Grade 12 - Comp. / Exponential annals, logarithms, convexity 10 exercises (100% corrected)

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x01234567891011f(x123456ABCf 1. Study of functions E.7572 Consider the function C defined on the interval 5 ; 60 by: C ( x ) = e 0.1 x + 20 x 1 Let C denote the derivative of the function C . Show that, for all x ∈ 5 ; 60 : C ( x ) = 0.1 · x · e 0.1 · x − e 0.1 · x − 20 x 2 2 Consider the function f defined on 5 ; 60 by: f ( x ) = 0.1 · x · e 0.1 · x − e 0.1 · x − 20 a Show that the function f is strictly increasing on 5 ; 60 . b Show that the equation f ( x )=0 has a unique solution ¸ in 5 ; 60 . c Give a bound for the unit of ¸ . d Deduce the sign table of f ( x ) on 5 ; 60 . 3 Deduce the variation table of C on 5 ; 60 . 4 Using the previous variation table, determine the number of solutions to the following equations : a C ( x ) = 2 b C ( x ) = 5 E.6983 Part A Consider the function f defined on the interval − 20 ; 20 by: f ( x )= − 2 x +30 · e 0.2 · x − 3 1 a Show that f ( x )= − 0.4 x +4 · e 0.2 x − 3 for all real numbers x in the interval − 20 ; 20 . b Draw up a table of variations for the function f on the interval − 20 ; 20 . Specify the exact value of the maximum of f . 2 a Show that, on the interval − 20 ; 20 , the equation f ( x )= − 2 has a unique solution ¸ . b Give a bound for ¸ with amplitude 0.1 . 3 A computer algebra system gives the following results : 1 Derive − 10 x + 200)e 0.2 x − 3 − 2 x + 30 e 0.2 x − 3 2 Derive − 2 x + 30 e 0.2 x − 3 − 0.4 x + 4 e 0.2 x − 3 3 Derive − 0.4 x + 4 e 0.2 x − 3 − 0.08 x + 0.4 e 0.2 x − 3 Answer the following two questions using the results pro-vided by the software: a Calculate the exact value of : 15 10 f ( x ) d x . b Determine the largest interval over which the function f is convex and specify the x-coordinate of the inflec-tion point. Part B A ski resort wants to open a new slope to the public. The re- lief of this slope is modeled below by the representative curve C f of the function f defined in part A on the interval 0 ; 10 . Point B represents the start of the new trail and point A represents the ski resort where the finish line is located The actual x represents the horizontal distance, expressed in km , from the ski resort, and f ( x ) represents the altitude, ex-pressed in km . The slope of the piste at point M is defined as the slope of the tangent to the curve C f at point M . For example, a slope of 15 % at a point on the piste corresponds to a slope of 15 100 =0.15 . 1 The elevation difference between the starting point and the end point of a ski slope is called the slope’s verti-cal drop. Calculate the vertical drop of this new slope, rounded to the nearest meter. 2 The ski resort must determine the difficulty of this new slope based on its slope. The slope will be classified as black, i.e. very difficult, if at least one section of the slope has a gradient greater than or equal to 40 % /. The slope will be classified as red, i.e. difficult, if at least one section of the slope has a gradient strictly be-tween 25 % and 40 % (and no section with a gradient greater than or equal to 40 % ) . If all sections of the slope have a gradient less than or equal to 25 % , then the slope will be classified as blue, i.e., easy Determine the difficulty level of this new slope. Justify your answer. E.7696 Consider the function f defined by: f ( x )=2 x 2 · ln( x ) on 0.2 ; 10 and note C f its representative curve in a plane reference frame. The aim of this exercise is to prove that the curve C f admits on 0.2 ; 10 a single tangent passing through the origin of the reference frame. Note f the derivative function of the function f . 1 Show that for x ∈ 0.2 ; 10 : f ( x )=2 x · 2 · ln( x )+1 2 Let a be a real from 0.2 ; 10 , show that the tangent to the curve C f at the point of abscissa a has equation : y =2 a · 2 · ln a +1 · x − 2 a 2 · ln a +1 . 3 Then answer the problem posed. https://chingmath.fr chapExoCorrec/7572 sacados/7572 Liban Mai 2013 5 points chapExoCorrec/6983 sacados/6983 x01234567891011f(x123456ABCf chapExoCorrec/7696 sacados/7696
