1.07s Parametric and implicit differentiation

5 The curve \(C\) has equation \(2 x ^ { 3 } + 3 x ^ { 2 } y - 3 y ^ { 3 } - 16 = 0\).

1. Find the coordinates of the point \(A\) on \(C\) at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) and \(x \neq 0\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

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 \(t\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$

7 The curve \(C\) has equation \(x ^ { 2 } + 2 x y - 4 y ^ { 2 } + 20 = 0\). Show that if the tangent to \(C\) at the point \(( x , y )\) is parallel to the \(x\)-axis then \(x + y = 0\). Hence find the coordinates of the stationary points on \(C\), and determine their nature.

5 The positive variables \(x\) and \(y\) are related by $$y = x ^ { 2 } + 2 \ln ( x y )$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when both \(x\) and \(y\) are equal to 1 .

5 The point \(P ( 2,1 )\) lies on the curve with equation $$x ^ { 3 } - 2 y ^ { 3 } = 3 x y$$ Find

1. the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(P\),
2. the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(P\).

10 The curve \(C\) has equation \(x ^ { 3 } + y ^ { 3 } = 3 x y\), for \(x > 0\) and \(y > 0\). Find a relationship between \(x\) and \(y\) when \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\). Find the exact coordinates of the turning point of \(C\), and determine the nature of this turning point.

6 A curve has cartesian equation \(y ^ { 2 } - x ^ { 2 } = 4\).

1. Verify that $$x = t - \frac { 1 } { t } , \quad y = t + \frac { 1 } { t } ,$$ are parametric equations of the curve.
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - 1 ) ( t + 1 ) } { t ^ { 2 } + 1 }\). Hence find the coordinates of the stationary points of the curve. Section B (36 marks)

3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\). \begin{figure}[h] \end{figure}

1. Show that the values of \(t\) where the curve cuts the \(y\)-axis are \(t = 0 , \pm \frac { 1 } { \sqrt { 3 } }\). Write down the corresponding values of \(y\).
2. Find the radius and centre of curvature when \(t = \frac { 1 } { \sqrt { 3 } }\). The arc of the curve given by \(0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }\) is denoted by \(C\).
3. Find the length of \(C\).
4. Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).

3

1. It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = ( 2 x + 1 ) \ln ( x + y )$$ and $$y ( 0 ) = 2$$ Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.1\), to obtain an approximation to \(y ( 0.1 )\), giving your answer to three decimal places.
2. It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) \ln ( x + y )$$ and \(y = 2\) when \(x = 0\).
   1. Use implicit differentiation to find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\), giving your answer in terms of \(x\) and \(y\).
   2. Hence find the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(y ( x )\). Give your answer in an exact form.
   3. Use your answer to part (b)(ii) to obtain an approximation to \(y ( 0.1 )\), giving your answer to three decimal places.
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10 A curve has parametric equations \(x = t + \frac { 2 } { t }\) and \(y = t - \frac { 2 } { t }\), for \(t \neq 0\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
2. Explain why the curve has no stationary points.
3. By considering \(x + y\), or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.

5 A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
2. Find the equation of the normal to the curve at the point ( 1,2 ).

9 A particle moves in the \(x - y\) plane so that at time \(t\) seconds, where \(t \geqslant 0\), its coordinates are given by \(x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }\).

1. Explain why the path of the particle never crosses the \(x\)-axis.
2. Determine the exact values of \(t\) when the path of the particle intersects the \(y\)-axis.
3. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }\).
4. Hence find the coordinates of the particle when its path is parallel to the \(y\)-axis.

7 A curve has equation \(2 x ^ { 3 } + 6 x y - 3 y ^ { 2 } = 2\).
Show that there are no points on this curve where the tangent is parallel to \(y = x\).

12 A curve has parametric equations \(x = \frac { 1 } { t } , y = 2 t\). The point \(P\) is \(\left( \frac { 1 } { p } , 2 p \right)\).

1. Show that the equation of the tangent at \(P\) can be written as \(y = - 2 p ^ { 2 } x + 4 p\). The tangent to this curve at \(P\) crosses the \(x\)-axis at the point \(A\) and the normal to this curve at \(P\) crosses the \(x\)-axis at the point \(B\).
2. Show that the ratio \(P A : P B\) is \(1 : 2 p ^ { 2 }\). \section*{END OF QUESTION PAPER}

3

1. A circle is defined by the parametric equations \(x = 3 + 2 \cos \theta , y = - 4 + 2 \sin \theta\).
   1. Find a cartesian equation of the circle.
   2. Write down the centre and radius of the circle.
2. In this question you must show detailed reasoning. The curve \(S\) is defined by the parametric equations \(x = 4 \cos t , y = 2 \sin t\). The line \(L\) is a tangent to \(S\) at the point given by \(t = \frac { 1 } { 6 } \pi\). Find where the line \(L\) cuts the \(x\)-axis.

