Grade 11 - STMG / Derivative number and second-degree function 47 exercises (100% corrected)

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x-4-3-2-101234y-2-1123AB -4-3-2-1234I-2-123JO 1. Reminders: affine functions E.11341 Consider the linear function defined by the relation: f ( x ) = − 1 ; 25 x + 0 ; 25 In the coordinate system below, consider the two points A and B shown below and denote C f as the line representing the function f : 1 a Justify that the points A and B belong to the line C f . b Draw the line C f representing the function f . 2 a Give an approximate value for the x-coordinate of the unique point C of C f with − 0 ; 5 as its y-coordinate. b Justify algebraically that the antecedent of the number − 0 ; 5 is 0 ; 6 . E.7370 Consider the function f defined on R by the relation: f ( x ) = 0.75 x + 1.25 1 Complete the table of values below : x − 3 − 2 − 1 0 1 2 f ( x ) 2 Represent the curve C f of the function f in the coordi-nate system below : E.7371 Proposition: If the slope is positive, the linear function is increasing. If the slope is negative, the linear function is decreas-ing. 1 Consider the function f defined on R by the relation: f ( x ) = 3 · x + 4 ; 5 a Solve the equation : f ( x )=0 b Complete the sign table for the function f : x −∞ + ∞ f ( x ) 2 Consider the function g defined on R by the relation: g ( x ) = − 2 · x + 0 ; 5 a Solve the equation : g ( x )=0 b Complete the sign table for the function g : x −∞ + ∞ g ( x ) E.7372 In each case, consider the line ( d ) pass-ing through points A and B . Determine the slope of each line ( d ) . a A (1 ; 2) ; B (5 ; 3) b A ( − 1 ; 2) ; B (3 ; − 2) c A (0 ; 1) ; B (2 ; 5) d A (5 ; 1) ; B (1 ; 4) https://chingmath.fr chapExoCorrec/11341 sacados/11341 x-4-3-2-101234y-2-1123AB chapExoCorrec/7370 sacados/7370 -4-3-2-1234I-2-123JO chapExoCorrec/7371 sacados/7371 chapExoCorrec/7372 sacados/7372
(d1AB(d2-6-4-20246-224J -3-2-12345I-4-3-2-12JO E.7373 Proposition: Let f be a linear function passing through points A ( x A ; y A ) and B ( x B ; y B ) . The slope m of function f has the value : m = y B − y A x B − x A The following graph shows three lines represented in an or-thonormal coordinate system O ; I ; J : Consider the two points A ( − 2 ; 3) and B (4 ; 0) belonging to the line ( d 1 ) : 1 a Show that the slope of the line ( d 1 ) is − 1 2 . b Determine the reduced equation of the line ( d 1 ) . 2 Determine the reduced equation of the line ( d 2 ) . E.7374 Consider the function f defined by the relation is : f ( x ) = 0.25 · x 2 − 0.5 · x − 2 In the plane provided with an orthonormal reference frame O ; I ; J , we note C f the representative curve of the func-tion f : 1 a Plot the straight line ( d ) whose equation is : y = 0.5 · x − 3 b What special feature does the line ( d ) have in relation to the curve C f ? 2 a Draw the straight line (Δ) whose equation is : y = − 1.5 · x − 3 b What special feature does the line (Δ) have in relation to the curve C f ? E.7420 Complete the sign tables below : x-3 x-3 E.11342 Complete the tables below : 1 x −∞ + ∞ x + 5 − 2 x − 8 x +5 − 2 x − 8 2 x −∞ + ∞ x − 1 4 − x − x − 1 ( x − 1)(4 − x )( − x − 1) https://chingmath.fr chapExoCorrec/7373 sacados/7373 (d1AB(d2-6-4-20246-224J chapExoCorrec/7374 sacados/7374 -3-2-12345I-4-3-2-12JO chapExoCorrec/7420 sacados/7420 chapExoCorrec/11342 sacados/11342
x-3-2-10123y-11CfCgABCD <0Aucune racine01racine−b2·a>02racines−b−2·a;−b2·a <00>0¸et˛sontlesdeuxracinesa>0a<0x−∞∞x−∞∞−x−∞∞−b/2a0x−∞∞−b/2a0−−x−∞∞αβ00−x−∞∞αβ00−− E.7421 Consider the following algebraic expres-sion : 2 x + 7 x + 3 − 4 x + 4 2 x + 1 1 Reduce the previous expression to the same denomina-tor. 2 Draw up the sign table for this expression. 