IITM BS DS Quiz Prep

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You

can u help me prep for quiz2 iitm bs ds

• courses this term: maths1/ct/python/stats1 (until week 8)
• first make sure you go through each and every concepts yk go through each courses one by one for not cutting corners yk
• second thing make sure u create patterns of all courses like courses/each & every weeks/ every topics/easy/medium/hard yk so that i have nicer grasp in everything yk & dissect each patterns in multiple steps completely with solutions or abstract or general things yk what i mean mun
• third thing if i would have doubts or wouldnt understand anything then i would lyk and i would practice myself from my shit or would ask u to help yk
• idk anythin else mun as of now

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ChatGPT

IITM BS DS — Quiz 2 Prep System

TL;DR

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1. The system we're going to use

Layer A — Concept

Example: conditional probability.

Layer B — Notation / representation

Layer C — Recognition

Layer D — Pattern

"Given that something has already happened, find the probability of another event."

Layer E — Procedure

Layer F — Variations

Layer G — Traps

Layer H — Difficulty

• Easy: direct recognition/application
• Medium: multiple concepts or transformation
• Hard: unfamiliar wording, indirect reasoning, multi-step composition

Layer I — Generalization

Layer J — Independent retrieval

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2. Canonical Quiz 2 syllabus

A. Mathematics for Data Science I

Week 1 — Set Theory

Core concepts

• Number systems
  • Natural numbers
  • Integers
  • Rational numbers
  • Real numbers
• Sets
• Set notation
• Set operations
  • Union
  • Intersection
  • Difference
  • Complement
• Relations
• Types of relations
• Functions
• Types of functions

Pattern families

1. Identify number-system membership.
2. Perform set operations.
3. Determine whether a relation has a given property.
4. Determine whether a mapping is a function.
5. Identify function type.

1. Multiple set operations.
2. Translate word descriptions → sets.
3. Determine properties of a relation.
4. Determine domain/range from a representation.
5. Function classification from mappings/graphs.

1. Combine set reasoning + relation reasoning.
2. Infer function properties indirectly.
3. Work backwards from constraints.
4. Questions where the representation itself is the trap.

Week 2 — Coordinate System & Straight Lines

• Cartesian/rectangular coordinate system
• Coordinates
• Slope
• Parallel lines
• Perpendicular lines
• Representations of a line
• General equation of a line
• Straight-line fitting

• Two points → find line.
• Point + slope → find line.
• Two lines → determine parallel/perpendicular.
• Equation → extract slope/intercept.
• Data points → straight-line fit.

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3. Maths Week 3 — Quadratic Functions

• Quadratic function
• Standard form
• Vertex
• Axis of symmetry
• Maximum/minimum
• Slope
• Quadratic equation
• Roots
• Discriminant
• Graph interpretation

Pattern M3-1 — Find roots

Pattern M3-2 — Determine number/type of roots

Pattern M3-3 — Find maximum/minimum

• $a>0$ → minimum
• $a<0$ → maximum

Pattern M3-4 — Recover quadratic from information

• vertex + another point
• roots + another point
• three points

Pattern M3-5 — Graph ↔ algebra

• roots
• vertex
• direction
• approximate equation

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4. Maths Week 4 — Algebra of Polynomials

Concepts

• Polynomial expressions
• Addition
• Subtraction
• Multiplication
• Division
• Polynomial algorithms
• Roots / $x$-intercepts
• Multiplicity
• End behaviour
• Turning points
• Polynomial graphing
• Constructing polynomials

Pattern families

• Add/subtract polynomials.
• Multiply.
• Evaluate polynomial.
• Identify degree.
• Identify leading coefficient.

• Polynomial division.
• Factorization.
• Roots → polynomial.
• Polynomial → graph characteristics.

• Multiplicity + graph behaviour.
• Determine unknown coefficients from constraints.
• Reconstruct polynomial from roots/points.
• Reason about end behaviour without fully expanding.

Important mental model

"some algebraic expression."

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5. Maths Week 5 — Functions

Concepts

• Function as mapping
• Domain
• Range
• Vertical line test
• Horizontal line test
• One-to-one behaviour
• Exponential functions
• Composition
• Inverse functions

Patterns

• Is this graph a function?
• Is this function one-to-one?
• Compute $f(g(x))$.
• Compute $g(f(x))$.
• Determine domain restrictions.
• Find inverse.
• Verify inverse.
• Distinguish function from inverse relation.

Classic trap

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6. Maths Week 6 — Logarithmic Functions

Concepts

Patterns

1. Evaluate logs.
2. Simplify logarithmic expressions.
3. Convert log ↔ exponential form.
4. Solve exponential equations.
5. Solve logarithmic equations.
6. Determine whether a solution is valid.
7. Interpret logarithmic graphs.

Major trap

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7. Maths Week 7 — Sequences, Limits & Continuity

Concepts

• Function of one variable
• Graphs
• Tangents
• Sequences
• Sequence limits
• Function limits
• Continuity

Pattern families

• Evaluate sequence terms.
• Identify convergence/divergence.
• Direct limit calculation.

• Algebraic manipulation before limits.
• Graph-based limit questions.
• One-sided behaviour.
• Continuity determination.

• Piecewise functions.
• Parameter selection for continuity.
• Limit + graph reasoning.
• Distinguishing:

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8. Maths Week 8 — Derivatives, Tangents & Critical Points

• Differentiability
• Derivative
• Computing derivatives
• L'Hôpital's rule
• Tangents
• Linear approximation
• Critical points
• Local maxima/minima. citeturn1view0

Core object

• instantaneous rate of change
• slope of tangent
• local behaviour

Patterns

Derivative computation

Tangent

Linear approximation

Critical points

Local extrema

L'Hôpital

= \lim_{x\to a}\frac{f'(x)}{g'(x)}$$ when the conditions for the rule hold. --- # 9. Computational Thinking This course is particularly important because IITM explicitly describes CT as learning programming concepts through **manual execution**, rather than simply writing code. citeturn1view2 That means our CT prep will heavily emphasize: > **trace → predict → abstract → construct → debug** --- ## CT Week 1 Official topics: variables, initialization, iterators, filtering, datatypes, flowcharts and sanity of data. citeturn1view2 Patterns: - Variable tracing - State changes - Initialization - Iterating through data - Filtering - Datatype identification - Flowchart interpretation - Detecting invalid/insane input Example abstract pattern: ```text value ← initial_value for each item: update value output value ``` You should eventually be able to look at this and immediately ask: > What is the state? > What changes it? > What remains invariant? > What is the final state? --- # 10. CT Week 2 - Iteration - Filtering - Selection - Pseudocode - Finding maximum/minimum - AND Patterns: ### Accumulation ```text total ← 0 for each x: total ← total + x ``` ### Maximum ```text best ← first item for each x: if x > best: best ← x ``` ### Filtering ```text if condition: keep/process x ``` ### Conjunction $$A\land B$$ Both must be true. --- # 11. CT Week 3 - Multiple non-nested iterations - Three-prizes problem - Procedures - Parameters - Side effects - OR This is where we start thinking in **reusable computational units**. Pattern: $$\text{input}\rightarrow\text{procedure}\rightarrow\text{output}$$ We'll distinguish: - parameter - argument - local state - returned result - side effect And: $$A\lor B$$ means at least one condition is true. --- # 12. CT Week 4 - Nested iterations - Birthday paradox - Binning This is a major pattern jump. Nested loop structure: ```text for each x: for each y: do something ``` Conceptually: $$n\times n$$ potential pairwise interactions. ### Birthday-paradox pattern Not merely a probability question. It's a **pair-generation / collision-detection** pattern. ### Binning Map continuous/discrete values into categories: $$x\rightarrow\text{bin}(x)$$ This idea becomes useful throughout data science. --- # 13. CT Week 5 — Lists Officially: lists and insertion sort. citeturn1view2 Patterns: - Access element - Modify element - Traverse list - Search - Insert - Delete - Count - Find max/min - Sort ### Insertion sort Mental model: > Maintain a sorted prefix; insert the next element into its correct position. We will trace this **by hand**, not just memorize code. --- # 14. CT Week 6 — Tables & Dictionaries Official syllabus calls this "Table, Dictionary." citeturn1view2 Patterns: ### Table Think: $$\text{row}\times\text{column}$$ ### Dictionary Think: $$\text{key}\rightarrow\text{value}$$ Typical questions: - Lookup - Update - Count frequencies - Map identifiers to information - Represent relationships - Translate table ↔ dictionary representation Frequency counting is a particularly important reusable pattern: ```text for each item: frequency[item] += 1 ``` --- # 15. CT Week 7 — Graphs & Matrices Officially: - Graphs - Matrices. citeturn1view2 Core graph vocabulary: - vertex/node - edge - adjacency - path - degree - directed/undirected relationships Matrix representation introduces another abstraction: $$A_{ij}$$ can encode whether/how node $i$ is related to node $j$. --- # 16. CT Week 8 — Adjacency Matrix & Edge-Labelled Graphs This is where representation becomes important. ### Adjacency matrix For a graph with $n$ vertices: $$A\in\mathbb{R}^{n\times n}$$ or, for a simple unweighted graph, commonly: $$A_{ij}\in\{0,1\}$$ depending on whether an edge exists. ### Edge-labelled graph Edges carry additional information: $$u\xrightarrow{\text{label}}v$$ The label could represent: - weight - distance - cost - type - relationship The important CT skill: > **Same underlying problem, different representation.** --- # 17. Programming in Python Officially, Weeks 1–8 are: 1. Introduction to algorithms 2. Conditionals 3. Conditionals continued 4. Iterations and ranges 5. Iterations and ranges continued 6. Basic collections 7. Basic collections continued 8. Basic collections continued. citeturn1view3turn0search5 And here's where I want to be particularly strict: **CT pattern → Python implementation → Python-specific behaviour.** --- ## Python W1 — Algorithms Concepts: - Python execution model - Variables - Expressions - Values - Types - Assignment - Input/output - Basic algorithmic thinking Patterns: ```python x = ... y = ... z = ... ``` Trace the state after every line. --- # 18. Python W2–3 — Conditionals Core structures: ```python if condition: ... elif condition: ... else: ... ``` Patterns: ### Binary decision $$P\rightarrow A/B$$ ### Multiple mutually exclusive cases $$P_1,P_2,\ldots,P_n$$ ### Compound condition ```python if A and B: ``` ### Alternative condition ```python if A or B: ``` Important distinction: ```python and ``` versus ```python or ``` and nested conditionals. We'll also train **truth-table thinking**, not just syntax. --- # 19. Python W4–5 — Iterations & Ranges Core structures: ```python for x in ...: ``` and ```python while condition: ``` Patterns: - fixed repetition - condition-controlled repetition - counting - accumulation - filtering - searching - maximum/minimum - nested iteration - range generation Important: ```python range(start, stop, step) ``` has an **exclusive stop**. This is one of the classic Python traps. --- # 20. Python W6–8 — Basic Collections This is a large chunk. ### Lists ```python [x1, x2, x3] ``` Patterns: - indexing - traversal - modification - append - insertion - deletion - membership - slicing - aggregation ### Tuples ```python (x, y) ``` Think: > ordered collection with different mutability semantics. ### Dictionaries ```python { key: value } ``` Think: $$key\rightarrow value$$ ### Collection patterns We'll train: - frequency counting - lookup - grouping - transformation - filtering - aggregation - nested collections - iteration over collection structures --- # 21. Statistics for Data Science I Stats is probably the course where **interpretation traps** will matter most. The official Weeks 1–8 are clearly defined. citeturn1view1 --- ## Stats W1 — Introduction & Types of Data Concepts: - Data - Cases/observations - Variables - Categorical data - Numerical data - Descriptive statistics - Inferential statistics - Scales of measurement Measurement scales: - Nominal - Ordinal - Interval - Ratio Pattern: > Given a variable → identify its type/measurement scale. The important skill is understanding **what mathematical operations make sense** for the variable. --- # 22. Stats W2 — Categorical Data Concepts: - Frequency distribution - Relative frequency - Graphical representation - Mode - Median for categorical variables - Appropriate graph selection - Graph interpretation Core relationship: $$\text{relative frequency} = \frac{\text{frequency}}{\text{total}}$$ Patterns: - raw categorical data → frequency table - frequency → relative frequency - table → graph - graph → interpretation - identify misleading/inappropriate graph --- # 23. Stats W3 — Numerical Data This is a **large pattern family**. ### Central tendency - Mean - Median - Mode Mean: $$\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i$$ ### Position - Quartiles - Percentiles ### Dispersion - Range - Variance - Standard deviation - IQR $$IQR=Q_3-Q_1$$ ### Five-number summary Patterns: - calculate summary - choose appropriate summary - compare datasets - detect effect of outliers - interpret spread - reconstruct information from summary statistics --- # 24. Stats W4 — Association Between Two Variables Officially includes categorical association, contingency tables, numerical association, scatterplots, covariance, Pearson correlation and point-biserial correlation. citeturn1view1 ### Categorical × categorical Contingency tables. ### Numerical × numerical Scatterplot. ### Covariance Conceptually: > Do two variables tend to move together? ### Pearson correlation $$-1\le r\le1$$ Interpret: - sign → direction - magnitude → strength of linear association ### Point-biserial correlation Useful when: - one variable is binary/categorical - one is numerical ### Major trap **Correlation ≠ causation.** And: > $r\approx0$ does not mean "no relationship whatsoever"; it primarily indicates little/no **linear** association. --- # 25. Stats W5 — Counting Concepts: - Addition rule - Multiplication rule - Factorials Factorial: $$n!=n(n-1)(n-2)\cdots1$$ ### Addition rule Use when alternatives are mutually exclusive: $$N=N_1+N_2+\cdots$$ ### Multiplication rule Use for sequential choices: $$N=N_1N_2\cdots$$ The real skill is **recognizing whether a problem is additive or multiplicative**. --- # 26. Stats W6 — Permutations & Combinations Permutation: $$P(n,r)=\frac{n!}{(n-r)!}$$ Combination: $$C(n,r)=\binom nr =\frac{n!}{r!(n-r)!}$$ ### Recognition rule Ask: > **Does order matter?** If yes → permutation-type reasoning. If no → combination-type reasoning. This becomes a major pattern classifier. --- # 27. Stats W7 — Probability Concepts: - Random experiment - Sample space - Event - Probability - Probability properties Core axioms: $$0\le P(A)\le1$$ $$P(S)=1$$ For disjoint events: $$P(A\cup B)=P(A)+P(B)$$ General addition: $$P(A\cup B) = P(A)+P(B)-P(A\cap B)

