Math notes / Calculus

Calculus · 100% marks notes

Calculus full marks come from a fixed pipeline: domain → continuity/diff → tool → evaluate. Run the same pipeline for limits, AOD, integrals, DE, and area so most stems finish in 60–90 seconds with controlled accuracy.

Goal Bank ~28–32 /100 Math (~full Calculus share)Weight ~28–30% of Math tags (2020–2026)Speed 60–90sShare 28.5%Trend Falling share

All Math notes Algebra Calculus Coordinate Trig Vectors · 3D

Jump to topicLimits AOD Integrals DE

Advanced Calculus rewards process over memory. If you can run the 60–90s pipeline cold, Limits/AOD/Integrals/DE/Area all feel like one subject. The sections below expand that pipeline into a full working system — not only quick bullets. Calculus full marks are classification discipline, not longer algebra.

Topic-wise on this page

Calculus · chapter details

Same content as each topic page — brief, paper asks, methods, formulas, and families — so you can study here first, then open a full topic only when you need samples or trends.

01 · Gatekeeper family — every Calculus paper uses this logic

Limits · Continuity · Differentiability

Full topic page →

PromiseFull marks here mean: one-sided limits table → continuity vs differentiability as separate verdicts → simplify before L’Hôpital → never apply L’Hôpital without 0/0 or ∞/∞.

Aligned to JEE Advanced differential calculus: limits, continuity, differentiability, L’Hospital, Rolle and Lagrange MVT. Piecewise/floor/|x| stems are won by LHL/RHL/LHD/RHD tables, not by smooth formulae. Limits/continuity Advanced is tables and legal L’Hôpital — not vague continuity talk.

01. Limit exists iff LHL=RHL. Continuity needs limit = f(a). Differentiability needs LHD=RHD (finite). Differentiable ⇒ continuous; converse false (|x| at 0).

02. Standard limits (sinx/x, etc.) and algebraic cancellation beat illegal L’Hôpital. State the indeterminate form explicitly when you use L’Hôpital.

03. Rolle/MVT: check hypotheses (continuity on closed, differentiability on open, f(a)=f(b) for Rolle) before concluding existence of c.

How Advanced asks this

Continuity/differentiability of piecewise f at a join
Floor / absolute-value behaviour on an interval
Limits needing rationalisation or standard forms
MC on Rolle/MVT hypotheses
NUM value of a limit

Traps that leak marks

L’Hôpital without indeterminate form
Claiming differentiability at |x| kink
Ignoring endpoints on closed intervals
Mixing continuity of f with continuity of f′
L’Hôpital on a determinate quotient
Claiming differentiability from two-sided limit of f alone
Applying MVT without verifying continuity on [a,b] and differentiability on (a,b)

Must cover (this topic)

L’Hôpital and continuity of composites

60–90s process

0–15s — Mark type: limit / continuity / differentiability / MVT. List candidate points.
15–35s — For continuity/diff: write LHL, RHL, f(a); then LHD, RHD if needed.
35–75s — Evaluate. Use standard limits, rationalisation, or L’Hôpital after stating form.
75–90s — Check endpoints on closed intervals; do not claim differentiability without continuity logic where relevant.

Methods that change the score

This chapter decides whether you trust your Calculus. These methods force the LHL/RHL table and ban illegal L’Hôpital — exactly what Advanced tests.

Method 01 · Table method

Makes continuity vs differentiability feel like bookkeeping, not philosophy.

Do this: LHL/RHL/f(a)/LHD/RHD in a tiny table at each break.

Method 02 · Simplify then L’Hôpital

Protects you from illegal L’Hôpital — Advanced designs those traps on purpose.

Do this: Cancel factors / use conjugates / standard limits first.

Method 03 · Hypothesis checklist

Rolle/MVT MC becomes a tick list — conclusions without hypotheses get rejected.

Do this: For Rolle/MVT MC, tick each hypothesis true/false.

Method 04 · Indeterminate stamp

Forces you to say 0/0 or ∞/∞ out loud before differentiating a quotient.

Do this: Write ‘0/0’ or ‘∞/∞’ before L’Hôpital.

Quick formulas

Limit existence: lim x→a f = L iff LHL = RHL = L — Two-sided limits
Continuity at a: f(a) defined, lim exists, lim = f(a) — Continuity checks
Derivative definition: f′(a)=lim h→0 [f(a+h)−f(a)]/h — First principles / one-sided derivatives
Standard limit: lim θ→0 sinθ/θ = 1 (radians) — Trig limits
L’Hôpital: If 0/0 or ∞/∞, lim f/g = lim f′/g′ (when latter exists) — Indeterminate quotients only
Rolle: f cont [a,b], diff (a,b), f(a)=f(b) ⇒ ∃c f′(c)=0 — Existence of horizontal tangent
Lagrange MVT: f cont [a,b], diff (a,b) ⇒ ∃c f′(c)=[f(b)−f(a)]/(b−a) — Mean slope realisation
Squeeze theorem: g≤f≤h and lim g=lim h=L ⇒ lim f=L — Oscillatory / boundable limits

Common question families

Piecewise continuity / differentiability

Looks like: f defined differently on intervals; ask at join / floor / |x|

Fast model: Tiny table: LHL, RHL, f(a), then LHD, RHD — separate verdicts.

