1.07s Parametric and implicit differentiation

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with equation $$y ^ { 3 } - x ^ { 2 } + 4 x ^ { 2 } y = k$$ where \(k\) is a positive constant greater than 1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) lies on \(C\).
   Given that the normal to \(C\) at \(P\) has equation \(y = x\), as shown in Figure 2,
2. find the value of \(k\).

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 6 t - 3 \sin 2 t \quad y = 2 \cos t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$ The curve meets the \(y\)-axis at 2 and the \(x\)-axis at \(k\), where \(k\) is a constant.

1. State the value of \(k\).
2. Use parametric differentiation to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t$$ where \(\lambda\) is a constant to be found. The point \(P\) with parameter \(\mathrm { t } = \frac { \pi } { 4 }\) lies on \(C\).
   The tangent to \(C\) at the point \(P\) cuts the \(y\)-axis at the point \(N\).
3. Find the exact \(y\) coordinate of \(N\), giving your answer in simplest form. The region bounded by the curve, the \(x\)-axis and the \(y\)-axis is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
4. 1. Show that the volume of this solid is given by $$\int _ { 0 } ^ { \alpha } \beta ( 1 - \cos 4 t ) d t$$ where \(\alpha\) and \(\beta\) are constants to be found.
   2. Hence, using algebraic integration, find the exact volume of this solid.

2. A curve \(C\) has the equation $$x ^ { 3 } + 2 x y - x - y ^ { 3 } - 20 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find an equation of the tangent to \(C\) at the point \(( 3 , - 2 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

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4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = \sqrt { 3 } \sin 2 t \quad y = 4 \cos ^ { 2 } t \quad 0 \leqslant t \leqslant \pi$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 3 } \tan 2 t\), where \(k\) is a constant to be found.
2. Find an equation of the tangent to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants.

8. \begin{figure}[h] \end{figure} Figure 1 shows a curve \(C\) with polar equation \(r = a \sin 2 \theta , 0 \leqslant \theta \leqslant \frac { \pi } { 2 }\), and a half-line \(l\).
The half-line \(l\) meets \(C\) at the pole \(O\) and at the point \(P\). The tangent to \(C\) at \(P\) is parallel to the initial line. The polar coordinates of \(P\) are \(( R , \phi )\).

1. Show that \(\cos \phi = \frac { 1 } { \sqrt { 3 } }\)
2. Find the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and \(l\).
3. Use calculus to show that the exact area of \(S\) is $$\frac { 1 } { 36 } a ^ { 2 } \left( 9 \arccos \left( \frac { 1 } { \sqrt { 3 } } \right) + \sqrt { 2 } \right)$$

4. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with polar equation $$r = 2 \cos 2 \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$ The line \(l\) is parallel to the initial line and is a tangent to \(C\). Find an equation of \(l\), giving your answer in the form \(r = \mathrm { f } ( \theta )\).

5. $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0$$

1. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\). Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0.5\) at \(x = 0\),
2. find a series solution for \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
   5. \(y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 2 y = 0\)
   1. Find an expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) in terms of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } , \frac { \mathrm {~d} y } { \mathrm {~d} x }\) and \(y\).

9. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 4 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l\) is the normal to the ellipse at the point \(P\).

1. Show that an equation for \(l\) is $$5 x \sin \theta - 4 y \cos \theta = 9 \sin \theta \cos \theta$$ The point \(F\) is the focus of \(E\) that lies on the positive \(x\)-axis.
2. Determine the coordinates of \(F\). The line \(l\) crosses the \(x\)-axis at the point \(Q\).
3. Show that $$\frac { | Q F | } { | P F | } = e$$ where \(e\) is the eccentricity of \(E\).
   

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8. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Determine the eccentricity of \(E\)
2. Hence, for this ellipse, determine
   1. the coordinates of the foci,
   2. the equations of the directrices. The point \(P\) lies on \(E\) and has coordinates \(( 3 \cos \theta , 2 \sin \theta )\). The line \(l _ { 1 }\) is the tangent to \(E\) at the point \(P\)
3. Using calculus, show that an equation for \(l _ { 1 }\) is $$2 x \cos \theta + 3 y \sin \theta = 6$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\) The line \(l _ { 1 }\) intersects the line \(l _ { 2 }\) at the point \(Q\)
4. Determine the coordinates of \(Q\)
5. Show that, as \(\theta\) varies, the point \(Q\) lies on the curve with equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = \alpha x ^ { 2 } + \beta y ^ { 2 }$$ where \(\alpha\) and \(\beta\) are constants to be determined.
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2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).