Pattern families

• direct probability
• complement
• union
• intersection
• mutually exclusive events
• sample-space counting
• probability from combinatorics

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28. Stats W8 — Conditional Probability

• Conditional probability
• Multiplication rule
• Independence
• Law of total probability
• Bayes' theorem

$$P(A\mid B) = \frac{P(A\cap B)}{P(B)}$$ $$P(A\cap B) = P(A\mid B)P(B)$$ $$P(A\cap B)=P(A)P(B)$$ $$P(A\mid B)=P(A)$$ $$P(A\mid B) = \frac{P(B\mid A)P(A)} {P(B)}$$

The pattern we're going to hammer

evidence $B$

probability of underlying condition $A$.

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29. The cross-course pattern map

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30. Our difficulty taxonomy

🟢 Easy — Recognition

Find the slope through $(1,2)$ and $(3,6)$.

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🟡 Medium — Transformation

A line is perpendicular to another line and passes through a given point. Find its equation.

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🔴 Hard — Composition

A probability problem requires counting → conditional probability → Bayes.

A Python question requires nested iteration → dictionary accumulation → condition filtering.

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⚫ Adversarial

• unusual wording
• misleading representation
• hidden constraint
• boundary case
• distractor information
• inverse reasoning
• subtle Python semantics

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31. Every pattern will eventually have this anatomy

Pattern

Given evidence, infer the probability of the underlying cause.

Step 1 — Identify

Step 2 — Translate

Step 3 — Determine requested direction

Step 4 — Recognize inversion

Step 5 — Expand denominator

$$P(B) = P(B\mid A)P(A) + P(B\mid A^c)P(A^c)$$

Step 6 — Substitute

Step 7 — Sanity check

Step 8 — Generalize

prior → mechanism → observed evidence → posterior

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32. How we'll actually study from here

Phase 1 — Scope lock

• Maths 1
• CT
• Python
• Stats 1

Phase 2 — Concept inventory

Week → topic → sub-concepts → notation → intuition → recognition → patterns → easy → medium → hard → traps → solved examples → independent problems

Phase 3 — Pattern library

text
MATHS
├── W1 Sets
│   ├── Set operations
│   │   ├── Direct
│   │   ├── Multi-operation
│   │   └── Word → symbolic
│   ├── Relations
│   └── Functions
│
├── W2 Lines
│   ├── Slope
│   ├── Equation
│   ├── Parallel
│   └── Perpendicular
│
...

Phase 4 — Retrieval

"give me one hint"

"why is my approach wrong?"

"I don't understand this concept."

Phase 5 — Mixed Quiz 2 simulation

"This is a Bayes question."

Here's a question.

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33. One important thing I want to preserve

$$\text{Sets} \rightarrow \text{Functions} \rightarrow \text{Graphs} \rightarrow \text{Limits} \rightarrow \text{Derivatives}$$ $$\text{Counting} \rightarrow \text{Probability} \rightarrow \text{Conditional Probability} \rightarrow \text{Bayes}$$ $$\text{CT} \rightarrow \text{Algorithms} \rightarrow \text{Iteration} \rightarrow \text{Collections} \rightarrow \text{Graphs}$$

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34. What I need from you eventually

• IITM lecture transcripts
• lecture notes
• GrPA questions
• weekly assignments
• Quiz 1 paper
• previous quiz questions
• course PDFs

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The operating rule from now on

"Maths W3"

Teach/revise every relevant W3 concept, then build its pattern taxonomy, demonstrate representative solutions, expose traps, and give you problems to solve independently.

"I don't understand this"

"give me questions"

"quiz me"

"I got this wrong"

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You

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ChatGPT

Quiz 2 Prep — Session 1

Course order

1. Maths 1 — W1 → W8
2. Stats 1 — W1 → W8
3. CT — W1 → W8
4. Python — W1 → W8
5. Mixed cross-course revision
6. Quiz 2 pattern drills
7. Full mock(s)

Mathematics 1 — Week 1

Sets, Number Systems, Relations & Functions

"Ah, this is that pattern."

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0. The Week 1 Map

$$\boxed{\text{Number Systems}} \rightarrow \boxed{\text{Sets}} \rightarrow \boxed{\text{Relations}} \rightarrow \boxed{\text{Functions}}$$

Numbers are objects → sets collect objects → relations connect objects → functions impose a special kind of connection.

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1. Number Systems

1.1 Natural numbers

$$\mathbb N=\{1,2,3,\ldots\}$$ $$\mathbb N_0=\{0,1,2,3,\ldots\}$$

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1.2 Integers

$$\mathbb Z=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\}$$

• negative integers
• zero
• positive integers

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1.3 Rational numbers

$$\frac pq$$ $$p,q\in\mathbb Z,\qquad q\neq0$$ $$\frac12,\quad -\frac73,\quad 5,\quad 0.25$$ $$5=\frac51$$ $$\mathbb Z\subseteq\mathbb Q$$

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1.4 Irrational numbers

$$\sqrt2,\quad \pi,\quad e$$

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1.5 Real numbers

$$\mathbb R=\mathbb Q\cup\{\text{irrational numbers}\}$$ $$\boxed{ \mathbb N\subseteq\mathbb Z\subseteq\mathbb Q\subseteq\mathbb R }$$

Recognition pattern

Which of these numbers belongs to $\mathbb Q$?

Can I express it as $\frac pq$?

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2. Set Theory

2.1 What is a set?

$$A=\{1,2,3,4\}$$ $$2\in A$$

$2$ is an element of $A$.

$$7\notin A$$

$7$ is not an element of $A$.

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3. Set Representation

3.1 Roster / listing form

$$A=\{2,4,6,8\}$$

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3.2 Set-builder form

$$A=\{x\in\mathbb N:x<5\}$$

The set of all natural numbers $x$ such that $x<5$.

$$A=\{1,2,3,4\}$$

Pattern

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4. Empty Set

$$\varnothing$$ $$\{\}$$ $$\varnothing\neq\{\varnothing\}$$

$\varnothing$

$\{\varnothing\}$

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5. Cardinality

How many elements does the set contain?

$$|A|$$ $$A=\{2,4,6,8\}$$ $$|A|=4$$ $$|\varnothing|=0$$ $$|\{\varnothing\}|=1$$

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6. Subsets

$$A=\{1,2,3\}$$ $$B=\{1,2\}$$ $$B\subseteq A$$

$B$ is a subset of $A$.

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6.1 Proper subset

$$B\subseteq A$$ $$B\neq A$$ $$B\subset A$$

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6.2 Important pattern

$$B\subseteq A$$

"Does $B$ look smaller?"

Is every element of $B$ contained in $A$?

$$A=\{1,2,3,4\}$$ $$B=\{2,4\}$$ $$B\subseteq A$$ $$C=\{2,5\}$$ $$5\notin A$$ $$C\not\subseteq A$$

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7. Set Equality

$$A=\{1,2,3\}$$ $$B=\{3,2,1\}$$ $$A=B$$ $$\{1,1,2,2,3\}=\{1,2,3\}$$

Contrast

text
[1, 2, 3]

text
[3, 2, 1]

$$\{1,2,3\}=\{3,2,1\}$$

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8. Power Set

$$\mathcal P(A)$$ $$A=\{a,b\}$$ $$\varnothing,\{a\},\{b\},\{a,b\}$$ $$\mathcal P(A) = \{\varnothing,\{a\},\{b\},\{a,b\}\}$$ $$|A|=n$$ $$\boxed{|\mathcal P(A)|=2^n}$$

Why?

included / not included.

$$2\times2\times\cdots\times2=2^n$$

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9. Set Operations

$$A=\{1,2,3,4\}$$ $$B=\{3,4,5,6\}$$

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9.1 Union

Everything in $A$ or $B$.

$$A\cup B$$ $$A\cup B=\{1,2,3,4,5,6\}$$

Mental model

$$\boxed{\text{OR}}$$

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10. Intersection

Things common to both.

$$A\cap B$$ $$A\cap B=\{3,4\}$$

Mental model

$$\boxed{\text{AND}}$$

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11. Difference

$$A-B$$

Elements in $A$ that are not in $B$.

$$A-B=\{1,2\}$$ $$A-B\neq B-A$$ $$B-A=\{5,6\}$$

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12. Complement

$$A^c=U-A$$ $$U=\{1,2,3,4,5,6\}$$ $$A=\{1,2,3\}$$ $$A^c=\{4,5,6\}$$

Important

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13. The Core Set Pattern

$$A=\{1,2,3,4\}$$ $$B=\{3,4,5,6\}$$

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14. Set Identities

Commutativity

$$A\cup B=B\cup A$$ $$A\cap B=B\cap A$$ $$A-B\neq B-A$$

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Associativity

$$(A\cup B)\cup C=A\cup(B\cup C)$$ $$(A\cap B)\cap C=A\cap(B\cap C)$$

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Distributivity

$$A\cap(B\cup C) = (A\cap B)\cup(A\cap C)$$ $$A\cup(B\cap C) = (A\cup B)\cap(A\cup C)$$

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15. De Morgan's Laws

$$\boxed{(A\cup B)^c=A^c\cap B^c}$$ $$\boxed{(A\cap B)^c=A^c\cup B^c}$$

NOT (A OR B) = NOT A AND NOT B

NOT (A AND B) = NOT A OR NOT B

• Statistics
• CT
• Python conditionals

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16. Relations

$$A=\{1,2,3\}$$ $$B=\{a,b\}$$ $$A\times B$$ $$A\times B= \{ (1,a),(1,b), (2,a),(2,b), (3,a),(3,b) \}$$ $$|A\times B|=|A||B|$$

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17. Why Ordered Pairs Matter

$$(1,a)\neq(a,1)$$ $$\{1,a\}=\{a,1\}$$

Set: order irrelevant
Ordered pair: order matters

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18. Relation

$$R=\{(1,a),(2,b),(3,a)\}$$

text
A                  B

1 ───────────────→ a
2 ───────────────→ b
3 ───────────────→ a

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19. Function

Every input has exactly one output.

$$f:A\to B$$

text
input → exactly one output

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20. Function Recognition Pattern

text
1 → a
2 → b
3 → a

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text
1 → a
1 → b
2 → a

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text
1 → a
2 → b
3

Golden rule

$$\boxed{\text{Every input → exactly one output}}$$

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21. Domain, Codomain & Range

$$f:A\to B$$

Domain

$$A$$

Codomain

$$B$$

Range

$$f:\{1,2,3\}\to\{a,b,c,d\}$$ $$1\to a$$ $$2\to b$$ $$3\to a$$ $$\text{Domain}=\{1,2,3\}$$ $$\text{Codomain}=\{a,b,c,d\}$$ $$\text{Range}=\{a,b\}$$

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22. One-to-One / Injective

Different inputs always produce different outputs.

$$f(a)=f(b)\Rightarrow a=b$$ $$1\to a$$ $$2\to b$$ $$3\to c$$ $$1\to a$$ $$2\to a$$

Mental model

No two inputs collide.

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23. Onto / Surjective

Every element of the codomain is actually hit.

$$A=\{1,2,3\}$$ $$B=\{a,b,c\}$$ $$1\to a,\quad2\to b,\quad3\to c$$ $$f:A\to B$$

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24. Bijective

$$\boxed{\text{injective + surjective}}$$

no collisions + nothing left unused.

