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WMA11/01 Jun 2023, Question 8: Equation of the Tangent Line

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 10 marks

The question, in short

Show full working without relying on a calculator: find the tangent line to y = (1/4)x³ − 8x^(−1/2) at P(4, 12), and integrate f′(x) = (1/4)x³ − 8x^(−1/2) through P to find f(x).

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

Part (a) · 5 marks

Find the tangent to y = (1/4)x³ − 8x^(−1/2) at P(4, 12), giving the answer as ax + by + c = 0 with integer coefficients.

  1. 1dy/dx = (3/4)x² + 4x^(−3/2)
  2. 2At x = 4: dy/dx = (3/4)(16) + 4/8 = 12 + 1/2 = 25/2
  3. 3Tangent: y − 12 = (25/2)(x − 4)
  4. 42y − 24 = 25x − 100
  5. 525x − 2y − 76 = 0

Answer25x − 2y − 76 = 0

Part (b) · 5 marks

Given f′(x) = (1/4)x³ − 8x^(−1/2) and that the curve passes through P(4, 12), find f(x) with coefficients in simplest form.

  1. 1Integrate: f(x) = (1/4)(x⁴/4) − 8x^(1/2)/(1/2) + c
  2. 2f(x) = x⁴/16 − 16x^(1/2) + c
  3. 3f(4) = 12: 256/16 − 16 × 2 + c = 12, so 16 − 32 + c = 12
  4. 4c = 28
  5. 5f(x) = x⁴/16 − 16√x + 28

Answerf(x) = x⁴/16 − 16√x + 28

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