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# HL History - 1
The course code for this page is **CHW3MZ**.
## Command terms
The following terms are regularly used by IB and have specific meanings:
- **Analyse**: Break down an idea into its essential elements
- **Compare**: Identify and justify the similarities between ideas
- **Contrast**: Identify and justify the differences between ideas
- **Discuss**: Provide a balanced review with a range of justified opinions or conclusions
- **Evaluate**: Appraise an argument with strengths and limitations
- **Examine**: Consider an argument or concept, revealing its assumptions and interrelations
- **To what extent**: Consider the merits and demerits of an idea or argument with justified opinions or conclusions
## Historiography
Historiography is the study of how history is developed and how historians have viewed history. It helps develop the ability to view events from multiple perspectives and reveals how others might view events in multiple perspectives. The **historical method** outlines the process of making history from start to finish:
- **Questions are asked** about any of the following from controversies, new perspectives, and gaps in history:
- major/minor long-/short-term **causes** of an event
- the **nature** of an event — the purpose of the event, the status of the event
- the immediate/long-term/short-term **impacts** of an event
- the **ethics** of an event
- a hypothetical **counterfactual** version of an event
- **Data is collected** to help answer the questions
- **Data is analysed** to identify patterns and trends via statistics, make judgements using OPCVL, and make comparisons of the main message (content) and PERMS (political, economical, religious, military, social)
- **Arguments are created** with a thesis and critical analyses per argument
- and the **findings are communicated** via any means, including essays, opinion paragraphs, debates, presentations, music, etc.
### Lenses
There are seven main schools of thought historians usually fall under that attempt to explain why history happens.
- The **great people** lens views history with the perspective that individuals create changes, and without those individuals history would be drastically altered. This school focuses on their individual motivations, relationships, etc.
- The **structuralist** lens views history and events as caused by changes in economic structures, technology, ideology, and social norms.
- The **decisionist** lens views history with the perspective that it is the decisions of people that are the cause of events.
- The **Marxist** lens focuses on class conflict — the exploiter and the exploitee and how their conflicts affect history.
- The **Toynbee** lens regards the response of people to crises as the largest cause to historical events.
- The **post-modern** lens states that history reflects the time period it was written in and the intent of the author. This school asserts that there is no historical truth and focuses on how history is shaped and manipulated to serve the agendas and needs of various civilisations.
- The **feminist** lens states that history should reflect the experience of both genders equally. This school particularly focuses on the treatment of and lives of women.
### Historical concepts
There are six key historical concepts that should be taken under consideration in the analysis of history.
- **Change**: how people and events create change, the differences before and after an event, and the nature and pace of the change
- **Continuity**: what doesn't change and to what extent things don't change
- **Causation**: why an event occurred (social, economic, political, etc.), and what factor(s) caused it
- **Consequence**: the long- and short-term effects of an event on society
- **Significance**: why the event matters and the importance of some details over others
- **Perspective**: the different perspectives, why there are different perspectives, and how those perspectives affect the interpretation of history
### Data collection
The abundance of **diverse** data allows for greater perspectives to be gleaned and thus more solutions to a problem. Data and their sources can fall under one of three categories:
- **Primary** data is from an original source with no sources under it — e.g., newspapers, memoirs, photographs, diaries, etc.
- **Secondary** data is an interpretation of a primary source(s) with its own argument — e.g., articles, thesises, documentaries, etc.
- **Tertiary** data compiles secondary sources and should only be used for context — e.g., wikis, textbooks, encyclopedias, etc.
### Logic
!!! definition
- **Logic** is the study of rules of inference and the analysis of arguments.
- A **conclusion** is a proposition that follows all others.
- An **inference** is a connection that acts as a logical leap between a premise and a conclusion.
- **Logically consistent** statements follow the three laws of thought and do not contradict.
- **Logically contradicting** statements do not follow the three laws of thought.
- The **validity** of a statement is its correctness of reasoning via the laws of thought.
- A **sound** argument is of valid form and has a true premise.
- An **argument** is a simple statement or disagreement that attempts to reach a conclusion by proving something true with evidence. Good arguments are sound, valid, clear, and avoids hasty conclusions.
**Aristotle** laid the foundations for the principles of formal logic via the three laws of thought.
- **Law of identity**: everything is identical with itself.
- **Law of noncontradiction**: contradictory statements cannot both be true.
- **Law of excluded middle**: any proposition must either be true or false.
!!! example
- Identity: A football is a football.
- Noncontradiction: If water and oil do not mix, and substance A mixes in water, it must not be oil.
- Excluded middle: The Nintendo Switch must either be or not be a potato.
**Deductive** arguments connect a general statement to a more specific statement based on laws, rules, and/or widely accepted principles.
!!! example
As monkeys like bananas and Lucy is a monkey, Lucy must like bananas.
**Inductive** arguments connect a specific statement to a more general statement based on **empiric** data.
!!! example
As three of the eight billion humans on Earth are mortal, all humans must be mortal.
### Logical fallacies
- An **ad hominem** argument attacks the arguer or anything else instead of the argument.
- e.g., *"You're a Nintendo fanboy; of course you think that."*
- Arguments that **appeal to authority** use the opinion of an authority on a topic is used as evidence to support an argument.
- e.g., *"The President of the United States said that we should inject disinfectant into ourselves, so it must be a good idea!"*
- Arguments that **appeal to emotion** manipulate the recipient's emotions typically via loaded language to win an argument.
- e.g., *"Those island devils have robbed us of our sleep at night — they must be eradicated!"*
- Arguments that **appeal to force** use threats to win an argument.
- e.g., *"I'm right, aren't I?" said the jock, flexing her biceps threateningly.*
- Arguments that **appeal to ignorance** assert a proposition is true because it has not been proven false.
- e.g., *"My laptop must secretly have chips in it that no one can detect from aliens because we don't know if there* aren't *undetectable chips in there."*
- **Bandwagoning or herding** arguments assert that a conclusion is true because it is accepted by most people. This is a result of confirmation bias.
- e.g., *"Ma, everyone else is jumping off that bridge, so why shouldn't I?"*
- A fallacy of **accident** wrongly applies a general rule to a specific exception.
- e.g., *Since surgeons cut people with knives and cutting people with knives is a crime, surgeons are criminals.*
- An argument that **begs the question** has circular reasoning by having premises that assume its conclusion.
- e.g., *Acid must be able to eat through your skin because it is corrosive.*
- **Cherry picking** occurs when evidence that supports the conclusion is pointed out while those that contradict the conclusion are ignored or withheld.
- e.g., *"Look at these perfect cherries — their tree must be in perfect condition!"*
- A fallacy of **converse accident** wrongly applies a specific exception to a general rule.
- e.g., *As the Nintendo Switch, a game console, is portable, all game consoles must be portable.*
- A **complex question**, also known as a trick question, embeds a proposition that is accepted when a direct answer is given to the question.
- e.g., *Have you stopped abusing children yet?*
- Arguments with a **false cause** incorrectly assume a cause to an effect.
- e.g., imagining correlation implies causation.
- **Hasty generalisations** appear in inductive generalisations based on insufficient evidence.
- e.g., *Since the first seven odd numbers are prime or square, all odd numbers must be prime or square.*
- Arguments that **miss the point** provide an irrelevant conclusion that fails to address the issue of the question.
- e.g., *"Is it allowed?" "It should be allowed because it's nowhere near as bad as alcohol."*
- A **non sequitur** is an invalid argument that does not follow the laws of thought.
- e.g., *All humans are mammals. Whales are mammals. Therefore, whales are humans.*
- A **no true Scotsman** (appeal to purity) fallacy takes a generalisation and doubles down to protect it by excluding counterexamples typically via emotionally charged language.
- e.g., *"Although your father is a Scotsman and dances, no* true *Scotsman would dance."*
- Arguments with **recency bias** put greater importance on recent data over historic data.
- e.g., *As GameStop's stock has risen over the past few days dramatically, it will continue to do so.*
- **Red herrings** change the issue of subject away from the original question.
- e.g., *You should support the new housing bill. We can't continue to see people living in the streets; we must have cheaper housing.*
- A **straw man** argument misrepresents the opposing position by making their arguments sound more extreme.
- e.g., *"We should relax laws on immigration." "The instant we let millions of people through our border is when our country falls."*
## Causes of the Chinese Civil War
### Decline of the Manchu Qing Dynasty
**— Long-term structural political**
In the Qing dynasty, from 1861 to 1908, Empress **Cixi** ruled China as an autocrat. Corruption was rampant in Beijing and officials could not control warlords in remote regions. Under Cixi, China became weak and was easily influenced by foreign powers.
### Foreign involvement
**— Long-term structural political**
The influence of foreign powers increased outrage among citizens at the inability of the government to do things and led to greater internal dissent.
The **opium wars** starting from 1839 were two armed conflicts in China between Western powers and the Qing dynasty. Both were won easily by the West due to their superior, more modern military technology. This resulted in China signing a series of what came to be known as "**unfair treaties**" starting with the Treaty of Nanjing in 1842, which ended the First Opium War and gave Hong Kong to Britain "in perpetuity" among other trading concessions.
The **First Sino-Japanese War** (also known as the War or Jiawu) in 1894–1895 also resulted in Japan's easy victory due to obsolete Chinese military technology.
!!! example
During the First Sino-Japanese War, Cixi took military money and spent it on palace renovations, demonstrating the corruption in and ineffectiveness of the regime.
### Outdated agricultural practices and limited industrial development
**— Long-term structural economic**
!!! context
In 1900, Japan and the United States were major industrial powers and both were steadily modernising.
China's rulers believed that Westerners were barbarians and that nothing could be learned from industrialisation. This contributed to their weak military strength as they fell further behind other countries.
!!! example
By 1914, only ~6 000 km of rail was laid in China while the US had laid ~225 000 km.
Additionally, the population boom meant that demand for food increased, but outdated agricultural practices and technologies could not keep up, resulting in famine.
### Tradition and class structure
**— Long-term structural social**
China operated under a **patriarchy** and had traditions and practices which were thought to be even at the time to be cruel and outdated especially for women.
!!! example
The life expectancy of a city in China (Shenyang, Daoyi) from 1792 to 1867 was less than 40 years for men and about 30 for women.
**Submission to the Qing dynasty** was traditional as well — the Chinese could not marry Manchus nor live in Manchuria.
!!! example
Men were required to wear a queue (a long pigtail behind a shaven forehead) to show subservience to the Manchu Empire.
The **class structure** in China did not change for hundreds of years and old traditions and practices persisted into the 1900s.
- The land peasants worked on belonged to local landlords.
- 80% of the population remained peasants.
- Landlords took a large amount of crops as rent and the government also took a large portion as taxes.
- The burdens from landlords and the government were compounded with natural disasters such as floods and droughts, resulting in famine.
- Landlords could taken peasant women as they wished, force peasants to perform extra duties, and beat them if they were questioned.
### Internal dissent
**— Long-term structural political**
The **Taiping Civil War** from 1850 to 1864 was one of the bloodiest wars ever and the largest conflict of the 19th century. Anti-Manchurian sentiment was high as the people did not like that the Qing dynasty was ruled not by themselves. At the time, they could not marry Manchus, could not settle in Manchuria, and men were required to wear queues as a reminder of submission to Qing rule.
The **Boxer Rebellion** from 1899 to 1901 was caused by an organisation known as the Boxers (due to many of their members practising Chinese martial arts) desiring an end to foreign control in China. To do so, they violenced across northern China targeting foreign property, Christian missionaries, and Chinese Christians. Although initially supported by Cixi, support was split in the country and eventually she accepted help from foreign armies to end the rebellion.
!!! example
The slogan of the Boxers was to "support the Qing government and exterminate foreigners".
### Introduction of progressive ideas and rise of revolutionaries
Sun Yixian
### 1911 revolution
failure of revolution
aftermath of revolution
### Warlords
**— Short-term**
warlords
### Rise of revolutionaries 2: electric boogaloo
**— Short-term**
first united front
screw this history too hard
## Consolidation and maintenance of power in China
### Use of legal methods
In the aftermath of the Chinese Civil War, political pluralism was adopted for the purposes of stability and unification. At the **Chinese People's Political Consultative Conference (PCC)** on 21 September 1949:
- [Mao Zedong](https://en.wikipedia.org/wiki/Mao_Zedong) (Chinese: 毛泽东) was elected Chairman of the People's Republic of China,
- [Zhou Enlai](https://en.wikipedia.org/wiki/Zhou_Enlai) (Chinese: 周恩来) was appointed first premier (prime minister) by Mao,
- a de facto constitution in the form of the Organic Law was adopted, and
- China was divided into six regions with each region under two civilian and two military officers to maintain control
<img src="/resources/images/ccp-hierarchy.png" width=700>(Source: Kognity)</img>
#### Xinjiang
The **reunification campaigns** from 1950 to 1953 consolidated Mao's power across a divided China and allowed him to gain control of its borders.
!!! background
The region of [Xinjiang](https://en.wikipedia.org/wiki/Xinjiang) followed the Qing dynasty, but was semi-independent during the rule of the warlords and the Republic of China. With a population consisting largely of the [Uyghur](https://en.wikipedia.org/wiki/Uyghurs) minority ethnic group as opposed to the [Han](https://en.wikipedia.org/wiki/Han_Chinese) that made up 98% of the population in China, the province was ruled by a coalition government made of local leaders and **Guomindang (GMD)** members at the time of the establishment of the PRC.
The **Chinese Communist Party (CCP)** was concerned that Xinjiang would start a separatist movement against them or become part of the USSR. Upon the CCP's army, the **People's Liberation Army (PLA)** moving into Xinjiang, provincial authorities pledged allegiance to the CCP and the province was fully under their control by 1951. It is referred to as the "Peaceful Liberation of Xinjiang" in Chinese historiography.
In 1955, the region became an [autonomous region](https://en.wikipedia.org/wiki/Autonomous_regions_of_China) of China, granting it a local government and more legislative rights than other provinces.
#### Controlling the population
All citizens were required to belong to a self-sufficient entity known as a *danwei* (Chinese: 单位), or **work unit**. Permission from the work unit was needed to marry or have children. Under the threat of punishment if policy was not followed, everyone was assigned a home a food to eat along with others in their work unit. Regardless of their size, all work units were obligated to provide or share adequate facilities — schools, housing, health care, etc.
A system of **household registration** to identify citizens known as *hukou* (Chinese: 户口) recorded the birth, death, and movement of people as well as their family members, connecting identifiable information to their location of permanent residence. Whether someone was given better benefits by the government was determined by their agricultural status (agricultural or non-agricultural — i.e. rural or urban, non-agricultural was better) and it was virtually impossible to switch to the other. During industrialisation from 1955 onward, Mao used this system to control rural-to-urban migration by way of a certificate on one's registration required to be able to move to urban area.
Public **records** per person known as *dang'an* (Chinese: 档案) contained personal information such as:
- employment records
- physical characteristics
- family background with photos
- transcripts and school records
- achievements, mistakes, and self-criticisms
- political activity, and more
The government and work unit could access these files, instilling a culture of fear into the citizens as the most intrusive form of surveillance.
#### 1954 Constitution
The [constitution](https://en.wikipedia.org/wiki/1954_Constitution_of_the_People%27s_Republic_of_China) was based on the PCC and Organic Law. It established rights such as equality of citizens andbetween Han and minority groups, prohibiting racial discrimination and oppression. It also set up a legal system where all citizens had the right to a fair trial by judges appointed by the government, but this was not genuinely implemented until after Mao.
### Use of force
#### Hundred Flowers Campaign
From 1956 to 1957, the CCP encouraged citizens to express their thoughts of the regime in the [Hundred Flowers Campaign](https://en.wikipedia.org/wiki/Hundred_Flowers_Campaign) (Chinese: 百花齐放), inviting intellectuals to criticise their policies.
!!! quote "Mao Zedong, 1956:"
The policy of letting a hundred flowers bloom and a hundred schools of thought contend is designed to promote the flourishing of the arts and the progress of science.
Initially, people were afraid of being arrested publicly and executed after the events of the Three- and Five- Antis campaigns, but eventually began to provide criticism. By May 1957, millions of letters were openly criticising the government. Rallies in the streets and posters and articles in magazines protested against corruption and censorship.
!!! example
Students at Peking University created a "Democratic Wall" on which they criticised the CCP with posters and letters over their control over intellectuals, the harshness of campaigns against counter-revolutionaries, low living standards, economic corruption and privileges among members of the party.
Mao abandoned the campaign in June 1957 and then began the [Anti-Rightist Campaign](https://en.wikipedia.org/wiki/Anti-Rightist_Campaign) in July where those who provided criticism were not persecuted. It is controversial whether Mao was genuinely surprised by the extent of the criticism or whether the campaign was to identify enemies of the CCP.
!!! definition
In China, a **rightist** was someone who favoured capitalism over communism, but eventually became a label for anyone who disagreed with Mao.
During the Anti-Rightist Campaign from 1957 to 1959, Mao attacked his critics in the Hundred Flowers Campaign, forcing them to take back what they said. Led by [Deng Xiaoping](https://en.wikipedia.org/wiki/Deng_Xiaoping) (Chinese: 邓小平), at least 550 000 citizens were declared rightists — most of them were sent to be re-educated in the countryside via labour reform. The result was that Mao accumulated more power over the party and over China.
#### Force in the Cultural Revolution
!!! background
The [**Cultural Revolution**](https://en.wikipedia.org/wiki/Cultural_Revolution) from 1966 to 1976 was declared by Mao to remove all capitalist and intellectual elements from China. In reality, it was Mao's attempt to remove his rivals in the government — [Liu Shaoqi](https://en.wikipedia.org/wiki/Liu_Shaoqi) (Chinese: 刘少奇) and Deng Xiaoping — believing their successful policies would damage his reputation and ideology, and to revive his cult of personality after the events of the Great Leap Forward.
The "Four Olds" — old ideas, culture, customs, and habits — were denounced and books were burnt, porcelain destroyed, museums ransacked, and heritage sites destroyed.
!!! example
More than 70% of Beijing's cultural artifacts were destroyed in August–September 1966.
Because religion and intellectuals could change people's minds, Mao directed temples, shrines, and religious statues to be destroyed. Intellectuals such as teachers were tortured, beaten to death, and sent to prison. Priests and the clergy were imprisoned and denounced — religious worship was banned entirely. [Red Guards](https://en.wikipedia.org/wiki/Red_Guards) comprising students from elementary to high school carried out humiliations and attacks as per the social norm because Mao had ordered them to.
!!! info
The **Little Red Book** was a pocket-sized collection of Mao's thoughts, assembled by his propaganda minister, [Lin Biao](https://en.wikipedia.org/wiki/Lin_Biao) (Chinese: 林彪). Published during the Cultural Revolution, it was viewed as the source of all truth and more than a billion copies circulated, second only to the Holy Bible. Members of the Red Guard were required to carry the book with them at all times, and its contents were viewed as the source of all truth.
In addition, landlords, rich peasants, counter-revolutionaries, rightists, and capitalists were all targeted. Due to the vague nature of thse labels, anyone could accuse anyone of anything. An environment of fear formed as people reported suspected enemies, with some citizens reporting neighbours as revenge, and children even reported and condemned their parents. The population was effectively mobilised to act as a **secret police** without actual secret police. Casualties from the Cultural Revolution are difficult to measure and range from thousands to millions.
!!! example
People were accused and persecuted for crimes such as forgetting a quote from the Little Red Book or owning a Western instrument.
#### Tibet
Declaring that Tibet was part of China, on 7 October 1950, the PLA invaded east Tibet under the banner of liberating them from Western imperialist powers, although the Tibet government recorded only ten foreigners in the country. As a poor country with little infrastructure and communications, the poorly equipped and trained Tibetan army of 8 000 was outmatched by the 40 000 sent by the PLA. On 23 May 1951, the CCP imposed on Tibet the [Seventeen Point Agreement for the Peaceful Liberation of Tibet](https://en.wikipedia.org/wiki/Seventeen_Point_Agreement) that confirmed Chinese soverignity over the province.
