Grade 11 / Derivative numbers 79 exercises corrected

a
ChingQuizz : 10 exercises available for Quizz assessment : 1. Reminders: manipulating regular expressions E.8366 Establish the identities below : a 2 x + 5 x + 1 = 7 · x + 2 x · x + 1 b 3 x + 3 − 2 5 · x − 1 = 13 x − 9 x + 3 5 x − 1 E.6002 Establish the following equations : a x x + 1 − 2 x − 3 x − 1 = 3 − x 2 x − 1 · x + 1 b x + 1 3 x − 2 − x + 2 3 x + 1 = 5 3 x − 2 3 x + 1 E.10592 Simplify the expressions below into quotient form where : the numerator is expanded and reduced the denominator is written as a product. a x 2 · x − 1 + 4 x + 1 b x + 3 x 2 + 1 − 4 x + 3 E.10588 Establish the following equality: 1 x + 1 + 2 2 x − 1 x = 4 x + 1 x · ( x + 1) · (2 x − 1) E.10593 Establish the following equations : a 1 x + 4 − 1 x + 4 x 2 = 16 x 2 · x + 4 b 1 x + 1 x + 1 + 1 ( x + 1) 2 = 2 x 2 + 4 x + 1 x · x + 1 2 E.6003 Express each of the algebraic ex-pressions below as a quotient où where the numerator and denominator are products of factors of the first degree : a 3 x x + 2 + x + 2 1 − x b x 2 + x + 3 x + 1 − x 2 + 4 x x + 2 2. Preliminary work on obtaining derived numbers E.8368 Consider the function f defined on R by: f ( x ) = x 2 + x + 1 Establish the following equalities: a f (1+ ı ) = ı 2 + 3 · ı + 3 b f (2+ h ) = h 2 + 5 · h + 7 où h ∈ R E.6060 Let f be the function defined by the relation: f ( x ) = x 2 − 3 · x + 2 a Establish, for any h ∈ R , the identity: f (2+ h ) = h 2 + h b For h ∈ R , determine a simplified expression for f (1+ h ) . E.9552 Consider the function g defined on R by: g ( x ) = x + 1 x 2 + 1 For h ∈ R , establish the following equality: g (1+ h ) − g (1) = − h · h + 1 h 2 + 2 · h + 2 E.10587 Let f be the function defined on 2 ; + ∞ by the relation: f ( x ) = x − 2 + 3 1 Consider the expression A defined by: A = f (4+ h ) − f (4) a Justify that, for h = − 3 , the expression A is not de-fined. b Justify that, for any h ∈ − 3 2 ; 3 2 , the expression A is defined. 2 For any h ∈ − 3 y 2 ; 3 2 , establish the equality: A = h h + 2 + 2 E.8367 Let f be the function defined on 1 2 ; + ∞ by the relation: f ( x ) = 2 x − 1 − 3 Establish , for any h ∈ − 3 2 ; + ∞ , the equality: f ( h +2) − f (2) = 2 · h 2 h + 3 + 3 3. Approach to tangents and directrix of affine functions https://chingmath.fr quizzChap/937 chapExoCorrec/8366 sacados/8366 chapExoCorrec/6002 sacados/6002 chapExoCorrec/10592 sacados/10592 chapExoCorrec/10588 sacados/10588 chapExoCorrec/10593 sacados/10593 chapExoCorrec/6003 sacados/6003 chapExoCorrec/8368 sacados/8368 chapExoCorrec/6060 sacados/6060 chapExoCorrec/9552 sacados/9552 chapExoCorrec/10587 sacados/10587 chapExoCorrec/8367 sacados/8367
E.2291 Proposition: let f be an affine function whose represen-tative curve passes through the two points A ( x A ; y A ) and B ( x B ; y B ) . The directing coefficient m of the function f has the value : m = y B − y A x B − x A Determine the directing coefficients of the straight lines ( d 1 ) and ( d 2 ) shown below : E.9560 Determine the slopes of the straight lines ( d 1 ) and ( d 2 ) shown below : E.4713 Consider the function f defined by the relation is : f ( x ) = 1 4 · x 2 − 1 2 · x − 2 In the plane provided with an orthonormal reference frame O ; I ; J , note C f the representative curve of the function f : 1 a Draw the line ( d ) whose equation is : y = 1 2 · x − 3 b Draw the line (Δ) whose equation is : y = − 3 2 · x − 3 2 What feature do the straight lines ( d ) and (Δ) have rel-ative to the curve C f ? E.5937 Consider the function f defined on R by the relation: f ( x ) = x 2 − 2 · x − 1 The plane is given an orthonormal coordinate system O ; I ; J . The curve C f representative of f is given below. Note ( d ) and (Δ) respectively, the tangents to the curve C f at the point of abscissa 1 2 and 2 . 1 Determine the coordinates of the points on the curve C f having 1 2 and 2 as abscissa, respectively. 2 a Graphically, give the slope-intercept formof the line ( d ) . b Determine the slope-intercept formof the line (Δ) . E.6878 The curve C representative of a function in a reference frame 0 ; I ; J is given below : 1 a Draw the tangent ( d ) to the curve C at the point of abscissa 4 . b Give the slope of the line ( d ) . 2 a Draw the tangent (Δ) to the curve C at the point of abscissa 1 . b Give the slope of the line (Δ) . 4. Tangent fields E.4654 Below is represented, in the reference frame ( O ; I ; J ) , the curve C f and four of its tangents : https://chingmath.fr chapExoCorrec/2291 sacados/2291 -4-3-2-1234I-12JO(d1(d2 chapExoCorrec/9560 sacados/9560 -4-3-2-1234I-12JO(d2(d1 chapExoCorrec/4713 sacados/4713 -5-4-3-2-12345I-3-2-12JOCf chapExoCorrec/5937 sacados/5937 -4-3-2-1234I-3-2-1JOCf(d chapExoCorrec/6878 sacados/6878 -3-2-123456I-123JO chapExoCorrec/4654 sacados/4654
