eng: set up essay checklist

Reviewed-on: https://git.eggworld.tk/eggy/eifueo/pulls/7

README.md

@@ -2,6 +2,21 @@
 
 A "competitor" of sorts to magicalsoup/highschool.
 
-Please note that the clone link is incorrect; it should be `https://git.eggworld.tk/eggy/eifueo.git`.
+The LaTeX formatting in this repository uses `$...$` for inline math, and `$$...$$` for multi-line math. MathJax is used to render this LaTeX.
 
-The LaTeX formatting in this repository uses `$...$` for inline math, and `$$...$$` for multi-line math.
+Admonitions can be added with documentation available [here](https://squidfunk.github.io/mkdocs-material/reference/admonitions/#usage).
+
+## Dependencies
+
+ - `mkdocs`
+ - `mkdocs-material`
+ - `mkdocs-material-extensions`
+ - `python-markdown-math`
+
+## Build instructions
+
+MkDocs is used to build the site.
+
+```
+mkdocs build
+```

docs/eng3uz.md

@@ -134,6 +134,11 @@ The course code for this page is **ENG3UZ**.
  - 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)
 
+## Essay analysis
+
+
+
 ## Resources
 
  - [Analysis of a Poem](/resources/g11/central-asserion-1.pdf)
+ - [Essay Analysis](/resources/g11/essay-analysis.pdf)

docs/mhf4u7.md

@@ -5,16 +5,446 @@ The course code for this page is **MHF4U7**.
 ## 4 - Statistics and probability
 
 !!! note "Definition"
-    - **Descriptive statistics:** The use of methods to organise, display, and describe data by using various charts and summary methods to reduce data to a manageable size.
+    - **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 $\lceil\sqrt{\text{# of elements}}\rceil$.
+	- 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., $0.001$.
+
+### 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
+    $$
+    \lim_{x \to 0} \frac{x}{(x+1^\frac{1}{3}-1} \\
+    \text{let } (x+1)^\frac{1}{3} \text{ be } y \\
+    x + 8 = y^3 \\
+    x = y^3 - 8, \text{as } x \to 0, y \to 2 \\
+    \lim_{y \to 2} \frac{y-2}{y^3 - 8} \\
+    = \lim_{y \to 2} \frac{(y-2)(y^2 + 4y + 4)}{(y^3-8)(y^2 + 4y + 4)} \\
+    = \lim_{y \to 2} \frac{1}{y^2 + 4y + 4} \\
+    = \frac{1}{2^2 + 4(2) + 4} \\
+    = \frac{1}{16}
+    $$
+
+### 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:
+    $$
+    f´(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\
+    = \lim_{h \to 0} \frac{2(x+h)^2 + 6 - (2x^2 - 6)}{h} \\
+    = \lim_{h \to 0} \frac{4xh+2h^2}{h} \\
+    = \lim_{h \to 0} 4x+2h \\
+    f´(x)=4x
+    $$
+
+### Derivative rules
+
+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:
+$$f´(x) = nx^{n-1}$$
+
+### Drawing derivative functions
+
+If the slope of a tangent is:
+
+ - positive/negative, that value on the derivative graph is also positive/negative, respectively
+ - zero (e.g., linear equations), 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.
 
 ## Resources
 
@@ -22,3 +452,5 @@ The course code for this page is **MHF4U7**.
  - [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)
+ - [Course Pack Unit 2: Limits and Rate of Change](/resources/g11/s1cp2.pdf)
+ - [TI-84 Plus Basic Calculator Functions](/resources/g11/ti-84-plus.pdf)

docs/sch3uz.md

@@ -2,17 +2,370 @@
 
 The course code for this page is **SCH3UZ**.
 
