A selection of free mathematics resources (calculators, graphing calculator, exercises, games, quizzes, course reminders) that allow you to draw and manipulate the usual mathematical functions.

## 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) :

 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

 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.

## Real functions : games and quizzes

• Quiz on the calculation of the derivative of a function (real functions): This quiz on mathematical functions allows you to practice using the techniques of calculating derivatives.
• Quiz derivative of the exponential function (real functions): This quiz on the exponential function allows you to practice using the techniques of calculating derivatives.
• Quiz derivative of logarithm function (real functions): This quiz on the logarithm function allows you to practice using the techniques of calculating derivatives.
• Quiz on finding function antiderivatives (real functions): This quiz on mathematical functions allows you to practice using the techniques of finding antiderivatives.
• Quiz solving equations with one unknown (real functions): This quiz on equations allows to practice solving different types of equations with one unknown.
• Quiz solving first degree equations (real functions): This quiz on first degree equations allows to practice solving simple equations with one unknown.
• Quiz solving equations of the second degree (real functions): This quiz on the equations of the second degree allows to practice the methods of resolution based on the use of the discriminant.