1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
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The curve \(C\) has equation $$13 x ^ { 2 } + 13 y ^ { 2 } - 10 x y = 52$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\) and \(y\), simplifying your answer.
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7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.

3. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 8 \cos \theta , 6 \sin \theta )\).

1. Using calculus, show that an equation for \(l\) is $$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
   The point \(M\) is the midpoint of \(A B\).
2. Determine a Cartesian equation for the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(a x ^ { 2 } + b y ^ { 2 } = c\) where \(a , b\) and \(c\) are integers.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > b > 0$$ The line \(l\) is a normal to \(E\) at a point \(P ( a \cos \theta , b \sin \theta ) , \quad 0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta - b y \cos \theta = \left( a ^ { 2 } - b ^ { 2 } \right) \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
2. Show that the area of the triangle \(O A B\), where \(O\) is the origin, may be written as \(k \sin 2 \theta\), giving the value of the constant \(k\) in terms of \(a\) and \(b\).
3. Find, in terms of \(a\) and \(b\), the exact coordinates of the point \(P\), for which the area of the triangle \(O A B\) is a maximum.

5. The ellipse \(E\) has equation $$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The point \(P ( a \cos \theta , b \sin \theta )\) is a general point on the ellipse \(E\).

1. Write down the value of \(a\) and the value of \(b\). The line \(L\) is a tangent to \(E\) at the point \(P\).
2. Show that an equation of the line \(L\) is given by $$3 y \sin \theta + x \cos \theta = 3$$ The line \(L\) meets the \(x\)-axis at the point \(Q\) and meets the \(y\)-axis at the point \(R\).
3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is given by $$k \operatorname { cosec } 2 \theta$$ where \(k\) is a constant to be found. The point \(M\) is the midpoint of \(Q R\).
4. Find a cartesian equation of the locus of \(M\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

5. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The point \(P ( 4 \sec \theta , 3 \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\), lies on \(H\).

1. Show that an equation of the normal to \(H\) at the point \(P\) is $$3 y + 4 x \sin \theta = 25 \tan \theta$$ The line \(l\) is the directrix of \(H\) for which \(x > 0\) The normal to \(H\) at \(P\) crosses the line \(l\) at the point \(Q\). Given that \(\theta = \frac { \pi } { 4 }\)
2. find the \(y\) coordinate of \(Q\), giving your answer in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers to be found.
   

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2. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 6 \cos \theta , 5 \sin \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)

1. Use calculus to show that an equation of \(l\) is $$6 x \sin \theta - 5 y \cos \theta = 11 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\). The point \(R\) is the foot of the perpendicular from \(P\) to the \(x\)-axis.
2. Show that \(\frac { O Q } { O R } = e ^ { 2 }\), where \(e\) is the eccentricity of the ellipse \(E\).

3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).

1. Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Show that the equation of the normal to \(C\) at the point with parameter \(t\) is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
4. Find the cartesian equation of the envelope of the normals to \(C\).
5. The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).

3 The curve \(C\) has parametric equations \(x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }\), for \(t \geqslant 0\).

1. Show that \(\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }\). Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 2\).
2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Show that the curvature at a general point on \(C\) is \(\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }\).
4. Find the coordinates of the centre of curvature corresponding to the point on \(C\) where \(t = 1\).

6 Fig. 6 shows the curve \(\mathrm { e } ^ { 2 y } = x ^ { 2 } + y\). \begin{figure}[h] \end{figure}

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x } { 2 \mathrm { e } ^ { 2 y } - 1 }\).
2. Hence find to 3 significant figures the coordinates of the point P , shown in Fig. 6, where the curve has infinite gradient. Section B (36 marks)

7 Fig. 7 shows the curve defined implicitly by the equation $$y ^ { 2 } + y = x ^ { 3 } + 2 x ,$$ together with the line \(x = 2\). \begin{figure}[h] \end{figure} Fig. 7 Find the coordinates of the points of intersection of the line and the curve.
Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). Hence find the gradient of the curve at each of these two points.

3. A curve has the equation \(x = y ^ { 2 } - 3 \ln 2 y\).

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { 2 y ^ { 2 } - 3 }$$
2. Find an equation for the tangent to the curve at the point where \(y = \frac { 1 } { 2 }\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

4

1. Given that \(x = ( 4 t + 9 ) ^ { \frac { 1 } { 2 } }\) and \(y = 6 \mathrm { e } ^ { \frac { 1 } { 2 } x + 1 }\), find expressions for \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} t }\) when \(t = 4\), giving your answer correct to 3 significant figures.