In 1959, an [uprising in Tibet](https://en.wikipedia.org/wiki/1959_Tibetan_uprising) led to the mass arrests of Tibetans and increased social and religious control. The spiritual leader of Tibetan Buddhism, the [Dalai Lama](https://en.wikipedia.org/wiki/Dalai_Lama), fled to India and some Buddhist practices were forbidden. The Tenth [Panchen Lama](https://en.wikipedia.org/wiki/Panchen_Lama) under the Dalai Lama wrote the [70 000 Character Petition](https://en.wikipedia.org/wiki/70,000_Character_Petition) addressed to the Chinese government, denouncing the abusive policies such as mass imprisonment and the high prisoner death rate in Tibet under communist rule. Mao rejected the claims and arrested him.
#### Guangdong purge
Guangdong (Chinese: 广东), also known as Canton, is a province in southern China close to Hong Kong and Macau and had held an economically important port that was the headquarters of the GMD. During the **reunification campaigns**, an estimated 28 000 people were executed as it was purged of Nationalist forces.
#### Antis campaigns
The [**Three-Anti Campaign**](https://en.wikipedia.org/wiki/Three-anti_and_Five-anti_Campaigns) in 1951 denounced waste, corruption, and bureaucratic inefficiency. Its targets were CCP members, former GMD members, and bureaucratic officials not in the party. Confessions were required for the people denounced in public trials. An estimated 5% of government officials were purged.
!!! example
One thousand officers were denounced in the first month — some for minor offenses like the use of an American car and some more serious such as the use of government money to purchase luxury furniture in offices.
The **Five-Anti Campaign** in 1952 denounced bribery, theft of state property, tax evasion, state property theft, chating on government contracts, and stealing economic intelligence. Its targets were the bourgeoisie, merchants, industrialists, and the rest of the capitalist class. The campaign sent a wave of fear in the bourgoisie, not helped by the Chinese encouraged to support the campaign and denounce people they suspected to be guilty. Those convicted also had to confess their crimes in public trials, and an estimated 450 000 private businesses were convicted.
The results of the campaigns were large-scale purges in the CCP and bourgeoisie, the ceoncept of class struggle devevloped in China, and it showed the population that it challenging the regime would be futile.
### Charisma and propaganda
Mao was seen as the figurehead of China and his image was proimently on display throughout China. His Little Red Book was distributed everywhere, and his charisma was a form of control over the population via his **cult of personality**. There was no distinction between Mao the person, the government, and China in the people's eyes. He received little backlash after the events of the Great Leap Forward and most of the population genuinely mourned his death.
!!! example "Example: "Long Live the Chinese Communist Party that Chairman Mao Personally Founded", April 1973"
<img src="/resources/images/mao-propaganda.jpeg" width=500>(Source: Kognity)</img>
**Propaganda** was under the control of the CCP's Central Propaganda Department up to the Cultural Revolution. Its aims were to spread Chinese ideology and the idea of Maoism against capitalism, indoctrinate the Chinese population, and reinforcce political messages to enhance Mao's cult of personality.
!!! example
The **Combat Illiteracy Campaign** from 1950 to 1956 sent simple reading material all over China including remote places to increase literacy while spreading ideology and propaganda.
From the 1940s to the 1950s, propaganda was generally colourfully visual because not many could read or write and it was cheaper and quicker to produce. Posters portrayed Mao as a god-like figure and all-powerful, symbolised as the father of the nation. They were put up everywhere in publicc spaces, poor citizens used them to decorate their homes, and Mao's picture was always in newspapers, stamps, and pins.
After 1949, radios and loudspeakers were also extensively used, broadcasting government propaganda and national anthems constantly.
**Role models** of real or fictitious people were used in propaganda to teach the population how to behave.
!!! example
Lei Fang was a fictitious soldier who died at 22 and was depicted as a happy and positive person despite experiencing a difficult childhood because his immediate family died at an early age. Mao's propaganda minister, Lin Biao, promoted his image by pushing his diary with positive comments about Mao. In 1953, Mao encouraged everyone to learn the "Lei Feng spirit", and the "Learn from Lei Feng" campaign launched during the Cultural Revolution aimed at getting people to obey and be loyal to Mao.
#### Thought Reform Movement
At the same time of the Antis campaigns, the [**Thought Reform Movement**](https://en.wikipedia.org/wiki/Thought_reform_in_China) from 1951 to 1952 was aimed to get the citizens to accept Marxism-Leninism and Maoism. Teachers and college staff were ordered to become Marxist-Leninists, and intellectuals who studied abroad were forced to confess as "implementers of the imperialist cultural invasion". School curricula were restructured, and propaganda and indoctrination were heavily used to change the citizens.
#### Cult of personality
Mao's cult of personality reached its peak during the Cultural Revolution. The people were drawn in by his promises after a decade of wars and corruption after the fall of the Qing dynasty. As the son of a poor peasant who worked hard to improve his social status, he maintained his image of being connected to the land and the peasants.
!!! example
As a publicity stunt, at the age of 72 years, he swam through the Yangtze River in July 1966 to prove that he was fit to rule. Propaganda at the time claimed he swam nearly 15 kilometres in just over an hour.
#### Education
To create citizens supportive of the state, the CCP controlled the curriculum, reading material, and other information that students were exposed to. Schools effectively became indoctrination centres.
!!! example
- Chinese textbooks were censored.
- The Little Red Book was the primary literary text.
- Elementary education focused on rote (memorisation via repetition) over critical thinking.
- Secondary education focused on testing, exams, physical education, and in practice it was mostly children of high ranking party members and government officials who attended.
**Language reforms** resulted in a new form of Mandarin that allowed for people averywhere in the country to communicate. This standardisation made Mandarin easier to learn but caused local languages to disappear.
The **Socialist Education Movement** from 1964 to 1966 had the goal to cleanse politics, the economy, organisation, and ideology as the "Four Cleanups". Intellectuals were sent to the countryside to be re-educated by peasants. They still attended school, but also worked in factories and with the peasants.
#### Propaganda in the Cultural Revolution
During the Cultural Revolution, it was required that a portrait or sculpture of Mao was present in each home.
Mao called for class struggle in all educational institutions, and education as an ideal was condemned. Peasants and industrial workers were made teachers and pupils, and they were encouraged to criticise their teachers, who were forced to wear dunce caps and were paraded around. Schools and universities closed down and were made into barracks for Red Guards.
!!! info
130 million young people stopped attending school.
The [**Down to the Countryside Movement**](https://en.wikipedia.org/wiki/Down_to_the_Countryside_Movement) beginning in 1968 when the Red Guard was becoming violent and difficult to control had Mao order them to return to schools, with more violent radical groups being forcibly suppressed by the PLA. Urban students were sent to the countryside to experience peasant life. This ripped millions of families apart and many did not go back to school or university when they were allowed to return home.
!!! info
From 1968 to 1976, 17 million young people were sent to rural areas.
#### Arts
Mao's wife, [Jiang Qing](https://en.wikipedia.org/wiki/Jiang_Qing) (Chinese: 江青) was made the chief of new Chinese culture and the "cultural purifier". She imposed censorship on anything that did not meet the criteria of "revolutionary purity".
**Statues** of Mao were erected in front of state offices, universities, and schools.
Traditional Chinese **operas** were replaced by those focused on the proletariat overthrowing class enemies. The more famous ones have political overtones with communist and/or revolutionary themes.
Folk **music** was made modern and an attempt was made to put traditional Chinese music on equal footing. Rousing songs that appealed to the masses were composed, and Western music was entirely banned — symphonic and classical music fell in this category as they were associated with elitism and the West. Mao's poems were put into choral and classical music — the Red Guard sang lyrics derived from the Little Red Book.
**"The East is Red"** was a revolutionary song that was the *de facto* national anthem during the Cultural Revolution. It was played through loudspeakers everywhere at dawn and dusk, sung by students at the beginning of the first class of each day, and shows began with this song.
!!! quote ""The East is Red""
The east is red, the sun is rising,
China has brought forth a Mao Zedong.
He works for the people's welfare.
Hurrah, he is the people's great saviour.
Chairman Mao loves the people,
He is our guide,
To build a new China,
Hurrah, he leads us forward!
The Communist Party is like the sun
Wherever it shines, it is bright.
Wherever there is a Communist Party,
Hurrah, there the people are liberated!
#### Historians
???+ quote "Yan Yen (poet):"
As a result of the Cultural Revolution, you could say the cultural trademark of my generation is that we had no culture.
???+ quote "Michael Lynch:"
Cultural terrorism — Result of the Cultural Revolution: Cultural wasteland. Artists who resisted the revolution were sent to re-educational labour camps, where they were brutally treated. Pianists and string players were made to scratch at the ground so they would never be able to play well again.
### Nature, extent, and treatment of opposition
Mao never had any direct opposition.
#### Struggle sessions
Struggle sessions were a form of public humiliation where people **self-criticise** to find the mistakes they made and to free themselves from error. It was used in the USSR from the 1920s and by Mao during the Yan'an Rectification Movement in 1941, the Anti-Rightist Campaign, and the Cultural Revolution.
#### Purges
**Landlords** were publicly humiliated after land reform and tried then executed in their villages until there were no more landlords. Red Guards looted and destroyed homes, attacking landlords and local officials.
!!! info
An estimated 2-3 million landlorrds were killed.
After Mao's retreat from the Great Leap Forward, in 1966, he decided to reorganise the party by "eliminating members who had taken a capitalist road". In 1968, Liu Shaoqi and Deng Xiaoping were denounced and stripped of their positions. Liu was struggled against and humiliated while the propaganda campaign accused him of being a traitor. He died in November 1969 in prison. Deng was also struggled against and publicly shamed but put under house arrest with his wife, later sent to Jiangxi for re-education through labour.
#### Laojiao and Laogai
[**Laojiao**](https://en.wikipedia.org/wiki/Re-education_through_labor) (Chinese: 劳教)), or re-education through labour, involved sending prisoners to labour camps. Designed to re-educate intellectuals, the goal was for prisoners to live and work with farmers and workers. They were also required to attend political classes where they denounced themselves and criticised their own thinking to realign their thoughts with communism. Those who committed smaller crimes that did not warrant capital punishment were sent here.
[**Laogai**](https://en.wikipedia.org/wiki/Laogai) (Chinese: 劳改), or reform through labour, is a lake in *Avatar: The Last Airbender*. Located in Ba Sing Se in the Earth Kingdom, it hid an underground prison where those interned were brainwashed. It is also compared to the USSR's [*gulag*](https://en.wikipedia.org/wiki/Gulag). As internment camps for criminals, they were built in areas with extreme wewather and interns were forced to perform hard labour such as digging dithes and building roads under extremely poor conditions.
!!! quote
The Earth King has invited you to Lake Laogai.
### Extent of authoritarian control
As China is still under authoritarian control and many numbers and facts are inaccurate or unknown, it is difficult to obtain concrete information. However, it appears very **totalitarian**.
The Chinese followed **Confucian** philosophy, so social harmony was very important to them. Individuals were expected to accept their position in society and respect authority and the hierarchy. This led to less opposition and greater conformity. Unlike in the West, where the state is viewed as the oppressor, the state was viewed in China as family and the protector of civilisation.
**Maoism** was described as "Marxism adapted to Chinese conditions". It held the following differences compared to Marxism:
1. Peasants are the **agents of change** as opposed to urban workers, and they are moldable via social engineering. Mao used this to set up his own cult that would keep others in line.
3. All revolutions require constant renewal, and permanent, **constant revolution** is required to prevent counter-revolution. Mao related stability to dangerous bureaucracy and privileged classes such as imperial China and the USSR under Khrushchev. He used this to justify constant revolution such as the Hundred Flowers Campaign and Cultural Revolution, creating a culture where violent upheaval was a regular way of life.
### Historians
???+ quote "Liang Heng, age 12, recount of late 1967 after 11 million young people travelled to Beijing for Mao:"
If there was a single thing that meant ecstasy to everyone in those days, it was seeing Chairman Mao. Ever since I had been in Peking, the possibility had been in the back of my mind, and, like every other Red Guard, I would have laid down my life for the chance… On May 1st Peng Ming was planning to go with a small group to conduct performances of Revolutionary songs at the Summer Palace during the day…and I was sometimes asked to carry drums and other instruments, so I went with Peng Ming’s group to the park. We were completely unprepared for what happened. In the middle of singing a song that Peng Ming had composed himself, we heard the great news: Chairman Mao was in the park! Gathering our instruments together hastily, we ran gasping to the spot, but it was too late. He was gone. All that remained of him was the touch of his hand on the hands of a few who had been lucky enough to get close to him. But we didn’t leave in disappointment. That trace of precious warmth in the palms of others seemed to us a more than adequate substitute for the real thing. Those Chairman Mao had touched now became the focus of our fervor. Everyone surged toward them with outstretched arms in hopes of transferring the sacred touch to their own hands. If you couldn’t get close enough for that, then shaking the hand of someone who had shaken the hands with Our Great Saving Star would have to do.
## Foreign policy of China
!!! definition
**Sinocentrism** is the idea that China is the cultural, political, and/or economic centre of the world.
Mao was concentrated on maintaining and consolidating his power in China, but he supported the idea of an **international communist revolution** and focused on re-establishing China's position as a great power in the world.
After [Joseph Stalin](https://en.wikipedia.org/wiki/Joseph_Stalin)'s death, Mao was seen by many to be the leader of the communist world.
!!! example
In 1964, China detonated its first nuclear bomb, claiming its necessity for defense and in opposing the "US imperialist policy of nuclear blackmail and nuclear threats". Originally advocating for the abolishment of nuclear weapons but blocked by the US, evidence seems to suggest that Mao was quite willing to use nuclear weapons, believing that China's population would recover quickly.
### Sino-Soviet relations
China's relationship with the USSR started off well enough but steadily deteriorated by the 1960s.
#### Treaty of Friendship, Alliance, and Mutual Assistance
On 16 December 1949, Mao took his first trip abroad to Moscow only a few months after the establishment of the PRC. He was not met with great enthusiasm — Soviet leaders would meet him but would not drink or eat lunch with him, and there were no celebrations upon his arrival. He was in essence treated like a minor politician from a small communist country.
The visit lasted three months and culminated in the signing of the treaty on 14 February 1950. In the [Sino-Soviet Treaty of Friendship, Alliance, and Mutual Assistance](https://en.wikipedia.org/wiki/Sino-Soviet_Treaty_of_Friendship,_Alliance_and_Mutual_Assistance), the USSR:
- recognised the People's Republic of China as the legitimate government of China
- lent $300 million over five years to aid economic and logistic recovery from a decade of warfare
- sent assistance from 11 000+ Soviet consultants and experts to be paid for by the PRC
#### Korean War
!!! background
From 1910 to August 1945, Korea was occupied by imperial Japan. After the end of World War II, the USSR and the US agreed to temporarily divide Korea along the 38th parallel and established a communist government in the north and a democratic government in the south, respectively.
<center><img src="/resources/images/korea-map.png" width=350></img></center>
(Source: Kognity)
Under a desire to unify Korea under communism, North Korean leader [Kim Il-Sung](https://en.wikipedia.org/wiki/Kim_Il-sung) asked Stalin for approval to attack South Korea but was denied due to the [Berlin Blockade](https://en.wikipedia.org/wiki/Berlin_Blockade) at the time. His approval was granted later in April 1950 and the surprise attack launched on 25 June 1950, capturing important cities such as Seoul.
In response, the [United Nations Security Council](https://en.wikipedia.org/wiki/United_Nations_Security_Council) declared North Korea as the aggressor and sent troops from 15 countries led by the US to restore peace under American general [Douglas MacArthur](https://en.wikipedia.org/wiki/Douglas_MacArthur), successfully retaking the 38th parallel.
!!! info
- The UN motion to send troops only succeeded because the Soviet delegate with [veto powers](https://en.wikipedia.org/wiki/United_Nations_Security_Council_veto_power) was absent as a protest against UN refusal to accept the PRC as the legitimate government of China.
- The United States, South Korea, and other nations sent 350 000, 400 000, and 50 000 troops, respectively.
!!! background
- North Korea made significant contributions to the CCP during their liberation of mainland China.
- Sino-American relations during this time period were especially poor due to the Truman administration declaring their support for the Republic of China on Taiwan as the "main China".
When UN forces crossed the Yalu River on the China-Korea border, Mao felt China's security was at stake and also recognised an opportunity to assert power. Additionally, concern over border security with a hostile east due to a revived Japan, a desire to replace the Soviet influence in North Korea with their own, and Stalin pressing Mao to assist in the war led China to intervene.
In October 1950, the [Chinese People's Volunteers](https://en.wikipedia.org/wiki/People%27s_Volunteer_Army) — in actuality a group of forces from the main People's Liberation Army under a different name to avoid official war with the US — deployed 500 000 troops push the UN troops back to the 38th parallel, resulting in a stalemate back at status quo by 1951. Talks lasted two years with US President [Eisenhower](https://en.wikipedia.org/wiki/Dwight_D._Eisenhower) threatening the use of nuclear weapons should they drag on until an [**armistice**](https://en.wikipedia.org/wiki/Korean_Armistice_Agreement) was signed on 27 July 1953.
???+ info
**Casualties in the Korean War**
<img src="/resources/images/korean-war-casualties.png" width=500>(Source: Kognity)</img>
From China's perspective, the war was both a success and a failure. Mao propagandised the war as a total success in their aim to "Resist America and Defend Korea".
| Success | Failure |
| --- | --- |
| Mao gained considerable prestige for being able to fight the US to a standstill | Heavy casualties — Mao's eldest son was killed in an air raid |
| North Korea remained communist | Sino-American relations deteriorated further, and China faced a total embargo from the US |
| China preserved its Manchurian border where its heavy industry was concentrated | The USSR-lent military equipment had to be repaid |
#### Sino-Soviet split
Stalin and Mao's relationship was tense, and the relationship between the two countries deteriorated much faster under [Nikita Khrushchev](https://en.wikipedia.org/wiki/Nikita_Khrushchev).
!!! background
The border between the Soviet Union and China was determined by many treaties signed by various officials over many years. This left many gray areas where both countries claimed soverignity.
**Border conflicts** in 1969 between the two countries led to seven months of unofficial conflicts and border clashes over various islands and rivers. At this point in time, both countries had nuclear weapons. Tensions persisted until September 1969 when the Chinese Premier and Soviet Minister of Foreign Affairs met in Beijing in September 1969.
**Personality and ideological conflicts** between the leaders and countries worsened relations. Mao acted like an obedient student and never openly contradicted Stalin while he was in power but was often annoyed by his level of control over the CCP. Stalin was annoyed by Mao's attitude but needed a strong ally.
!!! example
- In 1921, Mao believed that the rural population would lead China to revolution while Stalin interpreted that the proletariat could only be urban workers.
- After the end of World War II, Stalin requested Mao to work with the GMD, but Mao decided to wipe out the party and take power instead.
In 1956, after Stalin's death, Khrushchev gave a [secret speech](https://en.wikipedia.org/wiki/On_the_Cult_of_Personality_and_Its_Consequences) denouncing his rule. Mao was alarmed by the brutality of these attacks and interpreted the speech as criticism against him and his own leadership in China.
!!! definition
- **Détente** between the US and Soviet Union was the relaxation of strained relations between the two countries.
- **Revisionism** in this context is the betrayal of original revolutionary ideas.
- **Peaceful coexistence** is the belief that both capitalist and communist nations can exist together without war.
In 1957, Khrushchev organised a [conference in Moscow](https://en.wikipedia.org/wiki/1957_International_Meeting_of_Communist_and_Workers_Parties), inviting all communist states including China. Mao complained about Khrushchev's [revisionism](https://en.wikipedia.org/wiki/Revisionism_(Marxism)) and [peaceful coexistence](https://en.wikipedia.org/wiki/Peaceful_coexistence)/[detente](https://en.wikipedia.org/wiki/D%C3%A9tente) approaches with the United States. Mao believed that it was the duty of communists to conduct class warfare and that the Soviet Union was being too soft on the West by making concessions — they were not fit to lead the communist world. He thought that a final violent conflict was needed with capitalism.
In 1958, Khrushchev was invited to visit China and Mao treated him with disdain, aiming to make his visit unpleasant: the Soviet delegation was placed in a hotel with no AC, Mao invited Khrushchev to swim in his private pool — aware of his inability to swim, and refused any proposals for military cooperation and defense initiatives. In response, Khrushchev pulled most advisors out from China and removed all of them by 1960.
