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◈ CyberSecurity · 3rd Trimester · Mid Term Prep
1ST YEAR
STUDY MODE
▶ LINEAR ALGEBRA  ·  HACKING LAB  ·  ELECTRONICS  ·  CULTURAL STUDIES  ·  POLITICAL SCIENCE  ·  CALCULUS 2      ▶ LINEAR ALGEBRA  ·  HACKING LAB  ·  ELECTRONICS  ·  CULTURAL STUDIES  ·  POLITICAL SCIENCE  ·  CALCULUS 2     
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1ST YEAR STUDENT 3RD TRIMESTER · 2026 6 SUBJECTS
// SUBJECT MODULES
𝛁
Linear Algebra4 WEEKS LOADED
⚡
Hacking Lab3 LECTURES LOADED
⬡
Electronics2 LESSONS LOADED
◈
Cultural Studies4 WEEKS LOADED
◉
Political Science3 WEEKS LOADED
∫
Calculus 24 LECTURES LOADED
Linear Algebra
MATH-201 · 1ST YEAR · WEEKS 1–4 · CH. 1 & 2
4 WEEKS
4WEEKS
14TOPICS
1BIG THM
IMTKEY RESULT
// WEEK 1 — CH.1: LINEAR EQUATIONS (§1.1–1.4)
// 1.1–1.4 CORE CONCEPTS
DEFINITION
Linear System & Solution
System of m equations in n unknowns x₁…xₙ. A solution is a list (s₁…sₙ) satisfying all equations. The collection of all solutions = solution set.
a₁₁x₁ + a₁₂x₂ + … + a₁ₙxₙ = b₁ ⋮ aₘ₁x₁ + aₘ₂x₂ + … + aₘₙxₙ = bₘ → matrix form: Ax = b
KEY FACT
Consistent vs Inconsistent
INCONSISTENT
⟺ NO SOLUTION
Row gives 0 = c, c≠0
CONSISTENT
⟺ ≥1 SOLUTION
Either 1 or ∞ many
DEFINITION
Row Echelon Form (REF) & RREF
REF: leading entry = 1, each pivot right of previous, zero rows at bottom.
RREF: REF + pivot is the ONLY nonzero in its column.
REF: [1 2 3] RREF: [1 0 3] [0 1 4] [0 1 4] [0 0 1] [0 0 1]
ALGORITHM
Row Reduction Steps
1. Find leftmost nonzero column → pivot column
2. Swap to get nonzero at top, scale to make pivot = 1
3. Zero out all entries below pivot (EROs)
4. Repeat for submatrix → gives REF
5. Zero out entries above each pivot → gives RREF
EROs: Rᵢ ↔ Rⱼ | c·Rᵢ→Rᵢ (c≠0) | Rᵢ+c·Rⱼ→Rᵢ
THEOREM
3 Cases (Existence & Uniqueness)
d = n − rank(A) (free parameters) Case 1: inconsistent row (0…0|c), c≠0 → NO SOLUTION Case 2: d = 0, rank=n → UNIQUE SOLUTION Case 3: d ≥ 1, rank<n → ∞ MANY SOLUTIONS
// WEEK 2 — CH.1: SOLUTION SETS, INDEPENDENCE (§1.5–1.7)
// 1.5 SOLUTION SETS OF LINEAR SYSTEMS
DEFINITION
Homogeneous System
A system Ax = 0 (all constants = 0) is called homogeneous. It is ALWAYS consistent — trivial solution x = 0 always exists.
Ax = 0 → trivial solution: x = (0, 0, …, 0) always works Nontrivial solutions exist ⟺ system has FREE VARIABLES ⟺ rank(A) < n
KEY FACT
Parametric Vector Form
When free variables exist, write solution as a parametric vector form: x = p + t·v (or multiple params). Here p is a particular solution and v spans the solution set.
Example — solution set of Ax=0: x = t·v₁ + s·v₂ (t, s ∈ ℝ) — passes through origin Solution set of Ax=b (nonhomogeneous, consistent): x = p + t·v₁ + s·v₂ — translated parallel to homogeneous solution set
THEOREM 1.6
Nonhomogeneous Solution Structure
If p is any particular solution to Ax = b, then every solution has the form:
x = p + xₕ where xₕ is any solution to the homogeneous system Ax = 0
Geometric picture: solution set of Ax=b is a translate of solution set of Ax=0.
// 1.6 APPLICATIONS OF LINEAR SYSTEMS
APPLICATION
Network Flow & Balancing
Set up equations for each node: flow in = flow out. Write as linear system, row-reduce. Free variables = adjustable flows.
APPLICATION
Balancing Chemical Equations
Assign unknown coefficients to each molecule. For each element, write: atoms in = atoms out. Solve the linear system for the coefficients.
// 1.7 LINEAR INDEPENDENCE
DEFINITION
Linear Independence / Dependence
Vectors {v₁, v₂, …, vₚ} are linearly independent if the only solution to c₁v₁ + c₂v₂ + … + cₚvₚ = 0 is c₁=c₂=…=cₚ=0 (trivial).
They are linearly dependent if a nontrivial solution exists (some cᵢ ≠ 0).
Test: put vectors as columns → [v₁ v₂ … vₚ]x = 0 Row reduce. If only trivial solution → INDEPENDENT If free variable exists → DEPENDENT
KEY THEOREMS
Linear Dependence Facts
1. A set with the ZERO VECTOR is always dependent. 2. A set of 2 vectors is dependent ⟺ one is a scalar multiple of the other. 3. If p > n (more vectors than entries): ALWAYS dependent. (More columns than rows → free variable guaranteed) 4. {v₁,…,vₚ} dependent ⟺ at least one vector is a linear combination of the others (the preceding ones).
// WEEK 3 — CH.1&2: TRANSFORMATIONS, MATRIX OPS (§1.8–2.1)
// 1.8 LINEAR TRANSFORMATIONS
DEFINITION
Linear Transformation
A transformation T: ℝⁿ → ℝᵐ is linear if for all u, v ∈ ℝⁿ and scalars c:
1. T(u + v) = T(u) + T(v)   (additivity)
2. T(cu) = cT(u)   (homogeneity)
Consequence: T(0) = 0 always T(cu + dv) = cT(u) + dT(v) (superposition)
KEY IDEA
Matrix Multiplication = Linear Transformation
Every matrix multiplication T(x) = Ax defines a linear transformation. The matrix A IS the transformation — it tells you where each basis vector goes.
