Grade 11 / Exponential 89 exercises (100% corrected)

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ChingQuizz : 5 exercises available for Quizz assessment : 1. Introduction (tangent fields) E.3580 Consider a function f verifying the two conditions below : f (0) = 1 ; f ( x ) = f ( x ) for all x ∈ R Justify that the curves below are not the curve C f represen-tative of the function f . E.8547 In a plane equipped with a coordinate system, a ˇ tangent field ı is a set of vectors associated with each point in the plane. r105-0 A curve in a tangent field is a curve for which, at each point, the vector associated with the tangent field is a di-rection vector of its tangent. The tangent field below is constructed on the relation f ( x )= x : at the point with abscissa 2 , the curve has a tangent with slope 2 : 1 In the plane equipped with a coordinate system O ; −→ i ; −→ j , consider the four vectors shown below : Consider a function f that is differentiable on R and let C f be its representative curve a The function f passes through the point A 1 , and the tangent ( T ) to the curve C f at A 1 has the vector −→ u 1 as its direction vector. Give the slope of the tangent ( T ) . The function f passes through point A 2 and the tan-gent ( T ) to the curve C f at A 2 has vector −→ u 2 as its direction vector. Give the slope of the tangent ( T ) . The function f passes through the point A 3 and the curve C f has a tangent at A 3 , whose direction vector https://chingmath.fr chapExoCorrec/3580 sacados/3580 -4-2024246ijC1 -6-4-202246ijC2 chapExoCorrec/8547 sacados/8547 r105-0 122 -2-1I2JOA6−u6A5−u5A2−u2A1−u1A4−u4A3−u3
is −→ u 3 . Give the derivative of the function f at 0 ; 7 . b Which of the proposed relationships can be conjectured for the function f ? f ( x )= x f ( x )=2 · x f ( x )= f ( x ) f ( x )=2 · f ( x ) c For all k ∈{ 1 ; 2 ; ::: ; 6 } , the function f passes through the point A k and has a tangent whose direction vector is −→ u k . Draw a possible representation of the curve C f . 2 Below is a tangent field in the plane with a coordinate system : a Draw the representation of the curve C g of a function g that satisfies this tangent field and passes through the point B . b Draw the graph of the curve C h of a function h that satisfies this tangent field and passes through the point C . E.8403 In the plane provided with a refer-ence frame O ; I ; J , consider the curve C f of a function f defined and derivable on R shown below : On the interval − 4 ; 1 : On the interval − 2 5 ; 9 5 : 1 a Draw the tangent to the curve C f at the point of abscissa 0 and give the slope of this tangent. b Compare the numbers f (0) and f (0) . 2 Similarly, compare the numbers f ( − 1) and f ( − 1) . 3 Graphically, establish the equality below : f (1) = f (1) ; f 1 2 = f 1 2 . 2. Introduction (Euler method) E.8402 Consider the plane provided with an orthonormal reference frame O ; I ; J : 1 Consider the function g , piecewise affine, defined on R and whose representative curve C g is given below : Establish the equalities below : a g ( − 1) = g ( − 1) b g (0) = g (0) c g (1) = g (1) The following video shows that there are an infi-nite number of piecewise linear functions verifying these three properties. 2 Consider the function h defined on R and whose repre-sentative curve C h is given below : Establish the equalities below : a h ( − 1) = h ( − 1) b h (0.5) = h (0.5) c h (1) = h (1) https://chingmath.fr BC chapExoCorrec/8403 sacados/8403 -4-3-2-1IJOCf I234JOCf chapExoCorrec/8402 sacados/8402 -4-3-2-12I234JOCg r173-1 -4-3-2-12I234JOCh
La function h , shown above, verifies for any n ∈ Z such that − 8 n 4 , the relation: h n 2 = h n 2 E.8549 Part A - no 1 The function f , shown below in a reference frame, is a contin-uous function where on each of the intervals i ; i +1 ( i ∈ Z ) , the function f is an linear functionwhose slope is f ( i ) . 1 a Give the value of f (0) and the slope of the restriction of the function f on the interval 0 ; 1 b Give the value of f (1) and the slope of the restriction of the function f on the interval 1 ; 2 2 Determine the value of f ( − 1) so that f (0)=1 and the restriction of the function f on the interval − 1 ; 0 be an linear functionof directrix f ( − 1) . 3 Correctly complete the curve plot C f . Part B - no 1 / 2 The function g ,shown below in a reference frame, is a continuous linear functionwhere, on each of the intervals 1 2 · i ; 1 2 · i +1 ( i ∈ Z ) , the function g is an linear function-whose slope is g 1 2 · i . 1 a Give the value of g (0) and the directrix of the re-striction of the function g on the interval 0 ; 1 2 b Give the value of g 1 2 and the slope of the restriction of the function f on the interval 1 2 ; 1 2 Determine the value of g − 1 2 so that g (0)=1 and the restriction of the function f on the interval − 1 2 ; 0 is an linear functionwith directrix g − 1 2 . 