Grade 12 / Logarithms 70 exercises (including 68 corrected)

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-4-3-2-1234I-2-1234JOCeC ChingQuizz : 5 exercises available for Quizz assessment : 1. Introduction E.3709 Let f be the function defined by on 0 ; 1 : f ( x ) = x − 2 √ x + 1 Show that for all x belonging to the interval 0 ; 1 : f ◦ f ( x ) = x E.3877 In the plane provided with the orthonor-mal reference frame O ; I ; J , consider the curves C e and C representative of the exponential and logarithmic functions : 1 Graphically determine the values of the following two im- ages : exp ◦ ln (1) ; ln ◦ exp (0) 2 Using the first bisector of the plane, determine, if possi-ble, the value of the following images : a exp ◦ ln (2) b exp ◦ ln (3) c exp ◦ ln ( − 1) d ln ◦ exp ( − 1) e ln ◦ exp (1) f ln ◦ exp (1.5) E.3878 Solve the following equations : a e x = 3 b e 3 x − 1 = 4 c 2 · e x = − 2 d ln( x ) = 2 e ln(2 · x +1) = 5 f ln x + 3 2 = 3 E.3879 1 a Solve each of the following inequations : 3 x + 1 > 0 ; − x + 2 > 0 b Determine the definition set of the functions f and g : f ( x ) = ln(3 x +1) ; g ( x ) = ln( − x +2) 2 Determine the definition set of the following functions : a h ( x ) = ln x 2 d j ( x ) = ln e x − 1 e k ( x ) = ln e x − e − x f ‘ ( x ) = 1 ln( x ) − 1 2. Algebraic manipulations E.3880 Express each of the following numbers in the form : a + b · ln 2 a;b ∈ R a ln(4) b ln 2 2 c ln(6) − ln 3 2 d ln 2 · e 2 e ln 2 e 3 f ln e 5 E.3881 Simplify the following expressions : a ln e 2 · x b ln 2 · e 3 x − 1 c 2 · ln x + ln x d ln( x +1) − ln( x − 1) E.6874 Establish the following equations : a ln 16 + ln 4 = 6 · ln 2 b ln 7 5 = − ln 5 7 c ln 3 2 − ln 9 8 = ln 4 3 d ln 8 ln 2 = 3 E.6875 Consider the function f defined by: f ( x ) = ln e x +e − x 1 Justify that the function f is defined on R . 2 On R , establish the identity: f ( x )= x +ln 1+e − 2 x 3 Deduce that the function f is positive on R + . E.6876 1 What can be said about the image of two opposite num-bers by the exponential function? 2 What can be said about the image of two inverse numbers by the logarithm function? https://chingmath.fr chapExoCorrec/3709 sacados/3709 chapExoCorrec/3877 sacados/3877 -4-3-2-1234I-2-1234JOCeC chapExoCorrec/3878 sacados/3878 chapExoCorrec/3879 sacados/3879 chapExoCorrec/3880 sacados/3880 chapExoCorrec/3881 sacados/3881 chapExoCorrec/6874 sacados/6874 chapExoCorrec/6875 sacados/6875 chapExoCorrec/6876 sacados/6876
E.3882 Consider the function f defined by: f ( x ) = ln x + x 2 +1 + ln − x + x 2 +1 1 Consider the function u defined by: u ( x ) = ln − x + x 2 +1 By a disjunction of cases, establish that the function u is defined on R . 2 Admit that the function f is defined on R . Establish that the function f is the null function. E.5174 Let f be the function defined on 0 ; + ∞ by: f ( x )= x +e − x Consider the sequence u n n 1 with positive terms defined by: u 1 = 0 ; u n +1 = f ( u n ) = u n + e − u n We admit that for any positive number x , we have the rela-tion : ln(1+ x ) x 1 Deduce that, for any non-zero natural number n : ln( n +1) ln( n ) + 1 n 2 Demonstrate that, for any non-zero natural number n : f ln( n ) = ln( n ) + 1 n 3 Demonstrate by recurrence that, for any non-zero natu- ral number n : ln n u n 4 Deduce the limit of the sequence u n n 1 . E.4232 Consider the function f defined on R by: f ( x ) = ln e x +2 · e − x The curve ( C ) representative of the function f in a reference frame. Note ( d ) the straight line of equation y = x 1 Show that, for any real x : f ( x ) = x + ln 1+2 · e − 2 · x 2 Study the relative position of the curve C f and the straight line ( d ) . E.10387 Consider the function f defined on R ∗ + by: f ( x ) = 6 × ln(2 x ) We define the function g on R ∗ + by: g ( x ) = f (2 x ) 1 Establish that for all x belonging to the interval R ∗ + , we have : g ( x ) − f ( x ) = ln(64) 2 In an orthonormal frame of reference, consider the two curves C f and C g representative of the functions f and g respectively. What interpretation can be given to the question 1 in relation to the curves C f and C g . 3. Logarithmic equations and inequalities E.3883 For each question, specify the solution set of the equation and then solve it: a ln(5 x +1) = ln( x − 1) b 2 · ln(3 − x ) = ln(2) c ln(3 x +1) = 5 d 3 · e 2 x − 1 = 2 e ln x 2 − ln x − 2 = 0 f e 2 x + 4 · e x − 1 = 0 E.3884 Solve the following two systems of equa-tions in R : a 2 · ln x − 3 · ln y = − 11 ln x + ln y = 2 b ln(2 · x + y ) = 0 ln( x ) + ln( y ) = 1 E.5899 1 Solve in − 1 ; + ∞ , the inequation : ( E ): ln( x +1) ln x 2 +1 2 Solve in − 3 ; 2 , the inequation : ( F ): ln(4 − 2 x ) < ln( x +3) E.5897 Consider the inequation : ( E ): ln( x +1) ln(2 − x ) 1 Give the largest part I of R over which the two expres-sions ln( x +1) and ln(2 − x ) are defined. 