Outside the high school program / Derivatives 69 exercises (including 64 corrected)

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1. Derivative numbers E.2312 Let f be defined on R by the relation: f ( x )=4 · x 2 − 4 · x − 3 1 Calculate the derivative number of the function f in 2. 2 Determine the equation of the tangent to the curve C f at the point of abscissa 2. 2. Left and right derivatives E.3483 Consider the function f whose image of a number x is defined by the relation: f ( x ) = 1 2 · x + 2 · x 2 + 2 x + 1 + 1 1 Determine the definition set of the function f . 2 a Establish the following equality for any real number h : f ( − 1+ h ) − f ( − 1) h = h + 3 · h 2 2 · h b Deduce the value of the following two limits: lim x ↦→ 0 − f ( − 1+ h ) − f ( − 1) h lim x ↦→ 0 + f ( − 1+ h ) − f ( − 1) h 3 What can you say about the appearance of the curve C f at abscissa − 1 ? (draw the curve using a calculator) E.3484 In the plane provided with a reference frame O ; I ; J , consider the curve C f representative of a function f . Here is its representation : 1 Justify that the function f is not derivable for the ab-scissa of the point C . 2 a Graphically, determine the following two limits: lim h ↦→ 0 − f − 2+ h − f − 2 h ; lim h ↦→ 0 + f − 2+ h − f − 2 h b Is the function f derivable in − 2 ? 3 Determine, graphically, the value of the numbers derived from the function f at the abscissas of the points B and D . E.3485 We denote f the square root function ; this function is defined over R + . 1 Consider x ∈ R ∗ + et h ∈ R tel that x − h ; x + h ⊂ R + . a Establish the following limit: lim h ↦→ 0 x + h − x h b Determine the value of the number derived from the function f en x ∈ R ∗ + . 2 a Determine the value of the following limit: lim x ↦→ 0 + x x b What can be said about the derivability of the function f in 0? E.3508 1 Justify the non-derivability in 0 of the following func-tions : a f ( x ) = x b g ( x ) = | x | c h ( x ) = x 2 + x 2 Consider the function j defined on R by the relation: j ( x ) = x ·| x | Justify that the function j is derivable in 0 . https://chingmath.fr chapExoCorrec/2312 sacados/2312 chapExoCorrec/3483 sacados/3483 chapExoCorrec/3484 sacados/3484 -4-3-2-1234I-2-12JOABCDCfCf chapExoCorrec/3485 sacados/3485 chapExoCorrec/3508 sacados/3508
E.3533 Consider the function f defined on R whose image of a number x is given by the relation: f ( x ) = 1 2 · x · cos(2 · x ) 1 a Determine the equation of the tangent ( T ) to the curve C f at the point with abscissa 0 . b Justify that ( T ) is also the tangent to the curve C f at the point with abscissa ı . c Study the relative position of C f and ( T ) . 2 Consider the line ( d ) with equation : y = − 1 2 · x . a Prove that the line ( d ) is tangent to the curve C f at several points. b Study the relative position of ( d ) and C f . 3 a Complete the following table of values : x ı 2 ı 3 ı 4 ı f ( x ) b Plot the curve C f on R + 4 a Study the parity of the function f . b Plot the curve C f on R − 3. Limited development E.3504 1 Let f be a function defined in 0 such that : f ( x ) = a · x + b + x · " ( x ) where a ∈ R , b ∈ R et lim x ↦→ 0 " ( x )= 0 . Show that the function f is derivable in 0 . 2 Let g be a function defined in 0 whose image of x is de-fined by: g ( x ) = 3 − 2 x + x 2 a Justify, without performing any calculations, that the function g is derivable at 0 . b Give the value of the number derived from the function f in 0. 3 Let h be a function defined in 0 whose image of x is de-fined by: h ( x ) = − 3 x 3 + 3 x 2 + x − 2 x + 1 a Establish the following equality: h ( x )=3 x − 2 − 3 · x 3 x +1 b Deduce the equation of the tangent to the curve C h at the point of abscissa 0 . 4. Second-degree polynomials: derivative functions E.7648 Definition: Let f be a quadratic function defined by the expression : f ( x ) = a · x 2 + b · x + c where a , b , and c are three real numbers with a =0 . We call the derivative of the function f , the function f defined by: f ( x ) = 2 a · x + b Copy and complete the table below to obtain the expression of the function f derived from the function f : f ( x )= a · x 2 + b · x + c a b c f ( x )=2 a · x + b − 2 · x 2 − x + 1 0 ; 25 · x 2 + x − 1 x 2 − x − 4 · x 2 − 2 E.4667 Give the derivatives of the following polynomial functions : 1 f : x ↦−→ − 3 x + 2 2 g : x ↦−→ 4 x 2 − 4 3 h : x ↦−→ 2 x 2 + 3 x 4 j : x ↦−→ 5 x − 2 x 2 5 k : x ↦−→ − 2 x 2 + 2 x 6 ‘ : x ↦−→ (3 x + 11)(4 − x ) https://chingmath.fr chapExoCorrec/3533 sacados/3533 -3ı-2ı-1ı01ı2ı3ı−1.5ı−1ı−0.5ı0.5ı1ı1.5ı chapExoCorrec/3504 sacados/3504 chapExoCorrec/7648 sacados/7648 chapExoCorrec/4667 sacados/4667
