Outside the high school program / Exponentials and logarithms with base a 50 exercises (including 27 corrected)

a
1. Rational powers E.3913 Solve the following equations in R ∗ + : a x 3 = 5 b x 6 = 100 c ( x + 2) 4 = 5 d x 1 3 = 2 e x 5 2 = 6 f ( x + 1) 2 3 = 2 E.3914 Solve the following inequations : a x 1 2 > 5 b x 3 4 3 c ( x + 2) 2 3 1 E.3915 Write each of the expressions below as a rational power: a x 2 · x 1 3 b x 1 2 · x c x 1 4 x 1 5 e x 2 · x 3 4 x 5 · x 1 4 f 3 x 5 · x 1 3 E.3917 Determine the expressions of the deriva-tive functions of each of the following functions : a f ( x ) = 3 · x 2 3 − 1 b g ( x ) = 2 4 √ x c h ( x ) = e x · x 1 4 d j ( x ) = 3 √ x + 1 · ( x + 1) 3 E.3916 1 Using reasoning by recurrence, establish the following equality for any non-zero natural number n : x n − 1 = x − 1 · x n − 1 + x n − 2 + · · · + x + 1 2 Consider the number A defined by: A = 1 3 5 − 1 Determine an expression for the number A defined by a quotient whose denominator is an integer. 2. Exponentials and logarithms with base a E.3918 1 a Establish the following equality: 2 ln15 =e ln2 · ln15 b Compare, without the aid of a calculator, the following two numbers : 2 ln 15 ; 4 ln 5 2 a Establish the following equality: 9 3 4 = e 3 2 · ln 3 b Deduce the comparison of the following two integers : 9 3 4 ; 3 4 3 E.3221 The exercise includes an appendix, to be submitted with the copy The purpose of this problem is to study, for x and y distinct elements of the interval 0 ; + ∞ , the solution pairs of the equation ( E ): x y = y x and, in particular, the pairs consisting of integers. 1 Show that the equation ( E ) is equivalent to : ln x x = ln y y 2 Let h be the function defined on the interval 0 ; + ∞ by: h ( x ) = ln x x The curve C representing the function h is given in the appendix ; x 0 is the abscissa of the maximum of the func-tion h on the interval 0 ; + ∞ a Recall the limit of the function h at + ∞ and determine the limit of the function h at 0 . b Calculate h ( x ) , where h denotes the derivative of the function h ; find the variations of the function h . Determine the exact values of x 0 and h ( x 0 ) . c Determine the intersection of the curve C with the x-axis. 3 Let – be an element of the interval 0 ; 1 e . Prove the existence of a unique real number a in the in-terval 1 ; e and a single real number b from the interval e ; + ∞ such as h ( a )= h ( b )= – . Thus, the pair ( a ; b ) is a solution of ( E ) . 4 Consider the function s which, for any real number a in the interval ]1 ; e [ , associates the unique real number b from the interval ] e ; + ∞ [ such that h ( a )= h ( b ) (we will not attempt to express s ( a ) in terms of a ) . By reading the graph only and without justification, an- https://chingmath.fr chapExoCorrec/3913 sacados/3913 chapExoCorrec/3914 sacados/3914 chapExoCorrec/3915 sacados/3915 chapExoCorrec/3917 sacados/3917 chapExoCorrec/3916 sacados/3916 chapExoCorrec/3918 sacados/3918 chapExoCorrec/3221 sacados/3221 France Septembre 2004 6 points
2468101214161820I-8-6-4-224JO swer the following questions : a What is the limit of s when a approaches 1 by upper values? b What is the limit of s when a approaches e from below? c Determine the variations of the function s . Draw up the table of variations of s . 5 Determine the pairs of distinct integers that are solutions to ( E ) . E.3919 Solve the following equations : a 5 x = 3 b 6 3 − x = 5 c 3 x = 2 x d 5 x +1 = 2 3 · x e 3 × 2 x − 1 = 3 2 · x +1 f 2 x + 2 x +1 = 5 x E.3926 Solve the following inequations : a 3 x 2 b 0.2 x < 3 c 2 5 x · 3 2 x e 10 3. Algebraic manipulations: exponential E.2032 Give the result of the following calcula-tions : a ln e 5 b ln e 5 · e − 2 d e ln 5 e e ln 2+ln 1 2 f e ln 2 e ln 3 g ln 10 000 + ln 100 h ln 100 ln 1 000 c ln 2e 3 3 + ln 8e 2 3 E.2015 For each of the four statements, say whether it is true or false, justifying the choice made. Each question is marked out of one point, with the following rule : An unjustified answer earns no points. A wrong answer does not deduct any points. 1 The function f defined on R by f ( x )=e − 2 · x +1 is decreas-ing on R . 2 The equation e x 1+e x = 4 3 has only one solution in R . 3 The sequence defined, for any natural number n , by: u n = 1 + 1 2 + · · · + 1 2 n tends to 2 when n tends to + ∞ . 4 For any real number x , we have : 1.01 x < 1 000 000 . 4. Algebraic manipulations: exponential and logarithm E.2038 Simplify the following expressions : a ln(2 · x 2 + x ) + ln 1 x b ln x 7 − ln x 2 c 4 · ln x − 2 · ln x E.116 Simplify the writing of the following ex-pressions : a e x · e 2 x b e 2 x − 1 4 c e ln x d e − ln x e e x · e ln x f ln e x g ln x · e x h ln e x 2 E.69 Solve the following equations on R . Remem-ber to use the various algebraic properties of these functions : a e x + 1 1 b e 2 · x +1 > 0.5 c e 2 · x < 2 · e x d ln(2 · x + 1) > − 0.5 e ln( x 2 ) < 3 f ln(2 · x ) > ln x + 1 g 2 x 3 https://chingmath.fr 2468101214161820I-8-6-4-224JO chapExoCorrec/3919 sacados/3919 chapExoCorrec/3926 sacados/3926 sacados/2032 sacados/2015 sacados/2038 sacados/116 sacados/69
