Grade 12 / Continuity 26 exercises (100% corrected)

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-4-3-2-1234I-2-123JO -4-3-2-1234I-4-3-2-1234JO 1. Continuity at one point E.3537 Below is given the representative curve C f of the function f ; this curve admits a vertical tangent at the point of abscissa 1.5 : 1 Graphically determine the set of definition of the func-tion f . 2 Graphically determine the derivability set of the function f . 3 Determine the following limits: a lim x ↦→− 2 − f ( x ) b lim x ↦→− 2 + f ( x ) c lim x ↦→ 1 − f ( x ) d lim x ↦→ 1 + f ( x ) 4 a How many numbers are there a such that : lim x ↦→ a − f ( x ) = lim x ↦→ a + f ( x ) b What graphical peculiarity do points with abscissa a verifying such a condition have? E.3538 The integer part of 5.5 is 5; to extend, this notion to the set of real numbers (and particularly to negative numbers) , we define the integer part of a number x , which we note E ( x ) , as follows : ˇ E ( x ) is the largest integer in the set of integers less than or equal to x ı 1 a Suppose x = − 2.5 , justify that E ( x )= − 3 . b Complete the following table : x − 4.5 − 4 − 3.9 − 3.2 − 3 E ( x ) c Justify the following framing for any real number x : E ( x ) x <E ( x ) + 1 2 a Consider the orthogonal reference frame O ; I ; J below ; graph the function E on − 4 ; 4 b What special features does this curve have? E.5825 Consider the function f defined on R ∗ by the relation: f ( x ) = | x | x 1 Justify that the function f is not continuous at 0 . 2 Determine the following two limits: lim x ↦→ 0 − x =0 f ( x ) ; lim x ↦→ 0+ x =0 f ( x ) E.3542 Consider the function f defined by: f ( x ) = 3 x − 1 6 x 2 − 11 x + 3 1 Determine the set D f of definition of the function f . 2 Is it possible to extend by continuity the function f at 1 3 so that it is continuous at this value? That is, is there a number a such that the following function g is continuous in 1 3 : g ( x ) = 3 x − 1 6 x 2 − 11 x + 3 pour x ∈D f g 1 3 = a pour x = 1 3 https://chingmath.fr chapExoCorrec/3537 sacados/3537 -4-3-2-1234I-2-123JO chapExoCorrec/3538 sacados/3538 -4-3-2-1234I-4-3-2-1234JO chapExoCorrec/5825 sacados/5825 chapExoCorrec/3542 sacados/3542
−51510354-6-1-13xVariationdef −4243−92-1xVariationdef 234567I-2-123456JO E.3539 1 Consider the function f whose image of a number x is defined by the relation: f ( x ) = x 2 + 1 − 1 x a Determine the definition set of the function f . b Determine the value of the following two limits: lim x ↦→ 0 − f ( x ) ; lim x ↦→ 0 + f ( x ) c Can we say that the function f is continuous in 0 ? 2 Consider the function g defined on R by: g ( x ) = f ( x ) for x =0 g (0) = 0 Justify that the function g is continuous in 0 . 2. Sign table without intermediate value theorem E.2927 1 Let n be any non-zero natural number. Consider the function f defined on R + by the relation: f ( x ) = 1 + x n − 1 − n · x Establish the direction of variation of the function f on R + . 2 Establish the inequality below for any positive number x and any strictly positive natural number n : 1 + x n 1 + n · x The inequality established in the last question is called: Bernoulli inequality 3. Intermediate value theorem E.3543 Consider a function f which admits the following table of variations : 1 Justify that the function f cancels twice on its defining set. 2 Let m be a real number. Discuss as a function of the value of m the number of solutions to the equation f ( x )= m . E.3569 Consider a function f defined on the interval − 4 ; 4 admitting the following table of variations : 1 Justify that the function f cancels once on its defining set. 2 Let m be a real number. Discuss as a function of the value of m the number of solutions to the equation f ( x )= m . E.6239 Consider the function f defined, con- tinuous and strictly increasing on the interval 0 ; + ∞ such that : lim x ↦→ 0 + f ( x ) = −∞ ; lim x ↦→ + ∞ f ( x ) = + ∞ We call Γ its representative curve in an orthogonal reference frame O ; −→ i ; −→ j of the plane. 