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WMA11/01 Oct 2019, Question 11: Integrating with Two Conditions

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

The question, in short

Curve y = f(x) passes through P(4, 32/3), with f″(x) = 4/√x − 3 and f′(x) = 5 at P. Find the tangent at P and f(x).

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

Part (a) · 2 marks

Find the tangent line at P(4, 32/3) with gradient 5, giving it as y = mx + c with m and c found.

  1. 1Tangent through (4, 32/3) with gradient 5: y − 32/3 = 5(x − 4)
  2. 2y = 5x − 20 + 32/3 = 5x − 28/3

Answery = 5x − 28/3

Part (b) · 8 marks

Integrate f″(x) = 4/√x − 3 twice, using f′(4) = 5 and P(4, 32/3) to determine f(x).

  1. 1f′(x) = ∫(4x^(−½) − 3) dx = 8x^(½) − 3x + k
  2. 2f′(4) = 5: 16 − 12 + k = 5, so k = 1
  3. 3f′(x) = 8x^(½) − 3x + 1
  4. 4f(x) = ∫(8x^(½) − 3x + 1) dx = (16/3)x^(3/2) − (3/2)x² + x + d
  5. 5f(4) = 32/3: 128/3 − 24 + 4 + d = 32/3, so d = −12
  6. 6f(x) = (16/3)x^(3/2) − (3/2)x² + x − 12

Answerf(x) = (16/3)x^(3/2) − (3/2)x² + x − 12

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Also searched as: WMA11 October 2019 · WMA11/01 · wma11 oct19 · IAS Maths Pure Mathematics 1 October 2019 · question 11.
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