0/0 limit

Looks like: Quotient or difference that vanishes at the point

Fast model: Simplify / rationalise / standard limits FIRST; L’Hôpital only after stamping 0/0 or ∞/∞.

Rolle / MVT hypothesis MC

Looks like: Which theorem applies / existence of c

Fast model: Tick hypotheses one-by-one; conclusion only if all hold.

∞/∞ or e-limit / standard forms

Looks like: lim (1+u)^{1/u}; lim (a^x−1)/x; forms that become 1^∞ after rewrite

Fast model: Rewrite to exp(lim ln) or standard (e^x−1)/x → 1; stamp the form first.

Sample preview · 0/0 limit

Evaluate lim x→0 (√(1+x)−1)/x.

Answer: 1/2

Simplify first; L’Hôpital only after stamping 0/0.

More samples on full topic page →

Open full Limits notes (samples · trends · joins) →

02 · High — tangents, monotonicity, maxima/minima every year

Applications of Derivatives

Full topic page →

PromiseAOD full marks: simplify → differentiate → critical points + sign chart (and endpoints on closed intervals) → translate to tangent/normal/increasing language.

Syllabus applications: tangents & normals, increasing/decreasing, maxima/minima, Rolle/MVT links, implicit differentiation. Sign charts beat vague graphing under time pressure. AOD under the clock is simplify → differentiate → sign chart → endpoints.

01. Tangent: slope f′(x₀), equation y−y₀=f′(x₀)(x−x₀). Normal slope is negative reciprocal when f′≠0.

02. Extrema: solve f′=0 (and undefined points in domain), use first-derivative sign chart or second-derivative test when f′′≠0.

03. On [a,b], global max/min compare critical values with f(a), f(b).

How Advanced asks this

Equation of tangent/normal
Intervals of increase/decrease
Local/global maxima-minima (often with parameter)
Implicit differentiation
Rate / related rates style (when appears)

Traps that leak marks

Missing endpoints
Critical points outside domain
Forgetting negative reciprocal for normal
Using f′′ test when f′′=0
Reporting a local extremum as global without endpoint check
Normal slope as f' instead of −1/f' (when f'≠0)
Differentiating before algebraic simplification on a monster expression

Must cover (this topic)

Rolle / Lagrange MVT geometric meaning

60–90s process

0–15s — Name ask: tangent / monotonicity / extrema / implicit.
15–35s — Compute f′ (and f′′ only if needed). Find critical points in domain.
35–75s — Sign chart of f′ or second-derivative test; form tangent/normal if asked.
75–90s — Translate to required language; check normal slope −1/m.

Methods that change the score

AOD is translation: calculus → geometry/language. These methods make sign charts and point-slope automatic so extrema and tangents stop feeling ‘graphical and vague’.

Method 01 · Simplify then differentiate

Cuts chain-rule explosions that steal time and create sign errors.

Do this: Rewrite logs/products before chain-rule explosions.

Method 02 · Sign chart over memory

Extrema become a table, not a vague graph story.

Do this: Critical points → test f′ on either side.

Method 03 · Endpoint discipline

Global max/min on [a,b] stop leaking marks at the boundary.

Do this: On [a,b], always compare f(a), f(b) with critical values.

Method 04 · Point-slope geometry

Tangent/normal become mechanical — that is how they should feel.

Do this: Derivative gives slope; geometry finishes with point-slope form.

Quick formulas

Tangent: y−y₀ = f′(x₀)(x−x₀) — Tangent at a point
Normal: slope = −1/f′(x₀) if f′(x₀)≠0 — Normal line
Product / quotient / chain: (uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²; (f∘g)′=f′(g)g′ — All routine derivatives
Implicit: dy/dx = −F_x/F_y — F(x,y)=0
First derivative test: f′ changes +→− local max; −→+ local min — Sign chart extrema
Second derivative test: f′(c)=0, f′′(c)>0 local min; f′′(c)<0 local max — When f′′ easy and nonzero
Monotonicity: f′>0 increasing; f′<0 decreasing (on interval) — Increase/decrease intervals

Common question families

Tangent / normal at a point

Looks like: Equation of tangent or normal

Fast model: Point + m=f′(x₀) → point-slope; normal uses −1/m.

Max/min / monotonicity

Looks like: Increasing intervals, local/global extrema

Fast model: f′=0 critical points → sign chart; on [a,b] also compare endpoints.

