Real functions : Reminder

Real functions definition

A Real function from A to B is defined by giving :

Odd and even functions.

The calculator can be used to determine whether a function is even or odd.

Graphical representation of real functions

A representative curve of a numerical function f is the set of points with coordinates M(x; y), where y represents the image of x by f. Here, for example, is the graphical representation of the function f defined by `f(x)=x^2-3` obtained with the calculator .

Graphical representation of an even function.

In an orthogonal reference frame, when a function is even, the y-axis is an axis of symmetry of its graphical representation.

Graphical representation of an odd function

In an orthogonal frame of reference, when a function is odd, the origin O is a center of symmetry of the graphical representation.

Increasing and decreasing functions

f is a function and I is an interval contained in its set of definitions.

Calculating the derivative of a function

Usual formulas to use for the calculation of the derivative of a function

Table of derivatives of common functions

It is also necessary to know differentiated the usual functions which are in the following table (the differential calculator can help you) :

Table of derivatives of common functions

By applying these formulas and using this table, it is possible to calculate the derivative of any function. These are the calculation methods that the calculator uses to find the derivatives of functions.

Equation of the tangent to a curve at a point

C is the representative curve of a function f derivable at a point a. The tangent to C at the point A(a;f(a)) is the straight line through A whose directing coefficient is `f'(a)`.
An equation of the tangent to C at point A(a;f(a)) is :
`y = f(a) + f'(a)(x-a)`.

Increasing and decreasing functions and differential calculus.

Let f be a differentiable function on an interval I.

Calculating the antiderivatives of a function

Formulas for calculating antiderivatives

Table of antiderivatives functions
antiderivative(`k;x`)`kx + c`
antiderivative(`x`)`x^2/2 + c`
antiderivative(`x^n`)`x^(n+1)/(n+1) + c`
antiderivative(`1/x^n`)`-1/((n-1)*x^(n-1)) + c`
antiderivative(`abs(x)`)`x/2 + c`
antiderivative(`"arccos"(x)`)`x*arccos(x)-sqrt(1-(x)^2) + c`
antiderivative(`"arcsin"(x)`)`x*arcsin(x)+sqrt(1-(x)^2) + c`
antiderivative(`"arctan"(x)`)`x*arctan(x)-1/2*ln(1+(x)^2) + c`
antiderivative(`"ch"(x)`)`sh(x) + c`
antiderivative(`cos(x)`)`sin(x) + c`
antiderivative(`"cotan"(x)`)`ln(sin(x)) + c`
antiderivative(`"coth"(x)`)`ln(sh(x)) + c`
antiderivative(`exp(x)`)`exp(x) + c`
antiderivative(`ln(x)`)`x*ln(x)-x + c`
antiderivative(`log(x)`)`(x*log(x)-x)/ln(10) + c`
antiderivative(`"sh"(x)`)`ch(x) + c`
antiderivative(`sin(x)`)`-cos(x) + c`
antiderivative(`sqrt(x)`)`2/3*(x)^(3/2) + c`
antiderivative(`tan(x)`)`-ln(cos(x)) + c`
antiderivative(`"th"(x)`)`ln(ch(x)) + c`
The following conventions are used in the antiderivative integral table: c represents a constant.

The calculator allows to obtain an antiderivative for many usual functions.

Polynomial functions

A polynomial (also called a polynomial function) is a function defined in `RR` which can be written as `x -> a_n*x^n+...+ a_(n-1)*x^(n-1)+...+a_1*x+a_0` where n is a natural number and `a_0,a_1,...,a_n` are real numbers.
If `a_n!=0`, then n is the degree of the polynomial, which can be obtained with the polynomial degree calculator.

Among polynomials, some have been particularly studied, such as polynomials of degree 2. A polynomial of degree 2 is often called a trinomial of degree 2. Thanks to special calculation methods based on the discriminant, it is possible to find the roots of a trinomial (solution of the second-degree equation) .

As with all functions, it's possible to plot the representative curve of a trinomial function. This curve is called a parabola.

Other families of functions

Other remarkable functions include trigonometric functions, which are widely used in many fields.