1 Show that, for any natural number n , the equation f ( x )= n admits a single solution in 0 ; + ∞ . We denote ¸ n this solution. 2 On the appendix page, we have plotted Γ in the refer-ence frame O ; −→ i ; −→ j . Place the numbers ¸ 0 , ¸ 1 , ¸ 2 , ¸ 3 , ¸ 4 and ¸ 5 on the x-axis, leaving the construction lines visible. https://chingmath.fr chapExoCorrec/3539 sacados/3539 chapExoCorrec/2927 sacados/2927 chapExoCorrec/3543 sacados/3543 −51510354-6-1-13xVariationdef chapExoCorrec/3569 sacados/3569 −4243−92-1xVariationdef chapExoCorrec/6239 sacados/6239 234567I-2-123456JO
4. Exponential functions and the intermediate value theorem E.4302 Let f be the function defined on the interval 0 ; + ∞ by: f ( x ) = 1 x 2 · e 1 x 1 Demonstrate that, the derivative function of the function f is expressed, for any strictly positive real x , by: f ( x ) = − 1 x 4 · e 1 x · 2 · x + 1 2 Determine the sign of f and deduce the table of varia-tions of f on the interval 0 ; + ∞ . 3 Demonstrate that the equation f ( x )=2 has a unique so-lution noted ¸ belonging to the interval 0 ; + ∞ and give the approximate value of ¸ rounded to the hundredth. E.5528 Let g be the function defined on 0 ; + ∞ by: g ( x ) = x e x − e + e − 2 1 Let g be the function derived from the function g . Cal-culate g ( x ) for any real x of 0 ; + ∞ . Verify that the second derivative function g  is defined on 0 ; + ∞ by: g  ( x ) = (2 + x )e x 2 Deduce the variations of the function g on 0 ; + ∞ . 3 Establish that the equation g ( x )=0 admits a unique so-lution ¸ in the interval 0 ; + ∞ . Determine an approximate value of ¸ to the nearest 10 − 1 . 4 Deduce the variations of the function g on 0 ; + ∞ . 5. Study functions: exponentials E.3666 The aim of this question is to prove that there is a unique real solution to the equation : ( E ) : e 2 x + 4 · x · e x − 4 · e x − 4 = 0 To do this, consider the function defined on R by: f ( x ) = e 2 x + 4 · x · e x − 4 · e x − 4 . 1 Show that for any x belonging to −∞ ; 0 : e 2 x − 4 < 0 ; 4 · e x ( x − 1) < 0 2 Deduce that the equation ( E ) has no solution in the in-terval −∞ ; 0 . 3 Show that the function f is strictly increasing on the interval 0 ; + ∞ 4 Demonstrate that the equation ( E ) admits a unique so-lution in the interval 0 ; + ∞ . We denote ¸ this solution. Give a 10 − 2 amplitude frame of a . E.107 Let R denote the set of real num-bers, and consider the function f defined on R by: f ( x ) = x · e x − 1 + 1 Let C denote its graph in an orthonormal coordinate system O ; −→ i ; −→ j . Part A : Analysis of the Function 1 Find the limit of f as −∞ . What can we deduce about the curve C ? 2 Find the limit of f as + ∞ . 3 Assume that f is differentiable on R , and let f denote its derivative. Show that, for any real number x : f ( x )=( x +1) · e x − 1 4 Examine the variations of f on R and construct its vari-ation table on R . Part B: Finding a Specific Tangent Let a be a strictly positive real number. The goal of this part is to determine whether there exists a tangent to the curve C at the point with abscissa a that passes through the origin of the coordinate system. 1 Let T a denote the tangent to C at the x-coordinate a . Find an equation for T a . 2 Prove that a tangent to C at a point with abscissa a that is strictly positive passes through the origin of the coordinate system if and only if a satisfies the equality: 1 − a 2 · e a − 1 = 0 3 Prove that 1 is the unique solution on the interval 0 ; + ∞ of the equation 1 − x 2 · e x − 1 = 0 Hint: In this question, any attempt at a solution, even if incomplete, will be taken into account in the grading. 