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25. The Function Taxonomy

text
                    FUNCTIONS
                       │
             ┌─────────┴─────────┐
             │                   │
        Injective?           Surjective?
             │                   │
             └─────────┬─────────┘
                       │
                  Both = Bijective

• Injective: no output collision.
• Surjective: no unused codomain element.
• Bijective: both.

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26. Week 1 Pattern Bank — First Version

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27. Now I want you to do something

"yeah yeah makes sense"

Diagnostic — Week 1

Q1 — Number systems

$$-7,\quad 0,\quad \frac34,\quad \sqrt2,\quad \pi$$

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Q2 — Sets

$$A=\{1,2,3,4,5\}$$ $$B=\{2,4,6\}$$ $$A\cap B$$ $$A\cup B$$ $$A-B$$

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Q3 — Subsets

$$\{1,3\}\subseteq\{1,2,3,4\}?$$ $$\{1,5\}\subseteq\{1,2,3,4\}?$$

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Q4 — Power set

$$|A|=5$$ $$|\mathcal P(A)|?$$

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Q5 — Cartesian product

$$A=\{1,2\}$$ $$B=\{x,y,z\}$$ $$|A\times B|?$$

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Q6 — Function recognition

text
1 → a
2 → b
3 → a

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Q7 — Function recognition

text
1 → a
1 → b
2 → c

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Q8 — Domain / codomain / range

$$f:\{1,2,3\}\to\{a,b,c,d\}$$ $$f(1)=a,\qquad f(2)=b,\qquad f(3)=a$$

• Domain
• Codomain
• Range
• Is it injective?
• Is it surjective?
• Is it bijective?

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Q9 — Slightly nastier

$$A=\{1,2,3,4\}$$ $$B=\{3,4,5\}$$ $$C=\{4,5,6\}$$ $$A\cap(B\cup C)$$

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Q10 — Pattern recognition

"A function has domain $\{1,2,3,4\}$ and codomain $\{a,b,c\}$. Each element of the domain maps to exactly one element of the codomain, and every codomain element is used."

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$$\boxed{\text{Concept gap / Pattern gap / Execution error / Careless error / Correct}}$$

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You

• set is easy understndable but doubts are
• in union do we have duplicate numbers or things ig not
• iis there formula for cartesian product example for q5 it would be 1x+1y+1z... like it is it & what is small a vs A' kinda it notation without explaination is weird in text or by teacher
• what happens if not injective or surjective mun ig not function
• start next thing ig mun

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ChatGPT

Week 1 — Your doubts, cleaned up

1. Union does not keep duplicates

$$A=\{1,2,3\}$$ $$B=\{2,3,4\}$$ $$A\cup B=\{1,2,3,4\}$$ $$\{1,2,3,2,3,4\}$$

Mental model

Take everything that appears in either set, but represent each distinct element once.

$$\{1,1,2,2,3\}=\{1,2,3\}$$

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2. Cartesian product has a formula

$$A=\{1,2\}$$ $$B=\{x,y,z\}$$ $$A\times B = \{ (1,x),(1,y),(1,z), (2,x),(2,y),(2,z) \}$$ $$2\times3=6$$ $$\boxed{|A\times B|=|A|\cdot|B|}$$

Your "1x + 1y + 1z..." intuition

For every element of $A$, pair it with every element of $B$.

$$a\in A$$ $$(a,b)$$ $$b\in B$$ $$4\times7=28$$

Important trap

$$(1,x)\neq(x,1)$$ $$A\times B\neq B\times A$$

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3. Uppercase $A$ vs lowercase $a$

$$A$$ $$a$$ $$A=\{1,2,3\}$$ $$2\in A$$

• $A$ = the entire set
• $2$ = one element inside it

$$a\in A$$

$a$ is some element belonging to $A$.

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4. What is $A'$?

$$A'$$

A prime

$$A'$$ $$A^c$$ $$A'$$ $$A'$$

everything outside $A$ within the universal set.

$$U=\{1,2,3,4,5\}$$ $$A=\{1,2,3\}$$ $$A'=\{4,5\}$$

This is why context matters

"$A'$ always means complement."

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5. "If not injective or surjective, is it not a function?"

$$\boxed{\text{Every input has exactly one output}}$$

---

Function test

Does every input have exactly one output?

Is it injective?

Is it surjective?

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Example 1 — Function but not injective

$$1\rightarrow a$$ $$2\rightarrow a$$ $$3\rightarrow b$$ $$1\rightarrow a$$ $$2\rightarrow a$$

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Example 2 — Function but not surjective

$$f:\{1,2,3\}\rightarrow\{a,b,c,d\}$$ $$1\rightarrow a$$ $$2\rightarrow b$$ $$3\rightarrow a$$

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Example 3 — Function, neither injective nor surjective

$$1\rightarrow a$$ $$2\rightarrow a$$ $$3\rightarrow b$$ $$\{a,b,c,d\}$$

• Function? Yes
• Injective? No
• Surjective? No
• Bijective? No

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6. The hierarchy you should memorize

function → injective → surjective

text
                  FUNCTION
                     │
       ┌─────────────┴─────────────┐
       │                           │
   Injective?                  Surjective?
       │                           │
       └─────────────┬─────────────┘
                     │
              Both = Bijective

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7. One correction to my previous teaching approach

Symbol → pronunciation → meaning → example → distinction from similar symbols

$$\in$$ $$2\in A$$

2 is an element of $A$.

$$\subseteq$$ $$B\subseteq A$$

every element of $B$ is also an element of $A$.

$$2\in A$$ $$\{2\}\subseteq A$$

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Moving on

$$\boxed{\text{What it means}} \rightarrow \boxed{\text{When to use it}} \rightarrow \boxed{\text{Why it works}} \rightarrow \boxed{\text{Pattern}} \rightarrow \boxed{\text{Examples}}$$

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You

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ChatGPT

Maths 1 — Week 2

Coordinate System & Straight Lines

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1. The Week 2 mental model

$$\boxed{\text{Points}} \rightarrow \boxed{\text{Distance}} \rightarrow \boxed{\text{Slope}} \rightarrow \boxed{\text{Line equation}} \rightarrow \boxed{\text{Parallel / perpendicular}}$$

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2. Cartesian Coordinate System

$$(x,y)$$

What do they mean?

• $x$ = horizontal coordinate
• $y$ = vertical coordinate

$$P=(3,5)$$

Move 3 units horizontally and 5 units vertically.

---

2.1 Origin

$$(0,0)$$

---

3. Four Quadrants

$$(3,4)$$ $$(-3,4)$$ $$(-3,-4)$$ $$(3,-4)$$

Pattern

Look only at the signs.

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4. Distance Between Two Points

$$P_1=(x_1,y_1)$$ $$P_2=(x_2,y_2)$$ $$\boxed{ d= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }$$

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Why?

$$\Delta x=x_2-x_1$$ $$\Delta y=y_2-y_1$$ $$d^2=(\Delta x)^2+(\Delta y)^2$$ $$d=\sqrt{(\Delta x)^2+(\Delta y)^2}$$

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5. Distance Pattern

$$(1,2),\quad(4,6)$$

Step 1

$$\Delta x=4-1=3$$

Step 2

$$\Delta y=6-2=4$$

Step 3

$$d=\sqrt{3^2+4^2}$$ $$d=5$$

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6. Slope

How much $y$ changes when $x$ changes.

$$m$$ $$\boxed{ m=\frac{y_2-y_1}{x_2-x_1} }$$ $$m=\frac{\Delta y}{\Delta x}$$

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7. What Does Slope Actually Mean?

$$m=2$$

For every increase of $1$ in $x$, $y$ increases by $2$.

$$\Delta x=1 \Rightarrow \Delta y=2$$ $$m=\frac12$$

increase $x$ by $2$ → increase $y$ by $1$.

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8. Positive vs Negative Slope

Positive

$$m>0$$

text
      /
     /
    /
   /

Negative

$$m<0$$

text
\
 \
  \
   \

Zero

$$m=0$$

text
────────────

Undefined

text
│
│
│
│

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9. Why Is Vertical Slope Undefined?

$$x_2=x_1$$ $$x_2-x_1=0$$ $$\frac{\Delta y}{0}$$ $$\boxed{\text{vertical line → undefined slope}}$$

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10. Slope Pattern

$$P_1=(2,3)$$ $$P_2=(5,9)$$ $$m= \frac{9-3}{5-2}$$ $$m=\frac63=2$$

$y$ rises by 2 for every 1-unit increase in $x$.

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11. Equation of a Straight Line

$$\boxed{y=mx+c}$$

• $m$ = slope
• $c$ = $y$-intercept

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12. What Is the $y$-intercept?

$$x=0$$ $$y=mx+c$$ $$x=0$$ $$y=c$$ $$\boxed{c=\text{$y$-intercept}}$$

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13. Example

$$y=3x+2$$ $$m=3$$ $$c=2$$

• slope = $3$
• $y$-intercept = $2$

$$(0,2)$$

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14. Finding a Line from a Point + Slope

Find the line with slope $3$ passing through $(2,5)$.

$$y=mx+c$$ $$y=3x+c$$ $$5=3(2)+c$$ $$5=6+c$$ $$c=-1$$ $$\boxed{y=3x-1}$$

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15. Point-Slope Form

• slope $m$
• point $(x_1,y_1)$

$$\boxed{ y-y_1=m(x-x_1) }$$ $$y-5=3(x-2)$$ $$y-5=3x-6$$ $$y=3x-1$$

Pattern

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16. Finding a Line from Two Points

$$(1,2)$$ $$(4,8)$$

Step 1 — Find slope

$$m=\frac{8-2}{4-1}$$ $$m=2$$

Step 2 — Use either point

$$y-2=2(x-1)$$

Step 3 — Simplify

$$y-2=2x-2$$ $$\boxed{y=2x}$$

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17. General Form of a Line

$$\boxed{Ax+By+C=0}$$ $$2x+3y-6=0$$ $$3y=-2x+6$$ $$y=-\frac23x+2$$ $$m=-\frac23$$ $$c=2$$

Pattern

Rearrange into $y=mx+c$.

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18. Parallel Lines

$$m_1=m_2$$ $$y=2x+3$$ $$y=2x-7$$ $$m=2$$ $$\boxed{\text{parallel}}$$

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19. Perpendicular Lines

$$\boxed{m_1m_2=-1}$$ $$m_2=-\frac1{m_1}$$ $$m_1=2$$ $$m_2=-\frac12$$ $$2\left(-\frac12\right)=-1$$

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20. The "Negative Reciprocal" Pattern

flip the fraction and change the sign.

$$3\rightarrow-\frac13$$ $$\frac25\rightarrow-\frac52$$ $$-\frac47\rightarrow\frac74$$ $$m=0$$

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21. Parallel vs Perpendicular — Don't Mix Them

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22. A More Interesting Pattern

Find the equation of the line passing through $(2,3)$ and perpendicular to $y=4x-7$.

Step 1 — Identify existing slope

$$m_1=4$$

Step 2 — Perpendicular slope

$$m_2=-\frac14$$

Step 3 — We have point + slope

$$y-y_1=m(x-x_1)$$ $$y-3=-\frac14(x-2)$$

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23. Another Pattern: Is a Point on a Line?

$$y=2x+3$$

Is $(4,11)$ on the line?

$$x=4$$ $$2(4)+3=11$$ $$\boxed{\text{Yes}}$$

General pattern

Substitute $x=a$ and see whether the result equals $b$.

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24. Week 2 Pattern Bank — First Version

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25. One subtle thing to understand

$$\boxed{\text{Two points}} \rightarrow \boxed{\text{differences}} \rightarrow \boxed{\text{slope}} \rightarrow \boxed{\text{line}}$$ $$(x_1,y_1),(x_2,y_2)$$ $$\Delta x=x_2-x_1$$ $$\Delta y=y_2-y_1$$ $$m=\frac{\Delta y}{\Delta x}$$ $$y-y_1=m(x-x_1)$$

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26. Quick check before we continue

A

$$m=0$$

B

$$m=3$$

C

$$y=5x+1$$ $$y=5x-100$$

D

$$(2,5)$$ $$(6,13)$$

E

F — pattern recognition

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You

• what is mx is it mdelta and delta is kewl way to remember y2-y1 typeshit thing else it would be insanely fucked memory if not practiced enough yk nice shortcut and pythagorus parallel or analogy or is m = slope & x = x lol ykwhat i mean
• how did quoted thing became c=2 & m= -2/ & y=-2/3x+2 mun is it something tacit thing or normla stuff where did x & +2 went & i didnt got parallel lines quesion either ig i got it but y =mc+c slope same but could c affect parallel
• solveall the example if u want before starting next topic also start next topic in current outptu thingy onyl btw i got the theory but practice would make it concreate in head yk

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ChatGPT

Maths 1 — Week 2 Patch + Week 3 Start

1. First: what the hell is $mx$?

$$y=mx+c$$ $$mx=m\times x$$

• $m$ = slope
• $x$ = $x$-coordinate / input
• $c$ = $y$-intercept

$$m=3$$ $$y=3x+2$$ $$y=3\times x+2$$

Important notation distinction

$$mx$$ $$m\times x$$ $$\Delta x$$

change in $x$.

$$\boxed{\Delta x=x_2-x_1}$$ $$\boxed{\Delta y=y_2-y_1}$$ $$m=\frac{y_2-y_1}{x_2-x_1}$$ $$\boxed{m=\frac{\Delta y}{\Delta x}}$$

slope = change in $y$ / change in $x$

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2. Your Pythagoras analogy is actually useful

$$d=\sqrt{(\Delta x)^2+(\Delta y)^2}$$ $$\boxed{\text{horizontal change}} \quad+\quad \boxed{\text{vertical change}} \quad\rightarrow\quad \boxed{\text{diagonal distance}}$$ $$a^2+b^2=c^2$$ $$a=\Delta x$$ $$b=\Delta y$$ $$c=d$$

"Ah, two coordinate differences are the perpendicular sides of a right triangle."

