Topic Brief: Exponential Growth and Decay

Provenance: written 2026-09-29. Primary source: research/veridian/WMA13-verified-facts.md (§1 spec item 4.4; §5 "3. Exponentials and logarithms" and "4. Differentiation"; §9 lesson-audit log), grepped specifically for this topic before anything below was written. Checked against what already exists in this hub so this stays additive rather than repetitive: codex-deep-dive-exponential-growth-and-decay.md (full worked mechanics for five real questions — not re-derived here), signal-frequency-and-boundaries.md (the paper-wide frequency table, whose spec-4.4 row this brief starts from but does not just restate), forge-exam-technique-guide.md and codex-formula-and-method-bank.md (method skeletons, cross-referenced not repeated), content/veridian/question-bank/wma13.ts (one existing marked solution, Oct 2024 Q8(c)), and content/veridian/curriculum/wma13/exponential-growth-and-decay.ts (the lesson itself, read in full). No new primary PDFs were opened for this pass — every citation below traces to a finding already independently verified in one of those documents.

#SIGNAL Topic Master Brief: Exponential Growth and Decay (WMA13, spec 4.4)

#What the spec actually asks for

Spec item 4.4 (paraphrased from the spec PDF via WMA13-verified-facts.md line 45): understand and use exponential growth and decay, including the "initial" = value-at-t=0 convention, exploring behaviour for large t, judging whether a model's predicted range of values is sensible, and knowing $\frac{d}{dx}[a^x]=a^x\ln a$ without it being in the formula booklet.

#Frequency: seen in 6 of 13 series checked

Of the 13 WMA13 series in this course's archive (8 read start-to-finish with full mark schemes and examiner reports, 5 more checked for specific topics only — full list in signal-frequency-and-boundaries.md), a genuine spec-4.4 event — differentiating or reasoning about an exponential growth/decay model, not just solving an exponential equation — is located in 6: Jan 2022 (Q8), Jun 2023 (Q7), Jan 2024 (Q3 and Q5), Jan 2025 (Q5), Oct 2024 (Q4 and Q8), Jun 2026 (Q5(c)).

One additive point not stated elsewhere in this hub: Oct 2024 is the only series checked where this topic surfaces twice in the same paper as two separately-marked events — Q4(c) (differentiate a bacteria-growth model recovered from a log-linear equation, 2 marks) and Q8(c) (differentiate a sum of two exponential decay terms to find a maximum, 5 marks). A candidate cannot assume one exponential-modelling appearance uses up the paper's quota for this topic.

#The real mark-tariff pattern

Where the host question's full mark breakdown is documented, the shape repeats: a cheap 1-mark "read the model" opening (state the initial value; state the long-run limit), then a calculus- or algebra-heavy tail carrying most of the marks.

• Jun 2026 Q5 — total 7 marks (4 + 1 + 2); the differentiation part (c) is only 2 marks, because it is a single-term derivative evaluated at a stated $t$.
• Oct 2024 Q8 — total 12 marks across parts (a)–(e); the differentiate-and-find-a-maximum part (c) alone carries 5 (M1 A1 dM1 M1 A1), and the examiner report names it "This penultimate question" — the second-to-last of the paper's 9.
• Jan 2022 Q8, parts (a)–(c) as documented — 8 marks (4 + 1 + 3); the differentiation part (c) is 3.
• Jun 2023 Q7, parts (a)–(c) as documented — 8 marks (3 + 2 + 3); 5 of those 8 marks sit in the two calculus/algebra parts (b) and (c).

Position in the paper: across the six series, the host question number is Q3, Q4, Q5, Q7 or Q8 — never Q1 or Q2 (those slots are consistently numerical methods or functions, per the paper-wide table). Where the paper's total question count is independently confirmed (Oct 2024, 9 questions), this topic's biggest instance sits second-to-last. Treat this as "usually mid-to-late, often one of the two largest questions on the paper," not as a fixed slot — the exact number moves by series.

