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Local and Global Extrema, Critical Points, and Saddle Points

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Extremos locales

Definición. Sea f : C → R una función definida en un abierto C ⊆ R n. -Diremos que el punto a ∈ C es un máximo relativo (local) estricto de f si f (a) > f (x) para todo x perteneciente a un entorno reducido de a en C. -Diremos que el punto a ∈ C es un mínimo relativo (local) estricto de f si f (a) < f (x) para todo x perteneciente a un entorno reducido de a en C. -Diremos que el punto a ∈ C es un máximo relativo (local) de f si f (a) > f (x) para todo x perteneciente a un entorno de a en C. -Diremos que el punto a ∈ C es un mínimo relativo (local) de f si f (a) ≤ f (x) para todo x perteneciente a un entorno de a en C. A estos máximos y mínimos los llamaremos extremos locales de f en C.

Extremos

... Continue reading "Local and Global Extrema, Critical Points, and Saddle Points" »

Key Aspects of English Language Instruction

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Unit 10: English Spelling and Phonology

Index:

  • English Spelling
  • Orthographic Codes in EFL (Rules)
  • Sound-Grapheme Relationships: Vowels, Consonants, Silent Letters
  • Teaching the Written Code
  • Games and Tasks

1. English Spelling

Study of language from two perspectives:

  • Diachronic: Study of language over time.
  • Synchronic: Study over a specific period in history.

English is the result of the influence of different languages: French, Greek, Latin, Anglo-Saxon (Latin alphabet).

The Great Vowel Shift was a massive change affecting long vowels in the 15th-18th centuries.

Examples: 'enquiry' vs. 'inquiry', American English 'colour' vs. 'color'.

2. Orthographic Codes in EFL (Rules)

Requires good visual memory and the ability to associate forms and meanings.

Rules:

  • Consonant
... Continue reading "Key Aspects of English Language Instruction" »

Persuasive Speeches: Brutus and Antony in Julius Caesar

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Rhetorical Appeals in Julius Caesar

Brutus's Use of Rhetoric

Ethos: Establishing Credibility

Brutus spoke in prose, not blank verse. This was the everyday language of the common people. By speaking this way, Brutus effectively conveyed, “Hey, I’m one of you.”

He further established his credibility by stating, “Believe me for mine honor.” With this, Brutus asserted that the audience should trust and believe him due to his established reputation as an honorable person.

In essence, Brutus utilized Ethos in his speech to tell his audience, “Believe me...I’m a good guy, I’m one of you.”

Logos: Appealing to Logic

Brutus posed a rhetorical question: “Would you rather that Caesar be alive and you be slaves?” He used this to logically... Continue reading "Persuasive Speeches: Brutus and Antony in Julius Caesar" »

Literary Foundations: Beowulf, Poe, and Critical Terminology

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Beowulf: An Epic Poem

Beowulf is a foundational work of English literature, an epic poem with a rich history and profound themes.

Key Characteristics of Beowulf

  • Genre: Epic Poem
  • Publication: Circa 11th century in England (manuscript)
  • Composition: Believed to have been composed between 700 and 750 CE
  • Original Language: Old English
  • Oral Tradition: First told orally and passed on from poet to poet before being written down.
  • Author: Anonymous
  • Setting: Scandinavia
  • Literary Device: Extensive use of alliteration
  • Preservation: The original manuscript is preserved in the British Museum.

Core Values in Beowulf

  • Courage
  • Pride
  • Loyalty

Significance of Beowulf

  1. It is the longest poem written in Old English.
  2. It is considered the first example of poetry in a European language
... Continue reading "Literary Foundations: Beowulf, Poe, and Critical Terminology" »

Joshua's Conquest of Canaan and Israel's Tribal Inheritance

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Joshua's Conquest of Canaan: Campaigns and Tribal Inheritance

Understanding Nefesh and Neshama in Context

The concepts of Nefesh and Neshama are often discussed in relation to the nature of life and soul, particularly in the context of the commands given during the conquest:

  • Nefesh: Associated with animals; refers to the basic life force. In the context of conquest, it implies the killing of those who stood in the way.
  • Neshama: Associated with humans; refers to the higher soul or spirit. In the context of conquest, it implies the command to eliminate all human inhabitants of Canaan.