3 Deduce the solutions of the inequation : 2 x + 7 x + 3 4 x + 4 2 x + 1 E.11556 1 Consider the function f defined on R by the relation: f ( x ) = 4 · x − 3 a Solve the equation : f ( x )=0 b Complete the sign table for the function f : x −∞ + ∞ f ( x ) 2 Consider the function g defined on R by the relation: g ( x ) = − 3 · x + 1 ; 8 a Solve the equation : g ( x )=0 b Complete the sign table for the function g : x −∞ + ∞ g ( x ) E.11557 In the plane equipped with a coordinate system, consider the line (Δ) representing the linear function : f ( x )=4 x − 1 Which of the points below belong to the line (Δ) ? a A ( − 3 ; − 13) b B (6 ; 23) c C (2 ; 8) d D (0 ; − 2) E.11558 Consider two linear functions f and g satisfying the following equalities f (1)= − 0 ; 75 ; f ( − 2)=1 ; g ( − 1)= − 1 ; g (1)=0 ; 5 In the plane equipped with a coordinate system, consider the lines C f and C g representing the functions f and g : For each of these lines, we have highlighted two of their points as well as a vector indicating the direction of the line. 1 Determine the slope of the function f . 2 Determine the slope of the function g . 2. Reminders: second degree E.7503 Proposition: The roots of a polynomial are the values that cancel out this polynomial. For a second-degree polynomial a · x 2 + b · x + c , the number of existing roots depends on the discriminant : Solve the following equations : a 2 · x 2 − 6 · x + 4 = 0 b 4 · x 2 − 9 · x + 5 = 0 c x 2 + 4 · x + 5 = 0 d 3 · x 2 + 6 · x + 3 = 0 e − x 2 + x − 1 = 0 f − 4 · x 2 − 9 · x − 5 = 0 E.7509 Proposal: The sign chart for a quadratic polynomial de-pends on the sign of the coefficient of the quadratic term and the sign of the discriminant. The six possibilities are shown below : Draw the sign chart for each of the following expressions : a − x 2 + 4 · x + 5 b x 2 − 4 · x + 5 c x 2 + x − 6 d − 4 · x 2 − 4 · x + 8 e 2 · x 2 + 12 · x + 18 f − 4 · x 2 − 6 · x − 2 E.7642 Draw up the sign table for each of the second-degree polynomials below : a − x 2 − 8 · x − 7 b 3 · x 2 + 6 · x − 9 https://chingmath.fr chapExoCorrec/7421 sacados/7421 chapExoCorrec/11556 sacados/11556 chapExoCorrec/11557 sacados/11557 chapExoCorrec/11558 sacados/11558 x-3-2-10123y-11CfCgABCD chapExoCorrec/7503 sacados/7503 <0Aucune racine01racine−b2·a>02racines−b−2·a;−b2·a chapExoCorrec/7509 sacados/7509 <00>0¸et˛sontlesdeuxracinesa>0a<0x−∞∞x−∞∞−x−∞∞−b/2a0x−∞∞−b/2a0−−x−∞∞αβ00−x−∞∞αβ00−− chapExoCorrec/7642 sacados/7642
-4-3-2-1234I-2-123JOCf x-1012y-2-112Cf −203.........xVariationdef −114.........xVariationdeg -3-2-10123456-11234Cf(d3(d2AB 3. Reminders: reading graphs E.7587 Consider the graph C f of the function f defined on the interval − 3.5 ; 3.5 in the coordinate system O ; −→ i ; −→ j : 1 Using the graph, complete the table of values below : x − 3 0 1 2 4 f ( x ) 2 Solve the following equations graphically: a f ( x ) = 0 b f ( x ) = 2 3 Graphically, solve the inequalities: a f ( x ) 0 b f ( x ) 2 E.7588 We provide the graph with a reference point and consider the function f defined on the interval − 1.5 ; 2.5 , whose representative curve C f is shown opposite. 1 Complete the table of values below graphically: x − 1 0 0.5 1 1.5 f ( x ) 2 Graphically solve the equations : a f ( x ) = 0 b f ( x ) = 2 3 Graphically solve the inequalities: a f ( x ) 0 b f ( x ) 2 4. Reminders: table of variations E.7601 1 Consider the function f defined on the interval − 2 ; 3 by the expression : f ( x ) = 0.5 · x 2 − x + 2 Below is the table of variations of the function f where some information has not been given : 2 Consider the function g defined on the interval − 1 ; 4 by the expression : g ( x ) = − x 2 + 2 · x − 1 Below is given the table of variations of the function g where some information has not been given : 5. Introduction E.7510 Consider the second-degree function f defined for any real number x by the relation: f ( x ) = 0.25 · x 2 − x + 1.5 https://chingmath.fr chapExoCorrec/7587 sacados/7587 -4-3-2-1234I-2-123JOCf chapExoCorrec/7588 sacados/7588 x-1012y-2-112Cf chapExoCorrec/7601 sacados/7601 −203.........xVariationdef −114.........xVariationdeg chapExoCorrec/7510 sacados/7510 -3-2-10123456-11234Cf(d3(d2AB