#The recurring context and notation

Every one of the 6 confirmed instances is dressed as a named real-world scenario — none is a bare "$N=Ae^{kt}$, find $\frac{dN}{dt}$" prompt with no story: antibiotic-concentration decay (Jan 2022), a bacteria population from a log-linear equation (Oct 2024 Q4) and two bacteria populations crossing (Jun 2023), a cooling model with an impossible target (Jan 2024 Q5) and a charity-fundraising total checked against a target (Jan 2024 Q3), a second temperature-decay model (Jan 2025), a horse's heart rate (Oct 2024 Q8), and household waste-recycling uptake (Jun 2026). A drill set built entirely on abstract "$N_0a^t$" prompts has not covered how this topic is actually asked.

The model itself is not always handed over ready-made. In 3 of the 6 series (Jan 2022, Oct 2024, Jun 2026) the candidate must first convert a stated or graphed log-linear relationship into $N=ab^t$ form before any differentiation can start — a distinct, earlier skill from the differentiation itself. In the other 3 (Jun 2023, Jan 2024, Jan 2025) the exponential form is given directly. Notation also varies rather than staying fixed: discrete-rate forms ($1.4^{-t}$, $b^t$) appear in Jan 2022 and Jun 2026; continuous $e^{kt}$ forms appear in Jun 2023, Jan 2024, Jan 2025 and Oct 2024's Q8. A candidate drilled on only one notation is not fully prepared.

#Recurring errors, and how well-attested each one actually is

• Substituting a value instead of differentiating, when asked for a rate: confirmed with verbatim examiner wording in 3 of 6 series (Jan 2022 Q8(c), Jun 2023 Q7(b), Jan 2024 Q5(c)), with a related slip separately reported in Oct 2024 Q4(c).
• Losing the chain-rule sign on a negative exponent ($a^{-t}$ or $e^{-kt}$): confirmed in 2 series during differentiation itself (Jan 2022 Q8(c), Jan 2024 Q5(c)), plus a weaker third instance where the sign was lost during rearrangement rather than differentiation (Jan 2025 Q5(c)).
• Calculator-only solving with no algebra shown: named directly in the Oct 2024 Q8 examiner report as capping the score at 2 of 5 marks.
• Applying the power rule to a variable exponent (treating $2.24^t$ as a fixed power): one confirmed instance, Oct 2024 Q4(c) — newly found, not yet attested across multiple series.
• Two different "is this value possible" answers exist for spec 4.4's range-of-values clause: a hard impossibility (Jan 2024 Q5(d), a target below the model's floor) and a plain substitution check with a "yes" (Jan 2024 Q3(c)). Practising only one shape does not cover the other.

#If this topic appears, here's what to expect

A multi-part, named real-world scenario, likely one of the paper's two largest questions and not among the first two. Expect a free mark or two for reading the model (initial value, long-run limit), then marks that require actually differentiating — sometimes at a stated $t$, sometimes by setting the derivative to zero for a maximum, and, at least once on record (Oct 2024), by combining two differently-rated exponential terms first. Watch the sign if the exponent is negative, differentiate algebraically rather than reaching for a solver, and check whether the exponential form was handed to you or has to be built first from a log-linear statement or graph. A closing "is this plausible" part may ask you to show a target is unreachable, or simply to check it against a stated time — know both.

#Sources

• research/veridian/WMA13-verified-facts.md — every series/question citation above.
• codex-deep-dive-exponential-growth-and-decay.md, signal-frequency-and-boundaries.md, forge-exam-technique-guide.md, codex-formula-and-method-bank.md (this folder) — cross-checked so this brief adds the cross-series position/tariff/context synthesis and the Oct 2024 double-appearance finding, rather than repeating their own worked detail or headline count.
• content/veridian/curriculum/wma13/exponential-growth-and-decay.ts, content/veridian/question-bank/wma13.ts — checked for existing coverage before writing.

No likelihood, probability or grade-boundary figure is stated anywhere above — every count is "seen in N of the M series actually checked."