Numbers are the universal language of mathematics. From simple counting of objects to the most complex equations of modern physics, numbers accompany us in every aspect of daily life and scientific research. But how did the different types of numbers we use come about? And why do we need so many of them?

A second-degree inequality is an algebraic expression that establishes an order relation between two terms containing a second-degree variable. It can be written in the form:

\[ a x^2 + bx + c \leq 0 \quad \text{or} \quad a x^2 + bx + c \geq 0 \]

where \( a \) and \( b \) are real coefficients with \( a \neq 0 \) and \( x \) is the unknown variable. We speak of a strict inequality if

\[ a x^2 + bx + c < 0 \quad \text{or} \quad a x^2 + bx + c > 0 \]

This table collects the fundamental rules of differentiation.

The circle is the locus of points in the plane that are at a constant distance from a fixed point, called the center. This constant distance is called the radius. The circle is a closed curve, symmetric with respect to its center, and is a particular case of a degenerate conic obtained by intersecting a circular cone with a plane perpendicular to the axis of the cone.

The parabola is the locus of points in the plane for which the distance to a fixed point (focus) equals the distance to a fixed line (directrix). It has an axis of symmetry that passes through the focus and is perpendicular to the directrix. It is an open curve that is symmetric with respect to its own axis, with numerous applications in physics and geometry.

The study of the relative positions of two lines is one of the fundamental topics in plane analytic geometry. Understanding how two lines can be positioned in the Cartesian plane allows us to accurately classify all possible geometric situations: from intersection at a single point to complete superposition.

The projection of a point onto a line represents one of the fundamental concepts in analytic geometry. Given a point \(P(x_0, y_0)\) and a line \(r: ax + by + c = 0\), the orthogonal projection of \(P\) onto \(r\) is that point \(H\) on the line which achieves the minimum Euclidean distance from \(P\). Geometrically, \(H\) is the foot of the perpendicular dropped from \(P\) to the line \(r\).

A first-degree inequality is an algebraic expression that establishes an order relation between two terms containing a linear variable. It can be written in the form:

\[ a x + b \leq 0 \quad \text{or} \quad a x + b \geq 0 \]

where \( a \) and \( b \) are real coefficients with \( a \neq 0 \) and \( x \) is the unknown variable. We speak of a strict inequality if

\[ a x + b < 0 \quad \text{or} \quad a x + b > 0 \]

Bernoulli's inequality, stated by the Swiss mathematician Jacob Bernoulli in 1689, is of fundamental importance because it allows us to establish upper and lower bounds for exponential and polynomial functions.

Theorem. (Bernoulli's Inequality). Let \(x \in \mathbb{R}\) such that \(x \geq -1\). Then for every \(n \in \mathbb{N}\), the following inequality holds:

\[ (1 + x)^n \geq 1 + nx \]

Proof. We proceed by induction on the natural number \(n\).

Numerical sequences and limits of sequences are fundamental concepts in mathematical analysis. Understanding the behavior of a sequence when \(n \to \pm \infty\) is crucial for determining whether a sequence \( a_n \) is convergent, divergent, or irregular.

The Stolz-Cesàro theorem provides a useful tool for computing the limit of a ratio of sequences. It is particularly useful when the denominator tends to infinity and the computation of the limit is not immediate. This theorem represents a generalization of Cesàro's criteria and is often used to simplify the verification of sequence convergence.

Cauchy's theorem is a fundamental result that extends Lagrange's theorem by introducing a relationship between two functions.

On this page we will see how to calculate the derivative of the logarithm with base \( b > 0 \) using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \):

\[ \lim_{h \to 0}\frac{\log_b(x + h) - \log_b(x)}{h}, \quad \lim_{x \to x_0}\frac{\log_b(x) - \log_b(x_0)}{x - x_0} \]

Monotonic sequences (both increasing and decreasing) enjoy a very important property: they always have a limit, finite or infinite. This result, known as the monotonic sequence limit theorem, tells us precisely that an increasing sequence converges to its supremum, while a decreasing sequence converges to its infimum.

