1.07s Parametric and implicit differentiation

5 A curve has parametric equations \(x = 1 + u ^ { 2 } , y = 2 u ^ { 3 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\).
2. Hence find the gradient of the curve at the point with coordinates \(( 5,16 )\).

3 Fig. 3 shows an ellipse with parametric equations \(x = a \cos \theta , y = b \sin \theta\), for \(0 \leqslant \theta \leqslant 2 \pi\), where \(0 < b \leqslant a\).
The curve meets the positive \(x\)-axis at A and the positive \(y\)-axis at B . \begin{figure}[h] \end{figure}

1. Show that the radius of curvature at A is \(\frac { b ^ { 2 } } { a }\) and find the corresponding centre of curvature.
2. Write down the radius of curvature and the centre of curvature at B .
3. Find the relationship between \(a\) and \(b\) if the radius of curvature at B is equal to the radius of curvature at A . What does this mean geometrically?
4. Show that the arc length from A to B can be expressed as $$b \int _ { 0 } ^ { \frac { \pi } { 2 } } \sqrt { 1 + \lambda ^ { 2 } \sin ^ { 2 } \theta } d \theta$$ where \(\lambda ^ { 2 }\) is to be determined in terms of \(a\) and \(b\).
   Evaluate this integral in the case \(a = b\) and comment on your answer.
5. Find the cartesian equation of the evolute of the ellipse.

2. \begin{figure}[h] \end{figure} Figure 1 shows the curve defined by the equation $$y ^ { 2 } + 3 y - 6 \sin y = 4 - x ^ { 2 }$$ The point \(P ( x , y )\) lies on the curve.
The distance from the origin,\(O\) ,to \(P\) is \(D\) .

1. Write down an equation for \(D ^ { 2 }\) in terms of \(y\) only.
2. Hence determine the minimum value of \(D\) giving your answer in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-04_2266_53_312_1977}

10 A curve has equation \(x = ( y + 5 ) \ln ( 2 y - 7 )\).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of y .
2. Find the gradient of the curve where it crosses the y -axis.

12 The parametric equations of a curve are given by \(x = 2 \cos \theta\) and \(y = 3 \sin \theta\) for \(0 \leq \theta < 2 \pi\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). The tangents to the curve at the points P and Q pass through the point \(( 2,6 )\).
2. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2 \sin \theta + \cos \theta = 1\).
3. Find the values of \(\theta\) at the points \(P\) and \(Q\).

13 In this question you must show detailed reasoning. Find the exact values of the \(x\)-coordinates of the stationary points of the curve \(x ^ { 3 } + y ^ { 3 } = 3 x y + 35\).

5 A curve is defined by the parametric equations $$\begin{aligned} & x = 4 \times 2 ^ { - t } + 3 \\ & y = 3 \times 2 ^ { t } - 5 \end{aligned}$$ 5

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 } \times 2 ^ { 2 t }\) 5
2. Find the Cartesian equation of the curve in the form \(x y + a x + b y = c\), where \(a , b\) and \(c\) are integers.

12 A curve \(C\) has equation $$x ^ { 3 } \sin y + \cos y = A x$$ where \(A\) is a constant. \(C\) passes through the point \(P \left( \sqrt { 3 } , \frac { \pi } { 6 } \right)\) 12

1. Show that \(A = 2\) 12
2. (i) Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - 3 x ^ { 2 } \sin y } { x ^ { 3 } \cos y - \sin y }\) 12 (b) (ii) Hence, find the gradient of the curve at \(P\).
   12 (b) (iii) The tangent to \(C\) at \(P\) intersects the \(x\)-axis at \(Q\).
   Find the exact \(x\)-coordinate of \(Q\).

12 The equation of a curve is $$( x + y ) ^ { 2 } = 4 y + 2 x + 8$$ The curve intersects the positive \(x\)-axis at the point \(P\).
12

1. Show that the gradient of the curve at \(P\) is \(- \frac { 3 } { 2 }\)

   12	Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   	[2 marks]
   	\(\_\_\_\_\)

14 The curve \(C\) is defined for \(t \geq 0\) by the parametric equations $$x = t ^ { 2 } + t \quad \text { and } \quad y = 4 t ^ { 2 } - t ^ { 3 }$$ \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-26_691_608_541_717} 14

1. Find the gradient of \(C\) at the point where it intersects the positive \(x\)-axis.
   14
2. (i) The area \(A\) enclosed between \(C\) and the \(x\)-axis is given by $$A = \int _ { 0 } ^ { b } y \mathrm {~d} x$$ Find the value of \(b\).
   14 (b) (ii) Use the substitution \(y = 4 t ^ { 2 } - t ^ { 3 }\) to show that $$A = \int _ { 0 } ^ { 4 } \left( 4 t ^ { 2 } + 7 t ^ { 3 } - 2 t ^ { 4 } \right) \mathrm { d } t$$ 14 (b) (iii) Find the value of \(A\).

