Grade 12 - Comp. / Neperian logarithm function 32 exercises (100% corrected)

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012345678910-1123456Cf 01e∞−∞01∞xVariationdef 1. Derivatives E.7003 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = 3 · x − x · ln x Assume that f is derivable on the interval 0 ; + ∞ and de-note its derivative function by f . Which of the following three answers is correct? a f ( x ) = 3 − 1 x b f ( x ) = 3 − ln x c f ( x ) = 2 − ln x E.7009 Let g be the function defined on the interval 0 ; + ∞ by: g ( x ) = x + 1 · ln( x ) Which of the following answers is correct? a g ( x ) = 1 x b g ( x ) = 1 + ln( x ) c g ( x ) = − 1 x 2 d g ( x ) = 1 + 1 x + ln( x ) E.7392 From the four proposed answers, only one is correct. Specify which one, justifying your answer. Let f be the function defined on 0 ; + ∞ by: f ( x ) = x · ln( x ) − x We note f its derivative function. Then we have : a f ( x ) = 0 b f ( x ) = ln( x ) c f ( x ) = 1 x − 1 d f ( x ) = 1 x − x E.7745 Of the four propositions, only one is correct. The answer must be justified : Let the function f be defined on 0 ; + ∞ by f ( x )= ln x x and f its derivative function. We have : a f ( x ) = ln x − 1 x 2 b f ( x ) = 1 − ln x x 2 c f ( x ) = 1 x 2 d f ( x ) = 1 + ln x x 2 E.7023 This exercise is a multiple-choice questionnaire. For each of the following questions, only one of the four answers pro-vided is correct. No justification is required. A correct answer earns one point. An incorrect answer or no answer earns or loses no points Let the function f be defined for all strictly positive real num-bers x by: f ( x ) = 5 − x + 2 · ln x The graph C of the function f is shown below, along with T , the tangent to the graph C at the point A with abscissa 4 . 1 Let f be the derivative of f , then we have : a f ( x ) = − 1 + 2 x b f ( x )= − 2 · ln x +(5 − x ) · 2 x c f ( x ) = − x + 2 x d f ( x ) = 4 + 2 x 2 An equation of T is : a y = 1 2 · x + 5.7 b y =5.7 · x − 1 2 c y = − 1 2 · x + 1 + 2 · ln 4 d y = − 1 2 · x +3+2 · ln4 2. Study of functions E.7646 Definition - proposition : the function logarithm neperian , denoted ln is defined on the interval 0 ; + ∞ and admits the table of variations : Furthermore, its derivative function has the expression : ln x = 1 x https://chingmath.fr chapExoCorrec/7003 sacados/7003 chapExoCorrec/7009 sacados/7009 Extrait Liban Mai 2016 chapExoCorrec/7392 sacados/7392 chapExoCorrec/7745 sacados/7745 chapExoCorrec/7023 sacados/7023 012345678910-1123456Cf chapExoCorrec/7646 sacados/7646 01e∞−∞01∞xVariationdef
00,511,525101520 12307ln2−ln30xVariationdeB(x 1 Draw up the sign table for the neperian logarithm func-tion. 2 Consider the function f defined on 0 ; + ∞ by: f ( x ) = x − ln( x ) a Determine the expression of the function f derived from the function f . b Establish the sign table of the function f c Specify the directions of variation of the function f on 0 ; + ∞ . E.7029 Instruction : For each of the following five statements, in-dicate whether it is true or false and justify your answer. One point is awarded for each correct answer with a valid justification. Answers without justification will not be con-sidered. No points will be deducted for unanswered questions. Consider the function f defined on the interval 0 ; + ∞ by: f ( x ) = x · ln x − x + 1 Statement 1: The function f is increasing on the interval 0 ; 1 . Statement 2: The function f is convex on the interval 0 ; + ∞ . Statement 3: For all x belonging to the interval 0 ; + ∞ : f ( x ) 50 E.7433 Consider the function f defined on the interval 0 ; 10 by: f ( x ) = − x · ln x + 2 · x + 1 1 Calculate f ( x ) . 