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WMA11/01 Jan 2022, Question 6: Integration: Undoing Differentiation

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

The question, in short

Given f′(x) = (x + 3)²/(x√x) for x > 0 and P(4, 20) on curve C, find the gradient at P, the tangent line there, and f(x).

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

Part (ai) · 1 mark

Find the value of the gradient of C at the point P.

  1. 1f′(4) = (4 + 3)²/(4√4)
  2. 2= 49/8

Answer49/8

Part (aii) · 3 marks

Using the gradient at P, find the tangent to C at P in the form ax + by + c = 0 with integer coefficients.

  1. 1y − 20 = (49/8)(x − 4)
  2. 28y − 160 = 49x − 196
  3. 349x − 8y − 36 = 0

Answer49x − 8y − 36 = 0

Part (b) · 7 marks

Find f(x) by integrating f′(x), using P to determine the constant, and simplify the result.

  1. 1f′(x) = (x² + 6x + 9)/x^(3/2)
  2. 2= x^(1/2) + 6x^(−1/2) + 9x^(−3/2)
  3. 3f(x) = (2/3)x^(3/2) + 12x^(1/2) − 18x^(−1/2) + c
  4. 4At (4, 20): 20 = (2/3)(8) + 12(2) − 18(1/2) + c = 16/3 + 24 − 9 + c
  5. 5c = −1/3
  6. 6f(x) = (2/3)x^(3/2) + 12x^(1/2) − 18x^(−1/2) − 1/3

Answerf(x) = (2/3)x^(3/2) + 12x^(1/2) − 18x^(−1/2) − 1/3

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