Grade 12 / Annales exponentielles, logarithmes, integrals 14 exercises (including 13 corrected)

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xxyy-5-4-3-2-12345I-3-2-12345JOCfCg 1. Exponential functions E.5844 Let f be the differentiable func-tion defined on the interval 0 ; + ∞ by: f ( x ) = e x + 1 x 1 Study of an auxiliary function : a Let g be the derivative function, defined on 0 ; + ∞ by: g ( x ) = x 2 · e x − 1 Study the direction of variation of the function g . b Show that there exists a single real a belonging to 0 ; + ∞ such that g ( a )=0 . Show that a belongs to the interval 0.703 ; 0.704 . c Determine the sign of g ( x ) at 0 ; + ∞ . 2 Study of function f : a Determine the limits of the function f in 0 and in + ∞ . b Let f be the derivative function of f on the interval 0 ; + ∞ . Show that for any strictly positive real: f ( x ) = g ( x ) x 2 c Deduce the direction of variation of the function f and draw up its table of variations on the interval 0 ; + ∞ . d Demonstrate that the function f admits the real num-ber: m = 1 a 2 + 1 a e Justify that : 3.43 <m< 3.45 . E.5850 Consider the functions f and g defined for any real x by: f ( x ) = e x ; g ( x ) = 1 − e − x The representative curves of these functions in an orthogonal plane, noted respectively C f and C g are provided in the figure below : Part A These curves appear to admit two common tangents. Draw these tangents as best you can on the figure above. Part B In this part, we admit the existence of these common tan-gents. We note D one of them. This line is tangent to the curve C f at the point A of abscissa a and tangent to the curve C g at the point of abscissa b . 1 a Express as a function of a the slope of the tangent to the curve C f at the point A . b Express as a function of b the slope of the tangent to the curve C g at the point B . c Deduce that : b = − a . 2 Demonstrate that the real a is a solution of the equation : 2( x − 1)e x + 1 = 0 Part C Consider the function ’ defined on R by: ’ ( x ) = 2( x − 1)e x + 1 1 a Calculate the limits of the function ’ in −∞ and + ∞ . b Calculate the derivative of the function ’ , then study its sign. c Draw up the table of variations of the function ’ on R . Specify the value of ’ (0) . 2 a Show that the equation ’ ( x )=0 admits exactly two solutions in R . b Note ¸ the negative solution of the equation ’ ( x )=0 and ˛ the positive solution of this equation. Using a calculator, give the values of ¸ and ˛ rounded to the hundredth. https://chingmath.fr chapExoCorrec/5844 sacados/5844 chapExoCorrec/5850 sacados/5850 xxyy-5-4-3-2-12345I-3-2-12345JOCfCg
2345678I2345JO MPL¸ Part D In this part, we demonstrate the existence of these common tangents, which we admitted in part B . We note E the point on the curve C f of abscissa ¸ and F the point on the curve C g of abscissa − ¸ ( ¸ is the real number defined in part C ) . 1 Show that the straight line ( EF ) is tangent to the curve C f at the point E . 2 Show that ( EF ) is tangent to C g at the point F . E.6934 The director of a zoo wants to build a slide for the pandas. He makes the following diagram of this slide in cavalier perspective. Here’s the diagram: Part A : Modeling The profile of this slide is modeled by the curve C represent-ing the function f defined on the interval 1 ; 8 by: f ( x ) = a · x + b · e − x où a and b are two natural numbers. The curve C is plotted below in an orthonormal reference frame whose unit is the meter. 1 We want the tangent to the curve C at its point of ab-scissa 1 to be horizontal. Determine the value of the integer b . 2 We want the top of the slide to be between 3.5 and 4 meters high. Determine the value of the integer a . Part B: A visitor amenity It is assumed in the following that the function f introduced in part A is defined for any real x ∈ 1 ; 8 by: f ( x ) = 10 · x · e − x The toboggan retaining wall will be painted by an artist on one side only, hatched on the diagram at the start of the ex-ercise. On the quotation he proposes, the latter asks for a flat rate of 300 euros increased by 50 euros per square meter painted. 