Real functions : Reminder

Real functions definition

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

• A: starting set
• B: arrival set
• and a correspondence allowing to associate to any element x of A, one element y of B at most.

Odd and even functions.

• A function is even in `RR` if for any `x in RR` f(x)=f(-x)
• A function is odd in `RR` if for `x in RR` f(-x)=-f(x)

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.

• To say that f is strictly increasing on I means that for all real numbers u and v of the interval I, the inequality u > v implies f(u) > f(v).
• To say that f is strictly decreasing on I means that for all real numbers u and v in the interval I, the inequality u > v implies f(u) < f(v).

Calculating the derivative of a function

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

• Formula for calculating the derivative of a function sum : (u+v)' = u'+v'
• Formula for calculating the derivative of a function product : (uv)' = u'v+uv'
• Formula for calculating the derivative of a function multiplied by a constant : (ku)' = ku'
• Formula for calculating the inverse derivative of a function : `(1/v)'` = `-(v')/v^2`
• Formula for calculating the derivative of the ratio of two functions : `(u/v)'` = `(u'v-uv')/v^2`
• Formula for calculating the derivative of the chain rule : `(u@v)'= v'*u'@v`

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

derivative(`k;x`)	`0`
derivative(`x`)	`1`
derivative(`x^n`)	`n*x^(n-1)`
derivative(`sqrt(x)`)	`1/(2*sqrt(x))`
derivative(`abs(x)`)	`1`
derivative(`"arccos"(x)`)	`-1/sqrt(1-(x)^2)`
derivative(`"arcsin"(x)`)	`1/sqrt(1-(x)^2)`
derivative(`"arctan"(x)`)	`1/sqrt(1-(x)^2)`
derivative(`ch(x)`)	`sh(x)`
derivative(`cos(x)`)	`-sin(x)`
derivative(`""cotan""(x)`)	`-1/sin(x)^2`
derivative(`"coth"(x)`)	`-1/(sh(x))^2`
derivative(`exp(x)`)	`exp(x)`
derivative(`ln(x)`)	`1/(x)`
derivative(`log(x)`)	`1/(ln(10)*x)`
derivative(`sh(x)`)	`ch(x)`
derivative(`sin(x)`)	`cos(x)`
derivative(`tan(x)`)	`1/cos(x)^2`
derivative(`th(x)`)	`1/(ch(x))^2`

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.

• f is increasing on I if, and only if, its derivative is strictly positive for all x of I.
• f is decreasing on I if, and only if, its derivative is strictly negative for all x of I.
• f is constant on I if, and only if, its derivative cancels for all x of 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.