Starter decks
AP starter flashcard decks
Core vocabulary for 23 AP courses we're still building out. Each set covers the terms that show up first — open any course in the app to study them as flashcards and generate practice questions.
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📈 AP Microeconomics
10 starter terms · Core Concepts
Scarcity The fundamental economic problem: unlimited wants but limited resources, forcing choices and trade-offs
Opportunity Cost The value of the next-best alternative given up when a choice is made
Law of Demand As price rises, quantity demanded falls (inverse relationship), ceteris paribus
Law of Supply As price rises, quantity supplied rises (direct relationship), ceteris paribus
Equilibrium Price The price where quantity demanded equals quantity supplied — the market clears
Price Elasticity of Demand Measures how responsive quantity demanded is to a price change: %ΔQd ÷ %ΔP
Consumer Surplus Difference between what buyers are willing to pay and what they actually pay
Marginal Cost The additional cost of producing one more unit of output
Perfect Competition Market with many firms, identical products, no barriers to entry; firms are price takers
Monopoly Single seller with high barriers to entry; sets price above marginal cost, creating deadweight loss
🌐 AP Comparative Government
10 starter terms · Foundations
Regime The fundamental rules and norms of politics — how power is gained and exercised (democracy vs. authoritarianism)
Legitimacy The right to rule as accepted by the governed; sources include tradition, charisma, and rational-legal authority
The Six Core Countries UK, Russia, China, Mexico, Nigeria, and Iran — the comparison set for the entire course
Parliamentary System Executive (PM) chosen from and accountable to the legislature — e.g., the United Kingdom
Unitary vs. Federal Unitary: power concentrated in central government (UK, China, Iran). Federal: divided between levels (Russia, Mexico, Nigeria)
Authoritarian Regime Power concentrated in a leader/small elite not constitutionally responsible to the public (China, Russia, Iran)
Civil Society Organizations outside the state (unions, churches, NGOs) that can check government power
Rentier State State deriving most revenue from resource exports rather than taxation — e.g., Iran and Nigeria with oil
Supreme Leader (Iran) Iran's highest authority: controls military, judiciary, and media; sits above the elected president
Politburo Standing Committee China's top decision-making body — roughly seven members at the apex of the Communist Party
🖼️ AP Art History
10 starter terms · Key Works & Terms
Formal Analysis Interpreting a work through line, color, composition, scale, and material rather than only subject matter
Contrapposto Weight-shift stance in figures (one leg bearing weight) pioneered in Classical Greek sculpture
Parthenon Athens, 447–432 BCE; Iktinos & Kallikrates. Doric temple to Athena embodying ideal proportion
Chiaroscuro Dramatic modeling with light and dark — perfected by Caravaggio and Baroque painters
Linear Perspective Renaissance system (Brunelleschi) creating depth with orthogonals converging to a vanishing point
Sistine Chapel Ceiling Michelangelo, 1508–1512. Frescoes of Genesis scenes; High Renaissance mastery of the human form
Ukiyo-e Japanese 'pictures of the floating world' woodblock prints — e.g., Hokusai's Great Wave (c. 1830)
Impressionism 1870s French movement capturing fleeting light with visible brushwork — Monet, Renoir, Morisot
Cubism Picasso & Braque fracturing form into geometric planes with multiple simultaneous viewpoints
Benin Bronzes Brass plaques and heads from the Kingdom of Benin (Nigeria) honoring obas — royal power and ancestry
🎼 AP Music Theory
10 starter terms · Fundamentals
Major Scale Pattern Whole–Whole–Half–Whole–Whole–Whole–Half; the basis of Western tonality
Interval The distance between two pitches, named by number and quality (e.g., perfect fifth, minor third)
Triad Three-note chord built in thirds: root, third, fifth. Qualities: major, minor, diminished, augmented