xxyy-5-4-3-2-10123-4-3-2-11Cf(d1(d2 1 The straight line ( d 1 ) is an affine function with reduced equation : ( d 1 ) : y = a · x + b where a and b are two derived numbers. a Give the coordinates of the points A and B . b Determine the directing coefficient of the line ( d 1 ) . c Deduce the expression of the reduced equation of the line ( d 1 ) . 2 a Choose a point C of the line ( d 2 ) and give its coor-dinates. b Determine the reduced equation of the line ( d 2 ) . 6. Derivative function E.7643 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.7742 Copy and complete the table below to obtain the expression of the function f derived from the func-tion f : f ( x )= a · x 2 + b · x + c a b c f ( x )=2 a · x + b 2 · x 2 − 3 · x + 1 − 2 · x 2 + 0 ; 25 · x − 1 2 · x 2 − 3 3 · x 2 + 2 · x E.7551 Give the expression of the functions f of the second degree functions f defined below : a f ( x ) = 2 · x 2 + x − 1 b f ( x ) = 4 · x 2 + 8 · x + 4 c f ( x ) = x 2 − 3 · x + 3 d f ( x ) = − 2 · x 2 + 2 e f ( x ) = − x 2 − 3 · x + 4 f f ( x ) = 0.4 x 2 + x − 4 E.7552 1 Consider the second-degree function f defined by: f ( x ) = 3 · x 2 − x + 1 a Determine the expression of the function f derivative of the function f . b Calculate the following images by the function f : f (2) f (1) f (0.5) f (0) 2 Consider the second-degree function f defined by: f ( x ) = − x 2 + 2 · x − 4 a Determine the expression of the function f derivative of the function f . b Calculate the following images by the function f : f (1) f (0) f ( − 2) f ( − 0.5) E.7553 Consider the quadratic function f whose representative curve is given in the coordinate system below : 1 a The line ( d 1 ) is the tangent to the curve C f at the point with coordinates (1 ; − 0.5) . Determine the slope of the line ( d 1 ) . b The line ( d 2 ) is the tangent to the curve C f at the point with coordinates ( − 2 ; − 2) . Determine the slope of the line ( d 2 ) . 2 The expression of the function is defined by: f ( x ) = 0.5 · x 2 + x − 2 a Determine the expression of the function f derived from the function f . b Calculate the following images using the function f : f (1) f ( − 2) https://chingmath.fr chapExoCorrec/7643 sacados/7643 chapExoCorrec/7742 sacados/7742 chapExoCorrec/7551 sacados/7551 chapExoCorrec/7552 sacados/7552 chapExoCorrec/7553 sacados/7553 xxyy-5-4-3-2-10123-4-3-2-11Cf(d1(d2
C1x-3-2-10123y-112 C2x-3-2-10123y-2-11 C3x-3-2-10123y-2-11 C4x-3-2-10123y-112 x-10123y-3-2-11CfABC x-5-4-3-2-10y1234Cf E.7705 Consider the function f defined on ; by the relation: f ( x ) = − 0.5 · x 2 − 0.25 · x + 1.5 Note C f the representation of the function f in a reference frame. 1 a Solve the equation f ( x )=0 b Of the four curve representations below, only one is the C f curve. Which is it? Justify your answer. 2 a Determine the expression of the function f derived from the function f . b Of the four propositions below, only one is correct. Which is it? Justify your answer. f (2) = − 2.25 f (2) = − 1.75 f (2) = − 1.25 f (2) = − 0.75 E.11616 On considère la fonction f du second degré définie par : f ( x ) = − x 2 + 2 · x − 4 1 Déterminer l’expression de la fonction f dérivée de la fonction f . 2 Calculer les images suivantes par la fonction f : f (1) f (0) f ( − 2) f ( − 0 ; 5) E.11697 Donner l’expression des fonctions f des fonctions f du second degré définies ci-dessous : 1 f ( x ) = 5 · x 2 − x + 5 2 f ( x ) = − 3 · x 2 − 3 · x + 1 7. Tangent: finding the equation E.7564 Consider the function f defined on − 1 ; 3 by the relationship : f ( x ) = 0.5 · x 2 − x − 2 The representative curve of the function f in the plane with a reference point is given below : 1 Determine the expression of the function f derivative of the function f . 2 Note T the tangent to the curve C f at the point of abscissa 2 . a Determine the coordinates of the point A of abscissa 2 of the curve C f . b Determine the directing coefficient of the tangent ( T ) . c Determine the reduced equation of the tangent ( T ) . 