OCR MEI Further Numerical Methods 2020 November — Question 2 5 marks

Exam BoardOCR MEI
ModuleFurther Numerical Methods (Further Numerical Methods)
Year2020
SessionNovember
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicNumerical integration
TypeTrapezium rule applied to real-world data
DifficultyModerate -0.5 This is a straightforward polynomial interpolation problem requiring students to set up and solve a system of three linear equations by substituting the given points into p(x) = axΒ² + bx + c. While it involves algebraic manipulation and solving simultaneous equations, it's a standard textbook exercise with no conceptual difficulty or novel insight requiredβ€”just methodical application of a well-defined procedure.
2 Fig. 2 shows 3 values of \(x\) and the associated values of a function, \(\mathrm { f } ( x )\). \begin{table}[h]
\(x\)125
\(\mathrm { f } ( x )\)516.676.6
\captionsetup{labelformat=empty} \caption{Fig. 2}
\end{table} Find a polynomial \(p ( x )\) of degree 2 to approximate \(\mathrm { f } ( x )\), giving your answer in the form \(p ( x ) = a x ^ { 2 } + b x + c\), where \(a\), \(b\) and \(c\) are constants to be determined.
Question 2:
AnswerMarks
2(π‘₯π‘₯ β€’2)(π‘₯π‘₯ β€’5) (π‘₯π‘₯ β€’1)(π‘₯π‘₯ β€’5)
(1 β€’2)(1 β€’5)Γ—5+ ( 2 β€’1)(2 β€’5) Γ—16.6+
(π‘₯π‘₯ β€’1)(π‘₯π‘₯ β€’2)
(5 β€’1)(5 β€’2) Γ—76.6
isw
2
AnswerMarks
[𝑝𝑝(π‘₯π‘₯) =]2.1π‘₯π‘₯ +5.3π‘₯π‘₯ β€’2.4M1
A1
A1
A1
AnswerMarks
A11.1a
1.1
1.1
1.1
AnswerMarks
1.1use of Lagrange formula with 3
terms
all substitutons correct soi
a = 2.1
b = 5.3
AnswerMarks
c = β€’2.4M0 if x and y values
interchanged
[5]
AnswerMarks
21.49683286384 
Question 2:
2 | (π‘₯π‘₯ β€’2)(π‘₯π‘₯ β€’5) (π‘₯π‘₯ β€’1)(π‘₯π‘₯ β€’5)
(1 β€’2)(1 β€’5)Γ—5+ ( 2 β€’1)(2 β€’5) Γ—16.6+
(π‘₯π‘₯ β€’1)(π‘₯π‘₯ β€’2)
(5 β€’1)(5 β€’2) Γ—76.6
isw
2
[𝑝𝑝(π‘₯π‘₯) =]2.1π‘₯π‘₯ +5.3π‘₯π‘₯ β€’2.4 | M1
A1
A1
A1
A1 | 1.1a
1.1
1.1
1.1
1.1 | use of Lagrange formula with 3
terms
all substitutons correct soi
a = 2.1
b = 5.3
c = β€’2.4 | M0 if x and y values
interchanged
[5]
2 | 1.49683286384
2 Fig. 2 shows 3 values of $x$ and the associated values of a function, $\mathrm { f } ( x )$.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$x$ & 1 & 2 & 5 \\
\hline
$\mathrm { f } ( x )$ & 5 & 16.6 & 76.6 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{table}

Find a polynomial $p ( x )$ of degree 2 to approximate $\mathrm { f } ( x )$, giving your answer in the form $p ( x ) = a x ^ { 2 } + b x + c$, where $a$, $b$ and $c$ are constants to be determined.

\hfill \mbox{\textit{OCR MEI Further Numerical Methods 2020 Q2 [5]}}
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