- A curve has the equation
$$2 \sin 2 x - \tan y = 0$$
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
- Show that the tangent to the curve at the point \(\left( \frac { \pi } { 6 } , \frac { \pi } { 3 } \right)\) has the equation
$$y = \frac { 1 } { 2 } x + \frac { \pi } { 4 }$$
- continued
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3cf64017-e982-4165-9885-8524aaabdf84-06_433_812_246_479}
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\caption{Figure 1}
\end{figure}
Figure 1 shows the curve with parametric equations
$$x = a \sqrt { t } , \quad y = a t ( 1 - t ) , \quad t \geq 0$$
where \(a\) is a positive constant. - Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\). The tangent to the curve at \(A\) meets the \(y\)-axis at the point \(B\) as shown.
- Show that the area of triangle \(O A B\) is \(a ^ { 2 }\).