2 Solve the inequation ( E ) . E.3885 Determine the solution set for each of these inequalities, then solve them : E.4192 A container holds a gas made up of two kinds of particles : 75 % of A particles and 25 % of B particles. The A particles are radioactive and spontaneously transform into B particles ; each A particle yields a B particle as it transforms. We note p ( t ) the proportion of A particles in the gas. Thus, at instant t =0 , we have p (0)=0.75 . More generally, if t is expressed in years, we have : p ( t ) = 0.75 · e − λ · t où – is a real constant. The half-life of particles of type A is equal to 5 730 years. 1 Calculate – ; we’ll take a decimal approximation to 10 − 5 by default. 2 After how many years 10 % will particles of type A have transformed into particles of type B ? 3 Determine the value of t for which there will be as many particles of type A as particles of type B (round off to unity) . https://chingmath.fr chapExoCorrec/3882 sacados/3882 chapExoCorrec/5174 sacados/5174 chapExoCorrec/4232 sacados/4232 sacados/10387 chapExoCorrec/3883 sacados/3883 chapExoCorrec/3884 sacados/3884 chapExoCorrec/5899 sacados/5899 chapExoCorrec/5897 sacados/5897 chapExoCorrec/3885 sacados/3885 chapExoCorrec/4192 sacados/4192 Extrait France Septembre 2004
E.6254 Consider the function f defined on 0 ; + ∞ by: f ( x ) = 5 · e − x − 3 · e − 2 x + x − 3 Note C f the graphical representation of the function f and D the straight line with equation y = x − 3 in an orthogonal plane. The curve C f is assumed to lie above the line ( D ) on the interval 0 ; + ∞ . We note M the point of abscissa x of the curve C f , N the point of abscissa x of the line D and we are interested in the evolution of the distance MN . 1 For all x in the interval 0 ; + ∞ , determine the expres-sion for the distance MN as a function of x . We’ll note g this function. 2 Let g be the derivative function of the function g on the interval 0 ; + ∞ . For any x of the interval 0 ; + ∞ , calculate g ( x ) . 3 Show that the function g has a maximum on the interval 0 ; + ∞ to be determined. Give a graphical interpretation. E.10389 Solve the inequation : ln x + 1 ln 2 x − 3) 4. Power equations and inequalities E.3886 Solve, in Z , the following inequalities: a 5 n 2 b 0.1 n 2 c ln 2 n < e 2 d ln 2 n − ln 3 n 2 E.5898 1 Consider the sequence u n geometric with first term 5 and reason 2 3 . a What is the limit of the sequence u n ? b Determine the smallest natural integer n realizing the inequality: u n < 0.01 2 Consider the sequence v n geometric of first term 2 and reason 4 5 . Let S n be the sum of the ( n +1) first terms of the sequence v n : S n = v 0 + v 1 + · · · + v n a Justify that the sequence S n is increasing. b Express the sum S n as a function of rank n . c Justify that from a certain rank, all terms of the se-quence S n belong to the interval 9.9 ; 10 . d Determine the smallest natural integer n verifying S n ∈ 9.9 ; 10 . E.3288 Consider the function f defined on 1 ; + ∞ by: f ( x ) = 1 + 1 x 10 1 Study the direction of variation and limit in + ∞ of the function f . 2 Show that there exists in the interval 1 ; + ∞ a single real number ¸ such that f ( ¸ )=1.9 . 3 Determine the natural number n 0 such that : n 0 − 1 ¸ n 0 . 4 Show that, for any natural number n greater than or equal to 16 , we have : 1 + 1 n 10 1.9 . E.5204 A company manufactures articles. Following a series of inspections, an item is found to have at least one defect with a probability of 0.0494 . A small retailer places an order for items with this company. The stocks are large enough for the selection of these items to be assimilated to a successive and independent draw. 1 Ordering 25 items, calculate, to within 10 − 3 , the prob-ability that there are more than 2 faulty items in his order. 2 He wants the probability of having at least one defective item in his order to remain below 50 % . Determine the maximum value of the number n of items he can order. 5. Equations and inequalities with change of variables E.5846 Solve the equation and inequation be-low : a e 2 x + 2 · e x − 3 = 0 b e 2 x + e x − 2 < 0 E.6872 Solve the inequation : 2 · e 2 · x +6 · e x − 8 < 0 https://chingmath.fr chapExoCorrec/6254 sacados/6254 sacados/10389 chapExoCorrec/3886 sacados/3886 chapExoCorrec/5898 sacados/5898 chapExoCorrec/3288 sacados/3288 chapExoCorrec/5204 sacados/5204 Extrait d'Antilles-Guyanne Juin 2003 chapExoCorrec/5846 sacados/5846 chapExoCorrec/6872 sacados/6872
-4-3-2-101234-11ijBA E.3705 Let f be the function defined by the relation: f ( x ) = 2e 2 x − e x e 2 x − e x + 1 1 Determine the set D f of definition of the function f . 2 Let f be the derivative function of the function f . Show that the function f admits the expression : f ( x ) = − e x · e 2 x − 4 · e x + 1 e 2 x − e x + 1 2 3 a Study the sign of the polynomial x 2 − 4 x +1 . b Deduce that the function f admits the following table of signs : x −∞ a b + ∞ f ( x ) − 0 + 0 − We’ll specify the values of a and b . 4 a Determine the limit of f in −∞ and in + ∞ . b Draw up the table of variations of the function f . The approximate values of f ( a ) and f ( b ) will be specified. E.1255 Let f be the function defined on R by: f ( x )=1 − 4 · e x e 2 · x +1 Note C its representative curve in an orthogonal reference frame O ; −→ i ; −→ j . On the graph below, we have drawn the curve C . It intersects the x-axis at points A and B . The purpose of this part is to demonstrate some properties of the function f that can be conjectured from the graph. 