3 Correctly complete the curve plot C g . Note: the parts A and B represent the first two steps in constructing the curve of the exponential function by Euler’s method. To see the successive stages of this construction and their ˇ convergences ı towards a single curve, you can watch this animation : E.8412 In the plane provided with a refer-ence frame O ; I ; J , consider the curve C f of a function f defined on R : 1 Below is given the curve C f on − 4 ; 1 : a Draw the tangent to the curve C f at the point of ab-scissa 0 . Compare the numbers f (0) and f (0) b Compare the numbers f ( − 1) and f ( − 1) . 2 Below, is given the curve C f on − 2 5 ; 9 5 : a Compare the numbers f (1) and f (1) . b Compare the numbers f 1 2 and f 1 2 . 3. Introduction C E.8413 Consider a function f defined on R whose representative curve C f is given in the reference frame O ; I ; J orthonormal below : https://chingmath.fr chapExoCorrec/8549 sacados/8549 -4-3-2-12I23JOCf -4-3-2-12I23JOCg r558-0 chapExoCorrec/8412 sacados/8412 -4-3-2-1IJOCf I234JOCf chapExoCorrec/8413 sacados/8413
We admit that the function f is derivable on R and verifies the three properties below : f (0) = 1 ; f ( x ) = f ( x ) f ( x ) > 0 for all x ∈ R . Conjecture : 1 Consider the point A on the curve C f of abscissa 1 . a Place the point A and draw the tangent ( T ) to the curve C f at the point A . b Give the abscissa of the point B of intersection of the tangent ( T ) with the x-axis. 2 Repeat question 1 with point B abscissa − 1 . 3 What conjecture canthe abscissa of the point of contact of a tangent to the curve and the abscissa of the point of intersection of this tangent with the abscissa axis? 4 Check your conjecture with a third point on the curve C f chosen at random. Proof : Let a be any real number. 5 Establish that the tangent ( T ) to the curve C f at the point of abscissa a admits as slope-intercept form ( T ) : y = f ( a ) · x − a + 1 6 Establish the conjecture made in question 3 . Extension: 7 Using dynamic geometry software, check whether the rep-resentative curves of the square, inverse and square root functions verify this property. E.3342 The number of atoms in a radioac-tive source tends to decrease with time. We note N ( t ) the number of nuclei at time t . Observing this phenomenon over a variation in time, noted Δ t , the number of atoms also ex-perienced a variation, noted Δ N ( t ) . The following formula has been established : Δ N ( t ) N ( t ) = − – · Δ t where – is a constant depending solely on the nature of the radioactive source observed. 1 a La half-life of Radon-220 is 56 s . Determine the value of the constant – in the case of Radon-220, rounded to 10 − 4 . b We start with a sample containing 240 g containing ap-proximately 6.02 × 10 23 noyaux radon. Determine the time to wait for the observed quantity to weigh : 120 g ; 60 g 2 a Establish the following equality: Δ N ( t ) Δ t = − – · N ( t ) b What does the quantity Δ N ( t ) Δ t represent for the func-tion N ? Assuming that the function N is derivable as a function of time t , derive the relationship : N ( t ) = − – · N ( t ) 4. Exponential functions E.5499 We admit the existence of a func-tion f defined on R verifying the two conditions : f = f ; f (0)=1 Show that this function f is unique. We’ll denote it exp . Indications : This demonstration is carried out in two stages : We show that any function verifying these two condi-tions cannot cancel. To do this, we consider the func-tion h defined by: h ( x ) = f ( x ) · f ( − x ) and we show that the function h is constant. Assuming that there are two functions f and g verify-ing these two conditions, consider the function j defined by: j ( x )= f ( x ) g ( x ) and we show that the function j is constant. More precisely: j ( x )=1 Properties used: The derivative of the product function u · v has the expression : u · v = u ( x ) · v ( x ) + u ( x ) · v ( x ) For a;b ∈ R , we have : exp( a · x + b ) = a · exp a · x + b https://chingmath.fr -4-3-2-1I23JOCf chapExoCorrec/3342 sacados/3342 chapExoCorrec/5499 sacados/5499
E.9751 Objective: The purpose of this exercise is to establish an algebraic property of the exponential function. Properties used: The exponential function is the only function that sat-isfies : f = f ; f (0) = 1 The derivative of the product function u · v is ex-pressed as : u · v = u · v + u · v For a;b ∈ R , we have : exp( a · x + b ) = a · exp a · x + b 1 To do this, we consider the function g to be differentiable and defined on R by: g ( x ) = exp x · exp − x a Show that for all x ∈ R , we have : g ( x ) = 0 b Justify that, for all x ∈ R : g ( x )=1 2 a Deduce from the previous property that the expo-nential function never becomes zero. That is, for all x ∈ R : exp( x ) = 0 b We will establish that : exp( − x ) = 1 exp( x ) ∀ x ∈ R Note: since the exponential function does not vanish and since f (0) > 0 , we deduce that the exponential function is strictly positive on R . Since the exponential function is strictly positive and sat-isfies f = f , we can deduce that the exponential function is strictly increasing on R . E.9752 Objective: The purpose of this exercise is to establish an algebraic property of the exponential