Energetski pojasevi u siliciju

Energy Bands

• When N silicon atoms form a crystal, electron valence interaction causes discrete energy levels to split into N distinct levels.
• These levels form continuous energy bands because they are closely spaced.
• The valence band is the lower band and is formed from 3s orbitals.
• The conduction band is the upper band and is formed from 3p orbitals.
• The space between the valence and conduction band is the energy gap $E_g$.
• Energy band diagrams plot energy (E) versus distance (x), showing the bottom edge of the conduction band ($E_c$) and the top edge of the valence band ($E_v$).
• The energy gap is $E_g = E_c - E_v$.
• Electrons will occupy the lowest available energy states.
• At absolute zero (0 K), electrons are in the valence band, making the conduction band empty.
• At higher temperatures, some electrons jump into the conduction band to create mobile electrons, also creating holes in the valence band.
• The number of electrons in the conduction band and holes in the valence band dictates the material conductivity.
• Insulators have a large $E_g$, such as diamond with $E_g = 5.5$ eV.
• Semiconductors have a small $E_g$, such as silicon with $E_g = 1.12$ eV.
• Conductors have overlapping bands, such as metals.

Direct and Indirect Band Gaps

• Band structure portrays the relationship between energy (E) and momentum (k) of electrons.
• The minimum energy of the conduction band and the maximum energy of the valence band are at the same value of k in direct band gap semiconductors, with GaAs as an example.
• The minimum energy of the conduction band and the maximum energy of the valence band occur at different values of k in indirect band gap semiconductors, exemplified by Si.
• Direct band gap semiconductors are efficient for optical devices because of direct electron recombination and photon emission.
• Indirect band gap semiconductors require phonon assistance in the recombination process, making them less efficient for light emission.

Electrons

• Electrons in the conduction band are mobile and contribute to electrical current.
• The concentration of electrons in the conduction band is denoted by 'n' (electrons/cm³).
• Electrons move in response to an electric field (drift) or a concentration gradient (diffusion).
• The effective mass ($m_n^*$) of an electron in a semiconductor is different from the free electron mass ($m_0$).
• The crystal lattice's periodic potential on electron motion is accounted for using $m_n^*$.

Holes

• Absence of an electron in the valence band creates a hole.
• Holes behave as positively charged particles.
• The concentration of holes in the valence band is denoted by 'p' (holes/cm³).
• Holes move in response to an electric field (drift) or a concentration gradient (diffusion).
• Hole effective mass ($m_p^*$) in a semiconductor differs from free electron mass ($m_0$).
• Effect of the periodic potential of the crystal lattice on the hole's motion is taken into account with $m_p^*$.

Effective Mass

• Electron and hole effective mass is an important parameter affecting mobility and other properties.

Intrinsic Semiconductor

• Made of a pure semiconductor crystal with no impurities.
• The number of electrons in the conduction band is equal to the number of holes in the valence band.
• $n = p = n_i$, where $n_i$ is the intrinsic carrier concentration.
• $n_i$ depends on the material and temperature.
  • Silicon: $n_i \approx 1.5 \times 10^{10} cm^{-3}$ at 300 K.
  • Germanium: $n_i \approx 2.5 \times 10^{13} cm^{-3}$ at 300 K.
  • Gallium Arsenide: $n_i \approx 1.8 \times 10^6 cm^{-3}$ at 300 K.
• Intrinsic carrier concentration increases with temperature.

Temperature Dependence of $n_i$

• Intrinsic carrier concentration $n_i$ varies with temperature as:
  • $n_i = \sqrt{N_cN_v}e^{-E_g/2kT}$
    • $N_c$ represents effective density of states in the conduction band.
    • $N_v$ represents effective density of states in the valence band.
    • $E_g$ represents the energy gap.
    • k represents Boltzmann's constant ($1.38 \times 10^{-23} J/K$ or $8.62 \times 10^{-5} eV/K$).
    • T represents temperature in Kelvin.
  • $N_c = 2(\frac{2\pi m_n^*kT}{h^2})^{3/2}$
    • h represents Plank's constant ($6.626 \times 10^{-34} J \cdot s$).
  • $N_v = 2(\frac{2\pi m_p^*kT}{h^2})^{3/2}$
• Exponential term $e^{-E_g/2kT}$ dominates temperature dependence of $n_i$.