1 The function f appears to be increasing on the interval 0 ; + ∞ . a Verify that for any real x : f ( x ) = 4 · e x · e 2 · x − 1 e 2 · x + 1 2 b Deduce the direction of variation of the function f on the interval 0 ; + ∞ 2 The straight line with equation x =0 seems to be an axis of symmetry of the curve C . Demonstrate that this conjecture is true. (Question outside 2012 program) 3 We denote by a the abscissa of point A and pose c =e a . a Demonstrate that the real c is a solution of the equa-tion : x 2 − 4 · x + 1 = 0 Deduce the exact value of a . b Give the sign of f ( x ) according to the values of x . E.5845 Solve the following inequalities: a e x + 3 e x − 1 > 0 b − e 2 x − e x + 2 > 0 6. Limits E.3888 Determine the value of the following limits: a lim x ↦→ + ∞ 2 · ln x + x b lim x ↦→− 3 + ln( x +3) − 3 · x c lim x ↦→ 0 + ln( x ) − 1 x d lim x ↦→ + ∞ ln e x +1 e lim x ↦→ 0 + e − 2 · ln x +1 f lim x ↦→ + ∞ ln x 2 − x 7. Comparative growth limits E.3889 Determine the following limits: a lim x ↦→ + ∞ ln x x b lim x ↦→ 0 + x · ln x c lim x ↦→ 0 ln(1+ x ) x d lim x ↦→ 0 + ln( x ) · ( x 2 + 2) e lim x ↦→ + ∞ x − 2 ln x f lim x ↦→ + ∞ x · ln 1+ 1 x https://chingmath.fr chapExoCorrec/3705 sacados/3705 chapExoCorrec/1255 sacados/1255 -4-3-2-101234-11ijBA chapExoCorrec/5845 sacados/5845 chapExoCorrec/3888 sacados/3888 chapExoCorrec/3889 sacados/3889
ijCaAf(a E.4305 The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . Consider the nu-merical function ’ defined on R by: ’ ( x ) = ln x 2 − 2 · x +2 − x 1 Determine the limit of ’ in −∞ . 2 a Show that, for any strictly positive real x , ’ ( x ) = x · 2 · ln x x + ln 1 − 2 x + 2 x 2 x − 1 b Deduce the limit of ’ in + ∞ 8. Derivatives E.3887 Determine the expression of the deriva-tive function of each of the following functions : a f ( x ) = x · ln x b g ( x ) = ln x + 1 x 2 + 1 c h ( x ) = ln 1 − x d j ( x ) = ln e x − 1 E.5148 Consider the function f defined on 0 ; + ∞ by: f ( x ) = x 2 · ln x The curve ( C ) representative of the function f in the plane provided with an orthonormal reference frame. 1 a Determine the limit of f in + ∞ . b Study the variations of f on 0 ; + ∞ 2 For this question, any trace of research, however incom-plete, will be taken into account in the assessment. Show that there is a unique tangent to the curve C pass-ing through O . Specify an equation of this tangent. 9. Derivatives and tangents E.4230 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = x · 1 − ln x 1 Justify that the function f is derivable on 0 ; + ∞ and that it admits as derivative the function f whose expres-sion is : f ( x ) = − ln x 2 Let a be a strictly positive real number. Consider the tangent ( T a ) at the point A of the curve C of abscissa a . a Determine, as a function of the real number a , the co-ordinates of the point A , the point of intersection of the straight line ( T a ) and the y-axis. b Explain a simple procedure for constructing the tan-gent ( T a ) . Construct on the graphical representation of the curve C given below, the tangent ( T a ) at the point A placed on the figure. https://chingmath.fr chapExoCorrec/4305 sacados/4305 chapExoCorrec/3887 sacados/3887 chapExoCorrec/5148 sacados/5148 chapExoCorrec/4230 sacados/4230 ijCaAf(a
E.1616 Consider the function f defined on 0 ; + ∞ by: f ( x ) = x + ln x x We note C the representative curve of the function f in the plane provided with an orthonormal reference O ; −→ i ; −→ j of graphic unit 3 cm . 1 Determine the limit in 0 of the function f . What is the graphical interpretation of this result? We admit that the function f admits as derivative the func-tion f whose expression is : f ( x ) = 1 + 1 − ln x x and that the function f is strictly increasing. Let ( d ) be the straight line with equation y = x . 2 a Determine the abscissa of the point A on the curve C at which the tangent ( T ) is parallel to the line ( d ) . b Determine the equation of the tangent ( T ) . 3 Using the calculator, draw the straight lines ( d ) and ( T ) and the curve C . 10. Function studies E.3897 The plane is referred to an or-thonormal reference frame O ; −→ i ; −→ j . Consider the function f defined on the interval 0 ; + ∞ by: f ( x )= − 3 − ln x +2 · ln x 2 Note C its representative curve. 1 a Solve in 0 ; + ∞ the equation : f ( x )=0 . (We can put: ln x = X ) . b Solve in 0 ; + ∞ the inequation : f ( x ) > 0 . 2 a Determine the limits of f in 0 and in + ∞ . b Calculate f ( x ) . c Study the direction of variation of f and draw up its table of variations. 3 Determine an equation of the tangent T to the curve ( C ) at the point of abscissa e 5 4 . 4 We propose to study the position of the curve ( C ) with respect to the straight line ( T ) . To do this, consider the function ’ , defined on 0 ; + ∞ by: ’ ( x ) = f ( x ) − 4 · e − 5 4 · x − 41 8 a Montrer que : ’ ( x )= 4 · ln x − 1 x − 4 · e − 5 4 then calculate ’  ( x ) . b Study the direction of variation of ’ on 0 ; + ∞ . Deduce that, for any x belonging to 0 ; + ∞ , we have : ’ ( x ) 0 . c Calculate ’ e 5 4 . For any x belonging to 0 ; + ∞ , determine the sign of ’ ( x ) . Deduce the position of the curve ( C ) with respect to the straight line ( T ) . 