In mathematical analysis, a sequence is a rule that associates to each natural number \( n \in \mathbb{N} \) an element \( a_n \) belonging to a set \( X \). In other terms, a sequence is a function defined on the set of natural numbers with values in \( X \).

Table of Contents

• Definition of Sequence
• Examples

Formally, a sequence is defined as a function:

The Lagrange Theorem, also known as the Mean Value Theorem, is a fundamental result in mathematical analysis. This theorem states that, given a function continuous on a closed interval \( [a, b]\) and differentiable on \( (a, b) \), there exists at least one point where the derivative coincides with the difference quotient between the endpoints of the interval. The proof is based on Rolle's Theorem and the construction of an auxiliary function.

Rolle's Theorem is a fundamental result applicable to continuous and differentiable functions. This theorem states that, if a function \( f \) is continuous on a closed interval \([a,b]\), differentiable on the open interval \((a,b)\) and takes the same value at the endpoints \( f(a) = f(b) \), then there exists at least one interior point \( \xi \in (a,b) \) where the derivative of the function vanishes, that is \( f'(\xi) = 0 \). This result has numerous applications, including the proof of the Mean Value Theorem.

The Weierstrass Theorem states that a continuous function defined on a closed and bounded interval necessarily attains a maximum value and a minimum value.

To thoroughly understand the properties of logarithms, we will start from their definition. From here, we will demonstrate step by step the main rules that allow us to simplify and manipulate logarithmic expressions. Each property will be accompanied by a solved exercise to put into practice what has been learned.

Let us see how to calculate the derivative of the sine and cosine functions, using the limit of the difference quotient and fundamental trigonometric identities. We demonstrate step by step that the derivative of \( \sin(x) \) is \( \cos(x) \) and that of \( \cos(x) \) is \( -\sin(x) \), justifying each step in a clear and formal manner.

The mode is one of the simplest and most useful measures of central tendency for understanding the distribution of a dataset. It represents the value that appears with the highest frequency within a dataset. Unlike the mean and median, the mode can be defined for categorical or discrete data and does not require the data to be ordered. In this sense, the mode provides a clear measure of what is "most common" in a dataset.

The arithmetic mean, also called simply the mean, is one of the most widely used measures of central tendency in statistics. It represents a way to synthesize a set of numerical data into a single value that can be considered as the "center" or the "midpoint" of a distribution. This measure is widely adopted in various contexts, from social sciences to economics, to describe data and find a mediation between the most extreme values of a set.

On this page we will see how to calculate the derivative of the natural logarithm using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:

\[ \lim_{h \to 0}\frac{\ln(x + h) - \ln(x)}{h}, \quad \lim_{x \to x_0}\frac{\ln(x) - \ln(x_0)}{x - x_0} \]

Limit operations are of fundamental importance because they allow us to calculate the limit of a sum, of a product, or of a quotient by deducing it directly from the limits of the individual sequences. These rules considerably simplify calculations, allowing us to analyze the behavior of complex functions without having to resort to more elaborate methods.

We begin with the derivative of the tangent \( f(x) = \tan(x) \). The limit of the difference quotient is

\begin{align} f'(x) &= \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \\ &=\lim_{x \to x_0} \frac{\tan(x) - \tan(x_0)}{x - x_0} \end{align}

Using the identity for the difference of tangents:

\[ \tan(x) - \tan(x_0) = \frac{\sin(x - x_0)}{\cos(x) \cos(x_0)} \]

Substituting this identity into the difference quotient, we obtain:

We have already calculated some derivatives of elementary functions using the limit of the difference quotient of the function \(f(x)\). Now we will see how to calculate - in a more general way - the derivative of the sum \((f + g )(x_0)\), the derivative of the product \((f \cdot g )(x_0)\), of the inverse function \(f ^{ - 1 }(x_0)\) and of the composite function \((f \circ g)(x_0)\).