Example:

• Calculate intrinsic carrier concentration of silicon at 300 K.
• Given: $E_g = 1.12 eV$, $m_n^* = 1.08 m_0$, $m_p^* = 0.56 m_0$.
• $N_c = 2.8 \times 10^{19} cm^{-3}$
• $N_v = 1.04 \times 10^{19} cm^{-3}$
• $n_i = \sqrt{N_cN_v}e^{-E_g/2kT}$
• $n_i = 1.5 \times 10^{10} cm^{-3}$

Fermi Level

• The Fermi level ($E_F$) is the energy at which the probability of finding an electron is 50%.
• The Fermi level in an intrinsic semiconductor is located near the middle of the energy gap.
• $E_F = \frac{E_c + E_v}{2} + \frac{3}{4}kT \ln(\frac{m_p^}{m_n^})$
• If $m_n^* = m_p^*$, then $E_F = \frac{E_c + E_v}{2}$.
• The Fermi level is an important concept for understanding the behavior of semiconductor devices.

Doped Semiconductor

• Impurities added to an intrinsic semiconductor creates extrinsic semiconductors.
• Doping can change the conductivity of a semiconductor in a drastic manner.
• There are two types of doping: n-type and p-type.

n-type Semiconductor

• n-type semiconductors are created via doping with donor impurities.
• Donor impurities have more valence electrons than the semiconductor atoms they replace, with phosphorus in silicon as an example.
• Due the extra electron easily ionizing, it becomes a free electron in the conduction band.
• Donor atoms become positively charged ions.
• The concentration of donor impurities is denoted by $N_D$ (donors/cm³).
• Electron concentration is much larger than hole concentration ($n >> p$) in n-type material.
• Electrons are the majority carriers, and holes are the minority carriers.

p-type Semiconductor

• p-type semiconductors are created by adding acceptor impurities.
• With boron in silicon as an example, acceptor impurities have fewer valence electrons than the semiconductor atoms they replace.
• The missing electron creates a hole in the valence band.
• Acceptor atoms become negatively charged ions.
• The concentration of acceptor impurities is denoted by $N_A$ (acceptors/cm³).
• Hole concentration far exceeds electron concentration ($p >> n$) in p-type material.
• Holes are the material's majority carrier, while electrons are the minority.

Charge Neutrality

• Semiconductors must be electrically neural.
• Meaning total positive charge must equal total negative charge.
• $p + N_D = n + N_A$
• In n-type material ($N_D >> n_i$), $n \approx N_D$.
• In p-type material ($N_A >> n_i$), $p \approx N_A$.

Equilibrium Carrier Concentrations

• Product of electron and hole concentrations is constant at equilibrium:
  • $np = n_i^2$
• In n-type material:
  • $n_n \approx N_D$
  • $p_n = \frac{n_i^2}{N_D}$
• In p-type material:
  • $p_p \approx N_A$
  • $n_p = \frac{n_i^2}{N_A}$

Fermi Level in Extrinsic Semiconductors

• Fermi level is closer to the conduction band in n-type semiconductors.
• $E_F = E_c - kT \ln(\frac{N_c}{N_D})$
• Fermi level is closer to the valence band in p-type semiconductors.
• $E_F = E_v + kT \ln(\frac{N_v}{N_A})$

Compensation

• Semiconductor contains both donor and acceptor impurities when compensation takes place.
• If $N_D > N_A$, material is n-type with $n \approx N_D - N_A$.
• If $N_A > N_D$, material is p-type with $p \approx N_A - N_D$.
• If $N_A = N_D$, the material is compensated, and the carrier concentrations are close to intrinsic.