1 The straight line ( T 1 ) is called: ˇ The tangent to the curve C f at the point of abscissa − 1.5 ı Name the other three straight lines in the same way. 2 a The tangent ( T 1 ) passes through the points A (0 ; 2.5) and B ( − 3 ; 1) . Determine the slope of the line ( T 1 ) . b Determine the slopes of the tangents ( T 2 ) , ( T 3 ) and ( T 4 ) . E.2313 Draw the representative curve of a function passing through all the points indicated and respect-ing the tangent represented at each of them : E.2311 Consider a function f defined on the interval − 4 ; 4 . Information about this function and its representative curve is given in the table below : x − 4 − 2 0 2 4 f ( x ) 2.5 − 2 2 0 − 1.5 Coefficient directeur de la tangente au point d’abscisse x − 2 0 0 − 1 5 4 In the graph below, draw a curve that satisfies the informa-tion given about the function f and its curve C f : 5. Growth rate E.2292 Definition: Let f be a function defined on an interval I and a , b two real numbers belonging to I . We call the rate of increase of the function f between a and b , the number t defined by: t = f ( b ) − f ( a ) b − a Note: In a reference frame and considering the curve C representative of the function f on I , the rate of acr-roissement of the function f between a and b cor-responds to the directrix of the curve between the points A and B with respective abscissas a and b : https://chingmath.fr -4-3-2-12345I-3-2-123JOCfAB(T1(T2(T3(T4 chapExoCorrec/2313 sacados/2313 -4-224I-22JO chapExoCorrec/2311 sacados/2311 IJO chapExoCorrec/2292 sacados/2292 IJOAB|xB−xA||yB−yA|tyB−yAxB−xA
In the reference frame ( O ; I ; J ) orthonormal opposite is given the curve C f representative of the function f . Consider the points A , B , C of the curve C f of abscissas 0 , 1 and 2 respectively 1 Place the points A , B and C and by graphical reading, give their coordinate. 2 Calculate the rate of in-crease of the function f : a between 0 and 2 b between 1 and 2 E.5945 Consider the plane provided with a reference frame O ; I ; J in which are represented : The curve C f representative of the function f ; Strings ( d 1 ) , ( d 2 ) , ( d 3 ) and ( d 4 ) to curve C f respectively at points of abscissa 0.5 and 5 , 1 and 5 , 2 and 5 , 3 and 5 . 1 a Determine the rate of increase of the function f be-tween 0.5 and 5 . b Determine the slopes of the strings ( d 2 ) , ( d 3 ) , ( d 4 ) to the curve C f (if necessary, rounded to the nearest thou-sandth) . 2 a Draw, with a ruler, the tangent ( T ) to the curve C f at the coordinate point (5 ; − 1) . b Give an approximate value for the slope of the tangent ( T ) . E.8385 Consider the function f defined on R by the relation: f ( x ) = 1 1 + x 2 In the reference frame O ; I ; J below, is given the represen-tative curve of the function C f : 1 Determine the rates of increase of f : a entre 1 and 3 b entre 1 and 2 c entre 1 and 4 3 2 Draw the tangent ( d ) to the curve C f at the point of abscissa 1 , then conjecture the value of the slope of the tangent ( d ) . 3 What can be said about the sequence of numbers : − 1 5 ; − 3 10 ; − 21 50 ; − 1 2 6. First notion of limits E.2295 First approach to the concept of limits and conver-gence: For a value (term of a sequence or image by a function) subject to the variation of a parameter (rank of the term or variation of the variable) when the latter ˇ stabilizes ı to-wards a value, we refer to convergence . In the case of convergence, we refer to value limit 1 a Enter the following code in your Python editor : x=1; for i in range(1,10): x=x/10 y=(x*x+x)/(2*x*x+3*x) print(x,"\t",y) b After executing this code and analyzing the console display, answer the following questions : The successive values taken by the variable x seem to stabilize towards which values? Same question for the successive values of the vari-able y ? https://chingmath.fr -123I-4-3-2-123JOCf chapExoCorrec/5945 sacados/5945 23456I-1234JO(d1(d2(d3(d4Cf chapExoCorrec/8385 sacados/8385 -1234IJOCf chapExoCorrec/2295 sacados/2295
