Pure Mathematics 3

Deep Dive: Exponential Growth and Decay

Provenance: written 2026-09-29. Primary source, read and grepped directly for this topic: research/veridian/WMA13-verified-facts.md (§1 spec table item 4.4, §3 formula-booklet transcription, §5 "3. Exponentials and logarithms" and "4. Differentiation", §7, §8, and the two dated exponential-growth-and-decay.ts audit-log entries in §9). Checked against what already exists, to keep this additive rather than repetitive: content/veridian/curriculum/wma13/exponential-growth-and-decay.ts (full header comment and every mechanism/teach/trap block read), and this folder's codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md (grepped for every existing spec-4.4 citation), plus content/veridian/question-bank/wma13.ts. Six real primary-source documents were then re-read directly, independently of the research file's own prose, from the local mirrors at ~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_{Past_Papers,Mark_Schemes,Examiner_Reports}.md: January 2022 Q8, June 2023 Q7, January 2024 Q3 and Q5, and October 2024 Q4 — the question text, mark scheme and (where quoted) examiner-report wording below for these five questions are transcribed from that direct re-read, not carried over from the research file's paraphrase; every numeric result quoted from them was independently recomputed with Python before being written in, shown inline. Jan 2025 Q5 and Oct 2024 Q8 are cited only via the already-verified quotations in WMA13-verified-facts.md and are flagged as such where they appear. The Jun 2026 Q5 constants ( $a\approx316$ , $b\approx1.19$ ) are taken from WMA13-verified-facts.md's own re-verified figures; the part (c) rate of change built from them below was never sourced by that document and is computed fresh here, flagged as independently derived rather than mark-scheme-verified.

#CODEX Deep-Dive: Exponential Growth and Decay (WMA13, spec 4.4)

#Scope, and how this relates to what already exists

Spec 4.4 reads, in substance (paraphrased from WMA13-verified-facts.md line 45, itself transcribed from the spec PDF, Issue 3, April 2019, p.21): understand and use exponential growth and decay; know that "initial" means the value when $t=0$ ; be ready to explore behaviour for large $t$ or judge whether a model's predicted range of values is sensible; a second, improved model may be asked for; and know and use $\frac{d}{dx}[a^x]=a^x\ln a$ .

Every clause of that is already taught in exponential-growth-and-decay.ts: the identity's derivation, the initial-value convention, floor/ceiling long-run behaviour, one real "no solution" domain case (Jan 2024 Q5(d)), a VERIDIAN-original second-improved-model question, the negative-exponent sign trap, and — added in a later pass — the real Oct 2024 Q8 two-exponential-term maximum. codex-formula-and-method-bank.md and forge-exam-technique-guide.md between them add the Oct 2024 Q4(c) power-rule-vs-exponent-rule error (as a named, sourced but not yet built-out finding) and the general dM1-dependency mechanics for the two-exponential-term case. None of that is repeated below except by short pointer.

What follows instead: (1) states the model's two equivalent real forms and the identity that connects them, as a single piece of theory not written out this way anywhere else in the archive; (2) works five real exam questions in short form that are not currently built into any WMA13 lesson, technique-hub document or question-bank entry — three of them (Jun 2023 Q7's full three parts, Jan 2022 Q8's full three parts, Jan 2024 Q3) newly re-verified against primary sources this pass, and one (Jun 2026 Q5(c)) filled in from a gap the research file itself names but never closes; (3) separates spec 4.4's "predicted range of values" clause into two genuinely different real question-shapes, which the archive's own material only illustrates with one of the two; and (4) corrects a real mislabelling in the research file, found by that primary-source re-read. Doubling time/half-life and the units-and-direction-word trap are both already fully built elsewhere and are not repeated (§7).