1 Let g be the function defined on 1 ; 8 by: g ( x ) = 10 · − x − 1 · e − x Determine the derivative function of the g function. 2 How much is the artist’s quote? Part C : a constraint to check For safety reasons, the maximum slope of the slide must be limited. Consider a point M on the curve C , with abscissa different from 1 . We call ¸ the acute angle formed by the tangent at M to C at the abscissa axis. The following figure illustrates the situation. Constraints dictate that the angle ¸ must be less than 55 degrees. 1 Let f be the derivative function of the function f on the interval 1 ; 8 . We admit that, for any x of the interval 1 ; 8 : f ( x )=10 · 1 − x · e − x Study the variations of the function f on the interval 1 ; 8 . 2 Let x be a real from the interval 1 ; 8 and let M be the point of abscissa x on the curve C . Justify that : tan ¸ = ⏐ ⏐ f ( x ) ⏐ ⏐ 3 Does the slide comply with the constraints imposed? https://chingmath.fr chapExoCorrec/6934 sacados/6934 2345678I2345JO MPL¸
234I-1JO E.3121 Part A : Study of an auxiliary function : The function d is defined on − 1 ; + ∞ by: d ( x ) = e x x +1 1 Calculate the derivative function d . Deduce the varia-tions of d . 2 Determine the limits of d in − 1 and in + ∞ . 3 Show that, for any x> − 1 : 0 <d ( x ) <e Part B: study of the function f In this part, we are interested in the function f defined on the interval − 1 ; + ∞ by: f ( x ) = x + 1 − e x x +1 We call ( C ) the representative curve of f in an orthonormal reference frame, the graphical unit being 5 cm . We denote by f and f  the first and second derivatives of f . 1 a For x ∈ − 1 ; + ∞ , calculate f ( x ) and f  ( x ) . Vérifier que : f  ( x ) = 2 x + 1 ( x + 1) 4 · e x x +1 Deduce the direction of variations of f . b Draw up the table of variations of f . On admettra que lim x ↦→− 1 f ( x )= lim x ↦→ + ∞ f ( x )=1 2 a Demonstrate that the equation f ( x )=0 admits on the interval − 1 ; + ∞ two solutions, one of which is 0. In the rest of the problem, we’ll note ¸ the non-zero solution. b Give an approximate value of ¸ to the nearest hun-dredth. 3 a Study the variations of f . b Calculate the limits of f at the bounds of its defining set. c Draw up the table of variations of f . Part C : Extension of function f into − 1 Consider the function g defined on − 1 ; + ∞ by: g ( − 1) = 0 g ( x ) = f ( x ) for all x> − 1 We call C the representative curve of the function g in the reference frame of Part B . 1 a Show that we can write: g ( x ) − g ( − 1) x − ( − 1) = 1 − 1 x · x x + 1 · e x x +1 b For x ∈ − 1 ; + ∞ , determine the limit when x tends to − 1 from x x +1 then from x x +1 · e x x +1 . c Deduce that g is derivable in − 1 and specify its derived number g ( − 1) . 2 Construct ( D ) and ( C ) . Specify the tangents to C at points of abscissa − 1 , ¸ , 0 . E.5855 Part A Let f be the function defined on R by: f ( x ) = x · e 1 − x 1 Verify that for any real x : f ( x ) = e × x e x 2 Determine the limit of the function f in −∞ . 3 Determine the limit of the function f in + ∞ . Interpret this limit graphically. 4 Determine the derivative of the function f . 5 Study the variations of the function f on R then draw up the table of variation. Part B For any non-zero natural number n , consider the functions g n and h n defined on R by: g n ( x ) = 1 + x + x 2 + · · · + x n h n ( x ) = 1 + 2 x + · · · + n · x n − 1 1 Verify that, for any real x : (1 − x ) · g n ( x ) = 1 − x n +1 We then obtain, for any real x =1 : g n ( x ) = 1 − x n +1 1 − x 2 Compare the functions h n and g n , g n being the deriva-tive of the function g n . Deduce, that for any real x =1 : h n ( x ) = n · x n +1 − n + 1 · x n + 1 1 − x 2 3 Let S n = f (1)+ f (2)+ ··· + f ( n ) , f being the function de-fined in part A . Using the results of part B , determine an expression for S n and then its limit when n tends to + ∞ . 2. Exponentials and sequences E.3158 The plane is referred to an or-thogonal reference frame O ; −→ i ; −→ j . Let the function f be defined on 0 ; + ∞ by: f ( x ) = e − x cos(4 x ) and Γ its representative curve plotted in the reference frame O ; −→ i ; −→ j : https://chingmath.fr chapExoCorrec/3121 sacados/3121 chapExoCorrec/5855 sacados/5855 chapExoCorrec/3158 sacados/3158 234I-1JO