Circle of Fifths Arrangement of keys by ascending fifths showing key-signature relationships (C→G→D adds sharps)
Roman Numeral Analysis Labeling chords by scale degree: uppercase = major (I, IV, V), lowercase = minor (ii, vi)
Cadence Types Authentic (V–I), plagal (IV–I), half (ends on V), deceptive (V–vi)
Voice Leading Smooth connection of chords: avoid parallel fifths/octaves, resolve tendency tones (7̂→1̂, 4̂→3̂)
Non-Chord Tones Notes outside the harmony: passing tone, neighbor tone, suspension, appoggiatura, anticipation
Secondary Dominant A chord tonicizing another degree — V/V (five of five) resolves to V
Modes Scales from rotating the major scale: Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, Locrian
📉 AP Precalculus
10 starter terms · Function Toolkit
Function A relation assigning exactly one output to each input; passes the vertical line test
Average Rate of Change Slope of the secant line: [f(b) − f(a)] ÷ (b − a) over interval [a, b]
Polynomial End Behavior Determined by the leading term: even degree → ends match; odd degree → ends oppose
Exponential Function f(x) = a·bˣ — constant PERCENT change per unit; b>1 growth, 0<b<1 decay
Logarithm The inverse of exponentiation: log_b(x) = y means bʸ = x
Radian Angle where arc length equals radius; 180° = π radians
Unit Circle Values cos θ = x-coordinate, sin θ = y-coordinate of the point at angle θ on the circle of radius 1
Sinusoidal Function f(x) = A·sin(B(x−C)) + D: amplitude A, period 2π/B, phase shift C, midline D
Inverse Function Swaps inputs and outputs: f⁻¹(f(x)) = x; graph reflects across y = x
Polar Coordinates Locating points by (r, θ) — distance from origin and angle from polar axis
⚡ AP Physics C: E&M
10 starter terms · Core Principles
Coulomb's Law F = kq₁q₂/r² — electrostatic force between point charges, along the line joining them
Electric Field Force per unit charge: E = F/q. Points away from positive charges, toward negative
Gauss's Law Flux through a closed surface equals enclosed charge over ε₀: ∮E·dA = Q/ε₀
Electric Potential Potential energy per unit charge: V = U/q; E = −dV/dx (field points downhill in potential)
Capacitance C = Q/V. Parallel plate: C = ε₀A/d. Stores energy U = ½CV²
Ohm's Law & Resistivity V = IR; resistance R = ρL/A depends on material and geometry
Kirchhoff's Rules Junction: currents in = currents out (charge conservation). Loop: ΣΔV = 0 (energy conservation)
Magnetic Force F = qv × B on charges, F = IL × B on wires — always perpendicular to velocity
Ampère's Law ∮B·dl = μ₀I_enc — relates magnetic field circulation to enclosed current
Faraday's Law EMF = −dΦ/dt: changing magnetic flux induces EMF; Lenz's law gives the minus sign
🇪🇸 AP Spanish Language
10 starter terms · Essential Structures
Pretérito vs. Imperfecto Pretérito: completed past actions (fui). Imperfecto: ongoing/habitual past (iba) — background & description
Subjuntivo Mood for doubt, desire, emotion: 'Espero que vengas.' Triggered by WEIRDO: Wishes, Emotions, Impersonal, Recommendations, Doubt, Ojalá
Ser vs. Estar Ser: identity, origin, time (DOCTOR). Estar: location, condition, progressive (PLACE)
Por vs. Para Por: cause, duration, exchange ('por la mañana'). Para: purpose, destination, deadline ('para mañana')
Object Pronoun Order Indirect before direct: 'Me lo dio.' Le/les → se before lo/la: 'Se lo dije'
Conditional + Si Clauses 'Si tuviera dinero, viajaría' — imperfect subjunctive + conditional for hypotheticals
Presentational Writing AP task: argumentative essay synthesizing three sources (article, chart, audio) in 40 minutes
Registro Formal (usted) vs. informal (tú) address — match register to audience on every AP task
Comparación Cultural Speaking task comparing a cultural feature of your community with one from a Spanish-speaking region
Conectores Transition words that boost scores: sin embargo, por lo tanto, en cambio, a pesar de, no obstante
📕 AP Spanish Literature
10 starter terms · Obras y Términos
Lazarillo de Tormes Anonymous, 1554. First picaresque novel — a poor boy's satirical account serving corrupt masters