2 Demonstrate that the function f admits a maximum on the interval 0 ; 10 . 3 Calculate the exact value of the maximum of the function f on this same interval. E.7019 Consider the function f defined on the interval 0 ; 1 ; 5 by: f ( x ) = 9 x 2 · 1 − 2 · ln x + 10 The representative curve of f is given below : 1 Show that f ( x )= − 36 · x · ln x where f denotes the deriva-tive of the function f on the interval 0 ; 1 ; 5 . 2 Study the sign of f ( x ) on the interval 0 ; 1 ; 5 . 3 Deduce from the previous question the variations of the function f on the interval 0 ; 1 ; 5 . 3. Intermediate value theorem E.7441 A company manufactures and sells electronic components. Its monthly production capacity is between 1 000 and 30 000 units. It is assumed that all production is sold. The profit in thousands of euros, realized for the production and sale of x thousand units, is given over the interval 1 ; 30 by: B ( x ) = − 0.5 · x 2 + 6 · x − 20 + 2 x · ln x 1 Show that : B ( x ) = − x + 8 + 2 · ln x where B is the derivative of B over the interval 1 ; 30 . 2 We assume that B  ( x )= − 1+ 2 x , where B  is the second derivative of B over the interval 1 ; 30 . Justify the table of variations below for the derivative function B over the interval 1 ; 30 . 3 a Show that the equation B ( x )=0 has a unique solu- tion ¸ over the interval 1 ; 30 . b Give an approximate value to the nearest thousandth of the value of ¸ . 4 Deduce the sign of B ( x ) over the interval 1 ; 30 , and give the table of variations of the profit function B over the same interval. 5 How many units must be produced, to the nearest unit, for the company to achieve maximum profit? What is this maximum profit (rounded to the nearest thousand dollars) ? https://chingmath.fr chapExoCorrec/7029 sacados/7029 chapExoCorrec/7433 sacados/7433 chapExoCorrec/7019 sacados/7019 00,511,525101520 chapExoCorrec/7441 sacados/7441 12307ln2−ln30xVariationdeB(x
E.7434 Consider the function f defined on the interval 1 ; 10 by: f ( x ) = 4 · e − 0.5 · x +1 + x − 1 We admit that the function f is derivable on the interval 1 ; 10 and we note f its derivative function. 1 Demonstrate that for any real number x in the interval 1 ; 10 , we have : f ( x ) = − 2 · e − 0.5 · x +1 + 1 2 a Show that on the interval 1 ; 10 , the equation f ( x )=0 has as its only solution the number: ¸ =2+2 · ln2 b We admit that the set of solutions on the interval 1 ; 10 of the inequation f ( x ) 0 is 2+2 · ln2 ; 10 . Deduce the variations of the function f on the interval 1 ; 10 . 4. Algebraic properties E.7701 Only one of the proposed answers is correct. No justification is required. The exact value of ln 10 · e 2 is : a 2 · ln 10 + 2 b 4.302 585 093 c ln 10 + 2 d 2 · ln 10 · e E.7697 Determine whether the proposi-tion is true or false and justify the answer: Proposition: we have equality: e 5 · ln2 × e 7 · ln4 =2 19 E.7744 Of the four propositions, only one is correct. Copy the correct proposition without justification : For any real x< 0 , the real number ln − 1 x is equal to : a ln x b − ln − x c − ln x d 1 ln x E.7846 Say whether the statement below is true or false. Justify the answer given. For any strictly positive real a : ln a 3 − ln a 2 = ln a 25 − ln a 24 5. Solving equations E.7519 Among the four propositions below, only one is correct. Which one? The solution of the equation x 23 =92 is equal to : a 4 b 1.2 c e ln(92) 23 d e ln(23) 92 E.7693 Consider the function f defined on the interval 1 ; 10 by: f ( x ) = 4 · e − 0.5 · x +1 + x − 1 We admit that the function f is derivable on the interval 1 ; 10 and we note f its derivative function. 