Data Visualization: Charts and Tools

Chapter 6 - Data Visualization

• Data visualization stands for the graphical representation of information and data.
• Tools for data visualization include Tableau, Power BI, Google Charts, and Grafana.

Benefits of Data Visualization

• Information and data can be understood quickly and easily.
• Trends and outliers may be found through data visualization.
• New questions and areas of exploration can be identified.

Charts

Categorical Data

Bar Chart

• Bar charts compare different categories using rectangular bars.
• Categories are displayed on one axis and values are displayed on the other.

Pie Chart

• Pie charts represent the proportion of different categories in a circle.
• The angle of each sector is proportional to its associated percentage.

Numerical Data

Line Chart

• Data points connected by lines are displayed on line charts.
• Line charts are useful for showing trends and changes over a continuous period.

Scatter Plot

• Scatter plots use dots to represent two different variables.
• Correlations and patterns may be revealed through scatter plots.

Histogram

• Histograms represent the distribution of continuous data.
• Data is grouped into bins, while each bar's is proportional to its frequency.

Other Chart Types

Heatmap

• Heatmaps use color-coding to represent data values in a matrix.
• Patterns and correlations can be identified using heatmaps.

Box Plot

• Quartiles, median, and outliers are all displayed in a box plot.
• The distribution of different groups can be compared with box plots.

Choosing the Right Chart Type

• Data and questions should be understood to produce the right chart type.
• Bar charts and pie charts are used for categorical comparison.
• Line charts are useful to see trends over time.
• Scatter plots and heatmaps are helpful for displaying the relationship between variables.
• For distributions, use histograms and box plots.

Best Practices for Effective Visualization

• It needs to be simple and clear.
• Colors must be appropriate.
• Clear labels and titles are needed.
• Misleading scales or distortions should be avoided.
• Audiences' data understanding needs be considerd.

Data Visualization Tools

Python Libraries

Matplotlib

• A comprehensive library can create static, interactive, and animated visualizations in Python.
• This library offers control over every aspect of the chart.

import matplotlib.pyplot as plt
## Sample data
x = [1, 2, 3, 4, 5]
y = [2, 3, 5, 7, 11]
## Create a line plot
plt.plot(x, y)
## Add labels and title
plt.xlabel('X-axis')
plt.ylabel('Y-axis')
plt.title('Simple Line Plot')
## Show the plot
plt.show()

Seaborn

• Built on top of Matplotlib.
• Offers a high-level interface that helps create informative statistical graphics.

import seaborn as sns
import matplotlib.pyplot as plt
## Sample data
data = {'Category': ['A', 'B', 'C', 'A', 'B', 'C'],
        'Value': [10, 15, 7, 12, 9, 13]}
import pandas as pd
df = pd.DataFrame(data)
## Create a bar plot
sns.barplot(x='Category', y='Value', data=df)
## Add title
plt.title('Bar Plot using Seaborn')
## Show the plot
plt.show()

Plotly

• Allows interactive, web-based visualizations.

import plotly.express as px
## Sample data
data = {'Fruit': ['Apples', 'Oranges', 'Bananas', 'Apples', 'Oranges', 'Bananas'],
        'Amount': [4, 1, 2, 2, 4, 5],
        'City': ['SF', 'SF', 'SF', 'Montreal', 'Montreal', 'Montreal']}
import pandas as pd
df = pd.DataFrame(data)
## Create a bar chart
fig = px.bar(df, x="Fruit", y="Amount", color="City", barmode="group")
## Show the plot
fig.show()

Other tools

Tableau

• Business intelligence tool for interactive data visualization.

Power BI

• Microsoft's business analytics service for visualization and analysis.

Google Charts

• Web-based tool for interactive charts and dashboards.

Chemical Kinetics

Reaction Rate

• For the reaction: $aA + bB \rightarrow cC + dD$
• a, b, c, and d are stoichiometric coefficients.
• Rate of reaction is $Rate = -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

Rate Law

• The rate law expresses the relationship between the rate of a reaction and the concentrations of the reactants: $Rate = k[A]^m [B]^n$
• k is the rate constant
• m is the order of the reaction with respect to A
• n is the order of the reaction with respect to B
• m + n is the overall order of the reaction.