15 The curve with equation $$x ^ { 2 } + 2 y ^ { 3 } - 4 x y = 0$$ has a single stationary point at \(P\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-26_656_1138_548_450} 15

1. Show that the \(y\)-coordinate of \(P\) satisfies the equation $$y ^ { 2 } ( y - 2 ) = 0$$ 15
2. Hence, find the coordinates of \(P\) [0pt] [2 marks]

9 In this question you must show detailed reasoning.
The curve \(x y + y ^ { 2 } = 8\) is shown in Fig. 9. \begin{figure}[h] \end{figure} Find the coordinates of the points on the curve at which the normal has gradient 2.

1 The curve \(C\) is defined parametrically by $$x = 2 \cos ^ { 3 } t \quad \text { and } \quad y = 2 \sin ^ { 3 } t , \quad \text { for } 0 < t < \frac { 1 } { 2 } \pi .$$ Show that, at the point with parameter \(t\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$

The curve \(C\) has equation $$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$

1. Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
   For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\).
2. Show that \(A _ { 1 } = 0\).
3. Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
   Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\).
4. Show that for \(n \geqslant 2\), $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
5. Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

1. Determine the exact value of the eccentricity of \(E\) The points \(P ( 4 \cos \theta , 3 \sin \theta )\) and \(Q ( 4 \cos \theta , - 3 \sin \theta )\) lie on \(E\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l _ { 1 }\) is the normal to \(E\) at the point \(P\)
2. Use calculus to show that \(l _ { 1 }\) has equation $$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$ The line \(l _ { 2 }\) passes through the origin and the point \(Q\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\)
3. Determine, in simplest form, the coordinates of \(R\)
4. Hence show that, as \(\theta\) varies, \(R\) lies on an ellipse which has the same eccentricity as ellipse \(E\)

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation $$x = 2 y ^ { 2 } + 5 y - 6$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\). The point \(P\) lies on the curve and is shown in Figure 1.
   Given that the tangent to the curve at \(P\) is parallel to the \(y\)-axis,
2. find the coordinates of \(P\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

1. Show that $$\frac { d y } { d x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$ where \(k\) is a constant to be found. The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
3. Show that \(C\) has Cartesian equation $$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leqslant x \leqslant q$$ where \(q\) is a constant to be found.

4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$3 x ^ { 2 } + 2 y ^ { 2 } - 4 x y + 8 ^ { x } - 11 = 0$$ The point \(P\) has coordinates ( 1,2 ).

1. Verify that \(P\) lies on \(C\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The normal to \(C\) at \(P\) crosses the \(x\)-axis at a point \(Q\).
3. Find the \(x\) coordinate of \(Q\), giving your answer in the form \(a + b \ln 2\) where \(a\) and \(b\) are integers.

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Find the exact value of \(t\) at the point on the curve where the gradient is 2 .

11 A curve has parametric equations given by $$x = 2 \sin \theta , \quad y = \cos 2 \theta$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin \theta\).
2. Hence find the equation of the tangent to the curve at \(\theta = \frac { 1 } { 2 } \pi\).
3. Find the cartesian equation of the curve.

5 The curve \(C\) has equation \(x ^ { 2 } + x y + y ^ { 2 } = 19\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 x - y } { x + 2 y }\).
2. Hence find the equation of the normal to \(C\) at the point \(( 2,3 )\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).

1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).

8 The parametric equations of a curve are given by $$x = \mathrm { e } ^ { t } - 2 t , \quad y = \mathrm { e } ^ { t } - 5 t$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Show that \(t = - \ln 2\) at the point on the curve where the gradient is 3 .

10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).

9 Find the equations of all the horizontal tangents to the curve with equation \(y ^ { 2 } = x ^ { 4 } - 4 x ^ { 3 } + 36\).