+## Designing a scientific investigation
+
+### Scope
+
+The scope of an experiment goes at the very beginning of it. It includes a general introduction to the topic of investigation as well as personal interest.
+
+### Research question
+
+The research question of an experiment is a hyper-focused and specific question related to the topic. It contains and asks about the effect of an **independent variable** on a **dependent variable**.
+
+### Background information and hypothesis
+
+!!! note
+    This section can instead be placed immediately before the research question depending on the experiment.
+
+In this section, scientific theories are provided to help the reader understand the rationale of the question, the design of the experiment, and data processing measures. If any theoretical/literature values are used, they should be introduced here.
+
+A hypothesis consists of a justified prediction of the expected outcome and should be integrated with any background information.
+
+### Variables
+
+!!! definition
+    - The **independent** variable is the variable that is explicitly changed to attempt to affect the dependent variable.
+    - The **dependent** variable is the variable that is directly monitored and measured in the experiment and is expected to change if the independent variable changes.
+    - **Controlled** variables (also known as "control variables") are variables that should be kept constant so they do not affect the dependent variable.
+
+The independent variable, dependent variable, and any controlled variables should be listed under this section.
+
+### Materials
+
+A list of materials and equipment should be listed here, as well as their precision. If a controlled variable needs to be measured, any instruments that would be used to do so should also be listed here.
+
+### Procedure
+
+A clear, detailed, and concise set of instructions written in *past tense* should be placed in this section as either a numbered list or descriptive paragraph. To reduce confusion, if a numbered list is used, referring directly to numbers should be avoided, and referring to numbers recursively must *never* happen. A procedure must include:
+
+ - a clear, titled, labelled, and annotated diagram
+ - instructions for recording data (including for controlled variables)
+
+If necessary, a "setup" section can be added as preparatory steps should not be listed in the main procedure.
+
+### Data collection
+
+Data should be collected in an organised and titled table that should be prepared before the experiment. The data table must include:
+
+ - units with uncertainty, typically in the table header
+ - *qualitative* data (quantitative data can be optional in some experiments)
+ - repeated data/controlled variables, typically in the title
+ - any relevant information should be listed under the title
+
+Only **raw data** prior to any processing or calculations, with the exception of averages, should be present in the data table.
+
+A data table should be as concise as possible, and redundancy should be minimised. In that vein, trial numbers should *not* be recorded unless that data is relevant.
+
+!!! example
+    **Table 1: Effect of Fat Content on Sugar Content in Ice Cream**
+    
+    | Fat Content (g ± 0.1 g) | Sugar Content (g ± 0.1 g) | Notes |
+    | --- | --- | --- |
+    | 2.0 | 5.1 | - strawberry ice cream |
+    | 0.1 | 2.3 | - mint chocolate chip ice cream |
+
+Whenever possible, data tables should *not* span multiple pages. If that is unavoidable, a new title with "…continued" and new column headers must be present at the top of each new page.
+
+### Data processing
+
+A single sample calculation showing all steps should be present and clearly explained. The rest of the data can be processed without describing any steps. A **single** graph may be included if needed.
+
+Some general rules include:
+
+ - units and uncertainties must be present in all calculations
+ - simple operations such as averages and conversions (e.g., g to kg) do not need to be explained
+ - the graph, if any, should span at least half of the page (ideally the full page) and should directly answer the research question
+
+A final, reorganised, and processed data table should be present here, showing only relevant information.
+
+### Conclusion and evaluation
+
+This section should be free of any new background information or calculations. It should, in sequence:
+
+ - summarise the results of the experiment without connecting it to the hypothesis
+ - identify whether the results of the experiment agree or disagree with the hypothesis
+ - evaluate 3–5 systematic errors (usually) present in the experiment, both in the procedure and in data collection/processing, in **decreasing** order of impact to the experiment
+
+The evaluation of systematic errors should include:
+
+ - a description of the error
+ - how the error affected the data
+ - how the error affected the final result
+ - how the error can be remedied with available school resources
 ## 11.1 - Uncertainties and errors in measurement and results
 
-!!! info
-    Please see [SL Physics](/sph3u7/#12-uncertainties-and-errors) for more information.
+Please see [SL Physics#Uncertainties and errors](/sph3u7/#12-uncertainties-and-errors) for more information.
 
 ## 11.2 - Graphical techniques
 
+When plotting a graph:
 
+ - plot the independent variable on the horizontal axis and the dependent variable on the vertical axis
+ - label the axes, ensuring that the labels include units
+ - choose an appropriate scale for each axis
+ - give the graph an appropriate title at the **bottom** in **title case**
+ - draw a line of best fit
+ - include all uncertainties in the form of error bars
+	- if the error bars are too small to see, it should be noted explanation below
+
+### Titles
+
+The title of a graph should clearly indicate what the graph represents under what conditions in **title case**, so that any onlooker should be able to identify the experiment. It should not include "vs." Any legends present should be located under the graph.
+
+??? example
+    "Effect of Cat Deaths on Suicides in New Zealand"
+
+### Error bars
+
+Please see [SL Physics#Error bars](/sph3u7/#error-bars) for more information.
+
+### Line of best fit
+
+Please see [SL Physics#Uncertainty of gradient and intercepts](/sph3u7/#uncertainty-of-gradient-and-intercepts) for more information.
 
