A-LevelMathematicsNumerical methods9709/21 · Oct/Nov 2021 · Q48 Marks
The curve with equation y = xe²ˣ + 5e⁻ˣ has a minimum point M.
- (a)Show that the x-coordinate of M satisfies the equation x = 1/3 ln 5 - 1/3 ln(1 + 2x).[5]
- (b)Use an iterative formula, based on the equation in part (a), to find the x-coordinate of M correct to 3 significant figures. Use an initial value of 0.35 and give the result of each iteration to 5 significant figures.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.
- [M1] Attempt use of product rule to differentiate xe²ˣ
- [A1] Obtain e²ˣ + 2xe²ˣ - 5e⁻ˣ
- [M1] Equate first derivative to zero and multiply by eˣ to obtain an equation involving e³ˣ
- [A1] Obtain e³ˣ(1 + 2x) = 5 or equivalent
- [A1 AG] Confirm given result x = 1/3 ln 5 - 1/3 ln(1 + 2x) with sufficient detail
- [M1] Use iteration process correctly at least once
- [A1] Obtain final answer 0.357
- [A1] Show sufficient iterations to 5sf to justify answer or show sign change in interval [0.3565, 0.3575]
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