3 a Give the coordinates of two points belonging to the tangent ( T ) . b Draw the tangent ( T ) in the above reference frame. E.7565 Consider the function f defined on − 5 ; − 1 by the relation: f ( x )= − 0.5 · x 2 − 3 · x − 1 The representative curve of the function f in the plane with a reference point is given below : 1 Determine the expression of the function f derivative of the function f . 2 Note T the tangent to the curve C f at the point of abscissa − 2 . a Determine the directing coefficient of the tangent C f . b Determine the reduced equation of the tangent ( T ) . 3 Draw the tangent ( T ) in the above reference frame. https://chingmath.fr chapExoCorrec/7705 sacados/7705 C1x-3-2-10123y-112 C2x-3-2-10123y-2-11 C3x-3-2-10123y-2-11 C4x-3-2-10123y-112 chapExoCorrec/11616 sacados/11616 chapExoCorrec/11697 sacados/11697 chapExoCorrec/7564 sacados/7564 x-10123y-3-2-11CfABC chapExoCorrec/7565 sacados/7565 x-5-4-3-2-10y1234Cf
x-10123y-3-2-11Cf x01234567y-11234Cf x-3-2-1012y-4-3-2-11234Cf E.7589 Consider the function f defined on − 1 ; 3 by the relation: f ( x ) = x 2 − x − 2.5 The graph of the function f in the plane with a coordinate sys-tem is shown below : 1 Determine the expression of the function f derived from the function f . 2 Let T be the tangent to the curve C f at the point with abscissa 1.5 . a Determine the coordinates of the point A with ab-scissa 1.5 on the curve C f . b Determine the slope of the tangent ( T ) . c Determine the reduced equation of the tangent ( T ) . 1 a Give the coordinates of two points belonging to the tangent ( T ) . b Draw the tangent ( T ) in the coordinate system above. 8. Tangent: using the formula E.7609 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.7610 Consider the function f defined for any number x belonging to the interval − 3 ; 1.5 by: f ( x ) = 0.5 · x 2 + 1.5 · x − 1 Let C f be the curve representing the function f in a coor-dinate system and A the point with abscissa 1 belonging to C f . 1 Give the coordinates of point A . 2 Determine the value of the derivative of function f at x =1 . 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/7589 sacados/7589 x-10123y-3-2-11Cf chapExoCorrec/7609 sacados/7609 x01234567y-11234Cf chapExoCorrec/7610 sacados/7610 x-3-2-1012y-4-3-2-11234Cf
x-2-1012345y-2-11Cf x-4-3-2-10123y-11Cf xyC1−2;5 xyC2−2;5 xyC3−2;5 −215-71-14xVariationdef −215-72-14xVariationdef E.7641 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 : E.7707 Consider the function f defined for any number x belonging to the interval − 2 ; 5 by: f ( x ) = 0.5 · x 2 + 0.25 · x − 0.75 Let C f be the curve representing the function f in a coordi-nate system and A be the point with abscissa 1 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 =1 . 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 : Indicate on your answer sheet the coordinates of the two points used to draw the tangent. 9. Sign of derivative number and variation E.7563 Let f be the quadratic function defined on − 4 ; 4 by the relation: f ( x )= x 2 +5 · x +1 1 a Determine the expression of the function f derived from the function f . b Solve the equation : f ( x )=0 c Complete the table of signs below : x − 4 4 f ( x ) 2 Let C be the curve representing the function f in the plane equipped with a coordinate system. Which of these three curves is the curve C : E.7566 Let f be the quadratic function defined on − 2 ; 5 by the relation: f ( x )= − x 2 +2 · x +1 1 a Determine the expression of the function f derived from the function f . b Solve the equation : f ( x ) = 0 c Complete the table of signs below : x − 2 5 f ( x ) 2 Which of the following tables of variations is the table of variations of the function f : a b https://chingmath.fr chapExoCorrec/7641 sacados/7641 x-2-1012345y-2-11Cf chapExoCorrec/7707 sacados/7707 x-4-3-2-10123y-11Cf chapExoCorrec/7563 sacados/7563 xyC1−2;5 xyC2−2;5 xyC3−2;5 chapExoCorrec/7566 sacados/7566 −215-71-14xVariationdef −215-72-14xVariationdef