function. Properties used: The exponential function is the only function that sat-isfies : f = f ; f (0) = 1 For all x ∈ R , we have : exp( − x ) = 1 exp( x ) The derivative of the product function u · v is ex-pressed as : u · v = u · v + u · v For a;b ∈ R , we have : exp( a · x + b ) = a · exp a · x + b 1 The purpose of this question is to show that for all real numbers x and y , we have : exp x + y = exp( x ) × exp( y ) To do this, we consider the function f defined by: f ( x ) = exp( x + a ) · exp( − x ) where a ∈ R a Show that for all x ∈ R , we have : f ( x ) = 0 b Show that for all x ∈ R , we have : f ( x ) = exp( a ) c Deduce that ∀ x;y ∈ R , we have : exp x + y = exp( x ) · exp( y ) 2 Deduce that ∀ x;y ∈ R , we have : exp( x − y ) = exp( x ) exp( y ) 5. With the calculator E.7474 Consider the two functions f and g defined on R by the relations : f ( x ) = 2 x − x 2 ; g ( x ) = − x + 2 In a frame of reference O ; I ; J , note C f and C g the repre-sentative curves of the functions f and g respectively. Using the calculator, determine the coordinates of the points of intersection of the curves C f and C g . E.7027 Consider the function defined on 0 ; 8 by: f ( x ) = 0 ; 4 20 · e − x + 1 + 0 ; 4 In a mountainous region, a company is studying a road project connecting villages A and B located at two different altitudes. The function f , defined in part A , models the profile of this road project. The variable x represents the horizontal dis-tance, in kilometers, from village A , and f ( x ) represents the associated altitude, in kilometers. The graphical representation C f of function f is given below. Indication : For each of the following statements, indicate whether the statement is true or false, justifying your an-swer. Statement 1 The altitude of village B is 0 ; 6 km . Statement 2 The difference in altitude between villages A and B is 378 meters, rounded to the nearest meter. 6. Introduction to algebraic properties https://chingmath.fr chapExoCorrec/9751 sacados/9751 chapExoCorrec/9752 sacados/9752 chapExoCorrec/7474 sacados/7474 chapExoCorrec/7027 sacados/7027 ABx012345678f(x0,20,40,60,81Cf
E.8414 In the plane provided with a ref-erence frame O ; I ; J orthonormal, we give the curve C f representative of the function f exponential: Conjecture : 1 With the accuracy possible by graphical reading, com-plete the table of values of the function f : x − 1.4 − 1 − 0.4 0 0.6 1 exp( x ) 2 Using the previous results, approximate the following products : exp( − 1) × exp( − 0.4) exp( − 1) × exp(0.6) exp(0) × exp(1) exp( − 0.4) × exp(1) 3 What conjecture can be made about the product exp( a ) · exp( b ) for any real a and b ? Towards the proof : We admit that the exponential function is non-zero on R and will establish only a special case of the proof : For any real number x , we have : exp 1+ x = exp(1) · exp( x ) ( ∗ ) Consider the function g defined on R by: g ( x ) = exp( x +1) exp( x ) 4 Determine the expression of the function g derived from the function g . 5 Deduce that the simplified form of the function g and establish the property ( ∗ ) . Extension: 6 Establish the property below for any x ∈ R : exp(2+ x ) = exp(2) × exp( x ) 7 What simplification of the expression exp( n ) × exp( x ) can be conjectured, for any natural number n and any real number x . E.8416 In the plane provided with a ref-erence frame O ; I ; J orthonormal, we give the curve C f representative of the function f exponential: Conjecture : 1 With the accuracy possible by graphical reading, com-plete the table of values of the function f : x − 1.9 − 1.2 − 0.8 − 0.1 0.4 0.7 1.1 exp( x ) 2 Using the previous results, give the values of the follow-ing products : exp( − 1.9) × exp(1.1) exp( − 1.2) × exp(0.4) exp( − 0.8) × exp(0.7) exp(0.4) × exp(0.7) 3 What conjecture can be made about the product exp( x ) · exp( y ) for any real x and y ? Proof : We admit that the exponential function never cancels at R . For any real number a , consider the function g a defined on R by: g a ( x ) = exp( x + a ) exp( x ) ( ∗ ) 4 Determine the expression of the function g derived from the function g . 5 Establish that for any real x , we have g a ( x )=exp( a ) , then establish the following identity for all real x and y : exp( x ) × exp( y ) = exp( x + y ) Extension: 6 Establish the property below for any x ∈ R : exp(3 · x ) = exp( x ) 3 7 Establish the property below for any x ∈ R : exp x 2 = exp( x ) https://chingmath.fr chapExoCorrec/8414 sacados/8414 -4-3-2-1I23JOCf chapExoCorrec/8416 sacados/8416 -4-3-2-1I23JOCf
