eng: set up essay checklist

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

@@ -257,10 +257,195 @@ The **instantaneous rate of change (IRoC)** at point $P(a, f(a))$ is represented
 
 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
 
  - [IB Math Analysis and Approaches Syllabus](/resources/g11/ib-math-syllabus.pdf)

docs/sch3uz.md

@@ -45,24 +45,27 @@ If necessary, a "setup" section can be added as preparatory steps should not be
 
 ### Data collection
 
-Data should be collected in an organised and titled table that should be prepared before the experiment. To reduce plagiarism, the data table must be verified by a teacher before leaving the lab space. After verification, **no new data** can be added. During an experiment that spans multiple days, this data must be verified every day. The data table must include:
+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
-    **Effect of Fat Content on Sugar Content in Ice Cream**
+    **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.
@@ -71,7 +74,7 @@ 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 a full page and should directly answer the research question
+ - 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.
 
@@ -100,8 +103,10 @@ 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
+ - 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
 
@@ -124,7 +129,7 @@ Please see [SL Physics#Uncertainty of gradient and intercepts](/sph3u7/#uncertai
 
 !!! 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** describes the decrease in the effective nuclear charge of an electron because of the repulsion of other electrons in lower-energy shells.
+    - **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.
 
@@ -190,7 +195,7 @@ $$2\text{Li}_\text{(s)} + 2\text{H}_2\text{O}_\text{(l)} \rightarrow 2\text{LiOH
 The reaction of a **halogen** with hydrogen gas always forms a hydride. For example, fluorine reacts with hydrogen gas such that:
 $$\text{Fl}_\text{(g)} + \text{H}_\text{2 (g)} \rightarrow 2\text{HFl}_\text{(g)}$$
 
-###Models
+### Models
 
 Please see [SL Physics#Models](/sph3u7/#models) for more information.
 
@@ -224,6 +229,143 @@ Down a period, the number of shells occupied by the electrons increases, so vale
     - 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

@@ -299,11 +299,11 @@ The area below an acceleration-time graph at a given time is equal to the change
 
 <img src="/resources/images/constant-acceleration.png" width=700>(Source: Kognity)</img>
 
- - $s=$ change in displacement during time interval $t$ (i.e., from $t=0$ to $t$)
+ - $s=$ change in displacement during time interval $t$ ($\Delta d$)
- - $u=$ initial velocity at time $t=0$
+ - $u=$ initial velocity ($v_1$)
- - $v=$ final velocity at time $t$
+ - $v=$ final velocity ($v_2$)
  - $a=$ constant acceleration
- - $t=$ time elapsed since $t=0$
+ - $t=$ 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:
 
@@ -314,9 +314,11 @@ By the formula of the gradient and the formula for the area underneath an accele
 
 ### 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 the horizontal and vertical axes, then recombined using the Pythagorean theorem such that $v^2 = v_x^2 + v_y^2$.
+**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.
 
-##2.2 - Forces
+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