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WMA11/01 Jun 2022, Question 7: Integration: Undoing Differentiation

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Given f′(x) = 2/√x + A/x² + 3 with f″(x) = 0 at x = 4, find A, then use f(x) = 8√3 at x = 12 to determine f(x) in simplest form.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 4 marks

Differentiate f′(x) to get f″(x), set it equal to 0 at x = 4, and solve for the constant A.

  1. 1f′(x) = 2x^(−1/2) + Ax⁻² + 3
  2. 2f″(x) = −x^(−3/2) − 2Ax⁻³
  3. 3f″(4) = 0: −1/8 − 2A/64 = 0
  4. 4A = −4

AnswerA = −4

Part (b) · 5 marks

Integrate f′(x) with the found A, apply the condition f(x) = 8√3 when x = 12, and give f(x) with each term simplified.

  1. 1f′(x) = 2x^(−1/2) − 4x⁻² + 3
  2. 2f(x) = 4x^(1/2) + 4x⁻¹ + 3x + c
  3. 3f(12) = 4√12 + 4/12 + 36 + c = 8√3
  4. 48√3 + 1/3 + 36 + c = 8√3, so c = −109/3
  5. 5f(x) = 4√x + 4/x + 3x − 109/3

Answerf(x) = 4√x + 4/x + 3x − 109/3

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