E.8415 In the plane provided with a ref-erence frame O ; I ; J orthonormal, we give the curve C f representative of the function f exponential: Conjecture : 1 With the accuracy possible by graphical reading, com-plete the table of values of the function f : x − 0.7 − 0.5 − 0.1 0 0.1 0.5 0.7 exp( x ) 2 Using the previous results, give the values of the follow-ing products : exp( − 0.7) × exp(0.7) exp( − 0.5) × exp(0.5) exp( − 0.1) × exp(0.1) exp(0) × exp(0) 3 Conjecture a relationship between the numbers exp( a ) and exp( − a ) for any real number a ? Proof : Consider the function g defined on R by: g ( x ) = exp( − x ) × exp( x ) 4 Determine the expression of the function g derived from the function g . 5 Based on the value of g (0) , deduce the simplified form of the function g and establish the previous conjecture. Also justify that the exponential function never cancels at R . Extension: 6 Justify that the exponential function takes its values in R ∗ + . 7. Algebraic properties E.3589 Proposition: for any x;y ∈ R and for any n ∈ Z , we have : exp x + y = exp x × exp y exp x − y = exp x exp y exp n · x = exp x n a exp(3) · exp(5) b exp( − 2) · exp(4) c 1 exp( − 5) d exp(5) 3 E.9737 Simplify the following expressions : a e 3 − 2 · e 5 b e 6 − e 3 e · e 2 E.3610 Simplify the following expressions : a e · e 2 x +1 b e 3 − 2 x · e x +5 c e 2 · x − 1 · e 3 − x E.9738 Simplify the following entries : a exp(2 x +4) × exp(3 − x ) b exp( x ) 2 exp(3 − 2 x ) E.3590 Proposition: for any x;y ∈ R and for any n ∈ Z , we have : e x + y = e x × e y e x − y = e x e y e n · x = e x n Simplify the following expressions : a e 3 · e 4 b e 4 · e − 4 c e 4 3 · e 4 d e 5 · e − 3 e − 2 e e 5 · e 6 f e 6 · e − 2 e − 4 E.9727 Simplify the following entries : a e 3 x +1 · e 2 − 2 x b e x − 2 e 3 − x c e − x − e 2 x + 1 e x E.9736 Simplify the following expressions : a 3 · e 5 x 4 − 2 · e 10 x 2 b e 9 x − 2 · e 3 x 3 E.9726 Simplify the following expressions : a e 2 · x − 1 2 e 7 · x − 2 b 3 · e − 2 · x + e x + 1 e 2 · x E.3611 Establish the following equations : a 2 + 3 · e x + e 2 x e 2 x = 2 · e − 2 x + 3 · e − x + 1 b 1 − e x e 2 x = e − 2 x − e − x https://chingmath.fr chapExoCorrec/8415 sacados/8415 -4-3-2-1I23JOCf chapExoCorrec/3589 sacados/3589 chapExoCorrec/9737 sacados/9737 chapExoCorrec/3610 sacados/3610 chapExoCorrec/9738 sacados/9738 chapExoCorrec/3590 sacados/3590 chapExoCorrec/9727 sacados/9727 chapExoCorrec/9736 sacados/9736 chapExoCorrec/9726 sacados/9726 chapExoCorrec/3611 sacados/3611
E.3608 Simplify the following expressions : a e 5 − e 4 2 − e 5 + e 4 2 b e 2 + e − 2 · e 2 − e − 2 E.9725 Simplify the following expressions : a e 3 x 2 − e 2 x · e 2 x + e − 2 2 b e 3 x 2 + e − 3 x 2 − e 3 x − e − 3 x 2 E.6873 Copy the identities below, filling in the blanks correctly: a e x + e − x = e x · : : : + : : : b e x x 2 = e ... : : : 2 c 1 + e − x = : : : + : : : e x d 1 + e x e 2 x = : : : + : : : e e 3 x − e x e 3 x + e 2 x = 1 − : : : 1 + : : : f e 16 x = e ... 2 E.2031 Simplify the following expressions : e x +e − x 2 2 − e x − e − x 2 2 E.9724 Establish the following equations : a e 3 x + 2 e 3 x − 1 = 1 + 2 · e − 3 x 1 − e − 3 x b e 3 x − e 2 x e 3 x + e 2 x = e 2 x − 1 e x + 1 2 8. Equations E.8417 Proposition: for all real numbers a and b a = b ⇐⇒ e a =e b Solve the following equations : a e 5 x +1 = e 2 x b e 3 x +1 = 1 c e 1 − 3 x e = 1 E.3593 Solve the following equations on R : a exp( x ) = e b exp( − x ) = 1 c exp(2 x − 1) = e d e x − e − x = 0 E.8419 Solve the equations : a e x · e 2 x − e 2 = 0 b e 3 x − 1 − 1 e 2 − x − e = 0 c x · e x − x = 0 E.3616 Solve the following equations : a e x + e − x = 0 b e 3 x +1 = e − 2 x +3 c e 2 x − 1 = 0 d x · e 2 x − 2 · e 2 x = 0 E.8418 Solve the following equations : a e x 2 +1 = e x b e x +1 2 = e x 2 +1 c e x 2 +1 + e x = 0 E.9739 Solve the following equations on R : a e x 2 + x = 1 b e x 2 +5 = e x +2 2 9. Equations E.3617 Proposition: the exponential function is defined on R and has the following table of variation: Solve the following inequalities: a e x < 1 b e − x > 0 c e − x > 1 d e 2 x − 1 0 E.3594 Solve the following inequalities on R : a exp( x ) < e b exp( − x ) 1 E.9740 Proposition: For any real numbers a and b : e a > e b ⇐⇒ a > b e a < e b ⇐⇒ a <b Note: this property comes from the strict increasing of the function f . Solve the following inequalities on R : a e 2 x − 4 1 b e 2 x − 1 < e x https://chingmath.fr chapExoCorrec/3608 sacados/3608 chapExoCorrec/9725 sacados/9725 chapExoCorrec/6873 sacados/6873 fichierPlus/6873/ chapExoCorrec/2031 sacados/2031 chapExoCorrec/9724 sacados/9724 chapExoCorrec/8417 sacados/8417 chapExoCorrec/3593 sacados/3593 chapExoCorrec/8419 sacados/8419 chapExoCorrec/3616 sacados/3616 chapExoCorrec/8418 sacados/8418 chapExoCorrec/9739 sacados/9739 chapExoCorrec/3617 sacados/3617 −∞01∞01e∞xVariationdeexp chapExoCorrec/3594 sacados/3594 chapExoCorrec/9740 sacados/9740
