1.07s Parametric and implicit differentiation

3 The parametric equations of a curve are $$x = \sin \theta , \quad y = \sin 2 \theta , \quad \text { for } 0 \leqslant \theta \leqslant 2 \pi .$$

1. Find the exact value of the gradient of the curve at the point where \(\theta = \frac { 1 } { 6 } \pi\).
2. Show that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4 x ^ { 4 }\).

5 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).

1. Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
2. Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\). Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure}
3. Find the volume of revolution produced, giving your answer in exact form.

6 The curve \(C\) is defined parametrically by $$x = 4 t - t ^ { 2 } \quad \text { and } \quad y = 1 - \mathrm { e } ^ { - t }$$ where \(0 \leqslant t < 2\). Show that at all points of \(C\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { ( t - 1 ) \mathrm { e } ^ { - t } } { 4 ( 2 - t ) ^ { 3 } }$$ Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 7 } { 4 }\) is $$\frac { 4 e ^ { - \frac { 1 } { 2 } } - 3 } { 21 }$$

6 A curve has equation $$( x + y ) \left( x ^ { 2 } + y ^ { 2 } \right) = 1$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 0,1 )\).

1 The variables \(x\) and \(y\) are such that \(y = - 1\) when \(x = 1\) and $$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).

7 It is given that $$x = t ^ { 2 } \mathrm { e } ^ { - t ^ { 2 } } \quad \text { and } \quad y = t \mathrm { e } ^ { - t ^ { 2 } }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t ^ { 2 } } { 2 t - 2 t ^ { 3 } }$$
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\).

4 The curve \(C\) has equation $$2 x y ^ { 2 } + 3 x ^ { 2 } y = 1$$ Show that, at the point \(A ( - 1,1 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - 4\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

3 The curve \(C\) has equation $$x y + ( x + y ) ^ { 3 } = 1$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 }\) at the point \(A ( 1,0 )\) on \(C\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

4 Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { 3 }\) at the point \(A ( 1 , - 2 )\) on the curve with equation $$y ^ { 3 } - 3 x ^ { 2 } y + 2 = 0$$ and find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = ( 2 - t ) ^ { \frac { 1 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$ Find

1. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\),
2. the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 4\),
3. the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the \(x\)-axis and the \(y\)-axis.

7 The curve \(C\) has parametric equations $$x = \sin t , \quad y = \sin 2 t , \quad \text { for } 0 \leqslant t \leqslant \pi .$$ Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\). Hence, or otherwise, find the coordinates of the stationary points on \(C\) and determine their nature.

6 A curve has equation \(x ^ { 2 } - 6 x y + 25 y ^ { 2 } = 16\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at the point \(( 3,1 )\). By finding the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 3,1 )\), determine the nature of this turning point.

3 A curve \(C\) has equation \(\tan y = x\), for \(x > 0\).

1. Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
2. Hence find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( 1 , \frac { 1 } { 4 } \pi \right)\) on \(C\).

4 A curve \(C\) has equation \(x ^ { 3 } - 3 x y + y ^ { 2 } = 4\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 0,2 )\) of \(C\).

1 The variables \(x\) and \(y\) are such that \(y = - 1\) when \(x = 0\) and $$\left( x + \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = y ^ { 2 } + x$$

1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 0\).
2. Find also the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 0\).

1 A curve \(C\) has equation \(\cos y = x\), for \(- \pi < x < \pi\).

1. Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \cot y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
2. Hence find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \pi \right)\) on \(C\).

The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.

1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
   (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
   A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A curve has equation \(x ^ { 3 } + x y ^ { 2 } - y ^ { 3 } = 3\).

1. Show that there is no point of the curve at which \(\frac { d y } { d x } = 0\).
2. Find the values of \(\frac { d y } { d x }\) and \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 1 , - 1 )\).

7 The curve \(C\) has equation $$x y + ( x + y ) ^ { 5 } = 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 5 } { 6 }\) at the point \(A ( 1,0 )\) on \(C\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

10 The curve \(C\) has equation $$y = x ^ { 2 } + \lambda \sin ( x + y ) ,$$ where \(\lambda\) is a constant, and passes through the point \(A \left( \frac { 1 } { 4 } \pi , \frac { 1 } { 4 } \pi \right)\). Show that \(C\) has no tangent which is parallel to the \(y\)-axis. Show that, at \(A\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 - \frac { 1 } { 64 } \pi ( 4 - \pi ) ( \pi + 2 ) ^ { 2 }$$

5 The curve \(C\) has equation $$x ^ { 2 } - x y - 2 y ^ { 2 } = 4 .$$ Show that, at the point \(A ( 2,0 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

4 It is given that $$x = t + \sin t , \quad y = t ^ { 2 } + 2 \cos t$$ where \(- \pi < t < \pi\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 t \sin t } { ( 1 + \cos t ) ^ { 3 } }$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) increases with \(x\) over the given interval of \(t\).

4 A curve has parametric equations $$x = 2 \sin 2 t , \quad y = 3 \cos 2 t$$ for \(0 < t < \frac { 1 } { 2 } \pi\). For the point on the curve where \(t = \frac { 1 } { 3 } \pi\), find the value of

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).

4 A curve has parametric equations $$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$ except for certain values of \(\theta\), which should be stated. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(\theta = \frac { 1 } { 4 } \pi\).

8 A curve \(C\) has equation \(x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0\). Show that, at stationary points on \(C , x = - 2 y\). Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the stationary points.