1.07m Tangents and normals: gradient and equations

5 A curve has equation \(y = \frac { 1 } { x ^ { 3 } } + 48 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the equation of each of the two tangents to the curve that are parallel to the \(x\)-axis.
3. Find an equation of the normal to the curve at the point \(( 1,49 )\).

9 The diagram shows part of a curve crossing the \(x\)-axis at the origin \(O\) and at the point \(A ( 8,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-5_547_536_497_760} The curve has equation $$y = 12 x - 3 x ^ { \frac { 5 } { 3 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 8,0 )\) is \(y + 8 x = 64\).
3. Find \(\int \left( 12 x - 3 x ^ { \frac { 5 } { 3 } } \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve from \(O\) to \(A\) and the tangents \(O P\) and \(A P\).

5 The point \(P ( 2,8 )\) lies on a curve, and the point \(M\) is the only stationary point of the curve. The curve has equation \(y = 6 + 2 x - \frac { 8 } { x ^ { 2 } }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the normal to the curve at the point \(P ( 2,8 )\) has equation \(x + 4 y = 34\).
3. 1. Show that the stationary point \(M\) lies on the \(x\)-axis.
   2. Hence write down the equation of the tangent to the curve at \(M\).
4. The tangent to the curve at \(M\) and the normal to the curve at \(P\) intersect at the point \(T\). Find the coordinates of \(T\).

4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.

1. 1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
   2. Find \(\int \sqrt { x } \mathrm {~d} x\).
   3. Hence find the area of the shaded region.
2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
   1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
   2. Find the area of the triangle \(A O B\).
3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.

7 A curve is defined, for \(x > 0\), by the equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$

1. Express \(\frac { x ^ { 8 } - 1 } { x ^ { 3 } }\) in the form \(x ^ { p } - x ^ { q }\), where \(p\) and \(q\) are integers.
2. 1. Hence differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence show that f is an increasing function.
3. Find the gradient of the normal to the curve at the point \(( 1,0 )\).

7 At the point \(( x , y )\), where \(x > 0\), the gradient of a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7$$

1. 1. Verify that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 4\).
      (1 mark)
   2. Write \(\frac { 16 } { x ^ { 2 } }\) in the form \(16 x ^ { k }\), where \(k\) is an integer.
   3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   4. Hence determine whether the point where \(x = 4\) is a maximum or a minimum, giving a reason for your answer.
2. The point \(P ( 1,8 )\) lies on the curve.
   1. Show that the gradient of the curve at the point \(P\) is 12 .
   2. Find an equation of the normal to the curve at \(P\).
3. 1. Find \(\int \left( 3 x ^ { \frac { 1 } { 2 } } + \frac { 16 } { x ^ { 2 } } - 7 \right) \mathrm { d } x\).
   2. Hence find the equation of the curve which passes through the point \(P ( 1,8 )\).

1

1. Write \(\sqrt { x ^ { 3 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
   (1 mark)
2. A curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { 2 } - \sqrt { x ^ { 3 } }$$
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find the equation of the tangent to the curve at the point where \(x = 4\), giving your answer in the form \(y = m x + c\).

6 A curve \(C\) has the equation $$y = \frac { x ^ { 3 } + \sqrt { x } } { x } , \quad x > 0$$

1. Express \(\frac { x ^ { 3 } + \sqrt { x } } { x }\) in the form \(x ^ { p } + x ^ { q }\).
2. 1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the normal to the curve \(C\) at the point on the curve where \(x = 1\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence deduce that the curve \(C\) has no maximum points. \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-7_1463_1707_1244_153}

5 The diagram shows part of a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-4_480_645_354_694} The curve is defined for \(x \geqslant 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. 1. Hence find the coordinates of the maximum point \(M\).
   2. Write down the equation of the normal to the curve at \(M\).
3. The point \(P \left( \frac { 9 } { 4 } , \frac { 27 } { 4 } \right)\) lies on the curve.
   1. Find an equation of the normal to the curve at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are positive integers.
   2. The normals to the curve at the points \(M\) and \(P\) intersect at the point \(R\). Find the coordinates of \(R\). \(6 \quad\) A curve \(C\), defined for \(0 \leqslant x \leqslant 2 \pi\) by the equation \(y = \sin x\), where \(x\) is in radians, is sketched below. The region bounded by the curve \(C\), the \(x\)-axis from 0 to 2 and the line \(x = 2\) is shaded. \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-5_441_789_466_612}

6 A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$

1. Express \(\frac { 12 + x ^ { 2 } \sqrt { x } } { x }\) in the form \(12 x ^ { p } + x ^ { q }\).
2. 1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the normal to the curve at the point on the curve where \(x = 4\).
   3. The curve has a stationary point \(P\). Show that the \(x\)-coordinate of \(P\) can be written in the form \(2 ^ { k }\), where \(k\) is a rational number.