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } + y ^ { 3 } - 10 x - 12 y - 5 = 0\) a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 10 - 2 x } { 3 y ^ { 2 } - 12 }\) At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis.
b. Find the exact coordinates of \(Q\).

16. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\) The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\) b. Show that an equation for \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$ c. The cartesian equation for the curve \(C\) can be written in the form $$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found. The straight line with equation $$y = \frac { 1 } { 3 } x + k$$ where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } - 2 x y + 3 y ^ { 2 } = 50\)

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x } { 3 y - x }\) The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km .
   The points \(P\) and \(Q\) represent points that are furthest west and furthest east of the origin \(O\), as shown in Figure 4. Using part (a),
2. find the exact coordinates of the point \(P\).
3. Explain briefly how to find the coordinates of the point that is furthest north of the origin \(O\). (You do not need to carry out this calculation).

1. The curve \(C\), in the standard Cartesian plane, is defined by the equation

$$x = 4 \sin 2 y \quad \frac { - \pi } { 4 } < y < \frac { \pi } { 4 }$$ The curve \(C\) passes through the origin \(O\)

1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the origin.
2. 1. Use the small angle approximation for \(\sin 2 y\) to find an equation linking \(x\) and \(y\) for points close to the origin.
   2. Explain the relationship between the answers to (a) and (b)(i).
3. Show that, for all points \(( x , y )\) lying on \(C\), $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { a \sqrt { b - x ^ { 2 } } }$$ where \(a\) and \(b\) are constants to be found.

1. The curve \(C\) has equation

$$x ^ { 2 } \tan y = 9 \quad 0 < y < \frac { \pi } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 18 x } { x ^ { 4 } + 81 }$$
2. Prove that \(C\) has a point of inflection at \(x = \sqrt [ 4 ] { 27 }\)

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

$$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\) The line \(l\) is the normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers.

16. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 2 \tan t + 1 \quad y = 2 \sec ^ { 2 } t + 3 \quad - \frac { \pi } { 4 } \leqslant t \leqslant \frac { \pi } { 3 }$$ The line \(l\) is the normal to \(C\) at the point \(P\) where \(t = \frac { \pi } { 4 }\)

1. Using parametric differentiation, show that an equation for \(l\) is $$y = - \frac { 1 } { 2 } x + \frac { 17 } { 2 }$$
2. Show that all points on \(C\) satisfy the equation $$y = \frac { 1 } { 2 } ( x - 1 ) ^ { 2 } + 5$$ The straight line with equation $$y = - \frac { 1 } { 2 } x + k \quad \text { where } k \text { is a constant }$$ intersects \(C\) at two distinct points.
3. Find the range of possible values for \(k\).

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

Solutions relying on calculator technology are not acceptable.
A curve has equation $$x ^ { 3 } + 2 x y + 3 y ^ { 2 } = 47$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\) The point \(P ( - 2,5 )\) lies on the curve.
2. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

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

$$x = t ^ { 2 } + 6 t - 16 \quad y = 6 \ln ( t + 3 ) \quad t > - 3$$

1. Show that a Cartesian equation for \(C\) is $$y = A \ln ( x + B ) \quad x > - B$$ where \(A\) and \(B\) are integers to be found. The curve \(C\) cuts the \(y\)-axis at the point \(P\)
2. Show that the equation of the tangent to \(C\) at \(P\) can be written in the form $$a x + b y = c \ln 5$$ where \(a\), \(b\) and \(c\) are integers to be found.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with parametric equations $$x = ( t + 3 ) ^ { 2 } \quad y = 1 - t ^ { 3 } \quad - 2 \leqslant t \leqslant 1$$ The point \(P\) with coordinates \(( 4,2 )\) lies on \(C\).

1. Using parametric differentiation, show that the tangent to \(C\) at \(P\) has equation $$3 x + 4 y = 20$$ The curve \(C\) is used to model the profile of a slide at a water park.
   Units are in metres, with \(y\) being the height of the slide above water level.
2. Find, according to the model, the greatest height of the slide above water level.