T: ℝⁿ → ℝᵐ defined by T(x) = Ax where A is m×n Domain = ℝⁿ, Codomain = ℝᵐ Range = set of all Ax = column space of A
// 1.9 MATRIX OF A LINEAR TRANSFORMATION
THEOREM 1.10
Standard Matrix of T
For any linear T: ℝⁿ → ℝᵐ, there exists a unique matrix A such that T(x) = Ax. This matrix is:
A = [T(e₁) T(e₂) … T(eₙ)] where e₁, e₂, …, eₙ are standard basis vectors of ℝⁿ. Just apply T to each basis vector → that's the column!
DEFINITION
Onto and One-to-One
ONE-TO-ONE
T(u)=T(v) ⟹ u=v
⟺ Ax=0 has
only trivial sol.
⟺ columns of A
linearly independent
ONTO
Every b in ℝᵐ
is hit by some x
⟺ A has a pivot
in every ROW
⟺ cols span ℝᵐ
// 2.1 MATRIX OPERATIONS
DEFINITION
Matrix Operations
Addition: A + B (same size, add entry-by-entry) Scalar mult: cA (multiply every entry by c) Multiplication: (AB)ᵢⱼ = row i of A · col j of B A is m×n, B is n×p → AB is m×p Transpose: (Aᵀ)ᵢⱼ = Aⱼᵢ (flip rows/cols)
IMPORTANT PROPERTIES
Matrix Algebra Rules
A(BC) = (AB)C (associative) A(B+C) = AB + AC (distributive) (A+B)C = AC + BC (AB)ᵀ = BᵀAᵀ (reverse order!) (ABC)ᵀ = CᵀBᵀAᵀ ✗ AB ≠ BA in general (NOT commutative!) ✗ AB = AC does NOT imply B = C ✗ AB = 0 does NOT imply A=0 or B=0
REMEMBER
Column View of AB
Each column of AB = A times the corresponding column of B.
AB = A[b₁ b₂ … bₚ] = [Ab₁ Ab₂ … Abₚ]
// WEEK 4 — CH.2: INVERSE, IMT, LU (§2.2–2.5)
// 2.2 INVERSE OF A MATRIX
DEFINITION
Invertible (Nonsingular) Matrix
An n×n matrix A is invertible if there exists an n×n matrix C such that AC = CA = I. Then C = A⁻¹ (unique). A non-invertible matrix is called singular.
If A invertible: Ax = b has unique solution x = A⁻¹b (A⁻¹)⁻¹ = A (AB)⁻¹ = B⁻¹A⁻¹ ← REVERSE ORDER (Aᵀ)⁻¹ = (A⁻¹)ᵀ
ALGORITHM
Computing A⁻¹ by Row Reduction
Augment A with identity, row reduce to RREF. If A reduces to I, the right side becomes A⁻¹.
[ A | I ] → row reduce → [ I | A⁻¹ ] If left side CANNOT reduce to I → A is SINGULAR (no inverse)
FORMULA
2×2 Inverse Formula
A = [a b] A⁻¹ = ___1___ · [ d -b] [c d] ad - bc [-c a] det(A) = ad - bc If det(A) = 0 → A is SINGULAR (not invertible)
// 2.3 INVERTIBLE MATRIX THEOREM (IMT)
THEOREM 2.8 — THE BIG ONE
Invertible Matrix Theorem (IMT)
For an n×n matrix A, ALL of the following are equivalent:
a) A is invertible b) A is row equivalent to Iₙ c) A has n pivot positions d) Ax = 0 has only the trivial solution e) Columns of A are linearly independent f) T(x)=Ax is one-to-one g) Ax = b has a solution for every b in ℝⁿ h) Columns of A span ℝⁿ i) T(x)=Ax is onto j) There exists C such that CA = I k) There exists D such that AD = I l) Aᵀ is invertible
⚡ If ANY ONE is true → ALL are true. If ANY ONE is false → ALL are false.
// 2.4 PARTITIONED MATRICES
DEFINITION
Block Matrix Multiplication
Matrices can be divided into blocks (submatrices). Multiplication works block-by-block, same rules as regular multiplication — as long as block sizes are compatible.
[ A₁₁ A₁₂ ] [ B₁ ] [ A₁₁B₁ + A₁₂B₂ ] [ A₂₁ A₂₂ ] [ B₂ ] = [ A₂₁B₁ + A₂₂B₂ ]
// 2.5 MATRIX FACTORIZATIONS — LU DECOMPOSITION
DEFINITION
LU Factorization
An m×n matrix A (with no row swaps needed) can be written as A = LU where:
L = m×m lower triangular matrix with 1s on diagonal
U = m×n upper triangular (REF of A)
A = LU To solve Ax = b using LU: Step 1: Solve Ly = b (forward substitution) → find y Step 2: Solve Ux = y (back substitution) → find x WHY useful: Factor A once, solve for many different b vectors cheaply!