5 Draw the curve ( C ) and the straight line ( T ) . (graphic unit : 2 cm ) 11. Study of functions and the intermediate value theorem E.4297 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = ln x 2 + 4 1 Study the direction of variation of the function f on the interval 0 ; + ∞ . 2 Let g be the function defined on the interval 0 ; + ∞ by: g ( x ) = f ( x ) − x a Study the direction of variation of the function g on the interval 0 ; + ∞ . b Show that on the interval 2 ; 3 the equation g ( x )=0 admits a single solution which will be denoted ¸ . Give the value of ¸ rounded to 10 − 1 . c Justify that the real number ¸ is the unique solution of the equation f ( x )= x . https://chingmath.fr chapExoCorrec/1616 sacados/1616 Extrait Antilles-Guyane Septembre 2010 chapExoCorrec/3897 sacados/3897 Extrait Antilles-Guyane Septembre 2001 chapExoCorrec/4297 sacados/4297
-123456I-12345678JOCMN E.5963 We call f the function defined on the interval I = − 2 ; + ∞ by: f ( x ) = 1 + x · ln( x +2) We denote ( C f ) the curve representing f in the orthonormal coordinate system O ; −→ i ; −→ j . Part A : Study of the function f 1 Study of the variations of the derivative f : a f denotes the first derivative of f and f  denotes the second derivative. Calculate f ( x ) then f  ( x ) for x belonging to the interval − 2 ; + ∞ . b Study the variations of f on the interval − 2 ; + ∞ . c Determine the limits of f at − 2 and + ∞ . 2 Study the sign of f ( x ) . a Show that on the interval − 2 ; + ∞ , the equation f ( x )=0 has a unique solution ¸ belonging to the in-terval − 0.6 ; − 0.5 . b Deduce the sign of f ( x ) according to the values of x . 3 Study of the variations of f : a Study the variations of the function f on the interval − 2 ; + ∞ . b Determine the limits of f at − 2 and + ∞ . c Draw up a table of variations of f . Using a calculator, indicate the coordinates of the point on the x-axis to the nearest ¸ to 10 − 3 . Part B Let x 0 be a real belonging to the interval − 2 ; + ∞ , we call T x 0 the tangent to the curve ( C f ) at the point of abscissa x 0 . Note, for x belonging to the interval − 2 ; + ∞ : d ( x ) = f ( x ) − f ( x 0 ) · x − x 0 + f ( x 0 ) 1 Study the variations of d : a Verify that, for any x belonging to the interval − 2 ; + ∞ : d ( x ) = f ( x ) − f ( x 0 ) b Using the growth of the function f , give the sign of d ( x ) according to the values of x . Deduce the varia-tions of d over the interval − 2 ; + ∞ . 2 Determine the relative position of ( C f ) and T x 0 . E.5149 1 Let f be the function defined on 0 ; + ∞ by: f ( x ) = x · e x − 1 a Determine the limit of the function f in + ∞ and study the direction of variation of f . b Demonstrate that the equation f ( x )=0 admits a sin-gle solution ¸ on the interval 0 ; + ∞ . Determine an approximate value of ¸ to the nearest 10 − 2 . c Determine the sign of f ( x ) depending on the values of x . 2 We note C the representative curve of the exponential function and Γ that of the natural logarithm function in the plane provided with an orthornormal reference frame O ; −→ i ; −→ j . The curves C f and Γ are given below : Let x be a strictly positive real number. Let M be the point of C with abscissa x and N the point of Γ with abscissa x . Recall that for any strictly positive real x : e x > ln( x ) . a Show that the length MN is minimal when x = ¸ . Give an approximate value of this length to the nearest 10 − 2 . b Using question 1 , show that : e α = 1 ¸ . Deduce that the tangent to C at the point of abscissa ¸ and the tangent to Γ at the point of abscissa ¸ are parallel. 12. Study of functions with study of an auxiliary function E.109 Consider the function f defined on 1 ; + ∞ by: f ( x ) = x − ln( x ) x 1 Let g be the function defined on 1 ; + ∞ by: g ( x ) = x 2 − 1 + ln( x ) Show that the function g is positive on 1 ; + ∞ . 2 a Show that, for any x of 1 ; + ∞ : f ( x ) = g ( x ) x 2 b Deduce the direction of variation of f on 1 ; + ∞ . https://chingmath.fr chapExoCorrec/5963 sacados/5963 chapExoCorrec/5149 sacados/5149 Extrait Antilles-Guyanne Juin 2011 -123456I-12345678JOCMN chapExoCorrec/109 sacados/109
E.5141 Part A Consider the function g defined on the interval 0 ; + ∞ by: g ( x )=2 x 3 − 1+2 · ln x 1 Study the variations of the function g on the interval 0 ; + ∞ . 2 Justify that there exists a single real ¸ such that g ( ¸ )=0 . Give an approximate value of ¸ , rounded to the hun-dredth. 3 Deduce the sign of the function g on the interval 0 ; + ∞ . Part B Consider the function f defined on the interval 0 ; + ∞ by: f ( x )=2 x − ln x x 2 Note C the representative curve of the function f in the plane, provided with an orthogonal reference frame O ; −→ i ; −→ j . 1 Determine the limits of the function f in 0 and in + ∞ . 2 Justify that f ( x ) has the same sign as g ( x ) . 3 Deduce the table of variations of the function f . 