Don Quijote (fragmentos) Cervantes, 1605. Idealism vs. realism; metafiction; the birth of the modern novel
Sor Juana Inés de la Cruz Mexican nun-poet ('Hombres necios') defending women's right to learn — colonial proto-feminism
Federico García Lorca 'La casa de Bernarda Alba' — honor, repression, and tragedy in rural Spain; symbolism of white/black
Gabriel García Márquez 'El ahogado más hermoso del mundo' — magical realism transforming a village through myth
Jorge Luis Borges 'El Sur', 'Borges y yo' — labyrinths, doubles, and the blurred line between reality and fiction
Realismo Mágico Fantastic events narrated as ordinary — a hallmark of the Latin American Boom
El Boom 1960s–70s explosion of Latin American fiction: García Márquez, Cortázar, Fuentes, Vargas Llosa
Métrica Verse analysis: syllable count, rima consonante/asonante, arte mayor/menor, the sonnet's 14 hendecasyllables
Tono y Voz Narrativa Analyze who narrates (primera/tercera persona, omnisciente) and attitude — required in every essay
🇫🇷 AP French Language
10 starter terms · Structures Essentielles
Passé Composé vs. Imparfait Passé composé: completed events (j'ai mangé). Imparfait: habitual/background (je mangeais)
Le Subjonctif After expressions of emotion, doubt, necessity: 'Il faut que tu fasses...' — irregular stems matter
Object Pronouns Order before verb: me/te/nous/vous → le/la/les → lui/leur → y → en ('Je le lui donne')
Si Clauses Si + présent → futur; Si + imparfait → conditionnel; Si + plus-que-parfait → conditionnel passé
Agreement (Accord) Past participle agrees with preceding direct object with avoir; with subject when using être
Le Conditionnel Politeness and hypotheticals: 'Je voudrais', 'Il aimerait' — future stem + imparfait endings
Faire Causatif 'Faire + infinitive' = to have something done: 'Je fais réparer ma voiture'
La Francophonie The French-speaking world (Sénégal, Québec, Maroc, Haïti...) — cultural comparisons on the exam
Registres de Langue Soutenu, courant, familier — match tu/vous and vocabulary to context on every task
Connecteurs Logiques pourtant, néanmoins, ainsi, en revanche, par conséquent — essential for the synthesis essay
🇩🇪 AP German Language
10 starter terms · Kernstrukturen
Four Cases Nominativ (subject), Akkusativ (direct object), Dativ (indirect object), Genitiv (possession)
Word Order: V2 Rule The conjugated verb is ALWAYS second in main clauses: 'Heute gehe ich ins Kino'
Subordinate Clauses Conjunctions like weil, dass, obwohl kick the verb to the END: '...weil ich müde bin'
Two-Way Prepositions in, an, auf, über, unter... + Akkusativ for motion (wohin?), Dativ for location (wo?)
Perfekt Spoken past: haben/sein + past participle. Sein with motion/change: 'Ich bin gefahren'
Konjunktiv II Hypotheticals and politeness: wäre, hätte, würde + infinitive — 'Ich würde gern kommen'
Adjective Endings Depend on case, gender, and article type: 'der große Mann', 'ein großer Mann'
Modal Verbs können, müssen, dürfen, sollen, wollen, mögen — conjugated verb + infinitive at clause end
Passiv werden + past participle: 'Das Haus wird gebaut' (process) vs. sein-Passiv (state)
Relative Clauses Relative pronoun matches gender/number of antecedent, case from its own clause: 'der Mann, den ich sah'
🇮🇹 AP Italian Language
10 starter terms · Strutture Fondamentali
Passato Prossimo vs. Imperfetto Passato prossimo: completed events (ho mangiato). Imperfetto: description/habit (mangiavo)
Essere vs. Avere (ausiliari) Essere with motion/reflexives (sono andato — agrees!); avere with transitive verbs
Il Congiuntivo After opinion, emotion, doubt: 'Penso che sia...' — presente, passato, imperfetto, trapassato
Pronomi Combinati Indirect + direct merge: mi + lo → me lo; gli/le + lo → glielo ('Glielo do')
Il Condizionale Politeness & hypotheticals: 'Vorrei un caffè'; conditional past for unrealized events
Periodo Ipotetico Se + congiuntivo imperfetto → condizionale: 'Se avessi tempo, verrei'
Ci e Ne Ci = there/about it ('Ci vado'); Ne = of it/them ('Ne ho due')
Imperativo Commands: tu form drops -i for -are verbs (Parla!); negative tu = non + infinitive (Non parlare!)