1 Demonstrate that for any real number x in the interval 1 ; 10 , we have : f ( x )= − 2 · e − 0.5 · x +1 +1 . 2 Show that on the interval 1 ; 10 , the equation f ( x )=0 admits for unique solution the number ¸ =2+2 · ln2 . E.7847 Say whether the statement below is true or false. Justify the answer given. The equation x · ln( x )=2 · ln( x ) admits exactly two solutions 2 and 1 on 0 ; + ∞ . 6. Equation solving and algebraic properties E.7692 Among the four propositions below, only one is correct. Which one? The exact solution of the equation 1 2 x = 3 10 is : a 1.74 b ln(10) − ln(3) ln(2) c − ln(3) ln(5) d 0.5 7. Solving inequalities: graphically E.7053 The curve ( C ) below is the graph-ical representation of a function f defined and derivable on the interval − 1 ; 2 . https://chingmath.fr chapExoCorrec/7434 sacados/7434 chapExoCorrec/7701 sacados/7701 Extrait Antilles-Guyane Septembre 2014 chapExoCorrec/7697 sacados/7697 chapExoCorrec/7744 sacados/7744 chapExoCorrec/7846 sacados/7846 chapExoCorrec/7519 sacados/7519 chapExoCorrec/7693 sacados/7693 chapExoCorrec/7847 sacados/7847 chapExoCorrec/7692 sacados/7692 chapExoCorrec/7053 sacados/7053
-1-0,500,511,520,511,522,533,5CfGHS Let f be the derivative function of the function f . The point G has coordinates (0 ; 2) . The point H has coordinates (1 ; 3) . The straight line ( GH ) is the tangent to the curve ( C ) at the point G . The curve ( C ) admits a horizontal tangent at the point S of abscissa ln2 . No justification is required. By graphical reading : 1 Give the values of f (0) and f (0) . 2 Solve on − 1 ; 2 the inequation f ( x ) 0 . 8. Solving inequalities E.7698 Justify that the inequation 1 − e x 2 − 1 0 has as its solution set the interval − 1 ; 1 . E.7691 Solve in 0 ; 10 the inequation : 100 · e − x − 0.5 0 E.7045 Which of the following four state-ments is true? The integers n solutions of the inequation 1 2 n < 0.003 are all integers n such that : a n 8 b n 9 c n 8 d n 9 E.7210 In a forest exploration, the trees cut in this forest are used for heating. The price of a stere of wood (unit of volume measuring wood) increases every year by 3 % . After how many years will the price of a stere of wood have doubled? E.7034 Reminder : N is the set of natural numbers. Solve the inequality in N : 12 000 − 2 000 × 0 ; 75 n 11 950 E.7424 From the four proposed answers, only one is correct. Specify which one, justifying your answer. The natural numbers n verifying the inequation : 6 × 0.95 n − 1 2 belong to the interval: a −∞ ; ln3 ln 5.7 b −∞ ; ln 0.5 0.95 c −∞ ; ln(0.5) ln 0.95 d ln(0.5) ln 0.95 ; + ∞ E.7746 Consider the sequence u n whose terms are defined for any natural number n by: u n = 700 − 100 × 0.7 n 1 Let n be a natural number. Show that u n 697 is equiv-alent to 0.7 n 0.03 . 2 To solve this inequation, we use the following algorithm: N ← 0 U ← 1 As long as U>0; 03 N ← N+1 U ← 0.7 × U Fint As long as Give the value of the variable N at the end of the execu-tion of this algorithm. 3 Find this result by solving the inequation : 0.7 n 0.03 . 9. Solving inequalities and studying functions https://chingmath.fr -1-0,500,511,520,511,522,533,5CfGHS chapExoCorrec/7698 sacados/7698 chapExoCorrec/7691 sacados/7691 chapExoCorrec/7045 sacados/7045 Extrait Antilles-Guyane Juin 2014 chapExoCorrec/7210 sacados/7210 chapExoCorrec/7034 sacados/7034 chapExoCorrec/7424 sacados/7424 chapExoCorrec/7746 sacados/7746 Extrait Asie Juin 2013
