . E.7343 Consider the function f defined on R by the relation: f ( x ) = 3 · x + 4 x 2 + 1 1 Draw up the table of variations of the function f on R . 2 a Justify that the equation f ( x )=1 has a unique solu-tion on the interval − 3 ; 1 3 . b Using a calculator, determine the approximate value to the nearest hundredth of this solution. E.7366 Consider the function f defined on R by the relation: f ( x ) = 4 · x + 3 x 2 + 1 1 a Establish that the function f , derived from the func-tion f , has the expression : f ( x ) = − 4 · x 2 − 6 · x + 4 x 2 + 1 2 b Draw up the table of variations of the function f on − 5 ; 5 2 a Justify that the equation f ( x )=3 has two solutions, denoted ¸ and ˛ , on the interval − 5 ; 5 . b Give approximate values of ¸ and ˛ to the nearest thousandth. E.7289 Consider the function f defined on R by the relation: f ( x ) = 2 · x + 1 x 2 + 2 1 a Justify that the function f derived from the func-tion f has the expression : f ( x ) = − 2 · x 2 − 2 · x + 4 x 2 + 2 2 b Draw up the table of variations of the function f on the interval − 3 ; 3 . 2 Deduce the number of solutions to the equation f ( x )=0 on the interval − 3 ; 3 E.7327 Consider the function f defined on R + by the relation: f ( x ) = x − 1 · x Using computer algebra software, we obtain the following re-sults, which can be used without proof : L1 f(x) :=(x-1)* √ (x) f ( x ) = ( x − 1) · x L2 g(x):= Derivative f(x) g ( x ) = 3 · x − 1 2 · x L3 Solve f(x)=0 x =1 L4 Solve g(x)=0 x = 1 3 1 Draw up the table of variations of the function f . 2 a Determine the images of 1 3 and 9 by the function f . b Deduce that the equation f ( x )=6 has a unique solu-tion on 1 3 ; 9 . Then, justify that this equation also has a unique solution on R + . c Using a calculator, determine the exact value of the unique solution of the equation f ( x )=6 . https://chingmath.fr chapExoCorrec/7056 sacados/7056 Extrait Liban Juin 2015 −3−101−6−1−24xVariationdef chapExoCorrec/7041 sacados/7041 Extrait Antilles-Guyane Juin 2016 −1123−22−1−0;5xVariationdef chapExoCorrec/7328 sacados/7328 chapExoCorrec/7343 sacados/7343 chapExoCorrec/7366 sacados/7366 chapExoCorrec/7289 sacados/7289 chapExoCorrec/7327 sacados/7327
x-3-2-1012y-3-2-112345Cf −4−1140;5−1;510;5xVariationdef -4-3-2-1234I-2-12JOC1 -4-3-2-1234I-2-12JOC2 -4-3-2-1234I-2-12JOC3 E.7365 Below is the representative curve C f of a function defined and differentiable on the interval − 3 ; 2 . Let f be the derivative of the function f . The point A with coordinates (0 ; 3) belongs to the curve C f . B is the point with abscissa 1 belonging to the curve C f . We have the following information : the function f is strictly decreasing on the intervals − 3 ; − 0.5 and 1 ; 2 and it is strictly increasing on − 0.5 ; 1 ; the function f is strictly decreasing on the intervals − 3 ; − 2 and 1 2 ; 2 and strictly increasing on − 2 ; 1 2 ; The line Δ with equation y =0.5 · x +3 is tangent to the curve C f at point A ; The tangent Δ to the curve C f at point B is parallel to the x-axis. Each answer must be justified. An unjustified answer will not earn any points. 1 Give the value of f (1) . 2 What is the sign of f (2) ? 3 Give the value of f (0) . 4 Give the number of solutions to the equation f ( x )=0.5 . 7. Derivative functions and the intermediate value theorem E.7326 Consider the function f defined and derivable on R and whose derivative function admits the sign table below : x −∞ 1 5 + ∞ f ( x ) − 0 + 0 − Furthermore, the equation f ( x )=0 admits the following set of solutions : S = 3 Draw up the sign table, justifying your approach. E.7294 Consider the function f defined on the interval − 4 ; 4 whose derivative function f admits the table of variations below : In a reference frame O ; I ; J orthogonal, are shown the rep-resentative curves C 1 , C 2 and C 3 of three functions. Of these curves only one is a representation of the function f . Which is it? Justify your answer. https://chingmath.fr chapExoCorrec/7365 sacados/7365 Extrait Asie Juin 2017 x-3-2-1012y-3-2-112345Cf chapExoCorrec/7326 sacados/7326 chapExoCorrec/7294 sacados/7294 −4−1140;5−1;510;5xVariationdef -4-3-2-1234I-2-12JOC1 -4-3-2-1234I-2-12JOC2 -4-3-2-1234I-2-12JOC3
00,20,40,60,810,20,40,60,81DyxCCE 8. Unclassified financial years E.7058 An audit firm was tasked with studying the distribution of salaries in two subsidiaries of a company, called A and B . For the study, salaries are ranked in ascending order. The auditing firm modeled the salary distribution using func-tion u for subsidiary A and function v for subsidiary B . Functions u and v are defined on the interval 0 ; 1 by: u ( x ) = 0 ; 6 · x 2 + 0 ; 4 · x ; v ( x ) = 0 ; 7 · x 3 + 0 ; 1 · x 2 + 0 ; 2 · x The representative curves C and C of the functions u and v are plotted below. 1 Determine the representative curve of the function u , jus-tifying your answer. 2 When x represents a percentage of employees, u ( x ) and v ( x ) represent the percentage of the total payroll shared by these employees in their respective subsidiaries. Example: for curve C , point E (0 ; 60 ; 0 ; 3072) means that 60 % of the employees with the lowest salaries share 30 ; 72 % of the total payroll. a Calculate the percentage of the total payroll shared by the 50 % employees of subsidiary A with the lowest salaries. b For the 50 % employees with the lowest salaries, which of the subsidiaries, A or B , distributes the largest share of the total payroll? c Which subsidiary appears to have the most unequal distribution of salaries? E.5558 Consider the function f defined on R by: f ( x ) = x · e x − e x − 1 1 Show that the function f is strictly increasing at 0 ; + ∞ 2 a Show that the equation f ( x )=0 has a unique solu-tion, denoted ¸ , in 0 ; 2 . b Using the calculator, give a unit frame for ¸ . 3 Deduce the sign table for f ( x ) on 0 ; + ∞ . https://chingmath.fr chapExoCorrec/7058 sacados/7058 00,20,40,60,810,20,40,60,81DyxCCE chapExoCorrec/5558 sacados/5558 Extrait de Liban Mai 2013