Summary : Let’s consider the function f defined on R ∗ + by: f ( x ) = x 2 + x 2 · x 2 + 3 · x The table of values for the function f shown opposite presents values of x that stabilize towards 0 . We can see that their images converge towards 1 3 . We can conjecture that ˇ the limit of f ( x ) , when x tends towards 0 , has a value of 1 3 ı and we note : lim x ↦→ 0 f ( x ) = 1 3 . 2 Using the previous program, conjecture the value of the following limits: a lim x ↦→ 0 x + 1 x + 2 b lim x ↦→ 0 x> 0 1 + x x c lim x ↦→ 0 3 x 2 + x x 2 E.10327 Below is the representative curve C f of the function f defined on the interval R \{− 2 ; 1 : 1 On the interval −∞ ; − 2 , conjecture the following lim-its : lim x ↦→ + ∞ f ( x ) ; lim x ↦→− 2 f ( x ) 2 On the interval − 2 ; 1 , conjecture the following limits: lim x ↦→− 2 f ( x ) ; lim x ↦→ 1 f ( x ) 3 On the interval 1 ; + ∞ , conjecture the following limits: lim x ↦→ 1 f ( x ) ; lim x ↦→ + ∞ f ( x ) E.2803 Consider three functions f , g and h three functions defined on R ∗ + by the relations : f ( x ) = x + 2 x 2 + 3 ; g ( x ) = 2 · x − 1 √ x ; h ( x ) = 2 · x 2 + x 5 · x 1 Check the accuracy of the following equalities: f (0.01)= 2.01 3.00101 ; g (0.0001)= − 0.9998 0.01 ; h (0.01)= 0.0102 0.05 2 The tables of values for the functions f , g and h are given below : x 1 0.1 0.01 0.001 0.0001 f ( x ) 3 4 2.1 3.01 2.01 3.000 1 2.00 1 3.000 001 2.000 1 3.000 000 01 x 1 0.01 0.000 1 0.000 001 g ( x ) 1 − 0.98 0.1 − 0.999 8 0.01 − 0.999 998 0.00 1 x 1 0.1 0.01 0.001 0.000 1 h ( x ) 3 5 0.12 0.5 0.010 2 0.05 0.001 002 0.005 0.000 100 02 0.000 5 Using a calculator, complete the table below using values rounded to the nearest ten-thousandth : 3 Conjecture the value of the following three limits: lim x ↦→ 0 f ( x ) ; lim x ↦→ 0 g ( x ) ; lim x ↦→ 0 h ( x ) E.2288 Consider the function f defined on R \ − 3 2 ; 0 by: f ( x ) = x 2 + x 2 · x 2 + 3 · x 1 a Show that : f (10 − 2 )= 101 302 b Show that : f (10 − 2 )= 1 3 + 1 906 2 For any natural number n ( n ∈ N ) : a Show that : f 10 − n = 10 n + 1 3 × 10 n + 3 b Show that : f 10 − n = 1 3 + 1 9 × 10 n + 6 3 a Using question 2 b , complete the table below : x 0.1 0.01 0.001 0.0001 0.00001 f ( x ) 1 3 + 1 96 1 3 + 1 906 b Deduct the value of the limit: lim n ↦→ + ∞ f 10 − n 7. Introduction to derivative numbers E.6059 https://chingmath.fr xf(x10,40,10,343750,010,33343708...0,0010,33334444...0,00010,33333444...0,000010,33333334... chapExoCorrec/10327 sacados/10327 -4-3-2-1234I-2-123JOCf chapExoCorrec/2803 sacados/2803 xf(xg(xh(x10;10;010;0010;00010;000010;000001 chapExoCorrec/2288 sacados/2288 chapExoCorrec/6059 sacados/6059
In a reference frame O ; I ; J , consider the representative curve of the function f defined on R by the relation: f ( x ) = x 2 − 4 x + 2 In this exercise, we will deter-mine the slope-intercept formof the tangent to the curve C f at the point A (1 ; − 1) . 1 a Establish the following equality: f ( x ) − f (1) x − 1 = x − 3 pour tout x ∈ R \{ 1 } b Deduce the slope of the chord at the curve C f pass-ing through the points A and B (2 ; − 2) . Check your answer graphically. 2 Consider the two functions u and v defined on R \{ 1 } by the relationships : u ( x ) = f ( x ) − f (1) x − 1 ; v ( x ) = x − 3 a Here are two tables of values for u and v : x 1.1 1.01 1.001 1.000 1 u ( x ) − 0.19 0.1 − 0.0199 0.01 − 0.001999 0.001 − 0.00019999 0.0001 x 1.1 1.01 1.001 1.000 1 v ( x ) − 1.9 − 1.99 − 1.999 − 1.9999 What can we say about the value of u ( x ) as the number x approaches the value 1 ? b Draw a straight line ( d ) with equation : ( d ) : y = − 2 x + 1 E.5946 Consider the curve C of the square root function given below in the orthonormal coordinate sys-tem O ; I ; J . 1 Draw the chord ( D ) of the curve C passing through the points with abscissa 2 and 4 . 2 Consider the function u defined by: u ( x ) = x − 2 x − 4 Justify that the slope of the chord ( D ) is u (2) . 3 Consider the function v defined by: v ( x )= 1 x +2 a Establish the equality u ( x )= v ( x ) for all x ∈ R \{ 4 } . b Are the functions u and v defined for x =4 ? 4 a Here is a table ˇ almost ı of values for each of the functions u and v : x 4.41 4.0401 4.004001 4.00040001 u ( x ) 0.1 0.41 0.01 0.0401 0.001 0.004001 0.0001 0.00040001 u ( x ) x 4.41 4.0401 4.004001 4.00040001 v ( x ) 1 4.1 1 4.01 1 4.001 1 4.0001 v ( x ) Complete these two tables with decimal values rounded to the nearest 10 − 5 . b What value can we conjecture for the limit: lim x ↦→ 4 x − 4 x − 4 ? 8. Calculation of derivative numbers (version 1): polynomial functions E.727 Definition: Let f be a function defined on an interval I and a a number belonging to I . The function f is said to be derivable at a if the following limit exists : lim x ↦→ a x = a f ( x ) − f ( a ) x − a Then this limit is called the derivative number in a of the function f and is noted f ( a ) . The number derived from the square function in 3 : 1 For x =3 , establish the following equality: x 2 − 3 2 x − 3 = x +3 2 Consider f the square function : a Using question 1 , determine the value of the limit: lim x ↦→ 3 f ( x ) − f (3) x − 3 b Give the value of the number derived in 3 of the square function. https://chingmath.fr -12345I-3-2-123JOCf chapExoCorrec/5946 sacados/5946 2345I2JO chapExoCorrec/727 sacados/727