−2157114xVariationdef −114.........xVariationde ... −203.........xVariationde ... −203.........xVariationde ... 010xVariationdef c E.7602 1 Consider the function f defined on the interval − 2 ; 5 by the expression : f ( x ) = 2 · x 2 + x + 1 a Determine the expression of the function f derivative of the function f . b Determine the value of f (2) . c Consider the table of variations below : Justify that this table of variations cannot be the table of variations of the function f . 2 Consider the function g defined on the interval − 4 ; 3 by the expression : g ( x ) = − x 2 + 2 · x − 1 a Determine the expression of the function g derivative of the function g . b Determine the value of g ( − 1) . c Consider the table of variations below : Justify that this table of variations cannot be the table of variations of the function g . E.11617 On considère la fonction g définie sur l’intervalle − 4 ; 3 par l’expression : g ( x ) = − x 2 + 2 · x − 1 1 Déterminer l’expression de la fonction g dérivée de la fonction g . 2 Déterminer la valeur de g ( − 1) . 3 On considère le tableau de variations ci-dessous : Justifier que ce tableau de variations ne peut pas être le tableau de variations de la fonction g . 10. Study of variations E.7603 Method : Let f be a function defined on an interval a ; b , whose derivative function we will denote by f : If f is positive on a ; b , then the function f is increas-ing : For x ∈ a ; b ; f ( x ) 0 = ⇒ , f is increasing. If f is negative on a ; b , then the function f is de-creasing : For x ∈ a ; b ; f ( x ) 0 = ⇒ , f is decreasing. Thus, if the function f has the following sign table : x a ¸ ˛ b f ( x ) − 0 + 0 − then the function f admits the table of variations : 1 Consider the function f defined on the interval 0 ; 10 defined by: f ( x ) = x 2 − 12 · x − 10 a Determine the expression of the function f derived from the function f . b Complete the table of signs below : x 0 10 f ( x ) c Complete the table of variations below : 2 Consider the function g defined on the interval − 3 ; 4 defined by: g ( x ) = − x 2 + 2 · x + 3 a Determine the expression of the function g derived from the function g https://chingmath.fr −2157114xVariationdef chapExoCorrec/7602 sacados/7602 −114.........xVariationde ... −203.........xVariationde ... chapExoCorrec/11617 sacados/11617 −203.........xVariationde ... chapExoCorrec/7603 sacados/7603 010xVariationdef 010xVariationdef
−34xVariationdeg 010xVariationdef b Complete the sign table below : x − 3 4 g ( x ) c Complete the table of variations below : E.7606 1 Consider the function f defined on the interval − 4 ; 2 by: f ( x ) = − 2 · x 2 − 4 · x + 3 a Give the expression of the function f derivative of the function f . b Study the sign of the function f on the interval − 4 ; 2 . c Draw up the table of variations of the function f on − 4 ; 2 . 2 Consider the function g defined on the interval 0 ; 5 by: g ( x ) = 6 · x 2 − 3 · x + 3 a Give the expression for the function g derivative of the function g . b Study the sign of the function g on the interval 0 ; 5 . c Draw up the table of variations of the function g on 0 ; 5 . E.7708 Consider the function f defined on the interval − 3 ; 3 by: f ( x ) = 2 · x 2 − 3 · x + 3 1 Give the expression of the function f derivative of the function f . 2 Study the sign of the function f on the interval − 3 ; 3 . 3 Draw up the table of variations of the function f on − 3 ; 3 . E.11634 Method : Let f be a function defined on an interval a ; b , whose derivative function we will denote by f : If f is positive on a ; b , then the function f is increas-ing : For x ∈ a ; b ; f ( x ) 0 = ⇒ , f is increasing. If f is negative on a ; b , then the function f is de-creasing : For x ∈ a ; b ; f ( x ) 0 = ⇒ , f is decreasing. Thus, if the function f has the following sign table : x a ¸ ˛ b f ( x ) − 0 + 0 − then the function f admits the table of variations : 1 Consider the function f defined on the interval 0 ; 10 defined by: f ( x ) = x 2 − 12 · x − 10 a Determine the expression of the function f derived from the function f . b Complete the table of signs below : x 0 10 f ( x ) c Complete the table of variations below : 2 Consider the function g defined on the interval − 3 ; 4 defined by: g ( x ) = − x 2 + 2 · x + 3 a Determine the expression of the function g derived from the function g b Complete the sign table below : x − 3 4 g ( x ) c Complete the table of variations below : https://chingmath.fr −34xVariationdeg chapExoCorrec/7606 sacados/7606 chapExoCorrec/7708 sacados/7708 chapExoCorrec/11634 sacados/11634 010xVariationdef 010xVariationdef
