OCR GCE AS Level Further Maths B (MEI) Numerical Methods (optional - choose any two of Y411/Y412/Y413/Y414) November 2021 question paper Y414

The table shows 3 values of x and the associated values of f (x).

Use Lagrange’s method to construct the interpolating polynomial of degree 2 for the values in the table, giving your answer in the form

ax² + bx + c

where a, b and c are constants to be determined.

Explain why it is not possible to use Newton’s forward difference interpolation formula for

these values.

The table shows 2 values of x and the associated values of (f x). The values of x are exact and the values of f(x) have been rounded to 3 decimal places.

Explain why your answer to part (a) is so large.

The table shows 3 values of x and the associated values of f (x).

Use the forward difference method to find two estimates of f ′ (2).

Explain why it is not possible to use the central difference method in this case.

Explain why it is generally preferable to use the central difference method instead of the

forward difference method.

The Newton-Raphson method is to be used to solve the equation.

On the copy of Fig. 4.1 in the Printed Answer Booklet, illustrate how the Newton-Raphson

method finds x₁ with a starting value of x₀ = 1.

The spreadsheet output in Fig. 4.2 shows some iterates, together with some further analysis, when the Newton-Raphson method is used with a starting value of x₀ = 0.5.

Determine whether α = 2.817921 correct to 6 decimal places.

Explain what the entries in the ratio column in Fig. 4.2 tell you about the order of convergence

of this sequence of estimates.

Fig. 4.3 shows the spreadsheet output when the Newton-Raphson method is used with a starting value of x₀ = 0.1.

Explain why #NUM! is displayed in cell J7.

The method of false position is used to find β. Some spreadsheet output showing the method being applied is shown in Fig. 4.4.

Determine the values which would be displayed in row 13 of the output, giving your answers

correct to 5 decimal places where appropriate.

State the value of β as accurately as you can, justifying the precision quoted.

The formula in cell P9 is = IF(S8 > 0,R8,P8).

Give a similar formula for cell N9.

midpoint rule and Simpson’s rule. Some of the spreadsheet output she has generated is shown below. Some of the values are missing.

	A	B	C	D	E	F
1	n	Tn	Mn	S2n	difference	ratio
2	1	2.888888889	2
3	2	2.444444444	2.162175402	2.256265083
4	4	2.303309923	2.224831272	2.250990823	-0.005274
5	8	2.264070598		2.250517186	-0.000474	0.0898
6	16	2.253905539	2.248772298	2.250483378	-3.38E-05	0.07138
7	32	2.251338918	2.250052312	2.250481181	-2.2E-06	0.06499

Use the information in the output to determine the values in cells

• D2,

• E3,

• F4,

• C5.

James is investigating relative error. He uses Eⁿ = 2.718ⁿ to approximate eⁿ, and calculates the relative error for various values of n. Some of his results are shown in the spreadsheet output in Fig. 6.1.

Give a suitable formula for cell D6.

Let y be the relative error when Eⁿ is used to approximate eⁿ, and x be the relative error when E is used to approximate e¹.

Write down a formula for y in terms of x which could be used to model the relationship

between y, x and n.

Determine whether the model holds for n = 5.

James tests the model again, this time using Fⁿ = 2.7ⁿ to approximate eⁿ. Some of his results are shown in the spreadsheet output in Fig. 6.2.

Identify a limitation of the model found in part (b).

James carries out some further analysis, which is shown in the spreadsheet output in Fig. 6.3.

Write - 2.592E-05 in standard mathematical notation.

The formula in cell E22 is = E18*$D$21.

This formula has been copied across row 22 so that equivalent formulae are in cells F22 and G22.

Explain the purpose of the $ symbols.

The equation f(x) = 0 where f(x) = 0.9^x × x² - x - 1 has a root α, such that α ≈ 1.87.

Use the central difference method with h = 0.005 to determine an estimate of g ′ (1.87).

Explain how your answer to part (a) would influence Sundip’s decision.

Sundip decides to use the iterative formula given above with x₀ = 2. Her spreadsheet output, together with some further analysis, is shown.

	N	O	P	Q
10	r	xr	difference	ratio
11	0	2
12	1	1.9245009	-0.0754991
13	2	1.8925878	-0.0319131	0.4227
14	3	1.8790715	-0.0135163	0.4235
15	4	1.8733418	-0.0057297	0.4239
16	5	1.8709120	-0.0024298	0.4241
17	6	1.8698814	-0.0010306	0.4241
18	7	1.8694442	-0.0004372	0.4242
19	8	1.8692588	-0.0001854	0.4242
20	9	1.8691801	-7.8660E-05	0.4242

Write down suitable cell formulae for

• cell P12,

• cell Q13.

Explain what the entries in column Q tell you about the order of convergence of the estimates

in column O.

Without doing any more calculations, state the value of α as accurately as you can, justifying

the precision quoted.

Use the relaxed iteration x_n+1 = (1 - λ)x_n + λg(x_n), with λ = 1.736 and x₀ = 2 to determine

the value of α correct to 8 decimal places.