1.07s Parametric and implicit differentiation

6.

1. Given \(x = \tan ^ { 2 } 4 y , 0 < y < \frac { \pi } { 8 }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a function of \(x\). Write your answer in the form \(\frac { 1 } { A \left( x ^ { p } + x ^ { q } \right) }\), where \(A , p\) and \(q\) are constants to
   be found.
2. The volume \(V\) of a cube is increasing at a constant rate of \(2 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the length of the edge of the cube is increasing when the volume of the cube is \(64 \mathrm {~cm} ^ { 3 }\).

3. A curve \(C\) has equation $$3 ^ { x } + 6 y = \frac { 3 } { 2 } x y ^ { 2 }$$ Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) with coordinates (2, 3). Give your answer in the form \(\frac { a + \ln b } { 8 }\), where \(a\) and \(b\) are integers.

13. A curve \(C\) has parametric equations $$x = 6 \cos 2 t , \quad y = 2 \sin t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 }$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \operatorname { cosec } t\), giving the exact value of the constant \(\lambda\).
2. Find an equation of the normal to \(C\) at the point where \(t = \frac { \pi } { 3 }\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are simplified surds. The cartesian equation for the curve \(C\) can be written in the form $$x = f ( y ) , \quad - k < y < k$$ where \(\mathrm { f } ( y )\) is a polynomial in \(y\) and \(k\) is a constant.
3. Find \(\mathrm { f } ( y )\).
4. State the value of \(k\).

1. Find an equation of the tangent to the curve

$$x ^ { 3 } + 3 x ^ { 2 } y + y ^ { 3 } = 37$$ at the point \(( 1,3 )\). Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
(6)

13. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 4 has parametric equations $$x = 1 + \sqrt { 3 } \tan \theta , \quad y = 5 \sec \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(y\)-axis at \(A\) and has a minimum turning point at \(B\), as shown in Figure 4.

1. Find the exact coordinates of \(A\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \lambda \sin \theta\), giving the exact value of the constant \(\lambda\).
3. Find the coordinates of \(B\).
4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y = k \sqrt { \left( x ^ { 2 } - 2 x + 4 \right) }$$ where \(k\) is a simplified surd to be found.

1. A curve \(C\) has equation

$$3 ^ { x } + x y = x + y ^ { 2 } , \quad y > 1$$ The point \(P\) with coordinates \(( 4,11 )\) lies on \(C\).
Find the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(P\). Give your answer in the form \(a + b \ln 3\), where \(a\) and \(b\) are rational numbers.

1. Given that

$$y = 8 \tan ( 2 x ) , \quad - \frac { \pi } { 4 } < x < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} y } = \frac { A } { B + y ^ { 2 } }$$ where \(A\) and \(B\) are integers to be found.

11. \begin{figure}[h] \end{figure} The curve \(C\) shown in Figure 3 has parametric equations $$x = 3 \cos t , \quad y = 9 \sin 2 t , \quad 0 \leqslant t \leqslant 2 \pi$$ The curve \(C\) meets the \(x\)-axis at the origin and at the points \(A\) and \(B\), as shown in Figure 3 .

1. Write down the coordinates of \(A\) and \(B\).
2. Find the values of \(t\) at which the curve passes through the origin.
3. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), and hence find the gradient of the curve when \(t = \frac { \pi } { 6 }\)
4. Show that the cartesian equation for the curve \(C\) can be written in the form $$y ^ { 2 } = a x ^ { 2 } \left( b - x ^ { 2 } \right)$$ where \(a\) and \(b\) are integers to be determined.

1. The curve \(C\) has equation

$$81 y ^ { 3 } + 64 x ^ { 2 } y + 256 x = 0$$

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

8. A curve has parametric equations $$x = t ^ { 2 } - t , \quad y = \frac { 4 t } { 1 - t } \quad t \neq 1$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer as a simplified fraction.
2. Find an equation for the tangent to the curve at the point \(P\) where \(t = - 1\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The tangent to the curve at \(P\) cuts the curve at the point \(Q\).
3. Use algebra to find the coordinates of \(Q\).

2. A curve \(C\) has the equation $$x ^ { 3 } - 3 x y - x + y ^ { 3 } - 11 = 0$$ Find an equation of the tangent to \(C\) at the point \(( 2 , - 1 )\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. A curve has equation

$$4 x ^ { 2 } - y ^ { 2 } + 2 x y + 5 = 0$$ The points \(P\) and \(Q\) lie on the curve.
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(P\) and at \(Q\),

1. use implicit differentiation to show that \(y - 6 x = 0\) at \(P\) and at \(Q\).
2. Hence find the coordinates of \(P\) and \(Q\).
   

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = t ^ { 2 } + 2 t , \quad y = t ^ { 3 } - 9 t , \quad t \in \mathbb { R }$$ The curve cuts the \(x\)-axis at the origin and at the points \(A\) and \(B\) as shown in Figure 3 .