Implicit differentiation

Looks like: F(x,y)=0; dy/dx or tangent to implicit curve

Fast model: dy/dx=−F_x/F_y; only compute second derivative if asked.

Rate / related rates / approximation

Looks like: dy/dt from dx/dt; linear approximation / differentials

Fast model: Chain rule: dy/dt = y′(x) dx/dt; Δy ≈ f′(x)Δx for approximation.

Sample preview · Tangent / normal at a point

Equation of tangent to y=x² at x=1.

Answer: y = 2x − 1

Derivative → point-slope → done.

More samples on full topic page →

Open full AOD notes (samples · trends · joins) →

03 · High — King property and area bounds decide many marks

Integrals · Properties · Area

Full topic page →

PromiseIntegrals full marks: property glance (King/even-odd) before expanding → technique if needed → for area, intersections first then ∫|f−g|.

Syllabus: indefinite/definite integrals, properties, FTC, substitution, parts, partial fractions, area under curves. Advanced definite questions often reward King’s rule more than hard antiderivatives. Integrals/area Advanced is properties and correct limits — antiderivatives finish the job.

01. King’s rule: ∫_a^b f(x)dx = ∫_a^b f(a+b−x)dx. Classic: ∫₀ᵃ f/(f(x)+f(a−x)) = a/2 when defined.

02. Even/odd about 0: ∫_{-a}^{a} even = 2∫₀ᵃ; odd = 0 (when exists).

03. Area is geometry+calculus: solve f=g for bounds, test which curve is above, integrate absolute difference on each piece.

How Advanced asks this

Definite value via property
Technique: sub / parts / partial fractions
Area between curves / curve and axis
MC on integral inequalities/properties
FTC differentiation of integral

Traps that leak marks

Wrong area limits
Dropping absolute value when curves cross
Expanding when property kills
Parts when a substitution is enough
Integrating without absolute value when curves cross
Using indefinite techniques and dropping +C on a definite ask (harmless) but forgetting limit substitution
Parts thrashing: wrong u choice repeating forever

Must cover (this topic)

Definite integral properties + FTC
Area under simple curves

60–90s process

0–15s — Definite vs indefinite vs area. Mark limits and symmetry.
15–35s — Try King/even-odd/period before expanding integrand.
35–75s — Integrate with chosen technique; split area intervals if curves cross.
75–90s — Check limit order and whether property assumptions hold (denominator ≠0).

Methods that change the score

Definite integrals in Advanced are often property contests. These methods put King/intersections before antiderivatives — that is how ugly integrals finish in under a minute.

Method 01 · Property before expansion

Ugly definite integrals become constants — this is the Advanced definite-integral mindset.

Do this: Spend 10s on King/even-odd before antiderivative struggle.

Method 02 · Intersections before area

Wrong limits waste the whole attempt; this method makes bounds non-negotiable.

Do this: Solve f=g; order roots; test above/below.

Method 03 · ILATE for parts

Stops thrashing between parts and substitution under pressure.

Do this: Choose u via Inverse/Log/Algebraic/Trig/Exp preference.

Method 04 · Partial fractions gate

Rationals get a clear recipe instead of random substitutions.

Do this: Factor denominator; decompose before integrating rational functions.

Quick formulas

King’s rule: ∫_a^b f(x)dx = ∫_a^b f(a+b−x)dx — Symmetric definite limits
Classic a/2: ∫₀ᵃ f/(f(x)+f(a−x)) dx = a/2 — Standard Advanced collapse
Even / odd: ∫_{-a}^{a} even = 2∫₀ᵃ; odd = 0 — Symmetric about 0
FTC evaluation: ∫_a^b f = F(b)−F(a) — Antiderivative known
Integration by parts: ∫ u dv = uv − ∫ v du — Product integrands (ILATE pick u)
Area: ∫_a^b |f−g| dx on proper bounds — Area between curves
Reverse limits: ∫_a^b f = −∫_b^a f — Limit order fixes
Standard subs: t=tan(x/2) / x=a sinθ / etc. when structure matches — Indefinite technique

Common question families

King / symmetry definite

Looks like: Ugly integrand on [0,a] or [a,b] with hidden symmetry

Fast model: I=∫f; substitute x→a+b−x (or a−x); add; solve I — before antiderivative struggle.

Area between curves

Looks like: Bounded area; curve and axis; two curves

Fast model: Solve intersections FIRST; test which is above; ∫|f−g| piecewise.

Technique pick (parts/sub/PF)

Looks like: Indefinite integral of product / rational / composite

Fast model: 10s classify: sub if inner derivative visible; parts if product; PF if proper rational.

Even/odd on [−a,a]

Looks like: Definite integral symmetric about 0

Fast model: Odd → 0 (if exists); even → 2∫₀ᵃ — check continuity at 0.

