1.07s Parametric and implicit differentiation

1. The curve \(C\) has equation

$$16 y ^ { 3 } + 9 x ^ { 2 } y - 54 x = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

6. \begin{figure}[h] \end{figure} Figure 2 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 determined.
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.
3. Find a cartesian equation of \(C\).

2. The curve \(C\) has equation $$3 ^ { x - 1 } + x y - y ^ { 2 } + 5 = 0$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(( 1,3 )\) on the curve \(C\) can be written in the form \(\frac { 1 } { \lambda } \ln \left( \mu \mathrm { e } ^ { 3 } \right)\), where \(\lambda\) and \(\mu\) are integers to be found.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 27 \sec ^ { 3 } t , y = 3 \tan t , \quad 0 \leqslant t \leqslant \frac { \pi } { 3 }$$

1. Find the gradient of the curve \(C\) at the point where \(t = \frac { \pi } { 6 }\)
2. Show that the cartesian equation of \(C\) may be written in the form $$y = \left( x ^ { \frac { 2 } { 3 } } - 9 \right) ^ { \frac { 1 } { 2 } } , \quad a \leqslant x \leqslant b$$ stating the values of \(a\) and \(b\).
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08f62966-2e63-4542-a10a-c6453a3215e7-10_581_1173_1628_475} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The finite region \(R\) which is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 125\) is shown shaded in Figure 3. This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Use calculus to find the exact value of the volume of the solid of revolution. \section*{Question 7 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 8. In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Let \(x\) be the mass of waste products, in kg , at time \(t\) minutes after the start of the experiment. It is known that at time \(t\) minutes, the rate of increase of the mass of waste products, in kg per minute, is \(k\) times the mass of unburned fuel remaining, where \(k\) is a positive constant. The differential equation connecting \(x\) and \(t\) may be written in the form $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( M - x ) , \text { where } M \text { is a constant. }$$
   1. Explain, in the context of the problem, what \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) and \(M\) represent. Given that initially the mass of waste products is zero,
   2. solve the differential equation, expressing \(x\) in terms of \(k , M\) and \(t\). Given also that \(x = \frac { 1 } { 2 } M\) when \(t = \ln 4\),
   3. find the value of \(x\) when \(t = \ln 9\), expressing \(x\) in terms of \(M\), in its simplest form. \section*{Question 8 continued}

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

$$x = 2 \sin t , \quad y = 1 - \cos 2 t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(t = \frac { \pi } { 6 }\)
2. Find a cartesian equation for \(C\) in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant \(k\).
3. Write down the range of \(\mathrm { f } ( x )\).

7. A curve is described by the equation $$x ^ { 2 } + 4 x y + y ^ { 2 } + 27 = 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). A point \(Q\) lies on the curve.
   The tangent to the curve at \(Q\) is parallel to the \(y\)-axis.
   Given that the \(x\) coordinate of \(Q\) is negative,
2. use your answer to part (a) to find the coordinates of \(Q\).

3. $$x ^ { 2 } + y ^ { 2 } + 10 x + 2 y - 4 x y = 10$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\), fully simplifying your answer.
2. Find the values of \(y\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

8. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = t - 4 \sin t , y = 1 - 2 \cos t , \quad - \frac { 2 \pi } { 3 } \leqslant t \leqslant \frac { 2 \pi } { 3 }$$ The point \(A\), with coordinates ( \(k , 1\) ), lies on the curve. Given that \(k > 0\)

1. find the exact value of \(k\),
2. find the gradient of the curve at the point \(A\). There is one point on the curve where the gradient is equal to \(- \frac { 1 } { 2 }\)
3. Find the value of \(t\) at this point, showing each step in your working and giving your answer to 4 decimal places.
   [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

1. The curve \(C\) has equation

$$2 x ^ { 2 } y + 2 x + 4 y - \cos ( \pi y ) = 17$$

1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(\left( 3 , \frac { 1 } { 2 } \right)\) lies on \(C\).
   The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
2. Find the \(x\) coordinate of \(A\), giving your answer in the form \(\frac { a \pi + b } { c \pi + d }\), where \(a , b , c\) and \(d\) are integers to be determined.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 4 \tan t , \quad y = 5 \sqrt { 3 } \sin 2 t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( 4 \sqrt { 3 } , \frac { 15 } { 2 } \right)\).

1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer as a simplified surd. The point \(Q\) lies on the curve \(C\), where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
2. Find the exact coordinates of the point \(Q\).

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

$$x = 3 t - 4 , y = 5 - \frac { 6 } { t } , \quad t > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) The point \(P\) lies on \(C\) where \(t = \frac { 1 } { 2 }\)
2. Find the equation of the tangent to \(C\) at the point \(P\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are integers to be determined.
3. Show that the cartesian equation for \(C\) can be written in the form $$y = \frac { a x + b } { x + 4 } , \quad x > - 4$$ where \(a\) and \(b\) are integers to be determined.

