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4PM1/01 Jan 2022, Question 11: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 9 marks

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The question, in short

Show the derivative of a given product involving e^(4x) and a quadratic, then use it to find the volume when region R is rotated 360° about the x-axis, to 2 significant figures.

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

Part (a) · 5 marks

Differentiate y = (e^(4x)/32)(8x² − 4x + 1) and show the result simplifies to x²e^(4x).

  1. 1Product rule: dy/dx = (e^(4x)/32)(16x − 4) + (4e^(4x)/32)(8x² − 4x + 1)
  2. 2= (e^(4x)/32)[16x − 4 + 32x² − 16x + 4]
  3. 3= (e^(4x)/32)(32x²)
  4. 4= x²e^(4x)

Answerdy/dx = x²e^(4x) shown

Part (b) · 4 marks

Using part (a), calculate the volume of the solid formed by rotating region R about the x-axis, giving the answer to 2 significant figures.

  1. 1V = π ∫ from −2 to 0 of (3xe^(2x))² dx = 9π ∫ from −2 to 0 of x²e^(4x) dx
  2. 2By (a): = 9π[(e^(4x)/32)(8x² − 4x + 1)] from −2 to 0
  3. 3= 9π[1/32 − (e^(−8)/32)(32 + 8 + 1)]
  4. 4= 9π(1 − 41e^(−8))/32 = 0.8714…
  5. 5V ≈ 0.87

Answer0.87

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