 ## 11.3 - Spectroscopic identification of organic compounds
 
+## 12 - Atomic structure
+
+!!! definition
+    - The **effective nuclear charge** ($Z_\text{eff}$) is the net positive charge (attraction to the nucleus) experienced by an electron in an atom.
+    - **Electron shielding** is decrease in the effective nuclear charge of an electron because of the repulsion of other electrons in lower-energy shells.
+
+**Atomic notation** is used to represent individual atoms or ions. It is written in the form $^M_Z \text{Symbol}^\text{Charge}$, where $M$ is the mass number of the particle and $Z$ is the atomic number of the particle.
+
+!!! example
+    - $^1_1 \text{H}^{+}$ is the atomic notation for the most common hydrogen ion.
+    - $^{16}_8 \text{O}^{2-}$ is the atomic notation for the most common oxygen ion.
+    - $^{20}_{10} \text{Ne}$ is the atomic notation for the most common neon atom.
+
+### Isotopes
+
+Isotopes are atoms of the same element but with different masses, or alternatively, atoms with the same number of protons but with different numbers of neutrons.
+
+**Radioisotopes** are isotopes that are unstable (will spontaneously decay, are radioactive). Unstable atoms **decay** (break down) into one or more different isotopes of a different element. The **half-life** of a radioisotope is the time it takes for 50% of a sample's atoms to decay.
+
+!!! warning
+    Radioisotopes are dangerous! They emit radiation, which is not at all good for human health in the vast majority of cases. However, there are also useful applications for radioisotopes today. For example, Cobalt-60 is used in radiation therapy to kill cancer tumours by damaging their DNA.
+
+### Atomic mass
+
+The mass of every atom is represented relative to 1/12th of a carbon-12 atom. This mass is either unitless or expressed in terms of **atomic mass units (amu or u)**. On the periodic table, the **relative atomic mass** ($A_r$) is shown, which is the sum of the masses of each isotope combined with their natural abundance (%abundance).
+
+$$A_r = \text{%abundance}×\text{mass number of isotope}$$
+$$m_a = \Sigma A_r$$
+
+When calculating the atomic mass from the graph from a **mass spectrometer**, the sum of the natural abundances of each isotope may not equal 100 or 1 (not in %abundance). In this case, calculation of %abundance will need to be done before solving for $m_a$.
+
+A mass spectrometer may also provide mass in the form of $M/Z$, which represents mass over charge. For the sake of simplicity, $Z=1$, so $M/Z$ represents the mass of a particle.
+
+### Atomic radius
+
+The atomic radius of an atom is the distance from the centre of the nucleus to approximately the outer boundary of the electron shell. This cannot be directly measured by scientists.
+
+### Ionisation energy
+
+The first ionisation energy of an element is the minimum amount of energy required to remove one mole of electrons from one mole of *gaseous* atoms to form a mole of gaseous ions, so that
+$$\text{Q}_\text{(g)} \rightarrow \text{Q}_\text{(g)}^+ + \text{e}^-$$
+
+Any subsequent ionisation energies of an element are the minimum amount of energy required to remove one *additional* mole of electrons. For example, the second ionisation energy would involve
+$$\text{Q}_\text{(g)}^+ \rightarrow \text{Q}_\text{(g)}^{2+} + \text{e}^-$$
+
+It requires vastly more energy to remove an electron from a filled valence shell compared to an unfilled valence shell.
+
+### Electron affinity
+
+The electron affinity of an atom is the amount of energy **required** or **released** to *add* an electron to a neutral *gaseous* atom to form a negative ion, such that
+$$\text{Q}_\text{(g)} + \text{e}^- \rightarrow \text{Q}^-_\text{(g)}$$ 
+
+If energy is released, the atom has a **negative** electron affinity, and will form a stable ion.
+
+If energy is required, the atom has a **positive** electron affinity, and will form an unstable ion (the ion will spontaneously decay).
+
+### Electronegativity
+
+The electronegativity of an atom represents the ability of that atom to attract a pair of electrons in a **covalent bond**. It ranges from $0$ to $4$ on the Pauling scale. As electronegativity increases, the atom more strongly holds on to the electrons in its covalent bond, so the pair of electrons in that bond spend more time around the atom with the higher electronegativity.
+
+### Reactivity
+
+The reactivity of an element is how "willing" it is to give up or gain electrons to fill its valence shell.
+
+The reaction of an **alkali metal** with water always forms a hydroxide and hydrogen gas. For example, lithium reacts with water such that:
+$$2\text{Li}_\text{(s)} + 2\text{H}_2\text{O}_\text{(l)} \rightarrow 2\text{LiOH}_\text{(aq)} + \text{H}_{2 (g)}$$
+
+The reaction of a **halogen** with hydrogen gas always forms a hydride. For example, fluorine reacts with hydrogen gas such that:
+$$\text{Fl}_\text{(g)} + \text{H}_\text{2 (g)} \rightarrow 2\text{HFl}_\text{(g)}$$
+
+### Models
+
+Please see [SL Physics#Models](/sph3u7/#models) for more information.
+
+### Periodic trends
+
+Some trends in the periodic table include:
+