#1. The model, in its two equivalent real forms

A quantity growing or decaying at a constant proportional rate can be written two ways, and both appear in real WMA13 papers:

Discrete-rate form: $N=N_0a^t$ (or $N=ab^t$ , $N=pq^{-t}$ — the letters vary by question), where $a$ is the multiplier per unit of $t$ . $a>1$ is growth, $0<a<1$ is decay, and a negative exponent ( $a^{-t}$ ) is a second, equally common way of writing decay with $a>1$ .
Continuous-rate form: $N=N_0e^{kt}$ , where $k$ is the instantaneous proportional rate. $k>0$ is growth, $k<0$ is decay.

The two are the same law in different notation, linked by the P3 formula booklet's own given identity, $e^{x\ln a}=a^x$ (WMA13-verified-facts.md §3, p.4 of the booklet): setting $x=t$ , $a^t=e^{t\ln a}$ , so the two forms match exactly when $k=\ln a$ (equivalently $a=e^k$ ). This is not a cosmetic choice by question-setters — real papers use both, and a candidate who has only practised one form is not fully prepared for spec 4.4. Confirmed discrete-form models: Jan 2022 Q8(c) ( $x=400\times1.4^{-t}$ ), Oct 2024 Q4 ( $N=100\times2.24^t$ ), Jun 2026 Q5 ( $H=ab^t$ ). Confirmed continuous-form models: Jan 2024 Q5 ( $T=10+8e^{-Bt}$ ), Jan 2025 Q5 ( $H=24+280e^{-0.05t}$ , cited via WMA13-verified-facts.md only), Jun 2023 Q7 (both populations, $N=Ae^{kt}$ and $N=60\,000e^{-0.6t}$ ), Oct 2024 Q8 ( $H=32+40e^{-0.2t}-20e^{-0.9t}$ ).

#2. The one identity spec 4.4 requires, and how to rebuild it under pressure

$\frac{d}{dx}[a^x]=a^x\ln a$ is on the spec's own "will not appear in the booklet" list (WMA13-verified-facts.md §3) — it must be memorised, not looked up. If it is ever forgotten under exam pressure, it rebuilds in two lines from material the spec does give: rewrite $a^x=e^{x\ln a}$ (the booklet identity above), then differentiate with the memorised chain-rule result $e^{kx}\to ke^{kx}$ , treating $\ln a$ as the constant $k$ : $\frac{d}{dx}\left[e^{x\ln a}\right]=(\ln a)e^{x\ln a}=a^x\ln a$ . This two-line rebuild, and the negative-exponent extension $\frac{d}{dx}[a^{-x}]=-a^{-x}\ln a$ , are already fully derived and drilled in exponential-growth-and-decay.ts and are not repeated here.

The one confirmed real error that is not about a sign at all, but about which rule applies. Oct 2024 Q4(c) (real question and mark scheme re-read directly for this pass — see §3.3 below) records candidates producing $\frac{dN}{dt}=100t\times2.24^{t-1}$ for $N=100\times2.24^t$ : the power rule ("bring down the exponent, reduce it by one"), applied to a term where the variable is the exponent and the constant is the base — exactly backwards from the shape the power rule is for. This is the only real, located WMA13 instance of this specific confusion, and it is structurally distinct from the negative-exponent sign trap: that error assumes the right rule was chosen and only its sign was mishandled, while this one names the wrong rule outright. Before differentiating any $a^{f(t)}$ -shaped term, name which symbol is fixed (the base) and which is varying (the exponent) — if the variable is in the exponent, the rule is always $a^x\to a^x\ln a$ , never $nx^{n-1}$ .

#3. Real sub-cases, independently re-verified against primary sources

#3.1 Recovering a rate constant from two data points, not from a stated formula — Jun 2023 Q7(a), real, full detail

Every existing worked example in exponential-growth-and-decay.ts gives the model's constants already fixed in the question. A real WMA13 question can instead give two data points and require solving for the rate constant itself. Jun 2023 Q7 (re-read directly from the primary past paper and mark scheme this pass) models a bacteria population by $N=Ae^{kt}$ ( $A,k>0$ ), stating there were 2500 bacteria at the start of the study and 10000 exactly 8 hours later, and asks for the exact value of $A$ and $k$ to 4 s.f. [3 marks: B1, M1, A1].