x051015202530y-4-2246 E.2030 Simplify the following entries : a ln x − ln(2 x ) b 2 ln x − ln x c ln 1 + 1 x + ln x E.2039 Solve the following equations : a e x = 2 b e x = − 1 c ln x = 5 d ln x = − 2 e ln(2 · x +1) = 5 f e 3 − 2 x = 2 g 3 x = 2 h x 2 · 2 − ln x = 0 E.2037 Give the results of the following calcula-tions : a e ln 5 b e 2 · ln 2 c e ln 2+ln 2 d e ln 9 e ln 2 e ln(e 2 · e 5 ) f ln 5 2 − ln 1 5 2 g ln 10 − 5 + ln 10 8 h ln 1 000 000 ln 1 000 5. Study of functions: exponential E.72 Consider the function f defined on R by: f ( x ) = 2 · x + 5 − e x 1 a Solve the inequation 2 − e x > 0 in R . b Calculate f (ln 2) . 2 a For any number x belonging to R , calculate f ( x ) . b Draw up the table of variations of the function f . 6. Study of functions: logarithm E.87 The curve ( C below is the graphical representation in the plane provided with an orthonormal ref-erence frame of a function f defined on the interval 0 ; + ∞ . Note f the derivative function of f . The straight line ( D ) is the tangent to the curve ( C ) at the point of abscissa 4 and is parallel to the x-axis. The y-axis is asymptotic to the curve. Part A : Graphical readings 1 a Give, by graphical reading, a value approximated to the nearest 0.5 of f (2) and f (20) . b Give the exact value of f (4) . 2 By graphical reading, give the table of variations of the function f as well as the sign of the derivative f . Part B: Algebraic checks It is assumed that f ( x ) is of the form a · x + b + c · ln x where a , b and c are three real numbers and ln denotes the natural logarithm function. 1 a Express f ( x ) in terms of a , c and x . b Since the maximum of f is obtained for x equal to 4, deduce a relationship between a and c . 2 a Knowing that the representative curve of f passes through the point A (1 ; 4.5) , give a relationship be-tween a and b . b We know that the point B ( e ; 7 − 0.5 · e ) belongs to the representative curve of f . Deduce a relationship be-tween a , b and c . 3 a Deduce from the previous questions that we have : f ( x ) = − 0.5 · x + 5 + 2 · ln x b Deduce the exact values of f (2) and f (20) and of the maximum of f . c Determine the limit in 0 of f . https://chingmath.fr sacados/2030 sacados/2039 sacados/2037 sacados/72 chapExoCorrec/87 sacados/87 Japon - Juin 2004 - 7 points - Obligatoire x051015202530y-4-2246
A3eaij E.81 On the interval I =[0.25 ; 10] , n con-sider the function f defined by: f ( x )=ln x + 2 x . Let ( C ) be its representative curve in the plane provided with an orthonormal reference frame O ; −→ i ; −→ j , the unit of length being 2 cm . 1 Determine the derivative function f of this function f and draw up the table of variations of the function f on I . 2 Calculate f (1) and f (1) and deduce the reduced equa-tion of the tangent ( T ) to ( C ) at its point of abscissa 1. 3 Draw the curve ( C ) and the tangent ( T ) 4 a Expand : ( x − 3)( x − 6) . b Solve in the interval the equation : f ( x )= 1 9 . c Justify then that the curve ( C ) admits two parallel tangents to the line of equation y = x 9 . Derivatives of common functions : F ( x ) x x 2 x 3 1 x ln x e x F ( x ) 1 2 x 3 x 2 − 1 x 2 1 x e x Conditions x =0 x> 0 E.70 Consider the numerical function f defined on the interval [1 ; 12] by: f ( x ) = x − 1 − 4 · ln x We denote C the curve representing the function f in an or-thonormal unit coordinate system : 1 cm . 1 a Calculate the derivative f of the function f . Ver-ify that, for all x in the interval [1 ; 12] , f ( x ) can be written as : f ( x ) = x − 4 x b Study the sign of f on the interval [1;12] and deduce the table of variations of f . c Determine an equation of the tangent (Δ) to the curve at its point of abscissa 1. 2 a Copy and complete the following table, giving values rounded to the nearest 0.1. x 1 2 3 4 6 8 10 11 12 f ( x ) b Trace the curve C and the straight line (Δ) dans the same mark on the sheet of graph paper provided. Form : The derivative of the function ln sur the interval ]0 ; + ∞ [ est the function which, at x associe 1 x . E.92 The aim of the exercise is to study the function f defined on the interval 1 ; 3 by: f ( x )= x 2 − ln x x . Recall that e is the number such that : ln e =1 . Let ( C ) be the representative curve of the function f in a O ; −→ i ; −→ j frame. The curve ( C ) is given in the appendix to be returned with the copy. This curve will be used to check the accuracy of some results, but should not be used to justify answers. Part I Consider the function u defined on the interval [1 ; 3] by: u ( x ) = x 2 − 2 + 2 · ln x 1 Let u be the derivative of the function u . Calculate u ( x ) . 2 Draw up the table of variations of the function u on the interval [1 ; 3] . 3 We admit the existence of a unique number a , belonging to the interval [1 ; 3] such that u ( a )=0 . Copy and complete the table below, indicating the sign of u ( x ) . x 1 a 3 u ( x ) 0 Part II 1 a Let f be the derivative of the function f . We admit that for any x of the interval [1 ; 3] : f ( x ) = u ( x ) 2 · x 2 where u is defined in part I . Depending on the values of x , determine the sign of f ( x ) over the interval [1 ; 3] . b Draw up the table of variations of the function f (we will not calculate f ( a ) ) . 2 Note A the point with coordinates 1 ; 1 2 . Show that the tangent ( T ) to the curve ( C ) at the point of abscissa e is parallel to the line ( OA ) . 3 Draw the line ( OA ) and the tangent ( T ) on the appendix to be returned with the copy. Place the point B with co- https://chingmath.fr chapExoCorrec/81 sacados/81 sacados/70 sacados/92 A3eaij