+ - atomic radius decreases when going across a period and increases when going down a group
+ - ionic radius decreases when going across a period for groups 1–13, then sharply increases and then increases for groups 15–17; it increases when going down a group
+ - electron affinity increases when going across a period and decreases when going down a group
+ - ionisation energy increases when going across a period and decreases when going down a group
+ - electronegativity increases when going across a period and decreases when going down a group
+ - reactivity of alkali metals increases when going down the group
+ - reactivity of halogens decreases when going down the group
+
+When explaining these trends in the periodic table, it is best to use the following basic concepts to build on to larger points.
+
+Across a period, the number of shells occupied by the electrons is the same but the number of protons in the nucleus increases. Therefore,
+
+ - the attraction of each electron to the nucleus (effective nuclear charge) increases as the number of protons increases
+ - shielding is unchanged as the number of electrons between the valence electrons and the nucleus is the same
+
+Down a period, the number of shells occupied by the electrons increases, so valence electrons are further from the nucleus. Therefore,
+
+ - the attraction of valence electrons to the nucleus decreases due to the increasing distance
+ - shielding increases due to the increasing number of electrons between the valence electrons and the nucleus
+
+!!! example
+    To explain why there is a trend of decreasing atomic radius across a period:
+
+    - As the number of protons and electrons increase together, but the number of electron shells does not change, the effective nuclear charge of each electron increases, while the effect of shielding remains unchanged.
+    - This increased effective nuclear charge reduces the atomic radius compared to other atoms before it.
+
+## 4.0 - Chemical bonding and structure
+
+A chemical bond consists of the strong electronic interactions of the **valence** electrons between atoms that hold the atoms closer together. This only occurs if the atoms would reduce their potential energy by bonding.
+
+!!! reminder
+    - Metal + metal = metallic bond
+    - Metal + non-metal = ionic bond
+    - Non-metal + non-metal = covalent bond
+
+!!! reminder
+    When drawing a Lewis **dot diagram**, covalent bonds must be represented as two adjacent dots. When drawing a Lewis **structure**, covalent bonds must be represented as lines connecting the atoms. 
+
+    If the process stage is required:
+
+    - Electrons destined to be shared must be encircled.
+    - Electrons to be transferred must have arrows pointing to their destination.
+    - x'es are used to represent additional electrons that have an unknown source.
+
+### Percentage ionic character
+
+Bonding is a spectrum. The percentage ionic character of a chemical bond shows roughly the amount of time valence electrons spend near an atom or ion in a bond. The difference between two elements' electronegativity (ΔEN) indicates how covalent and how ionic the bond **behaves**.
+
+If ΔEN is:
+
+ - less than 0.5, it behaves like a **pure covalent** bond
+ - between 0.5 and 1.7, it behaves like a **polar covalent** bond
+ - greater than 1.7, it behaves like an **ionic** bond
+
+## 4.1 - Ionic bonding and structure
+
+An ionic bond is the electrostatic attraction between oppositely charged **ions**. Electrons are transferred first, and then the bond forms via the attraction of the now-positive and negative ions. This reduces the potential energy of the ions and therefore increases their stability.
+
+!!! definition
+    **Electrostatic attraction** is the force of attraction between opposite charges.
+
+!!! warning
+    When expressing ionic bonds in a Lewis dot diagram, ions with charges of the same sign must *never* be placed next to one another.
+
+### Structure of ionic compounds
+
+Ionic compounds are composed of a **lattice structure** (crystalline structure) of ions of alternating charges. A **formula unit** is the lowest ratio of positive to negative ions.
+
+<img src="/resources/images/nacl-lattice.jpeg" width=700>(Source: Kognity)</img>
+
+!!! example
+    In sodium chloride, the ratio of positive sodium ions to negative chloride ions is always 1:1, so its formula is NaCl.
+
+In an ionic compound, the number of ions that each ion can touch is referred to as the **coordination number**. It is stated as "(cation)(anion) is (coordination number of cation):(coordination number of anion) coordinated".
+
+!!! example
+    In the diagram above, each sodium ion touches six chloride ions, and each chloride ion touches six sodium ions. Therefore, "NaCl is 6:6 coordinated".
+
+## 4.2 - Covalent bonding
+
+A covalent bond is the electrostatic attraction between pairs of valence electrons and nuclei. This causes atoms to "share" electrons instead of gaining or losing them. Covalent bonds form molecules, which in turn form molecular compounds (not covalent compounds).
+
+<img src="/resources/images/covalent-bond.png" width=700>(Source: Kognity)</img>