Consider also the function g defined on 0 ; + ∞ by: g ( x ) = e − x and we name C its representative curve in the reference frame O ; −→ i ; −→ j . 1 a Show that, for any real x belonging to the interval 0 ; + ∞ : − e − x f ( x ) e − x b Deduce the limit of f in + ∞ . 2 Determine the coordinates of the points common to the Γ and C curves. 3 We define the sequence u n on N by: u n = f n · ı 2 . a Show that the sequence u n is a geometric sequence. Specify the reason. b Deduce the direction of variation of the sequence u n and study its convergence. 4 a Show that, for any real x belonging to the interval 0 ; + ∞ : f ( x ) = − e − x · cos(4 x ) + 4 · sin(4 x ) b Deduce that the curves Γ and C have the same tangent at each of their common points. 5 Give an approximate value to 10 − 1 to the nearest excess of the directing coefficient of the line T tangent to the curve Γ at the point of abscissa ı 2 . Complete the graph given in the appendix, plotting T and C . 3. Logarithmic functions E.3179 1 Organized knowledge transfer Prerequisites: The neperian logarithm function is derivable on the interval 0 ; + ∞ [ ; Its derivative function is the inverse func-tion : x ↦−→ 1 x . ln(1) = 0 Show that for all strictly positive real numbers ¸ and x : ln( ¸ · x ) = ln( ¸ ) + ln( x ) 2 Use the previous result to show that : ln 1 a = − ln( a ) ; ln a b = ln( a ) − ln( b ) . for all strictly positive reals a and b . 3 We give : 0.69 ln2 0.70 and 1.09 ln3 1.10 . From this we can deduce frames for : ln 6 ; ln 1 6 ; ln 3 8 E.3898 Part A Let g be the function defined for any real number x in the interval 0 ; + ∞ by: g ( x )= x − x · ln x 1 Determine the limits of the function g in 0 and + ∞ . 2 Show that g is derivable on the interval 0 ; + ∞ and that : g ( x ) = − ln x 3 Draw up the table of variations of the function g . Part B Let u n be the sequence defined for all n ∈ N ∗ by: u n = e n n n 1 Conjecture, using the calculator: a the direction of variation of the sequence u n ; b the eventual limit of the sequence u n . 2 Let v n be the sequence defined for all n ∈ N ∗ by: v n = ln u n . a Show that : v n = n − n · ln n . b Using the part A , determine the direction of variation of the sequence v n . c Deduce the direction of variation of the sequence u n . 3 Show that the sequence u n is bounded. 4 Show that the sequence u n is convergent and determine its limit. https://chingmath.fr chapExoCorrec/3179 sacados/3179 Antilles-Guyane Juin 2006 3 points chapExoCorrec/3898 sacados/3898 Antilles Guyane Juin 2010 6 points
-6-5-4-3-2-12I-2-1234JO -0.4-0.20.20.40.60.81.21.41.6I-0.20.20.40.60.81.2JO E.5433 Consider the equation ( E ) of un-known x real: e x = 3 x 2 + x 3 Part A : graphical conjecture The graphbelow shows the representative curve of the expo-nential function and that of the function f defined on R by f ( x )=3 · x 2 + x 3 as displayed by a calculator in the same or-thogonal reference frame. Using the graph above, conjecture the number of solutions to the equation ( E ) and their framing by two consecutive integers. Part B: study of the validity of the graphical conjec-ture 1 a Study according to the values of x , the sign of x 2 + x 3 . b Deduce that the equation ( E ) has no solution on the interval −∞ ; − 1 . c Verify that 0 is not a solution of ( E ) . 2 Consider the function h , defined for any real number from − 1 ; 0 ∪ 0 ; + ∞ by: h ( x ) = ln 3 + ln x 2 + ln(1+ x ) − x Show that, at − 1 ; 0 ∪ 0 ; + ∞ , the equation ( E ) is equivalent to h ( x )=0 . 