<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x=\ln \left( \cos 2\theta \right)$, $y=\ln \left( \sin 2\theta \right)$,</p>
<p>where $0<\theta <\frac{\pi }{4}$.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Show that the gradient of $C$ is always negative for all points on $C$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-content"><span class="list-item-text">What can be said about the tangents to $C$ as $\theta \to 0$ and $\theta \to \frac{\pi }{4}$?</span></div><p style="text-align: right;">[2]</p></li><li><div class="list-item-content"><span class="list-item-text">Show that the tangent to $C$ at the point where $\theta =\frac{\pi }{8}$ is given by the equation $x+y+\ln 2=0$.</span></div><div class="list-item-paragraph">Hence state, with a reason, whether this tangent will meet $C$ again.</div><p style="text-align: right;">[4]</p></li></ol>

$x=\ln \left( \cos 2\theta \right)$, $y=\ln \left( \sin 2\theta \right)$,

Show that the gradient of $C$ is always negative for all points on $C$.

What can be said about the tangents to $C$ as $\theta \to 0$ and $\theta \to \frac{\pi }{4}$?

Show that the tangent to $C$ at the point where $\theta =\frac{\pi }{8}$ is given by the equation $x+y+\ln 2=0$.

Hence state, with a reason, whether this tangent will meet $C$ again.

Tangents and Normals to Parametric Curves

Applications of Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

2018 HCI Promo Q5

2018 HCI Promo Q5

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