Method: the initial-value convention (§ above) gives $A$ directly, since $e^{k\cdot0}=1$ regardless of $k$ — $A=2500$ needs no equation at all, only the substitution $t=0$ . The second data point then gives one equation in $k$ alone: $10\,000=2500e^{8k}\Rightarrow e^{8k}=4\Rightarrow k=\frac{\ln4}{8}$ . Independently recomputed: $k=0.173287\ldots$ , matching the real mark scheme's own " $k=$ awrt 0.1733." The real mark scheme explicitly also credits a simultaneous-equations route (dividing the two data-point equations to eliminate $A$ first), but requires correct log/index work either way — this is a genuine third technique for this topic, distinct from both the log-linear-graph method (spec 3.3, a different lesson) and the log-linear-proof method in §3.3 below, because there is no graph and no given linear log-equation here, only two raw $(t,N)$ pairs and the exponential form itself.

#3.2 Two independently modelled quantities crossing — Jun 2023 Q7(c), real, full detail; corrects the research file's own mislabelling

The same question's second population is given directly as $N=60\,000e^{-0.6t}$ . Part (b) asks for the rate of decrease exactly 5 hours in, to 3 s.f. [2 marks: M1, A1] — this is the exact question WMA13-verified-facts.md and the lesson's own commonWrongPath already cite for the central substitution-instead-of-differentiation error, and is not re-argued here. Independently recomputed: $\frac{dN}{dt}=-36\,000e^{-0.6t}$ , so at $t=5$ , $\frac{dN}{dt}=-36\,000e^{-3}\approx-1792$ , i.e. a decrease of "awrt 1790" — matching the real mark scheme exactly.

Part (c), genuinely new here, asks for the time $T$ when the two populations were equal [3 marks: M1, dM1, A1]. This sets $60\,000e^{-0.6t}=2500e^{kt}$ (using the $k\approx0.1733$ found in (a)) — structurally close to, but a real, distinct variant of, the two-exponential-term maximum technique already fully built for Oct 2024 Q8: there, one model's own derivative is set to zero; here, two entirely separate models' own values are set equal to each other, with growth on one side and decay on the other. The real mark scheme's method is the same underlying move both times — collapse to one exponential term, then take logs — done here by dividing both sides to isolate $e^{(k+0.6)t}=24$ , so $t=\frac{\ln24}{k+0.6}$ . Independently recomputed: $T\approx4.110$ , matching the real mark scheme's " $T=4.11$ " (3 s.f.) exactly.

A correction to the existing research base, found by this direct re-read. WMA13-verified-facts.md line 224 describes Jun 2023 Q7 as "a charity-fundraising exponential-growth model" and Jan 2024 Q3 as "a population/decay comparison model." Both the real question papers, re-read directly for this pass, show the opposite: Jun 2023 Q7 is the two-bacteria-populations question worked above (no charity content anywhere in it), and Jan 2024 Q3 (§4 below) is the one about money raised for a charity (no population comparison in it — it is a single growing quantity, not two compared). The underlying citations (series and question numbers) are otherwise accurate and every number attached to them elsewhere in this course's material is unaffected; only the one-line paraphrase describing what each question is about has the two swapped. Worth fixing at the source, not just here.

#3.3 Log-linear form converted to exponential form, immediately followed by a differentiation part — a recurring compound sub-case, seen in 3 of 13 series checked

Three real, separate series pair the same two skills in the same order: a question gives a stated linear relationship in $\log_{10}$ (not a graph — the equation itself), asks the candidate to show it converts to $N=ab^t$ form (a genuine "show that," not just stating values), then asks for a rate of change from that converted form. This is a compound sub-case distinct from both the pure log-linear-graph method (spec 3.3, reading a gradient and intercept off two plotted points — a different lesson's territory) and the pure differentiation drills already in this lesson (which start from an exponential form already given, never derived).