**Chinese meddling in Soviet international affairs** resulted in open defiance against the USSR. Unimpressed with de-Stalinisation, when Albania left the USSR in 1961, China supported them against the Soviet Union, pouring money into the country. In return, Albanian leader Enver Hoxha declared his support for Mao.
Further examples of disagreements during the Sino-Soviet split include:
- 1958: Mao wanted the USSR to use their first satellite, Sputnik-1, to aid revolutionary efforts, but Khrushchev refused to risk nuclear conflict.
- 1959: [China invaded Tibet](https://en.wikipedia.org/wiki/Annexation_of_Tibet_by_the_People's_Republic_of_China), but the USSR refused support and withdrew its support from the Chinese nuclear program by refusing to give them a prototype weapon.
- 1962: China disagreed with the USSR in backing down and making in a deal in the [Cuban Missile Crisis](https://en.wikipedia.org/wiki/Cuban_Missile_Crisis), wanting them to support third world countries in their fight against communism.
- 1963: China opposed the USSR signing of the [Partial Nuclear Test Ban Treaty](https://en.wikipedia.org/wiki/Partial_Nuclear_Test_Ban_Treaty) against nuclear weapons, sparking a fierce propaganda war.
### Cross-Strait relations
Mao never recognised Taiwan as an independent state.
In the [First Taiwan Strait Crisis](https://en.wikipedia.org/wiki/First_Taiwan_Strait_Crisis) from 1954 to 1955, the PLA bombed various islands near Taiwan and then seized the [Yijiangshan Islands](https://en.wikipedia.org/wiki/Yijiangshan_Islands) in a military conflict. This led to the [Formosa Resolution](https://en.wikipedia.org/wiki/Formosa_Resolution_of_1955) to be enacted by the US Congress — that US forces would defend Taiwan against any attack from the mainland.
In 1958, Mao ordered the PLA to attack the Taiwan-surrounding and -controlled islands of Kinmen and Matsu without discussion with the USSR. As the US prepared for war because of the Formosa Resolution, Mao stood down because he did not have USSR backup. In the aftermath, Khrushchev accused Mao of being a [Trotskyist](https://en.wikipedia.org/wiki/Trotskyism) who had lost all sense of reality.
### Sino-American relations
The US viewed China as an aggressive country with the objective of threatening the security of the non-communist states surrounding it. China viewed the US as their enemy.
!!! example
The [Red Scares](https://en.wikipedia.org/wiki/Red_Scare) in the US and the anti-American and anti-capitalist propaganda in China (e.g., "Death to the American imperialists") meant that the two countries could not easily communicate diplomatically in public.
To "contain" communism in China, the US signed several treaties such as the [Southeast Asia Treaty Organization](https://en.wikipedia.org/wiki/Southeast_Asia_Treaty_Organization) and the [ANZUS Treaty](https://en.wikipedia.org/wiki/ANZUS) to ward off China. Additionally, they gave the Chinese seat at the UN to Taiwan, pushed allies to avoid entertaining diplomatic relations with Taiwan, supported countries that felt threatened by China, encouraged the split between the USSR and China, and implemented a trade embargo.
From 1970 onward, the two countries began to grow closer — China wanted a new strong ally as Sino-Soviet relations were deteriorating and the US wanted a way out of the Vietnam War.
Taking advantage of [ping-pong diplomacy](https://en.wikipedia.org/wiki/Ping-pong_diplomacy), the exchange of ping-pong players between the US and China, US Secretary of State Henry Kissinger went secretly to meet Chinese Premier Zhou Enlai.
The [**Shanghai Communiqué**](https://en.wikipedia.org/wiki/Shanghai_Communiqu%C3%A9) signed between the two countries on 28 February 1972 was a statement issued by both countries during US President Richard Nixon's visit to China — the first visit to the PRC by any US President — and began the normalisation of relations between them. Nixon described the visit as a mission for peace with the goal to re-establish communications after a generation of hostility, and agreements were reached to expand cultural, educational, and journalistic contracts.
### Historians
???+ quote "*Mao: a Biography* - Ross Terill (revisionist and somewhat sympathetic to Mao), 1995:"
Mao knew little of the world outside China, and nothing of the capitalist world… Yet Mao took a lively interest in the world beyond the Soviet Bloc during the early 1960s… It was as if the split with Russia in 1960 took a burden off the back of Chinese diplomacy. Instead of being junior partner in someone else’s show, Mao made China its own one-man show on the broadening stage of the Third World.
???+ quote "*Mao: Profiles in Power* - Shaun Breslin, 1998:"
Mao’s main objective in all of his foreign policy initiative from 1949 to 1976 was to safeguard China’s borders and restore China to its rightful position on the world stage. Mao had a traditional Sinocentrism: the notion that China is the central place in the world, and that only those who recognise and accept Chinese superiority can be considered to be civilised.
???+ quote "*Mao Zedong* - Maurice Meisner (sympathetic to socialist ideology and goals), 2007:"
Mao’s foreign policy clothed itself in revolutionary rhetoric, but was conservatively cautious in substance, based on narrow calculation of China’s national self-interest… In Mao’s view the Soviet Union posed a greater danger to China than did the United States.
## Resources
- [IB History Syllabus](/resources/g11/ib-history-syllabus.pdf)
- [Textbook: Origins and Development of Authoritarian and Single-Party States](/resources/g11/textbook-authoritarian-states.pdf)
- [Textbook: Causes and Effects of 20th-Century Wars, Second Edition ](/resources/g11/textbook-cause-20-century-wars.pdf)
- [Textbook: The Move to Global War](/resources/g11/textbook-move-global-war.pdf)

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# HL English - 1
The course code for this page is **ENG3UZ**.
## Literary techniques/devices
### Description
- Allusion: A brief and indirect reference to a thing or idea of significance.
- e.g., *"Look at Einstein over there, thinking he's so smart."*
- Anecdote: A short and usually personal story used to support a point.
- e.g., *"You should all be grateful! When I was a young lad, there was this girl that I liked, so what did I do about it? I went straight-up told to her face that I loved her! Of course, she smacked me away as if I were a stray dog, but if I was able to stand up to someone like her, you young'uns should be able to send a simple text message!"*
- Hyperbole: An exaggeration for the purpose of emphasis.
- e.g., *"If you null pointer me one more time I am going to crush you into two billion pieces!"*
- Imagery: The use of words to describe concepts in a way that involves the five senses.
- e.g., *"My mother…how sweet and juicy her tender limbs are."*
- Metaphor: An implicit comparison between two unlike things without the use of "like" or "as", used especially to describe abstract ideas indirectly.
- e.g., *"My computer is an absolute potato."*
- Simile: An explicit comparison between two unlike things *with* the use of "like" or "as", used especially to describe abstract ideas indirectly.
- e.g., *"She's as stupid as an elephant!"*
- Litotes: An understatement by negating a very positive or negative expression.
- e.g., *"My marks aren't the best in the world."*
- Personification: The application of human concepts to non-humans, such as human-like speech. This is a common example of a metaphor.
- e.g., *The rock blinked at me with a cute face unlike any other I had seen before. "…Owo. Uwu?"*
- Adjectives and adverbs: Words that directly describe and modify nouns and verbs, respectively.
- e.g., *The traitorous man died painfully.*
### Sound
- Alliteration: A number of words with the same beginning sound that appear close together.
- e.g., *The dancing damsel dazzled the crowd.*
- Assonance: A number of words with the same vowel sound, not at the beginning, that appear close together.
- e.g., *The cat in the hat knows a lot about that!*
- Cacophony: The use of words and phrases that are harsh to the ear by using consonants that require pressure to to say (e.g., p, b, d, g, k, ch-, sh-).
- e.g., *How much wood could a woodchuck chuck if a woodchuck could chuck wood?*
- Catalexis: A line missing a syllable at the end or beginning.
- Consonance: A number of words with the same consonant sound, not at the beginning, that appear close together.
- e.g., *Shelley sells shells by the seashore.*
- Dialect: A regional variety of language with spelling, grammar, and pronunciation that differentiates a population from others around them.
- e.g., *"There's five of them dirty ducks flappin' out and about, sir!"*
- Euphony: The use of words and phrases that are pleasing to the ear by using long vowel sounds, harmonious constants (l, m, n, r, f, v), and soft consonants or semi-vowels (w, s, y, th, wh).
- e.g., *"The velvet…it's so lovely, and so very soft…"*
- Onomatopoeia: A word which imitates a sound effect.
- e.g., *The cat meowed as he looked on with adoring eyes that tore through my squealing heart.*
- Pun: A play on words that involve words with similar sounds but different meanings.
- e.g., *"Lettuce finish our salads."*
- Repetition: Using a phrase for emphasis multiple times within close proximity.
- e.g., *"I was too late. I was far too late. My dog…she tried her best."*
- Rhyme: A repetition of words whose end syllables sound similar.
- e.g., *Roses are red / Violets are blue / Your family is dead / And you will be too*
- Rhyme scheme: The rhyming pattern or structure at the end of each line of poetry.
- e.g., the above poem uses ABAB as its rhyme scheme.
- Rhythm: The usage of stressed and unstressed syllables to demonstrate patterns, especially in verses.
- Anapest: Two unstressed syllables followed by a stressed syllable (uu/).
- e.g., *"Understand?"*
- Dactyl: A stressed syllable followed by two unstressed syllables (/uu).
- e.g., *"Marvelous!"*
- Iamb: An unstressed syllable followed by a stressed syllable (u/).
- e.g., *To be, or not to be, that is the question.*
- Spondee: A stressed syllable followed by another stressed syllable (//).
- e.g., *"Rage, rage against the dying of the light."* ("Do not go gentle into that good night", Dylan Thomas)
- Trochee: A stressed syllable followed an unstressed syllable (/u).
- e.g., *Double, double, toil and trouble* (*Macbeth*, William Shakespeare)
### Organisation and Pace
- Blank verse: An non-rhyming verse with a regular rhythmic structure, most commonly iambic pentameter (10 syllables/line, unstressed-to-stressed pattern)
- e.g., *But, woe is me, you are so sick of late, / So far from cheer and from your former state.* (*Hamlet*, William Shakespeare)
- Enjambment: A single thought in a line of poetry using two lines.
- e.g., *The world / is a tomato.*
- Caesura: A pause in a poem used to emphasise a phrase.
- e.g., *I, am intellectual, make no speling errors whatsoever.*
- Form:
- Ballad: A poetic story set to music relying on simple language, commonly containing romance and tragedy.
- Concrete poem: A poem written, arranged, or "drawn" in a way that its visual appearance is most important in conveying meaning.
- Couplet: Two consecutive rhyming lines in a verse with the same rhythmic structure.
- Dirge: A somber song expressing grief or mourning.
- Dramatic monologue: A character's speech written in the form of poetry.
- Lyric: A collection of verses and choruses with a single speaker to create a non-narrative poem or song.
- Ode: A complex poem praising or glorifying an event or individual, typically formal and solemn.
- Quatrain: A verse or poem with four lines.
- Sonnet: A 14-line poem with a specific rhyme scheme in iambic pentameter (10 syllables/line, stressed-to-unstressed pattern).
- Free verse: A poem without rhythm or rhyme schemes.
- Genre: A work classified by its form, content, and style. Poetry, fiction, drama, and non-fiction are the four main genres of literature.
- Line: A typically straight sequence of characters that ends at the last character before a newline.
- Meter: The pattern of stressed and unstressed syllables in a verse or poem.
- Point of view: The perspective that a literary work is written from. First person usually focuses more on characters while third person perspectives focus more on plot.
- Stanza: A collection of four or more lines of poetry with with a fixed length, meter, or rhyme scheme, similar to a paragraph in prose. However, it is not paragraph.
- Sentence types: The use of simple, compound, complex, and compound-complex sentences.
- Sentence lengths: The number of characters or words present in a sentence.
- Syntax: A set of rules in a language regarding the order of words.
- e.g., *Die, he will,* versus *He will die.*
- Dialogue vs. narrative: The use of explicit speech versus a description of events.
### Meaning
- Abstract language: The use of words that connect intangible ideas or emotions as opposed to physical actions or things.
- e.g., *She was beautiful.*
- Allegory: The use of characters and events to describe an abstract idea to teach a moral lesson.
- e.g., *Animal Farm* by George Orwell, which uses farm animals and their actions to describe the Russian Revolution.
- Ambiguity: A statement or phrase whose meaning is unclear or can have multiple different meanings. This can usually be resolved via context. Strongly recommended to be used in poetry analysis.
- e.g., *Nintendo sold 22 million of them.*
- Colloquialism: The use of informal phrases or slang in writing.
- e.g., *"Pog!"*
- Concrete language: The use of words that reference physical actions or things as opposed to intangible ideas or emotions.
- e.g., *She had black hair that reached down to her neck.*
- Connotation: An implied meaning from a phrase that differs from its explicit and direct meaning.
- e.g., *I carried him for the entire assignment.*
- Contrast/Juxtaposition: A comparison between multiple ideas or things by placing them side by side.
- e.g., *It was the best of times, it was the worst of times…* (*A Tale of Two Cities*, Charles Dickens)
- Denotation: The literal meaning of a word.
- e.g., *"My old computer no longer functions."*
- Diction: The word choice used by a speaker or writer to create an effect.
- e.g., *"I shall take Her Majesty back to the carriage."*
- Epigram: A brief, interesting, memorable, and sometimes surprising or satirical statement.
- e.g., *"No one can make you feel inferior without your consent."* (Eleanor Roosevelt)
- Irony: An unexpected event or statement revealing what was believed to be true was false.
- e.g., *Thunder boomed as rain poured down all around the classroom. "Nice weather we're having today."* (Verbal)
- e.g., *The fire station burned down.* (Situational)
- e.g., *The cat fell over, its head completely severed. Meanwhile, Johnny whistled in his car, thinking of all the treats and belly rubs he would give his precious kitty.* (Dramatic)
- Jargon: Specialised terms that use actual words, which are usually unfamiliar to the layman.
- e.g., *p53 is responsible for holding the cell cycle at the G1/S checkpoint and will induce apoptosis if damage to the cell's deoxyribonucleic acid is too great.*
- Motif: An recurring object or idea used to explain the theme.
- e.g., In *To Kill a Mockingbird* by Harper Lee, The usage of life in Maycomb to show goodness and recurring images of gloomy areas supports the central idea of co-existence of good and evil.
- Oxymoron: The pairing of two opposing or contradictory words.
- e.g., *It was her only choice.*
- Paradox: A statement that appears to be self-contradictory at first glance, typically used to allow the reader to think over an idea differently.
- e.g., *All animals are equal, but some are more equal than others.* (*Animal Farm*, George Orwell)
- Pathetic fallacy: A form of personification in which inanimate objects in nature are given emotions.
- e.g., *The wind roared as the earth shook in fury at our insolence.*
- Symbol: The use of objects to represent an abstract idea.
- e.g., *I smiled at the group of people assembled before me. They smiled back. "With the power of friendship and determination, we can overcome anything!"*
- Tone: The attitude of a writer or character toward a subject or audience, typically through word choice.
- e.g., *"I don't want to! That candy is MINE, and no one is going to take it from ME! Mine, mine, MINE!"*
- Theme: The "main idea" or underlying meaning of a literary work, which can be given directly or indirectly.
- e.g., *"Never forget that* you are royalty, *and that hundreds of thousands of souls have suffered and perished so you could become what you are. By their sacrifices, you have been given the comforts you take for granted. Always remember them, so that their sacrifices shall never be without meaning."* (*Eon Fable*, ScytheRider)
## General writing
### Active and passive voice
Active writing involves having the subject of a sentence perform the action, while passive writing involves the subject receiving the action.
!!! example
Active: *Joey mangled the teacher.*<br>
Passive: *The teacher was mangled by Joey.*
### Redundancy
When possible, any words that do not add meaning should not be present.
## Essay writing
An essay is a relatively brief non-fiction piece of writing (can be read in one sitting) that is focused on one subject.
### Types of essays
**Compare and contrast** essays look for similarities and differences between two concepts, objects, or ideas. Arguments are either structured **subject by subject** or **point by point**.
**Cause and effect** essays attempt to establish a causal connection between ideas or events, in essence explaining why something happens/ed.
**Definitional** essays focus on defining a term, idea, or concept.
**Narrative** essays make a point by telling a non-fictional event in the structure of a short story in first person.
- They are usually told chronologically.
- They usually have a purpose/thesis that is stated in the opening sentence.
- Dialogue is permitted in narrative essays.
- They are written with vivid imagery and descriptions to involve the reader with the goal of relating in some way to the thesis.
- Much like fictional stories, they should have conflicts and events.
- They are usually written in first person.
## Essay analysis
### Thesis statement
The thesis statement of an essay is effectively its central assertion, and may appear in different places:
- An **initial** thesis appears within the first paragraph or so.
- A **delayed** thesis appears anywhere else, but often appears at the end of the essay.
- An **inferred** thesis is one that does not appear in the essay at all. It is instead up to the reader to glean an inferred thesis via inference.
### Purpose
Essays may be classified under two general categories — persuasive/argumentative or expository/informative. Persuasive essays argue to convince a reader to take their position, while expository essays aim to explain a topic without bias.
### Audience
The target audience of an essay and those that it would appeal to are useful in determining the point of an essay. Some factors that may indicate the audience include,
- the topic of the essay
- bias of the author
- diction/language used in the essay
- the use of jargon or slang
- the formality of the essay — formal essays are typically more organised and appeal to logic more than informal essays
- literary techniques
- the tone/attitude of the author of the essay via the use of emotionally charged words
## Resources
- [Analysis of a Poem](/resources/g11/central-assertion-1.pdf)
- [Essay Analysis](/resources/g11/essay-analysis.pdf)
- [Literary Criticism Overview](/resources/g11/literary-criticism.pdf)

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# SL French - 1
The course code for this page is **FSF3U7**.
Ahaha good luck with this I'm outta here
## Resources
- [Textbook: Oxford IB French B Course Companion](/resources/g11/textbook-french-b-second-edition.pdf) ([Answers](/resources/g11/textbook-french-b-second-edition-answers.pdf))

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# SL Math - Analysis and Approaches - 2
The course code for this page is **MCV4U7**.
## Integration
Integration is an operation that finds the **net** area under a curve, and is the opposite operation of differentiation. As such, it is also known as **anti-differentiation**.
The area under a curve between the interval of x-values $[a,b]$ is:
$$A=\lim_{x\to\infty}\sum^n_{i=1}f(x_i)\Delta x$$
which can be simplified to, where $dx$ indicates that integration should be performed with respect to $x$:
$$A=\int^b_a f(x)dx$$
While $\Sigma$ refers to a finite sum, $\int$ refers to the sum of a limit.
As integration is the opposite operation of differentiation, they can cancel each other out.
$$\frac{d}{dx}\int f(x)dx=f(x)$$
The **integral** or **anti-derivative** of a function is capitalised by convention. Where $C$ is an unknown constant:
$$\int f(x)dx=F(x)+C$$
When integrating, there is always an unknown constant $C$ as there are infinitely many possible functions that have the same rate of change but have different vertical translations.
!!! definition
- $C$ is known as the **constant of integration**.
- $f(x)$ is the **integrand**.