E.9730 Solve the following inequalities: a e x − e − x > 0 b x · e − x − 3 · e − x < 0 E.8421 Solve the following inequalities: a e x 2 − 3 x +5 < e b e [(3 x +1) 2 ] < 0 E.9731 Solve the following inequalities on R : a e 2 x + 3e x < 4 b e x + e − x < 2 Hint: we will identify these inequalities with polynomial inequalities of degree 2 . 10. Relative position of curves E.4233 Let f and g be the functions defined on the interval 0 ; + ∞ by: f ( x ) = x · e − x ; g ( x ) = x 2 · e − x We note C f and C g the graphical representations of the func-tions f and g in the plane provided with a reference frame O ; −→ i ; −→ j . 1 Graphically, conjecture the relative positions of the curves C f and C g . 2 a Factorize the expression f ( x ) − g ( x ) . b Deduce the relative positions of the curves C f and C g . E.7523 A company wishes to use a decorative motif for its communication. To realize this pattern, its shape is modeled using two func-tions f and g defined for any real x of 0 ; 1 by: f ( x ) = 1 − x · e 3 x ; g ( x ) = x 2 − 2 · x + 1 1 Verify that the points A and B of coordinates (1 ; 0) and (0 ; 1) respectively are points common to the curves C f and C g . 2 We admit that :for any x in 0 ; 1 : f ( x ) − g ( x ) = 1 − x e 3 x − 1 + x a Justify that for any x in 0 ; 1 : e 3 · x − 1 0 b Deduce that for any x in 0 ; 1 : e 3 · x − 1 + x 0 c Study the sign of f ( x ) − g ( x ) for any x in 0 ; 1 . 11. Derivatives E.9728 For each function, determine the expression of the derivative function : 1 f ( x ) = 2 · e x + x 2 2 g ( x ) = 1 4 · e 3 x +1 + e − 2 x E.8133 Consider the function f de-fined, for any positive real number t by: f ( t ) = a · e − t 5 + b where a; b ∈ R We admit that f (0)=1 000 and that f verifies the relation: f ( t ) + 1 5 · f ( t ) = 4 for all t ∈ R Determine the values of a and b , and give the expression of the function f E.7479 Proposition: let a and b be two real numbers and the function f defined by: f ( x ) = e ax + b The function f , derived from f , has the expression : f ( x ) = a · e ax + b For each function, determine the expression of the function f derived from the function f : a f ( x ) = e x b f ( x ) = e 2 · x c f ( x ) = e 3 − x 12. Derivatives and study of functions https://chingmath.fr chapExoCorrec/9730 sacados/9730 chapExoCorrec/8421 sacados/8421 chapExoCorrec/9731 sacados/9731 chapExoCorrec/4233 sacados/4233 2345678IJOCfCg chapExoCorrec/7523 sacados/7523 xxyy-0,200,20,40,60,811,2-0,20,20,40,60,811,21,41,61,822,22,42,6CgCf chapExoCorrec/9728 sacados/9728 chapExoCorrec/8133 sacados/8133 chapExoCorrec/7479 sacados/7479
E.9741 Consider the function f defined by: f ( x ) = 3 · e 1 − 2 x 1 Determine the expression of the function f , derivative of the function f . Then, deduce the sign of f on R . 2 Deduce the direction of variations of the fontion f on R . E.9742 For each of the two functions be-low defined on R , determine the expression of their derivative function and their direction of variation on R : 1 f ( x ) = 3 · e 5 x +1 2 g ( x ) = 2 − 3 · e − x E.9743 For each of the functions below de-fined on R , determine the expression of their derivative func-tion, then study their direction of variation on R : 1 f ( x ) = x − e x 2 g ( x ) = 6 x + 3 · e − 2 x E.10510 Consider the function f defined on R by: f ( x ) = e 4 x − 4 − 4 · e · x Determine the expression of their derivative function, then study their direction of variation on R : E.5754 Let the functions f and g be defined on R by the relations : f ( x ) = e 1+ x + e 1 − x 2 g ( x ) = e 1+ x − e 1 − x 2 In an orthonormal frame of reference O ; I ; J , we give the curves C and C respectively representative of the functions f and g : 1 Establish that, for any real number x : f ( x ) − g ( x ) > 0 2 Establish that, for any real x , we have : f ( x ) = g ( x ) ; g ( x ) = f ( x ) 3 Consider a any real number: a Justify that the slope-intercept formof the tangent ( T ) to the curve C at the point of abscissa a has the ex-pression : y = g ( a ) · x − a + f ( a ) b Justify that the slope-intercept formof the tangent ( T ) to the curve C at the point of abscissa a has the ex-pression : y = f ( a ) · x − a + g ( a ) 4 Justify that the tangents ( T ) and ( T ) are secant and deduce the abscissa of the point of intersection. E.5990 Let the functions f and g be de-fined on R by the relations : f ( x ) = e 1+ x + e 1 − x 2 g ( x ) = e 1+ x − e 1 − x 2 In an orthonormal frame of refer-ence O ; I ; J , we give the curves C and C respectively representa-tive of the functions f and g : Part A : study of the relative position of the two curves 1 Demonstrate that the C curve always lies above the C curve. Part B: study of a geometric locus Let a be any real number. Consider : the tangent ( T ) to the curve C at the point of abscissa a ; the tangent ( T ) to the curve C at the point of abscissa a ; We admit that the straight lines ( T ) and ( T ) are never par-allel. We note M their point of intersection. 2 a Give the expression of the slope-intercept formof the tangent ( T ) as a function of a . b Give the expression of the reduced tangent equation ( T ) as a function of a . 3 Determine the abscissa of the point at M . 4 a Determine the coordinates of M . b Justify that the point M belongs to the curve of one of the reference functions to be specified. E.4231 Let f be the function defined on R by: f ( x ) = e x The representative curve of the function f in an orthonormal reference frame O ; −→ i ; −→ j is called C f . 1 Let a be a real number. Show that the tangent to the curve C f at the point M of abscissa a intersects the x-axis at the point P of abscissa a − 1 . 