Jan 2022 Q8(a)–(c) (re-read directly this pass): $\log_{10}x=2.74-0.079t$ is to be shown as $x=pq^{-t}$ (a negative-exponent instance), giving $p$ to the nearest whole number and $q$ to 2 s.f. [4 marks: M1, A1, dM1, A1]. Independently recomputed: $p=10^{2.74}\approx550$ , $q=10^{0.079}\approx1.20$ — both matching the real mark scheme's "awrt 550" and "awrt 1.2" exactly. Part (b) [1 mark, B1] asks candidates to interpret $p$ in words; the real mark scheme credits any statement that $p$ is the amount of antibiotic present at the start (condoning "the dose given"). This is a genuinely simpler skill than Jun 2026 Q5(b)'s interpret-the-rate-constant task already documented in WMA13-verified-facts.md (which needs three separate components — direction, percentage, per-year — because $b$ is a multiplicative rate) — here $p$ is only the initial-value constant, so the mark is earned by the same "initial means $t=0$ " recognition spec 4.4 already names, applied to a constant that had to be identified from a converted form rather than reached by substituting $t=0$ directly. Worth distinguishing the two rather than conflating them: interpreting an initial-value constant and interpreting a multiplicative rate constant are different real skills, confirmed on two different series each testing only one of the two. Part (c) then switches to a second, separately stated dataset, $x=400\times1.4^{-t}$ , and asks for $\frac{dx}{dt}$ at $t=5$ to 2 s.f. [3 marks: B1, M1, A1] — independently recomputed, $\frac{dx}{dt}=-400\ln(1.4)\times1.4^{-5}\approx-25$ , matching "awrt $-25$ " exactly, and this is the same real question the negative-exponent sign trap is already sourced to elsewhere in this course's material (candidates who drop the sign score B1 M1 A0, per the real scheme's own note).

Oct 2024 Q4(a)–(c) (re-read directly this pass, and genuinely new — not built into any WMA13 document before this one): $\log_{10}N=0.35t+2$ models bacteria on a surface. (a) [1 mark, B1]: the initial value is read directly from the equation at $t=0$ without even converting form — $\log_{10}N=2\Rightarrow N=100$ ; the real mark scheme explicitly disallows crediting this from the converted form found in (b), only from the given equation itself. (b) [3 marks: M1, dM1, A1]: show $N=ab^t$ with $b$ to 2 d.p. — independently recomputed, $a=10^2=100$ exactly, $b=10^{0.35}\approx2.2387$ , rounding to $2.24$ , matching the scheme exactly. (c) [2 marks: M1, A1] — the question paper asks candidates to use the part (b) form to find the rate of growth exactly 5 hours after monitoring began; independently recomputed with the rounded $b=2.24$ , $\frac{dN}{dt}=100\ln(2.24)\times2.24^5\approx4548$ ; with the unrounded $b$ , $\approx4532$ ; the real mark scheme accepts "an answer in the range 4530 to 4550" — wide enough, confirmed by this recomputation, to cover both the rounded-through and exact-value routes without either being penalised. The examiner report (re-read directly) records candidates instead evaluating $N$ at $t=5$ — independently recomputed, $100\times2.24^5\approx5640$ , exactly the wrong number the report names — the same substitution-not-differentiation error already the central finding elsewhere in this topic, now with a second real, confirmed wrong-path number to set beside the "no marks" outcome; and separately records the power-rule confusion already stated in §2 above.

