1. Figure 1 shows the graphs of y = tan-l x and xy = 1 intersecting at the point A with x-coordinate. a) (i) Show that 1.1 < a < 1.2. (ii) Use linear interpolation to find an approximation for A, giving your answer to two decimal places. Figure 2 shows the tangent to the curve xy = 1 at A. This tangent meets the x-axis at B. The region between the arc OA, the line AB and the x-axis is shaded. b) Show that the x-coordinate of B is 2. (Marks available: 7) Answer Answer outline and marking scheme for question: 1 Give yourself marks for mentioning any of the points below: a) (i) The line intersect at a tan-l a =1. Let f(x) = a tan-l a -1 Then f(1.1) = -0.0837206 < 0 and f(1.2) = +0.0512696 > 0 Therefore the change of sign (i. e. the x value of a is between 1.1 and 1.2) (ii) Using linear interpolation to get a more accurate answer (to 2 decimal places). (4 marks) b) Rearranging the equation of the straight line, gives: so at point A, x = a, so: The equation to a tangent to the straight line is: or At B, y = 0 therefore x = 2a. (3 marks) (Marks available: 7) 2. The variables x and y satisfy the differential equation and y = 2 when x = 1. a) Use a local linear approximation to show that, when x = 1.02, b) By using the iterative equation find approximations for the values of y when x takes the values 1.02, 1.04 and 1.06, giving each value to three places of decimals. (Marks available: 6) Answer Answer outline and marking scheme for question: 2 Give yourself marks for mentioning any of the points below: a) Using the local linear approximation, gives: (2 marks) b) Substituting values of x into the iterative equation given gives: First value is 2.157 Second value is 2.316 Third value is 2.477 (4 marks) (Marks available: 6) 3. Figure 1 above shows sketches of the graphs of y = e-x and y = x and their intersection at x = α, where α is approximately 0.57. (i) Starting with x1 = 0.57, carry out the iteration xn+1 = e-xn up to and including x5, recording each value of xn to four decimal places as you proceed. (ii) Write down an estimate of the value of α to three decimal places. (Marks available: 4) Answer Answer outline and marking scheme for question: 3 Give yourself marks for mentioning any of the points below: Performing the iteration on the equation given, gives: x2 = 0.5655 x3 = 0.5681 x4 = 0.5666 x5 = 0.5675 Taking the results above, the closest approximation to α at 3d. p is 0.567. (Marks available: 4)