2 Let N be the orthogonal project of point M onto the abscissa axis. Show that : −−→ NP = −−→ i 13. Derivatives of a product E.7480 Proposition: let f be a function defined on an interval I by the product : f ( x ) = u ( x ) × v ( x ) where the functions u and v are defined and derivable over I . Then the function f is derivable on I and its derivative func-tion is defined by: f ( x ) = u ( x ) · v ( x ) + u ( x ) · v ( x ) For each function, determine the expression of the function f derived from the function f : a f ( x ) = x · e x b f ( x ) = 1 − 2 · x · e x E.9744 For each function, determine the expression of the function f derived from the function f : a f ( x ) = 3 − x · e x b g ( x ) = x + 1 e x https://chingmath.fr chapExoCorrec/9741 sacados/9741 chapExoCorrec/9742 sacados/9742 chapExoCorrec/9743 sacados/9743 chapExoCorrec/10510 sacados/10510 chapExoCorrec/5754 sacados/5754 chapExoCorrec/5990 sacados/5990 Epreuve pratique - 2009 IJOCCM chapExoCorrec/4231 sacados/4231 chapExoCorrec/7480 sacados/7480 chapExoCorrec/9744 sacados/9744
E.9733 Consider the function f defined by: f ( x ) = (2 · x − 1) · e x Give the definition set of the function f and the expression of the function f derivative of the function f . Hint: we’ll give the expression of f in factorized form. E.7431 Consider a function defined and derivable on the interval − 3 ; 2 . We denote f the deriva-tive function of the function f . We give the following infor-mation about the function f : f (0) = 3 ; f (1) = 0 ; f (0) = 0.5 We admit that there exist three real a , b , c for which the function f defined above is defined, for any x of − 3 ; 2 , by: f ( x ) = a · x 2 + b · x + c · e x + 5 . 1 Using one of the previous pieces of information, justify that c = − 2 . 2 Assume that the derivative function f is given, for any real x from − 3 ; 2 , by: f ( x ) = a · x 2 + 2 · a + b · x − 2 + b · e x Using the previous information, justify that b =2.5 then that a = − 1 . 14. Derivatives of a product and linear composition E.3592 Determine the expression of the fol-lowing derivative functions : E.9745 Determine the expression of the fol-lowing derivative functions : 1 h ( x ) = x · e x +1 2 j ( x ) = x 2 + 1 · e 3 x +1 E.7032 Consider the function f de-fined on R by: f ( x ) = x + 1 · e − 2 x +3 . Which of the following four statements is true? The function f is derivable on R and its derivative function f is given by: a f ( x ) = − 2 · e − 2 x +3 b f ( x ) = e − 2 x +3 c f ( x ) = − 2 x +3 e − 2 x +3 d f ( x ) = − 2 x − 1 e − 2 x +3 E.7522 Let f be a function defined on the interval 0 ; 5 by: f ( x ) = a · x − 2 · e − x où a is a real number. We admit that the function f is twice derivable on the interval 0 ; 5 . Recall that f denotes the derivative function of the function f and admit that : f (0) = − 2 ; f (0) = 10 1 Show that for any real x in the interval 0 ; 5 , we have : f ( x ) = − a · x + a + 2 · e − x 2 Deduce from previous questions that a =8 . 3 Give the expression of f ( x ) . 15. Derivatives of a product and study of functions E.5571 Let f be the function de-fined and derivable on the set of real numbers R such that : f ( x ) = ( x + 1) · e x 1 Using the calculator, conjeture the limits of f in + ∞ and in −∞ . 2 Let f be the derivative function of the function f on R . Show that for any real x : f ( x ) = ( x + 2)e x 3 Draw up the table of variations of f at R . E.9849 Consider the function f defined on R by: f ( x ) = − 6 · x 2 + 5 · x · e x 1 Establish that the function f , derived from the function f , has the expression : f ( x ) = − 6 · x 2 − 7 · x + 5 · e x 2 Draw up the sign table for the function f . 3 Give the variations of the function f on R . Hints: image values and boundary limits are not requested. https://chingmath.fr chapExoCorrec/9733 sacados/9733 chapExoCorrec/7431 sacados/7431 Extrait Asie Juin 2017 chapExoCorrec/3592 sacados/3592 chapExoCorrec/9745 sacados/9745 chapExoCorrec/7032 sacados/7032 chapExoCorrec/7522 sacados/7522 chapExoCorrec/5571 sacados/5571 Extrait Antilles-Guyanes Juin 2013 chapExoCorrec/9849 sacados/9849
E.9795 Let f be the function defined at R by f ( x ) = (2 x + 1) · e x . On the graph below is plotted the curve C f representative of the function f . 1 Determine the coordinates of any points of intersection of the curve C f with the x-axis. 2 Show that, for any real x , that : f ( x ) = 2 x + 3 e x 3 Draw up the sign table for f ( x ) at R , then specify the variations of f at R . 4 Consider the straight line ( T ) tangent to the curve C f at the point of abscissa 0 : a Determine the slope-intercept formof the tangent ( T ) . b Draw in the reference frame below the tangent ( T ) . c Justify graphically that, for any real x , we have : 2 x +1 e x 3 x +1 5 We note C g the representative curve of the function g defined by: g ( x ) = 2 x + 1 e 1 − x Study the relative positions of the curves C f and C g . 16. Derivatives of a product, linear composition and study of functions E.7524 We admit that the func-tion f is defined, for any real x of the interval − 2 ; 4 by: f ( x ) = x + 2 · e − x +1 Let f be the derivative function of f . 