ordinates ( a ; f ( a )) and the tangent to the curve ( C at the point B . E.117 We note : k = 20 ln10 ; p 0 = 20 × 10 − 6 . Consider the function f defined on [ p 0 ; + ∞ [ by: f ( x ) = k · ln 50 000 · x 1 a For any x belonging to the defining set, calculate f ( x ) . b Give the table of variations of the function f . 2 a Show that : f (10 · x )= k · ln(10)+ f ( x ) b Express f (100 · x ) in terms of f ( x ) . E.120 Consider the function f defined on R by: f ( x ) = ln x 2 · 3 − 2 · ln x 1 Show that for x belonging to 0 ; + ∞ : f ( x ) = 6 · ln x · 1 − ln x x 2 a Solve the following inequations : ln x 0 1 − ln X 0 b Deduce the sign of f on ]0 ; + ∞ [ as well as its table of variations of the function f . 3 Calculate the extremums of the function f on [0.75 ; 3] . E.90 Reminder : The natural logarithm function is denoted by ln ; a and b are strictly positive real numbers and n is a natural number: ln a · b = ln a + ln b ; ln a b = ln a − ln b ln a n = n · ln a Part A On the attached sheet to be handed in with your exam pa-per, we have plotted in an orthonormal coordinate system the curve ( C ) representing the natural logarithm function and the parabola ( P ) representing the natural logarithm function and the parabola ( P ) representing the function f defined on R by: f ( x ) = 2 · x 2 − 3 · x + 9 2 The function f is differentiable on R and we denote f as its derivative. 1 a Calculate f ( x ) for all real numbers x . b Deduce the table of variations of the function f . (The limits of f at infinity are not required.) c What are the exact coordinates of the point S , the vertex of the parabola ( P ) ? 2 Consider the function g defined on the interval ]0 ; + ∞ [ by: g ( x )= f ( x ) − ln x =2 · x 2 − 3 · x + 9 2 − ln x and let g be its derivative. a Show that, for any strictly positive real number x : g ( x ) = (4 · x + 1)( x − 1) x b Study the variation of the function g on the interval ]0 ; + ∞ [ . Justify that the minimum of g is equal to 7 2 . c Deduce that for any strictly positive real number x : f ( x ) − ln x > 0 . What property of the curves ( P ) and ( C ) , visible graphically, does the above result justify? Part B For any strictly positive real number x , let M be the point on the curve ( p ) with abscissa x and N be the point on the curve ( C ) with the same abscissa x . We thus have : MN = f ( x ) − ln x = g ( x ) . (the length MN is expressed in the graphic unit of the diagram on the attached sheet.) 1 Place points M and N on the diagram in the attached sheet when x =2 . 2 Show that when x = 3 4 , we have : MN = 27 8 + 2 · ln 2 − ln 3 . Give the value of MN rounded to two decimal places. 3 a Using part A, determine for which value of x the length MN is minimal. What is this length? b Draw the corresponding segment [ MN ] in red on the diagram on the attached sheet. 4 What is the limit of the length MN when x tends to-wards 0 (with x> 0 ) ? https://chingmath.fr sacados/117 sacados/120 chapExoCorrec/90 sacados/90
-12345678I-4-3-2-123456789JO -101234567891011-2-112345 E.85 1 Consider the function f defined and derivable on R de-fined by: f ( x )=2 · x 2 − 3 · x + 9 2 a Calculate f ( x ) for any real x b Give the exact coordinates of the point S vertex of the parabola 2 Consider the function g defined on the interval ]0 ; + ∞ [ by: g ( x ) = f ( x ) − ln x = 2 · x 2 − 3 · x + 9 2 − ln x The function g is derivable on 0 ;+ ∞ a Show that, for any strictly positive real x : g ( x ) = (4 · x + 1)( x − 1) x b Study the variations of the function g on the interval ]0 ; + ∞ [ . Justify that the minimum of g is equal to 7 2 c Deduce that on 0 ; + ∞ , we have : f ( x ) − ln x> 0 . What can you say about the representative curves of the f function and the neperian logarithm. E.102 Parts A and B can be processed indepen-dently of each other. part A The curve C below is the graphical representation in an or-thonormal frame of reference of a function f defined and deriv-able on the interval ]0 ; 10] . Let f be the derivative function of f on this interval. We specify that the line T is tangent to the curve C at the point A of coordinates (1 ; 2) and passes through the point with coordinates (0 ; 1) . 1 Answer the following two questions by graphical reading : a Give f (1) and f (1) , justifying the value of f (1) . b Read the solutions of the equation f ( x )=0 on the in-terval ]0 ; 10] . 2 We know that f ( x ) is of the form f ( x )=ln x + a x + b , where a and b denote two real numbers. a Calculate f ( x ) . b Using the values found for f (1) and f (1) in question 1 , calculate a and b . c Deduce the expression for f ( x ) . Part B We now know that the function f is defined on the interval ]0 ; 10] by: f ( x )=ln x + 2 x − 2 . 1 a Verify that for any real number x in the interval ]0 ; 10] : f ( x ) = x − 2 x Study the sign f ( x ) b We admit that the limit of f ( x ) when x tends towards 0 is + ∞ . Draw up the table of variations of the function f . Deduce the number of solutions to the equation : f ( x ) = 0 over the interval ]0 ; 10] . 2 Is the number 5 really a solution of the equation : f ( x ) = 0 ? https://chingmath.fr -12345678I-4-3-2-123456789JO sacados/85 sacados/102 -101234567891011-2-112345