Sample preview · King / symmetry definite

Evaluate I = ∫₀^{π/2} sin x / (sin x + cos x) dx.

Answer: π/4

Property before expansion on [0,a] uglies.

More samples on full topic page →

Open full Integrals notes (samples · trends · joins) →

04 · Medium–High — classification decides everything

Differential Equations

Full topic page →

PromiseDE full marks: rewrite standard form → stamp SEP / HOM / LIN → apply only that recipe → use IC → optionally differentiate to verify.

JEE Advanced focuses on first-order: variable separable, homogeneous, linear (IF method). Misclassification is the main failure mode — not integration skill. DE scoring is classification first; every wrong template starts with a wrong stamp.

01. Separable: dy/g(y)=f(x)dx. Homogeneous: dy/dx=f(y/x) → y=vx. Linear: dy/dx+P(x)y=Q(x) → IF=e^{∫P dx}.

02. Formation of DE / order-degree ideas appear; keep order = highest derivative order, degree = power of that derivative after clearing radicals.

03. After solving, a 10s differentiate-and-plug check catches algebra slips when time allows.

How Advanced asks this

Solve separable / homogeneous / linear DE
Particular solution with initial condition
Matching DE ↔ general solution
MC on which method applies
Order/degree identification

Traps that leak marks

Wrong class stamp
Missing product rule for y=vx
Dropping absolute value in ln
Using IF on non-linear forms
Treating homogeneous as separable without the v-sub
Wrong IF exponent (forgetting ∫P or the sign of P)
Applying IC before integrating both sides cleanly

Must cover (this topic)

Homogeneous, separable, and linear first-order DEs

60–90s process

0–15s — Rewrite dy/dx form. Note if IC given.
15–35s — Stamp SEP / HOM / LIN (one word on rough sheet).
35–75s — Apply matching recipe only. Integrate carefully; apply IC.
75–90s — Optional verify by differentiating. Check |r|-style domain issues for ln.

Methods that change the score

DE full marks = classification. These methods stamp SEP/HOM/LIN before any integration — misclassification is what actually loses marks, not hard integrals.

Method 01 · Classify first

Misclassification is the real DE killer — this stamp fixes the root cause.

Do this: One-word stamp before any integration.

Method 02 · IF mechanics

Linear DE becomes one product-rule recognition, not a mystery.

Do this: Write IF, multiply through, recognise derivative of product.

Method 03 · Homogeneous product rule

Catches the algebra leak that makes y=vx solutions wrong.

Do this: When y=vx, differentiate with product rule: v+x v′.

Method 04 · IC last

Keeps general solution clean so particular solution algebra does not tangle early.

Do this: Find general solution, then plug IC for C.

Quick formulas

Separable: ∫ dy/g(y) = ∫ f(x) dx — Variables separate cleanly
Homogeneous: y=vx ⇒ dy/dx = v+x dv/dx = f(v) — f(y/x) form
Linear IF: IF=e^{∫P dx}; d/dx(y·IF)=Q·IF — dy/dx+Py=Q
Order / degree: Order = highest derivative order; degree = power of that derivative — Classification MC
General solution: Contains arbitrary constants equal in number to order (typical first-order → one C) — Matching solution forms

Common question families

Linear first-order

Looks like: dy/dx + P(x)y = Q(x)

Fast model: Stamp LIN → IF=e^{∫P} → d/dx(y·IF)=Q·IF.

Homogeneous first-order

Looks like: dy/dx = f(y/x) or degree-balanced M,N

Fast model: Stamp HOM → y=vx → remember product rule v+x v′.

Separable

Looks like: g(y) dy = f(x) dx after rearrange

Fast model: Stamp SEP → split → integrate both sides → +C → IC.

Matching DE ↔ solution

Looks like: List of DEs vs general solutions

Fast model: Differentiate the proposed solution (or plug) independently for each list item.

Sample preview · Linear first-order

Solve dy/dx + (2/x)y = x (x>0).

Answer: y = x²/4 + C/x²

IF recipe only after LIN stamp.

More samples on full topic page →

Open full DE notes (samples · trends · joins) →

Syllabus Trends · 28.5% · Falling share Mixed joins

Official · 2026

Calculus syllabus map

Full Math syllabus →

Maps to Differential Calculus + Integral Calculus (including first-order DEs and area).

Must cover from syllabus

L’Hôpital and continuity of composites
Rolle / Lagrange MVT geometric meaning
Definite integral properties + FTC
Area under simple curves
Homogeneous, separable, and linear first-order DEs

Gaps / exclusions to watch

Implicit differentiation up to order two
Formation of ODEs from word/geometry stems

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Differential Calculus

high

Exam focusLimits with L’Hôpital, continuity, maxima/minima, MVT/Rolle, tangents.