ALGORITHM
Finding L and U
1. Row reduce A to U (REF) using only row replacements (Rᵢ + c·Rⱼ → Rᵢ, NO swaps, NO scaling) 2. L = identity with multipliers placed below diagonal: If you used "Rᵢ − c·Rⱼ → Rᵢ" to eliminate, then place c in position (i,j) of L. Check: multiply L × U = A
SUMMARY
Chapter 2 Big Picture
Square matrix A (n×n): Invertible? YES → unique solution, det≠0, full rank, cols independent NO → singular, det=0, free variables, cols dependent LU = efficient way to solve Ax=b for many b IMT = one theorem connecting ALL invertibility concepts
// MASTER CHEAT SHEET — ALL 4 WEEKS
FORMULAS
Must-Know Formulas
d = n − rank(A) # free parameters det(2×2) = ad − bc # for invertibility (AB)⁻¹ = B⁻¹A⁻¹ # reverse order! (AB)ᵀ = BᵀAᵀ # reverse order! A = LU # LU factorization x = p + xₕ # nonhomog solution = particular + homog
QUICK CHECKS
How to Classify a System / Matrix
→ Row reduce [A|b] to RREF Has row (0…0 | c), c≠0? → INCONSISTENT (no solution) rank(A) = n, no inconsist.? → UNIQUE solution rank(A) < n, consistent? → INFINITELY MANY (d free params) Square matrix n×n: rank = n? → INVERTIBLE, det≠0, cols independent, spans ℝⁿ rank < n? → SINGULAR, det=0, cols dependent, doesn't span ℝⁿ
DEFINITIONS RECAP
Key Terms Fast Reference
Homogeneous system Ax = 0 (always consistent, trivial sol = 0) Particular solution any ONE solution p to Ax = b Linear independence c₁v₁+…+cₚvₚ=0 only if all cᵢ=0 Standard matrix of T A = [T(e₁) T(e₂) … T(eₙ)] One-to-one T Ax=0 trivial only ↔ cols independent Onto T cols of A span codomain ↔ pivot every row Invertible matrix AC = CA = I ↔ ALL conditions in IMT true LU factorization A = LU, L lower triangular, U upper (REF)
IMT SHORTLIST
Invertible Matrix Theorem — Remember These
n×n matrix A is INVERTIBLE ⟺ each of: • n pivot positions (full rank) • Ax=0 only trivial solution • Columns linearly independent • Columns span ℝⁿ • T(x)=Ax is one-to-one AND onto • det(A) ≠ 0 • Aᵀ is invertible
GOTCHAS
Common Mistakes to Avoid
✗ AB ≠ BA (matrix mult is NOT commutative) ✗ AB=0 does NOT mean A=0 or B=0 ✗ AB=AC does NOT mean B=C (unless A invertible) ✗ (AB)⁻¹ ≠ A⁻¹B⁻¹ → correct: B⁻¹A⁻¹ ✗ More vectors than dimensions → ALWAYS linearly dependent Zero vector in set → ALWAYS linearly dependent A set of 1 nonzero vector → ALWAYS linearly independent
Hacking Lab
SEC-310 · 1ST YEAR · LECTURES 1, 2, 4
3 LECTURES
3LECTURES
6PENTEST STAGES
4OSINT TOOLS
WPA2FOCUS TOPIC
// LECTURE 1 — INTRO TO HACKING LAB · KALI LINUX · PENTEST STAGES
TERMINOLOGY
Basic Security Terms
White Hat — ethical hacker, secures systems with permission Black Hat — malicious hacker, harmful intent Gray Hat — no permission, but not necessarily malicious Vulnerability — weakness in a system Exploit — method to take advantage of a vulnerability Penetration Test — authorized simulated attack to find weaknesses
KEY CONCEPT
Offensive vs Defensive Security
OFFENSIVE
Penetration Testing
Red Teaming
Ethical Hacking
Simulate attacks → find holes
DEFENSIVE
Blue Teaming
Incident Response
Threat Hunting
Monitor → detect → respond
6 STAGES
Penetration Testing Methodology
1. RECONNAISSANCE — collect info (OSINT, WHOIS, Google dorking) 2. SCANNING — find open ports, services, vulns (Nmap, Nessus) 3. GAINING ACCESS — exploit vulns (Metasploit, Hydra, phishing) 4. MAINTAINING ACCESS — persist (backdoors, privilege escalation) 5. COVERING TRACKS — hide activity (clear logs, disable monitoring) 6. REPORTING — document findings + mitigation suggestions
Always authorized — without permission it's illegal.
TOOL
Kali Linux
Debian-based OS by Offensive Security. Pre-installed with 600+ security tools. Standard platform for pen testing.
Tool categories: Info Gathering · Vulnerability Analysis · Wireless Attacks Web Applications · Exploitation · Password Attacks Forensics · Sniffing & Spoofing · Reporting Key shortcuts: Ctrl+Alt+T open terminal Ctrl+Shift+T new terminal tab Ctrl+C interrupt command Ctrl+R search command history history show past commands ls > file.txt save directory list to file
RECON STAGE DETAIL
Reconnaissance Techniques
Passive (no direct contact): WHOIS, Google Dorking, social media, job postings Active (direct contact): Nmap scans, banner grabbing, DNS queries Example: Testing a bank → LinkedIn reveals they use Apache + AWS → WHOIS shows domain registration details → Google dork finds exposed config files
// LECTURE 2 — INFORMATION GATHERING · OSINT TOOLS
CONCEPT
Active vs Passive Reconnaissance
| Feature | Passive | Active | |------------------|--------------------------|---------------------------| | Target contact | None (no direct probes) | Direct interaction | | Detectability | Hard to detect | Easier to detect | | Data sources | Third-party databases | Direct target queries | | Examples | Google Dorking, WHOIS | Nmap, Shodan queries |
CONCEPT
OSINT — Open Source Intelligence
Collecting and analyzing publicly available information. Legal but must be used ethically. Data sources: search engines, social media, public databases, dark web forums.
TOOL
Shodan — "The Hacker's Search Engine"
Scans the internet for connected devices. Unlike Google it indexes devices and services — open ports, software versions, vulnerabilities.
Basic searches: webcam → find accessible cameras apache → Apache web servers port:22 → SSH services country:KZ → limit to Kazakhstan org:"Company" → specific organization Example: apache country:KZ port:80 Finds: webcams, routers, IoT, SCADA, databases, servers
TOOL
Google Dorking — Advanced Search Operators
filetype:pdf → specific file type inurl:login → "login" in URL intext:"password" → word in page content intitle:"index of" → directory listings (exposed!) site:example.com → search within one site link:example.com → pages linking to site Practical combos: filetype:pdf intext:"confidential" site:example.com inurl:admin filetype:txt intext:"email" inurl:"view/view.shtml" → exposed cameras
TOOL
theHarvester — Email & Subdomain Collector
Gathers emails, subdomains, IP addresses, usernames from search engines and public APIs.
Command syntax: theHarvester -d example.com -b google -d target domain -b data source (google, bing, linkedin…) Collects: email addresses · subdomains · hostnames · IPs
TOOL
WHOIS & Maltego
WHOIS (who.is): → Domain owner name, email, company → Registration + expiration dates → Hosting provider + IP addresses Maltego: → Visual graph: maps relationships between people, domains, IPs, companies → Uses "Transforms" to auto-query multiple sources → Start from 1 data point (email) → expand to full network map
// LECTURE 4 — WI-FI SECURITY · WPA2 CRACKING · AIRCRACK-NG
THEORY
Wi-Fi Standards (IEEE 802.11)
802.11n (Wi-Fi 4) — up to 600 Mbps, 2.4/5 GHz 802.11ac (Wi-Fi 5) — up to 6.77 Gbps, 5 GHz 802.11ax (Wi-Fi 6) — up to 10.7 Gbps, 2.4/5/6 GHz 802.11be (Wi-Fi 7) — up to 30 Gbps, 2.4/5/6 GHz (2024) Frames: Packet = Layer 3 (routing), Frame = Layer 2 (local delivery) Frame types: Management | Control | Data
SECURITY PROTOCOLS
OPN / WEP / WPA2 / WPA3 Comparison
OPN — No encryption. Data in plain text. INSECURE. WEP — RC4 cipher. Reused IVs. BROKEN. Crackable in minutes with aircrack-ng. WPA2 — AES/CCMP encryption. Still dominant. Attack: capture 4-way handshake → dictionary/brute-force Security depends entirely on PASSWORD STRENGTH. WPA3 — SAE (Dragonfly handshake) replaces PSK. Forward secrecy. Offline dictionary attacks blocked. Most secure current standard.