Integrated Rate Laws

• $[A]_t$ stands for concentration of A at time t
• $[A]_0$ stands for the initial concentration of A

Arrhenius Equation

• The Arrhenius equation describes the temperature dependence of the rate constant: $k = Ae^{-\frac{E_a}{RT}}$
• A is the pre-exponential factor
• $E_a$ is the activation energy
• R is the ideal gas constant (8.314 J/mol·K)
• T is the temperature in Kelvin

Biopotential Electrodes

Half-cell Potential

• Half-cell potential ($E_{hc}$) is the potential difference arises at the metal-electrolyte interface when a metal electrode is immersed in an electrolyte solution
• $E_{hc} = E_0 - \frac{RT}{nF} \ln(M^{n+})$
• $E_0$ is the standard electrode potential
• $R$ is the universal gas constant ($8.314 \frac{J}{mol \cdot K}$)
• $T$ is the absolute temperature in Kelvin
• $n$ is the valence of the metallic ions
• $F$ is the Faraday constant ($96,485 \frac{C}{mol}$)
• $M^{n+}$ is the activity of the metal ions in the solution

Electrode Polarization

Polarizable Electrodes

• Exhibit large variations in half-cell potential with small changes in current.
• Interface is easily polarized
• Little or no actual charge crosses the metal-electrolyte interface, the current being displacement current.
• Example: Stainless steel electrodes

Nonpolarizable Electrodes

• Exhibit small changes in half-cell potential with large changes in current.
• Interface is not easily polarized
• Faradaic current flow.
• Example: Ag-AgCl electrodes

Ag-AgCl Electrode

• Silver-silver chloride (Ag-AgCl) electrode consists of a silver wire coated with a layer of silver chloride immersed in a chloride-containing electrolyte solution.

Half-cell Potential

• $AgCl + e^- \rightleftharpoons Ag + Cl^-$
• $E_{AgCl} = E_0 + \frac{RT}{nF} \ln(a_{Cl^-})$
• $a_{Cl^-}$ is the activity of chloride ions.

Advantages

• Very stable
• Low polarization
• Low noise
• Easily manufactured in different shapes and sizes

Types of Electrodes

Microelectrodes

• Used for intracellular recordings
• Tip diameter is less than $5 \mu m$
• High impedance (1-100 $M\Omega$)
• Made of metal or glass

Metal Plate Electrodes

• Made of stainless steel, gold, platinum, or silver
• Applied to skin surface by conductive gel or paste
• Used for ECG, EEG, and EMG recordings

Suction Electrodes

• Held to the skin by suction
• Provide good electrical contact
• Used for ECG recordings

Floating Electrodes

• Make contact with the skin through a conductive gel or paste
• Reduce motion artifacts
• Used for long-term monitoring

Needle Electrodes

• Inserted directly into the muscle tissue
• Provide high-quality EMG recordings
• Can be painful and increase the risk of infection

Elaboración de Artículos Científicos

• MSc. Juan Carlos Osorio Gómez
• Biólogo, MSc en Acuicultura
• Medellin, Colombia
• 2016

Tabla de contenido

• ¿Por qué publicar?
• ¿Dónde publicar?
• ¿Cómo se estructura un artículo científico?
• ¿Cómo escribir cada sección de un artículo científico?
• Recomendaciones generales
• ¿Cómo se somete un artículo científico?
• ¿Qué hacer después de someter un artículo científico?

¿Por qué publicar?

• To publicize your research findings
• To contribute to the advancement of scientific knowledge.
• To gain recognition and prestige as a researcher.
• To meet the requirements of your postgraduate program.
• To improve your curriculum vitae.
• To obtain funding for future investigations.

¿Dónde publicar?

