OCR MEI Further Numerical Methods 2021 November — Question 3 7 marks

Exam BoardOCR MEI
ModuleFurther Numerical Methods (Further Numerical Methods)
Year2021
SessionNovember
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSign Change & Interval Methods
TypeSecant Method or False Position
DifficultyStandard +0.3 This is a straightforward question about the False Position method requiring interpretation of spreadsheet formulas and basic understanding of convergence. Part (a) is direct reading, (b) requires simple logic adaptation, (c) involves one application of the False Position formula, and (d) tests standard knowledge that ratios approaching a constant indicate linear convergence. All parts are routine for Further Maths students who have studied numerical methods.
Spec1.06a Exponential function: a^x and e^x graphs and properties1.09c Simple iterative methods: x_{n+1} = g(x_n), cobweb and staircase diagrams4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials
3 The method of False Position is used to find a sequence of approximations to the root of an equation. The spreadsheet output showing these approximations, together with some further analysis, is shown below.
CDEFGHIJ
4af(a)b\(\mathrm { f } ( b )\)\(x _ { \text {new } }\)\(\mathrm { f } \left( x _ { \text {new } } \right)\)differenceratio
51-1.8248217.28991.09547-1.80507
61.09547-1.80507217.28991.18097-1.754180.08551
71.18097-1.75418217.28991.25641-1.662460.075440.88229
81.25641-1.66246217.28991.32164-1.527810.065230.86458
91.32164-1.52781217.28991.37672-1.357060.055080.84439
101.37672-1.35706217.28991.42208-1.16420.045360.8236
111.42208-1.1642217.28991.45853-0.966160.036460.80376
121.45853-0.96616217.28991.48719-0.778250.028660.78598
131.48719-0.77825217.2899
14
The formula in cell D5 is \(\quad = \mathrm { SINH } \left( \mathrm { C5 } ^ { \wedge } 2 \right) - \mathrm { C5 } ^ { \wedge } 3 - 2\).
  1. Write down the equation which is being solved. The formula in cell C 6 is \(\quad = \mathrm { IF } ( \mathrm { H } 5 < 0 , \mathrm { G } 5 , \mathrm { C } 5 )\).
  2. Write down a similar formula for cell E6.
  3. Calculate the values which would be displayed in cells G13 and G14 to find further approximations to the root.
  4. Explain what the values in column J tell you about
Question 3:
AnswerMarks Guidance
3(a) sinhx² ‒ x³ ‒ 2 = 0
[1]
AnswerMarks Guidance
3(b) =IF(H5>0,G5,E5)
[1]
AnswerMarks Guidance
3(c) oe
1.48719×17.2899−2×‒0.77825
17.2899−−0.77825
awrt 1.50928
AnswerMarks
awrt 1.52603M1
A1
AnswerMarks
A13.1a
1.1
AnswerMarks
1.1may be implied by 1.509…
NB f(1.50928) = ‒0.6111 to 4 sf
[3]
AnswerMarks Guidance
3(d) the ratios are decreasing which suggests the
convergence is (slightly) faster than 1st order
the ratios are close to 1 which suggests the
AnswerMarks Guidance
convergence is slowB1
B12.2b
2.2ballow between 1st and 2nd order do not allow eg not first
order
[2]
3
2.0791668
1.7783346
1.7360141
1.7351281
1.7351277
Question 3:
3 | (a) | sinhx² ‒ x³ ‒ 2 = 0 | B1 | 1.1 | must see = 0
[1]
3 | (b) | =IF(H5>0,G5,E5) | B1 | 1.1 | or =IF(H5<0,E5,G5) | must see =
[1]
3 | (c) | oe
1.48719×17.2899−2×‒0.77825
17.2899−−0.77825
awrt 1.50928
awrt 1.52603 | M1
A1
A1 | 3.1a
1.1
1.1 | may be implied by 1.509…
NB f(1.50928) = ‒0.6111 to 4 sf
[3]
3 | (d) | the ratios are decreasing which suggests the
convergence is (slightly) faster than 1st order
the ratios are close to 1 which suggests the
convergence is slow | B1
B1 | 2.2b
2.2b | allow between 1st and 2nd order | do not allow eg not first
order
[2]
3
2.0791668
1.7783346
1.7360141
1.7351281
1.7351277
3 The method of False Position is used to find a sequence of approximations to the root of an equation. The spreadsheet output showing these approximations, together with some further analysis, is shown below.

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & C & D & E & F & G & H & I & J \\
\hline
4 & a & f(a) & b & $\mathrm { f } ( b )$ & $x _ { \text {new } }$ & $\mathrm { f } \left( x _ { \text {new } } \right)$ & difference & ratio \\
\hline
5 & 1 & -1.8248 & 2 & 17.2899 & 1.09547 & -1.80507 &  &  \\
\hline
6 & 1.09547 & -1.80507 & 2 & 17.2899 & 1.18097 & -1.75418 & 0.08551 &  \\
\hline
7 & 1.18097 & -1.75418 & 2 & 17.2899 & 1.25641 & -1.66246 & 0.07544 & 0.88229 \\
\hline
8 & 1.25641 & -1.66246 & 2 & 17.2899 & 1.32164 & -1.52781 & 0.06523 & 0.86458 \\
\hline
9 & 1.32164 & -1.52781 & 2 & 17.2899 & 1.37672 & -1.35706 & 0.05508 & 0.84439 \\
\hline
10 & 1.37672 & -1.35706 & 2 & 17.2899 & 1.42208 & -1.1642 & 0.04536 & 0.8236 \\
\hline
11 & 1.42208 & -1.1642 & 2 & 17.2899 & 1.45853 & -0.96616 & 0.03646 & 0.80376 \\
\hline
12 & 1.45853 & -0.96616 & 2 & 17.2899 & 1.48719 & -0.77825 & 0.02866 & 0.78598 \\
\hline
13 & 1.48719 & -0.77825 & 2 & 17.2899 &  &  &  &  \\
\hline
14 &  &  &  &  &  &  &  &  \\
\hline
\end{tabular}
\end{center}

The formula in cell D5 is $\quad = \mathrm { SINH } \left( \mathrm { C5 } ^ { \wedge } 2 \right) - \mathrm { C5 } ^ { \wedge } 3 - 2$.
\begin{enumerate}[label=(\alph*)]
\item Write down the equation which is being solved.

The formula in cell C 6 is $\quad = \mathrm { IF } ( \mathrm { H } 5 < 0 , \mathrm { G } 5 , \mathrm { C } 5 )$.
\item Write down a similar formula for cell E6.
\item Calculate the values which would be displayed in cells G13 and G14 to find further approximations to the root.
\item Explain what the values in column J tell you about

\begin{itemize}
  \item the order of convergence of this sequence of estimates,
  \item the speed of convergence of this sequence of estimates.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2021 Q3 [7]}}
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