3 a For any real x belonging to − 1 ; 0 ∪ 0 ; + ∞ , show that we have : h ( x ) = − x 2 + 2 x + 2 x ( x + 1) b Determine the variations of the function h . c Determine the number of solutions to the equation : h ( x ) = 0 and give a value rounded to the hundredth for each solution. d Conclude as to the conjecture of part A . E.5151 Part A - Studying the sign of a function Denote by f the function defined on the interval 0 ; + ∞ by: f ( x ) = x 2 + 4 · ln x 1 Determine the table of variations of the function f , spec-ifying the limits of f in 0 and in + ∞ . 2 Show that the equation f ( x )=0 admits one solution ¸ and only one in the interval 0 ; + ∞ . 3 Deduce the sign of f ( x ) depending on the values of the strictly positive real x . Part B - An approximate value of the real ¸ defined in Part A On the graph provided below, part of the representative curve of ( C ) of the function g defined on R by has been plotted : g ( x ) = e − 1 4 x 2 We defined the sequence u n by: u 0 =0.5 ; u n +1 = g ( u n ) for all n ∈ N . 1 Verify that ¸ is the unique solution of the equation : g ( x ) = x . 2 By means of the curve ( C ) and the straight line with equation y = x , plot the terms u 1 , u 2 and u 3 of the se-quence u n on the x-axis. What conjecture can be made about the convergence of the sequence u n ? 3 We admit that for any natural number n : u 2 n a u 2 n +1 . Using the calculator, determine the smallest integer n for which the first three decimal places of u n and u n +1 are identical. Deduce that 0.838 is an approximate value of ¸ to within 10 − 3 . Part C - A problem of distance We call (Γ) the representative curve, in an orthonormal frame, of the function ’ defined on the interval 0 ; + ∞ by: ’ ( x ) = 2 ln x The aim of this part is to show that among the points on the curve (Γ) , there is one and only one that is closer to the origin O than all the others. 1 Let M be a point on the curve (Γ) and x its abscissa. Express OM as a function of x . 2 a Let h be the function defined on the interval 0 ; + ∞ by: h ( x )= x 2 +4 · ln x 2 Study the variations of the function h . The part A may be used. b Deduce that there exists a single point A of the curve Γ such that for any point M of Γ , distinct from A , we have OM >OA . 3 Show that the straight line ( OA ) is perpendicular to the tangent T to the curve (Γ) at the point A . https://chingmath.fr chapExoCorrec/5433 sacados/5433 -6-5-4-3-2-12I-2-1234JO chapExoCorrec/5151 sacados/5151 -0.4-0.20.20.40.60.81.21.41.6I-0.20.20.40.60.81.2JO
4. Exponentials and logarithms E.3965 1 Consider the function f 1 defined on 0 ; + ∞ by: f 1 ( x ) = 2 · x − 2 + ln x 2 +1 a Determine the limit of f 1 in + ∞ . b Determine the derivative of f 1 . c Draw up the table of variations of f 1 . 2 Let n be a non-zero natural number. Consider the func-tion f n , defined on 0 ; + ∞ by: f n ( x ) = 2 · x − 2 + ln x 2 +1 n a Determine the limit of f n in + ∞ . b Demonstrate that the function f n is strictly increasing on 0 ; + ∞ . c Show that the equation f n ( x )=0 admits a unique so-lution ¸ n on 0 ; + ∞ . d Justify that, for any non-zero natural number n : 0 <¸ n < 1 3 Show that for any non-zero natural number n : f n ¸ n +1 > 0 4 Study of the suite ¸ n : a Show that the sequence ¸ n is increasing. b Deduce that it is converging. c Use the expression ¸ n =1 − ln ¸ 2 n +1 2 · n to determine the limit of this sequence. 5. A little further on E.3266 Consider the function f defined on R by: f ( x ) = 1 + e − x − 2e − 2 x and C its representative curve in a plane related to an or-thogonal reference frame O ; −→ i ; −→ j . (graphic units: 3 cm on the x-axis and 8 cm on the y-axis) 1 a Let the polynomial P be defined on R by: P ( X ) = 1 + X − 2 X 2 . Study the sign of P ( X ) . b Deduce the sign of f ( x ) on R . c What can we deduce from this for the curve C ? 2 Determine the limit of the function f in + ∞ . What can you deduce for the curve C ? 3 Check that f ( x )=e − 2 x · e 2 x +e x − 2 , then determine the limit of f in −∞ . 