Futuro Semplice Also expresses probability: 'Saranno le otto' = It's probably eight o'clock
Concordanza Adjectives agree in gender AND number: ragazza simpatica → ragazze simpatiche
🇨🇳 AP Chinese Language
10 starter terms · 核心语法 Core Grammar
了 (le) Marks completed action (我吃了饭) or change of state (下雨了) — NOT simply past tense
把 (bǎ) Construction Moves object before verb to emphasize result: 我把书放在桌子上 (I put the book on the table)
Measure Words (量词) Required between number and noun: 一本书, 三个人, 两张纸 — each noun class has its own
是…的 (shì...de) Emphasizes time/place/manner of a past event: 我是昨天来的 (It was yesterday that I came)
比 (bǐ) Comparisons A 比 B + adjective: 他比我高 (He is taller than me); never add 很 before the adjective
Tone Pairs Mandarin's four tones change meaning: mā 妈 (mother), má 麻 (hemp), mǎ 马 (horse), mà 骂 (scold)
结果补语 Result Complements Verb + result: 看见 (see-perceive), 做完 (do-finish), 听懂 (listen-understand)
虽然…但是 'Although... but' — Chinese uses BOTH halves: 虽然很贵，但是我要买
越来越 'More and more': 天气越来越热 (The weather is getting hotter and hotter)
Cultural Presentation AP task: 2-minute presentation on a Chinese cultural practice/product — festivals, tea, calligraphy
🇯🇵 AP Japanese Language
10 starter terms · 基本文法 Core Grammar
Particles は vs. が は marks topic (known info); が marks subject (new info/emphasis): 私は学生です vs. 誰が来た？
て-form Connects verbs and makes requests: 食べて (eat and.../please eat); basis for ています, てください
ています Ongoing action (食べています = is eating) or resulting state (結婚しています = is married)
Keigo (敬語) Honorific speech: sonkeigo (respect for others), kenjōgo (humility), teineigo (polite です/ます)
Verb Groups Group 1 (う-verbs: 飲む), Group 2 (る-verbs: 食べる), Irregular (する, 来る) — conjugation depends on group
Plain vs. Polite Form だ/dictionary form with friends; です・ます with strangers/superiors — register matters on the exam
たら Conditional 'If/when': 雨が降ったら、行きません (If it rains, I won't go)
Kanji + Two Readings Most kanji have on'yomi (Chinese-derived: 水 sui) and kun'yomi (native: 水 mizu)
あげる・くれる・もらう Giving/receiving verbs encode direction and social relationship — a signature AP grammar point
Text Chat Task AP task: respond to six text-chat prompts in 90 seconds each — typed in Japanese
🏺 AP Latin
10 starter terms · Grammar & Authors
The Syllabus Vergil's Aeneid (selections, Books 1–12) and Caesar's Dē Bellō Gallicō (Books 1, 4, 5, 6)
Ablative Absolute Noun + participle in ablative, grammatically independent: 'urbe captā' = with the city captured
Indirect Statement Accusative + infinitive after verbs of saying/thinking: 'Dīcit eum venīre' = He says that he is coming
Purpose Clause ut/nē + subjunctive: 'Venit ut vidēret' = He came in order to see
Cum Clauses cum + subjunctive: circumstantial (when), causal (since), concessive (although)
Dactylic Hexameter Epic meter: six feet of dactyls (—◡◡) and spondees (——); scan Vergil aloud
Pietas Aeneas's defining virtue: duty to gods, family, and fatherland — the moral engine of the Aeneid
Gerundive of Obligation -ndus + sum: 'Carthāgō dēlenda est' = Carthage must be destroyed
Subjunctive Sequence Primary tenses → present/perfect subjunctive; secondary → imperfect/pluperfect
Litotes & Chiasmus Rhetorical figures to identify: understatement by negation; ABBA word order
💬 AP Seminar
10 starter terms · QUEST Framework
QUEST The course framework: Question & Explore, Understand & Analyze, Evaluate multiple perspectives, Synthesize, Team/Transform/Transmit
Lens A perspective for examining an issue: cultural, economic, environmental, ethical, political, scientific, artistic
Line of Reasoning The logical sequence of claims and evidence that leads to a conclusion — graders trace it explicitly
Credibility (RAVEN) Evaluate sources by Reputation, Ability to see, Vested interest, Expertise, Neutrality
IMP (Individual Media Project)... IRR Individual Research Report: 1,200-word analysis of your team's research question
IWA Individual Written Argument: 2,000-word evidence-based argument from released stimulus materials
Claim Types Fact (verifiable), value (judgment), policy (should) — arguments blend all three
Counterargument Address opposing views fairly, then rebut with evidence — required for top rubric scores
Synthesis Combining multiple sources/perspectives into an original argument, not summarizing them in sequence
Plagiarism Policy Any uncited work scores zero on the entire performance task — cite everything, always
🔬 AP Research
10 starter terms · Research Methods
The Academic Paper 4,000–5,000 word original research paper — 75% of the AP score; presentation + defense is 25%
Gap in the Literature Your study must address something existing scholarship hasn't answered — the justification for your work
Research Question Focused, researchable, feasible in one year: narrow scope beats grand ambition
Literature Review Synthesizes existing scholarship by THEME (not source-by-source summary) to reveal the gap
Methodology Your systematic approach: qualitative, quantitative, or mixed methods — must align with the question
IRB Institutional Review Board approval required before any research involving human subjects
Validity vs. Reliability Validity: measures what it claims. Reliability: produces consistent results on repetition
Sampling Methods Random, stratified, convenience, snowball — each carries different bias risks you must acknowledge