1. Find the coordinates of point \(A\) and show that point \(B\) has coordinates ( 15,0 ).
2. Show that the equation of the tangent to the curve at \(B\) is \(9 x - 4 y - 135 = 0\) The tangent to the curve at \(B\) cuts the curve again at the point \(X\).
3. Find the coordinates of \(X\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. A curve \(C\) has equation

$$3 x ^ { 2 } + 2 x y - 2 y ^ { 2 } + 4 = 0$$ Find an equation for the tangent to \(C\) at the point ( 2,4 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)

14. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)

1. find the coordinates of \(P\). The line \(l\) is the normal to \(C\) at \(P\).
2. Show that an equation of \(l\) is \(y = x + 3.5\) The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
3. Show that the area of \(S\) is given by $$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
4. Hence, by integration, find the exact area of \(S\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.)

   END

2. A curve \(C\) has parametric equations $$x = \frac { 3 } { 2 } t - 5 , \quad y = 4 - \frac { 6 } { t } \quad t \neq 0$$

1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(t = 3\), giving your answer as a fraction in its simplest form.
2. Show that a cartesian equation of \(C\) can be expressed in the form $$y = \frac { a x + b } { x + 5 } \quad x \neq k$$ where \(a , b\) and \(k\) are integers to be found.

10. The curve \(C\) satisfies the equation $$x \mathrm { e } ^ { 5 - 2 y } - y = 0 \quad x > 0 , \quad y > 0$$ The point \(P\) with coordinates ( \(2 \mathrm { e } ^ { - 1 } , 2\) ) lies on \(C\).
The tangent to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and cuts the \(y\)-axis at the point \(B\).
Given that \(O\) is the origin, find the exact area of triangle \(O A B\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}]{a377da06-a968-438c-bec2-ae55283dae47-35_4_21_127_2042} L

3. A curve \(C\) has parametric equations $$x = \sqrt { 3 } \tan \theta , \quad y = \sec ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ The cartesian equation of \(C\) is $$y = \mathrm { f } ( x ) , \quad 0 \leqslant x \leqslant k , \quad \text { where } k \text { is a constant }$$

1. State the value of \(k\).
2. Find \(\mathrm { f } ( x )\) in its simplest form.
3. Hence, or otherwise, find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\)

4. The curve \(C\) has equation $$3 y \mathrm { e } ^ { - 2 x } = 4 x ^ { 2 } + y ^ { 2 } + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) on \(C\) has coordinates \(( 0,2 )\).
2. Find the equation of the normal to \(C\) at \(P\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
   (3)

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2. The curve \(C\) has equation $$y ^ { 3 } + x ^ { 2 } y - 6 x = 0$$

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

10. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \frac { 20 t } { 2 t + 1 } \quad y = t ( t - 4 ) , \quad t > 0$$ The curve cuts the \(x\)-axis at the point \(P\).

1. Find the \(x\) coordinate of \(P\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( t - A ) ( 2 t + 1 ) ^ { 2 } } { B }\) where \(A\) and \(B\) are constants to be found.
3. 1. Make \(t\) the subject of the formula $$x = \frac { 20 t } { 2 t + 1 }$$
   2. Hence find a cartesian equation of the curve \(C\). Write your answer in the form $$y = \mathrm { f } ( x ) , \quad 0 < x < k$$ where \(\mathrm { f } ( x )\) is a single fraction and \(k\) is a constant to be found.

2. A curve \(C\) has equation $$x ^ { 3 } - 4 x y + 2 x + 3 y ^ { 2 } - 3 = 0$$ Find an equation of the normal 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. \includegraphics[max width=\textwidth, alt={}, center]{c6bde466-61ec-437d-a3b4-84511a98d788-05_108_166_2612_1781}

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = 7 t ^ { 2 } - 5 , \quad y = t \left( 9 - t ^ { 2 } \right) , \quad t \in \mathbb { R }$$

1. Find an equation of the tangent to \(C\) at the point where \(t = 1\) Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The curve \(C\) cuts the \(x\)-axis at the points \(A\) and \(B\), as shown in Figure 3
2. 1. Find the \(x\) coordinate of the point \(A\).
   2. Find the \(x\) coordinate of the point \(B\). The region \(R\), shown shaded in Figure 3, is enclosed by the loop of the curve \(C\).
3. Use integration to find the area of \(R\).

6. 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 \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

10. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { 3 } , \frac { 1 } { 2 } \sqrt { 3 } \right)\)

1. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
2. Show that \(Q\) has coordinates \(( k \sqrt { 3 } , 0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { 3 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
3. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { 3 } + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. \includegraphics[max width=\textwidth, alt={}, center]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-39_61_29_2608_1886}