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3. Now the confusing $2x+3y-6=0$ example

$$2x+3y-6=0$$ $$y=-\frac23x+2$$ $$2x+3y-6=0$$

Step 1 — Move $2x$ to the other side

$$3y-6=-2x$$

Step 2 — Move $-6$

$$3y=-2x+6$$

Step 3 — Divide EVERYTHING by $3$

$$\frac{3y}{3} = \frac{-2x+6}{3}$$ $$y=-\frac23x+\frac63$$ $$\frac63=2$$ $$\boxed{y=-\frac23x+2}$$

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Why did the $x$ "go"?

$$-\frac23x$$

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Why did $+2$ appear?

$$6\div3=2$$ $$2x+3y-6=0$$ $$3y=-2x+6$$ $$y=-\frac23x+2$$

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4. Why does this tell us the slope?

$$y=mx+c$$ $$y=-\frac23x+2$$ $$\boxed{m=-\frac23}$$ $$\boxed{c=2}$$

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5. And yes — $c$ can absolutely be different for parallel lines

$$y=5x+1$$ $$y=5x-100$$ $$m=5$$ $$c_1=1$$ $$c_2=-100$$

Why doesn't $c$ affect parallelness?

$$\boxed{m=\text{tilt}}$$ $$\boxed{c=\text{vertical position}}$$ $$y=2x$$ $$y=2x+5$$ $$y=2x-100$$

---

Visual mental model

$$\text{same }m,\quad\text{different }c$$ $$\boxed{\text{parallel lines have the same slope}}$$ $$\boxed{\text{different intercepts}}$$

Important edge case

$$y=2x+3$$ $$y=2x+3$$

---

6. Let's solve all the Week 2 examples

Example 1 — Distance

$$(1,2),(4,6)$$

Differences

$$\Delta x=4-1=3$$ $$\Delta y=6-2=4$$

Pythagoras

$$d=\sqrt{(\Delta x)^2+(\Delta y)^2}$$ $$d=\sqrt{3^2+4^2}$$ $$d=\sqrt{9+16}$$ $$\boxed{d=5}$$

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Example 2 — Slope

$$(2,3),(5,9)$$ $$m=\frac{\Delta y}{\Delta x}$$ $$\Delta y=9-3=6$$ $$\Delta x=5-2=3$$ $$m=\frac63$$ $$\boxed{m=2}$$

$y$ increases by $2$ for every $1$ increase in $x$.

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Example 3 — Point + slope

$$m=3$$ $$(2,5)$$ $$y-y_1=m(x-x_1)$$ $$y-5=3(x-2)$$ $$y-5=3x-6$$ $$\boxed{y=3x-1}$$ $$y=3(2)-1=5$$

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Example 4 — Two points → line

$$(1,2),(4,8)$$

Step 1: slope

$$\Delta y=8-2=6$$ $$\Delta x=4-1=3$$ $$m=2$$

Step 2: point-slope

$$y-2=2(x-1)$$

Step 3: simplify

$$y-2=2x-2$$ $$\boxed{y=2x}$$

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Example 5 — Parallel line

$$y=2x+7$$ $$(3,4)$$

Step 1

$$m=2$$ $$m_{\text{new}}=2$$

Step 2

$$y-4=2(x-3)$$

Step 3

$$y-4=2x-6$$ $$\boxed{y=2x-2}$$ $$y=2x+7$$ $$y=2x-2$$

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Example 6 — Perpendicular line

$$y=4x-7$$ $$(2,3)$$

Step 1

$$m_1=4$$

Step 2

$$m_2=-\frac14$$

Step 3

$$y-3=-\frac14(x-2)$$ $$y-3=-\frac14x+\frac12$$ $$y=-\frac14x+\frac72$$ $$\boxed{y=-\frac14x+\frac72}$$ $$4\left(-\frac14\right)=-1$$

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Example 7 — Is the point on the line?

$$y=2x+3$$ $$(4,11)$$ $$y=2(4)+3$$ $$y=8+3$$ $$y=11$$ $$\boxed{(4,11)\text{ lies on the line}}$$

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7. Week 2 — The compact mental toolkit

Coordinates

$$(x,y)$$

Change

$$\Delta x=x_2-x_1$$ $$\Delta y=y_2-y_1$$

Distance

$$d=\sqrt{(\Delta x)^2+(\Delta y)^2}$$

Slope

$$\boxed{m=\frac{\Delta y}{\Delta x}}$$

Line

$$\boxed{y=mx+c}$$

Point + slope

$$\boxed{y-y_1=m(x-x_1)}$$

Parallel

$$\boxed{m_1=m_2}$$

Perpendicular

$$\boxed{m_1m_2=-1}$$ $$\boxed{m=\text{tilt}}$$ $$\boxed{c=\text{vertical position}}$$

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Week 3 — Quadratic Functions

$$\boxed{\text{Linear}} \rightarrow \boxed{\text{Quadratic}}$$ $$y=mx+c$$ $$\boxed{f(x)=ax^2+bx+c}$$ $$a\neq0$$

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8. What makes something quadratic?

$$x^2+3x+2$$ $$2x^2-7x+1$$ $$-5x^2+4$$ $$3x+2$$ $$x^3+2x$$

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9. Why does $a$ matter so much?

$$f(x)=ax^2+bx+c$$

If:

$$a>0$$

text
\       /
 \     /
  \___/

If:

$$a<0$$

text
  /‾‾‾\
 /     \
/       \

$$\boxed{a>0\rightarrow\text{cup}}$$ $$\boxed{a<0\rightarrow\text{cap}}$$

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10. Quadratic roots

$$f(x)=0$$ $$f(x)=x^2-5x+6$$ $$x^2-5x+6=0$$ $$(x-2)(x-3)=0$$ $$x=2$$ $$x=3$$ $$\boxed{2,3}$$

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11. Why does factoring work?

$$(x-2)(x-3)=0$$ $$x-2=0$$ $$x-3=0$$ $$x=2,3$$

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12. Quadratic Formula

$$\boxed{ x= \frac{-b\pm\sqrt{b^2-4ac}}{2a} }$$ $$ax^2+bx+c=0$$ $$\boxed{\Delta=b^2-4ac}$$

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13. What does the discriminant tell us?

$\Delta>0$

$\Delta=0$

$\Delta<0$

$$\boxed{ \begin{array}{c|c} \Delta>0 & 2\text{ real roots}\\ \Delta=0 & 1\text{ repeated real root}\\ \Delta<0 & 0\text{ real roots} \end{array} }$$

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14. Example

$$x^2-4x+3=0$$ $$a=1,\quad b=-4,\quad c=3$$ $$\Delta=(-4)^2-4(1)(3)$$ $$=16-12$$ $$=4$$ $$\Delta>0$$ $$x^2-4x+3=(x-1)(x-3)$$ $$x=1,3$$

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15. Vertex

$$\boxed{x_v=-\frac{b}{2a}}$$ $$\boxed{ \text{vertex} = \left( -\frac{b}{2a}, f\left(-\frac{b}{2a}\right) \right) }$$

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16. Example — Vertex

$$f(x)=x^2-4x+3$$ $$a=1,\quad b=-4$$ $$x_v=-\frac{-4}{2(1)}$$ $$x_v=2$$ $$f(2)=2^2-4(2)+3$$ $$=4-8+3$$ $$=-1$$ $$\boxed{(2,-1)}$$

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17. The Week 3 pattern map begins

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One thing I want you to notice

Week 2

$$\text{equation} \leftrightarrow \text{graph} \leftrightarrow \text{geometric meaning}$$

Week 3

$$\text{quadratic equation} \leftrightarrow \text{parabola} \leftrightarrow \text{roots} \leftrightarrow \text{vertex}$$

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You

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ChatGPT

What does $y$ actually do?

$$\boxed{y=mx+c}$$ $$x\quad\longrightarrow\quad\boxed{mx+c}\quad\longrightarrow\quad y$$ $$y=2x+3$$ $$x=0$$ $$y=2(0)+3=3$$ $$(0,3)$$ $$x=1$$ $$y=2(1)+3=5$$ $$(1,5)$$ $$x=2$$ $$y=2(2)+3=7$$ $$(2,7)$$

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So what does each part control?

$$\boxed{y=mx+c}$$

$x$

$m$

$$m=\frac{\Delta y}{\Delta x}$$ $$m=2$$ $$\Delta x=1 \Rightarrow \Delta y=2$$

$c$

$$y=m(0)+c=c$$ $$\boxed{c=y\text{-intercept}}$$

$y$

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The really useful mental model

$$x=0$$ $$y=c$$ $$y=2x+3$$ $$(0,3)$$ $$+1\text{ in }x \Rightarrow +2\text{ in }y$$ $$(0,3) \rightarrow (1,5) \rightarrow (2,7) \rightarrow (3,9)$$

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Why changing $c$ moves the line up/down

$$y=2x$$ $$y=2x+5$$ $$y=2x-3$$ $$m=2$$

First

$$x=0\Rightarrow y=0$$ $$(0,0)$$

Second

$$x=0\Rightarrow y=5$$ $$(0,5)$$

Third

$$x=0\Rightarrow y=-3$$ $$(0,-3)$$

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And this explains parallel lines

$$y=2x+1$$ $$y=2x+100$$ $$(2x+100)-(2x+1)=99$$ $$m_1=m_2=2$$ $$\boxed{\text{same slope + different intercept}=\text{distinct parallel lines}}$$

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One more subtle thing: $y$ isn't "controlled by $c$"

"$c$ controls $y$."

$$\boxed{y=mx+c}$$ $$y=2x+3$$

• $2x$ → contribution based on where you are horizontally
• $3$ → fixed vertical offset
• $y$ → total vertical position

$$\underbrace{2x}_{\text{variable contribution}} + \underbrace{3}_{\text{constant offset}} = \underbrace{y}_{\text{result}}$$

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And yeah, your Week 3 status

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You

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ChatGPT

Maths 1 — Week 4

Algebra of Polynomials

$$\boxed{\text{terms}} \rightarrow \boxed{\text{polynomial}} \rightarrow \boxed{\text{operations}} \rightarrow \boxed{\text{roots}} \rightarrow \boxed{\text{factorization}} \rightarrow \boxed{\text{graph}}$$

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1. What exactly is a polynomial?

$$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_2x^2+a_1x+a_0$$ $$3x^2+2x+7$$ $$5x^4-2x^3+x-9$$

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2. Parts of a Polynomial

$$f(x)=4x^3-7x^2+2x-9$$ $$4x^3,\quad -7x^2,\quad 2x,\quad -9$$

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3. Degree

$$4x^3-7x^2+2x-9$$ $$3$$ $$\boxed{\deg(f)=3}$$

Examples

$$7x^5+3x^2-1$$ $$9x^2+4x+1$$ $$3x+7$$ $$8$$

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4. Polynomial Classification by Degree

$$\boxed{\text{degree tells you the highest power}}$$

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5. What is NOT a polynomial?

