P9.3 Newton–Raphson

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Many candidates were successful across all parts of this question demonstrating a good understanding of the Newton Raphson method, but this question did start to discriminate between the least confident and those who were more careful in their processing. Incorrect derivatives were occasionally seen, and candidates should be advised to refer to the derivative of tan(kx) provided in the formula booklet. A common error was forgetting to differentiate the x to 1 or losing it entirely. There were some attempts at integrating and unfortunately some misread (or misinterpreted) the notation and attempted to find the inverse of the function. These responses were unable to score any marks in part (b), even if they went on to differentiate in part (c). Candidates who converted 1 tan 2 x to 1 sin 2 1 cos 2 x x were generally not successful at applying the quotient rule correctly. Candidates had varying degrees of success in part (c). The Newton-Raphson formula was generally well quoted or used, however, difficulty evaluating 2 3.7 sec 2 led to the accuracy mark being lost. Evidence of substitution into the formula was inconsistent across the cohort, with some clearly showing the substitution and some leaving it to the examiner to try to establish what they had done. Those that did not explicitly show the substitution of 3.7 in f(x) and ( ) f ' x often made calculation errors and thus scored no marks in this part. Candidates are encouraged to show their substitution into the formula (not just stating ( ) ( ) f 3.7 3.7 f 3.7 - ) for this reason. Those candidates who did not obtain a correct derivative in part (b) were still able to achieve the method mark for using the Newton-Raphson formula correctly with their derivative. Those that did not differentiate in part (b) were able to go on and score both marks in part (c) if they restarted the differentiation at this stage.

Common mistakes

• x₂ = 3.7 − f(3.7)/f′(3.7) in radians
• evaluating sec²(1.85) is a common slip
• awrt 3.672

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

Part (c) was answered well with many candidates scoring full marks. Amongst those who didn’t score full marks, many did not heed the demand in the question to “Show your method” and just quoted their calculator’s solution without any explanation whilst others just wrote down a formula but did not show an application with values embedded. Others made rounding mistakes such as giving a final answer of 4.805 or 4.81. Premature rounding of f (5) and f′(5) also led to a final answer of 4.81. A significant number of candidates erroneously used degrees but could score the method mark if appropriate working was shown. Some candidates were able to “recover” minimal working due to the accuracy of their answer 4.804...., and gained the two marks. Some candidates also evaluated f (5) incorrectly even though it was given in the question. It was disappointing to see students misquoting The Newton-Raphson formula, typically with f(x) and f′(x) the wrong way round, even though the formula was given in the formula book.

Common mistakes

• x₁ = 5 − f(5)/f′(5) with values shown
• f(x) and f′(x) the right way round
• radians
• awrt 4.80 (not 4.8, 4.805 or 4.81 from premature rounding)

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

This was a well-answered question with part (a) and part (c) providing most of the discrimination. In part (a), most students substituted a defined 3 2 f( ) 2 1 n n n x x x and a correct 2 f ( ) 6 2 n n n x x x into the Newton-Raphson formula 1 f( ) . f ( ) n n n n x x x x Many students used correct algebra to arrive at the given result. Some students moved directly from 3 2 1 2 2 1 6 2 n n n n n n x x x x x x to 3 2 1 2 4 1 6 2 n n n n n x x x x x without showing the correct intermediate step of rationalising the denominator. There were bracketing errors, sign errors and the error of omitting ‘ 1 nx ’ to write only the right-hand side of the Newton-Raphson formula.

Common mistakes

• f′(x) = 6x² + 2x
• common-denominator intermediate step must be shown before the given answer
• keep the x_n term
• bracket/sign errors

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

In part (b), nearly all students wrote down a correct 2 3 3 2 , . 4 3 x x

Common mistakes

• x₂ = 3/4, x₃ = 2/3

Mark scheme (whole question; this part is marked with the orange bar)

Examiner's comment

In part (c), only a few students gave an acceptable explanation that detailed why the Newton- Raphson method did not work for this question. Some of these students stated ‘because there is a stationary point at 0, x then the method could not work’ while others explained that a tangent to the curve at 0 x would never meet the x-axis. The explanation, ‘not being able to divide by zero in the formula’, by itself, was not sufficient to gain credit in part (c).

Common mistakes

• stationary point at x = 0 so the tangent is horizontal and never meets the x-axis
• "cannot divide by zero" or "f′(0) = 0" alone scores B0