Limitations Explicitly stating what your study cannot claim — strengthens rather than weakens the paper
PREP The defense: Present, Respond to questions, Engage the panel, Prove your process (3–4 oral defense questions)
🎨 AP 2-D Art & Design
10 starter terms · Portfolio & Principles
Portfolio Structure Sustained Investigation (15 images + writing, 60%) and Selected Works (5 finished pieces, 40%) — no timed exam
Sustained Investigation A year-long inquiry guided by your own question, showing practice, experimentation, and revision
Elements of Art Line, shape, color, value, texture, space, form — the raw vocabulary of every composition
Principles of Design Balance, contrast, emphasis, movement, pattern, rhythm, unity, variety, proportion
Inquiry Question Drives the portfolio: specific and visual ('How can distortion express anxiety?') beats vague themes
Documentation of Process Sketches, failed attempts, and revisions ARE scoring evidence — photograph everything
Color Theory Complementary contrast, analogous harmony, temperature, saturation — deliberate palettes score higher
Composition Strategies Rule of thirds, leading lines, framing, figure-ground relationships, cropping for tension
Writing Component 1,200 characters on your inquiry + 100 per image on materials, processes, and ideas
Artistic Integrity Work from your own photos/observation; cited appropriation only — plagiarized imagery voids the portfolio
🗿 AP 3-D Art & Design
10 starter terms · Portfolio & Form
Portfolio Structure Sustained Investigation (15 images showing works in the round) + Selected Works (5 pieces, 2 views each)
Form vs. Shape Shape is 2-D; form occupies real space with height, width, AND depth — the heart of 3-D design
Additive vs. Subtractive Additive: building up (clay, welding, assemblage). Subtractive: carving away (wood, stone, foam)
Positive & Negative Space The mass of the object AND the voids around/through it — Henry Moore made holes eloquent
Armature Internal skeleton (wire, wood) supporting clay or plaster during construction
Scale & Proportion Size relative to the human body changes meaning — Oldenburg's giant everyday objects
Texture & Surface Actual (touchable) vs. implied; patina, glaze, and finish decisions are scoring evidence
Balance in 3-D Physical AND visual: symmetrical, asymmetrical, cantilevered — gravity is part of the medium
Installation & Site Work that responds to its space; document multiple angles and viewer relationship
Craftsmanship Joints, seams, and finishing show intentionality — sloppy construction reads as unresolved ideas
✏️ AP Drawing
10 starter terms · Drawing Foundations
Portfolio Structure Same as other art portfolios: Sustained Investigation (15 images, 60%) + Selected Works (5, 40%)
Mark-Making The signature of AP Drawing: hatching, cross-hatching, stippling, gesture, smudging — visible evidence of hand
Value Structure Full range from darkest dark to lightest light; squint to check your value pattern reads
Gesture Drawing Quick 30-second to 2-minute studies capturing motion and weight before detail
Sighting & Measuring Using your pencil to compare proportions and angles from observation — draw what you SEE
Linear Perspective 1-point, 2-point, 3-point systems for constructing believable space
Atmospheric Perspective Distant objects: lighter value, less contrast, cooler color, softer edges
Foreshortening Compressing forms that project toward the viewer — the hardest observational skill
Composition & Cropping Where the figure meets the frame creates tension; avoid centering everything
Observational vs. Imaginative Colleges want to see BOTH: rigorous observation and inventive personal imagery
10 starter terms · SCOTUS & Foundational Docs
Marbury v. Madison (1803) Established judicial review — the Court can strike down laws violating the Constitution
McCulloch v. Maryland (1819) Implied powers via Necessary & Proper Clause; states cannot tax federal institutions
Brown v. Board (1954) Racial segregation in schools violates Equal Protection — overturned Plessy's 'separate but equal'
Baker v. Carr (1962) 'One person, one vote' — federal courts can hear redistricting cases (justiciability)
Tinker v. Des Moines (1969) Students don't shed free speech at the schoolhouse gate — symbolic speech protected
New York Times v. US (1971) Pentagon Papers: heavy presumption against prior restraint of the press
Federalist No. 10 Madison: a large republic controls factions better than a small democracy
Federalist No. 51 'Ambition must be made to counteract ambition' — separation of powers & checks and balances
Brutus No. 1 Anti-Federalist warning: the Constitution gives the central government dangerous, consolidating power
Letter from Birmingham Jail MLK: just vs. unjust laws; the moral duty of civil disobedience; critique of the 'white moderate'
∫ AP Calculus BC
54 starter terms · Derivatives
Derivative Definition f'(x) = lim(h→0) [f(x+h) - f(x)] / h. Rate of change of f at x. Slope of tangent line.
Power Rule d/dx[x^n] = n·x^(n-1). Most fundamental rule. Example: d/dx[x³] = 3x².
Product Rule (fg)' = f'g + fg'. Derivative of product is DIFFERENT from product of derivatives.
Quotient Rule (f/g)' = (f'g - fg') / g². Low·d-high minus high·d-low, over low squared.