Drift

• Motion of charge carriers due to an electric field constitutes drift.
• Drift velocity ($v_d$) is proportional to the electric field (E):
  • $v_d = \mu E$
  • $\mu$ is mobility (cm²/V·s).
• Electron mobility is denoted by $\mu_n$.
• Hole mobility is denoted by $\mu_p$.
• Drift current density (J) is given by:
  • $J = qnv_d$ (for electrons)
  • $J = qpv_d$ (for holes)
  • $J = q(n\mu_n + p\mu_p)E$ (total current density)
  • $q$ is elementary charge ($1.602 \times 10^{-19} C$).

Mobility

• Mobility depends on temperature and doping concentration.
• Lattice scattering (phonons) limits mobility at low doping concentrations.
• Ionized impurity scattering limits mobility at high doping concentrations.
• Increasing temperature decreases mobility due to increased lattice vibrations.
• Temperature dependence of mobility can be approximated as:
  • $\mu \propto T^{-m}$
    • m represents a material-dependent constant, such as m = 2.4 for Si.

Conductivity and Resistivity

• Conductivity ($\sigma$) measures how easily a material conducts electricity.
  • $\sigma = q(n\mu_n + p\mu_p)$
• Resistivity ($\rho$) is inverse of conductivity.
  • $\rho = \frac{1}{\sigma} = \frac{1}{q(n\mu_n + p\mu_p)}$
• In n-type material: $\sigma \approx qn\mu_n \approx qN_D\mu_n$
• In p-type material: $\sigma \approx qp\mu_p \approx qN_A\mu_p$

Diffusion

• Movement of charge carriers from a region of high concentration to a region of low concentration.
• Diffusion current density (J) is proportional to concentration gradient:
  • $J = qD_n\frac{dn}{dx}$ (for electrons)
  • $J = -qD_p\frac{dp}{dx}$ (for holes)
  • $D_n$ represents electron diffusion coefficient (cm²/s).
  • $D_p$ represents hole diffusion coefficient (cm²/s).

Einstein Relationship

• Diffusion coefficient and mobility are related by the Einstein relationship:
  • $\frac{D_n}{\mu_n} = \frac{D_p}{\mu_p} = \frac{kT}{q}$

Total Current

• The total current is the sum of the drift and diffusion currents.
• For electrons:
  • $J_n = qn\mu_nE + qD_n\frac{dn}{dx}$
• For holes:
  • $J_p = qp\mu_pE - qD_p\frac{dp}{dx}$

Generation

• Process by which electron-hole pairs are created.
• Electron-hole pairs are created via thermal energy through thermal generation.
• Electron-hole pairs are created when photons with energy greater than the band gap energy are absorbed through optical generation.

Recombination

• Electrons and holes annihilate each other in this process.
• Band-to-band recombination is when one electron in the conduction band binds with a hole in the valence band.
• Electrons and holes recombine through energy levels within the band gap (traps) through recombination via traps.
• Surface recombination occurs at the surface of the semiconductor.

Recombination Rate

• Number of electron-hole pairs that combine per unit time per volume.
• It is proportional to the excess carrier concentrations.
  • $R = \frac{\Delta n}{\tau_n} = \frac{\Delta p}{\tau_p}$
    • $\Delta n$ is excess electron concentration.
    • $\Delta p$ is excess hole concentration.
    • $\tau_n$ is electron lifetime.
    • $\tau_p$ is hole lifetime.
• Lifetime is average time that an excess carrier exists before recombining.

Direct Recombination

• Recombination rate in direct band gap semiconductors is given by:
  • $R = B(np - n_i^2)$
    • B represents the recombination coefficient.

Indirect Recombination (Shockley-Read-Hall)

• Recombination rate in direct band gap semiconductors is dominated by recombination through traps.
• The Shockley-Read-Hall (SRH) recombination rate is given by:
  • $R = \frac{np - n_i^2}{\tau_p(n + n_1) + \tau_n(p + p_1)}$
    • $\tau_p$ and $\tau_n$ are the hole and electron lifetimes, respectively.
    • $n_1 = n_i e^{(E_t - E_i)/kT}$
    • $p_1 = n_i e^{(E_i - E_t)/kT}$
    • $E_t$ is the energy level of the trap.
    • $E_i$ is intrinsic Fermi level.