Syllabus points

Limits and continuity (sum/difference/product/quotient); L’Hôpital
Continuity of composites; intermediate value property
Derivatives: sum/product/quotient/chain; standard function families
Tangents and normals; increasing/decreasing; second derivatives; max/min
Rolle’s and Lagrange’s MVT; geometric interpretation; implicit derivatives up to order two

Do / check

State domain/continuity before differentiating piecewise
Check endpoints separately for extrema on closed intervals

Unit 02

Integral Calculus

high

Exam focusDefinite integral properties, area, first-order DEs (homogeneous, separable, linear).

Syllabus points

Indefinite integrals of standard functions
Definite integrals as limit of sums; properties; FTC
Integration by parts, substitution, partial fractions
Area bounded by simple curves
Formation of ODEs; homogeneous first-order; separation of variables; linear first-order DE

Do / check

Use definite-integral properties before expanding
Identify DE type before applying a method

2020–2026

Calculus trends · why it matters

Full trends board →

28.5%Of Math tags

Falling shareTrend direction

Core · dailyStudy intensity

9.86Avg questions / year

Build a pipeline: limits/continuity → derivatives/AOD → DE → definite integrals/area.

21.7Floor marks (300 plan)

28.9Target marks

33.3Stretch marks

Year-wise appearance

2016

219

229

239

248

258

2610

Subtopic weight

AOD — 18 (26.1%)
Definite Integrals — 11 (15.9%)
Continuity/Differentiability — 9 (13%)
Functions — 7 (10.1%)
Limits — 7 (10.1%)
Differential Equations — 7 (10.1%)
Area — 6 (8.7%)
Differentiation — 4 (5.8%)

Often joins with

Coordinate Geometry (area/conics)
Trigonometry (integrals / equations)
Algebra (functions counting)

Type mix

Numerical — 27 (39.1%)
Single correct — 20 (29%)
Multi correct — 18 (26.1%)
Matching — 4 (5.8%)

Reading the pattern

Calculus accounts for ~28.5% of tagged math questions (2020–2026).
Trend signal: falling (early-window avg 11.3/yr vs recent avg 8.7/yr).
Appears in 7/7 tagged years; peak 2020 (16 Q), low 2024 (8 Q).
Dominant subtopics: AOD (18), Definite Integrals (11), Continuity/Differentiability (9), Functions (7).
Question-type mix: Numerical 39.1%, Single correct 29.0%, Multi correct 26.1%, Matching 5.8%.

Full Calculus subject analysis →

Cross-topic

Always find limits from geometry before writing ∫.

2020 year review →Mathematics notes →Quick card →Syllabus →

Read first

How to use these notes for 100%

Point 01

Why Calculus decides Math 100 with Algebra: it is a large share every year and joins heavily with Conics (area), Trig (limits/integrals), and Algebra (functions). Students who ‘know formulae’ still lose marks by differentiating before simplifying, using L’Hôpital illegally, or integrating area with wrong bounds.

Point 02

What ‘common knowledge’ means here: six universal gates that work on SC, MC, NUM, and MATCH. The paper changes the costume (floor function, e^x, ellipse area join) — your gate order should not change.

Point 03

How to read this page: lock the protocol first, then study each method with its worked idea, then drill sub-portions. Use worked patterns as exam-voice practice: the same rough-sheet skeleton you must write in the hall.

Point 04

Full marks mindset: correct process beats fast wrong calculus. A definite integral collapsed by property in 45s is better than a three-minute expansion that still misses a factor of 1/2.

Point 05

Depth upgrade: Limits → AOD → Integrals → DE is a skill chain. A weak LHL/RHL table leaks into AOD sign charts and DE classification under time pressure.

Point 06

Definite integrals in Advanced are often property contests — King/odd-even/intersection before antiderivative theatre.

How the paper asks

Single correct: continuity/differentiability at a point, a clean definite value, tangent slope, or DE with IC — distractors match skipped LHL/RHL or wrong IF.
Multi-correct: several statements about monotonicity, continuity, or integral inequalities; each statement is an independent T/F.
Numerical: integer answers from definite integrals, counts of roots via Rolle/IVT ideas, or evaluated extrema.
Matching: function ↔ property, DE ↔ solution form, integral ↔ value after property.
Mixed joins: Calculus + Conics (area between curves), Calculus + Trig (limit/integral identities), Calculus + Algebra (piecewise defined by floor/| |).

Scoring plan in the hall

Bank easy continuity and property-definite stems early — they are fast if you gate correctly.
Do not start a long area integral before writing intersection points; wrong limits waste the whole attempt.
If classification (Limit vs DE vs Area) is unclear in 20s, skip and return — forced tools create wrong answers.
Target high accuracy on attempted Calculus; leave only multi-join monsters for second pass.