Chain Rule (f∘g)' = f'(g(x))·g'(x). Derivative of composition. Critical for complex functions.
Exponential Rule d/dx[e^x] = e^x. d/dx[a^x] = a^x·ln(a).
Logarithmic Rule d/dx[ln(x)] = 1/x. d/dx[log_a(x)] = 1/(x·ln(a)).
Trigonometric Derivatives d/dx[sin(x)]=cos(x), d/dx[cos(x)]=-sin(x), d/dx[tan(x)]=sec²(x).
Inverse Trig Derivatives d/dx[arcsin(x)] = 1/√(1-x²), d/dx[arctan(x)] = 1/(1+x²).
Implicit Differentiation Differentiate both sides of equation; solve for dy/dx. Used when can't solve for y explicitly.
Related Rates Two quantities related by equation; differentiate with respect to time. Example: ladder sliding down wall.
Unit 2 Summary Derivative formulas: power, product, quotient, chain rules. Applies to exponential, logarithmic, trig functions. Implicit differentiation handles complex relationships.
Antiderivative F'(x) = f(x). Reverse of derivative. ∫f(x)dx = F(x) + C. Indefinite integral; +C accounts for constant.
Power Rule (Integration) ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ -1. Reverse of power rule for derivatives.
Exponential Integration ∫e^x dx = e^x + C. ∫a^x dx = a^x / ln(a) + C.
Logarithmic Integration ∫(1/x)dx = ln|x| + C. Absolute value accounts for negative x.
Trigonometric Integration ∫sin(x)dx = -cos(x) + C. ∫cos(x)dx = sin(x) + C. ∫sec²(x)dx = tan(x) + C.
Definite Integral ∫[a,b] f(x)dx = F(b) - F(a). Fundamental Theorem of Calculus. Represents signed area under curve.
Area Under Curve ∫[a,b] f(x)dx if f>0. If f<0, integral negative; use |∫| for area.
Riemann Sums Approximate integral using rectangles. Left, right, midpoint, or trapezoidal. Converges to definite integral as n→∞.
Substitution (u-substitution) Let u = g(x), du = g'(x)dx. Transform integral: ∫f(g(x))g'(x)dx = ∫f(u)du. Reverse of chain rule.
Integration by Parts ∫u dv = uv - ∫v du. Choose u, dv to simplify. Remember LIATE: Log, Inverse trig, Algebraic, Trig, Exponential.
Unit 4 Summary Antiderivatives reverse derivatives. Definite integrals represent area. Substitution and by-parts handle complex integrands.
Area Between Curves A = ∫[a,b] |f(x) - g(x)| dx. Find intersections; integrate difference.
Volume of Solids (Disk/Washer) V = π∫[a,b] R²(x) dx (disk) or π∫[a,b] (R² - r²) dx (washer). Rotate curve around axis.
Volume of Solids (Shell Method) V = 2π∫[a,b] x·f(x) dx. Integrate around vertical axis when washer method difficult.
Arc Length L = ∫[a,b] √(1 + (f'(x))²) dx. Length of curve from a to b.
Surface Area S = 2π∫[a,b] f(x)√(1 + (f'(x))²) dx. Area of surface of revolution.
Average Value f_avg = (1/(b-a))∫[a,b] f(x) dx. Height of rectangle with same area as curve.
Accumulation Functions F(x) = ∫[a,x] f(t) dt. F'(x) = f(x) (Fundamental Theorem). Accumulates area from a to x.
Differential Equations dy/dx = f(x,y). Solving yields y as function of x. Example: exponential growth dy/dx = ky.
Separation of Variables Rearrange: g(y) dy = f(x) dx. Integrate both sides. ∫g(y) dy = ∫f(x) dx + C.
Unit 5 Summary Integrals calculate area, volume, arc length, surface area. Differential equations model real-world growth/decay.
Parametric Curves x=f(t), y=g(t). Single parameter traces 2D curve. Common in physics (projectiles), engineering.
Slope of Parametric dy/dx = (dy/dt) / (dx/dt). Velocity components divided.
Arc Length (Parametric) L = ∫√[(dx/dt)² + (dy/dt)²] dt. Integrates speed over interval.
Polar Coordinates (r, θ) where r = distance, θ = angle. x = r cos θ, y = r sin θ.
Area in Polar A = (1/2)∫r² dθ from θ₁ to θ₂. Integrates sector areas.
Polar Derivatives dy/dx = (r' sin θ + r cos θ) / (r' cos θ - r sin θ) where r' = dr/dθ.
Rose Curves r = a sin(nθ) or r = a cos(nθ). n petals if n odd, 2n if n even. Symmetric patterns.
