1.07m Tangents and normals: gradient and equations

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$x = \frac { 2 y ^ { 2 } + 6 } { 3 y - 3 }$$

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) giving your answer as a fully simplified fraction. The tangents at points \(P\) and \(Q\) on the curve are parallel to the \(y\)-axis, as shown in Figure 4.
2. Use the answer to part (a) to find the equations of these two tangents.

6. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve with equation $$y = \sqrt { 4 x - 7 }$$ The line \(l\), shown in Figure 3, is the normal to the curve at the point \(P ( 8,5 )\)

1. Use calculus to show that an equation of \(l\) is $$5 x + 2 y - 50 = 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and \(l\).
2. Use algebraic integration to find the exact area of \(R\).

9. \begin{figure}[h] \end{figure} The curve shown in Figure 5 has equation $$x = 4 \sin ^ { 2 } y - 1 \quad 0 \leqslant y \leqslant \frac { \pi } { 2 }$$ The point \(P \left( k , \frac { \pi } { 3 } \right)\) lies on the curve.

1. Verify that \(k = 2\)
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\)
   2. Hence show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 \sqrt { x + 1 } \sqrt { 3 - x } }\) The normal to the curve at \(P\) cuts the \(x\)-axis at the point \(N\).
3. Find the exact area of triangle \(O P N\), where \(O\) is the origin. Give your answer in the form \(a \pi + b \pi ^ { 2 }\) where \(a\) and \(b\) are constants.

9.

1. Given that $$\frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \equiv x ^ { 2 } + P + \frac { Q } { x - 4 } \quad x > - 3$$ find the value of the constant \(P\) and show that \(Q = 5\) The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$g ( x ) = \frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \quad - 3 < x < 3.5 \quad x \in \mathbb { R }$$
2. Find the equation of the tangent to \(C\) at the point where \(x = 2\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{96948fd3-5438-4e95-b41b-2f649ca8dfac-28_876_961_1055_495} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a sketch of the curve \(C\).
   The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2\)
3. Find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found. \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-31_2255_50_314_34}
   

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   \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-32_106_113_2524_1832}

8. A curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \arcsin \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2 \quad - \frac { \pi } { 2 } \leqslant y \leqslant \frac { \pi } { 2 }$$

1. Sketch \(C\).
2. Given \(x = 2 \sin y\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { A - x ^ { 2 } } }$$ where \(A\) is a constant to be found. The point \(P\) lies on \(C\) and has \(y\) coordinate \(\frac { \pi } { 4 }\)
3. Find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
   (3)

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x \left( x ^ { 2 } - 4 \right) e ^ { - \frac { 1 } { 2 } x }$$

1. Find \(f ^ { \prime } ( x )\). The line \(l\) is the normal to the curve at \(O\) and meets the curve again at the point \(P\). The point \(P\) lies in the 3rd quadrant, as shown in Figure 3.
2. Show that the \(x\) coordinate of \(P\) is a solution of the equation $$x = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x } }$$
3. Using the iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x _ { n } } } \quad \text { with } x _ { 1 } = - 2$$ find, to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the \(x\) coordinate of \(P\).

5. Given that $$y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } \quad x \neq 1$$ show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( x - 1 ) ^ { 3 } }\), where \(k\) is a constant to be found.
(6)

1. The curve \(C\) has equation

$$y = \frac { 3 x - 2 } { ( x - 2 ) ^ { 2 } } , \quad x \neq 2$$ The point \(P\) on \(C\) has \(x\) coordinate 3
Find an equation of the normal to \(C\) at the point \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with equation $$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Using your answer to (a), find the exact coordinates of the stationary point on the curve \(C\) for which \(x > 0\). Write each coordinate in its simplest form.
   (5) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 3\)
3. Complete the table below with the value of \(y\) corresponding to \(x = 1\)

   \(x\)	0	1	2	3
   \(y\)	0		\(\frac { 3 } { 5 } \ln 5\)	\(\frac { 3 } { 10 } \ln 10\)

4. Use the trapezium rule with all the \(y\) values in the completed table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.

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

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

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}

3. The point \(P\) lies on the curve with equation \(y = \ln \left( \frac { 1 } { 3 } x \right)\). The \(x\)-coordinate of \(P\) is 3 . Find an equation of the normal to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
(5)

3. The curve \(C\) has equation $$x = 2 \sin y .$$

1. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
3. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.

2. A curve \(C\) has equation $$y = \mathrm { e } ^ { 2 x } \tan x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$

1. Show that the turning points on \(C\) occur where \(\tan x = - 1\).
2. Find an equation of the tangent to \(C\) at the point where \(x = 0\).

1. A curve \(C\) has equation

$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point \(A ( 0,4 )\) lies on \(C\).

1. Find an equation of the normal to the curve \(C\) at \(A\).
2. Express \(y\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 significant figures.
3. Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places.

4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.

1. The curve \(C\) has equation

$$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac { \pi } { 2 }\). Write your answer in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants.
   

4. The point \(P\) is the point on the curve \(x = 2 \tan \left( y + \frac { \pi } { 12 } \right)\) with \(y\)-coordinate \(\frac { \pi } { 4 }\). Find an equation of the normal to the curve at \(P\).

1. The curve \(C\) has equation

$$y = ( 2 x - 3 ) ^ { 5 }$$ The point \(P\) lies on \(C\) and has coordinates \(( w , - 32 )\).
Find

1. the value of \(w\),
2. the equation of the tangent to \(C\) at the point \(P\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   

1. The point \(P\) lies on the curve with equation

$$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The \(y\)-coordinate of \(P\) is 8 .

1. Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\).
2. Find the equation of the tangent to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants to be found.
   

4. (i) Differentiate with respect to \(x\)

1. \(x ^ { 2 } \cos 3 x\)
2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\) (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2. A curve \(C\) has equation $$y = \frac { 3 } { ( 5 - 3 x ) ^ { 2 } } , \quad x \neq \frac { 5 } { 3 }$$ The point \(P\) on \(C\) has \(x\)-coordinate 2. Find an equation of the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

7. $$f ( x ) = \frac { 4 x - 5 } { ( 2 x + 1 ) ( x - 3 ) } - \frac { 2 x } { x ^ { 2 } - 9 } , \quad x \neq \pm 3 , x \neq - \frac { 1 } { 2 }$$

1. Show that $$f ( x ) = \frac { 5 } { ( 2 x + 1 ) ( x + 3 ) }$$ The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The point \(P \left( - 1 , - \frac { 5 } { 2 } \right)\) lies on \(C\).
2. Find an equation of the normal to \(C\) at \(P\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) which has equation $$y = \mathrm { e } ^ { x \sqrt { 3 } } \sin 3 x , \quad - \frac { \pi } { 3 } \leqslant x \leqslant \frac { \pi } { 3 }$$

1. Find the \(x\) coordinate of the turning point \(P\) on \(C\), for which \(x > 0\) Give your answer as a multiple of \(\pi\).
2. Find an equation of the normal to \(C\) at the point where \(x = 0\)