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4PM1/01 Jun 2023, Question 7: The Product and Quotient Rules

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 10 marks

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The question, in short

For the curve y = √(e^(4x)/(2x − 3)), use small increments δx and δy to find a differential approximation, then apply it numerically at x = 2.5.

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Part (a) · 7 marks

Show that the small change δy is approximately (e^(2x)(4x − 7))/(2x − 3)^(3/2) times δx.

  1. 1y = e^(2x)/(2x − 3)^(1/2)
  2. 2Quotient rule: dy/dx = [2e^(2x)(2x − 3)^(1/2) − e^(2x)(2x − 3)^(−1/2)]/(2x − 3)
  3. 3Multiply top and bottom by (2x − 3)^(1/2): dy/dx = [2(2x − 3)e^(2x) − e^(2x)]/(2x − 3)^(3/2)
  4. 4= e^(2x)(4x − 7)/(2x − 3)^(3/2)
  5. 5δy ≈ (dy/dx) δx gives the result

Answerdy/dx = e^(2x)(4x − 7)/(2x − 3)^(3/2), so δy ≈ (e^(2x)(4x − 7))/(2x − 3)^(3/2) δx — shown

Part (b) · 3 marks

Given x = 2.5, estimate δy to 2 significant figures when x increases by 0.2%.

  1. 1δx = 0.2% of 2.5 = 0.005
  2. 2δy ≈ e⁵(10 − 7)/(5 − 3)^(3/2) × 0.005
  3. 3= 3e⁵/(2√2) × 0.005 ≈ 0.787
  4. 4δy ≈ 0.79

Answerδy ≈ 0.79

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