E.7014 Let f be the function defined on the interval 3 ; 13 by: f ( x ) = − 2 x + 20 − e − 2 x +10 1 Show that the derivative function f , of the function f , defined for any x of the interval 3 ; 13 , has the expres-sion : f ( x ) = 2 · − 1 + e − 2 x +10 2 a Solve in the interval 3 ; 13 the inequation : f ( x ) 0 b Deduce the sign of f ( x ) on the interval 3 ; 13 and draw up the table of variations of f on this interval. The values in the table will, if necessary, be rounded to 10 − 3 . E.7694 Consider the function g defined on the interval 1 ; 45 by: g ( x ) = − 20 · x + 5 · x · ln( x ) + 30 1 a We note g the function derived from the function g . Show that, for any x belonging to 1 ; 45 , we have : g ( x ) = − 15 + 5 · ln( x ) b Show that the inequation − 15+5 · ln( x ) 0 is equiva-lent to x e 3 . c Draw up the table of variations of the function g (val-ues will be rounded to the hundredth if necessary) . 2 a Show that the equation g ( x )=0 admits a single so-lution ¸ on the interval 1 ; 45 . b Give a frame for ¸ of amplitude 0.01 . c Deduce the sign of g ( x ) according to the values of x in the interval 1 ; 45 . 10. Solving inequalities and sequences E.7700 Consider the sequence u n de-fined for any natural number n by the relation: u n = − 17.5 × 0.92 n + 37.5 1 Determine the first two terms of this sequence. 2 Determine the limit of the sequence u n . 3 Will the terms of the sequence u n exceed the value 30 ? If so, for which rank will this value be exceeded the first time? E.7699 Consider the sequence u n de-fined for any natural number n by: u n = 2 + 3 · 1 2 n . We admit that the sequence u n is decreasing. 1 Determine the limit of the sequence u n . 2 Consider the following algorithm: n ← 0 u ← 5 As long as u − 2 10 − 6 n ← n+1 u ← 2+3 · 1 2 n End as long as Determine the value of the variable n at the end of the execution of this algorithm. 11. Problems E.7747 A company that produces recy-cled paper was founded in the year 2000 , and the table below shows the evolution of its production. Year 2000 2002 2004 2006 2008 2010 2012 Rank of the year 0 2 4 6 8 10 12 Production in tons 7 000 18 811 36 620 49 000 58 012 63 098 68 500 1 a Determine the percentage increase in production be-tween the years 2000 and 2012 . Give the rounded re-sult in the form a % where a is an integer. b Determine a positive real number that is the solution to the equation : x 12 =9.79 . Interpret this number in terms of the rate of change in the company’s produc- tion between years 2000 and 2012 . The rounded result will be given in the form b % , where b is an integer. 2 The company calls on a firm of experts to model the evo-lution of the company’s production in order to make a projection up to 2020 . The consulting firm proposes the function f defined on the interval 2 ; 20 by: f ( x ) = 27 131 · ln x + 0.626 · x 3 where x represents the rank of the year and f ( x ) the number of tons produced. a Let f be the derivative of the function f over the inter-val 2 ; 20 . Determine f ( x ) and then the variations of the function f over 2 ; 20 . b Using this model, can the company exceed a produc-tion of 90 000 tons of recycled paper before the year 2020 ? Justify your answer. https://chingmath.fr chapExoCorrec/7014 sacados/7014 Extrait de Liban Mai 2016 chapExoCorrec/7694 sacados/7694 chapExoCorrec/7700 sacados/7700 Extrait Antilles Guyane Juin 2014 chapExoCorrec/7699 sacados/7699 chapExoCorrec/7747 sacados/7747