1.07s Parametric and implicit differentiation

4 The equation of a curve is \(x ^ { 3 } + y ^ { 3 } = 9 x y\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 y - x ^ { 2 } } { y ^ { 2 } - 3 x }\).
2. Find the equation of the tangent to the curve at the point ( 2,4 ), giving your answer in the form \(a x + b y = c\).

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

1. Find the coordinates of the two points on the curve where \(x = 1\).
2. Show by differentiation that at one of these points the tangent to the curve is parallel to the \(x\)-axis. Find the equation of the tangent to the curve at the other point, giving your answer in the form \(a x + b y + c = 0\).

4 The parametric equations of a curve are $$x = 1 - \mathrm { e } ^ { - t } , \quad y = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { 2 t } - 1\).
2. Hence find the exact value of \(t\) at the point on the curve at which the gradient is 2 .

4 The parametric equations of a curve are $$x = 1 + \ln ( t - 2 ) , \quad y = t + \frac { 9 } { t } , \quad \text { for } t > 2$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( t ^ { 2 } - 9 \right) ( t - 2 ) } { t ^ { 2 } }\).
2. Find the coordinates of the only point on the curve at which the gradient is equal to 0 .

8 The equation of a curve is $$x ^ { 2 } + 2 x y - y ^ { 2 } + 8 = 0$$

1. Show that the tangent to the curve at the point \(( - 2,2 )\) is parallel to the \(x\)-axis.
2. Find the equation of the tangent to the curve at the other point on the curve for which \(x = - 2\), giving your answer in the form \(y = m x + c\).

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { t ( t + 2 ) } { 3 \mathrm { e } ^ { 2 t } }\).
2. Show that the tangent to the curve at the point \(( 1,3 )\) is parallel to the \(x\)-axis.
3. Find the exact coordinates of the other point on the curve at which the tangent is parallel to the \(x\)-axis.

6 The parametric equations of a curve are $$x = 1 + 2 \sin ^ { 2 } \theta , \quad y = 4 \tan \theta$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sin \theta \cos ^ { 3 } \theta }\).
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac { 1 } { 4 } \pi\), giving your answer in the form \(y = m x + c\).

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

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 3 } { 3 - 2 y }\).
2. Find the coordinates of the two points on the curve at which the gradient is - 1 .

4 The parametric equations of a curve are $$x = \ln ( 1 - 2 t ) , \quad y = \frac { 2 } { t } , \quad \text { for } t < 0$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t } { t ^ { 2 } }\).
2. Find the exact coordinates of the only point on the curve at which the gradient is 3 .

7 The equation of a curve is $$3 x ^ { 2 } - 4 x y + 2 y ^ { 2 } - 6 = 0$$

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

5 The parametric equations of a curve are $$x = \cos 2 \theta - \cos \theta , \quad y = 4 \sin ^ { 2 } \theta$$ for \(0 \leqslant \theta \leqslant \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 \cos \theta } { 1 - 4 \cos \theta }\).
2. Find the coordinates of the point on the curve at which the gradient is - 4 .

5 The parametric equations of a curve are $$x = 1 + \sqrt { } t , \quad y = 3 \ln t$$

1. Find the exact value of the gradient of the curve at the point \(P\) where \(y = 6\).
2. Show that the tangent to the curve at \(P\) passes through the point \(( 1,0 )\).

4 For each of the following curves, find the exact gradient at the point indicated:

1. \(y = 3 \cos 2 x - 5 \sin x\) at \(\left( \frac { 1 } { 6 } \pi , - 1 \right)\),
2. \(x ^ { 3 } + 6 x y + y ^ { 3 } = 21\) at \(( 1,2 )\).

3 A curve has equation $$3 \ln x + 6 x y + y ^ { 2 } = 16$$ Find the equation of the normal to the curve at the point \(( 1,2 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7 \includegraphics[max width=\textwidth, alt={}, center]{250b4df9-2646-4246-bb6d-2be92bf29598-3_553_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
3. Find the value of the gradient of the curve at \(B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{7e100be2-9768-4fcd-b516-c714e53b0665-3_453_650_258_744} The diagram shows the curve with parametric equations $$x = 3 \cos t , \quad y = 2 \cos \left( t - \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant t < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 3 } ( \sqrt { } 3 - \cot t )\).
2. Find the equation of the tangent to the curve at the point where the curve crosses the positive \(y\)-axis. Give the answer in the form \(y = m x + c\).

3 The parametric equations of a curve are $$x = ( t + 1 ) \mathrm { e } ^ { t } , \quad y = 6 ( t + 4 ) ^ { \frac { 1 } { 2 } }$$ Find the equation of the tangent to the curve when \(t = 0\), giving the answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

7 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717} The diagram shows the curve with parametric equations $$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant \theta < 2 \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
2. Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.

6 The parametric equations of a curve are $$x = 2 \mathrm { e } ^ { 2 t } + 4 \mathrm { e } ^ { t } , \quad y = 5 t \mathrm { e } ^ { 2 t }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places.
2. Find the gradient of the normal to the curve at the point where the curve crosses the \(x\)-axis.

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

1. Find the equation of the tangent to the curve at the point \(( - 1,3 )\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
2. Show that there is no point on the curve at which the gradient is \(\frac { 1 } { 2 }\).

5 A curve has parametric equations $$x = t + \ln ( t + 1 ) , \quad y = 3 t \mathrm { e } ^ { 2 t }$$

1. Find the equation of the tangent to the curve at the origin.
2. Find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places. [4]

4 Find the gradient of the curve $$4 x + 3 y \mathrm { e } ^ { 2 x } + y ^ { 2 } = 10$$ at the point \(( 0,2 )\).

7 The equation of a curve is \(x ^ { 2 } - 4 x y - 2 y ^ { 2 } = 1\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that the gradient of the curve at the point \(( - 1,2 )\) is \(- \frac { 5 } { 2 }\). [5]
2. Show that the curve has no stationary points.
3. Find the \(x\)-coordinate of each of the points on the curve at which the tangent is parallel to the \(y\)-axis.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 The parametric equations of a curve are $$x = 3 \sin 2 \theta , \quad y = 1 + 2 \tan 2 \theta$$ for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).

1. Find the exact gradient of the curve at the point for which \(\theta = \frac { 1 } { 6 } \pi\).
2. Find the value of \(\theta\) at the point where the gradient of the curve is 2 , giving the value correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{77672e56-a268-47b8-ab8b-cd84b4b3de4f-10_551_689_258_726} The parametric equations of a curve are $$x = 6 \sin ^ { 2 } t , \quad y = 2 \sin 2 t + 3 \cos 2 t$$ for \(0 \leqslant t < \pi\). The curve crosses the \(x\)-axis at points \(B\) and \(D\) and the stationary points are \(A\) and \(C\), as shown in the diagram.

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { 3 } \cot 2 t - 1\).
2. Find the values of \(t\) at \(A\) and \(C\), giving each answer correct to 3 decimal places.
3. Find the value of the gradient of the curve at \(B\).