E.4668 Give the derivatives of the following polynomial functions : 1 f : x ↦−→ 3 x + 2 2 g : x ↦−→ x 2 + 4 3 h : x ↦−→ x 2 + x 4 j : x ↦−→ x + 2 x 2 5 k : x ↦−→ 3 x 2 − 2 x 6 ‘ : x ↦−→ ( x + 1)(2 x − 4) 5. Second-degree polynomials: tangents E.7650 Proposition: Let f be a function f that is differentiable at a , and let C be the curve representing the function f in a coordinate system. The tangent to the curve C at the point with abscissa a has the reduced equation : y = f ( a ) · x − a + f ( a ) Consider the function f defined for any number x belonging to the interval 0 ; 7 by: f ( x ) = − 0 ; 5 · x 2 + 3 ; 5 · x − 3 Let C f be the curve representing the function f in a coor-dinate system and A the point with abscissa 2 belonging to C f . 1 Give the coordinates of the point A . 2 Determine the value of the derivative of the function f at x =2 . 3 Determine the reduced equation of the tangent ( T ) to the curve C f at point A . 4 Draw the tangent ( T ) in the coordinate system below : E.7651 Consider the function f defined for any number x belonging to the interval − 2 ; 5 by: f ( x ) = 0.25 · x 2 − 0.75 · x − 1 Let C f be the curve representing the function f in a coor-dinate system and A the point with abscissa 2 belonging to C f . 1 Give the coordinates of point A . 2 Determine the value of the derivative of function f at x =2 . 3 Determine the reduced equation of the tangent ( T ) to the curve C f at point A . 4 Draw the tangent ( T ) in the coordinate system below : https://chingmath.fr chapExoCorrec/4668 sacados/4668 chapExoCorrec/7650 sacados/7650 x01234567y-11234Cf chapExoCorrec/7651 sacados/7651 x-2-1012345y-2-11Cf
E.4839 Consider the function f whose image of a number x is defined by the relation: f ( x ) = 1 2 · x 2 + 3 · x + 1 In the plane provided with a reference frame O ; I ; J , con-sider the curve C f representative of the function f : Note ( d ) the tangent to the curve C f at the point of abscissa − 2 . 1 a Determine the image of the number − 2 by the func-tion f . b Determine the expression of the function f derived from the function f . c Determine the derivative number of the function f in − 2 . 2 Determine the reduced equation of the line ( d ) . 3 Draw tangent ( d ) in frame O ; I ; J . E.7477 Consider the function f defined by: f ( x ) = 2 · x 2 − 5 · x + 4 The plane is provided with a reference frame O ; I ; J and note C f its representative curve in the plane. 1 Determine the equation of the tangent ( T ) to the curve C f at the point of abscissa 1 2 . 2 Check the previous result using the calculator. E.2809 Consider the polynomial function f of degree 2 which satisfies the two conditions below : The set of zeros for the function f is : − 1 ; 1 2 f (1) = 5 Determine the developed and simplified expression of the function f . E.2840 Determine the second-degree polynomial function f that satisfies the following conditions : The image of 2 is − 2 . ( − 1 ; 1) ∈ C f The tangent at C f at the point with abscissa 1 has a slope of 2. 6. Polynomials: tangents E.2845 Consider the function f defined on R by the relation: f : x ↦−→ x 4 4 − x 3 + x 2 + x − 2 Here is the representative curve of the function f in the ref-erence frame ( O ; I ; J ) : We are going to show that there is a straight line ( d ) of di-recting coefficient 1 which is the tangent to the curve C f at two distinct points. 1 Solve equation : f ( x ) = 1 2 Determine the reduced equation of the line ( d ) . E.4729 Consider the function f whose image of a real number x is given by: f ( x ) = − x 4 + 2 x 3 + 3 x 2 − 5 x − 3 In a frame of reference O ; I ; J , note C f the representative curve of the function f and ( d ) the straight line of equation : y = − x + 1 Demonstrate that the straight line ( d ) is tangent to the curve C , which we’ll specify at two points whose coordinates we will specify (we can conjecture the abscissa of these points using the calculator) . https://chingmath.fr chapExoCorrec/4839 sacados/4839 -6-5-4-3-2-12I-4-3-2-1JOCf chapExoCorrec/7477 sacados/7477 sacados/2809 chapExoCorrec/2840 sacados/2840 chapExoCorrec/2845 sacados/2845 -2-1234I-3-2-12JOCf chapExoCorrec/4729 sacados/4729
E.4732 Consider the function f defined by the relation: f ( x ) = 1 2 · x 4 − x 3 − 3 2 · x 2 + x + 2 The plane is given a ref-erence frame O ; I ; J orthonormal in which is represented the curve C f representative of the function f : 1 a Draw the straight line ( d ) of equation : y = − x . b Make a conjecture about the coordi-nates in which the line ( d ) is a tan-gent to the curve C f . 2 Establish your conjectures from question 1 b . 7. Polynomials: variation problems E.89 Part A Consider the functions f and g defined on R by: f ( x ) = 1 10 · x 3 − 6 · x 2 + 120 · x ; g ( x ) = 40 · x . 1 a Calculate the derivative number f ( x ) and verify that : f ( x ) = 3 10 · ( x − 20) 2 . b Study the direction of variation of the function f on R . c Calculate f (10) , f (20) and f (40) . 