1 Show that, for any x in the interval − 2 ; 4 , we have : f ( x ) = − x + 1 · e − x +1 2 Study the sign of f ( x ) on the interval − 2 ; 4 , then draw up the table of variations of f on this interval. E.7518 We admit that the function f is defined by: f ( x ) = x 2 − 2 · x + 1 · e − 2 · x +6 1 Monter que f ( x )= − 2 · x 2 +6 · x − 4 · e − 2 · x +6 , où f de-notes the function derived from the function f . 2 Study the direction of variation of the function f on the interval 0.7 ; 6 and draw up the table of variations of the function f on the interval 0.7 ; 6 . Calculation of ordinates is not required. E.7571 Consider the function f de-fined on R whose representative curve C f is plotted below in an orthonormal frame. It is assumed that f is of the form f ( x )= b − x · e a · x où a and b denote two constants. We know that : The points A (0 ; 2) and D (2 ; 0) belong to the curve C f . The tangent to the curve C f at point A is parallel to the x-axis. Let f be the derivative function of f , defined on R . 1 By graphical reading, indicate the values of f (2) and f (0) . 2 Calculate f ( x ) . 3 Using the previous questions, show that a and b are so-lutions of the following system : b − 2 = 0 a · b − 1 = 0 4 Calculate a and b and give the expression for f ( x ) . https://chingmath.fr chapExoCorrec/9795 sacados/9795 -4-3-2-12I-1234JOCfCg chapExoCorrec/7524 sacados/7524 chapExoCorrec/7518 sacados/7518 chapExoCorrec/7571 sacados/7571 x-8-7-6-5-4-3-2-101234y-4-3-2-1123Cf
E.9848 The graph below represents, in a reference frame C f and C g functions f and g defined on R by: f ( x ) = x 2 · e − x ; g ( x ) = e − x 1 a Determine the coordinates of the intersection points of C f and C g . b Study the relative position of the curves C f and C g . 2 For any real number x from the interval − 1 ; 1 , consider the points M of coordinates ( x ; f ( x )) and N of coordi-nates ( x ; g ( x )) , and we denote d ( x ) the distance MN . We assume that : d ( x )=e − x − x 2 · e − x . We admit that the function d is derivable on the interval − 1 ; 1 and we note d its derivative function. a Show that : d ( x ) = e − x x 2 − 2 x − 1 b Deduce the variations of the function d on the interval − 1 ; 1 . c Determine the common abscissa x 0 of the points M 0 and N 0 allowing to obtain a maximum distance d ( x 0 ) , and give an approximate value to the nearest 0.1 of the distance M 0 N 0 . 17. Derivatives of a quotient E.3612 Proposition: let f be a function defined on an interval I by the product : f ( x ) = u ( x ) v ( x ) where the functions u and v are defined and derivable over I . Then the function f is derivable on I and its derivative func-tion is defined by: f ( x ) = u ( x ) · v ( x ) − u ( x ) · v ( x ) v ( x ) 2 Determine the expression of the derivative functions of each of the following functions : 1 f ( x ) = 1 1 − e x 2 g ( x ) = e x +1 2 · x + 1 E.8431 For each row, the table below gives the expression of the derivative function f of the function f : f ( x ) = e x x + 1 f ( x ) = x · e x x + 1 2 f ( x ) = 2 · x + 1 e x f ( x ) = − 2 · x + 1 e x f ( x ) = 3 · e x + 1 e x − 1 f ( x ) = − 4 · e x e x − 1 2 Check the veracity of each of the expressions f given. 18. Derivatives of a quotient and linear composition E.9732 Determine the expression of the derivative functions of each of the following functions : 1 h ( x ) = 1 − e − 2 x e x 2 j ( x ) = 1 − e − 2 x 1 + e 2 x E.7478 1 Consider the function f defined on R by: f ( x ) = 1 0.5 + 100 · e − x Let f be the derivative function of f on R . Show that, for any real x belonging to R , we have : f ( x ) = 400 · e x e x + 200 2 2 Consider the function g defined on R by: g ( x ) = 100 · e − x 0.5 + 100 · e − x Let g be the derivative function of g on R . Show that, for any real x belonging to R , we have : g ( x ) = − 200 · e x e x + 200 2 19. Derivatives of a quotient and study of functions https://chingmath.fr chapExoCorrec/9848 sacados/9848 Extrait Mars 2021 x-2-1012y2468CfCfMN chapExoCorrec/3612 sacados/3612 chapExoCorrec/8431 sacados/8431 chapExoCorrec/9732 sacados/9732 chapExoCorrec/7478 sacados/7478
E.9734 Consider the function f defined on R ∗ + by: f ( x ) = e x x 1 Show that the function f , derived from the function f , can be expressed as : f ( x ) = x − 1 · e x x 2 2 a On R ∗ + , draw up the sign table for the function f . b On R ∗ + , deduce the variation table for the function f . Note: both limits are accepted : lim x ↦→ 0 + f ( x ) = + ∞ ; lim x ↦→ + ∞ f ( x ) = + ∞ E.10501 Consider the function f defined on R by: f ( x ) = x − 8 − 4e x e x + 1 1 Establish that the function f derived from the function f is expressed as : f ( x ) = e x − 1 2 e x + 1 2 2 Deduce the variations of the function f . E.3225 Consider the function f de-fined on R by: f ( x )= x e x − x Note ( C ) its representative curve in the plane referred to the orthogonal reference frame O ; −→ i ; −→ j , the graphical unit is 2 cm on the x-axis and 5 cm on the y-axis. Part A Let g be the function defined on R by: g ( x )=e x − x − 1 . 