-10123456789-112345AB(TC E.99 Let f be the numerical function de-fined on the interval ]0 ; + ∞ [ by: f ( x ) = (ln x ) 2 · (3 − 2 · ln x ) Let ( C ) be its curve represented in the plane provided with an orthonormal reference frame O ; −→ i ; −→ j (unité graphique 2 cm ) 1 a Calculate the limit of f in 0 and graphically inter-pret the result. b Calculate the limit of f in + ∞ 2 f denoting the derivative of f on 0 ; + ∞ , we assume that : f ( x ) = 6 · ln x · (1 − ln x ) x : a Solve inequations : ln x 0 1 − ln x 0 b Deduce the sign of f ( x ) and the variations of f . c Calculate the extremums of f on the interval 0.75 ; 3 . 3 Draw up the table of variations of f . 4 Draw the curve ( C ) . E.82 The curve (Γ) below represents, in an orthonormal coordinate system, a function f defined on the interval [1 ; + ∞ [ . Let f denote the derivative of f on this interval. The line ( T ) is tangent to the curve (Γ) at the point A (1 ; 1) . The tangent to the curve (Γ) at the point B with abscissa e is parallel to the x-axis. 1 By graphical analysis : a Find the slope of the line ( T ) . b Find f (1) and f (e) . c Determine the real numbers x in the interval [1 ; + ∞ [ that satisfy f ( x ) 0 d By drawing the tangent to the curve (Γ) at the point C as accurately as possible, read the slope of this tan-gent. 2 Assume that the function f is defined on the interval [1 ; + ∞ [ by: f ( x )= x 2 · 2 − ln x . a Calculate the y-coordinate of point B with x-coordinate e . b Determine the x-coordinate of point C , the intersec-tion of curve (Γ) with the x-axis. 3 The derivative f of the function f is defined on the in-terval [1 ; + ∞ [ by: f ( x )= k · ln e x where k is a given real number. a Check the answer given for f (e) in question 1 b Determine the real number k given that f e 2 = − 1 2 . c Find the equation of the tangent line to the curve (Γ) at the point C . d Find the coordinates of the point of intersection of the line ( T ) and the tangent to the curve (Γ) at point C . 7. Exponential, logarithm and sequence E.2011 Part A Let f be the function defined on the interval [0 ; 10] par : f ( x ) = 55 · e 0.5 x 1 Give the approximate values rounded to the unit of the numbers f (1) , f (2) , f (3) et f (4) . 2 a Determine the derivative function of the function f . b Determine the table of variations of the function f on the interval [0 ; 10] . 3 Solve in the interval [0 ; 10] , the equation : f ( x ) = 3 000 Rounding to the nearest unit will be given for any solu-tions. Part B A statistical study allows us to consider the function f of the part A as a satisfactory model to describe the evolution, from 2000 to 2010, of the total power of wind turbines in-stalled in France. More precisely, it is assumed that for the year (2000+ x ) where x is a natural number, the total power of wind turbines installed in France, expressed in megawatts, is given by f ( x ) . Using this model and exploiting the results of Part A, answer the following questions, giving the necessary justifications. https://chingmath.fr sacados/99 chapExoCorrec/82 sacados/82 Antilles - 2003 - 7 points - obligatoire -10123456789-112345AB(TC sacados/2011 Antilles-Guyane - Septembre 2007 - 6 points
-4-3-2-1234I234JOCfCg -4-3-2-1012-11234Cf -4-3-2-1012-11234Cg -4-3-2-1012-11234Ch 1 What was the total power of wind turbines in 2001? 2 In what year should the total capacity of wind turbines exceed 3 000 megawatts? 3 Will we be able to reach a total output of 10 000 megawatts in 2010? 4 For any natural number n , we pose : u n =55 · e 0.5 n a Prove that the sequence ( u n ) is a geometric sequence with common ratio e 0.5 b In the model studied, the total power of wind turbines therefore increases by the same percentage each year. Give this percentage, rounding the rate to the nearest tenth. E.33 2003 - 6 points - Compulsory The aim of this exercise is to study the decay of carbon 14, a radioactive body, and its use for dating fossils or skeletons. Part A Let N 0 be the number of carbon-14 atoms at time t =0 Let N 1 be the number of carbon-14 atoms one century later Let N k be the number of carbon-14 atoms after k centuries, k a natural number. We know that the number of carbon-14 atoms decreases very slowly over time, by about 1.24 % per century. 1 Justify that the sequence ( N k ) is a geometric sequence of reason 0.9876 2 Express N k as a function of N 0 and the integer k 3 What is the limit of the sequence ( N k ) ? Justify. Part B Cosmic rays continuously produce carbon-14 in the atmo-sphere, which decays there very slowly, so that the carbon-14 level in the atmosphere remains constant. During their lifetime, animal and plant tissues contain the same proportion of carbon 14 as the atmosphere ; when they die, carbon 14 assimilation ceases and it decays under the conditions seen in part A . 1 A prehistoric human skeleton contains 5 % initial carbon 14. Justify that his age can be estimated at 24 000 years. 2 Admit that we can thus estimate the age of fossils that contain at least 1% of the initial carbon. Using properties of the neperian logarithm function, de-termine the maximum age that can be calculated. 8. Basic exponential a E.7472 Consider the following five functions defined on R : f : x ↦−→ 0.2 x ; g : x ↦−→ 0.95 x ; h : x ↦−→ 1 x j : x ↦−→ 1.5 x ; k : x ↦−→ 3 x 1 Using the calculator, conjecture the direction(s) of vari-ation of each of these functions. 