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WMA11/01 Jan 2019, Question 12: Integration: Undoing Differentiation

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 9 marks

The question, in short

Curve y = f(x), x > 0, passes through P(4, −2), with dy/dx = 3x√x − 10x^(−1/2). Find the tangent at P in y = mx + c form, then find f(x).

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

Part (a) · 4 marks

Find the gradient at P from the derivative, then the tangent line at P, giving y = mx + c with integer m and c.

  1. 1At x = 4: dy/dx = 3 × 4 × 2 − 10/2 = 24 − 5 = 19
  2. 2Tangent through (4, −2): y + 2 = 19(x − 4)
  3. 3y = 19x − 78

Answery = 19x − 78

Part (b) · 5 marks

Integrate the given derivative and use the point P(4, −2) to determine the constant, giving f(x).

  1. 1Write dy/dx = 3x^(3/2) − 10x^(−1/2)
  2. 2Integrate: f(x) = (6/5)x^(5/2) − 20x^(1/2) + c
  3. 3Use (4, −2): −2 = (6/5)(32) − 20(2) + c = 38.4 − 40 + c
  4. 4c = −0.4, so f(x) = (6/5)x^(5/2) − 20x^(1/2) − 2/5

Answerf(x) = (6/5)x^(5/2) − 20x^(1/2) − 2/5

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