Negative powers

$$x^{-1}$$

Fractional powers

$$x^{1/2}$$

Variable in denominator

$$\frac1x$$

Variable exponent

$$2^x$$ $$0,1,2,3,\ldots$$

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6. Adding Polynomials

$$f(x)=3x^2+2x+1$$ $$g(x)=5x^2-4x+7$$ $$f(x)+g(x)$$ $$(3x^2+2x+1)+(5x^2-4x+7)$$ $$(3x^2+5x^2)+(2x-4x)+(1+7)$$ $$\boxed{8x^2-2x+8}$$

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7. What are "like terms"?

$$3x^2,\quad7x^2$$ $$3x^2,\quad7x$$ $$3x^2,\quad7x^3$$ $$3x^2+5x^2=8x^2$$ $$3x^2+5x$$

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8. Subtracting Polynomials

$$f(x)=3x^2+2x+1$$ $$g(x)=5x^2-4x+7$$ $$f(x)-g(x)$$ $$(3x^2+2x+1)-(5x^2-4x+7)$$ $$3x^2+2x+1-5x^2+4x-7$$ $$\boxed{-2x^2+6x-6}$$

Common trap

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9. Multiplying Polynomials

$$(x+2)(x+3)$$ $$x(x+3)+2(x+3)$$ $$=x^2+3x+2x+6$$ $$\boxed{x^2+5x+6}$$

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10. Why distribution works

$$(a+b)(c+d)$$ $$a(c+d)+b(c+d)$$ $$ac+ad+bc+bd$$ $$\boxed{(a+b)(c+d)=ac+ad+bc+bd}$$

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11. Polynomial Evaluation

$$f(x)=2x^3-3x+5$$ $$f(2)$$

Replace every $x$ with $2$.

$$f(2)=2(2)^3-3(2)+5$$ $$=2(8)-6+5$$ $$=16-6+5$$ $$\boxed{15}$$

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12. A Very Important Connection

$$f(a)=0$$ $$f(x)=x^2-5x+6$$ $$f(2)=4-10+6=0$$ $$\boxed{2\text{ is a root}}$$ $$f(3)=9-15+6=0$$ $$3$$

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13. Factor Theorem

$$\boxed{ f(a)=0 \iff (x-a)\text{ is a factor of }f(x) }$$ $$f(x)=x^2-5x+6$$ $$f(2)=0$$ $$(x-2)$$ $$x^2-5x+6=(x-2)(x-3)$$

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14. Root ↔ Factor ↔ $x$-intercept

$$f(a)=0$$

Algebra

Factorization

Graph

$$\boxed{ \text{root} \leftrightarrow \text{factor} \leftrightarrow \text{$x$-intercept} }$$

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15. Example

$$f(x)=x^3-6x^2+11x-6$$ $$x=1$$ $$f(1)=1-6+11-6=0$$ $$x=1$$ $$(x-1)$$ $$\boxed{ (x-1)(x-2)(x-3) }$$ $$1,2,3$$ $$(1,0),(2,0),(3,0)$$

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16. Polynomial Division

$$(x-a)$$ $$x^2-5x+6$$ $$x-2$$ $$x-3$$ $$x^2-5x+6=(x-2)(x-3)$$ $$\boxed{ \text{find one root} \rightarrow \text{find one factor} \rightarrow \text{divide} \rightarrow \text{reduce degree} }$$

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17. Synthetic Division — The Intuition

$$f(x)=x^3-6x^2+11x-6$$ $$x=1$$ $$x-1$$ $$\boxed{ \text{cubic} \rightarrow \text{quadratic} }$$

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18. Remainder Theorem

$$x-a$$ $$\boxed{f(a)}$$ $$f(2)=7$$ $$x-2$$ $$7$$ $$f(2)=0$$ $$x-2$$

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19. Multiplicity

$$f(x)=(x-2)^2(x+1)$$ $$x=2$$ $$x=-1$$

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20. Why multiplicity matters

$$f(x)=(x-2)^2$$ $$x=2$$ $$(x-2)^2\ge0$$

---

$$f(x)=x-2$$

Odd multiplicity

Even multiplicity

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21. Polynomial End Behaviour

$$f(x)=2x^3-7x+1$$ $$2x^3$$

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22. Even Degree

$$f(x)=x^4+\cdots$$ $$x^4>0$$ $$\boxed{\text{both ends up}}$$ $$\boxed{\text{both ends down}}$$

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23. Odd Degree

$$f(x)=x^3+\cdots$$ $$x\to+\infty \Rightarrow x^3\to+\infty$$ $$x\to-\infty \Rightarrow x^3\to-\infty$$

Positive leading coefficient

Negative leading coefficient

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24. End-Behaviour Cheat Pattern

$$\boxed{\text{degree parity + sign of leading coefficient}}$$

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25. Constructing a Polynomial from Roots

$$x=2,\quad x=5$$ $$(x-2),(x-5)$$ $$f(x)=k(x-2)(x-5)$$ $$f(0)=10$$ $$10=k(0-2)(0-5)$$ $$10=10k$$ $$k=1$$ $$\boxed{f(x)=(x-2)(x-5)}$$ $$\boxed{f(x)=x^2-7x+10}$$

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26. Why do we need $k$?

$$f(x)=(x-2)(x-5)$$ $$g(x)=7(x-2)(x-5)$$ $$\boxed{\text{roots determine factors, but not necessarily the coefficient}}$$

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27. Week 4 Pattern Bank

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28. The big Week 4 mental model

$$\boxed{ \text{Polynomial} }$$ $$\boxed{ \text{expression} \leftrightarrow \text{factors} \leftrightarrow \text{roots} \leftrightarrow \text{graph} }$$ $$f(x)=(x-1)^2(x+3)$$

Roots

$$x=1,\quad x=-3$$

Multiplicity

Graph

Degree

$$2+1=3$$

Leading coefficient

$$x\to-\infty\Rightarrow f(x)\to-\infty$$ $$x\to+\infty\Rightarrow f(x)\to+\infty$$

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Where we are

$$\boxed{\text{W1 ✓}} \rightarrow \boxed{\text{W2 ✓}} \rightarrow \boxed{\text{W3 ✓}} \rightarrow \boxed{\text{W4 ✓}} \rightarrow \text{W5}$$

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You

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ChatGPT

Maths 1 — Week 5

Functions

$$\boxed{\text{Function}} \rightarrow \boxed{\text{Domain/Range}} \rightarrow \boxed{\text{Graph}} \rightarrow \boxed{\text{One-to-one}} \rightarrow \boxed{\text{Composition}} \rightarrow \boxed{\text{Inverse}}$$

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1. What is a function, really?

$$f:A\to B$$

$f$ takes inputs from $A$ and produces outputs in $B$.

$$\boxed{x\rightarrow f\rightarrow f(x)}$$ $$f(x)=2x+3$$ $$x=4$$ $$f(4)=2(4)+3=11$$ $$4\rightarrow11$$

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2. $f(x)$ Does NOT Mean $f\times x$

$$f(x)$$

the output of function $f$ when the input is $x$.

$$f\times x$$ $$f(x)=x^2+1$$ $$f(3)=3^2+1=10$$

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3. Domain

$$f(x)=\frac1x$$ $$x=0$$ $$\frac10$$ $$\boxed{\text{Domain}=\mathbb R\setminus\{0\}}$$

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4. Range

$$f(x)=x^2$$ $$\mathbb R$$ $$x^2\ge0$$ $$\boxed{\text{Range}=[0,\infty)}$$

Domain asks what can go in?

Range asks what can come out?

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5. Domain Restrictions — Pattern Recognition

Pattern A — Denominator

$$f(x)=\frac{1}{x-3}$$ $$x-3\neq0$$ $$x\neq3$$

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Pattern B — Even root

$$f(x)=\sqrt{x-2}$$ $$x-2\ge0$$ $$\boxed{x\ge2}$$

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Pattern C — Logarithm

$$f(x)=\log(x-4)$$ $$x-4>0$$ $$\boxed{x>4}$$

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6. Vertical Line Test

Every vertical line intersects the graph at most once.

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Example

$$y=x^2$$ $$x=y^2$$ $$x=4$$ $$y^2=4$$ $$y=2$$ $$y=-2$$ $$x=4$$

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7. Horizontal Line Test

Injective = different inputs cannot produce the same output.

$$\boxed{\text{not one-to-one}}$$

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8. Vertical vs Horizontal Line Test

Mental shortcut

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9. Example: $f(x)=x^2$

$$f(x)=x^2$$ $$f(2)=4$$ $$f(-2)=4$$ $$\boxed{\text{function but not injective}}$$

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10. Restricting the Domain Can Change Injectivity

$$\mathbb R$$ $$f(x)=x^2$$ $$x\ge0$$ $$2,-2$$

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11. Composition of Functions

$$f(x)=2x+1$$ $$g(x)=x^2$$ $$(f\circ g)(x)$$

Apply $g$ first, then apply $f$.

$$(f\circ g)(x)=f(g(x))$$ $$g(x)=x^2$$ $$f(g(x))=f(x^2)$$ $$f(t)=2t+1$$ $$\boxed{(f\circ g)(x)=2x^2+1}$$

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12. Composition Order Matters

$$(f\circ g)(x)=2x^2+1$$ $$(g\circ f)(x)$$ $$g(f(x))$$ $$f(x)=2x+1$$ $$g(2x+1)=(2x+1)^2$$ $$\boxed{(g\circ f)(x)=(2x+1)^2}$$ $$\boxed{f\circ g\neq g\circ f}$$

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13. Composition Mental Model

"Which formula do I multiply?"

$$x \xrightarrow{g} g(x) \xrightarrow{f} f(g(x))$$ $$\boxed{f\circ g=\text{g first, f second}}$$

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14. Exponential Functions

$$\boxed{f(x)=a^x}$$ $$a>0,\qquad a\neq1$$ $$2^x,\quad3^x,\quad\left(\frac12\right)^x$$

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15. Why Exponentials Are Special

$$f(x)=2x$$ $$f(x)=2^x$$ $$2^0=1$$ $$2^1=2$$ $$2^2=4$$ $$2^3=8$$ $$2^4=16$$ $$\boxed{\text{output doubles}}$$

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16. Growth vs Decay

If:

$$a>1$$ $$a^x$$ $$2^x$$

If:

$$0<a<1$$ $$\left(\frac12\right)^x$$ $$\boxed{ a>1\rightarrow\text{growth} }$$ $$\boxed{ 0<a<1\rightarrow\text{decay} }$$

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17. Important Exponential Properties

Same base, multiply

$$a^x a^y=a^{x+y}$$

Divide

$$\frac{a^x}{a^y}=a^{x-y}$$

Power of a power

$$(a^x)^y=a^{xy}$$

Zero exponent

$$a^0=1$$

Negative exponent

$$a^{-x}=\frac1{a^x}$$

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18. Exponential Equations

$$2^x=8$$ $$8=2^3$$ $$2^x=2^3$$ $$\boxed{x=3}$$

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Another example

$$3^{2x}=27$$ $$27=3^3$$ $$3^{2x}=3^3$$ $$2x=3$$ $$\boxed{x=\frac32}$$

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19. Inverse Functions

$$f(x)=2x+3$$ $$x \rightarrow 2x \rightarrow 2x+3$$ $$y \rightarrow y-3 \rightarrow \frac{y-3}{2}$$ $$f^{-1}(x)=\frac{x-3}{2}$$

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20. Why Does an Inverse Require Injectivity?

$$f(1)=4$$ $$f(2)=4$$

What is $f^{-1}(4)$?

$$1$$ $$2?$$ $$\boxed{\text{injectivity is required for an inverse function}}$$

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21. Finding an Inverse — Procedure

$$y=2x+3$$

Step 1 — Start with $y=f(x)$

$$y=2x+3$$

Step 2 — Swap $x$ and $y$

$$x=2y+3$$

Step 3 — Solve for $y$

$$x-3=2y$$ $$y=\frac{x-3}{2}$$ $$\boxed{f^{-1}(x)=\frac{x-3}{2}}$$

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22. Verify the Inverse

$$f(x)=2x+3$$ $$f^{-1}(x)=\frac{x-3}{2}$$ $$f^{-1}(f(x))$$ $$f(x)=2x+3$$ $$f^{-1}(2x+3) = \frac{(2x+3)-3}{2}$$ $$=\frac{2x}{2}$$ $$=x$$ $$\boxed{f^{-1}(f(x))=x}$$ $$f(f^{-1}(x))=x$$

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23. Graph of an Inverse

$$\boxed{y=x}$$ $$(x,y)$$ $$(y,x)$$ $$\boxed{ f\leftrightarrow f^{-1} \text{ is reflection across }y=x }$$

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24. Week 5 Pattern Bank

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25. The big Week 5 connection

$$\boxed{\text{Function}}$$

Input-output perspective

$$x\rightarrow f(x)$$

Set perspective

$$f:A\to B$$

Graph perspective

$$y=f(x)$$

Composition perspective

$$x\rightarrow g(x)\rightarrow f(g(x))$$

Inverse perspective

$$x\rightarrow f(x)\rightarrow f^{-1}(f(x))=x$$

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Current Maths 1 progress

$$\boxed{W1\ ✓} \rightarrow \boxed{W2\ ✓} \rightarrow \boxed{W3\ ✓} \rightarrow \boxed{W4\ ✓} \rightarrow \boxed{W5\ ✓} \rightarrow W6$$ $$\boxed{\text{exponential}\leftrightarrow\text{logarithm}}$$

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You

• is f(t) in it t placeholder or variable typeshit for g yk
• now next week mun

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ChatGPT

$$f(t)=2t+1$$

take the input → multiply by $2$ → add $1$.

$$f(x)=2x+1$$ $$f(t)=2t+1$$ $$f(z)=2z+1$$ $$f(5)=2(5)+1=11$$

Why did I use $t$?

