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WMA11/01 Oct 2022, Question 5: Integration: Undoing Differentiation

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

The question, in short

Given f′(x) = 12/√x + x/3 − 4 for x > 0 and P(9, 8) on curve C, find f(x), then find where the normal at P meets the y-axis.

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

Part (a) · 5 marks

Integrate f′(x) and use the point P(9, 8) to determine the constant, giving f(x) in simplest form.

  1. 1f′(x) = 12x^(−1/2) + x/3 − 4
  2. 2f(x) = 24x^(1/2) + x²/6 − 4x + c
  3. 3At (9, 8): 8 = 24(3) + 81/6 − 36 + c
  4. 48 = 72 + 13.5 − 36 + c ⇒ c = −83/2
  5. 5f(x) = 24√x + x²/6 − 4x − 83/2

Answerf(x) = 24√x + x²/6 − 4x − 83/2

Part (b) · 4 marks

Find the gradient of C at P, use the perpendicular gradient for the normal l, then find its y-axis crossing.

  1. 1f′(9) = 12/3 + 9/3 − 4 = 3
  2. 2Normal gradient = −1/3
  3. 3y − 8 = −1/3 (x − 9)
  4. 4x = 0: y = 8 + 3 = 11, so (0, 11)

Answer(0, 11)

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