Types of Journals

• Indexed journals: Journals that comply with a series of quality criteria and are included in bibliographic databases (Web of Science, Scopus, etc.).
• Non-indexed journals: Journals that do not comply with the quality criteria of the bibliographic databases.

Criteria for selecting a journal

• Thematic area: The journal should publish articles related to your research area.
• Impact factor: The impact factor measures how often a journal's articles are cited.
• Publication time: The time a journal takes to publish an article after acceptance.
• Publication cost: Some journals charge to publish articles.
• Visibility: The journal should be visible to your target audience.

¿Cómo se estructura un artículo científico?

• Title
• Authors and affiliation
• Abstract
• Keywords
• Introduction
• Materials and methods
• Results
• Discussion
• Conclusions
• Acknowledgements
• References

¿Cómo escribir cada sección de un artículo científico?

Title

• It must be concise, clear, and precise.
• It must reflect the article's content.
• It must be attractive to readers.
• Avoid using abbreviations and technical jargon.

Authors and affiliation

• Include all authors who have contributed significantly to the article.
• Indicate each author's affiliation (institution to which they belong).
• Indicate the corresponding author (person to whom communications will be directed).

Abstract

• It must be a summary of the article.
• It must include the main objectives, methods, results, and conclusions.
• It must be clear, concise, and precise.
• It must be understandable to readers unfamiliar with the topic.
• Maximum extension: 250 words.

Keywords

• They should be relevant terms for the article's topic.
• They should facilitate the article's search in bibliographic databases.
• Number of keywords: 3-5.

Introduction

• It must present the research problem.
• It must review the relevant literature.
• It must justify the importance of the research.
• It must establish the research objectives.

Materials and methods

• It must describe the materials and methods used in the research.
• It must be detailed enough for other researchers to replicate the study.
• It must include information about the experimental design, participants, instruments, and procedures.

Results

• It must present the research results.
• It must use tables and figures to illustrate the results.
• It must be objective and avoid interpreting the results.

Discussion

• It must interpret the research results.
• It must relate the results to the existing literature.
• It must discuss the implications of the results.
• It must acknowledge the limitations of the research.

Conclusions

• It must summarize the main findings of the research.
• It must highlight the importance of the research.
• It must suggest future lines of research.

Acknowledgements

• Thank people and institutions that have contributed to the research.

References

• All sources cited in the article should be included.
• It must follow a specific citation format (APA, MLA, Chicago, etc.).

Recomendaciones generales

• Write clearly, concisely, and precisely.
• Use formal and objective language.
• Avoid technical jargon and abbreviations.
• Revise the grammar and spelling carefully.
• Ask other researchers to review your article before submitting it.

¿Cómo se somete un artículo científico?

• Select an appropriate journal.
• Read the journal's instructions for authors.
• Prepare your article according to the journal's instructions.
• Submit your article to the journal through the online submission system.
• Wait for the journal's response.

¿Qué hacer después de someter un artículo científico?

• If your article is accepted, review the galley proofs and respond to the editors' questions.
• If your article is rejected, review the reviewers' comments and consider submitting it to another journal.

Matemáticas

Introducción

• Las matemáticas son el estudio de las relaciones entre cantidades, magnitudes y propiedades, y de las operaciones lógicas utilizadas para deducir cantidades, magnitudes y propiedades desconocidas.

Álgebra

Expresiones algebraicas

• Una expresión algebraica es una combinación de letras, números y signos de operaciones.
• Las letras suelen representar cantidades desconocidas y se denominan variables o incógnitas.
• Ejemplo: $3x^2 + 2x - 5$

Operaciones con expresiones algebraicas

• Sumar o restar expresiones algebraicas, se deben combinar los términos semejantes.
• Multiplicar expresiones algebraicas, se utiliza la propiedad distributiva.
• Dividir expresiones algebraicas, se utiliza la división larga o la división sintética.

