10   5. The table below shows corresponding values of x and y for y = log3 2x The values of y are given to 2 decimal places as appropriate. x 3 4.5 6 7.5 9 y 1.63 2 2.26 2.46 2.63 (a) Using the trapezium rule with all the values of y in the table, find an estimate for log3 3 9 2x x d ∫ (3) Using your answer to part (a) and making your method clear, estimate (b) (i) log3 10 3 9 2x x ( ) ∫ d (ii) log3 3 9 18x x d ∫ (3) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ Turn over 11   Question 5 continued _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ (Total for Question 5 is 6 marks) 12   6. y x O α P Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x) where f(x) = 8 sin  1 2 x     − 3x + 9            x > 0 and x is measured in radians. The point P, shown in Figure 2, is a local maximum point on the curve. Using calculus and the sketch in Figure 2, (a) find the x coordinate of P, giving your answer to 3 significant figures. (4) The curve crosses the x-axis at x = α , as shown in Figure 2. Given that, to 3 decimal places, f(4) = 4.274 and f(5) = −1.212 (b) explain why α must lie in the interval [4, 5] (1) (c) Taking x0 = 5 as a first approximation to α, apply the Newton-Raphson method once to f(x) to obtain a second approximation to α. Show your method and give your answer to 3 significant figures. 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