4 Draw the curve C in the reference frame O ; −→ i ; −→ j . We will take as units : 2 cm on the abscissa axis and 1 cm on the ordinate axis. E.3891 Consider the function f defined on 0 ; + ∞ by: f ( x ) = x + ln x x Note C the representative curve of the function f in the plane provided with an orthonormal reference O ; −→ i ; −→ j of graphic unit 3 cm . I- Study of an auxiliary function Consider the function f defined on 0 ; + ∞ by: g ( x ) = x 2 + 1 − ln x 1 Study the variations of g on 0 ; + ∞ . 2 Deduce the sign of g on 0 ; + ∞ . II- Study of the function f and plot of its representa-tive curve C 1 Determine the limit in 0 of the function f . What is the graphical interpretation of this result? 2 Determine the limit in + ∞ of f . 3 Let f be the function derived from the function f . Cal-culate f ( x ) for any real x of 0 ; + ∞ . 4 Deduce the direction of variation of f on 0 ; + ∞ , then draw up the table of variations of the function f . Consider the line D of equation y = x . 5 Determine the point A of the curve C at which the tan-gent T is parallel to the line D . 6 In the reference frame O ; −→ i ; −→ j , plot the straight lines D and T and the curve C . E.5959 Part A : Study of an auxiliary function g Let g be the function defined on the interval 0 ; + ∞ by: g ( x ) = 2 x 2 − x 2 + 1 · ln x 2 +1 1 By detailing the calculations performed, show that : g ( x ) = 2 x − 2 x · ln x 2 +1 2 Study the direction of variation of g on the interval 0 ; + ∞ . 3 Show that there is a single real, which we’ll note ¸ , in the interval e − 1 ; e 2 − 1 , such that g ( ¸ )=0 ; give the decimal approximation 10 − 2 to the nearest default of ¸ . 4 Deduce the sign of g ( x ) , for x belonging to the interval 0 ; + ∞ . Part B: Study of the function f The function f is defined on 0 ; + ∞ by: f (0) = 0 ; f ( x ) = ln 1+ x 2 x lorsque x =0 Its representative curve ( C f ) , in the plane referred to a refer-ence frame of origin O is given below : 1 a Show that : lim x ↦→ 0 f ( x ) x = 1 . Deduce that f is derivative in 0 and give the value of f (0) . b Verify that, for x strictly positive : f ( x ) = g ( x ) x 2 · 1 + x 2 Study the direction of variation of f on the interval 0 ; + ∞ . 2 a Montrer que, pour x 1 : 0 f ( x ) ln 2 x 2 x b Deduce the limit of f in + ∞ . 13. Study of function families E.5962 The plane is given an orthonormal reference frame O ; −→ i ; −→ j . Part A Let f be the function defined on 0 ; + ∞ by: f ( x ) = 1 + 2 · ln x x 2 Let ( C ) be the representative curve of f and let ( C ) be that of the function h defined on 0 ; + ∞ by: h ( x )= 1 x . 1 Calculate the derivative f of f and study the variations of f . 2 For any x of 0 ; + ∞ , we pose : g ( x ) = 1 − x + 2 · ln x a Study the variations of the function g . b Show that the equation g ( x )=0 admits a unique solu- https://chingmath.fr chapExoCorrec/5141 sacados/5141 Extrait du Liban Mai Juin 2012 chapExoCorrec/3891 sacados/3891 chapExoCorrec/5959 sacados/5959 Extrait Centres etrangers Juin 1999 chapExoCorrec/5962 sacados/5962 Extrait d'Asie Juin 2003
-12345I-2-123JOC3CkTk tion in each of the intervals 0 ; 2 and 2 ; 4 . Let ¸ be the solution belonging to 2 ; 4 . Give a frame for ¸ of amplitude 10 − 2 . 3 a Show that f ( x ) − 1 x = g ( x ) x 2 and deduce that ( C ) and ( C ) intersect at two points. b Show that, for any real x greater than or equal to 4 , the following double inequality is true : 0 <f ( x ) 1 x Part B Consider, for any integer n greater than or equal to 1 , the function f n defined on 0 ; + ∞ by: f n ( x ) = 1 + 2 ln x x 2 n 1 Calculate the derivative f n of the function f n . 2 Solve the equation f n ( x )=0 . Let x n be the solution to this equation. 3 Determine the limit of the sequence x n . E.5150 For any natural number n greater than or equal to 1 , we denote by f n the function defined by: f n ( x ) = x n · e − x Note C n its representative curve in an orthogonal reference frame O ; −→ i ; −→ j of the plane. On the graph below, a curve C k où k is a non-zero natural number, its tangent T k at the point of abscissa 1 and the curve C 3 . The straight line T k intersects the abscissa axis at the point A of coordinates 4 5 ; 0 . 1 a Determine the limits of the function f 1 in −∞ and in + ∞ . b Study the variations of the function f 1 and draw up the table of variations of f 1 . c Using the graph, justify that k is an integer greater than or equal to 2 . 2 a Show that for n 1 , all curves C n pass through the point O and another point whose coordinates will be given. b Verify that for any natural number n greater than or equal to 2 , and for any real x : f n ( x ) = x n − 1 · ( n − x ) · e − x 3 On the graph, the function f 3 seems to admit a maxi-mum reached for x =3 . Validate this conjecture with a demonstration. 4 a Show that the straight line T k intersects the x-axis at the point with coordinates k − 2 k − 1 ; 0 . b Deduce, using the data in the statement, the value of the integer k . E.5856 Part A Consider the family of functions f k defined for k ∈ N by: f k ( x )=( x + k ) · e x + x Note C k the representative curve of the function f k in an orthonormal frame. Which function of the family f k admits the straight line ( d ) of equation y = x − e − 2 as tangent at the point of abscissa − 2 . Part B 1 Consider the function g defined by: g ( x ) = ( x + 1) · e x + e − 2 . a Determine the expression of the function g . b Draw up the table of variations of the function g . 2 Deduce from the previous questions the relative position of the curve C 1 and the straight line ( d ) . https://chingmath.fr chapExoCorrec/5150 sacados/5150 -12345I-2-123JOC3CkTk chapExoCorrec/5856 sacados/5856