$$f(t)=2t+1$$ $$g(x)=x^2$$ $$f(t)=2t+1$$ $$f(g(x))$$ $$f(g(x)) = 2(g(x))+1$$ $$g(x)=x^2$$ $$\boxed{f(g(x))=2x^2+1}$$ $$\boxed{x,t,z,u,\ldots=\text{just variable/placeholder names}}$$

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Maths 1 — Week 6

Logarithmic Functions

$$\boxed{ b^x=y \iff \log_b(y)=x }$$

Exponentiation asks: "What number do I get?"
Logarithm asks: "What exponent did I need?"**

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1. What is a logarithm?

$$2^3=8$$ $$\log_2(8)=3$$

"log base 2 of 8 equals 3."

$$2^3=8$$ $$\boxed{\log_b(x)=y}$$ $$\boxed{b^y=x}$$

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2. The three pieces

$$\log_b(x)=y$$

• $b$ = base
• $x$ = argument
• $y$ = result/exponent

$$\log_b(x)=y \quad\Longleftrightarrow\quad b^y=x$$

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3. Examples

Example 1

$$\log_2(8)$$

$2$ raised to what power gives $8$?

$$2^3=8$$ $$\boxed{\log_2(8)=3}$$

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Example 2

$$\log_5(25)$$

$5$ raised to what power gives $25$?

$$5^2=25$$ $$\boxed{\log_5(25)=2}$$

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Example 3

$$\log_{10}(1000)$$ $$10^3=1000$$ $$\boxed{3}$$

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4. Why logarithms are inverse functions

$$f(x)=2^x$$ $$3\rightarrow8$$ $$f^{-1}(x)=\log_2x$$ $$8\rightarrow3$$ $$\boxed{ 2^3=8 \iff \log_2(8)=3 }$$

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5. Domain of a logarithm

$$\log_b(x)$$ $$\boxed{x>0}$$ $$0$$ $$\log_2(-4)$$

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6. What about the base?

$$\boxed{b>0,\qquad b\neq1}$$ $$\log_2x$$ $$\log_{10}x$$ $$\log_{\frac12}x$$ $$b=0$$ $$b=1$$

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7. Fundamental Log Rules

Multiplication

$$\boxed{ \log_b(xy)=\log_bx+\log_by }$$ $$\log_2(8\cdot4)$$ $$\log_2(8)+\log_2(4)$$ $$=3+2$$ $$=5$$ $$8\cdot4=32=2^5$$

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8. Division

$$\boxed{ \log_b\left(\frac{x}{y}\right) = \log_bx-\log_by }$$ $$\log_2\left(\frac84\right)$$ $$=\log_2(8)-\log_2(4)$$ $$=3-2$$ $$=1$$

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9. Power Rule

$$\boxed{ \log_b(x^k)=k\log_b(x) }$$ $$\log_2(8^2)$$ $$2\log_2(8)$$ $$=2(3)=6$$ $$8^2=64=2^6$$

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10. A Very Important Non-Rule

$$\log_b(x+y) = \log_bx+\log_by$$ $$\log_b(xy) = \log_bx+\log_by$$ $$\boxed{ \log_b(x+y)\neq\log_bx+\log_by }$$

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11. Exponential → Logarithmic Translation

$$3^4=81$$ $$\boxed{\log_3(81)=4}$$ $$10^{-2}=0.01$$ $$\boxed{\log_{10}(0.01)=-2}$$

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12. Logarithmic → Exponential Translation

$$\log_7(49)=2$$ $$\boxed{7^2=49}$$

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13. Solving a Simple Log Equation

$$\log_2(x)=5$$ $$2^5=x$$ $$\boxed{x=32}$$

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14. Slightly More Interesting

$$\log_3(x-1)=2$$ $$3^2=x-1$$ $$9=x-1$$ $$\boxed{x=10}$$ $$x-1>0$$ $$10-1=9>0$$

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15. Solving an Exponential Equation Using Logs

$$2^x=7$$ $$\log(2^x)=\log7$$ $$x\log2=\log7$$ $$\boxed{ x=\frac{\log7}{\log2} }$$

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16. Change of Base

$$\boxed{ \log_bx= \frac{\log_ax}{\log_ab} }$$ $$\boxed{ \log_bx= \frac{\ln x}{\ln b} }$$ $$\boxed{ \log_bx= \frac{\log x}{\log b} }$$

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17. Why Change of Base Works

$$y=\log_bx$$ $$b^y=x$$ $$\log_a(b^y)=\log_ax$$ $$y\log_ab=\log_ax$$ $$y=\frac{\log_ax}{\log_ab}$$ $$y=\log_bx$$ $$\boxed{ \log_bx=\frac{\log_ax}{\log_ab} }$$

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18. Graph of a Logarithm

$$y=\log_bx$$ $$x>0$$ $$x=0$$ $$0<b<1$$ $$\boxed{y=x}$$

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19. Week 6 Pattern Bank

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20. The key connection

$$\boxed{ \text{Exponentiation} \leftrightarrow \text{Logarithm} }$$ $$\boxed{ \text{Function} \leftrightarrow \text{Inverse function} }$$ $$\boxed{ b^x \quad\overset{\text{inverse}}{\longleftrightarrow}\quad \log_bx }$$

Current map

$$W1\ ✓ \rightarrow W2\ ✓ \rightarrow W3\ ✓ \rightarrow W4\ ✓ \rightarrow W5\ ✓ \rightarrow \boxed{W6\ ✓} \rightarrow W7$$

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You

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ChatGPT

Maths 1 — Week 7

Sequences, Limits & Continuity

Instead of asking "what is the value exactly here?", we start asking "what is the behaviour as we get closer and closer to here?"

$$\boxed{\text{Sequence}} \rightarrow \boxed{\text{Limit}} \rightarrow \boxed{\text{Function limit}} \rightarrow \boxed{\text{Continuity}} \rightarrow \boxed{\text{Tangent intuition}}$$

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1. Sequences

$$1,2,3,4,5,\ldots$$ $$1,\frac12,\frac13,\frac14,\ldots$$ $$a_1,a_2,a_3,\ldots$$

the $n$-th term of the sequence.

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2. $n$ Is Basically the Position

$$a_n=\frac1n$$ $$a_1=1$$ $$a_2=\frac12$$ $$a_3=\frac13$$ $$a_4=\frac14$$ $$n=1,2,3,\ldots$$

$n$ is the index/position variable here.

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3. Sequence as a Function

$$a_n=f(n)$$ $$\mathbb N$$ $$a_n=\frac1n$$ $$1,2,3,\ldots$$ $$1,\frac12,\frac13,\ldots$$

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4. What Is a Limit?

$$a_n=\frac1n$$ $$1,\frac12,\frac13,\frac14,\frac15,\ldots$$ $$a_n\rightarrow0$$ $$\boxed{ \lim_{n\to\infty}\frac1n=0 }$$

The limit of $1/n$ as $n$ approaches infinity is $0$.

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5. But Does the Sequence Ever Become 0?

$$\frac1n>0$$ $$\lim_{n\to\infty}\frac1n=0$$

Limit ≠ necessarily attained value

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6. Another Example

$$a_n=2+\frac1n$$ $$3,\quad2.5,\quad2.333\ldots,\quad2.25,\ldots$$ $$\frac1n\rightarrow0$$ $$2+\frac1n\rightarrow2$$ $$\boxed{ \lim_{n\to\infty} \left(2+\frac1n\right)=2 }$$

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7. Divergence

$$a_n=n$$ $$1,2,3,4,5,\ldots$$ $$a_n=(-1)^n$$ $$-1,1,-1,1,-1,1,\ldots$$

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8. Convergence

$$\frac1n\rightarrow0$$

Mental model

$$\boxed{\text{Converge}=\text{settle toward something}}$$ $$\boxed{\text{Diverge}=\text{doesn't settle toward a finite value}}$$

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9. Basic Limit Patterns

Constant

$$\lim_{n\to\infty}c=c$$

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Reciprocal

$$\lim_{n\to\infty}\frac1n=0$$

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Polynomial growth

$$\lim_{n\to\infty}n^2=\infty$$

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Negative reciprocal

$$\lim_{n\to\infty}-\frac1n=0$$

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10. Function Limits

$$f(x)=x+2$$ $$x\to3?$$ $$f(x)\to5$$ $$\boxed{ \lim_{x\to3}(x+2)=5 }$$

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11. What Does $x\to3$ Mean?

$$x=3$$

$x$ is getting arbitrarily close to $3$.

$$2.9,\quad2.99,\quad2.999,\ldots$$ $$3.1,\quad3.01,\quad3.001,\ldots$$

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12. Direct Substitution

$$\lim_{x\to2}(x^2+3x)$$ $$x=2$$ $$=2^2+3(2)$$ $$=4+6$$ $$\boxed{10}$$

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13. Why Do We Need Limits If We Can Just Substitute?

$$\frac00$$ $$\lim_{x\to2} \frac{x^2-4}{x-2}$$ $$\frac{2^2-4}{2-2} = \frac00$$

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14. Factor and Cancel

$$x^2-4=(x-2)(x+2)$$ $$\frac{x^2-4}{x-2} = \frac{(x-2)(x+2)}{x-2}$$ $$=x+2$$ $$\lim_{x\to2}(x+2)$$ $$=4$$ $$\boxed{ \lim_{x\to2} \frac{x^2-4}{x-2} =4 }$$

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15. The Interesting Part

$$x=2$$ $$\lim_{x\to2} \frac{x^2-4}{x-2}=4$$ $$\boxed{ \text{limit behavior near a point} \neq \text{function value at that point} }$$

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16. Left-Hand Limit

$$x\to a^-$$ $$x\to2^-$$ $$1.9,\;1.99,\;1.999,\ldots$$

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17. Right-Hand Limit

$$x\to a^+$$ $$x\to2^+$$ $$2.1,\;2.01,\;2.001,\ldots$$

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18. When Does a Two-Sided Limit Exist?

$$\lim_{x\to a}f(x)$$ $$\boxed{ \lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) }$$

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19. Example

$$\lim_{x\to2^-}f(x)=3$$ $$\lim_{x\to2^+}f(x)=5$$ $$\boxed{ \lim_{x\to2}f(x) \text{ does not exist} }$$

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20. Continuity

You can pass through $a$ without a break, jump, or hole.

Condition 1 — $f(a)$ exists

Condition 2 — The limit exists

$$\lim_{x\to a}f(x)$$

Condition 3 — They agree

$$\boxed{ \lim_{x\to a}f(x)=f(a) }$$ $$\boxed{ f\text{ is continuous at }a \iff \lim_{x\to a}f(x)=f(a) }$$

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21. The Three-Part Continuity Checklist

$$\boxed{ \begin{aligned} 1.&\quad f(a)\text{ exists}\\ 2.&\quad \lim_{x\to a}f(x)\text{ exists}\\ 3.&\quad \lim_{x\to a}f(x)=f(a) \end{aligned} }$$

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22. Example — Continuous Function

$$f(x)=x^2$$

Function value

$$f(2)=4$$

Limit

$$\lim_{x\to2}x^2=4$$

Compare

$$4=4$$ $$\boxed{\text{continuous at }x=2}$$

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23. Example — Removable Discontinuity

$$f(x)= \begin{cases} x+2,&x\neq2\\ 7,&x=2 \end{cases}$$ $$f(2)=7$$ $$f(x)=x+2$$ $$\lim_{x\to2}f(x)=4$$ $$\lim_{x\to2}f(x)\neq f(2)$$ $$\boxed{\text{not continuous at }2}$$

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24. Piecewise Functions

$$f(x)= \begin{cases} x+1,&x<2\\ 3x-3,&x\ge2 \end{cases}$$ $$x=2$$

Left

$$\lim_{x\to2^-}f(x) = 2+1 = 3$$

Right

$$\lim_{x\to2^+}f(x) = 3(2)-3 = 3$$ $$3$$ $$f(2)=3(2)-3=3$$ $$\boxed{\text{continuous at }2}$$

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25. Tangent Intuition

$$(x,f(x))$$ $$(x+h,f(x+h))$$ $$\frac{f(x+h)-f(x)}{h}$$ $$h\to0$$ $$\boxed{ \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }$$

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26. The Big Week 7 Connection

$$\boxed{\text{Sequence}}$$ $$\boxed{\text{Limit}}$$ $$\boxed{\text{Continuity}}$$ $$\boxed{\text{Tangent}}$$ $$\boxed{\text{Derivative}}$$

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27. Week 7 Pattern Bank

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28. One mental model I'd keep

$$\lim_{x\to a}f(x)$$

Zoom toward $a$ and observe what the outputs are converging toward.

$$\boxed{f(a)}$$ $$\boxed{\lim_{x\to a}f(x)}$$

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Maths 1 progress

$$W1\ ✓ \rightarrow W2\ ✓ \rightarrow W3\ ✓ \rightarrow W4\ ✓ \rightarrow W5\ ✓ \rightarrow W6\ ✓ \rightarrow \boxed{W7\ ✓} \rightarrow W8$$

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You

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ChatGPT

Maths 1 — Week 7 → Week 8

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Part A — Week 7 Exam-Style Patterns

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Pattern W7-EX1 — Direct limit

Find:

$\lim_{x\to2}(x^2+3x-1)$

Recognition

$$=2^2+3(2)-1$$ $$=4+6-1$$ $$\boxed{9}$$

Pattern

$$\boxed{\text{polynomial limit}\rightarrow\text{plug in}}$$

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Pattern W7-EX2 — $0/0$ trap

Evaluate:

$\lim_{x\to3}\frac{x^2-9}{x-3}$

$$\frac{9-9}{3-3} = \frac00$$ $$x^2-9=(x-3)(x+3)$$ $$\frac{x^2-9}{x-3}=x+3$$ $$\lim_{x\to3}(x+3)=6$$ $$\boxed{6}$$

Recognition pattern

$$\boxed{ \text{direct substitution}\rightarrow\frac00 \rightarrow\text{factor/cancel} }$$

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Pattern W7-EX3 — Limit vs function value

$$f(x)= \begin{cases} x+2,&x\neq3\\ 100,&x=3 \end{cases}$$

Find $f(3)$ and $\lim_{x\to3}f(x)$.