Why it saves time

Replaces vague graphing with a 30-second table.

Worked idea

f(x)=x³−3x: f′=3(x−1)(x+1); sign chart ⇒ local max at −1, local min at 1. The chart is the answer engine; memorising ‘cubic shape’ is optional.

When not to force it

On closed intervals, also compare endpoint values — critical points alone are incomplete.

Exam voice

Worked patterns (copy this thinking)

Practise writing these rough-sheet flows until they feel automatic.

Pattern 01

Piecewise continuity / differentiability table

Exam signalf defined differently on intervals, or floor/| |, ask continuity/diff at a point.

Thinking order

List candidate points (joins, integers for floor).
Compute LHL, RHL, f(a).
If continuous, compute LHD, RHD.
Conclude statement-wise for MC.

Rough sheet skeleton

Point a = …
LHL=… RHL=… f(a)=…
LHD=… RHD=…
Continuous? Differentiable?

Final check

Did not claim differentiability without continuity. One-sided derivatives compared with the same difference-quotient definition.

Pattern 02

Definite integral by King / symmetry

Exam signalUgly definite integrand on [0,a] that looks resistant to antiderivative.

Thinking order

Write I = ∫₀ᵃ f(x)dx.
Substitute x→a−x to get I = ∫₀ᵃ f(a−x)dx.
Add and simplify f(x)+f(a−x).
Solve for I.

Rough sheet skeleton

I = …
x→a−x ⇒ I = …
2I = …
I = …

Final check

Denominator never zero on the interval; constants pulled out carefully.

Pattern 03

Area between curves with intersections first

Exam signalFind area bounded by two curves / curve and axes; often joined with conics or trig.

Thinking order

Solve f=g for x (or y).
Order roots; test which function is above on each interval.
Integrate |f−g| piecewise if needed.
Add pieces.

Rough sheet skeleton

f=g ⇒ x=…
On [ , ] above = …
∫ …
Total area = …

Final check

Limits match intersection roots; no missing middle interval when curves cross twice.

Pattern 04

First-order DE classification → recipe

Exam signaldy/dx = … or Mdx+Ndy=0 style reduced to first-order; may include IC.

Thinking order

Rewrite standard form.
Stamp SEP / HOM / LIN.
Apply only that recipe.
Use IC if given; optionally differentiate to verify.

Rough sheet skeleton

Form: …
Class: …
Working: …
General / particular solution

Final check

IF exponent has correct sign; homogeneous substitution differentiated correctly (product rule).

Instant recall

Mental bank

Keep these cold — no derivation during the 60–90s window.

L’Hôpital only for 0/0 or ∞/∞ indeterminate forms (state it first)
Rolle / Lagrange MVT: hypotheses + geometric meaning
∫₀ᵃ f(x)/(f(x)+f(a−x)) dx = a/2 (when defined)
King: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
Linear DE: dy/dx + Py = Q → IF = e^{∫P dx}
Homogeneous first-order: y = vx, then separate
Separable: dy/g(y) = f(x)dx
Area: bounds from intersections; use |f−g| if needed
d/dx |x| = sgn(x) for x≠0; floor breaks at integers
Integration by parts: ∫u dv = uv − ∫v du
Table method before L’Hôpital
Sign chart before extrema claims
Classify DE before integrating

Flash IF, King property, and DE classes for 5–7 minutes before mocks.
Each line must be writable without a book. If not, it is a 100% leak.
When you use a bank line in practice, tick it (‘used King’, ‘used IF’) to build method awareness.

Inside Calculus

Sub-portion playbooks (detailed)

Each family: explanation, paper look, template, formulae, worked mini, traps, drill.

01 · Core every-year

Limits · continuity · differentiability

60s triggerSee piecewise/floor/| |/0/0 → LHL/RHL table or indeterminate check first.

What this family really tests

This family is the gatekeeper for the rest of Calculus. Advanced repeatedly tests whether you respect one-sided behaviour. Limits that look like L’Hôpital bait are often algebraic cancellations first. Continuity is not differentiability — write both conclusions separately for MC.

How it looks in the paper

Continuity/differentiability of piecewise f at a join
Floor/greatest-integer continuity on an interval
Limits needing rationalisation or standard expansions
MC on Rolle/MVT hypotheses

Template (write on rough)

Identify problematic points and domain.
For continuity: LHL, RHL, f(a).
For differentiability: LHD, RHD (only after continuity if you use that logic carefully).
For limits: simplify; only then L’Hôpital if 0/0 or ∞/∞.
MC: each statement gets its own T/F line.