-4-3-2-101234-2-112ijC1 -2-1012345-3-2-1123456ijCourbe2Courbe3Courbe1 -4-3-2-123456I23456JO E.5848 Given a real number k , consider the function f k defined on R by: f k ( x ) = 1 1 + e − k · x The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . Part A In this part, we choose k =1 . We have , for any real x : f 1 ( x ) = 1 1 + e − x The graphical representation C 1 of the function f 1 in the frame O ; −→ i ; −→ j is given below : 1 Determine the limits of f 1 ( x ) in + ∞ and −∞ and graph-ically interpret the results obtained. 2 Demonstrate that, for any real x : f 1 ( x )= e x 1+e x . 3 Let f 1 be called the derivative function of f 1 on R . Cal-culate, for any real x , f 1 ( x ) . Deduce the variations of the function f 1 on R . Part B In this part, we choose k = − 1 and we wish to draw the curve C − 1 representing the function f − 1 . For any real x , we call P the point of C 1 of abscissa x and M the point of C − 1 of abscissa x . Note K midpoint of segment [ MP ] . 1 Show that, for any real x : f 1 ( x )+ f − 1 ( x )=1 . 2 Deduce that the point K belongs to the line of equation y = 1 2 . 3 Draw the curve C − 1 on the marker below. E.5849 In all that follows, m denotes any real number. Part A Let f be the function defined and derivable on the set of real numbers R such that : f ( x ) = ( x + 1) · e x 1 Calculate the limit of f in + ∞ and in −∞ . 2 Let f be the derivative function of the function f on R . Show that for any real x : f ( x )=( x +2) · e x . 3 Draw up the table of variations of f at R . Part B We define the function g m on R by: g m ( x ) = x + 1 − m · e − x and we denote C m the curve of the function g m in a O ; −→ i ; −→ j reference frame of the plane. 1 a Demonstrate that : g m ( x )=0 if, and only if, f ( x )= m . b Deduce from A , without justification, the number of points of intersection of the curve C m with the x-axis as a function of the real m . 2 The curves C 0 are shown below, C e and C − e (obtained by taking for m the values 0 , e and − e ) , respectively. Identify each of these curves on the figure below, justify-ing. 3 Study the position of the curve C m with respect to the straight line D of equation y = x +1 according to the val-ues of the real m . E.6964 Let k be a strictly positive real. Consider the functions f k defined on R by: f k ( x ) = x + k · e − x Note C k the representative curve of the function f k in a plane with an orthonormal reference frame. Some curves C k for different values of k are shown below. For any strictly positive real k , the function f k admits a mini-mum at R . The value at which this minimum is reached is the abscissa of the point A k of the curve C k . It would seem that, for any strictly positive real k , the points A k are aligned. Is this the case? https://chingmath.fr chapExoCorrec/5848 sacados/5848 -4-3-2-101234-2-112ijC1 chapExoCorrec/5849 sacados/5849 -2-1012345-3-2-1123456ijCourbe2Courbe3Courbe1 chapExoCorrec/6964 sacados/6964 Extrait Liban Juin 2017 -4-3-2-123456I23456JO
-12345678I-3-2-1234JO 14. Suites E.3925 1 Consider the function g defined on 1 ; + ∞ by: g ( x ) = ln(2 x ) + 1 − x a This question requires the development of a certain ap-proach involving several steps. The clarity of the study plan, the rigor of the reasoning, and the quality of the writing will be taken into account in the evaluation. Prove that the equation g ( x )=0 has a unique solution on 1 ; + ∞ , denoted by ¸ . b Prove that : ln 2 · ¸ + 1 = ¸ 2 Let u n be the sequence defined by u 0 =1 and, for any natural number n , by: u n +1 = ln(2 · u n ) + 1 . Let (Γ) denote the curve of equation y =ln(2 · x )+1 in an orthonormal coordinate system O ; −→ i ; −→ j . This curve is given below : a Using the curve (Γ) , construct the first four terms of the sequence on the x-axis. b Prove that for any natural number n : 1 u n u n +1 3 c Prove that the sequence u n converges to ¸ . E.5861 Consider the sequence u n de-fined by: u 0 =1 ; u n +1 = 2 u n 1 a Demonstrate that, for any natural number n : 0 <u n 2 b Determine the direction of variation of the sequence u n . c Show that the sequence u n is convergent. The value of its limit is not asked. 2 Consider the sequence v n defined, for any natural num- ber n , by: v n =ln u n − ln2 a Demonstrate that the sequence v n is the geometric sequence of reason 1 2 and first term v 0 = − ln2 . b Determine, for any natural number n , the expression of v n as a function of n , then of u n as a function of n . c Determine the limit of the sequence u n . E.5167 Let f be the function defined on 0 ; + ∞ by: f ( x ) = x + e − x Let ( C ) be the representative curve of f in an orthonormal frame O ; −→ i ; −→ j . Part A 1 Study the variations of the function f on 0 ; + ∞ . 