A-LevelMathematicsNumerical methods9709/22 · Oct/Nov 2017 · Q59 Marks

The diagram shows the curve y = 4e⁻²ˣ and a straight line. The curve crosses the y-axis at the point P. The straight line crosses the y-axis at the point (0, 9) and its gradient is equal to the gradient of the curve at P. The straight line meets the curve at two points, one of which is Q as shown. [Figure 5.1]

(i)Show that the x-coordinate of Q satisfies the equation x = 9/8 - ½e⁻²ˣ.[6]
(ii)Use an iterative formula based on the equation in part (i) to find the x-coordinate of Q correct to 3 significant figures. Give the result of each iteration to 5 significant figures.[3]

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Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

[*M1] Obtain derivative of the form ke⁻²ˣ
[A1] State or imply gradient of curve at P is -8
[*DM1] Form equation of straight line through (0,9) with negative gradient
[A1] Obtain y = -8x+9 or equivalent
[DM1] Equate equation of curve and equation of straight line
[A1] Rearrange to confirm x = 9/8 - ½e⁻²ˣ
[M1] Use iterative process correctly at least once
[A1] Obtain final answer 1.07
[A1] Show sufficient iterations to 5 sf to justify answer or show sign change in interval (1.065,1.075)

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