1.07s Parametric and implicit differentiation

1. The curve \(C\) has equation

$$( x + y ) ^ { 3 } = 3 x ^ { 2 } - 3 y - 2$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P ( 1,0 )\) lies on \(C\).
2. Show that the normal to \(C\) at \(P\) has equation $$y = - 2 x + 2$$
3. Prove that the normal to \(C\) at \(P\) does not meet \(C\) again. You should use algebra for your proof and make your reasoning clear.

12. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = 6 \sin t \quad y = 5 \sin 2 t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis. \begin{enumerate}[label=(\alph*)] \item

1. Show that the area of \(R\) is given by \(\int _ { 0 } ^ { \frac { \pi } { 2 } } 60 \sin t \cos ^ { 2 } t \mathrm {~d} t\)
2. Hence show, by algebraic integration, that the area of \(R\) is exactly 20 \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e28350e9-5090-4079-97da-e669ef9a5a7a-34_451_570_1416_742} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Part of the curve is used to model the profile of a small dam, shown shaded in Figure 4. Using the model and given that

1. In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\) The line \(l\) is the tangent to \(C\) at \(P\)

1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
3. Use algebraic integration to find the exact area of \(R\).

1. The curve \(C\) has equation

$$p x ^ { 3 } + q x y + 3 y ^ { 2 } = 26$$ where \(p\) and \(q\) are constants.

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a p x ^ { 2 } + b q y } { q x + c y }$$ where \(a\), \(b\) and \(c\) are integers to be found. Given that
   • the point \(P ( - 1 , - 4 )\) lies on \(C\)
   • the normal to \(C\) at \(P\) has equation \(19 x + 26 y + 123 = 0\)
   • find the value of \(p\) and the value of \(q\).

1. The curve \(C\) has parametric equations

$$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\)
2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\)

1. A curve \(C\) is given by the equation

$$\sin x + \cos y = 0.5 \quad - \frac { \pi } { 2 } \leqslant x < \frac { 3 \pi } { 2 } , - \pi < y < \pi$$ A point \(P\) lies on \(C\).
The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis.
Find the exact coordinates of all possible points \(P\), justifying your answer.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

8 A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\). \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-07_492_924_415_242}

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(t\).
2. Hence show that the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at P is - 1 .
3. Find the time at which the particle is travelling in the direction opposite to that at P .
4. Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).

11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h] \end{figure}

1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
2. Determine the area of the triangle AOB .

15 You must show detailed reasoning in this question. The equation of a curve is $$y ^ { 3 } - x y + 4 \sqrt { x } = 4 .$$ Find the gradient of the curve at each of the points where \(y = 1\).

13 Determine the coordinates of the turning points on the curve with equation $$y ^ { 2 } + x y + x ^ { 2 } - x = 1 .$$

8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).

1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).

5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).

1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.

12 The diagram shows the curve with parametric equations \(x = \sin 2 \theta + 2 , y = 2 \cos \theta + \cos 2 \theta\), for \(0 \leqslant \theta < 2 \pi\). \includegraphics[max width=\textwidth, alt={}, center]{60e1e785-c34b-48ef-a63f-13a25fee186e-10_771_673_397_239}

1. In this question you must show detailed reasoning. Determine the exact coordinates of all the stationary points on the curve.
2. Write down the equation of the line of symmetry of the curve.

6

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
   2. \(y = x ^ { 2 } \tan x\).
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
   2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).

3

1. Given that \(x = \tan ( 3 y + 1 )\) :
   1. find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\);
   2. find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = - \frac { 1 } { 3 }\).
2. Sketch the graph of \(y = \tan ^ { - 1 } x\).

9 The diagram shows the curve with equation \(y = \sin ^ { - 1 } 2 x\), where \(- \frac { 1 } { 2 } \leqslant x \leqslant \frac { 1 } { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{0ab0e757-270b-4c15-9202-9df2f02dddf3-4_790_752_906_644}

1. Find the \(y\)-coordinate of the point \(A\), where \(x = \frac { 1 } { 2 }\).
2. 1. Given that \(y = \sin ^ { - 1 } 2 x\), show that \(x = \frac { 1 } { 2 } \sin y\).
   2. Given that \(x = \frac { 1 } { 2 } \sin y\), find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
3. Using the answers to part (b) and a suitable trigonometrical identity, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { 1 - 4 x ^ { 2 } } }$$

1. Given that

$$x = \sec ^ { 2 } y + \tan y ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos ^ { 2 } y } { 2 \tan y + 1 } .$$

4. (a) Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\sqrt { 1 - \cos x }\)
2. \(x ^ { 3 } \ln x\) (b) Given that $$x = \frac { y + 1 } { 3 - 2 y } ,$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

4. A curve has the equation \(x = y \sqrt { 1 - 2 y }\).

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point \(A\) on the curve has \(y\)-coordinate - 1 .
2. Show that the equation of tangent to the curve at \(A\) can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where \(p\) and \(q\) are integers to be found.

1. (a) Given that

$$x = \sec \frac { y } { 2 } , \quad 0 \leq y < \pi ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x \sqrt { x ^ { 2 } - 1 } } .$$ (b) Find an equation for the tangent to the curve \(y = \sqrt { 3 + 2 \cos x }\) at the point where \(x = \frac { \pi } { 3 }\).

4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$

1. 1. Find the gradient of the curve at the point where \(t = 0\).
   2. Find an equation of the tangent to the curve at the point where \(t = 0\).
2. Show that the cartesian equation of the curve can be written in the form $$y = \frac { x ^ { 2 } } { k } - \frac { k } { x ^ { 2 } }$$ where \(k\) is an integer.

5 A curve is defined by the parametric equations $$x = 8 t ^ { 2 } - t , \quad y = \frac { 3 } { t }$$

1. Show that the cartesian equation of the curve can be written as \(x y ^ { 2 } + 3 y = k\), stating the value of the integer \(k\).
   (2 marks)
2. 1. Find an equation of the tangent to the curve at the point \(P\), where \(t = \frac { 1 } { 4 }\).
   2. Verify that the tangent at \(P\) intersects the curve when \(x = \frac { 3 } { 2 }\).

4

1. A curve is defined by the equation \(x ^ { 2 } - y ^ { 2 } = 8\).
   1. Show that at any point \(( p , q )\) on the curve, where \(q \neq 0\), the gradient of the curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { q }\).
      (2 marks)
   2. Show that the tangents at the points \(( p , q )\) and \(( p , - q )\) intersect on the \(x\)-axis.
      (4 marks)
2. Show that \(x = t + \frac { 2 } { t } , y = t - \frac { 2 } { t }\) are parametric equations of the curve \(x ^ { 2 } - y ^ { 2 } = 8\).
   (2 marks)

4

1. A curve is defined by the parametric equations \(x = 3 \cos 2 \theta , y = 2 \cos \theta\).
   1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { k \cos \theta }\), where \(k\) is an integer.
   2. Find an equation of the normal to the curve at the point where \(\theta = \frac { \pi } { 3 }\).
2. Find the exact value of \(\int _ { - \frac { \pi } { 4 } } ^ { \frac { \pi } { 4 } } \sin ^ { 2 } x \mathrm {~d} x\).

6 A curve is defined by the equation \(2 y + \mathrm { e } ^ { 2 x } y ^ { 2 } = x ^ { 2 } + C\), where \(C\) is a constant. The point \(P \left( 1 , \frac { 1 } { \mathrm { e } } \right)\) lies on the curve.

1. Find the exact value of \(C\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
3. Verify that \(P \left( 1 , \frac { 1 } { \mathrm { e } } \right)\) is a stationary point on the curve.