2 Determine the limit of f in + ∞ . Part B Consider the sequence u n n 1 with positive terms defined by: u 1 = 0 ; u n +1 = f ( u n ) = u n + e − u n 1 Demonstrate that, for any positive real x : ln(1+ x ) x We can study the function g defined on 0 ; + ∞ by: g ( x ) = x − ln(1+ x ) 2 Deduce that, for any non-zero natural number n : ln( n +1) ln( n ) + 1 n 3 Demonstrate that, for any non-zero natural number n : f ln( n ) = ln( n ) + 1 n 4 Demonstrate by recurrence that, for any non-zero natu-ral number n : ln n u n 5 Deduce the limit of the sequence u n n 1 . E.6908 Let u n be the sequence defined by u 0 =1 and, for any natural number n : u n +1 = u n − ln u n 2 +1 We admit that the function f defined by: f ( x ) = x − ln x 2 +1 is increasing on R . 1 Show by recurrence that, for any natural number n , u n belongs to 0 ; 1 . 2 Study the variations of the suite u n . 3 Show that the sequence u n is convergent. 4 We note ‘ its limit, and admit that ‘ verifies equality: f ( ‘ ) = ‘ Deduce the value of ‘ . 15. Suites and sums of terms https://chingmath.fr chapExoCorrec/3925 sacados/3925 -12345678I-3-2-1234JO chapExoCorrec/5861 sacados/5861 chapExoCorrec/5167 sacados/5167 Extrait Liban Mai 2011 chapExoCorrec/6908 sacados/6908
E.3289 We propose to study the sequence ( u n ) of real numbers defined by: u 1 = 3 2 ; u n +1 = u n · 1 + 1 2 n +1 1 Show by recurrence that u n > 0 for any natural number n 1 . 2 Show by recurrence that for any natural number n 1 : ln u n = ln 1+ 1 2 + ln 1+ 1 2 2 + · · · + ln 1+ 1 2 n E.3293 Consider the sequence v with gen-eral term v n defined by: v n = ln n n +1 for any n ∈ N ∗ où ln denotes the neperian logarithm function. We define the sum S n for any non-zero natural number n by: S n = v 1 + v 2 + · · · + v n Show by recurrence that, for any natural number n : S n = − ln n +1 E.5961 Consider the function f defined on R by: f ( x ) = ln 1+e − x Part A 1 Determine the limit of f in −∞ , then the limit of f in + ∞ . 2 Study the direction of variation of f . Part B Here are two functions extracted from algorithms taking as argument a natural number greater than or equal to 1 : Algorithm 1 Function f(n) a ← 0 k ← 1 As long as ...... a ← a+ exp( - k) k ← k + 1 End As long as Return k Algorithm 2 Function g(n) a ← 0 For i from 1 to n b ← ln 1+ exp( − i) ... End To Return a 1 Consider the sequence a n defined by: a n = e − 1 + e − 2 + · · · + e − n pour tout n ∈ N ∗ . We admit that the sequence a n is increasing and con-verges to the value 1 e − 1 Complete the blanks in the algorithm 1 so that the value returned by the function f is the rank of the first term in the sequence the sequence a n realizing an approximate value of 1 e − 1 to within 10 − 4 when the value supplied as an argument is the value 4 . 2 We define the sequence S n defined on N ∗ by: S n = f (1) + f (2) + · · · + f ( n ) . Complete the algorithm 2 so that the value returned by the function g is the value of the n rank term when the rank value is supplied to the g function as an argument. Part C 1 Let u and v be the functions defined on 0 ; + ∞ by: u ( t ) = ln 1+ t − t ; v ( t ) = ln 1+ t − t + 1 2 · t 2 a Study the variations of u and v . b Deduce that, for any positive real number t , we have : t − 1 2 t 2 ln 1+ t t 2 Let n be a non-zero natural integer ( n 1) . Consider the number: S n = f (1) + f (2) + · · · + f ( n ) a Demonstrate that for any natural number n : 1 − e − n e − 1 − 1 2 × 1 − e − 2 n e 2 − 1 S n 1 − e − n e − 1 b We admit the following proposition : Proposition: Let u n , v n and w n be three sequences such that : u n v n w n for all n ∈ N converge respectively to the numbers k , ‘ and m . Then we have the frame : k ‘ m and we admit that the sequence ( S n ) has a real limit L . Show that : ⏐ ⏐ ⏐ L − 1 e − 1 ⏐ ⏐ ⏐ 1 2 · e 2 − 1 E.6909 Consider the sequence u n de-fined for any strictly positive integer n by: u n = 1 + 1 2 + 1 3 + · · · + 1 n − ln n 1 We denote by f the function defined on the interval 1 ; + ∞ by: f ( x ) = 1 x + 1 + ln x x + 1 a Draw up the table of variations of the function f . b Deduce the sign of the function f . 2 a Show that for any strictly positive integer: u n +1 − u n = f ( n ) b Deduce the direction of variation of the sequence u n . 16. Logarithms and probabilities E.4195 In a random game, the event G ˇ the game is gagnée ı has probability: P ( G ) = 23 180 1 A player repeats this game independently six times. Cal-culate the probability that he wins exactly two and give a value rounded to the nearest 10 − 2 . 2 What minimum number of games must a player play for the probability of winning at least one to be greater than 0.9 ? https://chingmath.fr chapExoCorrec/3289 sacados/3289 chapExoCorrec/3293 sacados/3293 chapExoCorrec/5961 sacados/5961 Inspire de sportif de haut-niveau Septembre 1999 chapExoCorrec/6909 sacados/6909 chapExoCorrec/4195 sacados/4195