2 The curve ( C f ) representative of the function f is plotted on the attached sheet to be handed in with the copy. a Determine an equation of the tangent T A to curve ( C f ) at the point A of abscissa 10. b Draw on the attached sheet the curve ( C g ) representa-tive of the function g . c Show that the curve ( C f ) , the curve ( C g ) and the straight line T A intersect at the point of abscissa 40. Deduce the tangent T A to be drawn on the attached sheet. Part B The cost expressed in euros of a production is a function of the number of units x manufactured is equal to f ( x ) where f is the function studied in part A . 1 Show that for x units produced and sold 40 euros each, the profit in euros is expressed by g ( x ) − f ( x ) . 2 a Graphically determine the solutions of the equation f ( x )= g ( x ) on the interval 0 ; 45 . Make any useful construction lines and check that the integer values read are solutions. b Determine the interval to which the number of units manufactured x must belong for the company to be profitable. https://chingmath.fr chapExoCorrec/4732 sacados/4732 -2-123I-3-2-123JO chapExoCorrec/89 sacados/89 Antilles - Juin 2002 - 6 points -20-1001020304050-2000-10001000200030004000
E.110 Part A Let g be the function defined on [ − 1 ; 8] by g ( x )= x 2 − 6 · x +5 and shown below 1 a Graphically solve the equation g ( x )=0 . b Deduce the sign of g ( x ) on the interval [ − 1 ; 8] . c Graphically solve the equation g ( x )= − 3 2 a Does the function g admit a minimum on [ − 1 ; 8] ? b Verify that : g ( x )=( x − 1)( x − 5) for x belonging to − 1 ; 8 . c Find the sign of g ( x ) using a table Part B Let f be the function defined on [ − 1 ; 8] by: f ( x ) = 0.2 · x 3 − 1.8 · x 2 + 3 · x + 4 . We call ( C ) its representative curve in the plane provided with an orthonormal reference frame O ; −→ i ; −→ j (unit of length 1 cm ) . 1 Calculate the derivative of f , denoted f . 2 Verify that f ( x )=0.6 · g ( x ) for any x of [ − 1 ; 8] ( g is the function studied in part A ) . Deduce the sign of f ( x ) and the table of variations of the function f on [ − 1 ; 8] . 3 Draw ( C ) in the reference frame O ; −→ i ; −→ j E.4538 Let f be the function defined by the relation: f ( x )= − x 3 +2 1 Solve the equation : f ( x )=10 2 Establish that the function f is decreasing on R E.74 Consider a game of boules, such as pétanque. A player throws a ball, and we’re interested in the ball’s trajectory. Let the function g be defined on the interval [0 ; 2] by: g ( x ) = − x 2 + 1.5 · x + 1 Where: x is the elapsed time, in seconds, from the moment the ball leaves the thrower’s hand ; g ( x ) represents, in meters, the distance (vertical) separat-ing the ground from the ball after x seconds have elapsed 1 The function g is represented by part of the curve given in the appendix. Plot in color the representative curve ( C g ) of the function g on the appendix sheet. 2 a Calculate g (0) . Describe in a sentence what the point of abscissa 0 represents. b Calculate g (1) . Describe in a sentence what the point of abscissa 1 represents. 3 Calculate g ( x ) , g denoting the derivative of the function g . 4 a Find the sign of g ( x ) according to the values of x , x ∈ 0 [ 2] . b Deduce the complete table of variations of the function g . c Explaining the method used, indicate at what instant the ball reaches its maximum height. d Explaining the method used, indicate at what instant the ball touches the ground. https://chingmath.fr sacados/110 Canada - 2002 - 8 points - obligatoire ij chapExoCorrec/4538 sacados/4538 sacados/74 -1-0.500.511.522.5-1-0.50.511.52
E.78 Consider the function f defined by: f ( x ) = 54 · x 3 − 2 · x 2 + x sur l’intervalle 0 ; 1 . 1 a Calculate f ( x ) where f is the derivative function of the function f on the interval [0 ; 1] . b Check that : f ( x ) = 54 · (3 · x − 1)( x − 1) for all x in the interval [0 ; 1] . c Study the sign of f ( x ) on the interval [0 ; 1] . 2 Give the maximum of f on the interval [0 ; 1] . For what value of x is it reached? 3 Copy and complete the following table with the values of f ( x ) rounded to the nearest 0.1. x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f ( x ) 6.9 4 Plot the graphical representation of the function f on the enclosed sheet of graph paper, using 10 cm as the abscissa and 1 cm as the ordinate. E.6619 Consider the function f defined on R by the relation: f ( x ) = a · x 2 + b · x + c where a ∈ R ∗ and b;c ∈ R . In a reference frame O ; I ; J orthonormal, note C f the rep-resentative curve of the function f and (Δ) the tangent to the curve C f at the point of abscissa 2 . 1 Determine the directing coefficient of the straight line connecting the points on the curve A and B on the curve C f with abscissas 1 and 3 respectively. 