4 Then give an equation for the tangent line in question. 6. Function studies E.3544 Consider the function f whose image of a number x is defined by the relation: f ( x ) = x 2 − 1 − 1 x + 1 Let’s note C f the representative curve of the function f . 1 a Determine the definition set of the function f . b Determine the limits of the function f at its bounds. c Does the curve C f admit any asymptotes? If so, spec-ify which ones. 2 a Determine the expression of the function f derived from the function f . b Draw up the table of variations of the function f on https://chingmath.fr chapExoCorrec/4302 sacados/4302 Extrait d'Asie Juin 2010 chapExoCorrec/5528 sacados/5528 chapExoCorrec/3666 sacados/3666 chapExoCorrec/107 sacados/107 Extrait de Antilles-Guyanes Juin 2012 chapExoCorrec/3544 sacados/3544
the interval 1 ; + ∞ . 3 a Justify that the function f cancels only once on the interval 1 ; + ∞ . b Draw the representative curve of the function f using your calculator; use the functions of your calculator to determine an approximate value of this zero of the function f . 7. Study functions with the second derivative E.3562 Consider the function f defined on R whose image of a number x is defined by the relation: f ( x ) = x − 1 + 2 x 2 + 1 1 Determine the expression of the derivative f of the func-tion f , as well as that of the second derivative f  . 2 a Study the sign of the function f  . b Deduce the table of variations of the function f . (An approximate value of the extremums will be sought using the calculator) 3 a Show that the function f cancels for x =1 and also at a number ¸ verifying the frame : 0.2 <¸ < 0.3 b Deduce the sign table of the function f . 4 a Determine the value of the following two limits: lim x ↦→−∞ f ( x ) ; lim x ↦→ + ∞ f ( x ) b Draw up the table of variations of the function f . (An approximate value of the extremums will be sought using the calculator) E.5813 Consider the function f defined on R whose image of a number x is given by the relation: f ( x ) = 5 x − 1 x 2 + x + 1 1 a Show that the function f admits as second deriva-tive, the function f  defined by: f  ( x ) = − 6 x · ( x + 1) x 2 + x + 1 3 b Draw up the sign table for the function f  . 2 a Deduce from question 1 , the table of variations of the function f . b Justify that the function f is positive at R . 3 Justify that the function f admits a unique zero on R . E.5812 Consider the function f defined on R whose image of a real number x is defined by the expression : f ( x ) = 3 · x 2 + 1 − 2 · x The aim of the exercise is to show that the function f admits a global maximum and to obtain an approximate value. 1 a Show that the second derivative of the function f admits for expression : f  ( x ) = 3 x 2 + 1 · x 2 + 1 b Establish the following two limits: lim x ↦→−∞ f ( x ) = − 5 ; lim x ↦→ + ∞ f ( x ) = 1 c Draw up the table of variations of the function f . 2 a Justify that the function f cancels once at R . b Note ¸ the unique solution of the equation : f ( x )=0 . Briefly justify that the number ¸ belongs to the inter-val 0.8 ; 0.9 . 3 a Draw up the sign table for the function f . b Justify that the function f admits a global minimum that is reached for x = ¸ . E.5235 Let g be the function defined on 0 ; + ∞ by: g ( x ) = x · e x − e + e − 2 1 Let g be the function derived from the function g . Cal-culate g ( x ) for any real x of 0 ; + ∞ . Verify that the second derivative function g  is defined on 0 + ∞ by: g  ( x ) = (2 + x )e x 2 Deduce the variations of the function g on 0 ; + ∞ . 3 Establish that the equation g ( x )=0 admits a unique so-lution ¸ in the interval 0 ; + ∞ . Determine an approximate value of ¸ to the nearest 10 − 1 . 4 Deduce the variations of the function g on 0 ; + ∞ . 