4. The curve \(C\) has equation $$4 x ^ { 2 } - y ^ { 3 } - 4 x y + 2 ^ { y } = 0$$ The point \(P\) with coordinates \(( - 2,4 )\) lies on \(C\).

1. Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). The normal to \(C\) at \(P\) meets the \(y\)-axis at the point \(A\).
2. Find the \(y\) coordinate of \(A\), giving your answer in the form \(p + q \ln 2\), where \(p\) and \(q\) are constants to be determined.
   (3)

8. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 4 shows a sketch of part of the curve \(C\) with parametric equations $$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P ( k , 8 )\) lies on \(C\), where \(k\) is a constant.

1. Find the exact value of \(k\). The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = k\).
2. Show that the area of \(R\) can be expressed in the form $$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$ where \(\lambda , \alpha\) and \(\beta\) are constants to be determined.
3. Hence use integration to find the exact value of the area of \(R\).

1. The curve \(C\) has equation

$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$

1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. Find the \(x\) coordinates of the two points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give exact answers in their simplest form.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point \(A\) lies on the curve \(C\). Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)

1. find the exact value of \(k\), giving your answer in a fully simplified form.
2. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$

1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
2. Find an equation of the normal to the curve at \(P\).
3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).

1. The hyperbola \(H\) has Cartesian equation \(x y = 25\)

The parabola \(P\) has parametric equations \(x = 10 t ^ { 2 } , y = 20 t\) The hyperbola \(H\) intersects the parabola \(P\) at the point \(A\)

1. Use algebra to determine the coordinates of \(A\) The point \(B\) with coordinates \(( 10,20 )\) lies on \(P\)
2. Find an equation for the normal to \(P\) at \(B\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be determined.
3. Use algebra to determine, in simplest form, the exact coordinates of the points where this normal intersects the hyperbola \(H\) (6)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola \(H\) has equation \(x y = 20\) The point \(P \left( 2 t \sqrt { a } , \frac { 2 \sqrt { a } } { t } \right) , t \neq 0\), where \(a\) is a constant, is a general point on \(H\)

1. State the value of \(a\)
2. Show that the normal to \(H\) at the point \(P\) has equation $$t y - t ^ { 3 } x - 2 \sqrt { 5 } \left( 1 - t ^ { 4 } \right) = 0$$ The points \(A\) and \(B\) lie on \(H\) The point \(A\) has parameter \(t = c\) and the point \(B\) has parameter \(t = - \frac { 1 } { 2 c }\), where \(c\) is a constant. The normal to \(H\) at \(A\) meets \(H\) again at \(B\)
3. Determine the possible values of \(C\)

1. The hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\).
The tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).

1. Determine, in terms of \(c\) and \(t\),
   1. the coordinates of \(A\),
   2. the coordinates of \(B\). Given that the area of triangle \(A O B\), where \(O\) is the origin, is 90 square units,
2. determine the value of \(c\), giving your answer as a simplified surd.

1. The parabola \(C\) has equation \(y ^ { 2 } = \frac { 4 } { 3 } x\)

The point \(P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)\), where \(t \neq 0\), lies on \(C\).

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$3 t x + 3 y = t ^ { 3 } + 2 t$$ The normal to \(C\) at the point where \(t = 9\) meets \(C\) again at the point \(Q\).
2. Determine the exact coordinates of \(Q\).

1. The parabola \(C\) has equation \(y ^ { 2 } = 36 x\)

The point \(P \left( 9 t ^ { 2 } , 18 t \right)\), where \(t \neq 0\), lies on \(C\)

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
2. Hence find the equations of the two normals to \(C\) which pass through the point (54, 0), giving your answers in the form \(y = p x + q\) where \(p\) and \(q\) are constants to be determined. Given that
   • the normals found in part (b) intersect the directrix of \(C\) at the points \(A\) and \(B\)
   • the point \(F\) is the focus of \(C\)
   • determine the area of triangle \(A F B\)

1. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 9\)

The point \(P\) with coordinates \(\left( 3 t , \frac { 3 } { t } \right)\), where \(t \neq 0\), lies on \(H\)

1. Use calculus to determine an equation for the normal to \(H\) at the point \(P\) Give your answer in the form \(t y - t ^ { 3 } x = \mathrm { f } ( t )\) Given that \(t = 2\)
2. determine the coordinates of the point where the normal meets \(H\) again. Give your answer in simplest form.

1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.

The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)

1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$ The parabola \(C\) has equation \(y ^ { 2 } = 6 x\) The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\)
2. Determine the exact coordinates of \(Q\) Given that
   • the point \(R\) is the focus of \(C\)
   • the line \(l\) is the directrix of \(C\)
   • the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
   • determine the exact length of \(Q S\)