2 a Determine the derivative number of the function f for x =2 . b What conclusion can you draw? 8. Derivative functions: square root E.4687 The table below shows you, for each line, the expression of the image of x by a function and the expression of the number derived in x of this function. Check the accuracy of the expression of the number derived in x : Fonction Image de x Nombre derivative en x f x 3 − 5 · x 2 + x − 3 3 · x 2 − 10 · x + 1 g 2 · x − 1 x 2 + x − 2 · x 2 − 2 · x − 1 x 2 · ( x + 1) 2 h ( x 2 − 3) · x 5 · x 2 − 3 2 · x j 3 · x − 2 2 − x 4 ( x − 2) 2 E.4692 Consider the function f defined on R + by the relation: f ( x ) = x · 1 2 · x − 1 In the coordinate system O ; I ; J orthonormalized below is the curve C f representing the function f : 1 We denote f as the derivative of the function f . Estab-lish the following equality: f ( x ) = 3 · x − 2 4 · x 2 a Give the coordinates of the point C f with 4 as its abscissa. b Give the value of the derivative of the function f at 4 . c Determine the equation of the tangent ( T ) to the curve C f at the point with abscissa 4 . d Draw the tangent ( T ) . https://chingmath.fr chapExoCorrec/78 sacados/78 Liban - 2004 - 7 points - obligatoire chapExoCorrec/6619 sacados/6619 chapExoCorrec/4687 sacados/4687 chapExoCorrec/4692 sacados/4692 -123456I-3-2-12345JO
E.8391 We provide the plane with a refer-ence frame O ; I ; J and consider the function g defined by: g ( x ) = x · x − 1 We note C g its representative curve in the plane. 1 Determine the equation of the tangent ( d ) to the curve C g at the point of abscissa 1 4 . 2 Check your results using the calculator E.7652 Consider the function f defined on R + by: f ( x ) = x − x 1 Determine the expression of the derivative function f of the function f . 2 a Determine the image and the number derived from the number 2 by the function f . b Determine the expression of the tangent ( T ) to the curve C representative of the function f at the point of abscissa 2 . 3 a Draw the curve C and the tangent ( T ) using your calculator. b Conjecture the relative position of the curve C and the tangent ( T ) . 9. Calculator and derivative E.4642 1 Consider the function f whose image of a real number x is defined by the relation: f ( x ) = 1 3 x 3 − 2 x 2 + 3 x − 1 Here are two screenshots of the graphical representation of this function on a calculator screen showing the local minimum and maximum of this function : Draw, without justification, the sign table for the deriva-tive f of the function f . 2 Using a calculator and without justification, draw up the table of signs associated with the derivatives of the fol-lowing functions : a g ( x ) = − 1 3 x 3 + x 2 + 3 x + 2 b h ( x ) = 1 3 x 3 + x 2 − 3 x + 2 c j ( x ) = 2 3 x 3 + x 2 − 4 x − 1 d k ( x ) = − 4 3 x 3 − 2 x 2 + 1 E.4643 Using the calculator and without justifi-cation, for each of the functions below, draw up the sign table of the associated derivative function : a f ( x ) = x 2 x 2 + 1 b g ( x ) = 4 x 2 + 2 x + 3 c h ( x ) = x 2 − x − 2 x − 3 d j ( x ) = x − 5 − x 2 + 4 x + 4 E.4518 Answer the following questions using the calculator: 1 Let f be the function defined by: f ( x ) = x 3 − 4 · x 2 + x + 2 Draw up the table of variations of the function f on the interval − 1 ; 4 . 2 Consider the affine function g defined by: g ( x ) = − 3 · x + 2 Determine the coordinates of the intersection points of the curves C f and C g . 10. Products E.4704 Determine the expression of the deriva-tive function associated with each of the functions below : 1 f : x ↦−→ x 2 − 3 x · x 5 + 3 x 2 + 1 2 g : x ↦−→ x 4 + 3 x 2 + 1 · x 3 − x 2 E.7147 Consider the two functions f and g de-fined below by: 1 f ( x ) = ( x 2 − 3)(2 − x 3 ) 2 g ( x ) = x 7 − 4 · x − 1 x 3 + 2 · x Determine the expressions of their derivative functions. E.8396 1 f : x ↦→ (2 · x 2 − 1)(4 · x − 1) 2 g : x ↦→ (5 · x 4 − x +1)(3 − 2 · x 2 ) E.4690 Consider the function f defined on R ∗ by: g ( x ) = 1 x · x 2 + 1 Establish the following equality: g ( x )= x 2 − 1 x 2 https://chingmath.fr chapExoCorrec/8391 sacados/8391 chapExoCorrec/7652 sacados/7652 sacados/4642 sacados/4643 chapExoCorrec/4518 sacados/4518 chapExoCorrec/4704 sacados/4704 chapExoCorrec/7147 sacados/7147 chapExoCorrec/8396 sacados/8396 chapExoCorrec/4690 sacados/4690