Ecuaciones

• Una ecuación es una igualdad entre dos expresiones algebraicas.
• Resolver una ecuación consiste en encontrar el valor o los valores de la variable que hacen que la igualdad sea verdadera.
• Ejemplo: $2x + 3 = 7$
• Para resolver esta ecuación, se puede restar 3 a ambos lados de la ecuación y luego dividir ambos lados de la ecuación entre 2:
  • $2x + 3 - 3 = 7 - 3$
  • $2x = 4$
  • $x = 2$

Geometría

Figuras geométricas

• Una figura geométrica es un conjunto de puntos.
• Las figuras geométricas más comunes son el punto, la línea, el plano, el triángulo, el cuadrado, el círculo y la esfera.

Área

• El área es la medida de la superficie de una figura geométrica.
• El área de un cuadrado se calcula multiplicando la longitud de un lado por sí misma. El área de un círculo se calcula multiplicando el número pi ($\pi$) por el radio al cuadrado.

Volumen

• El volumen es la medida del espacio ocupado por un objeto.
• El volumen de un cubo se calcula multiplicando la longitud de un lado por sí misma tres veces.
• El volumen de una esfera se calcula multiplicando 4/3 por el número pi ($\pi$) por el radio al cubo.

Trigonometría

Ángulos

• Un ángulo es la medida de la apertura entre dos líneas que se intersectan en un punto.
• Los ángulos se miden en grados o radianes.
• Un círculo completo tiene 360 grados o $2\pi$ radianes.

Funciones trigonométricas

• Las funciones trigonométricas son funciones que relacionan los ángulos de un triángulo con las longitudes de sus lados.
• Las funciones trigonométricas más comunes son el seno, el coseno y la tangente.

Teorema de Pitágoras

• El teorema de Pitágoras establece que en un triángulo rectángulo, el cuadrado de la hipotenusa (el lado opuesto al ángulo recto) es igual a la suma de los cuadrados de los otros dos lados.

Cálculo

Límites

• Un límite es el valor al que se acerca una función cuando la variable se acerca a un valor determinado.
• Los límites se utilizan para definir la continuidad, la derivada y la integral de una función.

Derivadas

• La derivada de una función es la pendiente de la recta tangente a la gráfica de la función en un punto determinado.
• Las derivadas se utilizan para encontrar los máximos y mínimos de una función, y para estudiar la variación de una función.

Integrales

• La integral de una función es el área bajo la gráfica de la función entre dos puntos determinados.
• Las integrales se utilizan para calcular áreas, volúmenes y probabilidades.

Research

Designing studies

Why Conduct Research?

• Add to the body of knowledge
• Dispel myths
• Improve practice
• Inform policy debates

Generating a Research Question

• What problem needs investigation?
• Consider feasibility
• Consider your values

Overcoming Challenges

• What if no one has studied your specific question?
  • Is it too narrow?
  • Can you broaden the question?
• What if there is too much research on the question?
  • Can you narrow the question?
  • Can you apply it to a new population of interest?

Types of Research Questions

• Descriptive: seeks to describe a phenomenon or document its characteristics
  • "What are the characteristics of students who are suspended?"
• Predictive: seeks to predict an outcome
  • "Does attending preschool predict high school graduation?"
• Explanatory: seeks to understand the relationship between two or more constructs
  • "Does parental involvement impact student achievement?"

Quantitative vs. Qualitative

Variables

Types of Variables

• Independent Variable
  • The presumed cause
  • The variable you manipulate
  • Predictor variable
• Dependent Variable
  • The presumed effect
  • The variable you measure
  • Outcome variable

Conceptual vs. Operational Definitions

• Conceptual Definition: dictionary definition of the concept
  • Example: Bullying is unwanted, aggressive behavior
• Operational Definition: how you will measure the concept
  • Example: Bullying is the number of times a student reports being called names or pushed in a week

Hypotheses

• A testable statement about the relationship between two or more variables
  • Based on theory or prior research
  • States the expected relationship between variables
  • Must be falsifiable
• Example: Students who attend preschool will have higher reading achievement in 3rd grade than students who do not attend preschool.

