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WMA11/01 Oct 2023, Question 7: Integration: Undoing Differentiation

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

The question, in short

For curve C with x > 0, given f′(x) = (4x² + 10 − 7x^(1/2)) ÷ (4x^(1/2)) and P(4, −1) on C, find the gradient at P, the normal at P, and f(x).

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Part (ai) · 1 mark

Find the gradient of C at the point P where x = 4.

  1. 1f′(4) = (4 × 16 + 10 − 7 × 2)/(4 × 2)
  2. 2= 60/8 = 15/2

Answer15/2

Part (aii) · 3 marks

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

  1. 1Normal gradient = −1/(15/2) = −2/15
  2. 2y + 1 = −(2/15)(x − 4)
  3. 315y + 15 = −2x + 8
  4. 42x + 15y + 7 = 0

Answer2x + 15y + 7 = 0

Part (b) · 6 marks

Integrate f′(x) and use the point P(4, −1) to determine the constant, giving f(x).

  1. 1f′(x) = x^(3/2) + (5/2)x^(−1/2) − 7/4
  2. 2Integrate: f(x) = (2/5)x^(5/2) + 5x^(1/2) − (7/4)x + c
  3. 3Use P(4, −1): (2/5)(32) + 5(2) − 7 + c = −1
  4. 464/5 + 3 + c = −1, so c = −84/5
  5. 5f(x) = (2/5)x^(5/2) + 5x^(1/2) − (7/4)x − 84/5

Answerf(x) = (2/5)x^(5/2) + 5x^(1/2) − (7/4)x − 84/5

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