1.07s Parametric and implicit differentiation

8 \includegraphics[max width=\textwidth, alt={}, center]{de2f8bf3-fd03-4199-9eb2-c9cbac4d4385-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve \(( 1,4 )\) and \(( 3,3 )\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).

8 \includegraphics[max width=\textwidth, alt={}, center]{bdc467f6-105e-4429-95c6-701eaa43deff-10_549_495_258_824} The diagram shows the curve with parametric equations $$x = 2 - \cos 2 t , \quad y = 2 \sin ^ { 3 } t + 3 \cos ^ { 3 } t + 1$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). The end-points of the curve are \(( 1,4 )\) and \(( 3,3 )\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 } \sin t - \frac { 9 } { 4 } \cos t\).
2. Find the coordinates of the minimum point, giving each coordinate correct to 3 significant figures.
3. Find the exact gradient of the normal to the curve at the point for which \(x = 2\).

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

1. Find the gradient of the curve at the point for which \(\theta = 1\) radian.
2. Find the value of \(\sin \theta\) at the point on the curve where the tangent is parallel to the \(y\)-axis.

5 A curve has equation $$y ^ { 3 } \sin 2 x + 4 y = 8$$ Find the equation of the tangent to the curve at the point where it crosses the \(y\)-axis.

3 Find the gradient of the curve with equation $$2 x ^ { 2 } - 4 x y + 3 y ^ { 2 } = 3$$ at the point \(( 2,1 )\).

6 The equation of a curve is \(x y ( x + y ) = 2 a ^ { 3 }\), where \(a\) is a non-zero constant. Show that there is only one point on the curve at which the tangent is parallel to the \(x\)-axis, and find the coordinates of this point.

6 The parametric equations of a curve are $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t$$ where \(a\) is a positive constant and \(0 < t < \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin t + y \cos t = a \sin t \cos t$$
3. Hence show that, if this tangent meets the \(x\)-axis at \(X\) and the \(y\)-axis at \(Y\), then the length of \(X Y\) is always equal to \(a\).

9 \includegraphics[max width=\textwidth, alt={}, center]{a74e4ddf-d254-45f3-bd9a-adf7cd53b3a6-4_611_895_255_625} The diagram shows the curve \(y = \sqrt { } \left( \frac { 1 - x } { 1 + x } \right)\).

1. By first differentiating \(\frac { 1 - x } { 1 + x }\), obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that the gradient of the normal to the curve at the point \(( x , y )\) is \(( 1 + x ) \sqrt { } \left( 1 - x ^ { 2 } \right)\).
2. The gradient of the normal to the curve has its maximum value at the point \(P\) shown in the diagram. Find, by differentiation, the \(x\)-coordinate of \(P\).

6 The equation of a curve is $$x \ln y = 2 x + 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { y } { x ^ { 2 } }\).
2. Find the equation of the tangent to the curve at the point where \(y = 1\), giving your answer in the form \(a x + b y + c = 0\).

5 The curve with equation $$6 \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { y } + \mathrm { e } ^ { 2 y } = c$$ where \(k\) and \(c\) are constants, passes through the point \(P\) with coordinates \(( \ln 3 , \ln 2 )\).

1. Show that \(58 + 2 k = c\).
2. Given also that the gradient of the curve at \(P\) is - 6 , find the values of \(k\) and \(c\).

5 The parametric equations of a curve are $$x = \ln ( \tan t ) , \quad y = \sin ^ { 2 } t$$ where \(0 < t < \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the equation of the tangent to the curve at the point where \(x = 0\).

6 The equation of a curve is \(3 x ^ { 2 } - 4 x y + y ^ { 2 } = 45\).

1. Find the gradient of the curve at the point \(( 2 , - 3 )\).
2. Show that there are no points on the curve at which the gradient is 1 .

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

5 For each of the following curves, find the gradient at the point where the curve crosses the \(y\)-axis:

1. \(y = \frac { 1 + x ^ { 2 } } { 1 + \mathrm { e } ^ { 2 x } }\);
2. \(2 x ^ { 3 } + 5 x y + y ^ { 3 } = 8\).

