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0580/43 Oct/Nov 2022, Question 9: Gradient of a Curve at a Point

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 17 marks

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The question, in short

Sketch three graphs showing axis intercepts, then differentiate a cubic, find its gradient at x = −1, and locate points where the tangent gradient is −28.

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

Part (ai) · 2 marks

Sketch 3x − 4y = 12, marking where it crosses both axes.

  1. 1x = 0: −4y = 12, y = −3; y = 0: 3x = 12, x = 4
  2. 2Draw a straight line through (0, −3) and (4, 0), positive gradient, marking −3 and 4

AnswerCorrect sketch of 3x − 4y = 12 with y = −3 and x = 4 indicated on axes

Part (aii) · 4 marks

Sketch y = x² − 3x − 4, marking intercepts with the axes.

  1. 1y = (x − 4)(x + 1): roots x = −1 and x = 4
  2. 2y-intercept (0, −4)
  3. 3U-shaped parabola through these points, minimum just right of the y-axis (at x = 1.5) below the x-axis

AnswerCorrect sketch of y = x² − 3x − 4 with (0, −4) indicated as y-intercept and x = −1 and x = 4 indicated as roots; minimum in fourth quadrant, not at x = 0

Part (aiii) · 2 marks

Sketch y = 6ˣ, marking any intercept with the axes.

  1. 16⁰ = 1, so the curve crosses the y-axis at (0, 1)
  2. 2Increasing curve, above the x-axis, approaching the x-axis for negative x

AnswerCorrect sketch of y = 6ˣ with y-intercept indicated at (0, 1)

Part (bi) · 2 marks

Differentiate y = 5 + 8x − (4/3)x³ to find dy/dx.

  1. 1Multiply by the power, then take 1 off the power: axⁿ → naxⁿ⁻¹
  2. 25 → 0
  3. 38x → 8
  4. 4−(4/3)x³ → 3 × (−4/3)x² = −4x²
  5. 5dy/dx = 8 − 4x²

Answer8 − 4x²

Part (bii) · 2 marks

Find the gradient of y = 5 + 8x − (4/3)x³ when x = −1.

  1. 1The gradient is the value of dy/dx
  2. 2From part (i): dy/dx = 8 − 4x²
  3. 3At x = −1: 8 − 4 × (−1)² = 8 − 4 = 4

Answer4

Part (biii) · 5 marks

Given tangent gradient −28, find coordinates of both points where the tangent touches the curve.

  1. 18 − 4x² = −28
  2. 24x² = 36, x² = 9, x = 3 or −3
  3. 3x = 3: y = 5 + 24 − 4/3 × 27 = −7
  4. 4x = −3: y = 5 − 24 + 36 = 17
  5. 5(3, −7) and (−3, 17)

Answer(3, −7) and (−3, 17)

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Also searched as: 0580/43/O/N/22 · 0580_w22_qp_43 · 0580 w22 43 · question 9.
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