Unit 5: Calculus

Topic 5.19: Maclaurin series (HL only) Questions

Practice 20 exam-style questions for IB Math AA SL Topic 5.19. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.

11 mark

2026

In ln(1 + x) = x − x²/2 + x³/3 − …, the coefficient of x³ is:

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21 mark

2026

To evaluate lim(x→0) (sin x − x)/x³, you should:

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31 mark

2026

The Maclaurin series of cos(x²) begins:

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4Write down1 mark

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Write down the Maclaurin series of

sin x

up to and including the term in

x^{3}

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51 mark

2026

The constant term (the term in x⁰) of a Maclaurin series equals:

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61 mark

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Which series is 1 − x²/2! + x⁴/4! − …?

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7Find1 mark

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Find the constant term of the Maclaurin series of

f

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8Find3 marks

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Find the Maclaurin series of f(x) = e^{-x^{2}} up to and including the term in x⁴.

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9Evaluate2 marks

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Evaluate

x \to 0 lim \frac{ln ( 1 + x )}{x}

using a Maclaurin series.

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10Evaluate3 marks

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Evaluate

x \to 0 lim \frac{e ^{2 x} - 1}{x}

using a Maclaurin series.

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111 mark

2026

The series for

e^{x}

· sin x, up to the term in x², is:

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12Find3 marks

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Find the Maclaurin series of

f (x) = sin (x^{2})

up to and including the term in x⁶.

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131 mark

2026

To find the Maclaurin series of

e^{5 x}

, the quickest method is:

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14Find3 marks

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Find the Maclaurin series of f(x) =

e^{- x}

up to and including the term in x³, and state f''(0).

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15Explain1 mark

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Explain why the approximation is best near

x = 0

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161 mark

2026

Multiplying x·cos x and keeping terms up to x³ gives:

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17Find3 marks

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Find the Maclaurin series of f(x) = ln(1 + 2x) up to the term in x³.

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18Find3 marks

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Find the first three non-zero terms of the Maclaurin series of f(x) = cos(2x).

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19Find3 marks

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Use the Maclaurin formula to find the series of f(x) = sin x up to the term in x⁵.

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201 mark

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lim(x→0) (tan x)/x, given tan x = x + x³/3 + …, equals:

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