5 A surface \(S\) is defined by \(z = f ( x , y )\), where \(f ( x , y ) = y e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }\).
- Find \(\frac { \partial f } { \partial x }\).
- Show that \(\frac { \partial f } { \partial y } = - \left( x ^ { 2 } y + 2 x y + 2 y - 1 \right) e ^ { - \left( x ^ { 2 } + 2 x + 2 \right) y }\).
- Determine the coordinates of any stationary points on \(S\).
Fig. 5.1 shows the graph of \(z = e ^ { - x ^ { 2 } }\) and Fig. 5.2 shows the contour of \(S\) defined by \(z = 0.25\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_686_822_244}
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\caption{Fig. 5.1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76f3559a-f3b3-4a21-878f-adb261dd1236-5_478_437_822_1105}
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\caption{Fig. 5.2}
\end{figure}
- Specify a sequence of transformations which transforms the graph of \(\mathrm { z } = \mathrm { e } ^ { - \mathrm { x } ^ { 2 } }\) onto the graph of the section defined by \(z = f ( x , 1 )\).
- Hence, or otherwise, sketch the section defined by \(z = f ( x , 1 )\).
- Using Fig. 5.2 and your answer to part (c), classify any stationary points on \(S\), justifying your answer.
You are given that \(P\) is a point on \(S\) where \(z = 0\).
- Find, in vector form, the equation of the tangent plane to \(S\) at \(P\).
The tangent plane found in part (e) intersects \(S\) in a straight line, \(L\).
- Write down, in vector form, the equation of \(L\).