8. Studies of functions with a sub-function: exponential E.3904 Part A : Let g be the function defined on [0;+ ∞ by: g ( x )=e x − x · e x +1 . 1 Determine the limit of g in + ∞ . 2 Study the variations of the function g . 3 Give the table of variations of g . 4 a Show that the equation g ( x )=0 admits on 0 ; + ∞ a single solution. We note ¸ this solution. b Using the calculator, determine a frame of amplitude 10 − 2 of ¸ . c Demonstrate that : e α = 1 ¸ − 1 5 Determine the sign of g ( x ) depending on the values of x . Part B: Let A be the function defined and derivable on 0 ; + ∞ such that : A ( x ) = 4 x e x + 1 https://chingmath.fr chapExoCorrec/3562 sacados/3562 chapExoCorrec/5813 sacados/5813 chapExoCorrec/5812 sacados/5812 chapExoCorrec/5235 sacados/5235 chapExoCorrec/3904 sacados/3904
1 Demonstrate that for any real x positive or zero, A ( x ) has the same sign as g ( x ) où g is the function defined in part A . 2 Deduce the variations of the function A on 0 ; + ∞ . E.3704 Let the function f be defined ex-plicitly on a certain part D of R by means of the formula : f ( x ) = x 2 e x − x − 2 The study of this function f is proposed. 1 We pose : g ( x )=e x − x − 2 . a Study the sign on R of the derivative g ( x ) . b Find, by factoring e x , the limit of g ( x ) when x ↦→ + ∞ . c Deduce that there are two real numbers, denoted a and b , with a<b , such that : g ( a ) = g ( b ) = 0 d Show that : − 2 <a< − 1 ; 1 <b< 2 2 a Determine in D the derivative of the function f and show that it can be put into the form : f ( x ) = x · u ( x ) g ( x ) 2 b Calculate the derivative u  of u and study its sign. c Deduce from question b the variations of u , as well as those of u and show that there exists a real, denoted c such that u ( c )=0 . Show that c< − 2 . d Finally find the sign of this function u and the sign of the derivative f ( x ) . 3 Draw up the varitaion table for f . Show that the function f admits a minimum m in an interval −∞ ; a . Show that : m = c · (2 − c ) c + 1 . 4 Determine the limits of f in −∞ , a , b , + ∞ . (For a and b , a distinction will be made between limits on the right and limits on the left) . 5 Gather all the previous results by drawing the shape of the graph of the function f . 9. Studies of functions with a sub-function E.3557 1 Consider the polynomial function P defined for any real x by: P ( x ) = 2 x 3 − 3 x 2 − 1 a Study the variations of P . b Show that the equation P ( x )=0 admits one and only one real root, ¸ , and that ¸ belongs to the interval 1.6 ; 1.7 2 Let D be the set of real numbers strictly greater than − 1 . Consider the numerical function f defined on D by: f ( x ) = 1 − x 1 + x 3 C is the representative curve of f in the orthonormal plane (unit 4 cm ) . a Study the variations of f (for this we’ll use the results from 1 ) . b Write an equation of the straight line (Δ) tangent to the curve ( C ) at the abscissa point 0 . Study the posi-tion of the curve ( C ) with respect to the straight line (Δ) in the interval − 1 ; 1 . c Show that the curve ( C ) lies above its tangent at the point of abscissa 1. Draw the curve ( C ) , the straight line (Δ) and the tan-gent to ( C ) at abscissa 1. E.6246 Consider the function f defined on R by the expression : f ( x ) = x 3 + 5 · x 2 + 2 x 2 + 2 Part A : study of an appendix function Let g be the polynomial function of degree 3 defined by: g ( x ) = x 3 + 6 x + 16 1 Determine the limits of the function g in −∞ and + ∞ . 2 Draw up the table of variations of the function g on R . 3 Justify that the function g cancels once and only once on R . Note ¸ the unique solution to the equation : g ( x ) = 0 . We’ll give an approximate value of ¸ to the nearest hun-dredth. 4 Deduce the sign of the function g on R . Part B: study of the function f 5 Determine the limits of the function f in + ∞ and −∞ . 