+
+Whether a covalent bond is **pure** or **polar** indicates how evenly the shared electrons are shared between the atoms.
+
+ - A pure covalent bond has both nuclei attracting the valence electrons fairly evenly, so the difference in electronegativity (ΔEN) is low.
+ - A polar covalent bond has both nuclei attracting the valence electrons unevenly, so the ΔEN is high.
+
+### Bonding capacity
+
+The **bonding capacity** of a non-metal describes the number of covalent bonds it can form. This can be calculated via:
+
+1. Finding the number of needed electrons by taking the sum of 8 times the number of atoms. Hydrogen should be multiplied by 2 instead.
+2. Finding the number of electrons present by taking the sum of the valence electrons present. Any ions should have electrons added equal to their positive charge as well.
+
+The number of covalent bonds required is then:
+$$\frac{\text{needed} - \text{have}}{2}$$
+
+The number of lone pairs (pairs of un-bonded electrons) left over is:
+$$\frac{\text{have} - 2 × \text{bonds required}}{2}$$
+
+### Dative covalent bonds
+
+Sometimes, one atom in a covalent bond may contribute both electrons in a shared pair.
+
+## 4.3 - Covalent structures
+
+### Formal charge
+
+There may be several correct ways to draw covalent bonds in Lewis structures and dot diagrams. Solving for the **formal charge** of each atom involved in a covalent bond can help identify the **best** structure to construct. The formal charge of an atom in a covalent bond represents the charge that that atom has. The sum of all formal charges in a covalently bonded compound is equal to the charge of the overall compound.
+
+The formal charge of an atom can be calculated using the following equation:
+$$\text{Formal charge} = \text{# of valence electrons of element} - \text{# of unpaired electrons} - \text{# of covalent bonds}$$
+
+To find the best structure for a covalently bonded compound, the **absolute value** of the formal charge of all atoms in that compound should be **minimised**. Positively charged atoms will even accept **dative covalent bonds** from other atoms with negative formal charges.
+
+!!! warning
+    Some elements want formal charges of zero so much that they break the octet rule. These elements are $\text{P, S, Cl, Br, I, and Xe}$.
+
+### Resonance structures
+
+Even when considering formal charges, there may still be multiple best Lewis structures when molecules or polyatomic ions contain double or triple bonds. These equivalent structures are known as resonance structures, and the number of possible resonance structures is equal to the number of different positions for the double/triple bond. Double-sided arrows are used to show that the forms are resonant.
+
+<img src="/resources/images/resonance-structure.png" width=700>(Source: Kognity)</img>
+
+The resonance structures of a compound show that none of the models is truly correct but instead the actual structure is somewhere **in between all of them**, and is **not** "flipping" between the various resonance structures.
+
+!!! warning
+    Even molecules such as $\text{SO}_2$ have resonance structures as the possible naive structures prior to involving formal charges are also considered to be resonant.
+
+The **resonance-hybrid** structure shows that the actual strength of all three bonds is equal and somewhere between a single and double bond.
+
+<img src="/resources/images/carbonate-delocalised.png" width=400>(Source: Wikipedia)</img>
+
+### Exceptions to the octet rule
+
+Atoms such as boron and beryllium ($\text{B}$ and $\text{Be}$) may form **incomplete octets** (less than 8 electrons) in their valence shell due to their status as **small metalloids** that form covalent bonds. In total, boron can sometimes need only 6 electrons while beryllium may have only 4 in their valence shells.
+
+<img src="/resources/images/bb-octet-exceptions.png" width=700>(Source: Kognity)</img>
+
+!!! example
+    $\text{BeCl}_2$ and $\text{BCl}_3$ exist.
+
+Some elements in period 3 and beyond follow the formal charge exception above and may form **expanded octets** (more than 8 electrons and up to 12) in their valence shell. These include the aforementioned $\text{P, S, Cl, Br, I, Xe}$, as well as $\text{Si}$.
+
+**Free radicals** are molecules that end up with an odd number of electrons in their valence shell and are *very* reactive. Because one electron can never pair up with another, it remains forever alone.
+
+??? example
+    $\text{NO}_2$ is a free radical as one of nitrogen's atoms cannot pair with anything even after the formation of a dative covalent bond from oxygen.
+
+### Factors affecting bond strength
+
+The strength of a bond is determined by the amount of energy required to break that bond (**bond energy**).
+
+The length of a bond (**bond length**) has an inverse relationship with the strength of that bond, as the attraction of electrons to nuclei decreases with distance.
+
+Multiple (double/triple) bonds are shorter than single bonds (a higher **bond order**) and are therefore stronger.
+
+## 4.4 - Intermolecular forces
+
+## 4.5 - Metallic bonding
+
 ## Resources
 