Directional vs. Non-Directional Hypotheses

• Directional Hypothesis: states the direction of the relationship
  • Example: Increased instructional time will lead to higher student achievement.
• Non-Directional Hypothesis: states that a relationship exists, but not the direction
• Example: Instructional time is related to student achievement.

Lecture 11

October 22, 2013

Protein Structure Determination

• X-ray Crystallography
• Nuclear Magnetic Resonance (NMR) Spectroscopy
• Cryo-Electron Microscopy (cryo-EM)

X-ray Crystallography

• Grow crystals of the protein
• Shine X-rays on the crystal
• X-rays diffract based on the position of atoms
• Compute electron density map
• Fit the protein sequence to the map

Resolution is Key

Lower Resolution

• 4-6Å
• Helices visible
• Backbone visible
• Side chains are not resolved

Higher Resolution

• 1-2Å
• Atomic Resolution
• See individual atoms
• Get detailed information about bonding geometry and charge

Growing Crystals is an Art

• Get pure protein
• Concentrate the protein
• Add salt or organic solvent to precipitate the protein
• Hope that crystals grow
• This can take days, weeks, months, or years

Nuclear Magnetic Resonance (NMR) Spectroscopy

• Good for small proteins (

Calculus

1. Limits and Continuity

1.1. The concept of limit

• Intuitive definition of limit:
  • $x \rightarrow a$ ($x$ approaches $a$)
  • $f(x) \rightarrow L$ ($f(x)$ approaches $L$)
• Formal definition of limit:
  • $\forall \epsilon > 0$, $\exists \delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$
• One-sided limits:
  • $\lim_{x \to a^+} f(x)$ (right-hand limit)
  • $\lim_{x \to a^-} f(x)$ (left-hand limit)
• Infinite limits and limits at infinity:
  • $\lim_{x \to a} f(x) = \infty$ or $-\infty$
  • $\lim_{x \to \infty} f(x) = L$ or $\infty$
• Limit laws:
  • Sum, difference, product, quotient, and power laws
• Techniques for evaluating limits:
  • Direct substitution
  • Factoring
  • Rationalizing
  • Using trigonometric identities
  • Squeeze Theorem

1.2. Continuity

• Definition of continuity at a point:
  • $f$ is continuous at $x = a$ if $\lim_{x \to a} f(x) = f(a)$
• Types of discontinuities:
  • Removable discontinuity
  • Jump discontinuity
  • Infinite discontinuity
• Continuity on an interval:
  • $f$ is continuous on an open interval $(a, b)$ if it is continuous at every point in $(a, b)$.
  • $f$ is continuous on a closed interval $[a, b]$ if it is continuous on $(a, b)$ and $\lim_{x \to a^+} f(x) = f(a)$ and $\lim_{x \to b^-} f(x) = f(b)$.
• Properties of continuous functions:
  • The sum, difference, product, and quotient of continuous functions are continuous (where the quotient is defined).
  • The composition of continuous functions is continuous.
• Intermediate Value Theorem:
  • If $f$ is continuous on $[a, b]$ and $k$ is any number between $f(a)$ and $f(b)$, then there exists a number $c$ in $(a, b)$ such that $f(c) = k$.

2. Derivatives

2.1. Definition of the derivative

• The derivative as the slope of a tangent line
• The derivative as the instantaneous rate of change
• Definition of the derivative:
  • $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$
• Differentiability and continuity:
  • If $f$ is differentiable at $x = a$, then $f$ is continuous at $x = a$.
  • The converse is not necessarily true.
• One-sided derivatives:
  • Right-hand derivative: $\lim_{h \to 0^+} \frac{f(x + h) - f(x)}{h}$
  • Left-hand derivative: $\lim_{h \to 0^-} \frac{f(x + h) - f(x)}{h}$

2.2. Basic differentiation rules

• Power rule:
  • $\frac{d}{dx}(x^n) = nx^{n-1}$
• Constant rule:
  • $\frac{d}{dx}(c) = 0$
• Constant multiple rule:
  • $\frac{d}{dx}[cf(x)] = cf'(x)$
• Sum and difference rules:
  • $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$
• Product rule:
  • $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$
• Quotient rule:
  • $\frac{d}{dx}[\frac{f(x)}{g(x)}] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$
• Chain rule:
  • $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

