A-LevelMathematicsNumerical methods9709/33 · Oct/Nov 2022 · Q88 Marks
The curve with equation y = x³/(eˣ - 1) has a stationary point at x = p, where p > 0.
- (a)Show that p = 3(1 – e⁻ᵖ).[3]
- (b)Verify by calculation that p lies between 2.5 and 3.[2]
- (c)Use an iterative formula based on the equation in part (a) to determine p correct to 2 decimal places. Give the result of each iteration to 4 decimal places.[3]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.
- M1 Use quotient or product rule
- A1 Obtain correct derivative in any form
- A1 Equate derivative at x = p to zero and obtain the given equation
- M1 Evaluate a relevant expression or pair of relevant pair of expressions at p = 2.5 and p = 3
- A1 Complete the argument with correct calculated values
- M1 Use the iterative formula Pn+1 = 3(1-e⁻ᵖⁿ) correctly at least once
- A1 Obtain final answer p = 2.82
- A1 Show sufficient iterations to 4 d.p.to justify 2.82 to 2 d.p., or show there is a sign change in the interval (2.815, 2.825)
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