Lesson 20: Final Revision and Formula Booklet

The last lesson is about converting the whole course into exam-day performance: what the formula booklet gives you (so you never waste memory on it), what it deliberately withholds (so you know what must live in your head), and how to structure the final revision campaign (AA assessment: all topics, Papers 1–3). The booklet is issued for every paper — SL and HL versions differ (the HL booklet adds the HL content) — and fluency with it is a skill: in the exam you should locate any result within seconds, because you revised *with* it open beside you.

What the booklet contains (use it, don't memorise it)

Organised as Prior Learning, then Topics 1–5, each split SL / additional HL. Highlights you will reach for constantly:

What is NOT in the booklet (this is the revision list)

The booklet gives results, never *judgement*. Its derivatives table lists the derivatives of standard functions only; composite and product structures are yours to recognise and unpick. Not given: completing the square; how to choose test points for inequalities; when to split an area integral; how to set up a regression or an optimisation. One caution: booklet contents are revised between editions (recent editions list the logarithm laws in Topic 1 — check yours), so the durable rule is to verify against your own session's booklet during revision, and to memorise the *locations*, not the contents. Also not given: command-term meanings, the three-significant-figures convention, radian discipline, and every proof method of Lessons 12 and 24. Those live in you or nowhere.

Notation fluency (misreading costs marks)

f′(x), f″(x) for derivatives; ∫…dx and ∫ₐᵇ for integrals; lim(x→a); Σ for sums; (n choose r) written C(n, r) or with the bracket notation — know both; P(A|B) "given"; x̄, s, σ for mean and standard deviations (know which your GDC reports vs what the question wants); |v| for magnitude, a · b dot product; arg z, |z|, z̄; uₙ, Sₙ; B(n, p), N(μ, σ²). A ten-minute drill: write each symbol from its spoken name — in the exam you will read questions faster than you ever practised if the notation is automatic.

The revision campaign: four phases

Phase 1 — Topic audit (2–3 weeks out). Work topic by topic through Lessons 1–16 of this course; for each, do five mixed questions under no time pressure but *with the booklet open*, and log every error in an error log with a one-line cause ("dropped inner derivative", "degree mode", "forgot to reject root"). Patterns in the log are your real syllabus gaps. Phase 2 — Paper craft (1–2 weeks out). Full Section A sets against the clock (≈1 min/mark); practise the salvage moves of Lesson 19 — "show that" gifts, follow-through, star-and-move-on. Revisit the error log daily for ten minutes. Phase 3 — Full papers (final week). At least two complete timed papers per paper type (SL: 1 and 2; HL: 1, 2, 3), sat in exam conditions — same time of day, no pauses, booklet and GDC exactly as exam day (radian mode checked). Phase 4 — The 48 hours. No new material. Re-read the error log, skim the formula booklet cover to cover (locations!), sleep properly, prepare the practical kit. Confidence at this point is a resource you built in phases 1–3.

Worked example: reading a "formula-booklet question"

Worked Example 1 — what the booklet does for you. "Find the exact value of cos(2θ) given sin θ = 3/5, θ in Q2." The booklet supplies cos 2θ = 1 − 2sin²θ (one of the three forms — *choosing* the sin-form because sin is given is your judgement, not the booklet's). Value: 1 − 18/25 = 7/25. The Q2 detail matters for the companion sin 2θ = 2 × (3/5) × (−4/5) = −24/25. Lesson: booklet + judgement + Lesson 6's quadrant discipline — all three, every time.

Worked Example 2 — an error-log entry, dissected. Error: "Solved 2cos²x + 3 sin x = 0, reported only x = 7π/6." Cause analysis: converted and factorised correctly but read only one family from sin x = −1/2 (missed 11π/6) and never wrote the rejection of sin x = 2. Prescription: a closing checklist for every solve — *families complete? illegal values rejected in writing? interval endpoints checked?* Three questions, pasted inside the error log's front cover, worth several marks per paper.

Common mistakes (revision phase)

• Revising by re-reading notes — passive review feels productive and is not; questions under mild pressure are the unit of revision.
• Ignoring the error log's pattern — redoing topics you already ace because they feel good.
• Learning the booklet by heart (wasted effort) or never opening it until exam day (wasted resource) — the middle path is location fluency.
• First timed paper sat the night before — timing is a skill needing repetitions, not a surprise to schedule.
• Abandoning Paper 1 skills because Paper 2 allows the GDC — half the SL grade (and the harder half of HL) is technology-free.