2 What similarities can be found between these different functions? E.7473 Consider the two functions f and g each admitting an expression of the form : f : x ↦−→ q x ; g : x ↦−→ q x where q; q ∈ 0 ; + ∞ The functions f and g are exponential functions with base q and q respectively. 1 By observing the direction of variation of each of the functions f and g . What can be said about the q basis of each? 2 Graphically and by the image of the number 1 , determine the expression of the functions f and g . E.7475 Consider the three functions f , g , and h , whose graphs are shown below in a coordinate system O ; I ; J : Only one of these three functions is a basic exponential func-tion q . Which one? Specify the value of its base q . 9. Introduction to exponential functions https://chingmath.fr chapExoCorrec/33 sacados/33 chapExoCorrec/7472 sacados/7472 chapExoCorrec/7473 sacados/7473 -4-3-2-1234I234JOCfCg chapExoCorrec/7475 sacados/7475 -4-3-2-1012-11234Cf -4-3-2-1012-11234Cg -4-3-2-1012-11234Ch
01234567891050100150200250300 123456789101112ABnun0111,7523,06335,35949,379516,413628,723750,265887,9649153,93710269,389 E.7471 Consider the sequence u n geometric with first term 1 and reason 1.75 defined for any natural num-ber n positive or zero. Below is a table of values for the first eleven terms of the sequence, rounded to the nearest thousandth, and their asso-ciated graphical representation : 1 a Give the value of the product u 2 × u 5 . Could the value of the result have been predicted? Jus-tify. b Without calculation, give the value of the product u 6 × u 3 . 2 On the dotted line has been drawn the curve connecting the points n ; u n . Let’s note C this curve and let’s note f the function having the curve C as its representation. a Give the value of the images : f (1) ; f (3) Verify that : f (1+3) = f (1) × f (3) . b By graphical reading, determine the images of the num-bers 4.5 and 6.5 . Verify the equality: f (2) × f (4.5) = f (6.5) E.7470 Consider the two sequences u n and v n defined for any positive integer n or zero by: u n = 2.6 n v 0 = 3 ; v n +1 = 1.8 · v n 1 Give the characteristic elements of the sequences u n and v n . 2 Complete the table of values, rounding to the nearest thousandth : n 0 1 2 3 4 5 u n v n E.2014 Consider the function f defined on the interval ]0 ; 3] by: f ( x ) = 2 · ln x − x 2 + 2 where ln denotes the neperian logarithm function. 1 Determine : lim x ↦→ 0 f ( x ) . 2 Show that for any real number x in the interval ]0 ; 3] : f ( x ) = 2 · (1 − x )(1 + x ) x Draw up the table of variations of f . 3 Note C the representative curve of the function f in the plane related to an orthogonal reference frame (graphic units: 5 cm on the x-axis and 1 cm on the y-axis) . a Specify the directing coefficient of the tangent T to the curve C at the point of abscissa 2. b Draw the curve C and the straight line T on a sheet of graph paper. c Using the graph, determine the number of solutions to the equation f ( x )=0 in the interval ]0 ; 3] . d Using a calculator, give the value, rounded to the tenth, of each of these solutions. https://chingmath.fr chapExoCorrec/7471 sacados/7471 01234567891050100150200250300 123456789101112ABnun0111,7523,06335,35949,379516,413628,723750,265887,9649153,93710269,389 chapExoCorrec/7470 sacados/7470 sacados/2014 France - Septembre 2006 - 6 points
2I2JO E.83 Reminders: a being a real constant, the function x ↦−→ ln( a · x ) has derivative function x ↦−→ 1 x . x and y being two strictly positive real numbers : ln( x · y ) = ln x + ln y ; ln x y = ln x − ln y x being a strictly positive real: exp(ln x )= x Sound manifests itself through variations in air pressure. The unit of measurement for air pressure is the Pascal. Air pressure is exerted on the eardrum of the human ear. For a pressure greater than or equal to 20 × 10 − 6 Pascals exerted on its eardrum, the human ear perceives a sound whose level is measured in decibels. We note : p 0 = 20 × 10 − 6 . For a pressure of p Pascals exerted on the eardrum, with p p 0 , the perceived sound level is f ( p ) decibels where : f ( p ) = 20 ln(20) · ln p p 0 That is : f ( p )= 20 ln(10) · ln(50000 · p ) 1 What is the perceived sound level for a pressure of 2 Pas-cals? of 0.2 Pascals? of 0.002 Pascals? 2 We note k = 20 ln 20 and I = p 0 ; + ∞ . Therefore, f is the function defined on the interval I by: f ( x ) = k · ln(50 000 · x ) . Let f be the derivative function of the function f on the interval I . a Specify the value of f ( p 0 ) b For any real x belonging to the interval I , calculate f ( x ) . Deduce the direction of variations of the func-tion f on the interval I . c Interpret the results of a and b in terms of pressure exerted on the eardrum and perceived sound level. 3 From a sound level of 120 decibels, pain is felt. Determine the pressure p corresponding to this sound level. 4 a Show that for any real x belonging to the interval I : f (10 · x ) = k · ln(10) + f ( x ) . We deduce that : f (10 · x )=20+ f ( x ) and we say that : ˇthe sound level increases by 20 decibels when the pres-sure exerted on the eardrum is multiplied by 10.