This site offers many resources for manipulating numerical functions: calculators, quizzes, games, online exercises. More specific resources for trigonometric functions are also available.

The resources concern the derivative of a function, the antiderivative of a function, the limits of a function, the values of a function, the graphical representation of a function from its algebraic expression, the values for which the function cancels itself (the zeros).

Real functions : calculators

• Absolute value : abs. The abs function calculates online the absolute value of a number.
• Indefinite integral calculator : antiderivative. The antiderivative calculator allows to calculate an antiderivative online with detail and calculation steps.
• Arccosine : arccos. The arccos function allows the calculation of the arc cosine of a number. The arccosine function is the inverse functions of the cosine function.
• Arcsine : arcsin. The arcsin function allows the calculation of the arc sine of a number. The arcsine function is the inverse function of the sine function.
• Arctangent : arctan. The arctan function allows the calculation of the arctangent of a number. The arctangent function is the inverse functions of the tangent function.
• Array of values of a function : array_values. The values calculator returns the table of values of a function obtained from an initial value and the difference between two consecutive values (steps).
• Hyperbolic cosine : ch. The function ch calculates online the hyperbolic cosine of a number
• Cosine : cos. The cos trigonometric function calculates the cosine of an angle in radians, degrees or gradians.
• Cosecant : cosec. The trigonometric function sec allows to calculate the secant of an angle expressed in radians, degrees, or grades.
• Cotangent : cotan. The cotan trigonometric function to calculate the cotangent of an angle in radians, degrees or gradians.
• Hyperbolic cotangent : coth. The coth function calculates online the hyperbolic cotangent of a number.
• Cube root : cube_root. The cube_root function calculates online the cube root of a number.
• Degree of a polynomial : degree. The degree function calculates online the degree of a polynomial.
• Derivative calculator : derivative. The derivative calculator allows steps by steps calculation of the derivative of a function with respect to a variable.
• Solve for x calculator : equation_solver. The equation solver allows to solve equations with an unknown with calculation steps : linear equation, quadratic equation, logarithmic equation, differential equation.
• Exponential : exp. The function exp calculates online the exponential of a number.
• Inequality calculator : inequality_solver. Inequality solver that solves an inequality with the details of the calculation: linear inequality, quadratic inequality.
• Integral calculator : integral. The integral calculator calculates online the integral of a function between two values, the result is given in exact or approximated form.
• Even or odd function calculator : is_odd_or_even_function. Calculator for determining whether a function is an even function and an odd function.
• Limit calculator : limit. The limit calculator allows the calculation of the limit of a function with the detail and the calculation steps.
• Napierian logarithm : ln. The ln calculator allows to calculate online the natural logarithm of a number.
• Logarithm : log. The log function calculates the logarithm of a number online.
• Secant : sec. The trigonometric function sec allows to calculate the secant of an angle expressed in radians, degrees, or grades.
• Hyperbolic sine : sh. The sh function allows to calculate online the hyperbolic sine of a number.
• Sine : sin. The sin trigonometric function to calculate the sine of an angle in radians, degrees or gradians.
• Square root : sqrt. The sqrt function allows to calculate the square root of a number in exact form.
• Tangent : tan. The tan trigonometric function to calculate the tangent of an angle in radians, degrees or gradians.
• Taylor expansion calculator : taylor_series_expansion. The taylor series calculator allows to calculate the Taylor expansion of a function.
• Hyperbolic tangent : th. The function th allows to calculate online the hyperbolic tangent of a number.
• Valuation of a polynomial : valuation. The valuation function allows to calculate the valuation of a polynomial online.

Real functions : games, quizzes and exercises

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.

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