A-LevelMathematicsNumerical methods9709/33 · May/June 2018 · Q48 Marks
The curve with equation y = (ln x)/(3+x) has a stationary point at x = p.
- (i)Show that p satisfies the equation ln x = 1 + 3/x.[3]
- (ii)By sketching suitable graphs, show that the equation in part (i) has only one root.[2]
- (iii)It is given that the equation in part (i) can be written in the form x = (3+x)/(ln x). Use an iterative formula based on this rearrangement to determine the value of 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 (May/June 2018). The mark or mark type is shown in brackets.
- M1 Use the quotient or product rule
- A1 Obtain correct derivative in any form
- A1 Equate derivative to zero and obtain the given equation
- B1 Sketch a relevant graph, e.g. y = ln x
- B1 Sketch a second relevant graph, e.g. y = 1 + 3/x, and justify the given statement
- M1 Use iterative formula x_(n+1) = (3+x_n)/(ln x_n) correctly at least once
- A1 Obtain final answer 4.97
- A1 Show sufficient iterations to 4 d.p. to justify 4.97 to 2 d.p. or show there is a sign change in the interval (4.965, 4.975)
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