### Integration rules
$$
\begin{align*}
&\int 1dx &= &&x+C \\
&\int (ax^n)dx, n≠-1 &=&&\frac{a}{n+1}x^{n+1} + C \\
&\int (x^{-1})dx&=&&\ln|x|+C \\
&\int (ax+b)^{-1}dx&=&&\frac{\ln|ax+b|}{a}+C \\
&\int (ae^{kx})dx &= &&\frac{a}{k}e^{kx} + C \\
&\int (\sin kx)dx &= &&\frac{-\cos kx}{k}+C \\
&\int (\cos kx)dx &= &&\frac{\sin kx}{k}+C \\
\end{align*}
$$
Similar to differentiation, integration allows for constant multiples to be brought out and terms to be considered individually.
$$
\begin{align*}
&\int k\cdot f(x)dx&=&&k\int f(x)dx \\
&\int[f(x)\pm g(x)]dx&=&&\int f(x)dx \pm \int g(x)dx
\end{align*}
$$
### Indefinite integration
The indefinite integral of a function contains every possible anti-derivative — that is, it contains the constant of integration $C$.
$$\int f(x)dx=F(x)+C$$
### Substitution rule
Similar to limit evaluation, the substitution of complex expressions involving $x$ and $dx$ with $u$ and $du$ is generally used to work with the chain rule.
$$
u=g(x) \\
\int f(g(x))\cdot g´(x)\cdot dx = \int f(u)\cdot du
$$
??? example
To solve $\int (x\sqrt{x-1})dx$:
$$
let\ u=x-1 \\
∴ \frac{du}{dx}=1 \\
∴ du=dx \\
\begin{align*}
\int (x\sqrt{x-1})dx &\to \int(u+1)(u^\frac{1}{2})du \\
&= \int(u^\frac{3}{2}+u^\frac{1}{2})du \\
&= \frac{2}{5}u^\frac{5}{2}+\frac{2}{3}u^\frac{3}{2}+C \\
&= \frac{2}{5}(x-1)^\frac{5}{2} + \frac{2}{3}(x-1)^\frac{3}{2} + C
\end{align*}
$$
### Definite integration
To find a numerical value of the area under the curve in the bounded interval $[a,b]$, the **definite** integral can be taken.
$$\int^b_a f(x)dx$$
$a$ and $b$ are known as the lower and upper **limits of integration**, respectively.
<img src="/resources/images/integration.png" width=700>(Source; Kognity)</img>
Regions **under** the x-axis are treated as negative while those above are positive, cancelling each other out, so the definite integral finds something like the net area over an interval.
If $f(x)$ is continuous at $[a,b]$ and $F(x)$ is the anti-derivative, the definite integral is equal to:
$$\int^b_a f(x)dx=F(x)\biggr]^b_a=F(b)-F(a)$$
As such, it can be evaluated manually by integrating the function and subtracting the two anti-derivatives.
!!! warning
If $u$-substitution is used, the limits of integration must be adjusted accordingly.
To find the total **area** enclosed between the x-axis, $x=a$, $x=b$, and $f(x)$, the function needs to be split at each x-intercept and the absolute value of each definite integral in those intervals summed.
$$A=\int^b_a \big|f(x)\big| dx$$
### Properties of definite integration
The following rules only apply while $f(x)$ and $g(x)$ are continuous in the interval $[a,b]$ and $c$ is a constant.
$$
\begin{align*}
&\int^a_a f(x)dx&=&&0 \\
&\int^b_a c\cdot dx&=&&c(b-a) \\
&\int^a_b f(x)dx&=&&-\int^b_a f(x)dx \\
&\int^c_a f(x)dx&=&&\int^b_a f(x)dx + \int^c_b f(x)dx
\end{align*}
$$
The **constant multiple** and **sum** rules still apply.
### Area between two curves
To find the area enclosed between two curves, the graph should be sketched if possible and their points of intersection determined to identify which parts of each function are on top of the other at any given time. An interval chart may be helpful. For each section, where $f(x)$ is always greater than $g(x)$ in the interval $[a,b]$:
$$A=\int^b_a [f(x)-g(x)]dx, f(x)\geq g(x)\text{ in } [a,b]$$
If the limits of integration are not given, they are the outermost points of intersection of the two curves.
### Volumes of solids of revolution
Shapes formed by rotating a line or curve about a fixed axis, such as cones, spheres, and cylinders are all known as **solids of revolution**. By splicing each shape into infinitely small disks, the cylinder volume formula can be used to find the volume of the solid.
$$
\begin{align*}
V&=\lim_{x\to 0}\sum^b_{x=a}\pi y^2 dx \\
&=\pi\int^b_a y^2 dx
\end{align*}
$$
The area between two curves can also be rotated to form a solid, in which case its formula is:
$$V=\pi\int^b_a \big[g(x)^2-f(x)^2\big]dx, g(x)>f(x)$$
## Probability
!!! definition
- $\cap$ is the **intersection sign** and means "and".
- $\cup$ is the **union sign** and means "or".
- $\subset$ is the **subset sign** and indicates that the value on the left is a subset of the value on the right.
- The **sample space** of an experiment is a list/set of all of the possible outcomes.
- An **event** is a subset of a sample space that contains outcomes that meet a particular requirement.
### Sets
A **set** is a collection of things represented with curly brackets that can be assigned to a variable.
!!! example
$A=\{0,1,2\}$ assigns the variable $A$ to a collection of numbers $0, 1, 2$.
The variable $U$ is usually reserved for the **universal set**: a set that contains all of the elements under discussion for a particular situation.
Where both $A$ and $B$ are sets:
- $A\cap B$ returns a new set with only objects that belong to both $A$ **and** $B$.
- $A\cup B$ returns a new set with only objects that are inclusively in either $A$ **or** $B$.
- $A\subset B$ is true only if all of the elements in $A$ are also in $B$.
- $A'$ or $A^c$ return the **complement** of a set: they return all elements in the universal set that are **not** in $A$.
- $n(A)$ returns the number of elements in set $A$.
An empty/**null** set contains no objects and is represented either as $\{\}$ or $\emptyset$.
Two sets are **disjoint** or **distinct** if they have no common elements between them.
!!! warning
Generally, unless specified otherwise, "between" should be inferred to mean "inclusively between".
### Probability rules
The probability of an event is represented by $P(A)$, where $A$ is the event.
$$P(A)=\frac{n(A)}{n(U)}$$
As event $A$ must be a subset of all possible outcomes $U$, where $1$ indicates that the event always happens and $0$ the opposite:
$$0\leq P(A)\leq 1$$
The **complement** of event $A$ is the probability that it does not happen. It is written as $A^c$, $A'$, or $\pu{not } A$.
$$P(A')=1-P(A)$$
Events $A$ and $B$ are **disjoint** or mutually exclusive if no outcomes between them are common and can never happen simultaneously. As such the probability of one of the events happening is equal to their sum.
$$
P(A\cup B)=P(A)+P(B) \\
P(A\cap B)=0
$$
Events $A$ and $B$ are **exhaustive** if their union includes all possible outcomes in the sample space: $A\cup B=U$.
$$P(A\cup B)=1$$
The **principle of inclusion and exclusion** forms a general rule for the union between two *independent* events:
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$
For independent events:
$$P(A\cap B)=P(A)\times P(B)$$
### Conditional probability
A vertical bar is used between two events to denote that the event on the left occurs knowing that the right has already occurred.
$$P(A|B) = \frac{P(A\cap B)}{P(B)}$$
!!! example
$P(A|B)$ is the probability of event $A$ given $B$ has occurred.
### Probability distributions and discrete random variables
The **discrete random variable**, $X$, represents a **quantifiable**, measurable, discrete quantity. The lowercase $x$ represents a possible value of $X$.
The probability that $X$ takes on any one of the specific possible outcomes is written as $P(X=x)$. The sum of the probability all possible outcomes must still remain $1$:
$$\Sigma P(X=x)=1$$
!!! example
In an experiment of tossing a coin twice, possible values of $X$ include $0,1,2$ so $x\in\{0, 1, 2\}$.
A **probability distribution** is a distribution of outcomes and their probabilities. Events/outcomes are placed on the top row while probability is provided on the bottom in the form of a fraction. Probability distributions can also be graphed with the outcomes on the x-axis and their probabilities on the y-axis with lines similar to a bar graph sitting on the grid lines to represent a probability..
!!! example
For the coin ross experiment in the previous example, where $X$ is the number of tails when tossing a coin twice:
| $x$ | $0$ | $1$ | $2$ |
| --- | --- | --- | --- |
| $P(X=x)$ | $\frac{1}{4}$ | $\frac{1}{2}$ | $\frac{1}{4}$ |
The **expected value** of an experiment or the "expectation of $X$" is the mean value of $X$ that is expected to be obtained over many trials. It is equal to the sum of the value of all outcomes multiplied by their probability.
$$
\begin{align*}
E(X)&=\Sigma P(X=x)x \\
&=\mu=x_1p_1+x_2p_2+...+x_kp_k
\end{align*}
$$
!!! warning
It is possible that the expected value will not be a value in the set, and the expected value should **not be mistaken** with the outcome with the highest probability.
### Binomial distribution
**Bernoulli trials** have a fixed number of trials that are independent of each other and identical with only two possible outcomes — a success or failure.
Where $r$ is the number of successes in a Bernoulli trial:
$$P(X=r)={n\choose r}p^rq^{n-r}$$
where ${n\choose r}=\frac{n!}{r!(n-r)!}$
A binomial distribution is a probability distribution of two possible events, a success or a failure. The distribution is defined by the number of trials, $n$, and the probability of a success, $p$. The probability of failure is defined as $q=1-p$.
$X\sim$ denotes that the random variable $X$ is distributed in a certain way. Therefore, the binomial distribution of $X$ is expressed as:
$$X\sim B(n, p)$$
In a binomial distribution, the expected value and **variance** are as follows:
$$
E(X)=np \\
Var(X)=npq
$$
On a graphing display calculator, where $r$ is the number of successes:
$$
\begin{align*}
P(X=r)&=\text{binompdf}(n,p,r) \\
P(X<r)&=\text{binomcdf}(n,p,r-1) \\
P(X\leq r)&=\text{binomcdf}(n,p,r) \\
P(a\leq X\leq b)&=\text{binomcdf}(n,p,b) - \text{binomcdf}(n,p,a-1)
\end{align*}
$$
### Normal distribution
Also known as **Gaussian distribution** or in its graphical form, a normal or bell curve, the normal distribution is a **continuous** probability distribution for the random variable $x$.
<img src="/resources/images/gaussian-distribution.png" width=700>(Source: Kognity)</img>
In a normal distribution:
- The mean, median, and mode are all equal.
- The normal curve is bell-shaped and symmetric about the mean.
- The area under the curve is equal to one.
- The normal curve approaches but does not touch the x-axis as it approaches $\pm \infty$.
From $\mu-\sigma$ to $\mu+\sigma$, the curve curves downward. $\mu\pm\sigma$ are the **inflection points** of the graph. It is expressed graphically as:
$$X\sim N(\mu,\sigma^2)$$
where
$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^\frac{-(x-\mu)^2}{2\sigma^2}$$
~68%, ~95%, and ~99.7% of the data is found within one, two, and three standard deviations of the mean, respectively.
### Standard normal distribution
The **standard normal distribution** has a mean of 0 and standard deviation of 1. The horizontal scale of the standard normal curve corresponds to **$z$-scores** that represent the number of standard deviations away from the mean. To convert an $x$-score to a $z$-score:
$$z=\frac{x-\mu}{\sigma}$$
A **Standard Normal Table** can be used to determine the cumulative area under the standard normal curve to the left of scores -3.49 to 3.49. The area to the *right* of the score is equal to $1-z_\text{left}$. The area *between* two z-scores is the difference in between the area of the two z-scores.
To standardise a normal random variable, it should be converted from the form $X\sim N(\mu,\sigma^2)$ to $Z\sim N(0,1)$ via the formula to convert between x- and z-scores.
The probability of a z-score being less than a value can be rewritten as phi.
$$P(z<a)=\phi(a)$$
Some z-score rules partially taken from probability rules:
$$
\begin{align*}
P(z>-a)&=P(z<a) \\
1-P(z>a)&=P(z<a)
\end{align*}
$$
On a graphing display calculator:
The `normalcdf` command can be used to find the cumulative probabilty in a normal distribution in the format $\text{normalcdf}(a,b,\mu,\sigma)$, which will solve for $P(a<x<b)$. $-1000$ is generally a sufficiently low value to solve for just $P(x<b)$.
## Vectors
Please see [SL Physics 1#1.3 - Vectors and Scalars](/sph3u7/#13-vectors-and-scalars) for more information.
One vector can be represented in a variety of methods. The algebraic form $(1, 2)$ can also be represented in the alternate algebraic forms $[1, 2]$ and $1\choose 2$.
Where $v$ is the vector, $A$ is the initial and $B$ is the terminal point of the vector, a vector can be identified by any of the following symbols:
- $\vec{AB}$
- $\vec{v}$
- $\boldsymbol{v}$ (bolded)
The special **zero vector** $\vec{0}$ is a vector of undefined direction and zero magnitude.
Vectors with the same magnitude but opposite directions are equal to one another except one is the negative of the other.
**Colinear** vectors are those that parallel with one another — that is, with identical or opposite directions. Vectors that are colinear must also be **scalar multiples** of each other:
$$\vec{u}=k\vec{v}$$
**Position** vectors are vectors where the initial point is at the origin — where the terminal point is $A$, a position vector can be written as $\vec{OA}$.
**Colinear points** are points that lie on the same straight line. If two colinear vectors that share a common point can be formed between three points, those points are colinear.
### Unit vector
The **unit vector** of a vector is a vector of the same direction as the original with a magnitude of $1$. It is denoted via a caret/hat.
$$\hat{v}$$
From the original vector $\vec{u}$, the unit vector $\hat{u}$ can be found by dividing by the magnitude of the vector.
$$\hat{u}=\frac{\vec{u}}{|\vec{u}|}$$
The **standard unit vectors** $\hat{i}$ and $\hat{j}$ are unit vectors designated to point in the directions of the positive x- and y-axes.
$$
\hat{i}=(1,0) \\
\hat{j}=(0,1)
$$
Any vector in two dimensions can be expressed as a sum of scalar multiples of the vectors.
$$
\begin{align*}
\vec{u}&=\vec{OP} \\
&=(a,b) \\
&=a\hat{i}+b\hat{j} \\
&={a\choose b} \\
|\vec{u}|&=\sqrt{a^2+b^2}
\end{align*}
$$
The angle between two vectors is the smaller angle formed when the vectors are placed **tail to tail**.
### Three-dimensional vectors
The additional standard unit vector $\hat{k}$ is used for the z-dimension.
$$
\begin{align*}
\vec{u}&=\vec{OP} \\
&=(a,b,c) \\
&=a\hat{i}+b\hat{j}+c\hat{k}
\end{align*}
$$
In general, the x-plane is the one in and out of the page, the y-plane left and right, and the z-plane up and down.
### Vector operations
Please see [SL Physics 1#Adding/subtracting vectors diagrammatically](/sph3u7/#addingsubtracting-vectors-diagrammatically) for more details. The sum of two vectors is known as the **resultant** while the negative (opposite direction) version of that vector is known as the **equilibrant**.
The sum of two vectors can also be solved diagrammatically by envisioning the head-to-tail as a parallelogram.
<img src="/resources/images/vector-parallelogram.png" width=700>(Source: Kognity)</img>
### Dot product
Also known as the scalar product, the dot product between two vectors returns a **scalar** value representing the horizontal displacement after multiplication. Wheree $\theta$ is the angle contained between the vectors $\vec{u}$ and $\vec{v}$ when arranged tail-to-tail:
$$\vec{u}\bullet\vec{v}=|\vec{u}||\vec{v}|\cos\theta$$
!!! note
This implies that vectors perpendicular to one another must have a dot product of zero.
Much like regular multiplication, dot products are:
- communtative: $\vec{u}\bullet\vec{v}=\vec{v}\bullet\vec{u}$
- distributive over vectors: $\vec{u}\bullet(\vec{v}+\vec{w})=\vec{u}\bullet\vec{v}+\vec{u}\bullet\vec{w}$
- associative over scalars: $(m\vec{u})\bullet(n\vec{v})=mn(\vec{u}\bullet\vec{v})$
- $m(\vec{u}\bullet\vec{v})=(m\vec{u})\bullet\vec{v}=(mv)\bullet\vec{u}$
When working with algebraic vectors, their dot products are equal to the products of their components.
$$\vec{u}\bullet\vec{v}=u_xv_x+u_yv_y$$
### Vector line equations in two dimensions
!!! definition
The **Cartesian** or **scalar** form of a line is of the form $Ax+By+C$.
The vector equation for a straight line solves for an unknown position vector $\vec{r}$ on the line using a known position vector $\vec{r_0}$ on the line, a direction vector parallel to the line $\vec{m}$, and the variable **parameter** $t$. It is roughly similar to $y=b+xm$.
$$\vec{r}=\vec{r_0}+t\vec{m},t\in\mathbb{R}$$
The equation can be rewritten in the algebraic form to be
$$[x,y]=[x_0,y_0]+t[m_1,m_2], t\in\mathbb{R}$$
The direction vector is effectively the slope of a line.
$$\vec{m}=[\Delta x, \Delta y]$$
For a line in scalar form:
$$\vec{m}=[B, -A]$$
To determine if a point lies along a line defined by a vector equation, the parameter $t$ should be checked to be the same for the $x$ and $y$ coordinates of the point.
!!! warning
Vector equations are **not unique** — there can be different position vectors and direction vectors that return the same line.
The **parametric** form of a line breaks the vector form into components.
$$
\begin{align*}
x&=x_0+tm_1 \\
y&=y_0+tm_2,t\in\mathbb{R}
\end{align*}
$$
The **symmetric** form of the equation takes the parametric form and equates the two equations to each other using $t$.
$$\frac{x-x_0}{m_1}=\frac{y-y_0}{m_2},m_1,m_2\neq 0$$
If one of the **direction numbers** $m_1$ or $m_2$ is zero, the equation is rearranged such that only one position component is on one side.
!!! example
Where $m_2=0$:
$$\frac{x-x_0}{m_1},y=y_0$$
### Vector line equations in three dimensions
There is little difference between vector equations in two or three dimensions. An additional variable is added for the third dimension.
The vector form:
$$\vec{r}=\vec{r_0}+t\vec{m},t\in\mathbb{R}$$
The parametric form:
$$[x,y,z]=[x_0,y_0,z_0]+t[m_1,m_2,m_3],t\in\mathbb{R}$$
The symmetric form:
$$\frac{x-x_0}{m_1}=\frac{y-y_0}{m_2}=\frac{z-z_0}{m_3}$$
### Intersections of vector equation lines
Two lines are parallel if their direction vectors are scalar multiples of each other.
$$\vec{m_1}=k\vec{m_2},k\in\mathbb{R}$$
Two lines are coincident if they are parallel and share at least one point. Otherwise, they are distinct.
If two lines are not parallel and in two dimensions, they intersect. To solve for the point of intersection, the x and y variables in the parametric form can be equated and the parameter $t$ solved.
In three dimensions, there is a final possibility should the lines not be parallel: the lines may be *skew*. To determine if the lines are skew, the x, y, and z variables of **two** parametric equations should be equated to their counterparts in the other vector as if they intersect. The resulting $t$ and $s$ from the first and second line respectively should be substituted into the third equation and an equality check performed. Should there not be a solution that fulfills the third equation, the lines are skew. Otherwise, they intersect.
### Vector projections
If two vectors $\vec{a}$ and $\vec{b}$ are placed tail-to-tail, the **component** of $\vec{a}$ in the direction of $\vec{b}$ is known as the **vector projection of $\vec{a}$ onto $\vec{b}$**. Represented by $Projection$, its magnitude is called the **scalar projection**.
$$Proj_\vec{b}\vec{a}=\biggr(\frac{\vec{a}\bullet\vec{b}}{|\vec{b}|^2}\biggr)\vec{b}$$
$$
\begin{align*}
|Proj_\vec{b}\vec{a}|&=\frac{\vec{a}\bullet\vec{b}}{|\vec{b}|} \\
&=|\vec{a}|\cos\theta
\end{align*}
$$
!!! warning
The magnitude of any projection is always **positive**. If $\cos\theta$ returns a negative value, it needs to be absed again.
Vector projections are applied in work equations — see [SL Physics 1](/sph3u7/#work) for more information.