4 a Let f be the function derived from the function f , calculate f ( x ) . b Show that f ( x ) has the sign that (4 − e x ) , then study the sign of f ( x ) . c Draw up the table of variations of f . We’ll show that the maximum is a rational number. 5 a Show that the curve C and the straight line D of equation y =1 have only one point of intersection A whose coordinates will be determined. b Study the position of the curve C relative to the line D . 6 Determine an equation of the tangent T to the curve C at the point A . 7 Draw the straight lines D and T , then the curve C . 6. Unclassified financial years E.3255 Part A Consider the function f defined on the interval 0 ; + ∞ by: f ( x ) = x + ln x We call Γ its representative curve in an orthogonal reference frame O ; −→ i ; −→ j of the plane. 1 a Determine the limits of the function f at the bounds of its interval of definition. b Show that the function f is strictly increasing on the interval 0 ; + ∞ . 2 a Show that, for any natural number n , the equation f ( x )= n admits a single solution in 0 ; + ∞ . We denote ¸ n this solution. So we have : for any natural number n , ¸ n +ln ¸ n = n b Below, we’ve plotted Γ in the O ; −→ i ; −→ j reference frame. Place the numbers ¸ 0 , ¸ 1 , ¸ 2 , ¸ 3 , ¸ 4 and ¸ 5 on the x-axis, leaving the construction lines visible. https://chingmath.fr chapExoCorrec/3965 sacados/3965 Amerique du Sud Novembre 2007 6 points chapExoCorrec/3266 sacados/3266 chapExoCorrec/3255 sacados/3255
234567I-2-123456JO CfCg-2-1012345-2-1123ij c Specify the value of ¸ 1 . d Demonstrate that the sequence ( ¸ n ) is strictly increas-ing. 3 a Determine an equation of the tangent Δ to the curve Γ at point abscissa 1. b Study the variations of the function h defined on 0 ; + ∞ by: h ( x ) = ln x − x + 1 Deduce the position of curve Γ relative to Δ . c Plot Δ on the graph above. Show that, for any non-zero natural number n : n + 1 2 ¸ n . 4 Determine the limit of the sequence ( ¸ n ) . Part B Consider a function g continuous, strictly increasing on 0 ; + ∞ and such that : lim x ↦→ 0 g ( x ) = −∞ et lim x ↦→ + ∞ g ( x ) = + ∞ . We admit that we can, as we did in part A , define on N a sequence ( ˛ n ) of real numbers such that g ( ˛ n )= n , and that this sequence is strictly increasing. 1 Course demonstration : Prerequisite : definition of a sequence tending to + ∞ . ˇA sequence tends to + ∞ if, for any real A , all terms of the sequence are, from a certain rank, greater than A ı Prove the following theorem : a non-major increasing se-quence tends to + ∞ 2 Show that the sequence ( ˛ n ) tends to + ∞ . E.8142 An advertiser wants to print the following logo on a T-shirt : He draws this logo using the curves of two functions f and g defined on R by: f ( x ) = e − x · − cos x + sin x + 1 et g ( x ) = − e − x cos x We admit that the functions f and g are derivable on R . Part A - Study of the function f 1 Justify that, for any x ∈ R : − e − x f ( x ) 3 e − x 2 Deduce the limit of f in + ∞ 3 Demonstrate that, for any x ∈ R : f ( x ) = e − x 2 cos x − 1 where f is the derivative function of f . 4 In this question, we study the function f on the interval − ı ; ı . a Determine the sign of f ( x ) for x belonging to the in-terval − ı ; ı . b Deduce the variations of f on − ı ; ı . Part B - Logo area Note C f and C g the graphical representations of the functions f and g in an orthonormal frame O ; −→ i ; −→ j . The graphic unit is 2 centimeters. These two curves are plotted below : 1 Study the relative position of the C f curve to the C g curve at R . 2 Let H be the function defined on R by: H ( x ) = − cos x 2 − sin x 2 − 1 · e − x We admit that H is a primitive of the function x ↦−→ sin x +1 · e − x on R . We note D the domain bounded by the curve C f , the curve C g is the straight lines of equation x = − ı 2 and x = 3 ı 2 . a Hatch the domain D on the graph above. b Calculate, in area units, the area of the domain D , then give an approximate value to the nearest 10 − 2 in cm 2 . https://chingmath.fr 234567I-2-123456JO sacados/8142 Antilles-guyanes Juin 2018 5 points CfCg-2-1012345-2-1123ij