E.4155 An urn contains 10 white balls and n red balls, n being a natural number greater than or equal to 2 . A player is made to draw balls from the urn. At each draw, all the balls have the same probability of being drawn. The player draws 20 times successively and with delivery a ball from the urn. The draws are independent. Determine the minimum value of the integer n so that the probability of obtaining at least one red ball during these 20 draws is strictly greater than 0.999 . E.6047 A player starts a video game and plays several successive games. It is assumed that : the probability of him winning the first game is 0.1 ; if he wins one game, the probability of winning the next is equal to 0.8 ; if he loses one game, the probability of winning the next is equal to 0.6 . We note, for any non-zero natural number n : G n the event ˇ the player wins the n -th partie ı ; p n the probability of the event G n . We therefore have : p 1 =0.1 1 Using the total probability formula, show that for any non-zero natural number n : p n +1 = 1 5 · p n + 3 5 . 2 Show by recurrence that for any non-zero natural number n : p n = 3 4 − 13 4 · 1 5 n 3 Determine the limit of the suite p n when n tends to + ∞ . 4 For what values of the natural integer n does one have : 3 4 − p n < 10 − 7 ? 17. Courses E.3890 Prerequisites: We assume that the derivative of the expo-nential function and the derivation formula for u ◦ v , as well as its conditions of use, are known. We assume that the function ln is differentiable on 0 ; + ∞ and that for all x of 0 ; + ∞ , we have : exp ln x = x Using these four arguments, show that the derivative of the function ln is the function defined on 0 ; + ∞ that associates x with 1 x . E.3963 In this exercise, candidates are asked to establish two course results by following the pro-posed approach. Reminder : the function ln is defined and differentiable on 0 ; + ∞ , positive on 1 ; + ∞ , and satisfies : ln 1 = 0 Pour tous réels strictement positifs x et y , ln( x · y ) = ln x + ln y Pour tout réel strictement positif x , ln( x ) = 1 x ln(2) ≈ 0 ; 69 à 10 − 2 près. Consider the function f defined on 0 ; + ∞ by: f ( x ) = x − ln x 1 Study the variations of f and deduce that f has a mini-mum on 0 ; + ∞ . 2 Deduce the sign of f and then, for all x> 1 : 0 < ln x x < x x 3 Deduce that : lim x ↦→ + ∞ ln x x = 0 . E.3280 Organized restitution of con-naissances Prerequisites: The neperian logarithm function is derivable on the in-terval 0 ; + ∞ ; Its derivative function is the inverse function : x ↦−→ 1 x . ln(1) = 0 Show that for all strictly positive real numbers ¸ and x : ln( ¸x ) = ln( ¸ ) + ln( x ) E.6091 For each of the following state-ments, indicate whether it is true or false and justify the an-swer chosen. Consider the function g defined on − 1 2 ; + ∞ by: g ( x ) = 2 x · ln 2 x + 1 Proposition 1: On − 1 2 ; + ∞ , the equation g ( x )=2 x has a unique solution e − 1 2 . Proposition 2: The slope of the tangent to the representative curve of the function g at the point of abscissa 1 2 is : 1+ln4 . E.3659 Prerequisite: Recall that : lim x ↦→ + ∞ e x x = + ∞ 1 Demonstrate that : lim x ↦→ + ∞ ln x x =0 . 2 Deduce that for any non-zero natural number n : lim x ↦→ + ∞ ln x x n = 0 https://chingmath.fr chapExoCorrec/4155 sacados/4155 chapExoCorrec/6047 sacados/6047 chapExoCorrec/3890 sacados/3890 Extrait Antilles-Guyane Septembre 2010 chapExoCorrec/3963 sacados/3963 Amerique du Sud Novembre 2008 chapExoCorrec/3280 sacados/3280 Extrait Antilles-Guyane Juin 2006 chapExoCorrec/6091 sacados/6091 chapExoCorrec/3659 sacados/3659 Centres etrangers Juin 2008
18. Unclassified financial years E.3653 Let f be the function defined on R by: f ( x )=e x − x − 1 1 Determine the sign of f (we can study the variations of the function f ) . Establish the following inequality for any non-zero natu-ral number n : e 1 n 1 + 1 n 2 Using the previous inequality, show that for any non-zero natural number n : 1 + 1 n n e E.9729 Let f be the function defined on R by: f ( x ) = x + 2 − 4 · e x e x + 3 We denote by C its representative curve in the plane referred to an orthonormal reference O ; −→ i ; −→ j of graphic unit 2 cm . 1 a Let f be the derivative function of f . Calculate f ( x ) and show that, for any real x , we have : f ( x ) = e x − 3 e x + 3 2 b Determine the directions of variation of f on R . 2 Determine the equation of the tangent ( T ) to the curve C at the point of abscissa 0 . E.101 1 Solve the following system, where u and v are real num-bers : u + 1 2 · v = 0 u − 1 4 · v = 3 2 2 Let f be a function defined on R by: f ( x ) = a · e x + b e x + 1 ( a and b real) Find the values of the reals a and b , knowing that the representative curve of the function f in a O ; −→ i ; −→ j passes through O and that the tangent to the curve at this point is parallel to the line (Δ) of equation y = 3 2 x − 2 . 3 Let g be the function defined on R by: g ( x )=e x − 2 e x +1 a Solve in R the equation : g ( x )=0 . b Solve in R the inequation : g ( x ) 1 https://chingmath.fr chapExoCorrec/3653 sacados/3653 Inspire de Nouvelle-Caledonie Novembre 2007 chapExoCorrec/9729 sacados/9729 chapExoCorrec/101 sacados/101 France - septembre 2001 - 4 points