E.10325 Consider the function f defined on R by: f ( x ) = 2 · x 2 − x + 1 1 Establish the identity below, for any x ∈ R \{ 1 : f ( x ) − f (1) x − 1 = 2 · x + 1 2 Deduce the value of the number derived in 1 from the function f . E.10598 Consider the function f defined on R by: f ( x ) = 4 · x 2 − 7 · x + 3 1 Establish the identity below for any x ∈ R \{ 2 } : f ( x ) − f (2) x − 2 = 4 x + 1 2 Deduce the value of the derivative number in 1 of the function f . E.10347 Consider the function f defined by: f ( x ) = − 3 · x 2 + 4 · x + 1 1 Show that for any x ∈ R \{− 2 , we have : f ( x ) − f (2) x − 2 = − 3 x − 2 2 a Deduct the value of the limit: lim x ↦→ 2 f ( x ) − f (2) x − 2 b Give the derivative number of the function f in 2 . E.10606 Consider the function f defined on R by: f ( x ) = 2 · x 2 − 5 · x + 3 1 Determine the reals a and b realizing the identity on R \{ 1 } : f ( x ) − f (1) x − 1 = a · x + b 2 Deduce the value of the derivative number f (1) of the function f in 1 . 9. Calculation of derivative (version 1): rational functions E.9553 1 For x =3 , establish the following equality: 1 x − 1 3 x − 3 = − 1 3 x 2 We note f the inverse function. a Determine limit: lim x ↦→ 3 f ( x ) − f (3) x − 3 b Give the value of f (3) . E.10599 Consider the function f defined on R \ − 1 3 : f ( x ) = 4 3 · x + 1 1 For any x ∈ R \ − 1 3 ; 1 , establish the identity: f ( x ) − f (1) x − 1 = − 3 3 · x + 1 2 a Deduct the limit: lim x ↦→ 1 f ( x ) − f (1) x − 1 b Give the value of the derivative number f (1) of the function f in 1 . E.10589 Consider the fonctin f defined on R \{ 2 } by: f ( x ) = 1 3 x − 6 1 Establish equality: f ( x ) − f (3) x − 3 = − 1 3 x − 6 2 a Determine the value of the limit: b Deduce the derivative number value of the function f in 3 . E.10326 Consider the function f defined on R \ − 1 2 by: f ( x ) = x + 1 2 · x − 1 1 Establish the identity below, for all x ∈ R \ − 2 ; − 1 2 : f ( x ) − f ( − 2) x − ( − 2) = 3 5 · 2 · x − 1 2 Deduce the value of the derivative at − 2 of the function f . E.10600 Consider the function f defined on R \ − 1 by: f ( x ) = 2 · x − 1 x + 1 1 Establish the identity for all x ∈ R \ − 2 ; − 1 : f ( x ) − f ( − 2) x − ( − 2) = − 3 x + 1 2 a Deduce the value of the limit: lim x ↦→− 2 f ( x ) − f ( − 2) x − ( − 2) b Give the value of the derivative of the function f at − 2 . 10. Calculation of derivative (version 1): with radicals https://chingmath.fr chapExoCorrec/10325 sacados/10325 chapExoCorrec/10598 sacados/10598 chapExoCorrec/10347 sacados/10347 chapExoCorrec/10606 sacados/10606 chapExoCorrec/9553 sacados/9553 chapExoCorrec/10599 sacados/10599 chapExoCorrec/10589 sacados/10589 chapExoCorrec/10326 sacados/10326 chapExoCorrec/10600 sacados/10600
E.9554 1 For x =3 , establish identity: x − 3 x − 3 = 1 x + 3 2 Note f the square root function. a Deduct the value of the limit: lim x ↦→ 3 f ( x ) − f (3) x − 3 b Give the value of f (3) . E.10601 Consider the function f defined on − 1 2 ; + ∞ by: f ( x ) = 2 · x + 1 1 Establish the identity for all x ∈ − 1 2 ; + ∞ \ 4 : f ( x ) − f (4) x − 4 = 2 2 · x + 1 + 3 2 a Determine the limit: lim x ↦→ 4 f ( x ) − f (4) x − 4 b Determine the value of the derivative f (4) of the func-tion f at 4 . 11. Calculation of derivative numbers (version 2): polynomial functions E.2822 Definition: Let f be a function defined on an interval I and a a number belonging to I . The function f is said to be derivable at a if the following limit exists : lim h ↦→ 0 h =0 f ( a + h ) − f ( a ) h Then this limit is called the number derived in a from the function f and is noted f ( a ) . Consider the function f defined by: f : x ↦−→ 3 · x 2 − 2 x 1 For any h ∈ R , establish the identity: f (2+ h ) = 3 · h 2 + 10 · h + 8 2 a Establish that : lim h ↦→ 0 f (2+ h ) − f (2) h = 10 b Give the value of f (2) . E.9605 Consider the function f defined on R by: f ( x ) = x 2 − 3 · x + 1 1 Let h be a non-zero real number. Show that : f (2+ h ) − f (2) h = h + 1 2 Deduce the value of f (2) E.10608 Consider the function f defined on R by: f ( x ) = − 3 · x 2 + 4 · x − 2 1 a Establish the identity below : f (1+ h ) − f (1) h = − 3 h − 2 b Give the value of the derivative number of the function f in 1 . 