WEP — BROKEN
Never use
WPA2 — Vulnerable
to weak passwords
WPA3 — Current
Best standard
KEY CONCEPT
The 4-Way Handshake (WPA2 Attack Target)
4 messages exchanged between AP and client to derive encryption keys. This handshake is what attackers capture to attempt offline password cracking.
Message 1: AP sends ANonce (random value) to client Message 2: Client generates SNonce, derives PTK, sends back Message 3: AP confirms keys, sends GTK (group key) Message 4: Client confirms → secure channel established Key hierarchy: PMK (Pairwise Master Key) — from password PTK (Pairwise Transient Key) — encrypts unicast traffic GTK (Group Temporal Key) — encrypts broadcast/multicast Attack: capture handshake → run dictionary attack against PMK
TOOLSET
Aircrack-ng Suite — Commands
airmon-ng start wlan0 → enable monitor mode → creates wlan0mon airmon-ng check kill → stop interfering processes airmon-ng stop wlan0mon → disable monitor mode airodump-ng wlan0mon → scan all nearby networks airodump-ng -c 6 --bssid AA:BB:CC:DD:EE:FF -w capture wlan0mon → capture traffic on specific AP aireplay-ng -0 5 -a AA:BB:CC:DD:EE:FF wlan0mon → send 5 deauth packets (force handshake) aircrack-ng capture.cap -w wordlist.txt → crack WPA2 with dictionary attack Key airodump-ng output fields: BSSID — MAC of access point PWR — signal strength (more negative = weaker) ENC — encryption type (OPN/WEP/WPA2/WPA3) CIPHER — CCMP (AES) or TKIP AUTH — PSK (personal) or MGT (enterprise) ESSID — network name (SSID)
WPA2 CRACK WORKFLOW
Step-by-Step Attack Flow
1. airmon-ng start wlan0 → monitor mode 2. airodump-ng wlan0mon → find target network 3. airodump-ng -c [CH] --bssid [BSSID] -w cap wlan0mon → capture traffic 4. aireplay-ng -0 10 -a [BSSID] wlan0mon → deauth client → force handshake 5. Wait for "WPA handshake" message in airodump-ng 6. aircrack-ng cap.cap -w /usr/share/wordlists/rockyou.txt → dictionary attack 7. If password found → KEY FOUND!
// CHEAT SHEET — HACKING LAB ALL LECTURES
QUICK REFERENCE
Pentest Stages — 1 Line Each
1. Recon → gather info (OSINT, WHOIS, Google dork) 2. Scanning → Nmap ports, Nessus vulns, banner grabbing 3. Access → Metasploit exploit / Hydra brute-force / phishing 4. Persistence → backdoor, privilege escalation, rootkit 5. Cover tracks→ clear logs (rm -rf /var/log/auth.log), disable IDS 6. Report → executive summary + vuln list + risk + mitigation
TOOLS SUMMARY
Key Tools Fast Reference
Shodan → search internet-connected devices/services theHarvester → theHarvester -d domain.com -b google WHOIS → domain owner, dates, hosting info (who.is) Maltego → visual relationship graph (entity mapping) Nmap → network/port scanner Airmon-ng → enable monitor mode on WiFi adapter Airodump-ng → capture WiFi packets, discover networks Aireplay-ng → inject packets (deauth attack) Aircrack-ng → crack WEP/WPA2 keys from captured handshake Metasploit → exploit framework Hydra → password brute-force tool Wireshark → deep packet inspection / traffic analysis
WIFI SECURITY
Encryption Hierarchy to Remember
WEP → BROKEN (RC4, reused IVs, crack in minutes) WPA → Transitional (TKIP, better than WEP, still weak) WPA2 → Standard (AES/CCMP, attack via handshake capture) WPA3 → Best (SAE/Dragonfly, forward secrecy, blocks offline attacks) WPA2 attack requires: Monitor-mode capable WiFi adapter Capture 4-way handshake Dictionary/wordlist (rockyou.txt) → Success depends on PASSWORD STRENGTH
GOOGLE DORK CHEATSHEET
Most Useful Operators
filetype:pdf intext:"confidential" → confidential PDFs site:target.com inurl:admin → admin panels filetype:txt intext:"email" → email lists inurl:"view/view.shtml" → exposed cameras "webcamXP" country:US → webcams in US (Shodan) port:3389 country:KZ → exposed RDP in KZ (Shodan) intitle:"index of" "parent directory" → open directories
Electronics
ENG-220 · 1ST YEAR · LESSONS 1 & 2
2 LESSONS
2LESSONS
7COMPONENTS
3CORE LAWS
AC/DCKEY TOPIC
// LESSON 1 — RESISTOR · CAPACITOR · INDUCTOR · DIODE · TRANSISTOR · IC
OVERVIEW
Electronics — Basic Components
Science of controlling electric energy using active components (transistors, diodes, ICs) and passive components (resistors, capacitors, inductors). Core 7: Resistor, Capacitor, Inductor, Diode, LED, Transistor, IC.
COMPONENT 1
Resistor
Passive 2-terminal component. Implements electrical resistance. Unit: Ohm (Ω). Named after Georg Simon Ohm.
Ohm's Law: V = I × R Series: Rₜ = R₁ + R₂ + … (current same I₁=I₂, voltages split) Parallel: 1/Rₜ = 1/R₁ + 1/R₂ + … (voltage same V₁=V₂, currents split) Units: 1 kΩ = 10³ Ω | 1 MΩ = 10⁶ Ω Power: P = V²/R = I²R = V·I FIXED types: Carbon Composite — cheap, high tolerance, good at high freq Film Resistor — tolerance ≤1%, up to 10 MΩ Wire Wound — 0.01–100 kΩ, high precision VARIABLE types: Rheostat — adjusts current (2-terminal) Potentiometer — 3-terminal voltage divider (volume knob, joystick) Thermistor — resistance changes with TEMPERATURE Varistor (VDR) — conducts more at HIGH VOLTAGE (surge protection) Photoresistor(LDR)— resistance decreases with LIGHT (street lights)
COMPONENT 2
Capacitor
Passive device: two plates + insulator (dielectric) between them. Stores energy as electric field. Unit: Farad (F).