11. Quotient E.4705 Determine the expression of the deriva-tive function associated with each of the functions below : 1 f : x ↦−→ x 2 − 3 x x 3 − 4 x + 1 2 g : x ↦−→ 5 x 2 − 1 E.5228 Determine the expression of the deriva-tive of each of the following functions : a f ( x ) = x + 1 3 · x + 1 b g ( x ) = 5 · x + 1 3 − 2 · x c h ( x ) = x 2 + 3 · x + 1 2 · x + 1 d j ( x ) = 3 − 2 · x x 2 − 3 · x − 1 E.4728 Determine the expression of the deriva-tives of the following functions : 1 f ( x ) = x 2 − 4 3 · x 2 − x + 4 2 g ( x ) = x 2 − 3 · x 2 · x − 4 E.94 Determine the expression of the derivative functions of each of the functions below : a f : x ↦−→ 2 − 2 x 5 x + 1 b g : x ↦−→ (3 x − 2)(2 x 2 + 1) c h : x ↦−→ 1 3 x + 1 d j : x ↦−→ (2 x 2 + 3 x ) · x E.2349 Consider the two functions f and g by: f ( x ) = (2 · x + 1)(3 · x 2 − x + 1) ; g ( x ) = 2 · x + 5 1 − 4 · x Determine the expression of the derivative function of each of these two functions. 12. Quotients and tangents E.4693 Consider the function f defined over R + by the relation: f ( x ) = 5 · x − 2 x 2 + 1 In the reference frame O ; I ; J orthonormé below is given the curve C f représentative of the function f : 1 Note f la derivative function of the function f . Estab-lish the following equality: f ( x ) = − 5 · x 2 + 4 · x + 5 x 2 + 1 2 2 a Give the coordinates of the point of C f ayant 1 pour abscissa. b Give the value of the number derived from the function f in 1 . c Determine the equation of the tangent ( T ) to the curve C f au point of abscissa 1 . d Trace the tangent ( T ) . E.6616 Consider the function f defined by: f ( x ) = 4 · x + 1 x 2 + 2 · x + 2 The graphical representation C f of the function f in the O ; I ; J orthonormal frame below is given : 1 Establish that the function f derivative of the function f has the expression : f ( x ) = 2 · 1 − x · 2 x + 3 x 2 + 2 · x + 2 2 2 a Determine the reduced equation of the tangent (Δ) to the curve C f at the point of abscissa − 1 . b Draw the straight line (Δ) in the above reference frame. 3 What feature do the tangents to the curve C f have at the points of abscissa − 3 2 and 1 ? Justify your answer. https://chingmath.fr chapExoCorrec/4705 sacados/4705 chapExoCorrec/5228 sacados/5228 chapExoCorrec/4728 sacados/4728 chapExoCorrec/94 sacados/94 chapExoCorrec/2349 sacados/2349 chapExoCorrec/4693 sacados/4693 -4-3-2-1234I-4-3-2-12JO chapExoCorrec/6616 sacados/6616 xxyy-4-3-2-1234I-4-3-2-1JO
E.6043 Consider the function f defined on R whose image of a real x is defined by the relation: f ( x ) = 3 · x + 4 4 · x 2 + 4 We note C f the representative curve of the function f in the reference frame O ; I ; J : 1 Demonstrate that the function f has as its derivative the function f whose expression is defined by: f ( x ) = − 4 · 3 · x 2 + 8 · x − 3 4 · x 2 + 4 2 2 a Determine the reduced equation of the tangent ( d ) to the curve C f at the point of abscissa − 1 2 . b Draw, in the above reference frame, the tangent ( d ) . 3 Justify that all tangents to the curve C , whose abscissa of the points of contact belong to the interval − 3 ; 1 3 , are associated with increasing affine functions. E.5229 Consider the function f defined on R ∗ + by: f ( x ) = 2 · x − 4 √ x The curve C f representative of the function f in the reference frame O ; I ; J below is given : 1 Show that the function f has as its derivative the func-tion f whose expression is : f ( x ) = x + 2 x · x 2 a Determine the reduced equation of the tangent ( T ) to the curve C f at the point of abscissa 4 . b Draw the tangent ( T ) . 13. Quotient and variations E.4849 Consider the function f defined on R by the relation: f ( x ) = 7 · x − 7 x 2 + 3 1 Determine the expression of the function f derivative of the function f . 2 Draw up the sign table for the function f . 3 Draw up the table of variations of the function f (the exact values of the local extremums should be given) . E.4882 Consider the function g defined on R \{− 1 } by the relation: f ( x ) = x 2 − 2 · x + 4 x + 1 1 Show that the function g derived from the function g has expression : f ( x )= x 2 +2 x − 6 ( x +1) 2 2 Draw up the table of variations of the function g . E.4847 Consider the function f whose image of a number x is defined by the relation: f ( x ) = 3 · x + 3 x 2 + 3 1 Draw up the sign table for the function f . 2 Show that the function f derived from the function f has the expression : f ( x ) = − 3 · x 2 − 6 · x + 9 x 2 + 3 2 3 a Draw up the sign table for the function f . b Draw up the table of variations of the function f . (ex-act values of local extremums should be given) E.4848 Consider the function f whose image of a real number x is given by the relation: f ( x ) = 2 − x x 2 + 5 1 Show that the function f derivative of the function f has the expression : f ( x ) = x 2 − 4 · x − 5 x 2 + 5 2 2 Draw up the sign table for the function f 3 Draw up the table of variations of the function f (note the exact value of the local extremums) . https://chingmath.fr chapExoCorrec/6043 sacados/6043 -4-3-2-1234I-12JOCf chapExoCorrec/5229 sacados/5229 2345678I-1234JOCf chapExoCorrec/4849 sacados/4849 chapExoCorrec/4882 sacados/4882 chapExoCorrec/4847 sacados/4847 chapExoCorrec/4848 sacados/4848
