A-LevelMathematicsNumerical methods9709/21 · Oct/Nov 2022 · Q45 Marks

By sketching a suitable pair of graphs on the same diagram, show that the equation e^(x/2) = x⁵ has exactly one real root.

(a)By sketching a suitable pair of graphs on the same diagram, show that the equation e^(x/2) = x⁵ has exactly one real root.[2]
(b)Use the iterative formula x_(n+1) = ⁵√(e^(x_n)/x_n) to determine the root 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 (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] Draw approximately correct sketch of y = e^(x/2) (with some curve in second quadrant as well as first.)
[B1] Draw approximately correct sketch of y = x⁵ and confirm one root.
[B1] Draw approximately correct sketch of y = 5lnx or y = ln x^(5)
[B1] Draw approximately correct sketch of y = x/2 and confirm one root (Must have intersection in the 4th quadrant.)
[M1] Use iteration process correctly at least once
[A1] Obtain final answer 0.9128 (answer required to exactly 4 s.f.)
[A1] Show sufficient iterations to 6 s.f. to justify answer or show sign change in the interval [0.91275, 0.91285]

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