On this page we will see how to calculate the derivative of the power function using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:

\[ \lim_{h \to 0}\frac{(x + h)^n - x^n}{h} \quad , \quad \lim_{x \to x_0}\frac{x^n - x_0^n}{x - x_0} \]

On this page we will see how to calculate the derivative of the exponential function using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:

\[ \lim_{h \to 0}\frac{a^{x + h} - a^x}{h} \quad , \quad \lim_{x \to x_0}\frac{a^x - a^{x_0}}{x - x_0} \]

The sign preservation theorem for sequences states that if a real sequence \( a_n \) converges to a limit \( L \neq 0 \), there exists an index \( N \) beyond which all terms of the sequence have the same sign as \( L \). In other words:

\[ \lim_{n\to\infty} a_n = L > 0 \, \implies \, \exists N \in \mathbb{N} \, : \, \forall n \geq N \, , \, a_n > 0 \]

If instead \( L < 0 \), then:

\[ \lim_{n\to\infty} a_n = L < 0 \, \implies \, \exists N \in \mathbb{N} \, : \, \forall n \geq N \, , \, a_n < 0 \]

An equation is quadratic if and only if it can be written in the following form:

\[ a x ^ 2 + b x + c = 0 \quad , \quad a \neq 0 \]

called the standard form. The real numbers \( a , b \) and \( c \) are called the quadratic coefficient, linear coefficient and constant term.

We can always assume that the quadratic coefficient is positive. Indeed, in the case where \( a < 0 \), we simply multiply both sides by \( -1 \) to reduce to the case \( a > 0 \).

The sign preservation theorem for functions states that if a real function \( f \) has a limit \( L \neq 0 \) as \( x \to x_0 \), then there exists a neighborhood of \( x_0 \) such that the function \( f(x) \) preserves the same sign as \( L \) for all values of \( x \) in that neighborhood (possibly excluding \( x_0 \)). In other words:

\[ \lim_{x\to x_0} f(x) = L > 0 \, \implies \, \exists \delta > 0 \, : \, \forall x \in (x_0 - \delta, x_0 + \delta) \setminus \{ x_0 \} \, , \, f(x) > 0 \]

A first-degree equation is a first-degree polynomial set equal to zero. Generally, an equation is first-degree if it can be written in standard form:

\[ ax + b = 0 \quad \text{with} \quad a \neq 0 \]

The part to the left of the equals sign is called the left-hand side, while the part to the right is called the right-hand side. 

Let \( a \neq 0 \) and let \( n \in \mathbb{N} \). The \( n \)-th power of \( a \), denoted by the symbol \( a^n \), is defined as the product of \( a \) by itself \( n \) times. Mathematically, this product is expressed as:

\[ a^n := \underbrace{a \cdot \ldots \cdot a}_{n \text{ times}} \]

The number \( a \) is called the base of the power, \( n \) is the exponent of the power.

Even functions and odd functions are distinguished by their symmetries: even functions are symmetric with respect to the y-axis, while odd functions are symmetric with respect to the origin. We will also explore the behavior of the sum of functions: the sum of two even functions is still an even function, just as the sum of two odd functions is still odd. Finally, we will see how to decompose a function into its even part and its odd part.

In this section, we will examine the steps for calculating the variance of a random variable that follows a Gamma distribution. Computing the variance requires determining certain moments of the distribution, particularly the second moment \(\mathbb{E}(X^2)\) and the first moment \(\mathbb{E}(X)\).

A function is a rule between two sets, which associates to each element of the first set (the domain) a unique element of the second set (the codomain). In this post, we will analyze the formal definition of domain, codomain, the range and the main properties such as injectivity, surjectivity and bijectivity.

A line is a primitive concept in Euclidean geometry, meaning it cannot be defined in more elementary terms but is taken as a fundamental entity. It is intuitively described as an infinite set of points arranged in a constant direction, extending indefinitely in both directions. In a Cartesian coordinate system, a line can be represented by a linear equation and is characterized by a slope that determines its inclination with respect to the x-axis.

An ellipse is the set of all points in a plane such that the sum of the distances from two fixed points, called foci, is constant. It has two axes of symmetry, called the major axis and the minor axis. It is a closed and symmetric curve with numerous applications in physics and geometry.