Jun 2026 Q5(a)–(c) (constants only, taken from WMA13-verified-facts.md's own re-verified figures, not re-read from the PDF by this pass): a log-linear graph (not a stated equation this time — the one real member of this trio that starts from a plotted line, not algebra) gives $H=ab^t$ with $a\approx316$ , $b\approx1.19$ (both already independently confirmed with sympy in the source document). Part (c), asking for $\frac{dH}{dt}$ at $t=5$ to 2 s.f. [2 marks], is explicitly named in the research file as real, in scope for spec 4.4, and "not cited further" there because it fell outside the sibling lesson's own spec-3.1–3.3 boundary — a real, acknowledged gap this document closes. Independently derived here, not sourced to a seen mark-scheme value (the research file did not transcribe one): using the exact constants $a=10^{2.5}$ , $b=10^{0.075}$ , $\frac{dH}{dt}=a(\ln b)b^t$ , so at $t=5$ , $\frac{dH}{dt}\approx316.23\times0.17269\times2.37137\approx130$ households per year, to 2 s.f. This value should be treated as this document's own worked computation from already-verified constants, not as a confirmed mark-scheme figure, until the actual Jun 2026 Q5(c) mark scheme is independently located and checked against it.

Why this compound pattern is worth naming as one thing: three series, four years apart at the extremes (Jan 2022 to Jun 2026), pair "convert a stated log-linear relationship (by algebra or by graph) into $N=ab^t$ form" with "then differentiate that form for a rate," in that order, every time. A candidate drilled only on differentiating an exponential model that is simply handed to them, never one they must first derive from a logarithmic statement, has not covered a technique that recurs across roughly a quarter of this document's real anchors for this topic.

#4. Two real, distinct shapes for spec 4.4's "predicted range of values" clause

The spec's own guidance — explore behaviour for large $t$ , or consider whether the predicted range of values is appropriate — is tested in the real archive in two structurally different ways, and only one of the two is currently built into exponential-growth-and-decay.ts.

Shape A: hard mathematical impossibility. Already fully sourced and taught (Jan 2024 Q5(d)): a target value would require a strictly positive exponential term to equal zero or a negative number, which is impossible for any real, finite $t$ — the correct answer states this impossibility directly (the real mark scheme credits "which is not possible" or "you cannot find the log of a negative number," and explicitly rejects the imprecise "logs cannot be negative"). Not repeated further here.

Shape B: a plausibility check by direct substitution, with no impossibility anywhere in it. Jan 2024 Q3 (re-read directly this pass; the charity-fundraising question mislabelled in §3.2 above) models total money raised, $\log_{10}D=1.04+0.38t$ . Part (a) [3 marks] converts this to $D=ab^t$ : independently recomputed, $a=10^{1.04}\approx10.96$ , $b=10^{0.38}\approx2.399$ , matching the real scheme exactly. Part (b) [2 marks] asks when $D=45\,000$ : solving $45\,000=10.96\times2.399^{T}$ gives $T\approx9.51$ months, matching "awrt 9.51" exactly. Part (c) [2 marks] then asks whether the charity, aiming for £350000 within 12 months, will meet that aim — answered not by solving an equation at all, but by substituting $t=12$ directly into the given log form: $\log_{10}D=1.04+0.38\times12=5.6$ exactly, against $\log_{10}(350\,000)\approx5.544$ ; since $5.6>5.544$ , $D(12)\approx398\,000>350\,000$ , so the model says yes, matching the real mark scheme's own conclusion (which accepts either the log-space comparison or the converted value, "awrt £400000," provided the comparison itself is shown).

Shape A and Shape B answer the same spec clause with opposite verdicts and opposite mechanisms — Shape A shows a target is never reachable by any real $t$ ; Shape B checks whether a target is reached by one specific, stated $t$ , and here the answer is a plain "yes." A lesson or drill set that only ever produces "no, impossible" as the answer to this clause has not covered the real range of what it actually asks for.