### Cross product
The cross product or **vector product** is a vector that is perpendicular of two vectors that are not colinear. Where $\vec{u}_1,\vec{u}_2,\vec{3}$ represent the x, y, and z coordinates of the position vector $\vec{u}$, respectively:
$$
\begin{align*}
\vec{u}\times\vec{v}&=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
\vec{u}_1 & \vec{u}_2 & \vec{u}_3 \\
\vec{v}_1 & \vec{v}_2 & \vec{v}_3
\end{vmatrix} \\
\\
&=-\hat{j}\begin{vmatrix}
\vec{u}_1 & \vec{u}_3 \\
\vec{v}_1 & \vec{v}_3
\end{vmatrix}
+\hat{i}\begin{vmatrix}
\vec{u}_2 & \vec{u}_3 \\
\vec{v}_2 & \vec{v}_3
\end{vmatrix}
+\hat{k}\begin{vmatrix}
\vec{u}_1 & \vec{u}_2 \\
\vec{v}_1 & \vec{v}_2
\end{vmatrix} \\
\\
&=[\vec{u}_2\vec{v}_3-\vec{u}_3\vec{v}_2,\vec{u}_3\vec{v}_1-\vec{u}_1\vec{v}_3,\vec{u}_1\vec{v}_2-\vec{u}_2\vec{v}_1]
\end{align*}
$$
Cross products are:
- anti-communtative: $\vec{u}\times\vec{v}=-(\vec{u}\times\vec{v})$
- distributive: $\vec{u}\times(\vec{u}+\vec{w})=\vec{u}\times\vec{v}+\vec{u}\times\vec{w}$
- associative over scalars: $m(\vec{u}\times\vec{v})=(m\vec{u})\times\vec{v}=(m\vec{v})\times\vec{u}$
The **magnitude** of a cross product is opposite that of the dot product. Where $\theta$ is the smaller angle between the two vectors ($0\leq\theta\leq180^\circ$):
$$|\vec{u}\times\vec{v}|=|\vec{u}||\vec{v}|\sin\theta$$
This is also equal to the area of a parallelogram enclosed by the vectors — where one is the base and the other is the adjacent side.
To determine the **direction** of a cross product, the right-hand rule can be used. Spreading the fingers out:
- the thumb is the direction of the first vector
- the index finger is the direction of the second vector
- the palm faces the direction of the cross product
### Applications of vector operations
A **triple scalar product** is the result of a cross product performed first then put in a dot product.
$$|\vec{c}\bullet(\vec{a}\times\vec{b})|$$
In a **parallelpiped**, or a three-dimensional shape with six faces each a parallelogram with an identical one opposite it, the volume is the triple scalar product of the distinct three vectors that make up its side lengths:
$$A=|\vec{c}\bullet(\vec{a}\times\vec{b})|$$
For an object moving at **constant velocity in 2D space**, where $\vec{s}$ is its displacement, $\vec{s}_0$ is its initial displacement at $t=0$, $t$ is the time elapsed, and $\vec{v}$ is its velocity:
$$\vec{s}=\vec{s}_0+t\vec{v}$$
**Torque** ($\vec{\tau}$ or $\vec{M}$) is the ability to rotate an object — effectively angular/rotational force — and is the cross product of the **outward-pointing radius vector** ($\vec{r}$) and the **force** vector ($\vec{F}$).
$$
\begin{align*}
\vec{\tau}&=\vec{r}\times\vec{F} \\
&=|\vec{r}||\vec{F}|\sin\theta
\end{align*}
$$
<img src="/resources/images/torque.svg" width=700>(Source: Wikimedia)</img>
The direction of the torque can be found using the **right-hand rule**.
**Force** and **velocity** are vectors with magnitude and direction. See [SL Physics 1#Force diagrams](/sph3u7/#force-diagrams) and [SL Physics 1#Velocity](/sph3u7/#velocity) for more information.
### Operations with vector components
If **Cartesian vectors** (see [SL Physics 1#Adding/subtracting vectors algebraically](/sph3u7/#addingsubtracting-vectors-algebraically) for more details) cannot be used, the **sine and cosine laws** can be used, which are, respectively:
Where $a$, $b$, and $c$ are the lengths of a triangle, and $A$, $B$, and $C$ are their angles opposite to them:
$$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$
$$c^2=a^2+b^2-2ab\cos C$$
### Vector planes
A **plane** ($\pi$) is a flat surface extending infinitely in all directions and can be represented by a parallelogram.
So long as any of the following are known, their plane can be constructed:
- Two intersecting or parallel lines
- A line and a point not on the line
- 3 non-colinear points
In each scenario, an initial point $r_0$ and two direction vectors $\vec{u},\vec{v}$ can be derived to form the equation for a plane:
$$\vec r = \vec r_0 + s\vec u + t\vec v,s,t\in\mathbb R$$
This can be expanded to form the parametric form of the equation:
$$
\begin{align*}
x&=x_0+su_1+tv_1 \\
y&=y_0+su_2+tv_2 \\
z&=z_0+su_3+tv_3,s,t\in\mathbb R
\end{align*}
$$
Where $A,B.C.D$ are all integers, the **scalar** or Cartesian equation of a plane in three dimensions can be expressed as follows:
$$Ax+By+Cz+D=0$$
!!! info
$[A,B,C]$ is the **normal direction vector** of a plane.
### Interactions of planes
A line intersects a plane if the dot product between the two is not zero, and the resulting scalar multiple found can be used to find the point of intersection. Otherwise, once the equations are substituted into each other, if the statement is true, the line and plane are **parallel and coincident**. Otherwise, they are parallel.
The shortest distance between two **skew lines** $L_1$ and $L_2$ is equal to:
$$
\begin{align*}
d&=|Proj_\vec{n}\vec{P_1P_2}| \\
&=\frac{|\vec{P_1P_2}\bullet(\vec m_1\times\vec m_2)|}{|\vec m_1\times\vec m_2|}
\end{align*}
$$
The shortest distance between a point $P(x_1,y_1,z_1)$ and plane $\pi: Ax+By+Cz+D=0$ is equal to:
$$d=\frac{|Ax_1+By_1+Cz_1+D|}{\sqrt{A^2+B^2+C^2}}$$
The shortest distance between two parallel planes is equal to:
$$d=\frac{|D_1-D_2|}{\sqrt{A^2+B^2+C^2}}$$
Two planes are parallel if their direction vectors are scalar multiples of each other:
$$\vec n_1 = k\vec n_2$$
If they are also coincident, the D-values will also be identical or equal to the same scalar multiple as the normal:
$$D_1=D_2$$
Otherwise, the planes intersect, the line along which is equal to the cross product between the two direction vectors.
$$\vec m=\vec n_1\times\vec n_2$$
An initial point vector can be solved by setting any of the variables ($x,y,z$) to zero and solving for the others. Alternatively, the parameter $t$ can be set equal to one of the variables instead and the parametric equation derived that way.
The **angle between two planes** is equal to the angle between their normal direction vectors, which can be determined using the dot product formula.
When looking at three planes:
If all three normals are scalar multiples:
- If all three $D$-values are those same scalar multiples, the planes are parallel and coincident and they have infinite points of intersection along the plane equation.
- Otherwise, there are no solutions and the planes are parallel and distinct and/or parallel and coincident for two.
If two normals are scalar multiples:
- If the two parallel planes are coincident with the same $D$-values, there will be a line of intersection much like solving for intersection between two planes.
- Otherwise, the two parallel planes are distinct, forming a Z-pattern with the third plane and so there is no solution.
If no normals are scalar multiples:
- If the triple scalar product of the three planes is equal to zero, the normal vectors are not coplanar and so there will be a point of intersection.
- Alternatively, by solving the scalar equations for the planes, if:
- the result is a contradiction (e.g., $0 = 3$), there is no solution
- the result is true with no variable (e.g., $0 = 0$), there are is an infinite number of solutions along a line
- the result contains a variable (e.g., $t = 4$), there is a single point of intersection at the parameter $t$.
## Matrices
A **matrix** is a two-dimensional array with rows and columns, represented by a capital letter and a grid denoted by square brackets.
$$
A=
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
$$
$A_{ij}$ represents the element in the $i$th row and the $j$th column.
A **coefficient matrix** contains coefficients of variables.
$$
A=
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
$$
An **augmented matrix** also contains constants, separated by a vertical line.
$$
A=
\left[\begin{array}{rrr|r}
1 & 2 & 3 & 5 \\
4 & 5 & 6 & 10
\end{array}\right]
$$
!!! example
The equation system
$$
x+2y-4z=3 \\
-2x+y+3z=4 \\
4x-3y-z=-2
$$
can be written as the matrix
$$
A=
\left[\begin{array}{rrr|r}
1 & 2 & -4 & 3 \\
-2 & 1 & 3 & 4 \\
4 & -3 & -1 & -2
\end{array}\right]
$$
### Gaussian elimination
Gaussian elimination is used to solve a system of linear relations, such as that of plane equations. It aims to reduce a matrix into its **row echelon form** shown below to solve for each variable.
$$
A=
\left[\begin{array}{rrr|r}
a & b & c & d \\
0 & e & f & g \\
0 & 0 & h & i
\end{array}\right]
$$
The following **row operations** can be performed on the matrix to achieve this state:
- swapping (interchanging) the position of two rows
- $R_a \leftrightarrow R_b$
- multiplying a row by a non-zero constant
- $AR_a \to R_a$
- adding/subtracting rows, overwriting the destination row
- $R_a\pm R_b\to R_b$
- multiplying a row by a non-zero constant and then adding/subtracting it to another row
- $AR_a + R_b \to R_b$
!!! example
In the matrix from the previous example, by performing $R_1\leftrightarrow R_2$:
$$
A=
\left[\begin{array}{rrr|r}
-2 & 1 & 3 & 4 \\
1 & 2 & -4 & 3 \\
4 & -3 & -1 & -2
\end{array}\right]
$$
$5R_1\to R_1$:
$$
A=
\left[\begin{array}{rrr|r}
-10 & 5 & 15 & 20 \\
1 & 2 & -4 & 3 \\
4 & -3 & -1 & -2
\end{array}\right]
$$
$10R_2+R_1\to R_1$:
$$
A=
\left[\begin{array}{rrr|r}
0 & 25 & -25 & 50 \\
1 & 2 & -4 & 3 \\
4 & -3 & -1 & -2
\end{array}\right]
$$
## Resources
- [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)
- [IB Math Analysis and Approaches Formula Booklet](/resources/g11/ib-math-data-booklet.pdf)
- [Textbook: Calculus and Vectors 12](/resources/g11/calculus-vectors-textbook.pdf)
- [Textbook: Oxford SL Mathematics](/resources/g11/textbook-oxford-math.pdf)
- [Course Pack Unit 1: Integration](/resources/g11/s2cp1.pdf) ([Annotated](/resources/g11/s2cp1-anno.pdf))
- [Course Pack Unit 2: Probability](/resources/g11/s2cp2.pdf)([Annotated](/resources/g11/s2cp2-anno.pdf))
- [Course Pack Unit 3: Vectors](/resources/g11/s2cp3.pdf) ([Annotated](/resources/g11/s2cp3-anno.pdf))
- [Course Pack Unit 4: Vector Applications](/resources/g11/s2cp4.pdf) ([Annotated](/resources/g11/s2cp4-anno.pdf))
- [Course Pack Unit 5: Planes](/resources/g11/s2cp5.pdf) ([Annotated](/resources/g11/s2cp5-anno.pdf))
- [TI-84 Plus Calculator Guide](/resources/g11/ti-84-plus.pdf)

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# SL Math - Analysis and Approaches - 1
The course code for this page is **MHF4U7**.
## Review
### Logarithm rules
The logarithm of a product can be rewritten as the sum of two logarithms.
$$\log_c(ab)=\log_c(a)+\log_c(b)$$
The logarithm of a quotient can be rewritten as the difference of two logarithms.
$$\log_c\biggr(\frac{a}{b}\biggr)=\log_c(a)-\log_c(b)$$
The exponentials of a logarithm can be brought down to be coefficients.
$$\log_c(a^n)=n\log_c(a)$$
Some simple values can be easily found.
$$
a^{\log_a(x)}=x \\
\log_a(a)=1 \\
\log_a(1)=0
$$
## 3 - Geometry and trigonometry
To find the result of a primary trig ratio, the related acute angle (RAA) should first be found before referring to the CAST rule to determine quadrants before identifying all correct answers in the domain.
### Circles
The equation below is true for every point on a circle with radius $r$.
$$x^2+y^2=r^2$$
The area of a **sector** requires knowledge of the radius and angle in **radians** that the sector encompasses.
$$A=\frac{r^2\theta}{2}$$
<img src="/resources/images/sector.png" width=500>(Source: Kognity)</img>
### Trigonometric identities
The **Pythagorean identity** relates the radius of a circle to its x and y components.
$$\sin^2\theta+\cos^2\theta=1$$
The **quotient identity** relates the side lengths of a right-angled triangle.
$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
The **double angle identities** can be used to convert one trig ratio to another.
$$
\sin 2\theta = 2\sin\theta\cos\theta \\
\cos 2\theta = 2\cos^2\theta-1 \\
\cos 2\theta = \cos^2\theta-\sin^2\theta \\
\cos 2\theta = 1-2\sin^2\theta \\
\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}
$$
### Euler's number
Euler's number $e$ is a constant irrational number represented as a special limit in calculus.
$$e=\lim_{x\to ∞}\biggr(1+\frac{1}{x}\biggr)^x$$
The inverse of $e^x$ is $\log_e(x)$, which is known as the **natural logarithm** and can be rewritten as $\ln(x)$ ("lawn x").
## 4 - Statistics and probability
!!! note "Definition"
- **Statistics:** The techniques and procedures to analyse, interpret, display, and make decisions based on data.
- **Descriptive statistics:** The use of methods to work with and describe the **entire** data set.
- **Inferential statistics:** The use of samples to make judgements about a population.
- **Data set:** A collection of data with elements and observations, typically in the form of a table. It is similar to a map or dictionary in programming.
- **Element:** The name of an observation(s), similar to a key to a map/dictionary in programming.
- **Observation:** The collected data linked to an element, similar to a value to a map/dictionary in programming.
- **Population**: A collection of all elements of interest within a data set.
- **Sample**: The selection of a few elements within a population to represent that population.
- **Raw data:** Data collected prior to processing or ranking.
### Sampling
A good sample:
- represents the relevant features of the full population,
- is as large as reasonably possible so that it decently represents the full population,
- and is random.
The types of random sampling include:
- **Simple**: Choosing a sample completely randomly.
- **Convenience**: Choosing a sample based on ease of access to the data.
- **Systematic**: Choosing a random starting point, then choosing the rest of the sample at a consistent interval in a list.
- **Quota**: Choosing a sample whose members have specific characteristics.
- **Stratified**: Choosing a sample so that the proportion of specific characteristics matches that of the population.
??? example
- Simple: Using a random number generator to pick items from a list.
- Convenience: Asking the first 20 people met to answer a survey,
- Systematic: Rolling a die and getting a 6, so choosing the 6th element and every 10th element after that.
- Quota: Ensuring that all members of the sample all wear red jackets.
- Stratified: The population is 45% male and 55% female, so the proportion of the sample is also 45% male and 55% female.
### Types of data
!!! note "Definition"
- **Quantitative variable**: A variable that is numerical and can be sorted.
- **Discrete variable**: A quantitative variable that is countable.
- **Continuous variable**: A quantitative variable that can contain an infinite number of values between any two values.
- **Qualitative variable**: A variable that is not numerical and cannot be sorted.
- **Bias**: An unfair influence in data during the collection process, causing the data to be not truly representative of the population.
### Frequency distribution
A **frequency distribution** is a table that lists categories/ranges and the number of values in each category/range.
A frequency distribution table includes:
- A number of classes, all of the same width.
- This number is arbitrarily chosen, but a commonly used formula is $\lceil1+3.3\log({\text{# of elements})}\rceil$.
- The width (size) of each class is $\lceil\frac{\text{max} - \text{min}}{\text{# of classes}}\rceil$.
- Each class includes its lower bound and excludes its upper bound ($\text{lower} ≤ x < \text{upper}$)
- The **relative frequency** of a data set is the percentage of the whole data set present in that class in decimal form.
- The number of values that fall under each class.
- The largest value can either be included in the final class (changing its range to $\text{lower} ≤ x ≤ \text{highest}$), or put in a completely new class above the largest class.
??? example
| Height $x$ (cm) | Frequency |
| --- | --- |
| $1≤x<5$ | 2 |
| $5≤x<9$ | 3 |
| $9≤x≤14$ | 1 |
For a given class $i$, the midpoint of that class is as follows:
$$x_{i} = \frac{\text{lower bound} + \text{upper bound}}{2}$$
### Quartiles
A **percentile** is a value indicates the percentage of a data set that is below it. To find the location of a given percentile, $P_k = \frac{kn}{100}$, where $k$ denotes the percentile number and $n$ represents the sample size.
A **decile** indicates that $n×10$% of data in the data set is below it.
!!! example
A score equal to or greater than 97% of all scores in a test is said to be in the *97th percentile*, or in the *9th decile*.
Quartiles split a data set into four equal sections.
- The **minimum** is the lowest value of a data set.
- The **first quartile** ($Q_1$) is at the 25th percentile.
- The **median** is at the 50th percentile.
- The **third quartile** ($Q_3$) is at the 75th percentile.
- The **maximum** is the highest value of a data set.
The first and third quartiles are the median of the **[minimum, median)** and **(median, maximum]** respectively.
!!! warning
When the median is equal to a data point in a set, it *cannot* be used to find $Q_1$ or $Q_3$. Only use the data below or above the median.
!!! warning
When working with grouped data given in ranges, the actual data is unavailable. The five numbers above are instead:
- The minimum value is now the lower class boundary of the lowest class.
- The first and third quartiles, as well as the median, are now found by guesstimating the value on a cumulative frequency curve.
- The maximum value is now the upper class boundary of the highest class. If the highest value is excluded (e.g., $90≤x<100$), it also must be excluded when representing data (e.g., open dot instead of filled dot).
- A specific percentile can be found by guesstimating the value on a cumulative frequency curve.
The **interquartile range (IQR)** is equal to $Q_3 - Q_1$ and represents the range where 50% of the data lies.
### Outliers
Outliers are data values that significantly differ from the rest of the data set. They may be because of:
- a random natural occurrence, or
- abnormal circumstances
Outliers can be ignored once identified.
There are various methods to identify outliers. For **single-variable** data sets, the **lower and upper fences** may be used. Any data below the lower fence or above the upper fence can be considered outliers.
- The lower fence is equal to $Q_1 - 1.5×\text{IQR}$
- The upper fence is equal to $Q_3 + 1.5×\text{IQR}$
### Representing frequency
A **stem and leaf plot** can list out all the data points while grouping them simultaneously.
A **frequency histogram** can be used to represent frequency distribution, with the x-axis containing class boundaries, and the y-axis representing frequency.
<img src="/resources/images/frequency-discrete.png" width=700>(Source: Kognity)</img>
!!! note
If data is discrete, a gap must be left between the bars. If data is continuous, there must *not* be a gap between the bars.
A **cumulative frequency table** can be used to find the number of data values below a certain class boundary. It involves the addition of a **cumulative frequency** column which represents the sum of the frequency of the current class as well as every class before it. It is similar to a prefix sum array in computer science.
??? example
| Height $h$ (cm) | Frequency | Cumulative frequency |
| --- | --- | --- |
| $1≤h<10$ | 2 | 2 |
| $10≤h<19$ | 5 | 7 |
A **cumulative frequency curve** consists of an independent variable on the x-axis, and the cumulative frequency on the y-axis. In grouped data, the values on the x-axis correspond to the upper bound of a given class. This graph is useful for interpolation (e.g., the value of a given percentile).
<img src="/resources/images/cumulative-frequency-curve.png" width=700>(Source: Kognity)</img>
A **box-and-whisker plot** is a visual representation of the **"5-number summary"** of a data set. These five numbers are the minimum and maximum values, the median, and the first and third quartiles.
<img src="/resources/images/box-and-whisker.png" width=700>(Source: Kognity)</img>
!!! warning
In the image above, the maximum and minimum dots are filled. If these values were to be excluded (e.g., the upper class boundary in grouped data is excluded), they should be unfilled instead.
### Measures of central tendency
The **mean** is the sum of all values divided by the total number of values. $\bar{x}$ represents the mean of a sample while $µ$ represents the mean of a population.
$$\bar{x}=\frac{\sum x}{n}$$ where $n$ is equal to the number of values in the data set.
In grouped data, the mean can only be estimated, and is equal to the average of the sum of midpoint of all classes multiplied by their class frequency.
$$\bar{x} = \frac{\sum x_i f_i}{n}$$ where $x_i$ is the midpoint of the $i$th class and $f_i$ is the frequency of the $i$th class.
The **median** is the middle value when the data set is sorted. If the data set has an even number of values, the median is the mean of the two centre-most values.