Must automate

Floor/greatest-integer continuity checklist
Composite continuity intuition
Standard limits you actually use (sinx/x, (1+x)^{1/x} ideas as needed)
Stating indeterminate form before L’Hôpital

Formula sheet for this family

lim θ→0 sinθ/θ = 1 (θ in radians)
L’Hôpital: only 0/0 or ∞/∞
Definition: f′(a)=lim h→0 [f(a+h)−f(a)]/h
Rolle: f cont [a,b], diff (a,b), f(a)=f(b) ⇒ f′(c)=0

Traps

Applying L’Hôpital without indeterminate form
Ignoring endpoint behaviour on closed intervals
Using derivative existence as a substitute for continuity checks in wording-heavy MC

Why marks leak

L’Hôpital without indeterminate form
Claiming differentiability at a corner
Ignoring endpoints on closed-interval continuity asks
Mixing continuity of f with continuity of f′

Drill

Daily: 2 piecewise continuity/diff + 1 limit. Weekly: 1 Rolle/MVT MC set with hypothesis checklist written. Open full Limits notes.

Open full Limits · continuity · differentiability notes →

Mini worked example

lim x→0 (√(1+x)−1)/x: multiply conjugate → lim x→0 1/(√(1+x)+1)=1/2. Students who jump to L’Hôpital waste time and sometimes differentiate incorrectly under stress.

02 · High · independent page

AOD · tangents · extrema

60s triggerSee max/min/increasing/tangent/normal → simplify, then f′ + sign chart.

What this family really tests

AOD is translation between calculus and geometry/language. Tangents need slope f′(x₀) and point-slope form. Extrema need critical points inside domain plus endpoint comparison on closed intervals. Implicit differentiation appears when F(x,y)=0 — keep d²y/dx² only if asked.

How it looks in the paper

Equation of tangent/normal at a point
Intervals of increase/decrease
Local/global maxima-minima (sometimes with parameter)
Implicit differentiation up to order two

Template (write on rough)

Simplify f if needed.
Compute f′ (and f′′ only if needed).
Critical points + sign chart / second derivative test.
On closed interval: compare endpoints too.
Translate to the geometric ask (tangent slope, normal, range).

Must automate

Chain/product/quotient fluency
Implicit differentiation (first order always; second when required)
Tangent: y−y₀=f′(x₀)(x−x₀)
Sign chart construction without graphing panic

Formula sheet for this family

(uv)′=u′v+uv′; (u/v)′=(u′v−uv′)/v²
d/dx f(g(x))=f′(g(x))g′(x)
For F(x,y)=0: dy/dx = −F_x/F_y (when F_y≠0)
Second derivative test: f′′>0 local min, f′′<0 local max (if f′=0)

Traps

Missing critical points from domain edges
Confusing increasing with f′≥0 vs f′>0 wording
Using f′′ test when f′′=0 (inconclusive)

Why marks leak

Missing endpoint comparison on [a,b]
Critical points outside domain kept by mistake
Normal slope as negative reciprocal forgotten
Sign chart skipped; ‘by inspection’ wrong

Drill

Daily: 1 extrema (with sign chart) + 1 tangent/normal NUM. Weekly: 1 implicit differentiation MC. Open full AOD notes.

Open full AOD · tangents · extrema notes →

Mini worked example

Tangent to y=x² at x=1: slope=2, point(1,1) ⇒ y−1=2(x−1). Normal slope=−1/2. The method is derivative → point-slope; do not re-derive from first principles in the exam.

03 · High · independent page

Integrals · properties · area

60s triggerSee definite integral → property glance; see area → intersections first.

What this family really tests

Integrals split into two exam skills: (1) property/FTC definite evaluation, (2) area as geometry with calculus. Indefinite technique (parts/sub/partial fractions) still matters, but Advanced definite questions often reward King property more than heavy antiderivatives. Area joins with curves — still intersections first.

How it looks in the paper

Definite integral value via property
FTC / differentiation under integral sign style (when appears)
Area between curves / curve and axis
MC comparing integral inequalities or properties

Template (write on rough)

Definite: check symmetry/King/even-odd before expanding.
If technique needed: pick sub / parts / PF deliberately.
Area: solve intersections; split intervals; integrate |f−g|.
Keep constants and limits tidy; avoid decimal mess.
MC: verify each integral claim independently.

Must automate

King property and a/2 classic form
By parts / sub / partial fractions quick selection
Area between two curves workflow
FTC: d/dx ∫_a^{g(x)} f = f(g(x))g′(x) (standard form)

Formula sheet for this family

∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx
∫₀ᵃ f/(f(x)+f(a−x)) dx = a/2
∫u dv = uv − ∫v du
Area = ∫_a^b |f−g| dx on proper bounds

Traps

Wrong limits for area
Dropping absolute value when curves cross
Forcing parts when substitution is enough

Why marks leak

Wrong area limits (guessed from figure)
Dropping absolute value when curves cross
Expanding integrand that property would kill
Arithmetic slips after a correct setup

Drill

Daily: 1 property definite + 1 area with written intersections. Weekly: 8-question property blitz in 12 minutes. Open full Integrals notes.

