OCR GCE AS Level Further Maths B (MEI) Numerical Methods (optional - choose any two of Y411/Y412/Y413/Y414) November 2020 question paper Y414

Fig. 1.1 shows the graph of y = f (x) for 0 ⩽ x ⩽ 1.

Fig. 1.2 shows three values of x and the associated values of f (x).

Draw a suitable straight line on the diagram in the Printed Answer Booklet to show how the

answer to part (b) is obtained using the central difference method.

These estimates, together with some further analysis, are shown in Fig. 1.3.

Write 3.139E-05 in standard mathematical notation.

Explain what the entries in column 4 of Fig. 1.3 demonstrate about the order

of convergence of the central difference method.

With reference to Fig. 2.1, comment on the values obtained in part (a).

Fig. 3 shows three values of x and the associated values of f(x).

Use Lagrange’s method to construct the interpolating polynomial of degree 2 for the data points in Fig. 3, giving your answer in the form

ax² + bx + c,

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

You are given that the values of x are exact, but the values of f (x) are rounded to 2 decimal places.

Calculate the range of possible values of a, giving your answers correct to 4 decimal places.

A small company leases a machine which they use for scanning, printing and photocopying. The cost of each job completed is displayed in pounds to the nearest tenth of a penny. A sample of these costs is shown in Fig. 4.

Sue models the situation by assuming that the cost of each job is rounded correct to 3 decimal places before being displayed.

One day the machine is used for 412 jobs. Sue recorded the cost displayed for each job and found that the total of the displayed values was £45.730.

Find the range of possible values of the true cost according to Sue’s model, giving your

answers correct to the nearest penny.

Roxanne believes that the cost of each job is chopped to 3 decimal places before being displayed.

Find the range of possible values of the true cost according to Roxanne’s model, giving your

answers correct to the nearest penny.

The company received an invoice for £45.92 for the 412 jobs carried out on the machine on that day.

Explain whether the company should use Roxanne’s model or Sue’s model for future

budgeting.

Write the definite integral Arthur is approximating in standard mathematical notation.

Write down a suitable cell formula for cell F7.

Arthur obtains further approximations using the trapezium rule with 2, 4, 8 and 16 strips and these are shown in Fig. 5.2, together with T₁.

Use values from Fig. 5.2 to obtain S₂, an estimate of the integral using one application of

Simpson’s rule, giving your answer correct to 6 decimal places.

Arthur calculates S₂ using his spreadsheet and obtains the value 1.930649.

Explain why this is different to the value obtained using your calculator.

On the copy of Fig. 6.2 in the Printed Answer Booklet, starting with x₀ = 1, show how the

iterative formula works to find x₁ and x₂.

With reference to Fig. 6.2, explain whether the iterative formula could be used to find β.

The iterative formula in part (a) is to be used to approximate the largest of the three roots α. The spreadsheet in Fig. 6.3 shows 13 of the first 15 iterates obtained using this formula together with some further analysis.

Use extrapolation to obtain an improved approximation to α, justifying the precision quoted.

Use the secant method with x₀ = 0.5 and x₁ = 0.8 to obtain x₂, a first approximation to the

root, giving your answer correct to 4 decimal places.

When x₀ = 0.5 and x₁ = 1 are used as the starting values, x₂ is found to be 4.4577 correct to 4 decimal places.

Explain why a small change in x₁ leads to such a significant change in the value obtained for

x₂.