In grouped data, the median class is the class of the $\frac{n+1}{2}$th value if the number of values in the class is odd or the $\frac{n}{2}$th value otherwise.
The **mode** is the value that appears most often.
!!! definition
- **Unimodal**: A data set with one mode.
- **Bimodal**: A data set with two modes.
- **Multimodal**: A data set with more than two modes.
- **No mode**: A data set with no values occurring more than once.
In grouped data, the **modal class** is the class with the greatest frequency.
### Measures of dispersion
These are used to quantify the variability or spread of the data set.
The **range** of a data set is simple to calculate but is easily thrown off by outliers.
$$R = \max - \min$$
The **variance** ($\sigma^2$) and **standard deviation** ($\sigma$) of a data set are more useful. The standard deviation indicates how closely the values of a data set are clustered around the mean.
$$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{n}}$$ where $f_i$ is the frequency of the $i$th class, $x_i$ is the midpoint of the $i$th class, $\bar{x}$ is the mean of the whole data set, and $n$ is the number of values in the data set.
For ungrouped data, assume $f_i = 1$.
In a typical bell-shaped distribution:
- 68% of data lie within 1 standard deviation of the mean ($\bar{x} ± \sigma$)
- 95% of data lie within 2 standard deviations of the mean ($\bar{x} ± 2\sigma$)
- 99.7% of data lie within 3 standard deviations of the mean ($\bar{x} ± 3\sigma$)
- any data outside 3 standard deviations of the mean can be considered outliers
!!! info
The **points of inflection** (when the curve changes direction) of a normal bell curve occur at $\bar{x} ± \sigma$.
### Data transformation
When performing an operation with a constant value to a whole data set:
| Operation | Effect on mean | Effect on standard deviation |
| --- | --- | --- |
| Addition/subtraction | Increased/decreased by constant | No change |
| Multiplication/division | Multiplied/divided by constant | Multiplied/divided by constant |
### Linear correlation and regression
!!! definition
- **Interpolation**: The prediction of values within the range of a data set.
- **Extrapolation**: The prediction of values outside the range of a data set. This tends to be less reliable than interpolation as it is unknown if the model is accurate outside of the range of the data set..
A scatter plot is used to help find trends and relationships between variables, which is primarily used to predict results not in the data set.
If there is a clear trend in the data, there is said to be a **correlation** between the independent and dependent variables.
- If the line has an upward trend, it has a positive correlation.
- If the line has a downward trend, it has a negative correlation.
The strength of the correlation ranges from none, weak, moderate, strong, and perfect, where the latter shows a line passing through all data points.
The line of best fit may not be linear. It may be quadratic, exponential, logarithmic, or there might not be a line of best fit at all. In the latter case, there is **no correlation**.
**Correlation does not imply causation**. There may be an external **confounding factor** which causes both trends, instead.
!!! example
If ice cream consumption increases as deaths from drowning increase, it does not mean that drowning causes people to eat more ice cream. The confounding factor of summer increases ice cream consumption and frequency of swimming, which leads to more people drowning.
To find the **regression line** (line of best fit), a mean data point is required. The mean data point is a new point located at the mean of all x- and y-coordinates, or $M = (\bar{x}, \bar{y})$. The regression line then is the line that passes through the mean point while minimising the *vertical* distance from every data point. This is most easily performed on a graphing display calculator (GDC), but can be calculated manually if needed.
The **least squares regression** is used to find the equation of a line that passes through the mean point for which the *square* of the vertical distance between the line and all data points (the residuals) is minimised for each point. It involves forming a line such that the sum of all residuals is $0$, and the sum of all residuals squared is minimised.
Alternatively, to manually guesstimate a linear line of best fit, a line can be drawn from the mean point to a point that best appears to lie on the line of best fit.
The **Pearson product-moment correlation coefficient** (more commonly known as *Pearson's $r$* or the *$r$-value*) quantifies the **correlation strength** of a line of best fit, or how well the line of best fit fits. This value is such that $-1≤r≤1$, where
- $r>0$ is a positive correlation
- $r<0$ is a negative correlation
- $|r|=1$ is a perfect correlation
- $0.7≤|r|<1$ is a strong correlation
- $0.3≤|r|<0.7$ is a weak to moderate correlation
- $0≤|r|<0.3$ is no correlation, so that no line of best fit can be drawn.
## 5 - Calculus
### Rate of change
The **average rate of change (ARoC)** between points $P(a, f(a))$ and $Q(a + h, f(a+h))$ is represented by the slope of the **secant line ($m_s$)**. Therefore, as slope is the difference in rise over the difference of run ($\frac{\Delta y}{\Delta x}$), the slope of the secant line can be expressed as
$$m_s = \frac{f(a+h)-f(a)}{h}, h ≠ 0$$
This is known as the **difference quotient**.
The **instantaneous rate of change (IRoC)** at point $P(a, f(a))$ is represented by the slope of the **tangent line ($m_T$)**. The slope of the tangent line can be found by finding the difference quotient with $h$ as a very small value, e.g., $0.001$.
!!! warning
The above method of finding the IRoC should be disregarded in favour of finding the derivative.
### Sequences
A sequence is a **function** with a domain of all positive integers in sequence, but uses subscript notation ($t_n$) instead of function notation ($f(x)$).
!!! reminder
- The **recursive** formula for a sequence is $t_n = t_{n-1} + 2$ where $t_1 = 1$.
- The **arithmetic** formula for a sequence is $t_n = 2n-1$.
If the sequence is infinite, as $n$ becomes very large:
- If the sequence continuously grows, it **tends to infinity**. (E.g., $a_n = n^2, n ≥ 1$)
- If the sequence gets closer to a real number and converges on it, it **converges to a real limit**, or is **convergent**. (E.g., $a_n = \frac{1}{n}, n ≥ 1$)
- If the sequence never approaches a number, it **does not tend to a limit**, or is **divergent**. (E.g., $a_n = \sin(n \pi)$)
### Limits
A **limit** to a function is the behaviour of that function as a variable approaches, **but does not equal**, another variable.
!!! example
$$\lim_{x \to c} f(x) = L$$
"The limit of $f(x)$ as $x$ approaches $c$ is $L$."
If the lines on both sides of a limit do not converge at the same point, that limit *does not exist*.
If the lines on both sides of a limit become arbitrarily large as $x$ approaches $a$, it approaches infinity.
$$\lim_{x \to a} f(x) = ∞$$
### One-sided limits
A positive or negative sign is used at the top-right corner of the value approached to denote if that limit applies only to the negative or positive side, respectively. A limit without this sign applies to both sides.
!!! example
- $\lim_{x \to 3^-} f(x) = 2$ shows that as $x$ approaches $3$ from the negative (usually left) side, $f(x)$ approaches $2$.
- $\lim_{x \to 3^+} f(x) = 2$ shows that as $x$ approaches $3$ from the positive (usually right) side, $f(x)$ approaches $2$.
- $\lim_{x \to 3} f(x) = 2$ shows that as $x$ approaches $3$ from either side, $f(x)$ approaches $2$.
If $\lim_{x \to c^-} f(x) ≠ \lim_{x \to c^+} f(x)$, $\lim_{x \to c} f(x)$ **does not exist**.
### Properties of limits
The following properties assume that $f(x)$ and $g(x)$ have limits at $x = a$, and that $a$, $c$, and $k$ are all real numbers.
- $\lim_{x \to a} k = k$
- $\lim_{x \to a} x = a$
- $\lim_{x \to a} [f(x) ± g(x)] = \lim_{x \to a} f(x) ± \lim_{x \to a} g(x)$
- $\lim_{x \to a} [f(x) \cdot g(x)] = [\lim_{x \to a} f(x)] [\lim_{x \to a} g(x)]$
- $\lim_{x \to a} [k \cdot f(x)] = k \cdot \lim_{x \to a} f(x)$
- $\lim_{x \to a} [f(x)]^2 = [\lim_{x \to a} f(x)]^2$
### Evaluating limits
When solving for limits, there are five central strategies used, typically in this order if possible:
#### Direct substitution
Substitute $x$ as $a$ and solve.
??? example
$$
\lim_{x \to 5} (x^2 + 4x + 3) \\
= 5^2 + 4(5) + 3 \\
= 48
$$
If **only** direct substitution fails and returns $\frac{0}{0}$, continue on with the following steps. If **only** the denominator is $0$, the limit **does not exist**.
#### Factorisation, expansion, and simplification
Attempt to factor out the variable as much as possible so that the result is not $\frac{0}{0}$, and then perform direct substitution.
??? example
$$
\lim_{x \to 1} \frac{x^2 - 1}{x-1} \\
= \lim_{x \to 1} \frac{(x + 1) (x - 1)}{x-1} \\
= \lim_{x \to 1} (x+1) \\
= 1 + 1 \\
= 2
$$
#### Rationalisation
If there is a square root, multiplying both sides of a fraction by the conjugate may allow direct substitution or factorisation.
??? example
$$
\lim_{x \to 0} \frac{\sqrt{1-x}-1}{x} \\
= \lim_{x \to 0} \frac{\sqrt{1-x}-1}{x} \cdot \frac{\sqrt{1-x}+1}{\sqrt{1-x}+1} \\
= \lim_{x \to 0} \frac{1-x - 1}{x\sqrt{1-x} + x} \\
= \lim_{x \to 0} \frac{1}{\sqrt{1-x} + 1} \\
= \frac{1}{\sqrt{1-0} + 1} \\
= \frac{1}{2}
$$
#### One-sided limits
There may only be one-sided limits. In this case, breaking the limit up into its two one-sided limits can confirm if the two-sided limit does not exist when looked at together.
#### Change in variable
Substituting a variable in for the variable to be solved and then solving in terms of that variable may remove a problem variable.
??? example
$$
\lim_{x \to 0} \frac{x}{(x+1^\frac{1}{3}-1} \\
\text{let } (x+1)^\frac{1}{3} \text{ be } y \\
x + 8 = y^3 \\
x = y^3 - 8, \text{as } x \to 0, y \to 2 \\
\lim_{y \to 2} \frac{y-2}{y^3 - 8} \\
= \lim_{y \to 2} \frac{(y-2)(y^2 + 4y + 4)}{(y^3-8)(y^2 + 4y + 4)} \\
= \lim_{y \to 2} \frac{1}{y^2 + 4y + 4} \\
= \frac{1}{2^2 + 4(2) + 4} \\
= \frac{1}{16}
$$
!!! note
If $\lim_{x \to a} \frac{f(x)}{g(x)}$ exists and direct substitution is not possible, $x - a$ *must* be a factor of both $f(x)$ and $g(x)$ so that the discontinuity can be removed. Therefore, $f(a) = 0$ and $g(a) = 0$.
### Limits and continuity
If a function has holes or gaps or jumps (i.e., if it cannot be drawn with a writing utensil held down all the time), it is **discontinuous**. Otherwise, it is a **continuous** function. A function discontinuous at $x=a$ is "discontinuous at $a$", where $a$ is the "point of discontinuity".
A **removable discontinuity** occurs when there is a hole in a function. It can be expressed as when either
$$
f(a) = \text{DNE or} \\
\lim_{x \to a} f(x) ≠ f(a)
$$
A **jump discontinuity** occurs when both one-sided limits have different values. It is common in piecewise functions. It can be expressed as when
$$\lim_{x \to a^-} f(x) ≠ \lim_{x \to a^+} f(x)$$
An **infinite discontinuity** occurs when both one-sided limits are infinite. It is common when functions have vertical asymptotes. It can be expressed as when
$$\lim_{x \to a} f(x) = ± ∞$$
Therefore, a function is only continuous at $a$ if all of the following are true:
- $f(a)$ exists
- $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$
- $\lim_{x \to a} f(x) = f(a)$
### Limits approaching infinity
As $x$ approaches infinity, $\lim_{x \to ∞} f(x)$ has only three possible answers.
By dividing both sides of a fraction by the $x$ variable of the highest degree, if $m$ is the degree of the denominator and $n$ is the degree of the numerator:
- If $m > n$, $\lim_{x \to ∞} f(x) = 0$
- If $m < n$, $\lim_{x \to ∞} f(x) = ± ∞$
- The sign of infinity can be found by evaluating the limit
- If $m = n$, $\lim_{x \to ∞} f(x) = \frac{a}{b}$, where $a$ and $b$ are the coefficients of the degree of the numerator and the denominator, respectively.
### Derivatives
A derivative function is a function of all **tangent slopes** in the original function. It can either be expressed in function notation as $f´(x)$ ("f prime of x") or in Leibniz notation as $\frac{dy}{dx}$. The process of finding a derivative of a function is known as **differentiation**.
!!! note
Although evaluating a derivative function in function notation is the usual $f´(5)$ to solve for when $x = 5$, Leibniz notation is stupid and requires the following (the vertical bar shown should be solid):
$$\frac{dy}{dx} \biggr|_{x=5}$$
If $f´(a)$ exists, the function is "differentiable at $a$" such that $f´(a^-) = f´(a^+)$. Functions are only differentiable at $a$ if the function is **continuous at $a$** and the tangent at $a$ is not vertical.
!!! example
Some examples of issues that can cause $f´(a)=\text{DNE}$ are vertical asymptotes and other discontinuities, vertical tangents, cusps, and corners. The last two cause $f´(a^-) ≠ f´(a^+)$.
### Finding derivatives using first principles
The first principles method of finding derivatives involves using simple algebra and limits. Taking the difference quotient and adding a limit of $h \to 0$:
$$f´(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$
results in the equation of the derivative function. Direct substitution of $h$ will result in an indeterminate form, so the equation should be manipulated to remove $h$ from the denominator typically via factoring.
??? example
Differentiating $f(x)=2x^2 + 6$ using first principles:
$$
f´(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\
= \lim_{h \to 0} \frac{2(x+h)^2 + 6 - (2x^2 - 6)}{h} \\
= \lim_{h \to 0} \frac{4xh+2h^2}{h} \\
= \lim_{h \to 0} 4x+2h \\
f´(x)=4x
$$
### Drawing derivative functions
If the slope of a tangent is:
- positive/negative, that value on the derivative graph is also positive/negative, respectively
- zero, that value on the derivative graph is on the x-axis
Points of inflection on the original function become maximum/minimum points on the derivative graph.
The derivative of a linear equation is always constant, and the derivative of a constant value is $0$..
### Derivative rules
These rules can be used in place of/to supplement finding derivative functions using first principles and are usually much faster to calculate. These rules assume that all of the functions involved are differentiable.
The degree of a derivative is always the degree of the original function$-1$.
The **power rule** applies to all functions of the form $f(x)=x^n,x \in \mathbb{R}$, such that:
$$f´(x) = nx^{n-1}$$
??? example
$$f(x) = x^5$$
$$f´(x) = 5x^4$$
The **constant multiple rule** applies to all functions of the form $f(x) = k·g(x)$, where $k$ is any real number, such that:
$$f´(x) = k·g´(x)$$
??? example
$$f(x) = 2x^2$$
$$f´(x) = 2·2x$$
$$f´(x) = 4x$$
The **sum rule** applies to all functions of the form $f(x) = g(x) + h(x)$ such that:
$$f´(x) = g´(x) + h´(x)$$
??? example
$$f(x) = 2x^2 + 3x$$
$$f´(x) = 4x + 3$$
The **product rule** applies to all functions of the form $f(x) = g(x)h(x)$ such that:
$$f´(x) = g´(x)h(x) + g(x)h´(x)$$
??? example
$$f(x) = (2x+5)(x-1)$$
$$f´(x) = 2(x-1) + (2x+5)·1$$
$$f´(x) = 4x + 1$$
The **extended product rule** applies to all functions of the form $f(x) = g(x)h(x)j(x)$ such that:
$$f´(x) = g´(x)h(x)j(x) + g(x)h´(x)j(x) + g(x)h(x)j´(x)$$
The **quotient rule** applies to all functions of the form $f(x) = \frac{g(x)}{h(x)}$ such that:
$$f´(x) = \frac{g´(x)h(x)-g(x)h´(x)}{[h(x)]^2}, h(x) ≠ 0$$
??? example
$$f(x) = \frac{2x+5}{x-1}$$
$$f´(x) = \frac{2(x-1) - (2x+5)·1}{(x-1)^2}$$
$$f´(x) = -\frac{7}{(x-1)^2}$$
The **chain rule** applies to all functions of the form $f(x) = g(h(x))$ such that:
$$f´(x) = g´(h(x)) · h´(x)$$
??? example
$$f(x) = (4x^2-3x+1)^7$$
$$f´(x) = 7(4x^2-3x+1)^6 (8x-3)$$
### Trigonometric derivative rules
$$
\frac{d}{dx}\sin x = \cos x \\
\frac{d}{dx}\cos x = -\sin x \\
$$
These primary derived rules can be used to further derive the derivatives of the other trignometric ratios:
$$
\frac{d}{dx}\tan x = \sec^2 x \\
\frac{d}{dx}\csc x = -\csc x\cdot\cot x \\
\frac{d}{dx}\sec x = \sec x\cdot\tan x
$$
The **chain rule** applies to trigonometric functions and will be applied recursively if needed.
!!! example
$$\frac{d}{dx}[\sin g(x)]^n = n[\sin g(x)]^{n-1}\cdot\cos x\cdot g´(x)$$
Trigonometric identities are not polynomial so values on an interval need to be determined by substituting values between vertical asymptotes and critical points.
### Extended derivative rules
For an **exponential function** where $f(x)=b^x,b≠0$ or $f(x)=b^{g(x)}$, respectively:
$$
f´(x)=b^x\cdot\ln(b) \\
f´(g(x))=b^{g(x)}\cdot\ln(b)\cdot g´(x)
$$
For a **logarithmic function** where $f(x)=\log_b(x)$ or $f(x)=\log_b(g(x))$, respectively:
$$
f´(x)=\frac{1}{\ln(b)\cdot x} \\
f´(x)=\frac{g´(x)}{\ln(b)\cdot g(x)}
$$
From the above base derivatives the derivatives for functions involving $e$ and the **natural logarithm** can be found:
$$
\frac{d}{dx}e^x=e^x \\
\frac{d}{dx}e^{g(x)}=e^{g(x)}\cdot g´(x) \\
\frac{d}{dx}\ln(x)=\frac{1}{x} \\
\frac{d}{dx}\ln(g(x))=\frac{g´(x)}{g(x)}
$$
This opens up the possibility of **logarithmic differentiation**, which is required for exponential or logarithmic functions with a variable base. The **natural logarithm** of both sides should be taken prior to differentiation and logarithmic rules applied to simplify the equation.
### Higher order derivatives
The **second derivative** of $f(x)$ is the derivative of the first derivative of $f(x)$, that is, $f´´(x)$.
The $n$th derivative of $f(x)$ is $f^{(n)}(x)$, and is the derivative of the $n-1$th derivative. It is written as $\frac{d^ny}{dx^n}$ in Leibniz notation.
!!! example
The second derivative of an object's position with respect to time is its acceleration. See [SL Physics A#Displaying motion](/sph3u7/#displaying-motion) for more information.
### Interval charts
To identify the positive or negative regions of an equation, an interval line or chart can be used. To do so:
1. Factor the equation as much as possible and identify the x-intercepts.
2. Place the x-intercepts on a line.
3. Find the sign of the end behaviour by taking the sign of the leading coefficient.
4. When crossing an x-intercept, if the degree of that factor is even, the sign stays the same; otherwise, it alternates.
5. Repeat for every other region.
### Implicit differentiation
Implicit differentiation differentiates both sides of an equation with respect to $x$ and solves for $\frac{dy}{dx}$ ($y´$). Note that if $y$ is isolated, this is effectively the same as explicit differentiation. When differentiating implicitly, it must be shown that the derivative of both sides with respect to x ($\frac{d}{dx}$) is being taken.
!!! warning
The **chain rule** must be applied when differentiating terms that contain $y$.
!!! example
$$
\frac{x^2}{4} + y^2 = 1 \\
\frac{2x}{4} + 2y · \frac{dy}{dx} = 0 \\
\frac{dy}{dx} = -\frac{\frac{x}{2}}{2y} \\
\frac{dy}{dx} = -\frac{x}{4y}
$$
### Related rates
When solving for questions that ask for rate of change related to other rates of change, ensure that:
- variables are defined
- equations are written in terms of derivates
- the equations are differentiated **with respect to time**
- apply derivative rules (especially the chain rule) to every variable that is not a constant (i.e., that changes with respect to time)
- substitute values only at the end
## 5.2 - Increasing and decreasing functions
- If $f´(x) > 0$ in the interval $[a,b]$, $f$ is **increasing** on $[a,b]$.