ı b Express, for any real x belonging to the interval I , f (100 · x ) as a function of f ( x ) and state the correspond-ing sound level property. E.3178 The plane is provided with an orthonormal reference frame O ; −→ i ; −→ j . We are interested in the functions f derivable on 0 ; + ∞ verifying the conditions : (1) : for any real x belonging to 0 ; + ∞ , f ( x ) = 4 − f ( x ) 2 (2) : f (0) = 0 We admit that there exists a unique function f simultaneously verifying (1) and (2) . The two parts can be treated independently. The appendix will be completed and handed in with the copy at the end of the test. Part A. Study of a sequence To obtain an approximation of the representative curve of the function f , we use Euler’s iterative method with a step equal to 0.2 . We thus obtain a sequence of points noted ( M n ) , abscissa x n and ordinate y n such that : x 0 =0 and for any natural number n , x n +1 = x n + 0.2 y 0 =0 and for any natural number n , y n +1 = − 0.2 · y 2 n + y n + 0.8 1 a The coordinates of the first points are recorded in the table below. n 0 1 2 3 4 5 6 7 x n 0 0.2 0.4 y n 0 0.8000 1.4720 Complete this table. Results will be given to the near-est 10 − 4 . b Place, on the graph below, the points M n for n natural number less than or equal to 7 . c From this graph, what can we conjecture about the direction of variation of the sequence ( y n ) and its con- https://chingmath.fr sacados/83 chapExoCorrec/3178 sacados/3178 2I2JO
23I-1JOCf vergence? 2 a For real x , let p ( x )= − 0.2 x 2 + x +0.8 . Show that if x ∈ [0 ; 2] then p ( x ) ∈ [0 ; 2] . b Show that for any natural number n : 0 y n 2 . c Study the direction of variation of the sequence ( y n ) . d Is the sequence ( y n ) convergent? Part B. Study of a function Let g be the function defined on 0 ; + ∞ by: g ( x ) = 2 · e 4 x − 1 e 4 x + 1 and C g its representative curve. 1 Show that the function g verifies conditions (1) and (2) . 2 a Show that C g admits an asymptote Δ whose equa-tion is given. b Study the variations of g on 0 ; + ∞ . 3 Determine the abscissa ¸ of the point of intersection of Δ and the tangent at C g at the origin. 4 Draw, in the reference frame of the appendix, the curve C g and the elements highlighted in the previous ques-tions of this part B . E.5388 Let f be the function defined on the interval 0 ; + ∞ par : f ( x ) = 1 + ln x x 2 and let C be the curve representing the function f in a coor-dinate system on the plane. The curve C is given below : 1 a Study the limit of f at 0 . b What is the value of lim x ↦→ + ∞ ln( x ) x ? Deduce the limit of the function f at + ∞ . c Deduce any asymptotes to the curve C . 2 a Let f be the derivative of the function f on the interval 0 ; + ∞ . Prove that, for any real number x belonging to the interval 0 ; + ∞ : f ( x ) = − 1 − 2 · ln x x 3 b Solve over the interval 0 ; + ∞ l’inéquation: − 1 − 2 · ln x > 0 Deduce the sign of f ( x ) on the interval 0 ; + ∞ . c Draw up a table of variations for the function f . 3 a Show that the curve C has a single point of intersec-tion with the x-axis, whose coordinates you will spec-ify. b Deduce the sign of f ( x ) on the interval 0 ; + ∞ . 4 For any integer n 1 , we denote I n the area, expressed in units of area, of the domain bounded by the x-axis, the curve C and the lines with equations x = 1 e and x = n . a Prove that : 0 I 2 e − 1 2 . We assume that the function F , defined on the interval 0 ; + ∞ par : F ( x ) = − 2 − ln( x ) x is a primitive of the function f on the interval 0 ; + ∞ . b Calculate I n based on n . c Study the limit of I n in + ∞ . Interpret the result graph-ically. E.5171 Let f be the function defined on 0 ; + ∞ par : f ( x ) = x + e − x Note ( C ) the representative curve of f dans an orthonormal frame O ; −→ i ; −→ j . Part A 1 Study the variations of the function f sur 0 ; + ∞ . 2 Determine the limit of f en + ∞ . 3 Show that ( C ) admet an oblique asymptote, an equa-tion of which will be specified. (Question outside the 2012 syllabus) . 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 − n pour all n ∈ N 1 Show that, for any real x positif, ln(1+ x ) x . We can study the function g defined on 0 ; + ∞ par : g ( x ) = x − ln(1+ x ) . 2 Deduce, that for any natural number n nonzero : ln( n +1) ln( n ) + 1 n . 3 Demonstrate that, for any natural number n non-zero : f ln( n ) = ln( n ) + 1 n . 4 Demonstrate by recurrence that, for any natural number n non-zero : ln( n ) u n . 5 Determine the limit of u n n 1 . In the remainder of the exercise, it is assumed that for any integer n greater than or equal to 2 : u n 1+ 1 2 + · · · + 1 n − 1 6 a Prove that, for any integer k greater than or equal to 2 , we have : 1 k k k − 1 1 x d x b From this, we can deduce that, for any integer n greater than or equal to 2 , we have : u n 1 + ln( n − 1) 7 For any integer greater than or equal to 2 , it has been shown that : ln( n ) u n 1 + ln( n − 1) Prove that the sequence u n ln( n ) n 2 converges to 1 . https://chingmath.fr chapExoCorrec/5388 sacados/5388 23I-1JOCf chapExoCorrec/5171 sacados/5171 Liban Mai 2011 7 points