E.4846 Consider the function f defined by the relation: f ( x ) = x 2 + 2 · x + 1 x − 1 1 Determine the definition set of the function f . 2 Let f be the derivative function of the function f . Es-tablish the following equality: f ( x ) = x 2 − 2 · x − 3 ( x − 1) 2 3 a Draw up the sign table for the function f on R \{ 1 } . b Draw up the table of variations of the function f . In-dicate the exact value of the two local extremums of this function. 4 Deduce from the set of previous questions the sign table of the function f . E.4845 Consider the function f defined by the relation: f ( x ) = 2 · x 2 − x − 1 2 · x + 2 1 Determine the definition set of the function f . 2 a Factor the polynomial: 2 · x 2 − x − 1 . b Deduce from the set of previous questions the sign ta-ble of the function f . 3 Let f be the derivative function of the function f . Établir l’égalité suivante : f ( x ) = 4 · x 2 + 8 · x (2 · x + 2) 2 4 a Draw up the sign table for the function f on R \{− 1 } . b Draw up the table of variations of the function f . In-dicate the exact value of the two local extremums of this function. E.4841 Consider the function f defined by the relation: f ( x ) = 3 x + 3 x 2 + 3 In a reference frame O ; I ; J orthonormal, we give the curve C f representative of the function f : 1 Justify that the function f is defined on R . 2 Using its graphical representation, draw up the table of variations of the function f on the interval − 6 ; 3 (by graphical reading, approximate values will be used) . 3 a Establish the following expression for the function f derived from the function f : f ( x ) = − 3 · x 2 − 6 · x + 9 ( x 2 + 3) 2 b Study the sign table of the function f on R . 4 What conjecture can be made between the sign of the derivative function f and the direction of variation of the function f ? E.4891 Consider the function f defined on R , whose image of a number x is given by the relation: f ( x ) = x 2 + 5 x + 4 x 2 + 4 1 a Give the factored form of the polynomial x 2 +5 x +4 . b Draw up the sign table for the function f on R . c Solve the inequality: 5 x x 2 +4 − 1 2 Let f be the derivative of the function f : a Establish that the function f can be expressed as : f ( x ) = 20 − 5 x 2 x 2 + 4 2 b Draw up the table of variations of the function f . (in-dicate the values of the local extrema) . 3 In an orthonormal coordinate system O ; I ; J , consider the curve C f representing the function f : a The equation of the line ( d ) is y = 3 5 · x + 3 5 . Justify that the line ( d ) is the tangent to the curve C f at the point with abscissa − 1 . b Determine the equation of the tangent (Δ) to the curve C f at the point with abscissa 0 . c Plot the line (Δ) on the graph. 14. Derivatives review E.3429 Consider a function f defined and deriv-able on the interval − 2 ; 7 whose representative curve C f is given below : https://chingmath.fr chapExoCorrec/4846 sacados/4846 chapExoCorrec/4845 sacados/4845 chapExoCorrec/4841 sacados/4841 -6-5-4-3-2-123I-2-12JOCf chapExoCorrec/4891 sacados/4891 -6-5-4-3-2-1234I-123JOCf(d chapExoCorrec/3429 sacados/3429
The tangents to the curve C f , at the abscissa points − 1 , 1 , 2 , 5 have been drawn on the above representation. 1 Determine the derivative number of the function f in − 1 and in 1 . 2 Determine the equations of the tangents to the curve C f at points of abscissa 2 and 5 . E.3314 Determine the expression of the deriva-tive functions associated with the functions below : a f ( x ) = − 5 x 3 + 2 x − 2 b g ( x ) = x · 5 x + 1 c h ( x ) = 3 x − 1 2 − x d j ( x ) = x 5 − 2 x E.3507 Determine the expression of the deriva-tive function for each of the following functions : a f ( x ) = 5 · x + 2 · 3 − x b g ( x ) = 3 x + 2 2 − x c h ( x ) = − 2 x 2 + 3 x + 2 · √ x d j ( x ) = 2 · x − 2 √ x E.3912 Some of the results, to be demon-strated, may not be legible on the screen of your graphing calculator. Consider the function f defined on 0 ; 1 ∪ 1 ; + ∞ by: f ( x ) = 10 · ( x − 8) x · ( x − 1) and we denote by ( C ) its representative curve relative to an orthonormal reference frame O ; −→ i ; −→ j . 1 a Determine the limits of f in 0 and in + ∞ . b Determine the limits of f when x tends to 1 by lower values and when x tends to 1 by higher values. c Deduce the asymptotes to the curve ( C ) . 2 a Determine the derivative f of the function f . b Show that f ( x ) cancels for ¸ =8+2 14 and for ˛ = 8 − 2 14 . c Draw up the table of variations of f (using the calcu-lator, give the local extremums rounded to the nearest tenth) . E.6803 In the plane equipped with an orthonormal coordinate system O ; −→ i ; −→ j , we denote by C u the curve representing the function u defined on the interval 0 ; + ∞ by: u ( x ) = a + b x + c x 2 where a , b , and c are fixed real numbers. The curve C u and the line D with equation y =1 are plotted on the graph below : Note that the curve C u passes through the points A (1 ; 0) and B (4 ; 0) and that the y-axis and the line D are asymptotes to the curve C u . 1 a Give the values of u (1) and u (4) . b Give lim x ↦→ + ∞ u ( x ) . Deduce the value of a . c Deduce that, for any strictly positive real number x : u ( x ) = x 2 − 5 · x + 4 x 2 2 We assume the existence of a function f defined on the in-terval 0 ; + ∞ satisfying admitting as its derivative the function u : f = u Draw up the table of variations of the function f . (no value will be indicated in the table) https://chingmath.fr -2-1234567I-2-123JOCf chapExoCorrec/3314 sacados/3314 chapExoCorrec/3507 sacados/3507 chapExoCorrec/3912 sacados/3912 chapExoCorrec/6803 sacados/6803 ijCuDAB