−34xVariationdeg Temps de vol(en dixième de seconde)02468101214161820222426Hauteur(en mètre)1020304050 E.11635 1 Consider the function f defined on the interval − 4 ; 2 by: f ( x ) = − 2 · x 2 − 4 · x + 3 a Give the expression of the function f derivative of the function f . b Study the sign of the function f on the interval − 4 ; 2 . c Draw up the table of variations of the function f on − 4 ; 2 . 2 Consider the function g defined on the interval 0 ; 5 by: g ( x ) = 6 · x 2 − 3 · x + 3 a Give the expression for the function g derivative of the function g . b Study the sign of the function g on the interval 0 ; 5 . c Draw up the table of variations of the function g on 0 ; 5 . 11. Problems E.7709 During a fireworks festival, a pyrotechnician prepares to launch rockets from a platform lo-cated at a height of 8 meters. He has two types of rockets, labeled A and B . Part A The height, in meters, reached by rockets of type A as a func-tion of their flight time x , in tenths of a second, is modeled by the curve below. Answer the following two questions with the accuracy allowed by the graph. 1 What height will the rocket reach after 0.7 seconds of flight? 2 For safety reasons, the rocket must explode at an altitude greater than 40 meters. Determine the time interval to which x must belong in order to satisfy this constraint. Part B We model the height, in meters, reached by B type rockets as a function of their flight time x , in tenths of a second, by the function f defined for any real number x belonging to the interval 0 ; 20 : f ( x ) = − 0.5 · x 2 + 10 · x + 8 As in the case of type A rockets, type B rockets must explode when they are at an altitude greater than or equal to 40 me-ters. We are trying to determine the interval in which x must be located to satisfy this constraint. 1 a Show that to satisfy the constraint, x must be the solution to the inequality: − 0.5 · x 2 +10 · x − 32 0 . b Draw up the sign table for the function that associates x with − 0.5 · x 2 +10 · x − 32 on the interval 0 ; 20 and then answer the question. 2 a For any real number x in the interval 0 ; 20 , calcu-late f ( x ) , where f is the derivative of f . b The pyrotechnician wants to know the slope of the tan-gent at the point with abscissa 0 on the curve repre-senting f . Give the slope coefficient sought. 3 For aesthetic reasons, the pyrotechnician wants to det-onate his B rockets when they reach their maximum height. How long before detonation should he program the flight time? https://chingmath.fr −34xVariationdeg chapExoCorrec/11635 sacados/11635 chapExoCorrec/7709 sacados/7709 Temps de vol(en dixième de seconde)02468101214161820222426Hauteur(en mètre)1020304050
Distance en mètres0123456Hauteur en mètres12345CfJPAB E.7711 We are interested in the trajec-tory of a basketball thrown by a player facing the backboard. This trajectory is modeled in the coordinate system below. In this coordinate system, the x-axis corresponds to the line passing through the player’s feet and the base of the back-board, and the unit on both axes is the meter. We assume that the initial position of the ball is at point J and that the position of the basket is at point P . The trajectory of the ball is represented by the curve C rep-resenting a function f . The coordinates of the ball are therefore ( x ; f ( x )) . 1 Graphical study Using the figure above, answer the following questions : a What is the height of the ball when x =0.5 m ? b Does the ball reach the height of 5.5 m ? 