  - [IB Chemistry Data Booklet](/resources/g11/ib-chemistry-data-booklet.pdf)

docs/sph3u7.md

@@ -4,6 +4,9 @@ The course code for this page is **SPH3U7**.
 
 ## 1.1 - Measurements in physics
 
+!!! reminder
+    All physical quantities must be expressed as a **product** of a magnitude and a unit. For example, ten metres should be written as $10 \text{ m}$.
+
 ### Fundamental units
 
 Every other SI unit is derived from the fundamental SI units. Memorise these!
@@ -20,7 +23,7 @@ Every other SI unit is derived from the fundamental SI units. Memorise these!
 
 ### Metric prefixes
 
-Every SI unit can be expanded with metric prefixes.
+Every SI unit can be expanded with metric prefixes. Note that the difference between many of these prefixes is $10^3$.
 
 !!! example
     milli + metre = millimetre ($10^{-3}$) m
@@ -97,10 +100,7 @@ The order of magnitude of a number can be found by converting it to scientific n
 
 ### Uncertainties
 
-Uncertainties are stated in the form of [value] ± [uncertainty]. A value is only as precise as its absolute uncertainty. Absolute uncertainty of **measurement** is usually represented to only 1 significant digit.
-
-!!! note
-    Variables with uncertainty use an uppercase delta for their uncertainty value: $a ± \Delta a$
+Uncertainties are stated in the form of $a±\Delta a$. A value is only as precise as its absolute uncertainty. Absolute uncertainty of a **measurement** is usually represented to only 1 significant digit.
 
  - The absolute uncertainty of a number is written in the same unit as the value.
  - The percentage uncertainty of a number is the written as a percentage of the value.
@@ -112,7 +112,7 @@ Uncertainties are stated in the form of [value] ± [uncertainty]. A value is onl
 To determine a measurement's absolute uncertainty, if:
 
  - the instrument states its uncertainty, use that.
- - an analog instrument is used, the last digit is estimated and appended to the end of the reported value. The estimated digit is uncertain by 5 at its order of magnitude.
+ - an analog instrument is used, half of the most precise reading is uncertain.
  - a digital instrument is used, the last reported digit is uncertain by 1 at its order of magnitude.
 
 !!! example
@@ -123,35 +123,37 @@ See [Dealing with Uncertainties](/resources/g11/physics-uncertainties.pdf) for h
 
 ### Error bars
 
-Error bars represent the uncertainty of the data, typically representing that data point's standard deviation, and can be both horizontal or vertical.
+Error bars represent the uncertainty of the data and can be both horizontal or vertical. They are almost always required for both the independent and dependent variables. A data point with uncertain values is written as $(x ± \Delta x, y ± \Delta y)$
 
 <img src="/resources/images/error-bars.png" width=600>(Source: Kognity)</img>
 
-!!! note
-    On a graph, a data point with uncertain values is written as $(x ± \Delta x, y ± \Delta y)$
+If the error bars of a data point are too small to see, note at the bottom of the graph that error bars are too small to see.
 
 ### Uncertainty of gradient and intercepts
 
 !!! note "Definition"
     - The **line of best fit** is the line that passes through **as many error bars as possible** while passing as closely as possible to all data points.
-    - The **minimum and maximum lines** are lines that minimise/maximise their slopes while passing through the first and last **error bars**.
+    - The **minimum and maximum lines** are lines that minimise/maximise their slopes while passing through as many **error bars** as possible.
 
 !!! warning
     - Use solid lines for lines representing **continuous data** and dotted lines for **discrete data**.
+    - The line of best fit may not be a straight line.
+    - Be wary and verify the results of a best fit line from software, as outliers and data trends may not be recognised by a computer.
+    - It is better to leave a data point in the graph compared to removing it entirely, when possible.
 