#5. What this document deliberately does not re-cover

The sum of two exponential terms with different rate constants, set to zero for a maximum (Oct 2024 Q8, $H=32+40e^{-0.2t}-20e^{-0.9t}$ ): fully derived, fully sourced, and built as a real marked solution in exponential-growth-and-decay.ts, content/veridian/question-bank/wma13.ts, and both technique-hub documents already. §3.2 above builds the real, additive variant of this technique (two separate models crossing, not one model's own turning point) rather than repeating the original.
Doubling time and half-life, $T=\frac{\ln2}{k}$ : already derived and drilled in the lesson's own flashcards and mechanism block, explicitly flagged there as beyond what spec 4.4 itself names — no real WMA13 question asking for a doubling or halving time was located anywhere in the archive, by this pass or the lesson's own header note, so it is not built out again here.
The units-and-direction-word trap on a computed rate: independently confirmed across four series already (Oct 2021 Q3(b), Jan 2022 Q4(c), Oct 2022 Q5(a), Jan 2025 Q2(a)) and fully built into the lesson's marked solution, trap taxonomy and a dedicated MCQ. Not re-argued.
Jan 2024 Q5(a)–(c) ( $T=10+8e^{-Bt}$ , $B\approx0.00639$ from a second data point, rate at $t=2\approx-0.0505\,^{\circ}\text{C min}^{-1}$ ) and Jan 2025 Q5 ( $H=24+280e^{-0.05t}$ , a sign slip during rearrangement rather than differentiation): both real and both already cited with verbatim mark-scheme/examiner-report wording in WMA13-verified-facts.md and the lesson itself; re-verified during this pass only to the extent needed to confirm §4's Shape A/B split (Jan 2024 Q5) and not reproduced further.

#6. Exam-angle synthesis: every real anchor this document independently verified

Seen in 6 of the 13 series in this course's WMA13 archive (forge-exam-technique-guide.md's own count, Oct 2020–Jun 2026): Jan 2022 (Q8, full detail, §1 and §3.3), Jun 2023 (Q7, full detail, §3.1–3.2), Jan 2024 (Q3 full detail §4; Q5 cited, §5), Jan 2025 (Q5, cited only, §5), Oct 2024 (Q4 full detail, new this pass, §2 and §3.3; Q8, cited only, §5), Jun 2026 (Q5(c), constants sourced/rate independently derived, §3.3).

Three further contexts the research file names for this general spec area — Oct 2021 Q3 and Q7, Jun 2022 Q4 — are out of this document's scope, not omissions: on the evidence transcribed in WMA13-verified-facts.md they are solve-for-a-value or log-linear-graph questions (spec 3.1–3.3) with no differentiation, rate-of-change or model-validity part attached, the same boundary exponential-growth-and-decay.ts itself draws against its sibling lesson. This document did not independently re-read those three primary sources to check for a spec-4.4 part that might have been missed; the "6 of 13" figure above should be read as this document's own confirmed floor, not a claim that the remaining 7 series contain nothing at all.

No likelihood, probability or grade-boundary figure is stated anywhere above — every count is "seen in N of the M series actually checked," per this course's standing rule.

#Sources

research/veridian/WMA13-verified-facts.md — spec text, formula-booklet transcription, and the Jan 2024 Q5 / Jan 2025 Q5 / Oct 2024 Q8 / Jun 2026 Q5 citations reused rather than re-verified.
~/Documents/SOL/Papers/IAL_Paper3_Complete/Master_Consolidated_Docs/Mathematics_Paper3_Past_Papers.md, Mathematics_Paper3_Mark_Schemes.md, Mathematics_Paper3_Examiner_Reports.md — re-read directly this pass for Jan 2022 Q8, Jun 2023 Q7, Jan 2024 Q3, and Oct 2024 Q4; every numeric claim sourced to these five questions above was independently recomputed with Python, not taken from the mark scheme's stated answer alone.
content/veridian/curriculum/wma13/exponential-growth-and-decay.ts — read in full to keep this document additive; its own mechanism, worked-chain, chain-drill, method-comparison and both marked-solution blocks are cited by name, not reproduced.
content/veridian/technique-hub/wma13/codex-formula-and-method-bank.md, forge-exam-technique-guide.md, signal-frequency-and-boundaries.md — checked for existing coverage (the Oct 2024 Q4(c) power-rule finding and the 13-series count both originate here) and cross-referenced rather than repeated.
content/veridian/question-bank/wma13.ts — checked for existing marked-solution ids (the Oct 2024 Q8 entry) before writing §5's pointer.