Standard +0.8 This is a multi-step Further Maths question requiring differentiation, Maclaurin series construction, and then cleverly choosing x=2 to approximate √5. The final part requires insight to work backwards from the given fraction to determine the appropriate x-value, which elevates it above routine Taylor series exercises.
4. (a) (i) Given that \(f ( x ) = \sqrt { 1 + 2 x }\), find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
(ii) Hence, find the first three terms of the Maclaurin series for \(\sqrt { 1 + 2 x }\).
(b) Hence, using a suitable value for \(x\), show that \(\sqrt { 5 } \approx \frac { 143 } { 64 }\). [0pt]
4. (a) (i) Given that $f ( x ) = \sqrt { 1 + 2 x }$, find $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$.\\
(ii) Hence, find the first three terms of the Maclaurin series for $\sqrt { 1 + 2 x }$.\\
(b) Hence, using a suitable value for $x$, show that $\sqrt { 5 } \approx \frac { 143 } { 64 }$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2024 Q4 [6]}}