Maclaurin Series — Mark Scheme Patterns

Mark Allocation Overview

Question Type	Total Marks	M marks	A marks	B marks
First principles (3 terms)	5	1	4	0
First principles (4 terms)	6–7	1–2	4–5	0
Substitution / composite	4–5	1	2–3	1
Integral approximation	2	1	1	0

Pattern: First Principles (5–7 marks)

• M1: Attempt to differentiate (at least 2 derivatives, correct method)
• A1: $f(0)$ correct
• A1: $f'(0)$ correct
• A1: $f''(0)$ correct and $\frac{f''(0)}{2!}x^2$ term
• A1: $f'''(0)$ correct and $\frac{f'''(0)}{3!}x^3$ term
• Optional A1: $f^{(4)}(0)$ correct

关键

如果导数计算正确但忘记除以阶乘（如 $f''(0)x^2$ 而非 $f''(0)x^2/2!$），通常扣 1 分。

Pattern: Substitution into Standard Series (4–5 marks)

• B1: Correctly states standard series (e.g. $e^u = 1+u+u^2/2!+\cdots$)
• M1: Substitutes $u = g(x)$ into standard series
• A1: Correct first non-zero term
• A1: Correct second non-zero term
• A1: Correct third term (if required)

Pattern: Composite Functions (4–5 marks)

• M1: Recognise need for $\ln(1+u)$ expansion or equivalent
• A1: Correct inner expansion (e.g. $\cosh x$ or $e^x$)
• A1: Correct first term
• A1: Correct second term

评分规律

对于 $\ln(\cosh x)$ 类型：

• 展开 $\cosh x$ 获 B1
• 代入 $\ln(1+u)$ 获 M1
• 每正确一项得 A1

Pattern: Integral Approximation (2 marks)

• M1: Integrates series term-by-term (at least 2 terms)
• A1: Correct numerical answer (usually given to 3 or 4 s.f.)

注意

积分近似通常为压轴小题，只需 2 分。不必展开太多项，只需足够得到精度要求。

Common MS Coding

Code	Meaning
M1	Method mark — correct differentiation or substitution
A1	Accuracy mark — correct term/coefficient
B1	Independent mark — stating standard series
ft	Follow-through — accept error from earlier step
AG	Answer given — must show full working