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WMA11/01 Jan 2020, Question 3: The Derivative as a Limit

Pearson Edexcel International AS Level Mathematics · Pure Mathematics 1 (WMA11) · Calculator · 6 marks

The question, in short

For the curve y = x² + 3x − 2 with point P(3, 16), work out the tangent gradient at P, the gradient of chord PQ where Q has x = 3 + h, and how these two results connect.

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

Part (a) · 2 marks

Work out the slope of the tangent line touching the curve at P(3, 16).

  1. 1dy/dx = 2x + 3
  2. 2At x = 3: gradient = 2(3) + 3 = 9

Answer9

Part (b) · 3 marks

Using Q at x = 3 + h on the curve, work out the slope of line PQ in terms of h, fully simplified.

  1. 1y at Q: (3 + h)² + 3(3 + h) − 2 = h² + 9h + 16
  2. 2Gradient PQ = (h² + 9h + 16 − 16)/(3 + h − 3) = (h² + 9h)/h
  3. 3= 9 + h

Answer9 + h

Part (c) · 1 mark

State briefly how the slope found in part (b) relates to the tangent slope found in part (a).

  1. 1As h gets smaller, Q moves towards P along the curve
  2. 2The chord gradient 9 + h tends to 9, which is the gradient of the tangent found in (a)

AnswerAs h → 0, the gradient of PQ (9 + h) tends to 9, the gradient of the tangent at P

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