2.3. Derivatives of Trigonometric, Exponential, and Logarithmic Functions

2.4. Implicit Differentiation

• When to use implicit differentiation
• Technique for finding $\frac{dy}{dx}$ when $y$ is defined implicitly as a function of $x$

2.5. Higher-Order Derivatives

• Finding $f''(x)$, $f'''(x)$, etc.
• Notation for higher-order derivatives

2.6. Applications of the Derivative

• Related rates:
  • Solving problems involving rates of change of related variables
• Linear approximation and differentials:
  • Using the tangent line to approximate the value of a function near a given point
  • Differentials as estimates of change
• L'Hôpital's Rule:
  • Using derivatives to evaluate limits of indeterminate forms (e.g., $\frac{0}{0}$, $\frac{\infty}{\infty}$)

3. Applications of Derivatives

3.1. Maximum and Minimum Values

• Critical points:
  • Finding critical points of a function (where $f'(x) = 0$ or $f'(x)$ does not exist)
• Absolute (global) maximum and minimum values:
• Finding the absolute maximum and minimum values of a function on a closed interval
• The Extreme Value Theorem:
  • If $f$ is continuous on a closed interval $[a, b]$, then $f$ has an absolute maximum value and an absolute minimum value on $[a, b]$.
• Local (relative) maximum and minimum values:
  • Using the first derivative test to find local maximum and minimum values
• The first derivative test:
  • Using the sign of $f'(x)$ to determine where $f(x)$ is increasing or decreasing
• The second derivative test:
  • Using the sign of $f''(x)$ to determine the concavity of $f(x)$ and to find local maximum and minimum values

3.2. Curve Sketching

• Using the first and second derivatives to analyze the shape of a curve
• Finding intervals where a function is increasing, decreasing, concave up, or concave down
• Finding points of inflection
• Sketching the graph of a function using information about its derivatives

3.3. Optimization Problems

• Setting up and solving optimization problems in various contexts (e.g., maximizing area, minimizing cost)

3.4. Mean Value Theorem

• The Mean Value Theorem:
  • If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a number $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$.

4. Integration

4.1. Indefinite Integrals

• Antiderivatives:
  • Finding antiderivatives of functions
• Basic integration rules:
  • Power rule for integration
  • Constant multiple rule
  • Sum and difference rules
• Indefinite integrals of basic functions:
  • Integrating polynomials, trigonometric functions, exponential functions, and logarithmic functions
• Integration by substitution (u-substitution):
  • Using u-substitution to evaluate indefinite integrals

4.2. Definite Integrals

• The definite integral as the area under a curve
• Riemann sums:
  • Approximating definite integrals using Riemann sums (left, right, midpoint)
• The limit definition of the definite integral:
  • $\int_a^b f(x) , dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x$
• Properties of definite integrals:
  • $\int_a^a f(x) , dx = 0$
  • $\int_a^b f(x) , dx = -\int_b^a f(x) , dx$
  • $\int_a^b [cf(x)] , dx = c \int_a^b f(x) , dx$
  • $\int_a^b [f(x) \pm g(x)] , dx = \int_a^b f(x) , dx \pm \int_a^b g(x) , dx$
  • $\int_a^c f(x) , dx + \int_c^b f(x) , dx = \int_a^b f(x) , dx$

4.3. The Fundamental Theorem of Calculus

• The Fundamental Theorem of Calculus, Part 1:
• If $f$ is continuous on $[a, b]$, then the function $F(x) = \int_a^x f(t) , dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$, and $F'(x) = f(x)$.
• The Fundamental Theorem of Calculus, Part 2:
  • If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$, then $\int_a^b f(x) , dx = F(b) - F(a)$.
• Using the Fundamental Theorem to evaluate definite integrals