3 The parametric equations of a curve are $$x = \ln ( 2 t + 3 ) , \quad y = \frac { 3 t + 2 } { 2 t + 3 }$$ Find the gradient of the curve at the point where it crosses the \(y\)-axis.

6 \includegraphics[max width=\textwidth, alt={}, center]{b2136f5d-0d66-4524-bb76-fcc4cb59150c-3_405_914_260_612} The diagram shows the curve \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 2 \left( x ^ { 2 } - y ^ { 2 } \right)\) and one of its maximum points \(M\). Find the coordinates of \(M\).

10 \includegraphics[max width=\textwidth, alt={}, center]{3eefd6c1-924c-4b7e-8d17-a2942fb48234-3_515_508_1105_815} The diagram shows part of the curve with parametric equations $$x = 2 \ln ( t + 2 ) , \quad y = t ^ { 3 } + 2 t + 3$$

1. Find the gradient of the curve at the origin.
2. At the point \(P\) on the curve, the value of the parameter is \(p\). It is given that the gradient of the curve at \(P\) is \(\frac { 1 } { 2 }\).
   1. Show that \(p = \frac { 1 } { 3 p ^ { 2 } + 2 } - 2\).
   2. By first using an iterative formula based on the equation in part (a), determine the coordinates of the point \(P\). Give the result of each iteration to 5 decimal places and each coordinate of \(P\) correct to 2 decimal places.

5 The parametric equations of a curve are $$x = a \cos ^ { 4 } t , \quad y = a \sin ^ { 4 } t$$ where \(a\) is a positive constant.

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Show that the equation of the tangent to the curve at the point with parameter \(t\) is $$x \sin ^ { 2 } t + y \cos ^ { 2 } t = a \sin ^ { 2 } t \cos ^ { 2 } t$$
3. Hence show that if the tangent meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\), then $$O P + O Q = a$$ where \(O\) is the origin.

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

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

4 The parametric equations of a curve are $$x = t + \cos t , \quad y = \ln ( 1 + \sin t )$$ where \(- \frac { 1 } { 2 } \pi < t < \frac { 1 } { 2 } \pi\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec t\).
2. Hence find the \(x\)-coordinates of the points on the curve at which the gradient is equal to 3 . Give your answers correct to 3 significant figures.

4 The parametric equations of a curve are $$x = \ln \cos \theta , \quad y = 3 \theta - \tan \theta ,$$ where \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\tan \theta\).
2. Find the exact \(y\)-coordinate of the point on the curve at which the gradient of the normal is equal to 1 . \includegraphics[max width=\textwidth, alt={}, center]{b00cefad-7c3c-4672-b309-f19aafab8b01-08_378_689_260_726} The diagram shows a semicircle with centre \(O\), radius \(r\) and diameter \(A B\). The point \(P\) on its circumference is such that the area of the minor segment on \(A P\) is equal to half the area of the minor segment on \(B P\). The angle \(A O P\) is \(x\) radians.

3 Find the gradient of the curve \(x ^ { 3 } + 3 x y ^ { 2 } - y ^ { 3 } = 1\) at the point with coordinates \(( 1,3 )\).

6 A curve has equation $$\sin y \ln x = x - 2 \sin y$$ for \(- \frac { 1 } { 2 } \pi \leqslant y \leqslant \frac { 1 } { 2 } \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Hence find the exact \(x\)-coordinate of the point on the curve at which the tangent is parallel to the \(x\)-axis.

5 The variables \(x\) and \(y\) satisfy the relation \(\sin y = \tan x\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\). Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \cos x \sqrt { } ( \cos 2 x ) } .$$

4 The equation of a curve is $$\sqrt { } x + \sqrt { } y = \sqrt { } a$$ where \(a\) is a positive constant.

1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. The straight line with equation \(y = x\) intersects the curve at the point \(P\). Find the equation of the tangent to the curve at \(P\).