2 Study of the function f The function f is defined on the interval 0 ; 6 by: f ( x ) = − 0.4 · x 2 + 2.2 · x + 2 a Calculate f ( x ) where f is the derivative of the func-tion f . b Study the sign f ( x ) and deduce the table of variations of f on the interval 0 ; 6 . c What is the maximum height reached by the ball dur-ing this throw? 3 Modification of the throw In reality, the backboard, represented by segment [ AB ] in the figure above, is located at a distance of 5.3 m from the player. Point A is at a height of 2.9 m and point B is at a height of 3.5 m . The player decides to modify his throw to try to bounce the ball off the backboard. He then makes two successive shots. In the first shot, the trajectory of the ball is modeled by the function g defined on the interval 0 ; 6 by: g ( x ) = − 0.2 · x 2 + 1.2 · x + 2 In the second throw, the trajectory of the ball is modeled by the function h defined on the interval 0 ; 6 by: h ( x ) = − 0.3 · x 2 + 1.8 · x + 2 For each of these two throws, determine whether or not the ball bounces off the backboard. E.7710 In 2012 , the manager of a beach-front brasserie offers a lunch menu for 9.80 e . At this price, he serves an average of 420 covers per week. This formula is so successful that he decides to increase his price in subsequent summers. He observes a slight decrease in the number of covers, but his formula remains profitable. 1 The average weekly number of covers based on the price x of the menu is : N ( x )= − 19 · x +604 The price x of the menu is expressed in euros. a Calculate the average weekly number of covers when the menu price is 11 e . b Calculate the weekly turnover achieved by the brasserie when the menu is priced at 11 e . c Let C ( x ) be the weekly turnover in euros for a menu price of x euros. Show that : C ( x )= − 19 · x 2 +604 · x . 2 Consider the function c defined on the interval 0 ; 25 by: C ( x )= − 19 · x 2 +604 · x a Determine the expression of the derivative function C of C . b Give the sign of C ( x ) on the interval 0 ; 25 . c Draw up the table of variations of the function C on the interval 0 ; 25 . 3 a At what menu price is the brewery’s weekly turnover at its highest? Round your answer to two decimal places. b At this price, what is the weekly turnover of the brasserie? Round your answer to the nearest whole number. https://chingmath.fr chapExoCorrec/7711 sacados/7711 Distance en mètres0123456Hauteur en mètres12345CfJPAB chapExoCorrec/7710 sacados/7710 Extrait Antilles-Guyane Juin 2017
123456789101112ABCDxRecetteCoûtBéné∏ce0020-2010299018528052030405060708090100 E.7712 A company manufactures a model of wooden furniture. It can produce a maximum of 100 pieces of furniture per day. For x piece of furniture manufactured and sold, the daily production cost (expressed per day) , noted C ( x ) , is given by: C ( x ) = 2.25 · x 2 − 6 · x + 20 Each piece of furniture is sold 299 e . The company is open five days a week. The company manager has created the following spreadsheet : 1 a Provide a formula that, when entered in cell B2 , , allows you to obtain the revenue based on the number of pieces of furniture manufactured and sold each day by copying down. b Provide a formula that, when entered in cell C2 , , al-lows you to obtain, by copying down, the cost based on the number of pieces of furniture manufactured and sold each day. c Calculate the values associated with cells B7 , C7 et D7 . 2 Show that the daily profit corresponds to the production and sale of x pieces of furniture ( x ∈ 0 ; 100 ) is given by: B ( x ) = − 2.25 · x 2 + 305 · x − 20 3 Calculate B ( x ) and give the table of variations of B on 0 ; 100 . 4 How many pieces of furniture must be produced and sold to achieve maximum daily profit? Determine the maximum profit the company can make over a four-week period? https://chingmath.fr chapExoCorrec/7712 sacados/7712 Antilles-Guyane Juin 2015 123456789101112ABCDxRecetteCoûtBéné∏ce0020-2010299018528052030405060708090100