Surface Recombination

• Recombination at the surface of a semiconductor is characterized by the surface recombination velocity (S).
• $R_s = S\Delta n$

Low-Level Injection

• When excess carrier concentration is smaller than majority carrier concentration.
• In n-type material: $\Delta n p_0$

Continuity Equation

• Describes the time and space dependence of carrier concentrations.
• For electrons:
  • $\frac{\partial n}{\partial t} = \frac{1}{q}\frac{\partial J_n}{\partial x} + G_n - R_n$
    • $G_n$ represents generation rate of electrons.
    • $R_n$ represents recombination rate of electrons.
• For holes:
  • $\frac{\partial p}{\partial t} = -\frac{1}{q}\frac{\partial J_p}{\partial x} + G_p - R_p$
    • $G_p$ represents generation rate of holes.
    • $R_p$ represents recombination rate of holes.

Steady State

• When carrier concentrations do not change with time.
• $\frac{\partial n}{\partial t} = 0$
• $\frac{\partial p}{\partial t} = 0$

Diffusion Length

• Average distance that a carrier diffuses before recombining.
• For electrons:
  • $L_n = \sqrt{D_n\tau_n}$
• For holes:
  • $L_p = \sqrt{D_p\tau_p}$

Example

• Consider long p-type semiconductor bar with steady-state excess electron concentration injected at one end ($x = 0$).
• Excess electron concentration decays exponentially with distance:
  • $\Delta n(x) = \Delta n(0)e^{-x/L_n}$

Chapter Summary

• Energy bands determine electrical properties of semiconductors.
• Electrons and holes are the carrier charges in semiconductors.
• Intrinsic semiconductors are pure, while extrinsic semiconductors are doped with impurities.
• Doping creates n-type or p-type material.
• Carrier transport occurs through drift and diffusion.
• Drift is caused by an electric field, while diffusion is caused by a concentration gradient.
• Generation and recombination are processes that create and annihilate electron-hole pairs.
• Continuity equation tracks the time and space reliance of carrier concentrations.

Algorithmic Complexity

Definition

• Measure of time and space needed by an algorithm for an input of a given size
• Indicates how fast the algorithm runs
• Described with Asymptotic notation, Big O being most common.

Importance

• Determines required resources to run a program
• Compares efficiency of different algorithms and determine if they are suitable for specific use cases and constraints
• Algorithm complexity doesn't determine the quality of the algorithm

Big O scale

• Describes upper bound of time or space complexity
• $O(1)$ - indicates Excellent bound
• $O(log n)$ - indicates Great bound
• $O(n)$ - indicates a Good bound
• $O(n log n)$ - indicates a Fair bound
• $O(n^2)$ indicates a Bad bound
• $O(2^n)$ indicates a Horrible bound
• $O(n!)$ indicates a Nightmare bound

Constant Time - O(1)

• The time required is independent of input size

def constant_time(items: list):
    return items

Logarithmic Time - O(log n)

• Time increased logarithmically as input size increases

def logarithmic_time(items: list, item: int):
    low = 0
    high = len(items) - 1

    while low  item:
            high = mid - 1
        else:
            low = mid + 1
    return None

Linear Time - O(n)

• Time required increases linearly as input size increases

def linear_time(items: list):
    for item in items:
        print(item)

Log-Linear Time - O(n log n)

• Time required increases linearly with a logarithmic factor

def log_linear_time(items: list):
    if len(items) = pivot]

    return quick_sort(left) + [pivot] + quick_sort(right)

Quadratic Time - O(n^2)

• Time required increases quadratically as input size increases

def quadratic_time(items: list):
    for item in items:
        for item2 in items:
            print(item, item2)

O(1) Space complexity

• The algorithm uses constant space

def constant_space(items: list):
    sum = 0
    for item in items:
        sum += item
    return sum

