Q: What does H2 Maths Notes (JC 1-2): 5.2) Maclaurin Series cover? A: Series derivation, standard expansions, and approximation error handling for H2 Maths Topic 5.2.
Before you revise Memorise the first four terms of the classic expansions (exsinxcosxln(1+x)(1+x)n
). Practise deriving them quickly so you can adapt to composite functions and substitution questions.
Status: SEAB H2 Mathematics (9758, first exam 2026) syllabus last checked 2025-11-29 (PDF last modified 2024-10-16). Topic 5.2 scope unchanged; Section A still contributes 70 marks per paper (Pure Mathematics).
Definition
Maclaurin series expands a differentiable function about x=0:
f(x)=f(0)+f′(0)x+2!f′′(0)x2+3!f(3)(0)x3+…
Truncate after the required power; remainder term bounds the error.
Substitute expressions: replace x with ax or x2 to adapt templates.
Multiply series: keep terms up to required power.
For composite functions, write as product of known series and collect like powers.
Example -- Expansion of e2xcosx
Start by substituting x↦2x into the ex template to obtain 1+2x+2x2+34x3+….
Use the standard cosx expansion 1−2x2+24x4−….
Multiply the truncated polynomials and keep terms up to x3: 1+2x+23x2+31x3+….
Approximations and Error Bounds
Remainder estimate: use next term magnitude when ∣x∣ is small.
For inequality bounds, apply alternating series test (error < next term).
State approximation interval explicitly (e.g. valid for ∣x∣<1).
Error bounded by next term magnitude 4x4≈0.000010.
Solving Equations with Series
Replace functions with truncated series to solve equations near 0.
Example: solve ex=1+kx for small x by matching coefficients.
Example -- Estimate solution
Solve e−x=1−2x for small x.
Expand e−x=1−x+2x2−6x3+….
Equate: 1−x+2x2≈1−2x
Simplify: −x+2x2=−2x
Solutions: x=0 or x=1. The Maclaurin truncation near x=0 highlights the trivial root; the non-zero solution lies farther out (approximately x≈1.59) and needs either more terms or a calculator/numerical check beyond the small-x
Calculators and Verification
Use graphing calculator (GC) series expansion (Series or taylor) to confirm terms before committing to final answers.
Always write manual working; the GC is for checking only.
When quoting decimal approximations, state the truncated polynomial clearly and show substitution.
Exam Watch Points
Keep factorial denominators exact-do not evaluate unless simplifying later.
State the remainder/order of approximation (e.g. “accurate up to x3”).
For composite functions, show substitution steps to avoid losing method marks.
Mention validity range when required (usually ∣x∣<1).
Practice Quiz
Check that you can derive, manipulate, and apply Maclaurin expansions-including error language-without relying on memory aids.
Quick Revision Checklist
Derive Maclaurin series directly from derivatives at zero.
Memorise and adapt the standard expansions efficiently.
Estimate errors using next-term bounds or alternating-series rules.
Apply series to approximation or equation-solving problems with clear justifications.
