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WMA11/01 Jan 2020, Question 11: Integrating with Two Conditions

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 11 marks

The question, in short

Given f″(x) = 6/√(x³) + x for x > 0 and point P(4, −50) with f′(4) = −4, find the normal line at P and the function f(x).

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

Part (a) · 3 marks

Find the equation of the normal to the curve at P(4, −50), giving it in the form y = mx + c.

  1. 1Tangent gradient is −4, so the normal gradient is 1/4
  2. 2Normal through (4, −50): y + 50 = ¼(x − 4)
  3. 3y = ¼x − 51

Answery = (1/4)x − 51

Part (b) · 8 marks

Integrate f″(x) twice using f′(4) = −4 and f(4) = −50 to determine f(x).

  1. 1Write f″(x) = 6x^(−3/2) + x
  2. 2Integrate: f′(x) = −12x^(−1/2) + ½x² + k
  3. 3Use f′(4) = −4: −6 + 8 + k = −4, so k = −6
  4. 4Integrate again: f(x) = −24x^(1/2) + (1/6)x³ − 6x + d
  5. 5Use f(4) = −50: −48 + 32/3 − 24 + d = −50, so d = 34/3
  6. 6f(x) = −24√x + (1/6)x³ − 6x + 34/3

Answerf(x) = −24√x + (1/6)x³ − 6x + 34/3

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