Open full Integrals · properties · area notes →

Mini worked example

Area under y=sin x from 0 to π is 2. Between y=sin x and y=0 on [0,π] is still ∫ sin=2. Students who integrate sin from 0 to π/2 only and double carelessly sometimes drop a sign — write the interval explicitly.

04 · Medium–High · independent page

Differential equations

60s triggerSee dy/dx or Mdx+Ndy → rewrite, then stamp SEP / HOM / LIN.

What this family really tests

Syllabus DE is first-order focused: formation ideas, homogeneous, separable, linear. Full marks = classification discipline. After solving, a 10-second differentiate-and-plug check catches many algebra errors when time allows.

How it looks in the paper

Solve separable / homogeneous / linear DE
Find particular solution with IC
Matching DE ↔ general solution form
MC on which method applies

Template (write on rough)

Rewrite in standard form.
Classify: separable / homogeneous / linear.
Apply matching recipe only.
Apply IC if given.
Optional: differentiate solution to verify.

Must automate

Separable integration + constant handling
y=vx for homogeneous (including dy/dx = v+x dv/dx)
Integrating factor for linear first-order
Recognising degree for homogeneous

Formula sheet for this family

Linear: dy/dx + P(x)y = Q(x), IF=e^{∫P dx}
Homogeneous: dy/dx = f(y/x) → y=vx
Separable: ∫ dy/g(y) = ∫ f(x) dx
Keep ln|·| unless domain guarantees sign

Traps

Calling a DE linear incorrectly
Losing absolute value in ln| |
Using IF when variables are not in linear form

Why marks leak

Wrong class stamp (linear vs homogeneous)
Missing product rule when differentiating y=vx
Dropping absolute value in ln
Forgetting +C or mishandling IC algebra

Drill

Daily: 1 DE rotating SEP/HOM/LIN. Weekly: 1 matching DE↔solution + 1 IC particular solution. Open full DE notes.

Open full Differential equations notes →

Mini worked example

dy/dx = (y/x)+(x/y) may be handled by v=y/x after checking homogeneous degree. Students who attempt linear IF here waste minutes — classification would have redirected immediately.

Habit

Common practice system

Same system every week — builds transferable speed.

Daily

30–35 min pipeline drill: 1 continuity/limit + 1 AOD + 1 integral or DE — each must show type name by 15s.
Speak the gate: Domain → Simplify → Tool → Check. Log any stem where you differentiated/integrated before simplifying.
Rewrite one wrong stem as a model rough sheet (LHL table / King steps / intersection list / DE class stamp).
Track late classification (>30s to choose tool) separately from arithmetic mistakes.

Weekly

1 property-integral blitz (8 questions, 12 minutes) — process score: property used? yes/no.
1 area set with mandatory written intersections before integrating.
1 mixed Calculus multi-correct using independent T/F on every option.
Error log: illegal L’Hôpital, wrong area limits, DE misclass — write one anti-trap line each.

Before every mock

Flash: IF formula, King/a/2 property, DE classes, MVT hypotheses — 6–8 minutes.
In paper: clear easy continuity and property-definite stems first to bank marks.
Promise: no area integral without intersection solve on rough sheet.

Mastery notes

Common Calculus practice means repeating the same gates until they are automatic — not random ‘more integrals’.
Two-week metric: % of Calculus stems where type was named within 15s. Aim ≥90%.
Tag misses as Classification / Gate-skip / Arithmetic / Careless. Most 100% failures are Gate-skip (property/intersections/DE class).
Once weekly, teach King property and linear IF aloud without notes. If you stumble, it is not exam-ready.

Full marks gate

100% checklist for Calculus

Tick these only when true on timed practice — not when “you know the chapter”.

Piecewise/floor stems always get LHL/RHL/LHD/RHD written.
No L’Hôpital without stating 0/0 or ∞/∞.
Every DE starts with a classification word (SEP/HOM/LIN).
Every area starts with intersection solve on rough sheet.
Definite integrals get a property glance before expansion.
AOD extrema always use a sign chart or equivalent check (plus endpoints on closed intervals).
I can write IF = e^{∫P dx} and apply it on a standard linear DE in ≤75s.
I can collapse a King-property definite integral without finding a hard antiderivative.
My weekly error log has anti-trap rules, not only ‘be careful’.
In the last timed Calculus set, accuracy ≥90% on attempted questions.
Illegal L’Hôpital never appears in my timed set error log.
Every DE stem is stamped SEP/HOM/LIN before ink.

Paper coverage for Calculus Syllabus map Trends Math · 300 plan

Also study

Other Math note pages

Algebra →Coordinate Geometry →Trigonometry →Vectors & 3D Geometry →Limits →AOD →Integrals →DE →