Circles & Limacon r = a (circle), r = a + b sin θ (limacon). Various polar curves from equations.
Lemniscate r² = a² cos(2θ) or r² = a² sin(2θ). Figure-eight shaped. Symmetric about axes.
Spiral r = aθ or r = ae^(bθ). Logarithmic spiral common in nature (shells, galaxies).
Unit 9 Summary Parametric curves: dx/dx via derivatives of components. Polar: r² dθ integration. Rose, limacon, spiral, lemniscate curves.
Taylor Series Σ f^(n)(a)/n! · (x-a)^n. Polynomial approximation around point a. Exact for polynomials.
Maclaurin Series Taylor series around a=0. Σ f^(n)(0)/n! · x^n. Special case of Taylor.
Common Maclaurin Series e^x = Σ x^n/n!, sin(x) = Σ (-1)^n x^(2n+1)/(2n+1)!, cos(x) = Σ (-1)^n x^(2n)/(2n)!
Radius of Convergence Power series Σ aₙ(x-c)^n converges for |x-c|<R. Found using ratio test.
Interval of Convergence Range of x values where series converges. May include endpoints (test separately).
Error in Taylor Polynomial Remainder R_n(x) = f(x) - P_n(x). Bounded by Lagrange error formula.
Lagrange Error Bound |R_n(x)| ≤ M|x-a|^(n+1)/(n+1)! where M = max|f^(n+1)| on interval.
Power Series Manipulation Add, multiply, differentiate, integrate power series term-by-term within convergence radius.
Binomial Series (1+x)^r = 1 + rx + r(r-1)x²/2! + ... Converges for |x|<1. Extends to non-integer r.
Unit 10 Summary Taylor/Maclaurin series approximate functions. Convergence radius determined by ratio test. Error bounded by Lagrange formula.
🎢 AP Physics 1
30 starter terms · Work Definition
Work Definition W = F·d·cos(θ); force applied over distance. W = F·Δx only if F parallel to motion. Units: joules (J).
Work-Energy Theorem Work done on object = change in kinetic energy. W_net = ΔKE = ½mv² - ½mv₀².
Kinetic Energy KE = ½mv². Energy of motion. Doubles velocity → 4x kinetic energy (depends on v²).
Potential Energy (gravitational) PE = mgh (taking reference point as h=0). Changes with height. Moving up increases PE.
Potential Energy (elastic) PE = ½kx² for spring; k = spring constant, x = displacement. Stores energy when stretched/compressed.
Conservation of Energy Total energy conserved (no non-conservative forces). KE + PE = constant. Can transform between types.
Non-Conservative Forces Friction, air resistance do negative work. Mechanical energy decreases. Heat released = work by friction.
Power P = W/t = work per time unit. Units: watts (W). Also P = F·v; force times velocity.
Energy Transfer in Collisions Elastic collision: KE conserved. Inelastic: KE lost (converts to heat, sound, deformation).
Unit 3 Summary Work changes energy. KE = ½mv², PE = mgh or ½kx². Energy conserved; only transforms between types.
Momentum p = mv; property of moving object. Vector (has direction). Larger mass or velocity → more momentum.
Impulse-Momentum Theorem Impulse = change in momentum. J = FΔt = Δp = m(v_f - v_i). Force applied over time changes momentum.
Conservation of Momentum Total momentum before collision = total momentum after (no external forces). p_before = p_after.
Elastic Collision Momentum and KE both conserved. Rarely real (ideal case). Example: billiard balls (approximately).
Inelastic Collision Momentum conserved but KE lost (converted to heat, sound, deformation). Common in reality.
Perfectly Inelastic Objects stick together after collision; maximum KE loss. Example: car crashes.
Collision Analysis Before: p₁ + p₂ = total. After: p₁' + p₂' = total (same). Solve for unknown velocities.
Recoil When one object pushes off another (explosion), momenta equal/opposite. Example: gun recoil when firing.
Center of Mass Point where mass concentrated; accelerates as if all mass there. F_net = M·a_cm for whole system.
Unit 4 Summary Momentum p=mv; impulse changes momentum. Collisions conserve momentum (elastic/inelastic). Useful for analyzing interactions.
Torque τ = r × F = rF sin(θ). Rotational analog of force. Causes angular acceleration. Units: N·m.
Moment of Inertia I = Σmr²; rotational mass. Resistance to rotational acceleration. Higher I = harder to spin. Depends on mass distribution.
Rotational Second Law τ_net = Iα. Torque = moment of inertia × angular acceleration. Analogous to F = ma.
Angular Acceleration (α) Rate of change of angular velocity. α = Δω/Δt. Rad/s². Caused by net torque.
Rotational Kinetic Energy KE_rot = ½Iω². Rotating object has energy depending on moment of inertia and angular velocity.