4.4. Techniques of Integration

• Integration by parts:
  • Using integration by parts to evaluate integrals of the form $\int u , dv$
• Trigonometric integrals:
  • Evaluating integrals involving trigonometric functions
• Trigonometric substitution:
  • Using trigonometric substitution to evaluate integrals involving radicals
• Partial fractions:
  • Using partial fractions to evaluate integrals of rational functions

4.5. Applications of Definite Integrals

• Area between curves:
  • Finding the area between two curves using definite integrals
• Volumes of solids of revolution:
  • Using the disk method, washer method, and cylindrical shells to find the volumes of solids of revolution
• Average value of a function:
  • Finding the average value of a function on an interval using a definite integral

5. Differential Equations

5.1. Basic Concepts

• Definitions of differential equations:
  • Ordinary differential equation (ODE)
  • Partial differential equation (PDE)
• Order and linearity of differential equations
• Solutions of differential equations:
  • General solution
  • Particular solution

5.2. First-Order Differential Equations

• Separable equations:
  • Solving separable differential equations by separating variables and integrating
• Linear equations:
  • Solving linear differential equations using an integrating factor

6. Infinite Sequences and Series

6.1. Sequences

• Definition of a sequence:
  • An ordered list of numbers
• Convergence and divergence of sequences:
  • Determining whether a sequence converges to a limit or diverges
• Limit laws for sequences
• Squeeze Theorem for sequences

6.2. Series

• Definition of a series:
• The sum of the terms of a sequence
• Convergence and divergence of series:
  • Determining whether a series converges to a finite sum or diverges
• Geometric series:
  • Identifying and evaluating geometric series
• Telescoping series:
  • Identifying and evaluating telescoping series
• The Integral Test:
  • Using the Integral Test to determine the convergence or divergence of a series
• Comparison Tests:
  • Using the Comparison Test and the Limit Comparison Test to determine the convergence or divergence of a series
• Alternating Series Test:
  • Using the Alternating Series Test to determine the convergence of an alternating series
• Absolute and Conditional Convergence:
  • Determining whether a series converges absolutely or conditionally
• Ratio Test:
  • Using the Ratio Test to determine the convergence or divergence of a series
• Root Test:
  • Using the Root Test to determine the convergence or divergence of a series

6.3. Power Series

• Definition of a power series:
• A series of the form $\sum_{n=0}^\infty c_n (x - a)^n$
• Radius and interval of convergence:
  • Finding the radius and interval of convergence of a power series
• Representing functions as power series:
  • Finding the power series representation of a function
• Taylor and Maclaurin series:
  • Finding the Taylor and Maclaurin series of a function
• Applications of Taylor and Maclaurin series:
  • Approximating functions using Taylor polynomials
  • Evaluating limits using Taylor series
  • Solving differential equations using Taylor series

Economía

• Economy is the study of how societies use scarce resources to produce valuable goods and services and distribute them among different individuals.

Escasez y eficiencia:

• Two key ideas in economics are scarcity and efficiency.
• Escasez: Goods are limited relative to desires.
• Eficiencia: Involves utilizing resources in the best possible way to satisfy the needs and desires of individuals.
• Economics seeks to maximize the use of limited resources.

Microeconomía y macroeconomía:

• The main areas are microeconomics and macroeconomics.

• Microeconomía: Deals with the behavior of individual entities such as markets, companies, and households.

• Macroeconomía: Examines the performance of the economy overall, including topics like inflation, unemployment, and economic growth.

Lógica económica :

• Economists analyze economic events scientifically by:
  • Observing economic events
  • Relying on statistical analysis

Economists need to be aware of common fallacies in economic reasoning:

• The post hoc fallacy
• The failure to maintain everything else constant
• The fallacy of composition

Economía positiva frente a economía normativa

• It is important to distinguish economia positiva ans normative when considering the economy:

• Economía positiva: Deals with facts and behavior in the economy.

• Economía normativa: Involves judgments of value and ethics.

La economía como ciencia

• Economics is a social science that uses the scientific method to understand economic phenomena. Economists develop theories,