- If $f´(x) < 0$ in the interval $[a,b]$, $f$ is **decreasing** on $[a,b]$.
- If $f´(x) = 0$ in the interval $[a,b]$, $f$ is **constant** on $[a,b]$.
- The points where $f´(x)=0$ are the **critical**/maximum/minimum points.
Functions only change whether they are increasing/decreasing/constant at the **critical points**/"relative extrema".
These points and whether the intervals between them increase/decrease can be found by using an **interval chart/line** using the first derivative.
!!! example
If $f(x)=\frac{2x-3}{x^2+2x-3}$:
- $f$ is decreasing on $(-∞, -3) \cup (-3, 0) \cup (3, ∞)$.
- $f$ is increasing on $(0, 1) \cup (1, 3)$.
### Extrema
Extrema are the maximum and minimum points in a function or an interval of a function. They must be at **critical points**—where $f(x)=0$ or $f(x)=\text{DNE}$, and may include the **boundary points** if looking for extrema in a given interval.
The highest and lowest point(s) of $f(x)$ are known as the **absolute** maximum/minimum of $f(x)$.
Any other **relative/local** maxima or minima are such that all of the points around that point are higher or lower.
**Fermat's theorem** states that if $f(c)$ is a local extremum, $c$ must be a critical number of $f$. Therefore, if $f$ is continuous in the closed interval $[a,b]$, the absolute extrema of $f$ must occur at $a$, $b$, or a critical number.
To find the extrema of a **continuous** function $f(x)$, where $x=a$ is a critical value, the **first derivative test** may be used with the assistance of an interval chart/line. If only an interval of a function is under consideration, the boundary points must be taken under consideration as well.
- If $f´(a)$ changes from positive to negative, there is a relative/local minimum at $x=a$.
- If $f´(a)$ changes from negative to positive, there is a relative/local maximum at $x=a$.
- If the sign is the same on both sides, there is no extrema at $x=a$.
- The greatest/least relative/local maximum/minimum is the absolute maximum/minimum.
Alternatively, the second derivative test may be used instead. At the critical point where $x=a$, a positive $f´´(a)$ indicates a **local minimum** while a negative $f´´(a)$ indicates a **local maximum**. If $f´´(x)=0$, the test is inconclusive and the first derivative test must be used.
!!! example
The absolute minimum of $f(x)=x^2$ is at $(0,0)$. There is no absolute maximum nor are there any other relative/local maximum/minimum points.
!!! warning
- There can be multiple absolute maxima/minima if there are multiple points that are both highest/lowest.
- If a function is a horizontal line, the absolute maximum and minimum for $x \in \text{domain}$ is $y$.
### Concavity
!!! definition
A **point of inflection** on a curve is such that $f´´(x)=0 \text{ or DNE}$ and the signs of $f´´(x)$ around the point change (e.g., positive to negative).
- An interval is **concave up** if it increases from left to right and tangent lines are drawn below the curve, so $f´´(x)>0$. It is shaped like a smile.
- An interval is **concave down** if it increases from left to right and tangent lines are drawn **above** the curve, and $f´´(x)<0$. It is shaped like a frown.
Changes in concavity only occur at points of inflection.
### Cusps
A cusp is a point on a continuous function that is not differentiable as the slopes on both sides approach -∞ and ∞. Concavity does not change at a cusp, but they may be considered for local extrema.
### Optimisation
To optimise for a maximum or minimum of a variable:
- Identify an equation with only one variable dependent on another
- Find the first derivative and identify critical points
- Use the second derivative test to identify if the critical point is a maximum or minimum
- Check constraints and throw away any inadmissible results
Diagrams with labelled variables may be helpful.
### Asymptote behaviour
The vertical asymptotes of a function are at values of $x$ that make the denominator of the simplified function $0$. The behaviour near them can be found using limits as $x$ approaches those points.
The horizontal asymptotes of a function can be found as $x$ approaches positive and negative infinity. To determine behaviour near them, the sign of $\lim_{xs \to ±∞} f(x) - L$, where $L$ is the y-coordinate of the asymptote. A positive limit indicates that $f(x)$ is above the asymptote while a negative limit indicates that $f(x)$ is below the asymptote.
### Curve sketching
- Determine the domain of the function and consider discontinuities (holes and asymptotes)
- Determine the y-intercept and if easy, x-intercepts
- Determine the behaviour near vertical and horizontal asymptotes
- Identify critical points by solving $f´(x)=0$ or $f´(x)=\text{DNE}$
- Use the first or second derivative tests to test critical points
- Identify points of inflection by solving $f´´(x)=0$ or $f´´(x)=\text{DNE}$ and test concavity on both sides of possible points
## Resources
- [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)
- [IB Math Analysis and Approaches Formula Booklet](/resources/g11/ib-math-data-booklet.pdf)
- [Calculus and Vectors 12 Textbook](/resources/g11/calculus-vectors-textbook.pdf)
- [Course Pack Unit 1: Descriptive Statistics](/resources/g11/s1cp1.pdf) ([Annotated](/resources/g11/s1cp1-anno.pdf))
- [Course Pack Unit 2: Limits and Rate of Change](/resources/g11/s1cp2.pdf) ([Annotated](/resources/g11/s1cp2-anno.pdf))
- [Course Pack Unit 3: Derivatives and Applications](/resources/g11/s1cp3.pdf) ([Annotated](/resources/g11/s1cp3-anno.pdf))
- [Course Pack Unit 4: Curve Sketching and Optimisation](/resources/g11/s1cp4.pdf) ([Annotated](/resources/g11/s1cp4-anno.pdf))
- [Course Pack Unit 5: Trigonometric, Exponential, and Logarithmic Functions](/resources/g11/s1cp5.pdf) ([Annotated](/resources/g11/s1cp5-anno.pdf))
- [TI-84 Plus Guide](/resources/g11/ti-84-plus.pdf)

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# SL Physics - 2
The course code for this page is **SPH4U7**.
## Magnetism
Magnets are objects with north/south dipoles that create a field around them. Although ferromagnetic substances can repel each other, paramagnetic substances are always attracted to a magnetic field. See [HL Chemistry#Physics properties of transition elements](/sch3uz/#physical-properties-of-transition-elements) for more details.
### Magnetic fields
<img src="/resources/images/magnetic-field.png" width=700>(Source: Kognity)</img>
Similar to electric and gravitational fields, magnetic fields (also known as **B-fields**) are drawn by their effect on a north pole. Since magnetic poles always appear in equal magnitude pairs, all magnetic field lines for a magnet must form closed loops from north to south **outside** and south to north **inside** the magnet. Much like electric field lines, magnetic field lines never touch
<img src="/resources/images/more-magnetic-fields.png" width=700>(Source: Kognity)</img>
Atoms in ferromagnetic materials are tiny magnets with **dipoles**. These dipoles act on neighbouring dipoles and can cause the whole object to align — this is known as an **electric domain**.
!!! note
Nickel, cobalt, or any alloy with nickel, cobalt, or iron can become magnetised this way.
**Unmagnetised** domains have dipoles pointing in random directions that are aligned when exposed to a magnetic field where they become **magnetised** domains. As such, bar magnets are always broken into smaller magnets, each with two poles — a monopole is impossible to create.
### Straight-line electromagnets
Moving electric charges produce magnetic fields. A circle filled with an "x" indicates that the current is moving away from the viewer in the third dimension while a dotted circle indicates it is moving toward the viewer.
These magnetic fields are centred on the conductor, are in a plane perpendicular to the conductor, and have decreasing magnetic field strength over distance.
<img src="/resources/images/straight-line-electromagnet.png" width=700>(Source: Kognity)</img>
The **right-hand rule** for straight-line conductors indicates that when the conductor would be grasped by the right hand, the thumb would point in the direction of current and the fingers pointing in the direction of the magnetic field.
<img src="/resources/images/right-hand-rule.png" width=700>(Source: Kognity)</img>
### Solenoid electromagnets
A **solenoid** is a conductor coil in a tight helix. Current passed through a solenoid will generate a **uniform magnetic field** inside the coil with a pattern identical to that of a bar magnet outside it.
<img src="/resources/images/selenoid-electromagnet.png" width=700>(Source: Kognity)</img>
The right-hand rule can be applied again to a solenoid to identify the direction of the north pole or direction of magnetic field in the coil:
<img src="/resources/images/selenoid-right-hand-rule.png" width=700>(Source: Kognity)</img>
### Properties of moving charges
As only moving electric charges generate magnetic fields, stationary electric charges are **unaffected** by external magnetic fields. Moving charges are affected by Newton's third law of motion — equal and opposite forces are exerted on the charge and the magnet. As such, where $q$ is the charge of the particle and $\vec{v}\times \vec{B}$ is the **cross product** (vector multiplication) of the velocity of the particle and the magnetic field strength in Teslas:
$$\vec{F_m}=q\vec{v}\times \vec{B}$$
**Magnetic field strength** ($B$) represents the force acting per unit current in a conductor of unit length perpendicular to the field with the unit Tesla ($\pu{T}$)
The **magnetic force** is always plane **perpendicular** to both $\vec{v}$ and $\vec{B}$. Just the magnitude of the force can be found by using the angle between the two vectors ($\theta$):
$$|F_m|=qvB\sin\theta$$
Regardless of $\theta$, the force vector is always perpendicular to both $B$ and $v$,
The above equation can be rearranged to find **electromagnetic** force in terms of current and wire length in a **uniform magnetic field**:
$$|F_{em}|=BIL\sin\theta$$
<img src="/resources/images/magnet-on-wire.png" width=700>(Source: Kognity</img>
The **right-hand-rule** can be used to determine the direction of force — the thumb points in the direction of current/velocity, the fingers point in the direction of the magnetic field, and the palm points in the direction of force. Alternatively, just three fingers can be used.
<img src="/resources/images/right-hand-rule-forces.png" width=700>(Source: Kognity)</img>
When two straight-line conductors with current flowing through them are brought together, they either mutually attract or repel. The ampere is defined based on the current required to flow through a scenario involving two parallel straight-line conductors.
<img src="/resources/images/parallel-current-conductors.png" width=700>(Source: Kognity)</img>
Inside a **uniform magnetic field**, charges move in **uniform circular motion** at a constant velocity. If the particle did not enter the field at a perfect right angle, some of the velocity is used to change the path of the particle to be in a spiral — still perfectly circular, but additionally moving in the third dimension perpendicular to the circle.
$$\Sigma F_c = F_m$$
## Nukes
### Atomic structure
!!! definition
- A **photon** is a particle that represents light.
- A **nucleon** is a subatomic particle in an atomic nucleus — that is, a proton or neutron.
- A **nuclide** is a nucleus with a specific number of protons and neutrons.
Please see [HL Chemistry#2.1 - Atoms](/sch3uz/#21-atoms) and [HL Chemistry 1#2.2 - Electrons in atoms](/sch3uz/#22-electrons-in-atoms) for more information.
An electron in an atom will only become excited if a photon of exactly the right amount of energy strikes it. That energy can be found using the formula:
$$\Delta E=hf$$
where $E$ is the energy of the photon at frequency $f$, and $h$ is Planck's constant ($\pu{6.63\times 10^{-34} Js}$ or $\pu{4.14\times10^{-15} eVs}$).
An electron that de-excites will emit a photon of that exact energy and thus frequency to return to its previous state.
### Binding energy
According to Einstein's theory of special relativity:
$$\Delta E=\Delta mc^2$$
**Neutrons** in the nucleus hold the protons together via **strong nuclear forces** that somewhat act like glue. An increase in neutrons increases the strong nuclear force. In smaller nuclei, $N=Z$, but in larger nuclei, $N>Z$ as more neutrons are required to keep the nucleus stable as the number of protons increases.
The mass of a stable nucleus is always less than the sum of the masses of the individual nucleons (the **mass defect**) as some of the mass is converted to energy during the formation of a nucleus. The energy used is known as the **binding energy** of a nucleus.
$$E_\pu{binding} = \pu{mass defect}\times c^2$$
As such, the binding energy of a nucleus is also the energy required to separate it completely into individual nucleons.
Atomic mass is measured relative to the mass of a carbon-12 atom, which is exactly 12 u (unified atomic mass units).
$$\pu{1 u}=\pu{1.661\times10^{-27} kg}=\pu{931.5 MeVc^-2}$$
A higher **binding energy per nucleon** results in more energy required to break it apart and thus it being more stable.
<img src="/resources/images/binding-energy-curve.png" width=700>(Source: Kognity)</img>
!!! note
It is required to know the general shape of the curve, that $~8.8 \pu{ MeV}$ is the maximum, and that the end boundaries are $0$ and $~7.5 \pu{ MeV}$. It is also required to know the elements at each of those points (hydrogen-1, iron-56/nickel-62, and uranium-238).
Since a greater binding energy per nucleon is more energetically favourable, nuclei to the right of iron-56 fission (split) while those to the left fuse (combine) to release energy — changes that would increase binding energy per nucleon are likely to occur because of this.
### Radioactive decay
!!! definition
- An **alpha particle** is a helium-4 nucleus (2 protons, 2 neutrons).
- A **beta particle** is an electron.
- A **gamma ray** is a photon.
Radioactivity is the emission of **ionising** (will make ions) radiation due to changes of a nucleus. It is **random** and spontaneous — it is unaffected by external factors such as other nuclei decaying.
**Nuclear equations** are similar to chemical equations but show how nuclei change in a nuclear process by tracking the atomic and mass numbers. A nuclear equation is balanced so that the sum of the atomic and the sum of the mass numbers on both sides are equal.
$$A\to B+C$$
**Alpha decay** occurs when the strong nuclear force is unable to hold a large nucleus together and emits an alpha particle. The alpha particle is positive and can barely penetrate paper. The two particles move in opposite directions with momentums equal in magnitude.
<img src="/resources/images/alpha-decay.png" width=700>(Source: Kognity)</img>
$$\ce{^A_Z N → ^{A-4}_{Z-2} N' + ^4_2 He}$$
$\ce{^4_2 He}$ may be replaced by $\ce{^4_2\alpha}$.
!!! example
Radium-226 alpha decays to radon-222.
**Beta-minus decay** ($\beta^-$) occurs when a neutron decays into a proton and releases a beta-minus particle (an electron and an electron antineutrino). It can penetrate up to 3 mm of aluminum. Where $\overline{v}_e$ is the antineutrino:
<img src="/resources/images/beta-minus-decay.png" width=700>(Source: Kognity)</img>
$$\ce{^1_0n → ^1_1p + ^0_{-1}e + ^0_0\overline{v}_e}$$
!!! note
The bar over the electron antineutrino identifies it as an **antiparticle**.
The beta-minus particle can be written explicitly over the electron as $^0_{-1}\beta$.
In terms of the mother nucleus, the reaction results in the mass number staying the same while the atomic number increases by one.
**Beta-plus decay** ($\beta^+$) occurs when a proton decays into a neutron and releases a **positron** (an antielectron with a positive charge) and an electron neutrino ($v_e$)
$$\ce{^1_1p → ^1_0n + ^0_1e + ^0_0v_e}$$
The positron can be written as a beta-plus particle as $^0_1\beta$.
In terms of the mother nucleus, the reaction results in the mass number staying the same while the atomic number decreases by one.
**Gamma decay** occurs when an excited nucleus transfers its energy to a high-energy photon with frequencies in the gamma region of the electromagnetic spectrum. There is no change in mass nor atomic number. A nuclide with an asterisk $*$ indicates it as excited. This emits *ionising radiation* which is not good for living beings.
<img src="/resources/images/gamma-decay.png" width=700>(Source: Kognity)</img>
$$\ce{^A_ZX^* → ^A_ZX + ^0_0\gamma}$$
### Detecting radiation
As radiation cannot be seen, it must be detected experimentally.
A **Geiger counter** utilises a gas-filled tube with a wire in the centre at high voltage. When a charged particle passes through, gas is ionised which cascade onto the wire to produce a pulse.
A **cloud chamber** contains vapour that turns into liquid droplets when ionising particles pass though, resulting in visible lines showing the path of the particles. A magnetic field can spiral the particle in a certain direction which allows for its charge to be identified.
### Half-life
The **half-life** of an element is the time required for half of the nuclides in a sample to decay — it is always the same no matter the number of initial nuclides.
As such, this means that the number of parent nuclei decreases by 50% of its current value each half-life.
<img src="/resources/images/half-life.png" width=700>(Source: Kognity)</img>
!!! example
**Radioactive dating** analyses the ratio between carbon-14 and carbon-12 to determine the age of plant nmatter. As the ratio of C-14 and C-12 in the atmosphere has been relatively comstant, living plants maintain that balance by constantly exchanging carbon. Dead plants' carbon-14 slowly decay at a known rate, so the ratio can be used to determine the time since the plant died.
### Nuclear reactions
A nuclear reaction occurs when a nucleus is hit by another nucleus or subatomic particle and a different nuclide is formed (nuclear **transmutation**). In such a collision, energy and momentum must be conserved. Generally, neutrons are most commonly used in these reactions as they are not affected by Coulomb force exerted by the protons or electrons.
The **reaction energy** $Q$ is the difference in mass between the initial and final states multiplied by $c$ squared. In the sample reaction $a+X → Y+b$:
$$Q=[(M_a + M_X) - (M_Y + M_b)]c^2$$
- If $Q$ is positive, the reaction is **exothermic** and will occur at any amount of initial kinetic energy.
- If $Q$ is negative, an initial kinetic energy equal to $Q$ is required (the activation energy).
Nuclear **fusion** occurs when two lighter nuclei combine into a heavier one, releasing energy in the process.
Nuclear **fission** occurs when a heavy nucleus splits into two lighter nuclei. Along with some excess neutrons, energy is released. The two split pieces are usually somewhat unequal.
### Fission in reactors
The energy release of nuclei is very large — the energy density per unit mass is much higher than any other conventional source.
As nuclei get smaller, their stablility increases as the number of neutrons also decreases, so excess neutrons can set off a chain reaction by reacting with more nuclei.
<img src="/resources/images/fission-chain.gif" width=700>(Source: Kognity)</img>
However, neutrons that are ejected often have too much energy and must be **moderated** to slow down to prevent a critical mass where the number of reactions is self-sustaining, leading to overheating and reactor meltdown. A **moderator** is a material surrounding fuel rods to slow down incoming neutrons — usually heavy water.
**Control rods** are also inserted into the reaction core to control the rate of reaction. These absorb the neutrons from the moderator and the amount absorbed can be adjusted by raising the rods partially up to all the way from the reactor.
Nuclear power is superior to other types of energy generation in that:
- it has a high power output due to high energy density
- there are large reserves of nuclear fuel on Earth
- there are no greenhouse gases emitted to generate power
Nuclear power has the following issues in that:
- waste is highly radioactive with long half-lives, rendering removal and storage of nuclear waste a major issue
- initial startup costs are expensive
- strict maintenance is required due to the risk of nuclear meltdown
- fissionable fuel can be recovered and used for destructive weapons
- mining uranium is unhealthy — miners are exposed to harmful radiation and waste material from mines deemed not pure enough is not easy to dispose
### Nuclear fusion
Nuclear fusion generates energy per unit mass an order of magnitude greater than can be achieved with fission. The sun takes hydrogen and fuses it into helium. Heavier stars can fuse elements up to iron-56.
$$\ce{4 ^1_1 H → ^4_2 He + 2 e^+ + 2 v_e + 2\gamma}$$
Nuclear fusion power has the following issues in that it is currently unsustainable for more than a few seconds because:
- the temperature required for the reaction is greater than 100 million degrees Celsius
- it requires more energy input to heat the sample than is obtained from the fusion reaction
- materials currently known cannot withstand the temperature making containment difficult — currently magnetic fields are used to hold the particles in place
## The Standard Model
<img src="/resources/images/standard-model.png" width=700>(Source: Kognity)</img>
### Elementary particles
An elementary particle is a subatomic particle that is not composed of other particles.