Series: 1/Cₜ = 1/C₁ + 1/C₂ → total C DECREASES (opposite to R!) Parallel: Cₜ = C₁ + C₂ → total C INCREASES Charging: plates accumulate charge, voltage rises toward supply Discharging: stored energy released back to circuit
COMPONENT 3
Inductor
Passive component: wire coil. Stores energy as magnetic field. Inductance ∝ number of turns in coil. Unit: Henry (H).
Core types (determines frequency range): Air Core — no core, Radio Frequencies (RF), minimal signal loss Ferromagnetic — iron/ferrite, higher inductance, high-freq losses Toroidal — ring-shaped, minimum EM interference, AC circuits Laminated Iron — thin steel sheets, blocks eddy currents, LOW freq / transformers Powdered Iron — distributed air gaps, high energy storage, low eddy loss Magnetic rule: LIKE poles REPEL, UNLIKE poles ATTRACT
COMPONENT 4
Diode
2-terminal semiconductor. Conducts current in ONE direction only. Terminals: Anode (+) and Cathode (−). The bar in the symbol marks the cathode.
Forward bias → current flows (anode+ to cathode−) Reverse bias → current blocked (open circuit) Uses: circuit protection (wrong polarity), rectification (AC→DC)
COMPONENT 5
LED — Light Emitting Diode
Diode that emits light when forward biased. Electrons recombine with holes → release photons. Color determined by semiconductor's band gap energy.
COMPONENT 6
Transistor
3-terminal semiconductor: amplify or switch signals. Terminals: Base (activates), Collector (+ output), Emitter (− reference). Types: NPN and PNP.
Small Base current → controls large Collector-Emitter current Modes: Cutoff (OFF) | Active (amplify) | Saturation (fully ON)
COMPONENT 7
Integrated Circuit (IC)
Millions/billions of transistors on one chip. The "brain" — receives input → produces output. Modern microprocessors: billions of transistors per square inch.
KEY RULE
Series vs Parallel Summary
SERIES PARALLEL Current: I₁ = I₂ (same) I = I₁ + I₂ (splits) Voltage: V = V₁ + V₂ (splits) V₁ = V₂ (same) Resistance: Rₜ = R₁ + R₂ 1/Rₜ = 1/R₁ + 1/R₂
// LESSON 2 — CURRENT · VOLTAGE · RESISTANCE · OHM'S LAW · AC/DC
FOUNDATION
Atomic Structure
Atom = protons(+) + neutrons(N) + electrons(−) Shells: Shell 1 = max 2e⁻ | Shell 2 = max 8e⁻ | Shell 3 = max 18e⁻ Valence electrons = outermost shell = easiest to remove = carry current Ion: atom with gained/lost electrons Lost e⁻ → positive ion (+) Gained e⁻ → negative ion (−) Conductors: ≤3 valence e⁻, low R (Cu, Ag, Au, Al) Semiconductors: 4 valence e⁻, medium R (Si, Ge) Insulators: ≥5 valence e⁻, high R (rubber, glass, plastic) Charge unit: Coulomb (C), symbol Q 1 Coulomb = charge of ~6.24 × 10¹⁸ electrons
CONCEPT
Current (I)
Rate of electron flow past a point. Current = flow. Unit: Ampere (A).
I = Q / t (charge per unit time) DC — Direct Current: flows ONE direction only Sources: batteries, solar cells AC — Alternating Current: periodically REVERSES direction Sources: generators, power grid Homes: reverses every 1/120 s → 60 Hz (KZ/EU: 50 Hz) Current carriers: Solids → free electrons Liquids → positive and negative ions (electrolyte) Gases → ionized atoms (e.g. neon bulb) Vacuum → electrons via thermionic emission
CONCEPT
Voltage (V)
Electric pressure / potential difference that pushes current. Unit: Volt (V) = 1 Joule/Coulomb. Also called EMF (electromotive force).
Types: AC voltage — from generators (mechanical → electric) DC voltage — from batteries (chemical → electric) Sources of voltage: Generator → mechanical energy → electric (most common) Battery/Cell → chemical energy → electric Thermocouple → heat → electric (Seebeck effect) Crystal → pressure → electric (piezoelectric) Solar cell → light → electric (photovoltaic)
CONCEPT
Resistance (R)
Opposition to current flow. Unit: Ohm (Ω). Depends on 4 factors.
4 factors: 1. Material type (copper vs rubber) 2. Length (longer → MORE resistance: R ∝ L) 3. Cross-section area (thicker → LESS resistance: R ∝ 1/A) 4. Temperature (usually hotter → more resistance) Superconductivity: R = 0 near absolute zero (−273°C)
CORE LAW
Ohm's Law + Power
V = I × R → I = V/R → R = V/I Power: P = V × I P = I² × R P = V² / R Units: V (volts), I (amps), R (ohms), P (watts)
CONCEPT
AC Waveforms & Key Parameters
Sine wave — standard AC shape (homes/factories) Square, Triangle, Sawtooth — other common waveforms Definitions: Cycle — one complete waveform repetition Period (T) — time for one cycle (seconds) Frequency — f = 1/T (unit: Hz = cycles/second) AC voltage values: Peak (Vp) — maximum value above zero Peak-to-peak — Vpp = 2 × Vp Average — Vav = 0.637 × Vp RMS (effective) — Vrms = 0.707 × Vp ← MOST IMPORTANT → RMS = equivalent DC value for same heating effect → "220V outlet" means 220V RMS Oscilloscope: instrument that displays waveform shape, measures amplitude, period and frequency visually
KIRCHHOFF'S LAWS
KCL — Kirchhoff's Current Law (1st Law)
The total current entering a junction (node) equals the total current leaving it. No charge is lost at a node.
ΣI_in = ΣI_out Example: I₁ →─┬─→ I₂ └─→ I₃ I₁ = I₂ + I₃ Think of it like water flow into a pipe junction: what goes in MUST come out — charge cannot pile up at a node.
KIRCHHOFF'S LAWS
KVL — Kirchhoff's Voltage Law (2nd Law)
The sum of all voltages around any closed loop in a circuit equals zero. Energy supplied = energy consumed.
ΣV = 0 (around any closed loop) Convention: going around the loop, voltage RISE (across source) → add (+) voltage DROP (across resistor)→ subtract (−) Example (simple series loop with battery + 2 resistors): +Vs − V_R1 − V_R2 = 0 Vs = V_R1 + V_R2 ← supply voltage = sum of drops Remember: Ohm's Law + KVL/KCL together solve ANY circuit!