O(n) Space complexity

• The Algorithm uses linear space

def linear_space(items: list):
    new_list = []
    for item in items:
        new_list.append(item * 2)
    return new_list

Calculus - Newton's Method

Questions

• Use Newton's method to approximate a root of various equations, iterating from $x_0$ to $x_2$

Questions

• In question 8, find the point for function f(x) to a minimum value based on $x_0$

Machine Learning for Algorithmic Trading

• Focuses on enabling systems to learn from data and is a subset of AI.
• Differs from Traditional Programming
  • Traditional Programming: Data + Program $\rightarrow$Output
  • Machine Learning: Data + Output $\rightarrow$ Program
• Learns patterns and relationships from data instead of programming it.

Types of Machine Learning

Supervised learning

• Training data is labeled.
• Algorithms learn to map inputs to outputs.
• Examples: Regression, Classification.

Unsupervised Learning

• Training data is unlabeled.
• Algorithms learn to find patterns and structure in the data.
• Examples: Clustering, Dimensionality Reduction.

Reinforcement Learning

• Algorithms learn to make decisions by interacting with an environment.
• Algorithms receive rewards or penalties for their actions.
• Goal: Maximize cumulative reward.

Advantages of Machine Learning in Trading

Pattern Recognition

• Identify complex patterns and relationships in financial data that humans may miss.

Adaptability

• Adapt to changing market conditions and new data.

Automation

• Automate trading strategies, reducing the need for manual intervention.

Improved Decision Making

• Enhance trading decisions with data-driven insights.

Challenges of Machine Learning in Trading

Overfitting

• Models may perform well on training data but poorly on new data.

Data Quality

• Financial data can be noisy and incomplete.

Computational Resources

• Training complex models can be computationally expensive.

Interpretability

• Some models are difficult to interpret, making it hard to understand why they make certain predictions.

Linear Regression

Description

• Models the relationship between a dependent variable and one or more independent variables
  • Using linear equation to observed data.

Use Cases

• Predicting stock prices
• Forecasting volatility.

Formula

$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 +... + \epsilon$ - $Y$ is the dependent variable - $X_i$ is the independent variables - $\beta_i$ are the coefficients - $\epsilon$ is the error term

Logistic Regression

Description

• Predicts probability of binary outcome (0 or 1).

Use Cases

• Predicting whether a stock price will go up or down.
• Classifying trading signals.

Equation

$P(Y=1) = \frac{1}{1 + e^{-(\beta_0 + \beta_1X_1 + \beta_2X_2 +...)}}$

Support Vector Machines (SVM)

Description

• Finds optimal hyperplane to separate data into different classes.

Uses Cases

• Classification, regression.

Key Concept

• Kernel Trick maps data into higher dimensional space.

Terminology

• Support Vectors are data points closest to the hyperplane.

Decision Trees

Description

• Builds a tree-like model to make decisions based on input features.

Use Cases

• Classification, regression.

Advantage

• Easy to interpret.

Disadvantage

• Prone to overfitting.

Random Forest

Description

• Ensemble of decision trees.

Use Cases

• Classification, regression.

Advantage

• Reduces overfitting, high accuracy.

Neural Networks

Description

• Complex models inspired by the structure of the human brain.

Use Cases

• Time series forecasting, pattern recognition.

Neural Network Types

Feedforward Neural Networks (FFNN)

Recurrent Neural Networks (RNN)

Long Short-Term Memory (LSTM)

Convolutional Neural Networks (CNN)

K-Means Clustering

Description

• Partitions data into k clusters based on similarity.

Use Cases

• Identifying market segments, grouping similar stocks.

Algorithm

Initialize k centroids.

Assign each data point to the nearest centroid.

Update centroids based on the mean of the data points in each cluster.

Repeat steps 2 and 3 until convergence.

Feature Engineering

• Process of selecting, transforming, and creating features from raw data to improve model performance.

Common Financial Features

Moving Averages

• Average price over a specific period.

Relative Strength Index (RSI)

• Measures magnitude of recent price changes to evaluate overbought or oversold conditions.

