even and odd functions
This type of exercise can be solved using the function : is_odd_or_even_function
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) :
| derivative(`k;x`) | `0` |
| derivative(`x`) | `1` |
| derivative(`x^n`) | `n*x^(n-1)` |
| derivative(`1/x`) | `-1/x^2` |
| derivative(`1/x^2`) | `-2/x^3` |
| 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
| 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 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.