APPLYING KCL & KVL
How to Use Kirchhoff's Laws — Steps
Step 1: Label all nodes (junction points) in the circuit Step 2: Assign current directions to each branch (guess is OK) If result is negative → actual direction was opposite Step 3: Apply KCL at each node: ΣI_in = ΣI_out Step 4: Identify all independent closed loops Step 5: Apply KVL around each loop: ΣV = 0 Step 6: Solve the system of simultaneous equations Tip: n nodes → (n−1) independent KCL equations Works for both DC and AC circuits
// CHEAT SHEET — ALL ELECTRONICS FORMULAS & FACTS
ALL FORMULAS
Must-Know Formulas
Ohm's Law: V = IR | I = V/R | R = V/I Power: P = VI = I²R = V²/R Current: I = Q/t Frequency: f = 1/T KCL (node): ΣI_in = ΣI_out (no charge lost) KVL (loop): ΣV = 0 (voltage drops = voltage rises) Resistors Series: Rₜ = R₁ + R₂ + R₃ Resistors Parallel: 1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ Capacitors Series: 1/Cₜ = 1/C₁ + 1/C₂ ← OPPOSITE to R! Capacitors Parallel: Cₜ = C₁ + C₂ AC waveform: f = 1/T Vpp = 2·Vp Vav = 0.637·Vp Vrms = 0.707·Vp (= effective/household value)
COMPONENTS
7 Components Quick Summary
Resistor → opposes current · V=IR · unit Ω Capacitor → stores electric field energy · unit Farad Inductor → stores magnetic field energy · unit Henry Diode → one-way current (anode→cathode only) LED → diode that emits light when forward biased Transistor → amplify/switch · 3 terminals: Base, Collector, Emitter IC → millions of transistors on chip · brain of circuit
COLOUR CODE
Resistor Colour Code
Value = (Band1)(Band2) × Multiplier ± Tolerance Digit: Black=0 Brown=1 Red=2 Orange=3 Yellow=4 Green=5 Blue=6 Violet=7 Grey=8 White=9 Multiplier: Gold=×0.1 Silver=×0.01 Tolerance: Gold=±5% Silver=±10% Brown=±1% Red=±2% Example: Brown-Black-Red-Gold = 10 × 100 ± 5% = 1kΩ ±5%
GOTCHAS
Common Mistakes
✗ Capacitor series ≠ Resistor series (formulas are REVERSED) ✗ Conventional current (+→−) ≠ electron flow (−→+) ✗ Vrms ≠ Vpeak — household "220V" means 220V RMS → Vpeak = 220/0.707 ≈ 311V DC = straight line, AC = sine wave, reverses direction Ohm's Law only valid for resistors (not diodes/LEDs) Transistor: Base controls Collector-Emitter current KCL: charge in = charge out at every node (no charge lost) KVL: sum of all voltages in any closed loop = 0 KCL + KVL + Ohm's Law → can solve ANY circuit
Cultural Studies
HUM-140 · 1ST YEAR · WEEKS 1–4 · MIDTERM REVIEW
4 WEEKS
4WEEKS
16TOPICS
4QUIZ SETS
SIGNSKEY THEME
// WEEK 1 — MORPHOLOGY OF CULTURE · LANGUAGE OF CULTURE
BASICS
What Is Culture?
Culture is the full range of learned human behavior, values, beliefs, language, and practices. It includes both ideas in people’s minds and material things created by society.
Non-material: values, norms, language, behavior Material: tools, buildings, clothing, art
MODEL
Three Elements of Culture
Mentifacts → ideas, beliefs, knowledge Sociofacts → family, law, institutions Artifacts → objects, technology, architecture
THEORY
Main Approaches
Symbolic theory → culture is a system of meanings Sociobiological theory → ideas spread like memes Functional view → culture helps people adapt and survive
FAST TERMS
Important Concepts
High culture → elite cultural patterns Popular culture → culture of the majority Mainstream culture→ values accepted by most of society Subculture → culture of a specific group Counterculture → group opposing dominant norms
// WEEK 2 — NOMADIC CULTURE OF KAZAKHSTAN · PROTO-TURKIC HERITAGE
CORE IDEA
What Makes Culture Nomadic?
Nomadic culture is based on pastoral life, mobility, and adaptation to степь, semi-desert, and mountain environments. Movement is a normal way of life, not a temporary exception.
Main traits: herding, seasonal movement, steppe worldview
TYPES
Forms of Nomadism
Nomadic → constantly moving Semi-nomadic → winter settlement + seasonal moves Semi-settled → part moves, part stays Yaylak → summer movement to highland pastures
HERITAGE
Proto-Turkic Spiritual World
Tengrism → Heaven, Earth, Man Fire cult → purification and protection Aruach → ancestor spirits Zher-Su → sacred earth-water forces
WHY IMPORTANT
Contribution of Nomads
Portable yurt, carts, felt culture, horse tradition, weapons technology, Silk Road exchange between East and West
// WEEK 3 — SEMIOTICS OF CULTURE · ANATOMY OF CULTURE
CODES
Types of Cultural Codes
Preliterate → oral tradition, myth, ritual Written → books, law, logic, history Screen → visual impact, mosaic perception Digital → hypertext, networks, interactivity
SEMIOTICS
What Is a Sign?
Semiotics studies how culture communicates through signs and symbols.