-10123456-3-2-1123CDA 2. Convexity E.6993 Let f be a function defined on the interval 0 ; 5 by: f ( x )= a · x − 2 · e − x , where a is a real number. Throughout this exercise, we assume that the function f is twice differentiable on the interval 0 ; 5 . The graph of the function C of the function f is shown below in a coordinate system with origin O . The curves C and D both pass through the point A (0 ; − 2) . The line D is tangent to the curve C at point A and has the equation : y =10 · x − 2 . Recall that f denotes the derivative of the function f . 1 Using the above information and without justifying the values of f (0) and f (0) , give 2 a Show that for any real number x in the interval 0 ; 5 , we have : f ( x ) = − a · x + a + 2 · e − x b Deduce from the previous questions that : a =8 c Give the expression of f ( x ) . 3 a Specify the sign of f ( x ) on the interval 0 ; 5 . You can make a table. b Deduce the table of variations of the function f on the same interval. c Solve the equation on the interval 0 ; 5 : f ( x )=0 . 4 Using computer algebra software, we obtained the follow-ing results : 1 g ( x ):=( − 8 ∗ x +10) ∗ exp ( − x ) → g ( x ) = − 8 x + 10 · e − x 2 Derive g ( x ) ; x → 8 · x − 18 · e − x 3 Solve [(8 ∗ x − 18) ∗ exp ( − x ) > 0 ; x ] → x > 9 = 4 Using these results : a Give the expression of f  , the second derivative of the function f . b Justify that the curve C has an inflection point, giving the exact value of the abscissa. 5 A company manufactures toasters. After conducting a study, its director finds that if the company manufac-tures x thousand toasters every day (where x is a real number in the interval 0 ; 5 ) , then the daily profit is given, in hundreds of thousands of euros, by the function f defined by: f ( x ) = 8 · x − 2 · e − x a How many toasters must the company manufacture in order to achieve maximum profit? b What is the value of this maximum profit, rounded to the nearest euro? E.7425 The two parts are linked Part A Consider the function f defined on the interval 0 ; 10 by: f ( x ) = 1 0.5 + 100 · e − x Let f be the derivative of f on the interval 0 ; 10 . 1 Show that, for any real number x in the interval 0 ; 10 , we have : f ( x ) = 100 · e − x 0.5 + 100 · e − x 2 Let f  be the second derivative of f on the interval 0 ; 10 . A computer algebra system provides the following expression for f  ( x ) : f  ( x ) = 100 · e − x · 100 · e − x − 0.5 0.5 + 100 · e − x 3 2 a Show that, in the interval 0 ; 10 , the inequality: 100 · e − x − 0.5 0 is equivalent to the inequality: x − ln 0.005 . b Deduce the sign table for the function f  on the inter-val 0 ; 10 . 3 We call C f the representative curve of f plotted on a co-ordinate system. Using question 2 , , show that the curve C f has an in-flection point denoted I , whose exact abscissa value will be specified. 4 Using the results from question 2 , , determine the in-terval over which the function f is concave. Part B Throughout this part, temperatures will be expressed in degrees Celsius, denoted o C . On December 12 , 2015 , COP21, the United Nations climate change conference, adopted the first universal climate agree-ment, known as the Paris Agreement, signed by 195 countries. This agreement confirms the goal that, by the year 2100 , the Earth’s temperature will not exceed 2 o C the temperature of the year 1900 . In this section, we model, using the function f from section A , , a possible temperature evolution that would allow the Paris Agreement target to be achieved. The representative curve C f of the function is plotted below, and I is its inflection point. On the x-axis, the year 1900 corresponds to 0 and one unit represents 25 years, so the year 1925 corresponds to 1 . On the y-axis, we have represented the number of degrees Celsius above the temperature of 1900 . https://chingmath.fr chapExoCorrec/6993 sacados/6993 -10123456-3-2-1123CDA chapExoCorrec/7425 sacados/7425 Liban Juin 2017
