1.07s Parametric and implicit differentiation

6 The parametric equations of a curve are $$x = \ln ( 2 + 3 t ) , \quad y = \frac { t } { 2 + 3 t }$$

1. Show that the gradient of the curve is always positive.
2. Find the equation of the tangent to the curve at the point where it intersects the \(y\)-axis.

3 The parametric equations of a curve are $$x = t + \ln ( t + 2 ) , \quad y = ( t - 1 ) \mathrm { e } ^ { - 2 t }$$ where \(t > - 2\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
2. Find the exact \(y\)-coordinate of the stationary point of the curve.

8 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } + 2 x y + 8 = 0\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
   The tangent to the curve at the point where \(x = 0\) and the tangent at the point where \(y = 0\) intersect at the acute angle \(\alpha\).
2. Find the exact value of \(\tan \alpha\).

7 The equation of a curve is \(x ^ { 3 } + 3 x ^ { 2 } y - y ^ { 3 } = 3\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } + 2 x y } { y ^ { 2 } - x ^ { 2 } }\).
2. Find the coordinates of the points on the curve where the tangent is parallel to the \(x\)-axis.

6 The parametric equations of a curve are \(x = \frac { 1 } { \cos t } , y = \ln \tan t\), where \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos t } { \sin ^ { 2 } t }\).
2. Find the equation of the tangent to the curve at the point where \(y = 0\).

5 The equation of a curve is \(x ^ { 2 } y - a y ^ { 2 } = 4 a ^ { 3 }\), where \(a\) is a non-zero constant.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x y } { 2 a y - x ^ { 2 } }\).
2. Hence find the coordinates of the points where the tangent to the curve is parallel to the \(y\)-axis. [4]

7 The equation of a curve is \(3 x ^ { 2 } + 4 x y + 3 y ^ { 2 } = 5\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x + 2 y } { 2 x + 3 y }\).
2. Hence find the exact coordinates of the two points on the curve at which the tangent is parallel to \(y + 2 x = 0\).

4 The parametric equations of a curve are $$x = \frac { \cos \theta } { 2 - \sin \theta } , \quad y = \theta + 2 \cos \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 - \sin \theta ) ^ { 2 }\).

10

1. Given that \(2 x = \tan y\), show that \(\frac { d y } { d x } = \frac { 2 } { 1 + 4 x ^ { 2 } }\).
2. Hence find the exact value of \(\int _ { \frac { 1 } { 2 } } ^ { \frac { \sqrt { 3 } } { 2 } } x \tan ^ { - 1 } ( 2 x ) \mathrm { d } x\).

4 The equation of a curve is \(\mathrm { ye } ^ { 2 \mathrm { x } } + \mathrm { y } ^ { 2 } \mathrm { e } ^ { \mathrm { x } } = 6\).
Find the gradient of the curve at the point where \(y = 1\).

8 \includegraphics[max width=\textwidth, alt={}, center]{8f81a526-783c-4321-b540-c9deccfee17b-12_639_713_262_715} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 3\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The point \(M\) on \(A B\) is such that \(M B = 2 A M\). The midpoint of \(F G\) is \(N\).

1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Find the position vector of \(P\), the foot of the perpendicular from \(D\) to the line through \(M\) and \(N\). [4]

4 The parametric equations of a curve are $$x = 1 - \cos \theta , \quad y = \cos \theta - \frac { 1 } { 4 } \cos 2 \theta$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right)\).

5 The parametric equations of a curve are $$x = t \mathrm { e } ^ { 2 t } , \quad y = t ^ { 2 } + t + 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - 2 t }\).
2. Hence show that the normal to the curve, where \(t = - 1\), passes through the point \(\left( 0,3 - \frac { 1 } { \mathrm { e } ^ { 4 } } \right)\).

6 The equation of a curve is \(2 y ^ { 2 } + 3 x y + x = x ^ { 2 }\).

1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 2 \mathrm { x } - 3 \mathrm { y } - 1 } { 4 \mathrm { y } + 3 \mathrm { x } }\).
2. Hence show that the curve does not have a tangent that is parallel to the \(x\)-axis.

3 The parametric equations of a curve are $$x = 3 - \cos 2 \theta , \quad y = 2 \theta + \sin 2 \theta$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta\).

5 \includegraphics[max width=\textwidth, alt={}, center]{77a45360-8e1d-4f4f-9830-075d832a14cf-08_334_895_258_625} The diagram shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta$$ for \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(- 2 \sin \theta \cos ^ { 3 } \theta\).
   The gradient of the curve has its maximum value at the point \(P\).
2. Find the exact value of the \(x\)-coordinate of \(P\).

9 The equation of a curve is \(y \mathrm { e } ^ { 2 x } - y ^ { 2 } \mathrm { e } ^ { x } = 2\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 y \mathrm { e } ^ { x } - y ^ { 2 } } { 2 y - \mathrm { e } ^ { x } }\).
2. Find the exact coordinates of the point on the curve where the tangent is parallel to the \(y\)-axis.

7 The equation of a curve is \(\ln ( x + y ) = x - 2 y\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x + y - 1 } { 2 ( x + y ) + 1 }\).
2. Find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis. \(\quad\) [3]

4 The parametric equations of a curve are $$x = 2 t - \tan t , \quad y = \ln ( \sin 2 t )$$ for \(0 < t < \frac { 1 } { 2 } \pi\).
Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot t\).

6 The parametric equations of a curve are $$x = \sqrt { t } + 3 , \quad y = \ln t$$ for \(t > 0\).

1. Obtain a simplified expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Hence find the exact coordinates of the point on the curve at which the gradient of the normal is - 2 .

2 The parametric equations of a curve are $$x = ( \ln t ) ^ { 2 } , \quad y = \mathrm { e } ^ { 2 - t ^ { 2 } }$$ for \(t > 0\).
Find the gradient of the curve at the point where \(t = \mathrm { e }\), simplifying your answer.

7 The equation of a curve is \(x ^ { 3 } + y ^ { 2 } + 3 x ^ { 2 } + 3 y = 4\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 x ^ { 2 } + 6 x } { 2 y + 3 }\).
2. Hence find the coordinates of the points on the curve at which the tangent is parallel to the \(x\)-axis. [5]

3 The equation of a curve is \(\ln ( x + y ) = 3 x ^ { 2 } y\).
Find the gradient of the curve at the point \(( 1,0 )\).

7 The parametric equations of a curve are $$x = 3 \sin 2 t , \quad y = \tan t + \cot t$$ for \(0 < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 } { 3 \sin ^ { 2 } 2 t }\). \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-10_2716_40_109_2009} \includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-11_2723_33_99_22}
2. Find the equation of the normal to the curve at the point where \(t = \frac { 1 } { 4 } \pi\). Give your answer in the form \(p y + q x + r = 0\), where \(p , q\) and \(r\) are integers.

5 The variables \(x\) and \(y\) are such that \(y = 0\) when \(x = 0\) and $$( x + 1 ) y + ( x + y + 1 ) ^ { 3 } = 1$$

1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 3 } { 4 }\) when \(x = 0\).
2. Find the Maclaurin's series for \(y\) up to and including the term in \(x ^ { 2 }\).