4 A curve has equation \(y = \frac { 1 } { x ^ { 2 } } + 4 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(P ( - 1 , - 3 )\) lies on the curve. Find an equation of the normal to the curve at the point \(P\).
3. Find an equation of the tangent to the curve that is parallel to the line \(y = - 12 x\).
   [0pt] [5 marks]

8 The point \(A\) lies on the curve with equation \(y = x ^ { \frac { 1 } { 2 } }\). The tangent to this curve at \(A\) is parallel to the line \(3 y - 2 x = 1\). Find an equation of this tangent at \(A\).
[0pt] [5 marks]

3 The diagram shows a curve with a maximum point \(M\). \includegraphics[max width=\textwidth, alt={}, center]{e183578a-29a8-4112-b941-06c8894ed078-06_512_867_354_589} The curve is defined for \(x > 0\) by the equation $$y = 6 x ^ { \frac { 1 } { 2 } } - x - 3$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the \(y\)-coordinate of the maximum point \(M\).
3. Find an equation of the normal to the curve at the point \(P ( 4,5 )\).
4. It is given that the normal to the curve at \(P\), when translated by the vector \(\left[ \begin{array} { l } k \\ 0 \end{array} \right]\), passes through the point \(M\). Find the value of the constant \(k\).
   [0pt] [3 marks]

7. \begin{figure}[h] \end{figure} Fig. 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x\).
The curve crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Factorise \(\mathrm { f } ( x )\) completely.
2. Write down the \(x\)-coordinates of the points \(A\) and \(B\).
3. Find the gradient of \(C\) at \(A\). The region \(R\) is bounded by \(C\) and the line \(O A\), and the region \(S\) is bounded by \(C\) and the line \(A B\).
4. Use integration to find the area of the combined regions \(R\) and \(S\), shown shaded in Fig.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
   \end{figure} Fig. 3 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0\). The point \(P\), on \(C\), has \(x\)-coordinate 1 and the point \(Q\) is the minimum turning point of \(C\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find the coordinates of \(Q\).
   3. Show that \(P Q\) is parallel to the \(x\)-axis.
   4. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).

1. The point \(A\) has coordinates ( 4,6 ).

Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),

1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
2. Find the coordinates of \(B\).
3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   

6

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
   2. \(y = x ^ { 2 } \tan x\).
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
   2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).

9 The sketch shows the graph of \(y = 4 - \mathrm { e } ^ { 2 x }\). The curve crosses the \(y\)-axis at the point \(A\) and the \(x\)-axis at the point \(B\). \includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-5_711_921_466_557}

1. 1. Find \(\int \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x\).
      (2 marks)
   2. Hence show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 - \mathrm { e } ^ { 2 x } \right) \mathrm { d } x = 4 \ln 2 - \frac { 3 } { 2 }\).
2. 1. Write down the \(y\)-coordinate of \(A\).
   2. Show that \(x = \ln 2\) at \(B\).
3. Find the equation of the normal to the curve \(y = 4 - \mathrm { e } ^ { 2 x }\) at the point \(B\).
4. Find the area of the region enclosed by the curve \(y = 4 - \mathrm { e } ^ { 2 x }\), the normal to the curve at \(B\) and the \(y\)-axis.

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 3 } - 1 \right) ^ { 6 }\).
2. A curve has equation \(y = x \ln x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln x\) at the point on the curve where \(x = \mathrm { e }\).

