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WMA11/01 Oct 2021, Question 10: Integration: Undoing Differentiation

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 7 marks

The question, in short

A curve y = f(x) for x > 0 has f′(x) = ax − 12x^(1/3) and f″(x) = 0 at x = 27, passing through (1, −8). Find a and then f(x).

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

Part (a) · 3 marks

Use the given second derivative condition at x = 27 to determine the constant a.

  1. 1f″(x) = a − 4x^(−2/3)
  2. 2f″(27) = 0: a − 4 × 27^(−2/3) = 0
  3. 327^(−2/3) = 1/9, so a = 4/9

Answera = 4/9

Part (b) · 4 marks

Using your value of a, integrate to obtain f(x), applying the point (1, −8) to find the constant.

  1. 1f(x) = ∫((4/9)x − 12x^(1/3)) dx
  2. 2= (2/9)x² − 12 × (3/4)x^(4/3) + c = (2/9)x² − 9x^(4/3) + c
  3. 3At (1, −8): 2/9 − 9 + c = −8
  4. 4c = 7/9, so f(x) = (2/9)x² − 9x^(4/3) + 7/9

Answerf(x) = (2/9)x² − 9x^(4/3) + 7/9

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