Particles currently thought to be elementary as of January 2021 include bosons, quarks, and leptons:
### Bosons/Force exchange particles
!!! definition
- **Virtual particles** are bosons that do not have infinite range.
Bosons are particles that carry/exchange forces between particles. The theory of exchange forces posits that all forces are due to particles exchanged between elementary particles. There are four types of bosons that can be roughly categorised by their effect.
**Gluons** ($\pu{g}$) are exchanged for matter to feel **strong nuclear** force: the strongest interaction between particles. These particles are heavy (120 MeV/c<sup>2</sup>) and short-lived, thus giving the force a very short range.
**Photons** ($\pu{\gamma}$) are exchanged for matter to feel **electromagnetic** force: the second strongest force responsible for magnetism and electric force that only act on charged particles. These particles have a rest mass of zero and travel for an infinite distance until they are absorbed.
The **W<sup>+</sup>, W<sup>-</sup>, and Z<sup>0</sup>** bosons are together responsible for the weak nuclear force and are the third strongest force. These particles have a heavy rest mass (80 GeV/c<sup>2</sup> for Ws, 91 GeV/c<sup>2</sup> for Z) and so are even more limited in range than gluons.
**Gravitons** are responsible for gravitational force: the weakest force. These particles are massless and so they have infinite range.
The Higgs field and Higgs boson are responsible for elementary particles obtaining their mass because of magical fields and rainbows.
### Quarks
!!! definition
- A **hadron** is any particle made of quarks.
- A **baryon** is any hadron made of three quarks. An **antibaryon** is any particle made of three antiquarks.
- A **meson** is a hadron made of exactly one quark and one antiquark involved in the **strong** interaction.
- A **fermion** is any particle with mass (hadrons or leptons)
Gluons (strong force) only interact with quarks, which are heavier, more tightly bound elementary particles. There are six quarks with different properties:
| Charge | | | |
| --- | --- | --- | --- |
| $\frac{2}{3}$e | up (u) | charm (c) | top (t) |
| $-\frac{1}{3}$e | down (d) | strange (s) | bottom (b) |
!!! reminder
e is the elementary charge ($\pu{1.6\times10^{-19} C}$).
- All quarks have a **baryon number** of $\frac{1}{3}$.
- All quarks have a **strangeness number** of 0 except for the strange quark, whose number is equal to -1.
- All quarks have their own respective **antiquark**: an antiparticle with opposite charge and baryon number but otherwise identical mass.
- The **quark confinement theory** states that a singular quark cannot be isolated from its hadron.
Every particle has its own **antiparticle** with the same properties but with opposite quantum numbers. In practice, this indicates that mass stays the same while baryon number, lepton number, and charge are opposite. When a corresponding quark and antiquark meet, annihilate each other to become energy. They are denoted by a bar over their letter.
!!! example
An up antiquark (also known as "u-bar") is written as ū.
!!! note
- Protons are composed of two up quarks and one down quark (uud).
- Neutrons are composed of one up quark and two down quarks (udd).
### Leptons
Leptons are lighter and more loosely bound elementary particles compared to quarks. They do not participate in the strong interaction. All leptons have a **lepton generation/family** which is based on their relative mass. A higher mass indicates a higher generation.
| Charge | Generation 1 (L<sub>I</sub>) | Generation 2 (L<sub>II</sub>) | Generation 3 (L<sub>III</sub>) |
| --- | --- | --- | --- |
| -1e | electron (e) | muon (µ) | tau ($\tau$) |
| 0 | electron neutrino ($\pu{v_e}$) | muon neutrino ($\pu{v_\mu}$) | tau neutrino ($\pu{v_\tau}$) |
- All leptons have a **lepton number** of 1.
### Elementary particle interactions
In any interaction, the following are true:
- **charge** is conserved
- the **baryon number** is always conserved
- the **lepton number** of each family is always conserved
- the **strangeness number** is always conserved in *strong and electromagnetic interactions*
!!! example
A lepton number of $\pu{L_{III} = 1}$ on one side becoming $\pu{L_{II} = 1}$ on the other is impossible as lepton family must be kept consistent during interactions.
## Feynman diagrams
A Feynman diagram provides a graphic representation of particle interactions to predict the outcome of a particle collision.
Generally, the time axis is left-to-right but can be specified to be otherwise. The following assumes time moves from left to right.
Fermions are represented by **straight lines with arrows**. Particles have their arrows pointing *forward* in time while antiparticles point backward (even though they still move in the direction of time).
<img src="/resources/images/fermion-feynman.png" width=700>(Source: Kognity)</img>
Bosons/force exchange particles are represented by wiggly lines with no arrow.
<img src="/resources/images/boson-feynman.png" width=700>(Source: Kognity)</img>
Particles only interact at a **vertex** where left refers to the state before the interaction while the right refers to the state afterward. A vertex must have one arrow pointing **toward** and one **away** from the vertex. Conservation laws apply at each vertex.
<img src="/resources/images/vertex-feynman.png" width=700>(Source: Kognity)</img>
Contents of hadrons must be shown. (See the last example for an example.)
### Feynman diagram examples
!!! example
An electron being repelled by another electron due to Coulomb repulsion:
<img src="/resources/images/electron-repulsion.png" width=700>(Source: Kognity)</img>
!!! example
Beta decay:
<img src="/resources/images/beta-decay-feynman.png" width=700>(Source: Kognity)</img>
!!! example
Some weak interaction that violates strangeness:
<img src="/resources/images/weird-feynman.png" width=700>(Source: Kognity)</img>
## Energy sources
!!! definition
- A **primary** energy source is one that is not transformed and used directly by the consumer, such as burning wood for heat.
- A **secondary** energy source is one that is converted from a primary source, such as electricity.
- **Proved reserves** are the resources that are known to be obtainable.
- **Production** are the actual reserves and placed in the market in a certain time interval.
- The **expectancy** of a product is the estimated time its proved reserves will last given its production (proved reserves ÷ production).
- The **specific energy** ($E_{SP}$) of a source is the energy obtained per unit mass of fuel (J/kg).
- The **energy density** ($E_D$) of a source is the energy obtained per unit volume (J/m<sup>3</sup>).
If a fuel source can be replenished with a "reasonable" amount of time — one human generation, or 50–100 years — it is considered to be **renewable**. The world still largely uses non-renewable energy sources.
<img src="/resources/images/renewable-energy-pie.png" width=700>(Source: Kognity)</img>
Electricity is the most common secondary energy source due to its convenience and ease of transport.
### Sankey diagrams
Sankey diagrams show the transfer of energy in a system via directed lines proportional to quantity of energy. Arrows pointing away indicate energy **degradation** — losses in energy transformation.
<img src="/resources/images/sankey-diagram.png" width=700>(Source: Kognity)</img>
To minimise electrical losses during transportation, high voltage, low resistance wires with high cross-sectional areas are used to reduce resistance to reduce power loss.
### Power generation
A moving magnetic field produces an electromotive force as alternating current via **induction** and is how the large majority of power generation is handled.
<img src="/resources/images/power-plant-generation.png" width=700>(Source: Kognity)</img>
In a nutshell, a source of thermal energy such as burning fossil fuels is used to boil water whose steam is then used to turn a turbine to generate power before condensing and repeating the cycle.
Coal and oil-powered power plants have efficiencies of around 40% while natural gas is slightly higher at 50% as the gas itself can be used to turn a turbine.
In **nuclear** power plants, the coolant fluid is instead used to turn a turbine.
<img src="/resources/images/nuclear-power-generation.png" width=700>(Source: Kognity)</img>
As only uranium-235 is fissile, but uranium-238 is significantly more common (99.3% U-238 to 0.7% U-235), uranium must first be enriched until the concentration of U-235 is ~3%. **Gaseous diffusion** is a form of enrichment by forming uranium hexafluoride gas and then spinning it in a centrifuge to force separation of U-238 and U-235 based on mass. The energy used in enriching uranium is substantial and should be included in Sankey diagrams.
Moderators are used to encourage fission as they slow down neutrons that are going too fast to fission (most) to a speed more suitable for fission.
### Wind
The kinetic energy of wind can be harnessed to generate power. As convection currents provide the greatest airflow near large bodies of water, wind farms are often constructed there. The wind turns rotor blades which turn a turbine to generate power. It is a source of clean and renewable energy.
Assuming **all wind kinetic energy** is converted to mechanical energy, where $P$ is the power generated, $A$ is the area of the circle that the blades spin around, $\rho$ is the density of the air, and $v$ is the speed of the wind in the direction of the blades:
$$P=\frac{1}{2}A\rho v^3$$
<img src="/resources/images/wind-generator.png" width=700>(Source: Kognity)</img>
| Advantages | Disadvantages |
| --- | --- |
| Renewable | Wind strength is inconsistent |
| Wind is widely available | Turbine blades may kill birds |
| Does not emit greenhouse gases | Many of them are needed to replace one fossil fuel plant, requiring lots of space so they don't interfere with each other |
### Hydro
In hydroelectric plants, a dam is often used to increase the height of a reservoir so that it falls and spins a turbine to generate power. As such, the energy generated is roughly equal to the gravitational potential energy of the water. Where $\Delta h$ is the **average height** of the water from the turbine:
$$E=mg\Delta h$$
<img src="/resources/images/hydro-generator.png" width=700>(Source: Kognity)</img>
During times of lower demand, dams often have a **pumped storage** system that pumps water back into the reservoir for use during higher demand.
### Tidal
A **tidal barrage** generates energy via the kinetic energy of water moving during changes in tide using a multi-directional turbine.
### Photovoltaic
!!! definition
**Intensity** is the power delivered per unit area (watts per square metre).
Photovoltaic (PV) cells are made of silicon doped with phosphorus and boron impurities to convert sunlight directly into electricity. Light from the sun frees electrons in the silicon to produce a current.
<img src="/resources/images/photovoltaic-generation.png" width=700>(Source: Kognity)</img>
The **solar constant** $S$ of $\pu{1.36\times10^3 W/m^2}$ determines the intensity of the sun's light that reaches the Earth. At different latitudes and between seasons, the intensity changes because the Earth is round and is tilted, respectively.
### Solar heating
Instead of converting between multiple forms of energy, solar heating directly converts the sun's energy to heat, increasing efficiency drastically. By using insulation, a black substance, and a glass top, the heat from the sun is trapped and absorbed into water where it is used to heat things.
<img src="/resources/images/solar-heating.png" width=700>(Source: Kognity)</img>
## Thermal energy transfer
!!! definition
- **Radiation** is the transfer of energy via electromagnetic waves emitted away from an object. No medium is needed.
- **Convection** is the transfer of thermal energy via another medium away from an object.
- **Conduction** is the transfer of thermal energy via physical contact.
### Black bodies
!!! definition
**Emissivity** ($e$) is a dimensionless value from 0 to 1 indicating the ability of an object to emit radiation relative to a black body (which has an emissivity of 1). Darker and duller surfaces have an emissivity closer to 1 while shinier and whiter ones are closer to 0.
All bodies with an absolute temperature will emit radiant energy in the form of electromagnetic waves. The temperature of the body determines the wavelengths and power of the radiation emitted. A **perfect emitter** has an emissivity of 1 and is known as a **black body**, absorbing all electromagnetic radiation.
Generally, as the temperature of a body goes down, its peak power density is reduced and its peak wavelength increases.
**Wien's displacement law** relates the temperature of a black body to the waves it emits. Where $\lambda_\text{max}$ is the peak wavelength in metres and $T$ is the temperature of the body in kelvin:
$$\lambda_\text{max}\times T=\pu{2.9\times10^{-3} m\cdot K}$$
The **Stefan-Boltzmann** law relates the specifications of a body to the power it emits. Where $P$ is the power emitted by the body, $A$ is its surface area, $T$ is its temperature, $e$ is its emissivity, and $\sigma$ is the Stefan-Boltzmann constant (equal to $\pu{5.67\times10^{-8} Wm^{-2}K^{-4}}$):
$$P=e\sigma AT^4$$
In problems where the environment temperature is **different** from the temperature of an object, there will be power loss. The net power emitted by a body will be:
$$P_\text{net}=e\sigma A(T_1^4-T_2^4)$$
The solar radiation reaching earth is equal to $\pu{S= 3.9\times10^{26} W}$ with the assumption that it is a black body.
### Albedo
Derived from $I=\frac{P}{A}$, the intensity at a point in space can be related to the power of the radiation emitted by the source ($P$) and the distance between the two ($d$):
$$I=\frac{P}{4\pi d^2}$$
!!! example
The solar constant is derived in this way by substituting $d$ as the distance from the Earth to the sun.
As Earth and most other planetary bodies are not flat disks pointed at the sun, in reality the sun's intensity is reduced to a quarter due to the formula for a sphere. Therefore, the power absorbed/incident to the Earth is equal to, where $S$ is the solar constant:
$$P_\text{in}=(1-\alpha)\frac{S}{4}A$$
**Albedo** ($\alpha$) is the ratio of power from incident rays reflected or scattered to the power absorbed by a body, ranging from 0 to 1. A black body has albedo 0. On average, Earth's albedo is equal to $0.3$ due primarily to the atmosphere but also clouds and ice.
$$\alpha=\frac{\text{energy scattered/reflected}}{\text{energy absorbed}}$$
Greenhouse gases are responsible for remaining increases in temperature. By absorbing and then re-emittng their natural frequencies of electromagnetic radiation (infrared for greenhouse gases), they delay the release of radiation back into space and heat up the atmosphere.
### Radiation absorption by greenhouse gases
All molecules have a natural frequency at which they absorb radiation from the electromagnetic spectrum and **resonate** at. The natural frequency of greenhouse gases is in the infrared region, which is what the Earth re-emits solar radiation as. Therefore, greenhouse gases absorb this radiation and them re-emit it in all directions, "trapping" some of the radiation. Resonance is also the phenomenon responsible for the protection from ultraviolet radiation by the ozone layer.
## Photoelectric effect
**Wave-particle duality** posits that everything can be described as either a particle or a wave, and that all particles will show some wave characteristics and all waves will show some particle characteristics, so really what the hell is happening?
The photoelectric effect is the phenomenon in which electrons are emitted when electromagnetic radiation hits a material. It was theorised that EM radiation travelled in **discrete** energy packets known as **quanta**, which held a defined amount of energy that could not be divided smaller. Where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength of the light:
$$E=\frac{hc}{\lambda}$$
!!! example
The application of the photoelectric effect in the real world has led to large advances in technology, such as in **photodiodes** in light sensors — semiconductors where electrons are freed by indicent light to raise the conductivity of the material, and **charged coupled devices** in cameras — the "film" of the modern digital camera.
It was later observed that for a given material, electrons were ejected when light shone on a surface only if the light's energy/frequency was greater than a certain threshold. Below that threshold, electrons were not ejected **regardless** of the intensity of the light.
Electrons have a "binding energy" that hold them to the nucleus of an atom. To release an electron from the nucleus, energy greater than that binding energy must be provided. This binding energy, known as the **work function** $W_0$, is therefore the minimum energy required for a surface to eject electrons.
As such, where $E_k$ is the kinetic energy of the ejected electron, $E_{ph}$ is the energy of the incident photon/radiation, and $W_0$ is the work function of the surface:
$$E_k=E_{ph}-W_0$$
Electrons are ejected and thus the photoelectric effect observed **only** if $E_k>0$. The equation above shows that the kinetic energy of an ejected electron is determined **only** by the wavelength/frequency of the incident radiation and *not* by the intensity of the light.
The intensity of incoming radiation effectively represents the number of photons striking per unit area of a surface, so while it does not affect whether electrons are ejected, it affects the **number** of electrons that are ejected *only if they are determined to be ejected*.
In a light frequency-kinetic energy graph,
- the x-intercept represents the **threshold frequency** $f_0$: the minimum frequency required to liberate electrons at all.
- the y-intercept represents the **work function** $-W_0$: the "binding energy" of the electrons.
### Momentum of photons
It was also observed that during photon-electron collisions that momentum and kinetic energy were conserved, further reinforcing the idea of wave-particle duality of light. Where $h$ is Planck's constant, $f_i$ and $f_f$ are the initial and final frequencies of the light, $m$ is the mass of the electron, and $v_{ei}$ and $v_{ef} are the initial and final velocities of the electron:
$$hf_i+\frac{1}{2}mv^2_{ei}=hf_f + \frac{1}{2}mv^2_{ef}$$
In general, the momentum of a photon is equal to, where $h$ is Planck's constant and $\lambda$ is the wavelength of the light:
$$p=\frac{h}{\lambda}$$
### Matter waves
Particles/waves cannot act like particles and waves at the same time. For a given observation, it adopts the property of one or the other. In reality, all particles exhibit wave properties *sometimes* and all waves exhibit particle properties *sometimes*. Each particle has a wave function that determines how likely it is to be somewhere at any point in time.
By equating the equations for momentum of photons and particles, the (de Broglie) wavelength of a particle can be determined. Where $\lambda$ is the wavelength of the particle, $m$ is its mass, $v$ is its velocity, and $h$ is Planck's constant:
$$\lambda=\frac{h}{mv}$$
Therefore, wavelengths of "particles" are only really significant for small masses at high speeds rather than large masses at lower speeds.
!!! example
A 50 kg mass moving at 16 m/s is has a wavelength many orders of magnitude smaller than a quark and will not display any observable wave behaviours.
The discovery of wave-particle duality has led to advancements in technology such as the scanning electron microscope.
## Special relativity
Einstein's theory of special relativity states that time and space are relative depending on the **frame of reference** of the observer, and light travels at the *same speed* of $\pu{3.0\times10^8 m/s}$ in a vacuum no matter how it is observed in all inertial frame of reference.
- An **inertial** reference frame is one in which the law of inertia in the frame holds true. Only frames of reference moving at a constant velocity or at rest are inertial, and the same laws of physics apply in all inertial frames of reference.
### Time dilation and length contraction
The faster an observer moves, to ensure that it appears to them that light travels at $c$, time slows down for the observer. Observers in inertial frames of reference will experience time at a slower rate — this phenomenon is known as time dilation.
For two **inertial** reference frames, where $t_s$ is the time observed between two events (stationary/**proper** time) at the same location and at rest relative to a stationary observer, $t_m$ is the time observed between two events in a different frame of reference (moving time), $v$ is the speed difference between the frames of reference, and $\gamma$ is the **Lorentz factor**:
$$
\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \\
t_m=\gamma t_s
$$
Space is also relative. An observer moving at a higher constant velocity will have space contract — this phenomenon is known as **length contraction**.
Where $L_s$ is the length/distance measured of an object at the same location and at rest relative to a stationary observer, and $L_m$ is the length/distance from a different reference frame:
$$L_m=\frac{L_s}{\gamma}$$
From different inertial reference points, there can never be agreement on whether events are simultaneous. Because time slows down for an observer moving faster, one event will also appear to happen first.
### Matter-energy equivalence
The total energy of an object in an inertial frame of reference relative to another is related to its velocity and its mass.
$$
E_t=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}} \\
E_k=E_t-E_\text{rest}
$$
At rest, the energy of an object is related to its mass.
$$E_\text{rest}=mc^2$$
## Resources
- [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf)
- [IB SL Physics Syllabus](/resources/g11/ib-physics-syllabus.pdf)
- [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf)
- [External: IB Physics Notes](https://ibphysics.org)

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# Theory of Knowledge
## Knowledge questions and claims
There are two types of knowledge **claims**:
- First-order claims: claims that are made in areas of knowledge or by individual knowers about the world
- Second-order claims: claims that are made about **knowledge** justified using TOK, usually involving an examination of the nature of knowledge
!!! example
- The sky is blue. (first-order)
- Logic is innate. (second-order)
Knowledge **questions** are questions that examine or engage with knowledge claims, such as by including any of the following phrases:
- How can we know that…
- …knowledge…
- How far is it justified…
!!! example
Is our knowledge in mathematics more certain than that of science?
## Knowledge themes and areas
The main areas of knowledge are:
- History
- Human sciences
- Natural sciences
- Mathematics
- Arts
The main themes of knowledge are:
- Knowledge and technology
- Knowledge and language
- Knowledge and politics
- Knowledge and religion
- Knowledge and indigenous studies
## Resources
- [External: TOK 2022](https://tok2022.weebly.com)