 <img src="/resources/images/error-slopes.png" width=700>(Source: Kognity)</img>
 
 The uncertainty of the **slope** of the line of best fit is the difference between the maximum and minimum slopes.
-$$m_{best fit} ± m_{max}-m_{min}$$
+$$m_{\text{best fit}} ± \frac{m_{\max}-m_{\min}}{2}$$
 The uncertainty of the **intercepts** is the difference between the intercepts of the maximum and minimum lines.
-$$intercept_{best fit} ± intercept_{max} - intercept_{min}$$
+$$\text{intercept}_{\text{best fit}} ± \frac{\text{intercept}_{\max} - \text{intercept}_{\min}}{2}$$
 
 
 ## 1.3 - Vectors and scalars
 
 !!! note "Definition"
     - **Scalar:** A physical quantity with a numerical value (magnitude) and a unit.
-    - **Vector:** A physical quantity with a numerical value (magnitude), a unit, and a **direction.**
+    - **Vector:** A physical quantity with a **non-negative** numerical value (magnitude), a unit, and a **direction.**
 
 ??? example
     - Scalar quantities include speed, distance, mass, temperature, pressure, time, frequency, current, voltage, and more.
@@ -169,22 +171,27 @@ $$|\vec{a}| = 1 \text{ m}$$
 ### Adding/subtracting vectors diagrammatically
 
 1. Draw the first vector.
-2. Draw the second vector with its *tail* at the *head* of the first vector.
+2. Draw the second vector with its tail at the head of the first vector.
 3. Repeat step 2 as necessary for as many vectors as you want by attaching them to the *head* of the last vector.
-4. Draw a new (**resultant**) vector from the *tail* of the first vector to the *head* of the last vector.
+4. Draw a new ("resultant") vector from the tail of the first vector to the head of the last vector.
 
 <img src="/resources/images/vector-add-direction.png" width=700>(Source: Kognity)</img>
 
-When subtracting a vector, **negate** the vector being subtracted by giving it an opposite direction and then add the vectors.
+When subtracting exactly one vector from another, repeat the steps above, but instead place the second vector at the **tail** of the first, then draw the resultant vector from the head of the second vector to the head of the first vector. Note that this only applies when subtracting exactly one vector from another.
+
+!!! example
+    In the diagram above, $\vec{b}=\vec{a+b}-\vec{a}$.
+
+Alternatively, for any number of vectors, negate the vector(s) being subtracted by **giving it an opposite direction** and then add the negative vectors.
 
 <img src="/resources/images/vector-subtract-direction.png" width=700>(Source: Kognity)</img>
 
 ### Adding/subtracting vectors algebraically
 
-Vectors can be broken up into two vectors (**"components"**) laying on the x- and y-axes via trigonometry such that the resultant of the two components is the original vector. This is especially helpful when adding larger (3+) numbers of vectors.
+Vectors can be broken up into two **component vectors** laying on the x- and y-axes via trigonometry such that the resultant of the two components is the original vector. This is especially helpful when adding larger (3+) numbers of vectors.
 $$\vec{F}_x + \vec{F}_y = \vec{F}$$
 
-!!! info "Reminder"
+!!! reminder
     The **component form** of a vector is expressed as $(|\vec{a}_x|, |\vec{a}_y|)$
 
 <img src="/resources/images/vector-simple-adding.png" width=700>(Source: Kognity)</img>
@@ -216,7 +223,7 @@ $$
 To find the resultant direction, use inverse tan to calculate the angle of the vector using the lengths of its components.
 
 $$
-\vec{c}_{direction} = \tan^{-1} \frac{c_y}{c_x}
+\theta_{c} = \tan^{-1}(\frac{c_y}{c_x})
 $$
 
 ### Multiplying vectors and scalars
@@ -228,6 +235,99 @@ $$\vec{v} × s = (|\vec{v}|×s)[\theta_{v}]$$
 !!! example
     $$3 \text{ m} · 47 \text{ ms}^{-1}[N20°E] = 141 \text{ ms}^{-1}[N20°E]$$
 