Function value

$$\boxed{f(3)=100}$$

Limit

$$x+2$$ $$\lim_{x\to3}f(x)=3+2=5$$ $$\boxed{f(3)=100,\qquad\lim_{x\to3}f(x)=5}$$ $$\boxed{f(a)\neq\lim_{x\to a}f(x)}$$

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Pattern W7-EX4 — Does the limit exist?

$$\lim_{x\to2^-}f(x)=4$$ $$\lim_{x\to2^+}f(x)=7$$

Find $\lim_{x\to2}f(x)$.

$$4\neq7$$ $$\boxed{\text{DNE}}$$

Recognition

$$\boxed{ L^-=L^+ \Rightarrow\text{two-sided limit exists} }$$ $$\boxed{ L^-\neq L^+ \Rightarrow\text{DNE} }$$

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Pattern W7-EX5 — Continuity parameter

$$f(x)= \begin{cases} x+3,&x<2\\ kx,&x\ge2 \end{cases}$$

Step 1 — Left-hand limit

$$\lim_{x\to2^-}f(x)=2+3=5$$

Step 2 — Right-hand limit

$$\lim_{x\to2^+}f(x)=2k$$ $$5=2k$$ $$\boxed{k=\frac52}$$

Recognition

$$\boxed{ \text{left limit} = \text{right limit} = f(a) }$$

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Pattern W7-EX6 — Sequence

Determine:

$\lim_{n\to\infty}\frac{3n+1}{n+5}$

$$\frac{3n+1}{n+5} = \frac{3+\frac1n}{1+\frac5n}$$ $$n\to\infty$$ $$\frac1n\to0$$ $$\boxed{3}$$

Pattern

$$\frac{\text{leading coefficient numerator}} {\text{leading coefficient denominator}}$$ $$\frac31=3$$

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Pattern W7-EX7 — Tangent preparation

$$f(x)=x^2$$ $$(2,f(2))$$ $$(2+h,f(2+h))$$ $$\frac{f(2+h)-f(2)}{h}$$ $$f(2+h)=(2+h)^2$$ $$=4+4h+h^2$$ $$f(2)=4$$ $$\frac{4+4h+h^2-4}{h}$$ $$=\frac{4h+h^2}{h}$$ $$=4+h$$ $$h\to0$$ $$4$$

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Week 7: What I actually want you to recognize

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Part B — Maths 1 Week 8

Derivatives, Tangents & Critical Points

$$\boxed{\text{Average change}} \rightarrow \boxed{\text{Secant}} \rightarrow \boxed{\text{Limit}} \rightarrow \boxed{\text{Tangent}} \rightarrow \boxed{\text{Derivative}} \rightarrow \boxed{\text{Optimization}}$$

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1. Average Rate of Change

$$y=f(x)$$ $$(x_1,f(x_1))$$ $$(x_2,f(x_2))$$ $$\boxed{ \frac{f(x_2)-f(x_1)} {x_2-x_1} }$$

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2. Why Isn't That the Derivative?

$$x_2=x_1+h$$ $$\frac{f(x_1+h)-f(x_1)}{h}$$ $$h\to0$$ $$\boxed{ f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }$$

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3. What Does $f'(x)$ Mean?

$$f'(x)$$

"f prime of x"

Geometric

$$\boxed{\text{slope of tangent}}$$

Physical

$$\boxed{\text{instantaneous rate of change}}$$

Mathematical

$$\boxed{\text{limit of secant slopes}}$$

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4. Example from First Principles

$$f(x)=x^2$$ $$f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}h$$ $$f(x+h)=(x+h)^2$$ $$=x^2+2xh+h^2$$ $$f'(x) = \lim_{h\to0} \frac{x^2+2xh+h^2-x^2}{h}$$ $$= \lim_{h\to0} \frac{2xh+h^2}{h}$$ $$= \lim_{h\to0}(2x+h)$$ $$\boxed{f'(x)=2x}$$

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5. Derivative Rules

Constant

$$\boxed{\frac{d}{dx}[c]=0}$$

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Power Rule

$$\boxed{ \frac{d}{dx}(x^n)=nx^{n-1} }$$ $$\frac{d}{dx}(x^2)=2x$$ $$\frac{d}{dx}(x^5)=5x^4$$ $$\frac{d}{dx}(x^{10})=10x^9$$

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6. Constant Multiple

$$\frac{d}{dx}[cf(x)] = cf'(x)$$ $$f(x)=7x^3$$ $$f'(x)=7(3x^2)$$ $$\boxed{21x^2}$$

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7. Sum Rule

$$\boxed{ (f+g)'=f'+g' }$$ $$f(x)=3x^4+2x^2-7x+9$$ $$f'(x) = 12x^3+4x-7$$ $$\frac{d}{dx}(9)=0$$

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8. Product Rule

$$f(x)=g(x)h(x)$$ $$\boxed{ (gh)'=g'h+gh' }$$ $$f(x)=x^2(x+1)$$ $$g=x^2,\qquad h=x+1$$ $$g'=2x$$ $$h'=1$$ $$f'(x) = (2x)(x+1)+x^2(1)$$ $$=2x^2+2x+x^2$$ $$\boxed{3x^2+2x}$$

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9. Quotient Rule

$$f(x)=\frac{g(x)}{h(x)}$$ $$\boxed{ f'(x) = \frac{g'h-gh'}{h^2} }$$

low d-high minus high d-low, over low squared

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10. Chain Rule

$$f(x)=g(h(x))$$ $$\boxed{ f'(x)=g'(h(x))h'(x) }$$

Differentiate the outside, keep the inside, then multiply by the derivative of the inside.

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11. Example — Chain Rule

$$f(x)=(3x+1)^5$$ $$u^5$$ $$5u^4$$ $$u=3x+1$$ $$3$$ $$f'(x)=5(3x+1)^4(3)$$ $$\boxed{15(3x+1)^4}$$

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12. Tangent Line

$$f(x)=x^2$$ $$x=2$$

Step 1 — Find point

$$f(2)=4$$ $$(2,4)$$

Step 2 — Find derivative

$$f'(x)=2x$$

Step 3 — Evaluate derivative

$$f'(2)=4$$ $$m=4$$

Step 4 — Point-slope equation

$$y-4=4(x-2)$$ $$\boxed{y=4x-4}$$

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13. Linear Approximation

$$\boxed{ f(x)\approx f(a)+f'(a)(x-a) }$$

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14. Why Does Linear Approximation Work?

$$\boxed{\text{locally curved}\approx\text{locally linear}}$$

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15. Example

$$\sqrt{4.1}$$ $$f(x)=\sqrt{x}$$ $$a=4$$ $$f(4)=2$$ $$f'(x)=\frac1{2\sqrt{x}}$$ $$f'(4)=\frac14$$ $$f(4.1) \approx f(4)+f'(4)(4.1-4)$$ $$=2+\frac14(0.1)$$ $$=2+0.025$$ $$\boxed{2.025}$$

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16. Critical Points

$$f'(x)=0$$

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17. Example

$$f(x)=x^2-4x+3$$ $$f'(x)=2x-4$$ $$f'(x)=0$$ $$2x-4=0$$ $$x=2$$ $$(2,-1)$$

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18. Local Maximum / Minimum

$$f'(x)>0$$ $$f'(x)<0$$ $$\text{increasing}\rightarrow\text{decreasing}$$ $$\boxed{\text{local maximum}}$$

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$$f'(x)<0$$ $$f'(x)>0$$ $$\text{decreasing}\rightarrow\text{increasing}$$ $$\boxed{\text{local minimum}}$$

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19. The Sign-Change Pattern

$$+\rightarrow- \quad\Rightarrow\quad \boxed{\text{maximum}}$$ $$-\rightarrow+ \quad\Rightarrow\quad \boxed{\text{minimum}}$$

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20. Example

$$f(x)=x^3-3x$$ $$f'(x)=3x^2-3$$ $$f'(x)=3(x^2-1)$$ $$=3(x-1)(x+1)$$ $$x=-1,\quad x=1$$

$x<-1$

$$f'(x)>0$$

$-1<x<1$

$$f'(x)<0$$

$x>1$

$$f'(x)>0$$ $$+\rightarrow-$$ $$-\rightarrow+$$

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21. L'Hôpital's Rule

$$\lim_{x\to a}\frac{f(x)}{g(x)}$$ $$\frac00$$ $$\boxed{ \lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)} }$$

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22. Example

$$\lim_{x\to0}\frac{\sin x}{x}$$ $$\frac00$$ $$\frac{d}{dx}(\sin x)=\cos x$$ $$\frac{d}{dx}(x)=1$$ $$\lim_{x\to0}\frac{\cos x}{1}$$ $$=\cos0$$ $$\boxed{1}$$

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23. Important L'Hôpital Trap

"There's a fraction, so I'll use L'Hôpital."

$$\frac00$$ $$\frac{\infty}{\infty}$$ $$\frac25$$ $$\frac25$$

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24. Week 8 Pattern Bank

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25. The entire Maths 1 Weeks 1–8 story

$$\text{Sets}$$ $$\text{Lines}$$ $$\text{Quadratics}$$ $$\text{Polynomials}$$ $$\text{Functions}$$ $$\text{Exponentials / Logs}$$ $$\text{Limits}$$ $$\text{Derivatives}$$ $$\boxed{ \text{algebraic objects} \rightarrow \text{functions} \rightarrow \text{behaviour} \rightarrow \text{change} }$$ $$f'(x)=0$$ $$x=-\frac{b}{2a}$$

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Maths 1 Theory Pass: COMPLETE

$$\boxed{ W1 ✓ \;W2 ✓ \;W3 ✓ \;W4 ✓ \;W5 ✓ \;W6 ✓ \;W7 ✓ \;W8 ✓ }$$

"theory gets it, practice makes it concrete."

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You

• what is h in w7/w8 & i get all topics but more practice would make concepts solid and few more doubts or like formulas retention yk maybe & also calculus is fun mun but tedious or time consuming math is fun but to reach at mental math stage about anything it requires training for while and maybe something i should create with code for practicing better yk like matiks apps or duolingo typeshit for becoming solid in concepts fundamentally with spaced repetition and few kewl techniques & coding/vibe coding is fun
• next start stats w1 time mun but before that u can do ur practice/pattern pass if u like munn

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ChatGPT

First: what the hell is $h$? 😭

$h$ = a small change in $x$

$$x+h$$

start at $x$, then move a little bit by an amount $h$.

$$\boxed{h=\text{change/increment in }x}$$

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In the limit

$$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$

Original point

$$x$$

Slightly shifted point

$$x+h$$

Change in output

$$f(x+h)-f(x)$$

Change in input

$$(x+h)-x=h$$ $$\frac{f(x+h)-f(x)}{h} = \frac{\text{change in output}} {\text{change in input}}$$ $$\boxed{\frac{\Delta y}{\Delta x}}$$ $$h\to0$$

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Why not just use $\Delta x$?

$$\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ $$\boxed{h\approx\Delta x}$$

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The coolest way to visualize it

$$f(x)=x^2$$ $$h=1$$ $$x=2$$ $$x+h=3$$ $$h=0.1$$ $$2\rightarrow2.1$$ $$h=0.01$$ $$2\rightarrow2.01$$ $$h=0.0001$$ $$2\rightarrow2.0001$$ $$h\to0$$

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And this connects Week 7 → Week 8 beautifully

Week 7

$$\frac{f(x+h)-f(x)}{h}$$

Week 8

$$\boxed{ f'(x)= \lim_{h\to0} \frac{f(x+h)-f(x)}{h} }$$ $$\boxed{ \text{average change} \xrightarrow{h\to0} \text{instantaneous change} }$$

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Now your bigger point: yes, practice is the missing layer

$$\boxed{ \text{Understand} \rightarrow \text{Recall} \rightarrow \text{Recognize} \rightarrow \text{Execute} }$$

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Maths 1 — Mini Pattern Pass

"Here is a problem."

