1.07l Derivative of ln(x): and related functions

5 The line with equation \(y = k x - k\), where \(k\) is a positive constant, is a tangent to the curve with equation \(y = - \frac { 1 } { 2 x }\). Find, in either order, the value of \(k\) and the coordinates of the point where the tangent meets the curve. [5]

4 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 } { \sqrt { x + 6 } } + \frac { 6 } { x ^ { 2 } }\) and that \(\mathrm { f } ( 3 ) = 1\). Find \(\mathrm { f } ( x )\).

7 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-3_646_841_593_651} The diagram shows part of the graph of \(y = \frac { 18 } { x }\) and the normal to the curve at \(P ( 6,3 )\). This normal meets the \(x\)-axis at \(R\). The point \(Q\) on the \(x\)-axis and the point \(S\) on the curve are such that \(P Q\) and \(S R\) are parallel to the \(y\)-axis.

1. Find the equation of the normal at \(P\) and show that \(R\) is the point ( \(4 \frac { 1 } { 2 } , 0\) ).
2. Show that the volume of the solid obtained when the shaded region \(P Q R S\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis is \(18 \pi\).

2 Find the gradient of the curve \(y = \frac { 12 } { x ^ { 2 } - 4 x }\) at the point where \(x = 3\).

9 A curve has equation \(y = \frac { 4 } { \sqrt { } x }\).

1. The normal to the curve at the point \(( 4,2 )\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(P Q\), correct to 3 significant figures.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).

1 A curve has equation \(y = \frac { k } { x }\). Given that the gradient of the curve is - 3 when \(x = 2\), find the value of the constant \(k\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { \sqrt { } ( 6 - 2 x ) }\), and \(P ( 1,8 )\) is a point on the curve.

1. The normal to the curve at the point \(P\) meets the coordinate axes at \(Q\) and at \(R\). Find the coordinates of the mid-point of \(Q R\).
2. Find the equation of the curve.

9 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_391_595_1978_774} The diagram shows part of the curve \(y = \frac { 6 } { 3 x - 2 }\).

1. Find the gradient of the curve at the point where \(x = 2\).
2. Find the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in terms of \(\pi\).

7 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_766_589_251_778} The diagram shows part of the curve \(y = 2 - \frac { 18 } { 2 x + 3 }\), which crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(A\) crosses the \(y\)-axis at \(C\).

1. Show that the equation of the line \(A C\) is \(9 x + 4 y = 27\).
2. Find the length of \(B C\).

5 The equation of a curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 3 x - 2 ) }\). Given that the curve passes through the point \(P ( 2,11 )\), find

1. the equation of the normal to the curve at \(P\),
2. the equation of the curve.

3 The straight line \(y = m x + 14\) is a tangent to the curve \(y = \frac { 12 } { x } + 2\) at the point \(P\). Find the value of the constant \(m\) and the coordinates of \(P\).

9 A function f is defined by \(\mathrm { f } ( x ) = \frac { 5 } { 1 - 3 x }\), for \(x \geqslant 1\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Determine, with a reason, whether \(f\) is an increasing function, a decreasing function or neither.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

11 \includegraphics[max width=\textwidth, alt={}, center]{fe4c3555-5736-48c4-b61a-9f6b9a1ee46e-4_598_789_255_678} The diagram shows the curve \(y = \sqrt { } ( 1 + 4 x )\), which intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). The normal to the curve at \(B\) meets the \(x\)-axis at \(C\). Find

1. the equation of \(B C\),
2. the area of the shaded region.

11 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-4_643_570_849_790} The diagram shows part of the curve \(y = \frac { 8 } { \sqrt { } x } - x\) and points \(A ( 1,7 )\) and \(B ( 4,0 )\) which lie on the curve. The tangent to the curve at \(B\) intersects the line \(x = 1\) at the point \(C\).

1. Find the coordinates of \(C\).
2. Find the area of the shaded region.

4 A curve has equation \(y = \frac { 4 } { ( 3 x + 1 ) ^ { 2 } }\). Find the equation of the tangent to the curve at the point where the line \(x = - 1\) intersects the curve.

