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WMA11/01 Jun 2024, Question 10: Integration: Undoing Differentiation

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

The question, in short

For curve C with y = f(x), x > 0, given f′(x) = 6x − (2x − 1)(3x + 2)/(2√x) and P(4, 12) on C, find the normal 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) · 4 marks

Work out the gradient of C at P, then give the perpendicular line through P as y = mx + c with integer m and c.

  1. 1f′(4) = 24 − (7)(14)/(2 × 2) = 24 − 24.5 = −½
  2. 2Normal gradient = −1/(−½) = 2
  3. 3y − 12 = 2(x − 4)
  4. 4y = 2x + 4

Answery = 2x + 4

Part (b) · 6 marks

Integrate the given derivative, using the point P to determine the constant, and simplify each term of f(x).

  1. 1(2x − 1)(3x + 2) = 6x² + x − 2
  2. 2(6x² + x − 2)/(2√x) = 3x^(3/2) + ½x^(½) − x^(−½)
  3. 3f′(x) = 6x − 3x^(3/2) − ½x^(½) + x^(−½)
  4. 4f(x) = 3x² − (6/5)x^(5/2) − (1/3)x^(3/2) + 2x^(½) + c
  5. 5At (4, 12): 12 = 48 − 192/5 − 8/3 + 4 + c, so c = 16/15
  6. 6f(x) = 3x² − (6/5)x^(5/2) − (1/3)x^(3/2) + 2x^(½) + 16/15

Answerf(x) = 3x² − (6/5)x^(5/2) − (1/3)x^(3/2) + 2x^(½) + 16/15

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