Durée en jours05101520Nombre de pucerons en milliers246 10. Annales - Term L? E.2013 Aphids invade a rose garden. La-dybugs, aphid predators, are introduced into this rose garden. After twenty days, the number of aphids can be estimated at 770, or 0.77 thousands. We are interested in the evolution of the number of aphids (ex-pressed in thousands) present in the rose garden as a function of the time elapsed since the introduction of the ladybugs. We denote f this function and t this duration. The unit of dura-tion is one day. When ladybugs are introduced, we therefore have t =0 . 1 Studies have shown that the number of aphids (expressed in thousands) as a function of the time t elapsed since the introduction of ladybugs, was modelled by the function f defined, for any real number t element of [0 ; 20] by: f ( t ) = (2 · t + 2) · e − kt where k is a constant positive real number. a What is the number of aphids at the time the ladybugs are introduced into this rose garden? b Determine the exact value of k and then one of its values approximated to the nearest thousandth. Throughout the rest of the exercise, consider the function f defined for any real number t element of [0 ; 20] by f ( t )= (2 · t +2)e − 0.2 t , correctly represents the evolution of aphid num-bers as a function of time t . Note f the derivative function of f and ( C ) f the representative curve of f in an orthogonal reference frame. 2 a Demonstrate that, for any real number t of [0 ; 20] : f ( t ) = ( − 0.4 · t + 1.6)e − 0.2 · t b How many days after, the introduction of predators will the number of aphids start to decrease? c Calculate f (0) . Use this derived number to calculate, without using a calculator, an approximation of the number of aphids present in the rose garden after one day. 3 The graph given in Appendix 2 is a drawing of ( C f ) . This graph is to be completed and returned with the copy. a Using the information given or obtained previously, place the units on the marker. b Aphids are no longer considered a problem once their number has fallen below 1 000 . Read graphically after how many days this threshold will be reached. leave the construction lines used for this read-ing visible. E.75 Part A Consider the functions f and g defined on the interval 1 ; 25 by: f ( x ) = 5 · x − 50 ; g ( x ) = exp x 100 − 100 1 a Determine the direction of variation of the function f on the interval 1 ; 25 . b Find the derivative g ( x ) . c Examine the direction of variation of the function g on the interval [1 ; 25] . d Plot the graphs of the functions f and g on a Cartesian coordinate system. The graphing units will be : 2 cm for 5 units on the x-axis. 1 cm for 20 units on the y-axis. Note: For the graph of the function g , we will limit our-selves to plotting the points whose x-coordinates lie between 1 and 10. Part B 2 Consider the function h defined on the interval 1 ; 25 by h ( x )=30 · ln x − 2 · x +10 . Let h denote the derivative of h . a Show that : h ( x )= 30 − 2 · x x . b Investigate the direction of variation of the function h on the interval 1 ; 25 . c Show that the function h has a maximum. Give the value at which this maximum is reached and the value of the function at that point. d Plot the graph of the function h on the same coordi-nate plane as in question 1 . Part C The earnings, expressed in thousands of euros, of three singers are a function of the number x of weeks that have elapsed since the simultaneous release of their albums and are given by f ( x ) , g ( x ) , and h ( x ) . 3 In this question, no justification is required. The task is simply to interpret the data. a Which function corresponds to the earnings of singer https://chingmath.fr chapExoCorrec/2013 sacados/2013 Durée en jours05101520Nombre de pucerons en milliers246 chapExoCorrec/75 sacados/75 Japon - 2003 - 8 points - Obligatoire
A , in whom the producers invested nearly 100 thou-sand euros and whose success was phenomenal after a few weeks of intense promotion? b Which function corresponds to the earnings of singer B , an unknown artist who achieved rapid success despite a lack of promotional investment before his popularity began to wane after fifteen weeks? c Using the questions in 1 et 2 , as a guide, describe the trend in the earnings of singer C . 4 By reading the graph : a Determine the number of weeks that elapse before singer C earns more than singer B . b Determine the number of weeks that pass before singer A earns more than singer B . E.76 Part A Consider the function g defined on R by: g ( x )=e x − 2 · x . 1 Calculate g ( x ) , where g denotes the derivative of g , and then construct the table of variations for g . 2 Deduce that for any real number x in R , g ( x ) > 0 . Part B Consider the function f defined on R by: f ( x )=e x − x 2 . 1 Find the limit of f as −∞ , then the limit of f as + ∞ . For the limit at + ∞ , note that for x not equal to zero, f ( x ) can be written as : x 2 · e x x 2 − 1 2 Compute f ( x ) , where f denotes the derivative of the function f , and then, using the part A , construct the variation table for f . 3 Assume that the equation f ( x )=0 has at least one solu-tion in R . a Compute f ( − 1) and f (0) . b Show that the solution to equation f ( x )=0 is unique and belongs to the interval − 1 ; 0 . c Using a calculator to compute f ( x ) for different val-ues of x , give the value of this solution rounded to the nearest 10 − 3 . Justify the value chosen. E.97 the following questions are taken from the 2006 Terminales L exams : 1 Let f be a function defined on I =[20 ; 150] by: f ( x ) = 2 · x + 13122 x a Show that on the interval I : f ( x ) = 2 x 2 · ( x − 81)( x + 81) Deduce that on the interval I , f ( x ) is of the sign of ( x − 81) . b Draw up the table of variations of the function f on the interval I . 