Angular Momentum L = Iω. Conservation: if τ_net = 0, then L constant. Figure skater: pull arms in → ω increases.
Lever Arm Perpendicular distance from pivot to force line. Longer lever arm → more torque. Same force, different arm = different torque.
Rolling Motion v_cm = ωr. Combines translation + rotation. KE_total = ½mv_cm² + ½Iω². Energy split between translation/rotation.
Equilibrium Rotational Στ = 0 (no rotation) and ΣF = 0 (no translation). Rigid body balanced.
Unit 7 Summary Torque causes angular acceleration (τ = Iα). Angular momentum conserved if no external torque. Rolling combines linear + rotational motion.
🧲 AP Physics 2
30 starter terms · Electric Forces & Potential
Coulomb's Law F = k·q₁q₂/r² where k = 8.99×10⁹ N·m²/C². Electric force between charges; repulsive if same sign, attractive if opposite.
Electric Field E = F/q = kQ/r². Force per unit positive charge. Vector; points away from positive, toward negative charge.
Electric Potential V = U/q = kQ/r. Energy per unit positive charge. Scalar. Equipotentials: points of same potential.
Electric Potential Difference ΔV = V_B - V_A. Work per charge moving from A to B. W = qΔV.
Relationship: E & V E = -dV/dx. Electric field is negative gradient of potential. Higher field = steeper potential change.
Uniform Electric Field E constant between parallel plates. V = Ed where d = plate separation. Field points from + to -.
Potential Energy U = kq₁q₂/r. Repulsive (q same sign) = positive U. Attractive (q opposite) = negative U.
Conductor vs Insulator Conductor: charges move freely (metal). Electric field = 0 inside. Insulator: charges fixed (rubber). Field passes through.
Electron Volts (eV) Energy unit: 1 eV = work to move 1 electron across 1 volt potential difference. 1 eV = 1.6×10⁻¹⁹ J.
Unit 3 Summary Coulomb's law governs electric force. Electric field: force per charge. Potential: energy per charge. V and E related.
Ohm's Law V = IR where V = voltage (volts), I = current (amperes), R = resistance (ohms). Describes electrical relationship.
Resistance R = ρL/A where ρ = resistivity, L = length, A = cross-section. Longer wire = higher R. Thicker wire = lower R.
Power P = IV = I²R = V²/R. Power dissipated as heat. Measured in watts (W).
Series Circuits Components in single loop. I same everywhere. V_total = V₁ + V₂ + ... R_total = R₁ + R₂ + ...
Parallel Circuits Components in branches. V same everywhere. I_total = I₁ + I₂ + ... 1/R_total = 1/R₁ + 1/R₂ + ...
Kirchhoff's Laws Junction rule: current in = current out. Loop rule: sum of voltages around loop = 0. Essential for circuit analysis.
EMF vs Terminal Voltage EMF (ε): theoretical voltage source. Terminal V: reduced by internal resistance. V_terminal = ε - Ir.
Capacitance C = Q/V where Q = charge, V = voltage. Capacitor stores charge. Unit: farad (F).
Capacitor Energy U = ½QV = ½CV² = ½Q²/C. Energy stored in electric field.
Dielectric Material between capacitor plates; increases capacitance. C = κC₀ where κ = dielectric constant.
Unit 4 Summary Circuits: Ohm's law (V=IR). Series: R adds, I same. Parallel: V same, 1/R adds. Capacitors store charge and energy.
Wave Properties Wavelength λ (distance between peaks), frequency f (cycles per time), period T = 1/f, amplitude A (max displacement).
Wave Speed v = fλ = λ/T. Depends on medium. Light in vacuum: 3×10⁸ m/s. Sound in air: ~340 m/s.
Transverse vs Longitudinal Transverse: oscillations perpendicular to wave motion (light, water). Longitudinal: oscillations parallel (sound).
Interference Constructive: peaks align (amplitude increases). Destructive: peak+trough cancel (amplitude decreases).
Double Slit Light passes two slits; creates interference pattern. Bright: path difference = nλ. Dark: path difference = (n+1/2)λ.
Diffraction Wave bends around obstacles/edges. Single slit: dark when sin θ = nλ/w. Limits resolution (can't distinguish close objects).
Doppler Effect Observed frequency changes if source moving. f' = f(v ± v_observer)/(v ± v_source). Blue-shift: approaching, red-shift: receding.
Standing Waves Interference of wave and reflection; nodes (no motion), antinodes (max motion). Fundamental + harmonics.
Unit 6 Summary Waves: v=fλ. Interference: constructive/destructive. Diffraction limits resolution. Doppler: motion changes frequency.
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