2 We note C f the representative curve of the function f in the plane provided with a reference frame. Determine the reduced equation of the tangent ( T ) to the curve C f at the point of abscissa 1 . E.10602 Consider the function f defined on R by: f ( x ) = 3 · x 2 − 4 · x + 1 1 For any h ∈ R ∗ , simplify the expression : f (1+ h ) − f (1) h 2 Deduce the derivative number f (1) of the function f in 1 . E.9555 Consider the function f defined on R by: f ( x ) = x 2 + 3 · x + 1 Determine the derivative number of the function f in − 1 . 12. Calculation of derivative (version 2): rational functions E.10591 Consider the function f defined on R \{− 2 by: f ( x ) = 6 x + 2 1 Establish that, for any h such that 1+ h ∈ R \{− 2 , we have : f (1+ h ) − f (1) = − 2 h h + 3 2 Determine the derivative number of the function f in 1 . E.10590 Consider the function f defined on R \ − 1 by: f ( x ) = − 3 x + 1 Determine the derivative number of the function f in 1 . E.8488 Consider the function f defined on R \{− 2 } by: f ( x ) = 2 · x + 1 x + 2 Determine the derivative number of the function f for x =1 . https://chingmath.fr chapExoCorrec/9554 sacados/9554 chapExoCorrec/10601 sacados/10601 chapExoCorrec/2822 sacados/2822 chapExoCorrec/9605 sacados/9605 chapExoCorrec/10608 sacados/10608 chapExoCorrec/10602 sacados/10602 chapExoCorrec/9555 sacados/9555 chapExoCorrec/10591 sacados/10591 chapExoCorrec/10590 sacados/10590 chapExoCorrec/8488 sacados/8488
E.10603 Consider the function f defined on R \ 5 2 by: f ( x ) = 2 · x + 1 2 · x − 5 1 For any h ∈ R ∗ \ − 1 2 , give the simplified form of the expression : f (3+ h ) − f (3) h 2 Deduce the derivative number f (3) of the function f in 3 . 13. Calculation of derivative numbers (version 2): with radicals E.9556 Consider the function f defined on R + by: f ( x ) = x · x Determine the derivative number of the function f in 4 . 14. Calculating derivative numbers and directing coefficients of tangents E.2810 Proposition: (admitted) Let f be a function defined on an interval I and derivable at a ∈ I . We note C f the rep-resentative curve of the function f in the plane provided with a reference frame. The curve C f admits a tangent at the point of abscissa a and this tangent has as its directing coefficient the number f ( a ) . Consider the function f defined on R by the relation: f : x ↦−→ 1 2 · x 2 − 2 · x + 1 1 a Show that, for any h ∈ R , we have : f (4+ h ) − f (4) = 1 2 · h 2 + 2 · h b Determine the value of the limit: lim h ↦→ 0 f (4+ h ) − f (4) h 2 In the plane provided with a reference frame ( O ; I ; J ) orthonormal, are plotted the curve C f representative of the function f and the tangent ( T ) to the curve C f at the point of abscissa 4 : Give the directing coefficient of the tangent ( T ) . E.9557 Consider the function f defined on − 1 ; + ∞ by: f ( x ) = 3 · x − 1 x + 1 In the plane provided with the reference frame O ; I ; J or-thonormal, are represented the curve C f representative of the function f and the tangent ( T ) to the curve C f at the point of abscissa 2 . 1 For any h ∈ − 1 ; 1 , establish the identity: f (2+ h ) − f ((2) h = 4 3 · h + 3 2 Determine the slope of the tangent ( T ) . E.9606 Consider the function f defined on R \{− 1 } by: f ( x ) = x + 3 x + 1 1 Establish that for any integer h such that h +1 =0 , we have : f (1+ h ) − f (1) h = − 1 h + 2 2 Deduce the derivative number of the function f in 1 . Note C f the representative curve of the function f in the plane with a reference frame. 3 Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 1 . https://chingmath.fr chapExoCorrec/10603 sacados/10603 chapExoCorrec/9556 sacados/9556 chapExoCorrec/2810 sacados/2810 IJOCf(T4 chapExoCorrec/9557 sacados/9557 IJO(TCf2 chapExoCorrec/9606 sacados/9606
E.2350 Consider the function f defined by: f ( x ) = 6 x + 1 x 2 − x + 2 1 Determine the definition set of the function f . 2 a Establish the following equality: f (1+ h ) − f (1) h = − 7 h + 5 2( h 2 + h + 2) b Deduce the value of the number derived from the func-tion f into 1 . 3 In the plane provided with the reference frame O ; I ; J , is given the representative curve C f of the function f : Justifying your approach, draw the tangent ( T ) to the curve C f at the point of abscissa 1 . 