2 Consider the function g defined on the interval ]0 ; + ∞ [ by: g ( x ) = 2 · x 2 − 3 · x + 9 2 − ln x The function g is derivable on the interval ]0 ; + ∞ [ and we note g its derivative function : a Show that, for any strictly positive real x : g ( x ) = (4 · x + 1)( x − 1) x . b Study the variations of the function g on the interval ]0 ; + ∞ [ 3 Let f be the numerical function defined on the interval ]0 ; + ∞ [ by: f ( x ) = (ln x ) 2 · (3 − 2 ln x ) a f denoting the derivative of f on ]0 ; + ∞ [ , we assume that f ( x ) = 6 · ln x · (1 − ln x ) x . ln x 0 1 − ln x 0 b Deduce the sign of f ( x ) and the variations of f . c Calculate the extremums of f on the interval [0.75 ; 3] 4 We denote by f the function defined on R by: f ( x ) = 2 · x + 5 − e x We denote f the derivative function of f . We call Γ its representative curve in an orthonormal ref-erence frame. a Solve in R the inequation of unknown x : 2 − e x > 0 b Calculate the exact value of f (ln 2) . c Determine the limits of f in −∞ and in + ∞ . d Calculate f ( x ) then draw up the complete table of variations of the function f . 5 Consider the function f defined on R by: f ( x ) = x + 3 e x − 1 e 2 x = x + 3 · e − x − e − 2 x And the function g defined on R by: g ( x ) = 1 − 3 · e − x + 2 · e − 2 x a Show that for any real x : g ( x )= (e x − 1)(e x − 2) e 2 x b Study the sign of g ( x ) depending on the values of x . c Show that, for any real x , f ( x )= g ( x ) . Deduce the table of variations of f at R . https://chingmath.fr chapExoCorrec/76 sacados/76 sacados/97
Instantten heures2345678IQuantité présente dans le sang (encm3)23JO E.88 Consider the function d defined on [0 ; + ∞ [ by: f ( x ) = 1000 · e x · ln(0.94) 1 Show that for any integer n : (0.94) n = e n · ln(0.94) 2 Give a value rounded to the nearest integer for : f 1 7 et f 365 7 . 3 a For any number x ∈ [0 ; + ∞ [ , calculate f ( x ) . b Give a value of ln(0.94) rounded to the tenth and de-duce the direction of variations of f on [0 ; + ∞ [ . 4 Show that the equation f ( x )=500 , verifies x = ln(0.5) ln(0.94) E.113 Parts A and B are independent Part A To perform a medical examination, a dose of 3 cm 3 of a drug substance is injected into the bloodstream by intramuscular injection of a patient at the time t =0 ( t is expressed in hours) . This then gradually passes into the bloodstream. Diffusion reaches its maximum after one hour. The curve in the appendix represents the amount of substance present in the blood at time t . 1 Construct on the attached sheet the tangent to the curve at the point of abscissa 2, knowing that its directing co-efficient is equal to ( − 0.9) . 2 From the graph comment on the evolution of the amount of drug substance contained in the blood. 3 In order to perform the test, the amount of drug sub- stance in the blood must be greater than or equal to 0.5 cm 3 . Graphically determine how much time is avail-able to perform this examination. Part B A patient was injected by intravenous injection 1 cm 3 of drug at time t =0 . The substance is immediately distributed in the blood and is then gradually eliminated. Experimentally, it is shown that the amount q ( t ) of substance present in the blood at time t is given by the relation q ( t )=e − 0.15 · t where t is expressed in hours. 1 What volume of this product remains after 90 minutes? 2 What volume of this product has the patient eliminated after half an hour? one hour? 3 We give q ( t )= − 0.15 · e − 0.15 · t where q denotes the func-tion derived from the function q . Study the variations of the function q on the interval [0 ; 9] then plot its graphical representation in an orthog-onal frame of reference, taking 2 cm as the abscissa and 10 cm as the ordinate. 11. Unclassified financial years E.100 Give, in each case, the definition set as well as the derivative function of each proposed function : 1 f ( x ) = 2 · ln x + 2 · x 2 f ( x ) = 3 · ln(5 − 3 · x ) + 2 3 f ( x ) = ( x + 1) · ln x E.103 Using the calculator, determine the follow-ing limits: a lim x ↦→ 0 + ln x x b lim x ↦→ + ∞ ln x x c lim x ↦→ 0 + x 2 − ln x d lim x ↦→ + −∞ x 2 − e x e lim x ↦→ + ∞ x 2 − e x f lim x ↦→ + ∞ x 2 · e x E.3774 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 relative to an orthonormal reference frame O ; −→ i ; −→ j graphical unit 2 cm . 1 a Prove that the line D 1 of equation y = x +2 is an asymptote to the curve C . . b Study the position of C relative to D 1 . 2 a We denote f as the derivative of f . Calculate f ( x ) and show that, for any real number x , we have : f ( x ) = e x − 3 e x + 3 2 b Study the variations of f on R and draw up a table of variations for the function f . 3 Determine the equation of the tangent D 2 to the curve C at the point with abscissa 0 . 4 We assume that the point I is the center of symmetry of the curve C . Draw the curve C , the lines D 1 and D 2 . Remember that the chosen graphic unit is 2 cm . https://chingmath.fr sacados/88 sacados/113 Instantten heures2345678IQuantité présente dans le sang (encm3)23JO chapExoCorrec/100 sacados/100 sacados/103 chapExoCorrec/3774 sacados/3774 Inspire de La Reunion Juin 2009