Rangsdesannées012345678910NombredeoCau-dessusdelatempératurede1900123I -123456I-3-2-123JO 1 a Calculate f (10) , rounding the result to two decimal places. b Deduce that in 2150 , with this model, the Paris Agree-ment target will be met. 2 a Using part A , determine the year corresponding to the x-coordinate of the inflection point I of curve C f . Round the result to the nearest whole number b Calculate, for that year, the number of additional de-grees Celsius compared to 1900 . 3 The rate of increase in degrees Celsius is called the rate of global warming. It is assumed that, starting from 1900 , the rate of global warming is modeled by the function f . a Is it true to say that after 2033 the Earth’s tempera-ture will decrease? Justify your answer. b Is it true to say that after 2033 the rate of global warm-ing will decrease? Justify your answer. 4 To save the islands threatened by rising sea levels, the Earth’s temperature must not exceed 1.5 o C the temper-ature of the year 1900 . Determine the year in which the Earth’s temperature will reach this threshold, according to this model. 3. Convexity and the intermediate value theorem E.7426 Let f be a function defined on the interval 0 ; 5 by: f ( x ) = a · x − 2 · e − x where a is a real number. Throughout this exercise, we assume that the function f is twice differentiable on the interval 0 ; 5 . The representative curve C of the function f is given below in a coordinate system with origin O . The curves C and D both pass through the point A (0 ; − 2) . The line D is tangent to the curve C at point A and has equa-tion y =10 x − 2 . Recall that f denotes the derivative of the function f . 1 Using the above information and without justifying the values of f (0) and f (0) , give 2 a Show that for any real number x in the interval 0 ; 5 , we have : f ( x ) = − a · x + a + 2 · e − x b Deduce from the previous questions that : a =8 . c Give the expression of f ( x ) . 3 a Specify the sign of f ( x ) on the interval 0 ; 5 . You can make a table. b Deduce the table of variations of the function f on this interval. c Solve the equation f ( x )=0 on the interval 0 ; 5 . 4 Using computer algebra software, we obtained the follow-ing results : 1 g(x):=(-8*x+10)*exp(-x) → g ( x ) := − 8 · x + 10 · e − x 2 Dériver[g(x),x] → 8 · x − 18 · exp( − x ) 3 Résoudre[(8*x-18)*exp(-x) > 0,x] → x > 9 4 Using these results : a Give the expression of f  , the second derivative of the function f . b Justify that the curve C has an inflection point, giving the exact value of the abscissa. 5 A company manufactures toasters. After conducting a study, its director finds that if the company manufactures x thousand toasters per day (where x is a real number in the interval 0 ; 5 ) , then the daily profit is given, in hun-dreds of thousands of euros, by the function f defined by: f ( x ) = 8 · x − 2 · e − x a How many toasters must the company manufacture in order to achieve maximum profit? b What is the value of this maximum profit? Give the value rounded to the nearest euro. https://chingmath.fr Rangsdesannées012345678910NombredeoCau-dessusdelatempératurede1900123I chapExoCorrec/7426 sacados/7426 -123456I-3-2-123JO
-1234567891011I-3-2-123JO E.7525 Part A Consider the function f defined and differentiable on the in-terval 1 ; 25 by: f ( x ) = 10 − e 0.2 · x +1 x A computer algebra system provides the following results that can be used : f(x) :10-e^(0.2*x+1)/x x ↦−→ 10 − exp : 0.2 · x + 1 x factorize(derive(f(x))) exp 0.2 · x + 1 · 1 − 0.2 · x x 2 factorize(derive(derive(f(x)))) exp 0.2 · x + 1 · − x 2 + 10 · x − 50 25 · x 3 1 Find the factorized expression of f ( x ) where f is the derivative of f . 2 Study the sign of f on the interval 1 ; 25 and draw up the table of variations of f on the interval 1 ; 25 . Round the values to three decimal places. 3 Consider the equation f ( x )=0 . a Show that the equation f ( x )=0 has no solution on the interval 1 ; 5 . b Show that equation f ( x )=0 has a unique solution ¸ on the interval 5 ; 25 . c Determine an amplitude range 10 − 2 for the solution ¸ . d Using one of the results given by the computer algebra software, justify that the function f is concave on the interval 1 ; 25 . Part B An agri-food company manufactures animal feed. We are interested in the profit made, in thousands of euros, corre-sponding to the production of x tens of tons of food We assume that this profit can be modeled by the function f studied in section A above. The minimum production is 10 tons, so x 1 . The answers to the following questions will be justified in part A . 1 What is the maximum profit in euros that the company can generate? For what quantity of food is this maximum profit ob-tained? 2 Determine, to the nearest ton, the maximum quantity of food that must be produced for the company to make a profit. 