15. Reduced tangent equation E.9558 Proposition: let f be a function defined on an interval I and derivable at a ∈ I . The tangent to the curve C f at the point of abscissa a has the slope-intercept form y = f ( a ) · x − a + f ( a ) Consider the function f defined on R by: f ( x ) = 0.1 · x 2 + 0.2 · x − 0.8 We give the curve C f representative of the function f in the O ; I ; J orthonormal frame below : 1 a For any h ∈ R , establish the identity: f (2+ h ) − f (2) h =0.1 · h +0.6 b Deduce the derivative number of the function f in 2 . 2 a Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 2 . b Draw the tangent ( T ) in the above reference frame. E.9607 Consider the function f defined on R by: f ( x ) = 2 · x 2 − 3 · x + 1 We note C f the representative curve of the function f in the reference frame O ; I ; J below 1 Establish that : f (1) = 1 2 Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 1 . E.9559 Consider the function f defined on the interval − 1 ; + ∞ by: f ( x ) = 3 · x + 1 2 · x + 2 In the plane provided with a reference frame O ; I ; J pro-vided with an orthonormal reference frame, is given the curve C f representative of the function f . 1 Demonstrate that : f (1) = 1 4 2 a Determine the slope-intercept formof the tangent ( T ) to the curve C f at the point of abscissa 1 . b Draw the tangent ( T ) in the reference frame below. https://chingmath.fr chapExoCorrec/2350 sacados/2350 -6-5-4-3-2-123456I-2-1234JOCf chapExoCorrec/9558 sacados/9558 -6-5-4-3-2-1234I-2-1JOCf chapExoCorrec/9607 sacados/9607 -3-2-1234I2JOCf chapExoCorrec/9559 sacados/9559 -2-123456I-12JOCf
E.10604 Consider the function f defined on R \{− 2 defined by: f ( x ) = 3 x + 2 1 a For any value of h ∈ R \{ 0 ; − 3 , determine the sim-plification of the expression : f (1+ h ) − f (1) h b Deduce the value of the derivative number f (1) of the function f in 1 . 2 In the plane with a reference frame O ; I ; J , note C f the representative curve of the function f . Determine the reduced equation ( T ) of the tangent to the curve C f at the point of abscissa 1 . E.2843 Let f be the function whose image of a real number x is defined by the relation: f ( x ) = 3 · x 2 − 2 x 2 + 1 1 a For any non-zero real number h , establish the equal-ity: f (1+ h ) − f (1) h = 5 · h + 10 2 · h 2 + 4 · h + 4 b Deduce the value of the derivative number f (1) of the function f in 1 . 2 Consider the plane with a reference frame O ; I ; J and note C f the representative curve of the function f . a Determine the equation of the tangent ( T ) to the curve C f at the point of abscissa 1 . b Draw the tangent ( T ) in the reference frame. 3 Determine the coordinates of the various points of inter-section of ( T ) and C f . E.9612 Note: By choosing a straight line in the plane, it is always possible to adjust the coefficients of a second-degree polyno-mial so that this line is a tangent at − 1 to the curve of the function : Consider the quadratic function f defined by: f ( x ) = a · x 2 + 2 · x + b where a;b ∈ R In the plane equipped with an orthonormal coordinate sys-tem, we denote C f the representative curve of the function f . The line ( T ) with reduced equation y = x +2 is the tangent to the curve C f at the point with abscissa − 1 . 1 For all h ∈ R ∗ , establish the identity: f ( − 1+ h ) − f ( − 1) h = a · h − 2 · a +2 2 Deduce the expression of the function f . E.10607 Consider the function f defined on R \{− 1 by: f ( x ) = 2 · x + 3 x + 1 1 a Determine the reals a and b realizing the identity: f (1+ h ) − f (1) h = − 1 a · h + b b Establish that : f (1) = − 1 4 2 Note C f the representative curve of the function f in the plane with a reference frame. Determine the reduced equation of the tangent ( T ) to the curve C f at the point of abscissa 1 . E.10609 The plane is given an orthonormal reference frame O ; I ; J and we consider : the function f defined by: f ( x ) = a x + b where a;b ∈ R and we note C f its representative curve. the straight line ( T ) representative of the function g de-fined by: g ( x ) = − 4 x + 4 Determine the values of a and b so that the straight line ( T ) is the tangent to the curve C f at the point of abscissa 2 . 16. Tangent and relative position E.2961 Consider the function f whose im-age of a number x is defined by the relation: f ( x ) = 2 · x 3 − 3 · x 2 − x + 1 In the plane provided with a reference frame ( O ; I ; J ) , we note C f the representative curve of the function f . We note ( T ) the tangent to the curve C f at the point of abscissa 1 . 1 a For any non-zero real number h , establish the iden-tity: https://chingmath.fr chapExoCorrec/10604 sacados/10604 chapExoCorrec/2843 sacados/2843 fichierPlus/2843/diapoCorrection.pdf -6-5-4-3-2-123456I-2-123JO chapExoCorrec/9612 sacados/9612 IJOAjustement de la valeura IJOAjustement de la valeurb chapExoCorrec/10607 sacados/10607 chapExoCorrec/10609 sacados/10609 chapExoCorrec/2961 sacados/2961