Saussure: signifier + signified Peirce: icon, index, symbol
MYTH
Myth as Cultural Meaning
Classical view → myth explains the world Barthes → myth turns meanings into ideology Functions → integrative, cognitive, compensatory
SPIRITUAL CULTURE
Morality and Early Beliefs
Animism, totemism, fetishism, pantheism, deism, monotheism Morality = socially formed system of norms, values, duty
// WEEK 4 — MEDIEVAL CENTRAL ASIA · TURKIC CULTURAL HERITAGE
URBAN LIFE
Medieval Turkic City
Citadel → ruler and fortress Shahristan → central residential zone Rabad → crafts and trade outside walls
CULTURE
Urban Development
Mosques, baths, bazaars, glazed ceramics, glassmaking, metalwork, coin minting, Silk Road trade
KEY FIGURES
Intellectual and Spiritual Giants
Yusuf Balasagun → Kutadgu Bilig Mahmud al-Kashgari → Divan lugat at-Turk Al-Farabi → The Virtuous City Khoja Akhmet Yassavi → Divani Hikmet
HISTORICAL NOTE
Resilience and Revival
Mongol invasion damaged many cities, but later trade and architecture revived under new political stability
// CULTURAL STUDIES — SHORT CHEAT SHEET
MUST REMEMBER
Fast Review
Culture = learned behavior + values + symbols + artifacts Mentifacts / Sociofacts / Artifacts High / Popular / Mainstream / Subculture / Counterculture Cultural diffusion = spread of traits between cultures Cultural lag = parts of culture change at different speeds
NOMADIC BLOCK
Kazakh & Turkic Heritage
Nomadism = mobility + herding + adaptation Tengrism = Heaven, Earth, Man Aruach = ancestor spirit Silk Road = exchange of goods and culture
SEMIOTICS BLOCK
Signs and Myth
Saussure → signifier/signified Peirce → icon / index / symbol Myth → message carrying cultural ideology
MEDIEVAL BLOCK
Cities and Thinkers
Citadel / Shahristan / Rabad Otrar, Taraz, Balasagun Al-Farabi, Balasagun, al-Kashgari, Yassavi
Political Science
POL-150 · 1ST YEAR · WEEKS 1–3
3 WEEKS
3WEEKS
24KEY TERMS
7THINKERS
MFTFOCUS TOPIC
// WEEK 1 — POLITICAL SCIENCE AS A SCIENCE
DEFINITION
What Is Politics?
Politics is the organization and use of power in society. It includes the ideas, institutions, and processes through which communities make decisions, resolve conflicts, and shape collective life.
Politics = power + conflict + rules + decision-making Question often asked: who gets what, when, and how?
DEFINITION
Political Science
Political science studies how power works, how institutions function, how decisions are made, and how political ideas affect everyday life. It connects politics with society, law, economics, culture, and technology.
CORE MODEL
5 Main Functions of Political Science
1. Descriptive → describes political reality 2. Explanatory → explains causes and patterns 3. Predictive → makes evidence-based forecasts 4. Normative → asks what is just / legitimate / fair 5. Practical → gives recommendations for policy and governance
METHODS
Quantitative vs Qualitative Methods
QUANTITATIVE
numbers, surveys, election data, statistics, experiments
QUALITATIVE
case studies, interviews, historical and comparative analysis
EXAMPLES
Common Research Methods
Case study → deep analysis of one country / leader / policy Survey → public opinion and political behavior Experiment → tests effects of a variable on behavior Comparison → contrasts systems, institutions, or outcomes
// WEEK 2 — DEVELOPMENT OF POLITICAL THOUGHT
BIG IDEA
Political Thought Across Civilizations
Political thought developed from ancient times to the modern era. It reflects ideas about power, justice, law, state, authority, morality, and governance in different civilizations.
ANCIENT CHINA
Three Major Traditions
Confucianism → moral leadership, tradition, hierarchy, family values Legalism → strict laws, harsh punishments, strong authority Daoism → harmony with nature, minimal interference, non-action (wu wei)
ANCIENT INDIA
Pragmatic and Moral Approaches
Arthashastra (Kautilya) → statecraft, diplomacy, economy, strategy Buddhism → ethics, nonviolence, welfare, just rule Charvaka → rationalism, rejection of religious dogma
ANCIENT GREECE
Key Thinkers of the Ancient West
Socrates → dialogue and questioning (maieutics) Plato → ideal state, philosopher-king, justice in ordered classes Aristotle → forms of government, polity as best practical regime
MIDDLE AGES & RENAISSANCE
From Theology to Political Realism
Augustine → state is necessary because humans are sinful Aquinas → politics should promote the common good and virtue Machiavelli → politics should be studied realistically, not morally Bodin → sovereignty is the absolute and perpetual power of the state
KAZAKHSTAN
Political Thought in Kazakhstan
Al-Farabi → virtuous city, moral and intellectual ruler Valikhanov → reform, secular education, equality before law Altynsarin → literacy, education, social progress Abai → truth, dignity, justice, wisdom
// WEEK 3 — MORAL FOUNDATIONS OF POLITICS
BIG QUESTION
Morality and Politics
Politics is not morally neutral. Some see politics mainly as power and interests, while others argue that morality is essential for justice, legitimacy, and public trust.
MFT
Moral Foundations Theory
Political behavior is shaped not only by reason, but also by moral intuitions. Different people prioritize different moral foundations, which influences their political views.
Core foundations often discussed: Care / Harm Fairness / Cheating Loyalty / Betrayal Authority / Subversion Liberty / Oppression
MORAL TRADITIONS
Three Traditions in Political Thought
Utilitarianism → right action maximizes happiness Marxism → class conflict, material conditions, means of production Social Contract→ people give up some freedom for order, security, legitimacy
POWER & OBEDIENCE
Socio-Political Experiments
Stanford Prison Experiment (Zimbardo, 1971) → structure and roles can shape abusive behavior Milgram Obedience Experiments → many people obey authority even when actions seem wrong Political lesson: institutions must limit power accountability matters good intentions alone are not enough
DISCUSSION CASES
Modern Political Problems
AI surveillance Encrypted messages and privacy Cybersecurity monitoring AI replacing jobs Big Tech data collection AI in courts Internet censorship Deepfakes in politics
// MASTER CHEAT SHEET — POLITICAL SCIENCE WEEKS 1–3
FAST RECAP
Must-Know Definitions
Politics → organization and use of power in society Political science → study of power, institutions, decisions, and political ideas Sovereignty → absolute and perpetual power of the state Social contract → agreement creating legitimate political authority General will → collective interest of the people Wu wei → non-action / minimal interference in governing
5 FUNCTIONS
Political Science Does 5 Things
Describe Explain Predict Evaluate (normative) Apply in practice
THINKERS
Who Said What?
Confucianism → moral leadership Kautilya → strategy and statecraft Plato → philosopher-king Aristotle → polity Augustine → state because of sinful nature Aquinas → common good Machiavelli → political realism Bodin → sovereignty Rousseau → social contract, general will Al-Farabi → virtuous city
EXAM MEMORY
Week 3 in One Minute
Morality matters in politics People react through moral intuitions Utilitarianism = greatest happiness Marxism = class and material relations Social contract = freedom exchanged for order Milgram + Stanford = power without limits leads to abuse
Calculus 2
MATH-202 · 1ST YEAR · LECTURES 1–4
4 LECTURES
4LECTURES
18+TOPICS
3KEY TESTS
TSCORE TOOL
// LECTURE 1 — INFINITE SERIES & CONVERGENCE
DEFINITION
Infinite Series & Partial Sums
An infinite series has the form a₁ + a₂ + a₃ + .... Its behavior is determined by the sequence of partial sums sₙ = a₁ + ... + aₙ. The series converges if the partial sums approach a finite limit; otherwise it diverges.