E.3312 This exercise is a multiple-choice questionnaire ; for each of the five questions, one and only one statement is correct. Indicate the question number on your copy and copy the correct statement without justifying your choice. Barème: Each question is awarded 1 point. An incorrect answer deducts 0.5 point. An unanswered question neither earns nor deducts any points. If the total points are negative, the mark awarded to the exercise is reduced to zero. Let f be the function defined on 4 ; + ∞ par : f ( x ) = − 2 x + 1 − 8 x − 4 and Γ sa representative curve in an orthonormal plane. 1 Another expression for f ( x ) est : a f ( x ) = − 2 x + 1 − 2 x − 1 b f ( x ) = 2 x 2 − 9 x + 12 4 − x c f ( x ) = 2 x 2 + 9 x − 2 x − 4 2 Let f be the derivative of f on 4 ; + ∞ . An expression of f ( x ) is : a f ( x ) = − 2 − 8 x − 4 2 b f ( x ) = (2 − x )( x − 6) ( x − 4) 2 c f ( x ) = − 2 x 2 + 16 x − 24 ( x − 4) 2 3 The curve Γ has as its asymptote : a the line with equation y =4 b the line with equation x =4 c the line with equation y =4 x E.3313 Let f be a function whose incom-plete table of variations is as follows ; we denote by f the function derived from the function f . Let f be defined on −∞ ; − 1 ∪ − 1 ; + ∞ by: f ( x ) = a · x + b + c x + 1 where a , b and c are real numbers. 1 Calculate f ( x ) as a function of a , b and c . 2 Using the information in the above table of variations, show that we have : a =1 ; b = − 1 ; c =4 3 Determine the missing limits in the variation table pro-vided. E.3509 Consider the function f defined on R by the relation: f : x ↦−→ 3 2 · x 4 + 3 · x 3 − 9 2 · x 2 − 5 · x + 6 Below, we give the representative curve of the function f in a reference frame ( O ; I ; J ) : The curve C f representative of this function admits a straight line ( d ) of directrix 1 as tangent at two points. Determine the equation of this line and the coordinates of these two points. E.6136 Consider the set F of couples ( x ; y ) of reals such that that x and y are the measures of the lengths of two sides of a right-angled triangle whose perimeter is equal to 1 . Consider the function f defined on the interval 0 ; 1 2 by: f ( x ) = 1 − 2 x 2 · 1 − x Verify that for any real x in the interval 0 ; 1 2 , the pairs ( x ; f ( x )) belong to F . 15. Derivability at a point E.3541 Consider the function f defined on R as follows : f ( x ) = x 2 + 3 x + 3 for all x ∈ −∞ ; − 1 f ( x ) = 2 x 2 + 1 pour tout x ∈ − 1 ; + ∞ 1 a Plot the function f using your calculator. b Make a conjecture about the continuity and derivabil-ity of the function f . 2 Justify that the function f is continuous in − 1 . 3 Justify that the function f is derivable in − 1 . https://chingmath.fr chapExoCorrec/3312 sacados/3312 chapExoCorrec/3313 sacados/3313 Bac ES Antilles −∞-3-11∞∞−∞-6:::2:::xVariationdef chapExoCorrec/3509 sacados/3509 -4-3-2-1234I-22468JO chapExoCorrec/6136 sacados/6136 chapExoCorrec/3541 sacados/3541
E.5081 Parts A and B are independent of each other. Part C involves the previous parts. Consider the plane provided with an orthonormal reference frame O ; I ; J . Part A Consider the function f defined by the relation: f ( x ) = 4 · x + 4 3 − x Note C f representative curve of the function f . 1 Determine the definition set of the function f . 2 Show that the tangent ( d ) to the curve C f au point of abscissa 1 admet for reduced equation : ( d ) : y = x + 1 3 Draw the table of variations of the function f on its set of definition. 4 Study the position of the curve C f relative to the line ( d ) . We will use the factorization : x 3 − x 2 − x + 1 = ( x − 1) 2 ( x + 1) Part B Consider the function g defined by the relation: g ( x ) = 3 · x − x 2 3 4 · x We denote C g the curve representing the function g . 1 Is the function g continuous on R ? 2 Show that the function g has as its derivative the func-tion g defined by the expression : g ( x ) = x · (6 − 5 x ) · (3 − x ) 2 4 3 How many horizontal tangents does the curve C g have? 4 Draw up a table of variations for the function g . (the values will not be indicated in the table of variations) Part C Consider the function h defined by the relation: h ( x ) = f ( x ) pour x ∈ − 1 ; 1 h ( x ) = g ( x ) pour x ∈ 1 ; + ∞ 1 Answer the following questions, giving reasons for your answers : a Is the function h continuous at 1 ? b Is the function h differentiable at 1 ? 