f (1+ h ) − f (1) h = 2 · h 2 + 3 · h − 1 b What is the directing coefficient of the tangent ( T ) ? Justify your approach. 2 Determine the equation of the tangent ( T ) to the curve C f . 3 a Determine the value of the reals a , b and c realizing the identity: f ( x ) + x = x − 1 · a · x 2 + b · x + c b Deduce the coordinates of the intersection points of the curve C f with the tangent ( T ) . 17. Derivative number: graphical reading E.7714 The graphical representation of a function f defined and derivable on R is plotted below-below, along with the respective tangents at abscissa points − 3 and 0 . Which of the following four answers is correct: a The number derived from f to 0 is − 1 b The number derived from f into − 1 is 0 c The number derived from f into − 3 is − 1 d The number derived from f into − 3 is 3 E.7713 Consider a function f defined for any strictly positive real x and whose representative curve C of the function f , as well as T , the tangent to the curve C at point A of abscissa 4 are shown below : Which of the following four statements is correct? On the interval 0 ; 10 , the number derived from the function f takes the value 0 : a Aucune fois b Une fois c Deux fois d More than two fois E.7715 Note: In this multiple-choice questionnaire, no justifica-tion is required. For each question, only one of the answers provided is correct. A correct answer is worth 0 ; 75 points. An incorrect answer or no answer does not result in any points being deducted or awarded. Write the question number and your answer on your answer sheet. In an orthonormal coordinate system, we are given the repre-sentative curve C f of a function f that is defined and differ-entiable on the interval − 1 ; 5 . Let f be the derivative of f . The curve C f passes through the point A (0 ; 1) and the point B with abscissa 1 . The tangent T 0 to the curve at point A passes through point C (2 ; 3) , and the tangent T 1 at point B is parallel to the x-axis. 1 The exact value of the derivative of the function f at 1 is : a 0 b 1 c 1 ; 6 d other answer 2 The exact value of the derivative of the function f at 0 is : a 0 b 1 c 1 ; 6 d other answer 3 The exact value of f (1) is : a 0 b 1 c 1 ; 6 d other answer https://chingmath.fr chapExoCorrec/7714 sacados/7714 -7-6-5-4-3-2-12345I-2-12345JOCf chapExoCorrec/7713 sacados/7713 012345678910-1123456CfA(T chapExoCorrec/7715 sacados/7715 -10123450,511,522,53CfACBT0T1
E.4772 Consider a function f defined and derivable on the interval − 4 ; 3 , and note f the deriva-tive function of f . The representative curve of f is the Γ curve given below. The curve Γ is assumed to have the following properties : The curve Γ passes through the point A (0 ; 5) ; The tangent in A to the curve Γ passes through the point B ( − 2 ; 4) ; The curve Γ admits a tangent parallel to the exe of ab-scissas at the point of abscissa 0.5 . Furthermore, the function f is strictly increasing on the inter-val − 4 ; 0.5 and strictly decreasing on the interval 0.5 ; 3 . 1 Place the points A and B and draw the tangent at A to the curve Γ . 2 Graphically determine the number of solutions to the equation f ( x )=3 , and give for each solution a frame by two consecutively integers. 3 a Give the derivative number of the function f in 0 (no justification required) . b On its defining set, how many times is the derivative number of the function f 0 . Justify your answer. c On what largest interval is the derivative number of the function f strictly negative. 18. Derivative numbers and square roots E.8386 Consider the function g defined by the relation: g : x ↦−→ 3 · x + 1 Show that : g (1) = 3 4 Reminder : g (1) = lim h ↦→ 0 g (1+ h ) − g (1) h E.2309 Consider the function f defined on − 1 2 ; + ∞ by the relation: f ( x ) = 2 · x + 1 1 a Establish the following equality: f (4+ h ) − f (4) h = 2 2 · h + 9 + 3 b Deduce the value of the derivative of the function f at 4 . Reminder : f (4) = lim h ↦→ 0 f (4+ h ) − f (4) h 2 a Give the coordinates of the point A on the curve C f with abscissa 4 . b Determine the reduced equation of the tangent ( d ) to the curve C f at the point with abscissa 4 . E.2836 Consider the function f whose im-age of x is defined by: f ( x ) = 5 − 2 · x 1 Determine the definition set of the function f . 2 a For any real number h verifying: h =0 ; − 2+ h ∈D f Establish equality: f ( − 2+ h ) − f ( − 2) h = − 2 9 − 2 · h + 3 b Deduce the value of the number derived from the func-tion f into − 2 . 