2 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{93ee09be-f014-4dd7-a8da-8646837b17a5-1_471_674_761_719}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure}
- Show algebraically that \(\mathrm { f } ( x )\) is an odd function. Interpret this result geometrically.
- Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \left( 2 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }\). Hence find the exact gradient of the curve at the origin.
- Find the exact area of the region bounded by the curve, the \(x\)-axis and the line \(x = 1\).
- \(( A )\) Show that if \(y = \frac { x } { \sqrt { 2 + x ^ { 2 } } }\), then \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\).
(B) Differentiate \(\frac { 1 } { y ^ { 2 } } = \frac { 2 } { x ^ { 2 } } + 1\) implicitly to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y ^ { 3 } } { x ^ { 3 } }\). Explain why this expression cannot be used to find the gradient of the curve at the origin.