June 2022 Paper 3 Q10

OCR MEICurrent spec5 marksDifferentiationNumerical Methods

10 In this question you must show detailed reasoning.

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Approximating the sine function” are reproduced below; the line numbers are those printed on the Insert.

Lines 12–14
Fig. C2.1 shows the curve \(y = \sin x\) and the quadratic curve which goes through the points \((0, 0)\), \(\left(\frac{\pi}{2}, 1\right)\) and \((\pi, 0)\). The equation of this curve is \(y = \dfrac{4x(\pi - x)}{\pi^2}\). Fig. C2.2 shows the curve \(y = \dfrac{4x(\pi - x)}{\pi^2} - \sin x\).

Graph on a grid, x from −2 to 5 and y from −2 to 2, showing y = sin x and the quadratic curve, which are very close together between x = 0 and x = π, both reaching about 1 near x = 1.6.
Fig. C2.1
Graph on a grid, x from −2 to 5 and y from −2 to 2, showing the difference curve: close to the x-axis between x = 0 and x = π with two small humps near x = 0.5 and x = 2.7, falling steeply outside that interval.
Fig. C2.2

Fig. C2.2 indicates that the curve \(y = \dfrac{4x(\pi - x)}{\pi^2} - \sin x\) has a stationary point near \(x = 3\).

  • Verify that the \(x\)-coordinate of this stationary point is between 2.6 and 2.7.
  • Show that this stationary point is a maximum turning point. [5]
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