3 Note C f the representative curve of the function f in an orthonormal reference frame. Determine the reduced equation of the tangent to the curve C f at the point of abscissa − 2 . E.6044 Consider the function f defined on − 5 2 ; + ∞ by the relation: f ( x ) = 2 · x + 5 Note C f the representative curve of the function f in an O ; I ; J orthonormal reference frame. Determine the equation of the tangent to the curve C f at the point of abscissa 2 . https://chingmath.fr chapExoCorrec/4772 sacados/4772 -4-3-2-123I-12345JO chapExoCorrec/8386 sacados/8386 chapExoCorrec/2309 sacados/2309 chapExoCorrec/2836 sacados/2836 chapExoCorrec/6044 sacados/6044
E.2837 1 Let f be the function defined by the relation: f : x ↦−→ − 2 x 2 + x + 1 a Determine the definition set of the function f . b Determine the value of the following limit: lim h ↦→ 0 f 1 2 + h − f 1 2 h c Deduce the equation of the tangent to the curve C f at the point of abscissa 1 2 . 2 Let g be the function defined by the relation: g : x ↦−→ − 2 x 3 + 3 x 2 + x − 1 a Determine the value of the following limit: lim x ↦→ 0 g (1+ h ) − g (1) h b Deduce the equation of the tangent to the curve C g at the point of abscissa 1 . 3 The representation of the representative curves of these two functions is given below : Draw the two tangents obtained previously in the above reference frame. E.2669 Consider the function f whose im-age of x is given by the relation: f ( x ) = x 2 − 3 · x + 1 1 Establish the following equality: f (3+ h ) − f (3) h = h + 3 h 2 + 3 · h + 1 + 1 2 a Deduce the value of the following limit: f (3) = lim h ↦→ 0 f (3+ h ) − f (3) h b Determine the reduced equation of the tangent to the curve C f at the point of abscissa 3. E.5177 Consider the function f whose image of x , for x ∈ [1 ; + ∞ [ , is defined by the relation: f ( x ) = ( x 2 − 4 · x + 3) 2 · x − 2 1 Give a simplified form of : f (1+ h ) h for h> 0 2 Deduce the derivative number of the function f in 1. 19. Unclassified financial years E.10612 Consider the function f defined on R by: f ( x ) = − 3 · x 2 + 10 x − 1 1 a Determine the value of a and b realizing the follow-ing identity for any h ∈ R ∗ : f (2+ h ) − f (2) h = a · x + b b Deduce the value of the derivative number of the func-tion f in 2 . 2 Consider the plane provided with an orthonormal refer-ence frame. Note C f the representative curve of the func-tion f and ( T ) the tangent to C f at the point of abscissa 2 . Determine the reduced equation of the tangent ( T ) . E.10613 Consider the function f defined on R by: f ( x ) = x 3 · x − 1 1 a Determine the value of a and b realizing the follow-ing identity for any h ∈ R ∗ : f (1+ h ) − f (1) h = − 1 a · x + b b Deduce the value of the derivative number of the func-tion f in 1 . 2 Consider the plane provided with an orthonormal refer-ence frame. Note C f the representative curve of the func-tion f and ( T ) the tangent to C f at the point of abscissa 1 . Determine the reduced equation of the tangent ( T ) . https://chingmath.fr chapExoCorrec/2837 sacados/2837 -2-123I-12JOCfCg chapExoCorrec/2669 sacados/2669 chapExoCorrec/5177 sacados/5177 chapExoCorrec/10612 sacados/10612 chapExoCorrec/10613 sacados/10613
E.10614 Consider the function f defined on R \{ 2 } by: f ( x ) = 1 2 − x 1 Let a and h be such that : a ∈ R \{ 2 } ; h ∈ R ∗ ; a + h ∈ R \{ 2 } Establish identity: f ( a + h ) − f ( a ) = h 2 − a 2 − h − a 2 The plane is given a reference frame and C f is the repre-sentative curve of the function f . Determine the reduced equations of the two tangents ( T ) and ( T ) to the curve C f having the y-intercept − 4 . E.10615 Consider the function f defined on R by: f ( x ) = x 3 1 Let a and h such that h ∈ R ∗ , establish the identity: f ( a + h ) − f ( a ) = h · h 2 + 3 · a · h + 3 · a 2 2 The plane is given a reference frame and C f is the repre-sentative curve of the function f . Determine the reduced equation of the single tangent ( T ) to the curve C f having 2 as the y-intercept. E.2925 Consider the function f , defined on − 1 5 ; + ∞ , whose image of a real number x is given by the relation: f ( x ) = 5 · x + 1 1 For x ∈D f and h ∈ R such that h =0 and ( x + h ) ∈D f , es-tablish the following equality: f ( x + h ) − f ( x ) h = 5 5 · x + 5 · h + 1 + 5 · x + 1 2 Deduce the expression of the derivative function of the function f . https://chingmath.fr chapExoCorrec/10614 sacados/10614 chapExoCorrec/10615 sacados/10615 chapExoCorrec/2925 sacados/2925