+## 2.1 - Motion
+
+!!! definition
+    - **Uniform motion**: Constant speed.
+    - **Position**: The location of an object relative to an origin (typically the position of the object at time zero).
+    - **Distance**: The scalar of the magnitude of the exact path taken by an object from an initial to a final position.
+    - **Displacement**: The vector of the shortest path from an initial to a final position.
+    - **Acceleration**: The vector of the rate of change of *velocity* over time.
+
+### Models
+
+A **scientific model** is a simplification of a system based on assumptions that predicts and/or explain observations for that system.
+
+!!! note "Definition"
+    - **System**: An object or a connected group of objects.
+    - **Point particle assumption**: An assumption that models a system as a blob of matter. It is more reliable if the size and shape of the object(s) do not matter much.
+
+### Velocity
+
+Velocity is the vector of the rate of change of *displacement* over time, and can be represented as $\frac{\Delta d}{\Delta t}$.
+
+The *average* velocity of an object is the velocity over an interval in time, calculated by finding the slope of the **secant** from the start and end position on a position-time graph.
+
+The *instantaneous velocity* of an object is the velocity at a specific moment in time, calculated by finding the slope of the **tangent** at that moment on a position-time graph.
+
+!!! definition
+    - A **secant** is a straight line which intersects two points on a curve.
+    - A **tangent** is a straight line that does not intersect a curve but "touches" it at exactly one point.
+
+### Displaying motion
+
+A **position-time graph** expands on the motion diagram by specifying a precise **position** value on the vertical axis in addition to time on the horizontal axis. The line of best fit indicates the object's speed, as well as if it is accelerating or decelerating.
+
+$s$ is commonly used in IB to represent displacement and $s_{0}$ represents the initial position (when $t=0$).
+
+<img src="/resources/images/position-time-graph.png" width=700>(Source: Kognity)</img>.
+
+The slope of the line in a position-time graph represents that object's velocity. If the slope is not linear, the object is not moving uniformly (at a constant speed).
+
+A **velocity-time graph** is similar to a position-time graph but replaces the position on the vertical axis with an object's velocity instead.
+
+<img src="/resources/images/velocity-time-graph.png" width=700>(Source: Kognity)</img>
+
+On a velocity-time graph, the slope represents that object's acceleration. If the slope is not linear, the object is not accelerating uniformly (accelerating at a constant rate).
+
+The area below a velocity-time graph at a given time is equal to the displacement (change in position, $\Delta d$) at that time, since $ms^{-1}×s=m$. When finding the displacement of an object when it is accelerating, breaking up the graph into a rectangle and a triangle then adding their areas will give the displacement.
+
+<img src="/resources/images/velocity-time-displacement.png" width=700>(Source: Kognity)</img>
+
+An **acceleration-time graph** is similar to a velocity-time graph but replaces the velocity on the vertical axis with an object's acceleration instead.
+
+The area below an acceleration-time graph at a given time is equal to the change in velocity ($\Delta v$) at that time.
+
+!!! note
+    If there is any instantaneous jump on a position-, velocity-, or acceleration-time graph (which is impossible in reality but may be used to simplify matters), a dashed line must be used to connect the two sides to ensure that the line remains a function.
+
+### Uniformly accelerated motion
+
+**Uniformly accelerated motion** is the constant acceleration in a **straight line**, or the constant increase in velocity over equal time intervals. The five key $suvat$ variables can be used to represent the various information in uniformly accelerated motion.
+
+### Kinematic equations
+
+<img src="/resources/images/constant-acceleration.png" width=700>(Source: Kognity)</img>
+
+ - $s=$ change in displacement during time interval $t$ ($\Delta d$)
+ - $u=$ initial velocity ($v_1$)
+ - $v=$ final velocity ($v_2$)
+ - $a=$ constant acceleration
+ - $t=$ time elapsed ($\Delta t$)
+
+By the formula of the gradient and the formula for the area underneath an acceleration time graph, the following formulas can be derived and are in the data booklet:
+
+ - $s=ut + \frac{1}{2}at^2$
+ - $v = u + at$
+ - $s = \frac{1}{2}(u+v)t$
+ - $v^2 = u^2 + 2as$
+
+### Projectile motion
+
+**Projectile motion** is uniformly accelerated motion that does not leave the vertical plane (is two-dimensional). Note that the two directions (horizontal and vertical) that the projectile moves in are independent of one another. This means that variables such as average velocity can be calculated by breaking up the motion into vector **components**, then finding the resultant vector.
+
+Projectiles move at a constant horizontal velocity and move at a uniformly accelerated velocity (usually $9.81 \text{ ms}^2 \text{ [down]}$).
+
+## 2.2 - Forces
+
+## 2.3 - Work, energy, and power
+
+## 2.4 - Momentum and impulse
+
+## 3.1 - Thermal concepts
+
+## 3.2 - Modelling a gas
+
 ## Resources
 
  - [IB Physics Data Booklet](/resources/g11/ib-physics-data-booklet.pdf)