6 a Determine the expression of the derivative f of the function f and show that it admits the expression : f ( x ) = x · g ( x ) x 2 + 2 2 b Draw up the table of variations of the function f . We’ll use the approximate value : f ( ¸ ) ≈ 2.36 7 In a frame of reference O ; I ; J orthonormal, consider the curve C f representative of the function f . a Determine the slope-intercept formof the tangent (Δ) to the curve C f at the point of abscissa − 1 . b Study the sign of the difference f ( x ) − − x +1 on R . Deduce the relative position of the curve C f relative to the tangent (Δ) . https://chingmath.fr chapExoCorrec/3704 sacados/3704 Concours Efrei Avril 2002 chapExoCorrec/3557 sacados/3557 chapExoCorrec/6246 sacados/6246 fichierPlus/6246/
E.3556 Consider the numerical functions of one real variable defined by: f : x ↦−→ 1 3 · x 2 + x + 1 x ; g : x ↦−→ 2 x 3 + x 2 − 1 1 Show that for any x =0 the numbers f ( x ) and g ( x ) have the same sign. 2 Study the variations of the function g on R . Deduce that the equation g ( x )=0 admits in R a unique solution ¸ with 0 <¸< 1 . (We won’t try to calculate ¸ ) 3 Draw up the table of variations of the function f . (we admit that f ( ¸ ) ≈ 0.87 ) We denote by ( C ) the graphical representation of the function f in an orthonormal frame (unit 3 cm ) , by I the point of ( C ) with abscissa − 1 and by J the point of ( C ) with abscissa +1 . 4 a Verify that the straight line ( IJ ) is the tangent at J to ( C ) . b Determine an equation of the tangent ( T ) in I at ( C ) . 5 Study the position of ( C ) relative to ( T ) . 6 Use the previous results to construct the curve ( C ) . (We’ll take 2 = 3 as the value of ¸ ) E.3563 Consider the function f defined on R by: f ( x ) = x 3 − 1 + 2 · x 2 1 Study the limits of f in + ∞ and in −∞ 2 Consider the function g defined on R by: g ( x ) = 3 · x · 2 x 2 + 1 − 2 a Justify that the function g is strictly increasing on R . b Show that there exists a single real ¸ such that g ( ¸ )= 0 . In addition, justify that : 0.5 <¸< 0.6 . c Draw up the sign table for the function g . 3 a Determine the expression of the derivative f of the function f . b Draw up the sign table for f . c Deduce the table of variations of the function f . (use approximate values obtained with calculator) . E.5102 1 Consider the function g defined on R by: g ( x ) = − 8 x 3 + 4 x − 4 a Draw up the table of variations of the function g . (approximate values will be used to complete the table) b Observing that g ( − 1)=0 , draw up the sign table for the function g . 2 Consider the function f defined on R by: f ( x ) = 2 x 2 + 1 2 x 2 + 2 x + 3 2 a Determine the definition set of the function f . b Determine the limits of f at the bounds of its defining set. c Establish that the function f admits as derivative the function f whose expression is given by the relation: f ( x ) = − 8 x 3 + 4 x − 4 (2 x 2 + 2 x + 3) 3 d Draw up the table of variations of the function f . (approximate values will be used to complete the table) e Deduce the sign table for the function f . 10. Dichotomy E.3534 Consider the function f defined by: f ( x ) = x 3 + 3 · x 1 a Draw up the table of variations of the function f . b Justify that the number 5 admits a single antecedent by the function f ; this number will be noted ¸ . 2 We set the value a 0 =0 and b 0 =2 . We wish to construct by the dichotomy method the two suites a n and b n adjacent and convergent to ¸ . a Complete the table below : a n c n b n f ( a n ) f ( c n ) f ( b n ) n =0 n =1 n =2 n =3 n =4 n =5 b How accurately do we obtain the value of ¸ using the table. https://chingmath.fr chapExoCorrec/3556 sacados/3556 Nancy-Metz 1987 chapExoCorrec/3563 sacados/3563 fichierPlus/3563/diapoCorrection.pdf chapExoCorrec/5102 sacados/5102 chapExoCorrec/3534 sacados/3534