"Hello student, this is a Week 7 continuity question."

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Q1

$$A=\{1,2,3,4\}$$ $$B=\{3,4,5\}$$ $$A\cup B$$ $$A\cap B$$

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Q2

$$(2,5),\quad(6,13)$$

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Q3

$$(3,7)$$

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Q4

$$f(x)=x^2-6x+5$$

1. roots
2. vertex
3. whether the vertex is a maximum or minimum

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Q5

$$f(x)=2x+3$$ $$g(x)=x^2$$ $$f(g(x))$$ $$g(f(x))$$

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Q6

$$\log_2(x-1)=3$$

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Q7

$$\lim_{x\to2} \frac{x^2-4}{x-2}$$

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Q8

$$\lim_{n\to\infty} \frac{5n+2}{n-7}$$

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Q9

$$f(x)=x^3-3x$$

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Q10

$$f(x)=x^2$$ $$f'(x)$$

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Q11 — recognition rather than calculation

$$\lim_{x\to3^-}f(x)=5$$ $$\lim_{x\to3^+}f(x)=5$$ $$f(3)=100$$

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Q12 — slightly evil

$$f(x)= \begin{cases} x+4,&x<2\\ kx+2,&x\ge2 \end{cases}$$

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Don't worry about doing these immediately

• Easy: direct recognition
• Medium: transformation
• Hard: composition
• Adversarial: notation/traps/unusual wording

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Your "mental math takes training" observation

"I understand how multiplication works"

$37\times24$ appears and your brain immediately decomposes it.

$$x^5$$

power rule → $5x^4$

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And your coding idea is actually interesting

procedural mathematics training engine

text
Maths 1
│
├── Functions
│   ├── Domain
│   ├── Range
│   ├── Injectivity
│   ├── Composition
│   └── Inverse
│
├── Limits
│   ├── Direct substitution
│   ├── 0/0
│   ├── Factorization
│   ├── One-sided
│   └── Continuity
│
└── Derivatives
    ├── Definition
    ├── Power rule
    ├── Product rule
    ├── Chain rule
    ├── Tangent
    └── Optimization

text
pattern: limit-factorization
difficulty: medium
skills:
  - algebraic-factorization
  - limit-recognition
  - cancellation

"You got 7/10."

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A potentially very good learning loop

$$\boxed{ \text{Learn} \rightarrow \text{Retrieve} \rightarrow \text{Solve} \rightarrow \text{Diagnose} \rightarrow \text{Revisit} \rightarrow \text{Generalize} }$$

Question A

$$\lim_{x\to2}\frac{x^2-4}{x-2}$$

Question B

$$\lim_{x\to5}\frac{x^2-25}{x-5}$$

Question C

$$\lim_{x\to a}\frac{x^2-a^2}{x-a}$$ $$2\rightarrow5\rightarrow a$$ $$\frac{x^2-a^2}{x-a} \rightarrow \frac{(x-a)(x+a)}{x-a} \rightarrow x+a$$

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Anyway — Stats 1 starts now

Statistics for Data Science I — Week 1

Introduction to Data & Types of Data

What exactly is a piece of data, and what kind of thing is it?

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1. What is Data?

Data = recorded information about observations.

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2. Observation vs Variable

Observation

Student A.

Variable

Age.

$$\boxed{ \text{rows}\approx\text{observations} }$$ $$\boxed{ \text{columns}\approx\text{variables} }$$

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3. Types of Variables

$$\boxed{ \text{Categorical} \quad\text{vs}\quad \text{Numerical} }$$

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4. Categorical Variables

• City
• Blood group
• Brand
• Gender category
• Yes/No
• Course
• Type of device

$$\text{City}\in \{ \text{Ahmedabad}, \text{Mumbai}, \text{Delhi} \}$$

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5. Numerical Variables

• Age
• Height
• Weight
• Income
• Temperature
• Number of courses
• Distance

$$\text{height}=178$$ $$178-170=8$$

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6. The Important Trap

text
Student ID = 104728

$$104728-103512$$

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7. Categorical Variables Have Subtypes

Nominal

• City
• Eye colour
• Browser
• Blood group

$$\text{Mumbai}<\text{Delhi}$$

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8. Ordinal

• Satisfaction:
  • dissatisfied
  • neutral
  • satisfied
• Education level:
  • school
  • undergraduate
  • postgraduate
• Rating:
  • poor
  • fair
  • good
  • excellent

$$\text{poor}<\text{fair}<\text{good}<\text{excellent}$$

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9. Numerical Variables — Another Useful Split

Discrete

$$0,1,2,3,\ldots$$

• number of siblings
• number of errors
• number of goals
• number of customers

$$2.73\text{ siblings}$$

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Continuous

• height
• weight
• time
• distance
• temperature

$$178.1,\quad178.12,\quad178.123,\ldots$$

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10. The Taxonomy

$$\boxed{\text{Data}}$$ $$\boxed{\text{Variables}}$$

Categorical

• Nominal
• Ordinal

Numerical

• Discrete
• Continuous

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11. Exam-Style Recognition

Question

A survey records the number of movies watched by each person last month.

$$\boxed{\text{Numerical + discrete}}$$

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Question

A survey records each person's favourite streaming platform.

$$\boxed{\text{Categorical + nominal}}$$

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Question

Customers rate a product as poor, average, good, or excellent.

$$\boxed{\text{Categorical + ordinal}}$$

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Question

A sensor records the temperature of a machine.

$$\boxed{\text{Numerical + continuous}}$$

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12. The deeper statistical idea

$$\bar{x}$$ $$\sigma$$

What kind of variable am I dealing with?

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13. Stats W1 Pattern Bank — First Version

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Current master progress

Maths 1

$$\boxed{W1-W8\text{ theory pass complete}}$$

Stats 1

$$\boxed{W1\text{ started}}$$

CT

Python

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You

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ChatGPT

Stats 1 — Weeks 2–4

• multiple weeks can fit into one response when they're naturally connected;
• still cover the concepts, notation, edge cases, and exam patterns;
• don't spend 3 responses saying the same thing in different words;
• practice will come back as a separate layer.

$$\boxed{\text{Describe data}} \rightarrow \boxed{\text{Summarize data}} \rightarrow \boxed{\text{Visualize data}}$$

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Week 2 — Measures of Central Tendency

Where is the "center" of my data?

$$\boxed{\text{Mean, Median, Mode}}$$

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1. Mean

$$x_1,x_2,\ldots,x_n$$ $$\boxed{ \bar{x}=\frac{x_1+x_2+\cdots+x_n}{n} }$$ $$\boxed{ \bar{x}=\frac1n\sum_{i=1}^n x_i }$$

Example

$$2,4,6,8$$ $$\bar{x}=\frac{2+4+6+8}{4}$$ $$=\frac{20}{4}$$ $$\boxed{5}$$

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2. Weighted Mean

$$\bar{x}_w = 0.2(80)+0.3(70)+0.5(90)$$ $$=16+21+45$$ $$\boxed{82}$$ $$\boxed{ \bar{x}_w= \frac{\sum w_ix_i}{\sum w_i} }$$

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3. Median

$$2,5,7,9,11$$ $$\boxed{7}$$ $$\boxed{\text{median}=x_{\frac{n+1}{2}}}$$

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Even number of observations

$$2,5,7,9$$ $$5,\quad7$$ $$\frac{5+7}{2}=6$$ $$\boxed{\text{median}=6}$$ $$\boxed{ \text{median} = \frac{x_{n/2}+x_{n/2+1}}2 }$$

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4. Mode

$$2,3,3,4,5,3,6$$ $$\boxed{\text{mode}=3}$$

• one mode;
• multiple modes;
• no unique mode.

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5. Mean vs Median — Why Have Both?

$$2,3,4,5,100$$ $$\bar{x}=\frac{114}{5}=22.8$$ $$4$$ $$\boxed{\text{mean is sensitive to outliers}}$$ $$\boxed{\text{median is more resistant to outliers}}$$

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6. When Should You Think Mean vs Median?

Roughly symmetric data

Strongly skewed data / outliers

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7. Exam Pattern — Central Tendency

Data: $4,7,7,8,9,100$

Mean

$$\bar{x} = \frac{4+7+7+8+9+100}{6}$$ $$=\frac{135}{6}$$ $$\boxed{22.5}$$

Median

$$7,8$$ $$\boxed{7.5}$$

Interpretation

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8. Week 2 Pattern Bank

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Week 3 — Measures of Dispersion

Where is the center?

Dataset A

$$49,50,51$$

Dataset B

$$10,50,90$$ $$\bar{x}=50$$ $$\boxed{\text{dispersion}}$$

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9. Range

$$\boxed{ \text{Range}=\max(x)-\min(x) }$$ $$10,20,30,50$$ $$50-10=40$$

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10. Why Range Is Weak

$$10,20,30,40,50$$ $$10,20,30,40,500$$

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11. Variance

$$\boxed{ \sigma^2 = \frac1N \sum_{i=1}^{N}(x_i-\mu)^2 }$$ $$\boxed{ s^2 = \frac1{n-1} \sum_{i=1}^{n}(x_i-\bar{x})^2 }$$

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12. Why Square the Differences?

$$5$$ $$3,7$$ $$-2,+2$$ $$\frac{-2+2}{2}=0$$ $$(-2)^2=4$$ $$(2)^2=4$$

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13. Variance Example

$$2,4,6$$ $$\bar{x}=\frac{12}{3}=4$$ $$-2,0,+2$$ $$4,0,4$$

Population variance

$$\sigma^2 = \frac{4+0+4}{3}$$ $$\boxed{\frac83}$$

Sample variance

$$s^2 = \frac{8}{3-1}$$ $$\boxed4$$

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14. Standard Deviation

$$\text{variance}\rightarrow cm^2$$ $$\boxed{ \sigma=\sqrt{\sigma^2} }$$ $$\boxed{ s=\sqrt{s^2} }$$ $$s^2=4$$ $$s=2$$

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15. Why Standard Deviation Is Intuitive

$$\bar{x}=50$$ $$s=5$$

Observations typically lie around several units away from the mean, with the standard deviation giving the scale of that spread.

"every observation is exactly 5 away."

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16. Computational Shortcut for Variance

$$\boxed{ \sigma^2=E[X^2]-(E[X])^2 }$$ $$\boxed{ \sigma^2= \frac{\sum x_i^2}{N} -\mu^2 }$$

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Example

$$2,4,6$$ $$\mu=4$$ $$E[X^2] = \frac{4+16+36}{3}$$ $$=\frac{56}{3}$$ $$\sigma^2 = \frac{56}{3}-16$$ $$= \frac83$$

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17. Week 3 Pattern Bank

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Week 4 — Percentiles, Quartiles & Box Plots

"What is the center?"

"Where does this observation sit relative to the rest?"

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18. Percentiles

90th percentile

$$90\text{th percentile}$$

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19. Quartiles

First quartile

$$\boxed{Q_1}$$

Second quartile

$$\boxed{Q_2}$$

Third quartile

$$\boxed{Q_3}$$ $$\boxed{ Q_1\rightarrow25\% }$$ $$\boxed{ Q_2\rightarrow50\% }$$ $$\boxed{ Q_3\rightarrow75\% }$$

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20. Interquartile Range

$$\boxed{ IQR=Q_3-Q_1 }$$

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21. Example

$$Q_1=20$$ $$Q_3=50$$ $$IQR=50-20$$ $$\boxed{30}$$

The central 50% spans 30 units.

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22. Box Plot

$$\boxed{ \min,\;Q_1,\;Q_2,\;Q_3,\;\max }$$ $$Q_1\rightarrow Q_3$$ $$Q_2$$

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23. Detecting Outliers with IQR

Lower fence

$$\boxed{ Q_1-1.5(IQR) }$$

Upper fence

$$\boxed{ Q_3+1.5(IQR) }$$

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24. Example

$$Q_1=10$$ $$Q_3=20$$ $$IQR=10$$ $$10-1.5(10)$$ $$=-5$$ $$20+1.5(10)$$ $$=35$$

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25. Box Plot → Distribution Shape

possible right skew.

possible left skew.

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26. Week 4 Pattern Bank

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27. The W2–W4 Big Picture

$$\boxed{\text{Where is the data?}}$$ $$\boxed{\text{How spread out is it?}}$$ $$\boxed{\text{Where does each observation sit?}}$$ $$\boxed{ \text{Center} \rightarrow \text{Spread} \rightarrow \text{Position} }$$ $$\boxed{ \text{Shape} \rightarrow \text{Relationship} \rightarrow \text{Inference} }$$

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Stats 1 Progress

$$\boxed{W1-W4\text{ theory pass}}$$

• W1: Types of data and variables
• W2: Mean, median, mode, weighted mean
• W3: Range, variance, standard deviation
• W4: Percentiles, quartiles, IQR, box plots, outliers

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