2 In the coordinate system below : We want to draw by hand the curve C h representing the function h observed using the calculator. To do this, we begin by: drawing the tangent ( d ) ; place the horizontal tangents of the curve C h . E.3600 A - Study of a function : Consider the function f defined by the relation: f ( x ) = ( x − 1) · x 2 − 1 1 Determine the domain D f of the function f . 2 a Study the following two limits: lim h ↦→ 0 + f (1+ h ) − f (1) h ; lim h ↦→ 0 − f ( − 1+ h ) − f ( − 1) h b Study the differentiability of the function f at − 1 and at 1 . 3 a Determine the expression of the function f derived from the function f . b Draw up the table of variations of the function f . B - Extension by continuity: Consider the function g defined on R as follows : g ( x ) = f ( x ) pour x ∈ −∞ ; − 1 ∪ 1 ; + ∞ g ( x ) = − x 3 + x 2 + x − 1 pour x ∈ − 1 ; 1 1 Justify that the function g is continuous on R . 2 Determine the set of differentiability of the function g . Justify your statements. 3 Draw up the table of variations of the function g . 4 Justify the existence of a unique number ¸ belonging to R that satisfies the following two conditions : g ( ¸ ) = 2 ; 2 <¸ < 2.1 16. Unclassified financial years E.77 A rectangle ABCD has perimeter 10 cm . Part A In this part, we pose AB = x (en cm ) .s 1 In which closed interval can the real x vary? 2 Express the area S ( x ) of the rectangle ABCD as a func-tion of x . Part B The function f is defined on R by: f ( x )= − x 2 +5 · x . 1 Using its derivative function f , draw up the table of variations of f . https://chingmath.fr chapExoCorrec/5081 sacados/5081 -2-12345I-2-123JO chapExoCorrec/3600 sacados/3600 chapExoCorrec/77 sacados/77 Liban - 2002 - obligatoire
2 Deduce the value of x for which the area of the rectangle ABCD is maximum. 3 a In the plane provided with an orthonormal frame of reference (unit: 2 cm on each axis) , draw the curve C representative of f on the interval [0 ; 5] . b On the same figure as at 3 a , draw the tangent to the curve C at its point of abscissa 1. E.84 We want to solve, in the set of real numbers R , the equation : x 3 − 2 · x 2 − 4 · x +5=0 . A - Graphical method 1 a Verify that the number 2 is not a solution to the equation. b Show that, for x =2 , the equation x 2 = 4 · x − 5 x − 2 is equiv-alent to the equation x 3 − 2 · x 2 − 4 · x +5=0 2 Let f be the function defined for any real x other than 2 by f ( x )= 4 · x − 5 x − 2 . Its representative curve H in an or-thonormal reference frame is given in the appendix, to be returned with the copy. a By graphical reading, indicate the direction of varia-tion of f on each of the intervals ] −∞ ; 2[ and ]2 ; + ∞ [ . b Determine the derivative f of f then justify the result read in the previous question. 3 Let g be the function defined on R by g ( x )= x 2 . Plot its representative curve P in the reference frame used for H . 4 By graphical reading, determine the number of solutions in R of the equation x 3 − 2 x 2 − 4 x +5=0 . Give the exact value or an approximate value to the near-est 10 − 1 of each of these solutions. B - Algebraic method 1 Verify that, for any real: ( x − 1)( x 2 − x − 5) = x 3 − 2 · x 2 − 4 · x + 5 . 2 Let h be the function defined on R by: h ( x )= x 2 − x − 5 . a Study the direction of variation of h . b Show that h 1 2 is the minimum value taken by h . c We pose : x = 1 2 + u . Express h 1 2 + u as a function of u ; factor the resulting expression. d Deduce the values of the real x for which h ( x )=0 . 3 Give the set of solutions in R of the equation : x 3 − 2 · x 2 − 4 · x + 5 = 0 . E.4881 In a reference frame, note C f the repre-sentative curve of the function f defined on R by the relation: f ( x ) = x 3 − 2 x 2 + 2 x + 1 Note ( T ) the tangent to the curve C f at the point of abscissa 1 . 1 Determine the reduced equation of the line ( T ) . 2 Determine the coordinates of the intersection points of C f and the line ( T ) . https://chingmath.fr chapExoCorrec/84 sacados/84 -10-8-6-4-20246810-8-6-4-224681012 chapExoCorrec/4881 sacados/4881
E.7089 Consider the function f defined by the relation is : f ( x ) = 1 4 · x 2 − 1 2 · x − 2 In the plane with an orthonormal reference frame O ; I ; J , note C f the representative curve of the function f : 1 a Plot the straight line ( d ) whose equation is : y = 1 2 · x − 3 b What is the name of the line ( d ) relative to the curve C f ? 2 a Draw the straight line (Δ) whose equation is : y = − 3 2 · x − 3 b What is the name of the line (Δ) relative to the curve C f ? https://chingmath.fr chapExoCorrec/7089 sacados/7089 -3-2-12345I-4-3-2-12JO