1 Study the variations of the function g on R . Deduce the sign of g . 2 Justify that for any x , (e x − x ) is strictly positive. Part B 1 a Calculate f ( x ) , f denoting the derivative function of f . b Study the directions of variation of f . 2 a Determine an equation of the tangent ( T ) to the curve ( C ) at the point of abscissa 0. b Using part A , investigate the position of the curve ( C ) relative to the straight line ( T ) . E.10503 Consider the function f defined on R by: f ( x ) = 3 x + 1 − 12 · e x e x + 1 1 Let f be the derivative function of the function f . Show that the function f admits an expression of the form : f ( x ) = 3 · g ( x ) 2 where g is a function defined on R . 2 Deduce the direction of variation of the function f on R . E.10504 Consider the function f defined on R by: f ( x ) = 2 x + 1 − 16 · e x 2 · e x + 1 Determine the variations of the function f . 20. Derivatives of a quotient, linear composition and study of functions E.10509 Consider the function f defined on R \{ 1 } by: f ( x ) = e 2 x x + 1 1 Establish that : f ( x ) = 2 x + 1 · e 2 x x + 1 2 2 Draw up the table of variations of the function f . 21. Towards differential equations E.8408 Consider the function f defined on R by the relation: f ( x ) = e 2 x +3 1 Show that, for any real number x , we have the relation: 2 · f ( x ) − f ( x ) = 0 2 a Which of the following expressions of a function g verifies the relation ( ∗ ) : g ( x ) = e 2 x +3 + 4 g ( x ) = e 8 x +12 g ( x ) = 4 · e 2 x +3 g ( x ) = e − 2 x − 3 b Give the expression of a third function h verifying the relation ( ∗ ) . E.9735 Consider the function f defined on R by: f ( x ) = x · e x 1 a Show that the function f , derived from the function f , by: f ( x ) = x + 1 · e x b Determine the expression of the function f  , derivative of the function f . 2 Consider the function g defined by: g = f  − 2 · f + f Justify that the function g is the null function. https://chingmath.fr chapExoCorrec/9734 sacados/9734 chapExoCorrec/10501 sacados/10501 chapExoCorrec/3225 sacados/3225 chapExoCorrec/10503 sacados/10503 chapExoCorrec/10504 sacados/10504 chapExoCorrec/10509 sacados/10509 chapExoCorrec/8408 sacados/8408 chapExoCorrec/9735 sacados/9735
E.8409 Consider the function f defined on R by the relation: f ( x ) = x + 1 · e 2 x Show that the function f verifies the relation: f  ( x ) − 4 · f ( x ) + 4 · f ( x ) = 0 E.8411 Consider the function f defined on R by the relation: f ( x ) = − x − 1 · e x Show that the function f verifies for any real number x : f  ( x ) − 3 · f ( x ) + 2 · f ( x ) = e x 22. Variations and differentiations E.8369 Consider the function f defined on R by the relation: f ( x ) = e x e x + 1 1 Determine the expression of the function f derived from the function f . 2 Draw up the table of variations of the function f on the interval 0 ; 10 . E.8372 Consider the function f defined on R by the relation: f ( x ) = 2 · e x − 1 e 2 x 1 Determine the expression of the function f derived from the function f . 2 Draw up the table of variations of the function f on the interval − 1 ; 10 . E.8370 Consider the function f defined on R by the relation: f ( x ) = e x e 2 x + 1 1 Determine the expression of the function f derived from the function f . 2 Draw up the table of variations of the function f on the interval − 1 ; 10 . E.8371 Consider the function f defined on R ∗ + by the relation: f ( x ) = e 2 x − 2 e x − 1 1 Determine the expression of the function f derived from the function f . 2 Draw up the table of variations of the function f on the interval 1 ; 10 . 23. Exponential functions and sequences E.288 Let a be any real number. Con-sider the sequence u n defined by: u 0 = a ; u n +1 = e 2 u n − e u n for any n ∈ N . Note that this equality can also be written as : u n +1 = e u n · e u n − 1 . Consider the function g defined for any real x by: g ( x ) = e 2 x − e x − x 1 Calculate g ( x ) and prove that, for any real x : g ( x ) = e x − 1 2 · e x + 1 2 Determine the variations of the function g and give the value of its minimum. 3 Noting that u n +1 − u n = g u n , study the direction of vari-ation of the sequence u n . 24. Unclassified financial years E.3340 Let f be a function defined on R verify-ing the relation: f ( x ) = x for all x ∈ R 1 Give at least two functions that verify this relationship. 2 The tangent field shown below is proposed : https://chingmath.fr chapExoCorrec/8409 sacados/8409 chapExoCorrec/8411 sacados/8411 chapExoCorrec/8369 sacados/8369 chapExoCorrec/8372 sacados/8372 chapExoCorrec/8370 sacados/8370 chapExoCorrec/8371 sacados/8371 chapExoCorrec/288 sacados/288 chapExoCorrec/3340 sacados/3340 -4-3-2-1234I2345JO
a Verify that each tangent represented on the line with equation x =2 has slope 2 . b Verify that for each tangent having for origin the coor-dinate point ( x ; y ) , its slope is x . 3 Now consider the function f that verifies the following two conditions : f (0) = 3 2 ; f ( x ) = x for all x ∈ R Draw the curve C f representative of the function f . https://chingmath.fr