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Numerical function exercise 1.

I/ g is a numerical function defined on ℝ -{1} by:  g(x)=x²-2x+ln |x-1| 

1-Study the change of the function g And calculate g(0) and g(2).

2-Conclude the sign of g(x) According to the value of x.

II/  f is a numerical function defined on ℝ -{1} by

Numerical functions

and (c)  Its graphic representation in a homogeneous and orthogonal coordinate system  (o;i,j) .

1-Prove that for every x of  ℝ -{1} Then:  
                                                                                                     g(x)/(x-1)²

2-
Prove that the curve (c) accepts two asymptotic lines, one of which is oblique (Δ), Then write an equation of (Δ).
3-Study the position of the curve (c) in relation to the straight line (Δ).

4-Prove that the curve (c) accepts two tangents (T) and (T') with the parallel of a straight line (Δ), then write the equations of  (T) and (T').

5- Prove that the point w(-1;1) is the center of symmetry of the curve (c).
6- Prove that the curve (c) accept two inflection point, And select them.
7-Create the two tangents (T) and (T') and the curve (c).
8-  function defined on ℝ* by:  
                                                                      
h(x)=x-ln|x|/x


and (Ch) Its graphic representation in the previous coordinate system.
prove that (Ch) is the image of (C) by Translation. And draw it.


Numerical function exercise 2.


g is a function defined on ]0;+∞[  by: g(x)=x²-2+lnx

1- Study the change of the function g on ]0;+∞[ .

2-Prove that the equation accepts a single solution α Where:
                                         1.31   < a <  1.32


3-Conclude the sign of g(x) on  ]0;+∞[ .

II)  We consider the function f defined on ]0;+∞[  by:
                                                                                 f(x)=x²+(2-lnx)²


1-Express the derivative function f in terms of g(x)

2- conclude the change of the function f  on  ]0;+∞[ .

Numerical function exercise 3.

f is a Polynomial function where:  

Polynomial function

1-Calculate f(-1) and then find the real numbers a , b, c  where:  

                                          f(x)=(x-1)(ax+bx+c)

2-numerical function defined on by:   
                                                                             
Polynomial function


and (D) straight line its equation y=2x+2
 a) Prove that the curve (Cg) Representative of the function g and the straight line (D) They have a common point A its range is zero.
b) Calculate g'(x) and write the equation of the tangent of (Cg) at the point A.
    
objectives:                                           

The objective of the first exercise: Study the change of the function and determination its sign; determination the derivative of a function; determination asymptotic lines and the position of the curve in relation to the straight line; determination the center of symmetry of the curve; determination the inflection point and Create the two tangents; determination the translation of the curve.

The objective of the second exercise:  Study the change of the function; Define an restriction to solve the function; determination its sign of the function; and conclude the change of the function from another function.
The objective of the third exercise: analysis Polynomial  to two factors; and determine the point of intersection of a curved with a straight line.


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