Moving Average Convergence Divergence (MACD)

• Shows the relationship between two moving averages of a price.

Bollinger Bands

• Volatility indicator using a moving average and standard deviations.

Volume

• Number of shares traded in a given period.

Volatility

• Measure of price fluctuations.

Metrics for Regression

Mean Squared Error (MSE)

• Average of the squared differences between predicted and actual values.

R-squared ($R^2$)

• Proportion of the variance in the dependent variable that is predictable from the independent variables
  • $R^2 = 1 - \frac{SSE}{SST}$
    • SSE is the sum of squared Errors
    • SST is the total sum of Squares

Metrics for Classification

Accuracy

• Proportion of correctly classified instances.

Precision

• Proportion of true positives out of all predicted positives.

Recall

• Proportion of true positives out of all actual positives.

F1-Score

• Harmonic mean of precision and recall.
  • $F1 = 2 * \frac{Precision * Recall}{Precision + Recall}$

Confusion Matrix

• Table showing performance of a classification model.

Implementing Machine Learning in Trading requires several steps

Data Collection and Preparation

• Gather historical financial data from reliable sources.
• Clean and preprocess the data (handle missing values, outliers).

Feature Engineering

• Create relevant features from the raw data

Model Selection

• Choose appropriate machine learning algorithms based on the problem type.

Training and Validation

• Split the data into training, validation, and test sets.
• Train the model on the training data.
• Tune hyperparameters using the validation set.

Backtesting

• Evaluate the model's performance on historical data.

Deployment

• Implement the model in a live trading environment.

Monitoring and Maintenance

• Continuously monitor the model's performance and retrain as needed.

Tools and Libraries

The Python Programming Language is popular due to its open source coding libraries

Pandas

• Offers data manipulation and analysis.

NumPy

• Offers numerical computing.

Scikit-learn

• Offers various machine learning algorithms.

TensorFlow

• Offers deep learning frameworks.

Keras

• Offers high-level neural networks API.

Matplotlib, Seaborn

• Provides Data visualization.

Machine learning offers powerful tools for algorithmic trading

• Careful planning, data preparation, and model evaluation are crucial for success.
• Continuous learning and adaptation are essential in the dynamic world of financial markets.

Prérequis Mathématiques

Defintions

Ensemble

• Collection d'objets.

Cardinal

• Nombre d'éléments.

Ensemble vide

• Ensemble ne contenant aucun élément noté $\emptyset$.

Opérations de base

Union

• $A \cup B = {x | x \in A \text{ ou } x \in B}$

Intersection

• $A \cap B = {x | x \in A \text{ et } x \in B}$

Différence

• $A \setminus B = {x | x \in A \text{ et } x \notin B}$

Complémentaire

• $\bar{A} = {x | x \notin A}$

Nummber sets

Entiers naturels

• $\mathbb{N} = {0, 1, 2, 3,...}$

Entiers relatifs

• $\mathbb{Z} = {..., -2, -1, 0, 1, 2,...}$

Nombres rationnels

• $\mathbb{Q} = {p/q | p \in \mathbb{Z}, q \in \mathbb{N^*}}$

Nombres réels

• $\mathbb{R}$: inclut $\mathbb{Q}$ et les irrationnels ($\sqrt{2}, \pi$, etc.)

Nombres complexes

• $\mathbb{C} = {a + bi | a, b \in \mathbb{R}, i^2 = -1}$

Defintions

Fonction

• Relation entre un ensemble de départ (domaine) et un ensemble d'arrivée (codomaine) telle que chaque élément du domaine est associé à un unique élément du codomaine.

Domaine

• Ensemble de toutes les valeurs d'entrée possibles pour lesquelles la fonction est définie.

Image

• Ensemble de toutes les valeurs de sortie possibles de la fonction.

Types de fonctions

Linéaire

• $f(x) = ax + b$

Polynômiale

• $f(x) = a_n x^n + a_{n-1} x^{n-1} +... + a_1 x + a_0$

Exponentielle

• $f(x) = a^x$

Logarithmique

• $f(x) = log_a(x)$

Trigonométrique

• sin(x), cos(x), tan(x), etc.

