This type of exercise can be solved using the function : derivative
A Real function from A to B is defined by giving :
The calculator can be used to determine whether a function is even or odd.
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 .
In an orthogonal reference frame, when a function is even, the y-axis is an axis of symmetry of its graphical representation.
In an orthogonal frame of reference, when a function is odd, the origin O is a center of symmetry of the graphical representation.
f is a function and I is an interval contained in its set of definitions.
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(`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.
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)`.
Let f be a differentiable function on an interval I.
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.
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 remarkable functions include trigonometric functions, which are widely used in many fields.