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0606/23 May/Jun 2025, Question 8: Connected Rates of Change and Small Changes

Cambridge IGCSE Additional Mathematics 0606 · Paper 23 (Calculator) · Calculator · 9 marks

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The question, in short

Differentiate y = e^(3x+2) tan x to estimate the small change in y near x = 0.1, then solve dy/dx = sin(3x + π) through (π/9, 4/3) to find y at x = 5π/12.

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Part (a) · 5 marks

Using calculus, find the approximate change in y = e^(3x+2) tan x when x increases from 0.1 to 0.1 + h for small h.

  1. 1Product rule: dy/dx = 3e^(3x+2) tan x + e^(3x+2) sec²x
  2. 2At x = 0.1: dy/dx = e^2.3(3 tan 0.1 + sec²0.1) ≈ 13.08
  3. 3δy ≈ (dy/dx) × δx = 13.08 × h
  4. 4δy ≈ 13.1h

Answer13.1h

Part (b) · 4 marks

Given dy/dx = sin(3x + π) and the curve through (π/9, 4/3), find the exact y-coordinate where x = 5π/12.

  1. 1y = −(1/3)cos(3x + π) + c
  2. 2At (π/9, 4/3): 4/3 = −(1/3)cos(4π/3) + c = 1/6 + c, so c = 7/6
  3. 3At x = 5π/12: y = −(1/3)cos(9π/4) + 7/6
  4. 4y = −(1/3)(√2/2) + 7/6 = (7 − √2)/6

Answer(7 − √2)/6

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