Opérations sur les fonctions

Addition

• $(f + g)(x) = f(x) + g(x)$

Soustraction

• $(f - g)(x) = f(x) - g(x)$

Multiplication

• $(f \cdot g)(x) = f(x) \cdot g(x)$

Division

• $(f / g)(x) = f(x) / g(x)$

Composition

• $(f \circ g)(x) = f(g(x))$

Fonctions spéciales

Valeur absolue

$|x| = \begin{cases} x, & \text{si } x \geq 0 \ -x, & \text{si } x < 0 \end{cases}$

Partie entière

• $\lfloor x \rfloor$ = plus grand entier inférieur ou égal à x.

Trigonométrie

Cercle trigonométrique

• Un cercle de rayon 1 centré à l'origine d'un plan cartésien.

Fonctions trigonométriques de base

Sinus (sin)

• Rapport du côté opposé à l'hypoténuse dans un triangle rectangle.

Cosinus (cos)

• Rapport du côté adjacent à l'hypoténuse dans un triangle rectangle.

Tangente (tan)

• Rapport du côté opposé au côté adjacent dans un triangle rectangle.
  • tan(x) = sin(x) / cos(x).

Identités trigonométriques importantes

• sin$^2$(x) + cos$^2$(x) = 1
• sin(2x) = 2sin(x)cos(x)
• cos(2x) = cos$^2$(x) - sin$^2$(x)

Valeurs remarquables

Limites

Définition

• La limite d'une fonction f(x) lorsque x approche a, notée $\lim_{x \to a} f(x) = L$, signifie que les valeurs de f(x) se rapprochent arbitrairement près de L lorsque x se rapprochent de a.

Propriétés

• $\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$
• $\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$
• $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$
• $\lim_{x \to a} [f(x) / g(x)] = \lim_{x \to a} f(x) / \lim_{x \to a} g(x)$, si $\lim_{x \to a} g(x) \neq 0$

Limites Importantes

• $\lim_{x \to 0} \frac{sin(x)}{x} = 1$
• $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$

Dérivées

Définition

• La dérivée d'une fonction f(x) en un point x, notée f'(x) ou $\frac{df}{dx}$, représente le taux de variation instantané de la fonction en ce point.
• $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Règles de Dérivation

• $(c)' = 0$ (c est une constante)
• $(x^n)' = nx^{n-1}$
• $(cf(x))' = cf'(x)$
• $(f(x) + g(x))' = f'(x) + g'(x)$
• $(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$ (règle du produit)
• $(f(x) / g(x))' = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$ (règle du quotient)
• $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ (règle de la chaîne)

Dérivées de Fonctions Élémentaires

• $(sin(x))' = cos(x)$
• $(cos(x))' = -sin(x)$
• $(tan(x))' = sec^2(x) = 1 + tan^2(x)$
• $(e^x)' = e^x$
• $(ln(x))' = \frac{1}{x}$

Applications des Dérivées

• Optimisation, and analyse de function.

Calcul Intégral

Intégrales Indéfinies

Définition

• Une intégrale indéfinie d'une fonction f(x), notée $\int f(x) dx$, est une fonction F(x) telle que F'(x) = f(x).
• $\int f(x) dx = F(x) + C$, où C est la constante d'intégration.

Règles d'Intégration

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, pour $n \neq -1$
• $\int \frac{1}{x} dx = ln|x| + C$
• $\int e^x dx = e^x + C$
• $\int sin(x) dx = -cos(x) + C$
• $\int cos(x) dx = sin(x) + C$

Techniques d'Intégration

• Intégration par substitution
• Intégration par parties
• Décomposition en éléments simples

Intégrales Définies

Définition

• Une intégrale définie d'une fonction f(x) sur un intervalle [a, b], notée $\int_{a}^{b} f(x) dx$, représente l'aire algébrique sous la courbe de f(x) entre a et b