9 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_849_565_1466_790} The diagram shows part of the curve \(y = 8 - \sqrt { } ( 4 - x )\) and the tangent to the curve at \(P ( 3,7 )\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the equation of the tangent to the curve at \(P\) in the form \(y = m x + c\).
3. Find, showing all necessary working, the area of the shaded region.

5 A function f is such that \(\mathrm { f } ( x ) = \frac { 15 } { 2 x + 3 }\) for \(0 \leqslant x \leqslant 6\).

1. Find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and use your result to explain why f has an inverse.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\), and state the domain and range of \(\mathrm { f } ^ { - 1 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{028c7979-6b24-42d0-9857-c616a169b2b2-18_510_410_260_863} The diagram shows part of the curve \(y = \frac { 4 } { 5 - 3 x }\).

1. Find the equation of the normal to the curve at the point where \(x = 1\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.
   The shaded region is bounded by the curve, the coordinate axes and the line \(x = 1\).
2. Find, showing all necessary working, the volume obtained when this shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

5 A curve has equation \(y = 3 + \frac { 12 } { 2 - x }\).

1. Find the equation of the tangent to the curve at the point where the curve crosses the \(x\)-axis.
2. A point moves along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of 0.04 units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\).

9 \includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-4_563_679_938_733} The diagram shows points \(A ( 0,4 )\) and \(B ( 2,1 )\) on the curve \(y = \frac { 8 } { 3 x + 2 }\). The tangent to the curve at \(B\) crosses the \(x\)-axis at \(C\). The point \(D\) has coordinates \(( 2,0 )\).

1. Find the equation of the tangent to the curve at \(B\) and hence show that the area of triangle \(B D C\) is \(\frac { 4 } { 3 }\).
2. Show that the volume of the solid formed when the shaded region \(O D B A\) is rotated completely about the \(x\)-axis is \(8 \pi\).

10 The equation of a curve is \(y = 3 + 4 x - x ^ { 2 }\).

1. Show that the equation of the normal to the curve at the point \(( 3,6 )\) is \(2 y = x + 9\).
2. Given that the normal meets the coordinate axes at points \(A\) and \(B\), find the coordinates of the mid-point of \(A B\).
3. Find the coordinates of the point at which the normal meets the curve again.

9 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_502_663_255_740} The diagram shows part of the curve \(y = \frac { 9 } { 2 x + 3 }\), crossing the \(y\)-axis at the point \(B ( 0,3 )\). The point \(A\) on the curve has coordinates \(( 3,1 )\) and the tangent to the curve at \(A\) crosses the \(y\)-axis at \(C\).

1. Find the equation of the tangent to the curve at \(A\).
2. Determine, showing all necessary working, whether \(C\) is nearer to \(B\) or to \(O\).
3. Find, showing all necessary working, the exact volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

10 A straight line has equation \(y = - 2 x + k\), where \(k\) is a constant, and a curve has equation \(y = \frac { 2 } { x - 3 }\).

1. Show that the \(x\)-coordinates of any points of intersection of the line and curve are given by the equation \(2 x ^ { 2 } - ( 6 + k ) x + ( 2 + 3 k ) = 0\).
2. Find the two values of \(k\) for which the line is a tangent to the curve. The two tangents, given by the values of \(k\) found in part (ii), touch the curve at points \(A\) and \(B\).
3. Find the coordinates of \(A\) and \(B\) and the equation of the line \(A B\).

9 The function f is defined for \(x > 0\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 2 x - \frac { 2 } { x ^ { 2 } }\). The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( 2,6 )\).

1. Find the equation of the normal to the curve at \(P\).
2. Find the equation of the curve.
3. Find the \(x\)-coordinate of the stationary point and state with a reason whether this point is a maximum or a minimum.

11 \includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-4_517_857_1594_644} The diagram shows part of the curve \(y = ( 1 + 4 x ) ^ { \frac { 1 } { 2 } }\) and a point \(P ( 6,5 )\) lying on the curve. The line \(P Q\) intersects the \(x\)-axis at \(Q ( 8,0 )\).

1. Show that \(P Q\) is a normal to the curve.
2. Find, showing all necessary working, the exact volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
   [0pt] [In part (ii) you may find it useful to apply the fact that the volume, \(V\), of a cone of base radius \(r\) and vertical height \(h\), is given by \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).]