7 A curve has equation \(y = 4 x \cos 2 x\).

1. Find an exact equation of the tangent to the curve at the point on the curve where $$x = \frac { \pi } { 4 }$$
2. The region shaded on the diagram below is bounded by the curve \(y = 4 x \cos 2 x\) and the \(x\)-axis from \(x = 0\) to \(x = \frac { \pi } { 4 }\). \includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_487_878_740_591} By using integration by parts, find the exact value of the area of the shaded region.
   (5 marks)
   \includegraphics[max width=\textwidth, alt={}]{b8614dd6-2197-40c3-a673-5bef3e3653a5-8_1275_1717_1432_150}

1

1. The curve with equation $$y = \frac { \cos x } { 2 x + 1 } , \quad x > - \frac { 1 } { 2 }$$ intersects the line \(y = \frac { 1 } { 2 }\) at the point where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 0 and \(\frac { \pi } { 2 }\).
   2. Show that the equation \(\frac { \cos x } { 2 x + 1 } = \frac { 1 } { 2 }\) can be rearranged into the form $$x = \cos x - \frac { 1 } { 2 }$$
   3. Use the iteration \(x _ { n + 1 } = \cos x _ { n } - \frac { 1 } { 2 }\) with \(x _ { 1 } = 0\) to find \(x _ { 3 }\), giving your answer to three decimal places.
2. 1. Given that \(y = \frac { \cos x } { 2 x + 1 }\), use the quotient rule to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence find the gradient of the normal to the curve \(y = \frac { \cos x } { 2 x + 1 }\) at the point on the curve where \(x = 0\).

6 The diagram shows the curve with equation \(y = \sqrt { 100 - 4 x ^ { 2 } }\), where \(x \geqslant 0\). \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_518_494_367_758}

1. Calculate the volume of the solid generated when the region bounded by the curve shown above and the coordinate axes is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis, giving your answer in terms of \(\pi\).
2. Use the mid-ordinate rule with five strips of equal width to find an estimate for \(\int _ { 0 } ^ { 5 } \sqrt { 100 - 4 x ^ { 2 } } \mathrm {~d} x\), giving your answer to three significant figures.
3. The point \(P\) on the curve has coordinates \(( 3,8 )\).
   1. Find the gradient of the curve \(y = \sqrt { 100 - 4 x ^ { 2 } }\) at the point \(P\).
   2. Hence show that the equation of the tangent to the curve at the point \(P\) can be written as \(2 y + 3 x = 25\).
4. The shaded regions on the diagram below are bounded by the curve, the tangent at \(P\) and the coordinate axes. \includegraphics[max width=\textwidth, alt={}, center]{a596af76-9680-4ccb-a512-5b2575414429-5_642_546_1800_731} Use your answers to part (b) and part (c)(ii) to find an approximate value for the total area of the shaded regions. Give your answer to three significant figures.

6 The diagram shows the curve \(y = \frac { \ln x } { x }\). \includegraphics[max width=\textwidth, alt={}, center]{33ca7e6d-b9eb-46be-b5b0-c5685212d7ff-4_586_1034_1612_513} The curve crosses the \(x\)-axis at \(A\) and has a stationary point at \(B\).

1. State the coordinates of \(A\).
2. Find the coordinates of the stationary point, \(B\), of the curve, giving your answer in an exact form.
3. Find the exact value of the gradient of the normal to the curve at the point where \(x = \mathrm { e } ^ { 3 }\).

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

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find an equation of the tangent to the curve \(y = x ^ { 3 } \ln x\) at the point on the curve where \(x = \mathrm { e }\).
   2. This tangent intersects the \(x\)-axis at the point \(A\). Find the exact value of the \(x\)-coordinate of the point \(A\).

2 A curve has equation \(y = 2 \ln ( 2 \mathrm { e } - x )\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find an equation of the normal to the curve \(y = 2 \ln ( 2 \mathrm { e } - x )\) at the point on the curve where \(x = \mathrm { e }\).
   [0pt] [4 marks]
3. The curve \(y = 2 \ln ( 2 \mathrm { e } - x )\) intersects the line \(y = x\) at a single point, where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 1 and 3 .
   2. Use the recurrence relation $$x _ { n + 1 } = 2 \ln \left( 2 \mathrm { e } - x _ { n } \right)$$ with \(x _ { 1 } = 1\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
   3. Figure 1, on the opposite page, shows a sketch of parts of the graphs of \(y = 2 \ln ( 2 \mathrm { e } - x )\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      [0pt] [2 marks] \section*{(c)(iii)} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-05_864_1284_1802_386}
      \end{figure}