IB Exam Experience

Question 1 [2 marks]. State the formula booklet result you would use, and the judgement it does not make for you, when differentiating f(x) = (3x + 1)⁶. *Notes:* the booklet gives d/dx [xⁿ] = nxⁿ⁻¹ (and the chain pattern via standard entries); it does not supply the chain rule's application — the inner derivative ×3 is your judgement: f′(x) = 18(3x + 1)⁵. Full credit needs both halves.

Question 2 [3 marks]. A candidate's error log shows the same error three times: "area integral across a sign change". Write the three-line protocol that fixes it, and apply it to y = x² − 2x on [0, 3] by finding where the split falls. *Notes:* protocol — find zeros inside the interval; split the integral there; take absolute values piecewise (any equivalent wording). Zeros at 0 and 2; split at x = 2; the curve is negative on (0, 2), positive on (2, 3).

Question 3 [4 marks]. Plan the final 90 minutes before a Paper 1 (no calculator) using only materials allowed in the room later: your error log and the formula booklet. Allocate the time across four activities with reasons. *Notes:* any sensible allocation anchored to: error-log top patterns (largest block — your personal highest-yield), booklet location sweep (medium — fluency check, especially Topics 4–5 entries used less often), notation drill (short), logistics/rest (short but explicit). Credit reasoning, not the specific numbers.

Extended task [9 marks]. A student has 10 study sessions before Paper 2. Their error log: chain rule ×5, binomial cdf/pdf confusion ×3, regression extrapolation comments ×2, radian mode ×2. (a) Rank the four error types by (frequency × marks-at-stake) and justify the top priority. [3] (b) Design sessions 1–4 to attack the top two error types, specifying activities. [4] (c) State how sessions 9–10 should differ in character from sessions 1–4, and why. [2] *Notes:* (a) chain rule first — highest frequency and it appears inside other topics (integration, optimisation); radian mode is cheapest to fix (a checklist habit, near-zero study cost for potentially paper-wide benefit) — accept either as top if justified by frequency × stakes reasoning. (b) sessions must be *active*: targeted question sets on composites in varied dress (powers, e^, ln, trig), immediate self-marking against the "inner derivative present?" check; binomial: pdf/cdf discrimination drills with "exactly/at most/at least" sorting and GDC keystroke fluency. (c) sessions 9–10 are full timed papers + error-log review only: integration under pressure, no new content — because late-stage gains come from exam craft and confidence, and new material risks displacing consolidated skill.

Closing word. You now own the course's full arc: functions to calculus, triangles to vectors, data to Proof. The papers reward exactly what these lessons practised — visible method, exact answers, and judgement. Good luck.

Going deeper: a final self-test across the whole course

Close the course with a diagnostic sweep — one fluent item per topic, answered with the booklet closed, then checked. Functions: state the inverse of f(x) = (x − 2)/(x + 5). Algebra: sum the series 3 + 6 + 12 + … to 10 terms. Trig: solve sin x = cos x on [0, 2π]. Vectors: angle between (1, 2, 2) and (2, −1, 2). Calculus: differentiate and integrate x e^x. (Answers: derivative e^x(x + 1); integral e^x(x − 1) + C). Probability: from a pack of cards, P(king | red). Statistics: interpret r = −0.15. Each answer below; any hesitation names its lesson for one final targeted pass.

Answers (check after attempting). Inverse: swap x and y in y = (x − 2)/(x + 5) and solve: x(y + 5) = y − 2 → xy + 5x = y − 2 → y(x − 1) = −2 − 5x → f⁻¹(x) = (5x + 2)/(1 − x) ✓. Series: geometric, r = 2: S₁₀ = 3(2¹⁰ − 1)/(2 − 1) = 3069. Trig: tan x = 1 → x = π/4, 5π/4. Vectors: dot = 2 − 2 + 4 = 4; magnitudes 3 and 3; cos θ = 4/9 → θ ≈ 63.6°. Cards: P(king | red) = 2/26 = 1/13. r = −0.15: very weak negative linear association — essentially no linear relationship.

Extra exam practice [2 marks]. Your booklet sweep stalls on one item: you cannot recall whether the volume of revolution about the x-axis is π∫y² dx or ∫πy dx. Resolve it without the booklet. *Notes:* dimensional reasoning — volume must scale like (length)³; y² dx has units of length³, y dx only length²; and the disc picture (radius y, thickness dx, volume πr² × thickness) confirms V = π∫y² dx. Deriving beats recalling when recall fails — a fitting final technique for the whole course.