Joshua's Campaigns and the Conquest of Canaan

The Book of Joshua details the Israelite conquest of the land of Canaan, led by Joshua, fulfilling the divine commands given... Continue reading "Joshua's Conquest of Canaan and Israel's Tribal Inheritance" »

Estats i Capitals d'Amèrica

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Estats i Capitals Americans

Amèrica del Nord

  • Canadà - Ottawa
  • Estats Units - Washington DC
  • Mèxic - Ciutat de Mèxic

Amèrica Central i el Carib

  • Antigua i Barbuda - Saint John's
  • Bahames - Nassau
  • Barbados - Bridgetown
  • Belize - Belmopan
  • Costa Rica - San José
  • Cuba - L'Havana
  • Dominica - Roseau
  • El Salvador - San Salvador
  • Grenada - Saint George's
  • Guatemala - Guatemala
  • Haití - Port-au-Prince
  • Hondures - Tegucigalpa
  • Jamaica - Kingston
  • Nicaragua - Managua
  • Panamà - Panamà
  • República Dominicana - Santo Domingo
  • Saint Kitts i Nevis - Basseterre
  • Saint Lucia - Castries
  • Saint Vincent i Grenadines - Kingstown
  • Trinitat i Tobago - Port-of-Spain

Amèrica del Sud

  • Argentina - Buenos Aires
  • Bolívia - Sucre
  • Brasil - Brasília
  • Xile - Santiago
  • Colòmbia - Bogotà
  • Equador - Quito
  • Guyana - Georgetown
  • Paraguai
... Continue reading "Estats i Capitals d'Amèrica" »

Modernist Literary Figures: Joyce, Mansfield, Woolf

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Key Characteristics of Modernist Novels

The literary works of the early 20th century often feature a deeply poetic tone and profound philosophical connotations. These elements are central to understanding the innovative approaches of authors like James Joyce, Katherine Mansfield, and Virginia Woolf.

James Joyce: Irish Modernist Pioneer

Born in Dublin in 1882 and passing away in Zurich in 1941, James Joyce was an influential Irish Catholic (Jesuit-educated) novelist and poet. His works are deeply shaped by two fundamental components: his Jesuit upbringing and the city of Dublin, both of which he sought to escape. He lived abroad in Trieste, Paris, and Zurich.

Notable Works by James Joyce

  • Dubliners (1914): A collection of 15 poignant sketches depicting
... Continue reading "Modernist Literary Figures: Joyce, Mansfield, Woolf" »

Key Figures of 19th and 20th Century British and American Literature

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British Literature: Traditional Styles and Norms

Joseph Conrad (19th/20th Century)

  • Background: Polish origin, served as a sailor for 24 years in the British Merchant Navy.
  • Themes: Works inspired by his journeys, featuring exotic and romantic landscapes.
  • Style: Analyzed individuals in exceptional and extreme circumstances; condemned the cruelty of colonialism.
  • Technique: Some stories are told from different points of view.

Rudyard Kipling (19th/20th Century)

  • Background: Lived in India until age 5, then sent to Britain for education.
  • Career: Worked as a combatant by age 16 and lived until 23.
  • Reception: Idealized by nationalists and criticized by imperialists.
  • Notable Work: The Jungle Book.

Virginia Woolf

  • Background: Born in London, educated at home in her
... Continue reading "Key Figures of 19th and 20th Century British and American Literature" »

Ancient Rome's Founding: Myths and History

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The Founding of Rome: Between Myth and Reality

The traditional date for the founding of the greatest city of the Western world was the product of guesswork by Roman writers of the late centuries BC, working backward from their own time. There were reports of kings of Rome in the early days before the last of them, Tarquin the Proud, was expelled and the Roman Republic was founded, which was believed to have happened in 510 BC. Allowing for the reigns of Tarquin's predecessors carried the calculation back some two to three hundred years. After ranging shots by various writers, the author Varro, greatly respected for his learning in the first century BC, settled on the year 753 BC, which became the accepted, official date. All subsequent dates... Continue reading "Ancient Rome's Founding: Myths and History" »

The History of 'Happy Birthday to You' Song

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The History of "Happy Birthday to You"

"Happy Birthday" had its origins in 1893, when Mildred Hill, a kindergarten teacher in Louisville, Kentucky, put together a simple little tune. Whether she composed it entirely out of her head, or was influenced by other "folk song" fragments, is not clear. But when her younger sister Patty Hill, also a teacher, added words, the groundwork was laid for a very pleasing song.

From Greeting to Celebration

They called it "Good Morning to All":

  • Good morning to you,
  • Good morning to you.
  • Good morning, dear children,
  • Good morning to all.

"Good Morning to All" was initially intended, and used, as a welcoming song to be sung by the teacher to the class each morning. But when it was published later in 1893, in a book of... Continue reading "The History of 'Happy Birthday to You' Song" »