∞ Σ aₙ converges ⇔ sₙ = Σₖ₌₁ⁿ aₖ has a finite limit L n=1 If sₙ → L, then Σ aₙ = L
THEOREM
Geometric Series
a + ar + ar² + ar³ + ... Converges if |r| < 1, and ∞ Σ arⁿ⁻¹ = a / (1 - r) n=1 Diverges if |r| ≥ 1
EXAMPLE
Telescoping Series
A telescoping series simplifies because many terms cancel. For example, 1 / (n(n+1)) = 1/n - 1/(n+1), so the series collapses after expansion.
∞ Σ 1 / (n(n+1)) = 1 n=1
TEST
n-th Term Test for Divergence
If lim aₙ ≠ 0 or the limit does not exist, then the series definitely diverges. This is a quick first check.
If lim aₙ ≠ 0 ⇒ Σ aₙ diverges
INTEGRAL TEST
Integral Test & p-Series
For positive, continuous, decreasing functions, the series and the improper integral behave the same. This gives the classic result for p-series.
∞ ∞ Σ 1/nᵖ behaves like ∫ 1/xᵖ dx n=1 1 p-series: Converges if p > 1 Diverges if 0 < p ≤ 1
// LECTURE 2 — POWER SERIES
DEFINITION
Power Series
A power series centered at 0 has the form Σ cₙxⁿ. Centered at a, it becomes Σ cₙ(x-a)ⁿ. It acts like an infinite polynomial.
∞ Σ cₙxⁿ or Σ cₙ(x-a)ⁿ n=0 n=0
KEY IDEA
Radius of Convergence
Every power series has a radius of convergence R. Inside |x-a| < R it converges absolutely. Outside it diverges. At endpoints, you must test separately.
|x-a| < R → converges |x-a| > R → diverges x = a ± R → check separately
EXAMPLE
Geometric Power Series
∞ Σ xⁿ = 1 + x + x² + ... = 1/(1-x), |x| < 1 n=0
THEOREM
Term-by-Term Differentiation
Inside the interval of convergence, you can differentiate power series term by term.
If f(x) = Σ cₙ(x-a)ⁿ, then f'(x) = Σ n cₙ (x-a)ⁿ⁻¹
THEOREM
Term-by-Term Integration
You can also integrate power series term by term, which is very useful for building new series such as ln(1+x).
∫ Σ cₙ(x-a)ⁿ dx = Σ cₙ(x-a)ⁿ⁺¹/(n+1) + C ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ... for -1 < x < 1
// LECTURE 3 — TAYLOR, MACLAURIN & APPLICATIONS
DEFINITION
Taylor & Maclaurin Series
The Taylor series of a function at x = a is built from derivatives at that point. The Maclaurin series is the special case a = 0.
∞ Σ f⁽ᵏ⁾(a)/k! · (x-a)ᵏ k=0 Maclaurin: a = 0
MUST KNOW
Standard Maclaurin Series
1/(1-x) = 1 + x + x² + x³ + ... eˣ = 1 + x + x²/2! + x³/3! + ... sin x = x - x³/3! + x⁵/5! - ... cos x = 1 - x²/2! + x⁴/4! - ... ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ...
APPLICATION
Binomial Series
(1+x)ᵐ = 1 + mx + m(m-1)/2! · x² + ... Converges absolutely for |x| < 1
APPLICATION
Estimate Integrals & Limits
Taylor series helps estimate nonelementary integrals and evaluate indeterminate forms by replacing complicated functions with simpler polynomial approximations.
Example ideas: • estimate ∫₀¹ sin(x²) dx • evaluate lim x→1 ln(x)/(x-1) • evaluate lim x→0 (sin x - tan x)/x³
FOURIER
Fourier Series Basics
A periodic function of period 2π can be represented using sines and cosines.
f(x) = a₀ + Σ (aₙ cos nx + bₙ sin nx) For period 2π: a₀ = (1/2π)∫₋π^π f(x)dx aₙ = (1/π)∫₋π^π f(x)cos(nx)dx bₙ = (1/π)∫₋π^π f(x)sin(nx)dx
// LECTURE 4 — DIFFERENTIAL EQUATIONS & COMPLEX NUMBERS
DEFINITION
Differential Equation
A differential equation contains an unknown function and its derivatives. If the function depends on one variable, it is an ODE; if it depends on several variables, it is a PDE.
ODE → one independent variable PDE → two or more independent variables
CLASSIFICATION
Order & Degree
The order is the highest derivative present. The degree is the power of the highest derivative after radicals/fractions are removed.
y' = 2x + 3 → 1st order y'' + 9y' + 3 = 0 → 2nd order
DEFINITION
Complex Number
A complex number has the form z = a + bi, where i² = -1. The real part is a, the imaginary part is b.
z = a + bi |z| = √(a² + b²) arg(z) = φ
TRIG FORM
Polar / Trigonometric Form
z = r(cos φ + i sin φ) Multiplication: multiply magnitudes, add arguments Division: divide magnitudes, subtract arguments
FORMULA
De Moivre's Theorem
If z = r(cos θ + i sin θ), then zⁿ = rⁿ(cos nθ + i sin nθ) Useful for powers and roots of complex numbers.
// CHEAT SHEET — CALCULUS 2
QUICK TESTS
Convergence Fast Check
1. Check lim aₙ. If not 0 → diverges 2. Geometric? Use |r| < 1 3. p-series? Σ1/nᵖ converges only if p > 1 4. Positive decreasing terms? Try Integral Test
POWER SERIES
Must Remember
Σ xⁿ = 1/(1-x), |x|<1 ln(1+x) = x - x²/2 + x³/3 - ... Differentiate / integrate term-by-term only inside interval of convergence Always test endpoints separately
MACLAURIN LIST
Top 5 Expansions
eˣ = 1 + x + x²/2! + x³/3! + ... sin x = x - x³/3! + x⁵/5! - ... cos x = 1 - x²/2! + x⁴/4! - ... 1/(1-x) = 1 + x + x² + ... ln(1+x) = x - x²/2 + x³/3 - ...
REMEMBER
Complex Numbers & DE
z = a + bi |z| = √(a²+b²) z = r(cosφ + i sinφ) zⁿ = rⁿ(cos nφ + i sin nφ) ODE → one variable PDE → several variables Order = highest derivative