4. Qcm E.6971 This exercise is an MCQ (multiple-choice questionnaire) . For each of the questions proposed, only one of the three answers is correct. Copy the question number and the cor-rect answer. No justification is required. A correct answer earns 1 point, a wrong answer or no answer earns or deducts no point. A multiple answer scores no points. Let f be a function defined and derivable on the interval 0 ; 10 whose representative curve C f is given below in a ref-erence frame with origin O . Recall that f denotes the function derived from the function f . 1 Le number of solutions on the interval 0 ; 10 of equation f ( x )=0 is equal to : a 1 b 2 c 3 2 Le real number f (7) is : a nul b strictement positif c strictement négatif 3 La function f is : a increasing on 0 ; 10 b croissante on 4 ; 7 c décroissante on 4 ; 7 4 On admits that for any x in the interval 0 ; 10 , we have : f ( x ) = ln( x ) − x 2 + 1 The curve C f admits on this interval a point of inflection : a d’abscisse 2.1 b d’abscisse 0.9 c d’abscisse 2 https://chingmath.fr chapExoCorrec/7525 sacados/7525 Antilles-Guyane Septembre 2017 chapExoCorrec/6971 sacados/6971 -1234567891011I-3-2-123JO
e2345I23456JOCAB E.7789 Note: This exercise is a multiple-choice quiz. (Multiple-choice questionnaire) . For each question, only one of the three answers is correct. Copy the question number and the correct answer. No jus-tification is required. A correct answer is worth 1 points, while an incorrect answer or no answer is worth neither points nor penalties. Multiple answers are not worth any points. Consider the function f defined on the interval 0 ; 5 ; 5 by: f ( x ) = 5 + 5 · ln x x Its graphical representation is the curve C given below in a coordinate system with origin O . We assume that the point A on the graph is the only inflec-tion point of the curve C on the interval 0 ; 5 ; 5 . Let B be the point on this curve with abscissa e . We assume that the function f is twice differentiable on this interval. Recall that f denotes the derivative of the function f and f  its second derivative. We assume that for all x in the interval 0 ; 5 ; 5 , we have : f ( x ) = − 5 · ln x x 2 ; f  ( x ) = 10 · ln x − 5 x 3 1 The function f is : a positive or zero on the interval 0 ; 5 ; 5 ; b negative or zero on the interval 1 ; 5 c negative or zero on the interval 0 ; 5 ; 1 . 2 The slope of the tangent to the curve C at point B is equal to : a − 5 e 2 b 10 e c 5 e 3 3 The function f is : a increasing on the interval 0 ; 5 ; 1 b decreasing on the interval 1 ; 5 c increasing on the interval 2 ; 5 4 The exact value of the x-coordinate of point A on curve C is equal to : a 1 ; 65 b 1 ; 6 c e 0 , 5 5 Let A be the area, measured in area units, of the plane region bounded by the curve C , the x-axis, and the lines with equations x =1 and x =4 . This area satisfies : a 20 A 30 b 10 A 15 c 5 A 8 5. Functions and probabilities E.7428 In this exercise, we consider the first digit of non-zero natural numbers, written in decimal no-tation. For example, the first digit of 2017 is 2 and the first digit of 95 is 9 . In certain circumstances, the first digit of a non-zero random number can be modeled by a random variable X such that for any integer c between 1 and 9 : P X = c = ln c +1 − ln c ln 10 This law is called Benford’s law. 1 What is the value of P X =1 ? 2 We want to examine whether Benford’s law is a valid model in two specific cases. a First case A statistical file from INSEE indicates the popula-tion of municipalities in France as of January 1 er 2016 (field : Metropolitan France and the overseas depart- ments of Guadeloupe, French Guiana, Martinique, and Réunion) . From this file, we can see that there are 36 677 inhab-ited municipalities. Among them, there are 11 094 mu-nicipalities whose population is a number beginning with the digit 1 . Does this observation seem compatible with the state-ment : " the first digit of the population of municipal-ities in France on January 1 er 2016 follows Benford’s law ı? b Second case For each candidate for the baccalaureate exam in the 2017 session, we consider their height in centimeters. We designate X as the random variable equal to the first digit of the height in centimeters of a randomly selected candidate. Does Benford’s law seem to you to be an appropriate law for X ? https://chingmath.fr chapExoCorrec/7789 sacados/7789 e2345I23456JOCAB chapExoCorrec/7428 sacados/7428