A-LevelMathematicsNumerical methods9709/22 · Feb/Mar 2020 · Q69 Marks
A curve has equation y = x³e⁻⁰.²ˣ where x ≥ 0. At the point P on the curve, the gradient of the curve is 15.
- (a)Show that the x-coordinate of P satisfies the equation x = (75e⁻⁰.²ˣ)/(15 + x)[4]
- (b)Use the equation in part (a) to show by calculation that the x-coordinate of P lies between 1.7 and 1.8.[2]
- (c)Use an iterative formula, based on the equation in part (a), to find the x-coordinate of P correct to 4 significant figures. Give the result of each iteration to 6 significant figures.[3]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.
- (a) Differentiate using the product rule
- (a) Obtain 3x²e⁰.²ˣ +0.2x³e⁰.²ˣ OE
- (a) Equate first derivative to 15 and rearrange to x = ...
- (a) Confirm x = (75e⁻⁰.²ˣ)/(15+x) AG – necessary detail needed
- (b) Consider sign of x – (75e⁻⁰.²ˣ)/(15+x) or equivalent for 1.7 and 1.8
- (b) Obtain -0.08... and 0.03... or equivalents and justify conclusion
- (c) Use iterative process correctly at least once Answer required to exactly 4 sf
- (c) Obtain final answer 1.771
- (c) Show sufficient iterations to 6 sf to justify answer or show a sign change in the interval [1.7705, 1.7715]
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