1.08b Integrate x^n: where n != -1 and sums

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6\) and the point \(( 9,2 )\) lies on the curve.

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point on the curve and determine the nature of the stationary point.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { ( 1 + 2 x ) ^ { 2 } }\) and the point \(\left( 1 , \frac { 1 } { 2 } \right)\) lies on the curve.

1. Find the equation of the curve.
2. Find the set of values of \(x\) for which the gradient of the curve is less than \(\frac { 1 } { 3 }\).

4

1. Differentiate \(\frac { 2 x ^ { 3 } + 5 } { x }\) with respect to \(x\).
2. Find \(\int ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\) and hence find the value of \(\int _ { 0 } ^ { 1 } ( 3 x - 2 ) ^ { 5 } \mathrm {~d} x\).

9 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { \sqrt { } x } - 1\) and \(P ( 9,5 )\) is a point on the curve.

1. Find the equation of the curve.
2. Find the coordinates of the stationary point on the curve.
3. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and determine the nature of the stationary point.
4. The normal to the curve at \(P\) makes an angle of \(\tan ^ { - 1 } k\) with the positive \(x\)-axis. Find the value of \(k\).

11 \includegraphics[max width=\textwidth, alt={}, center]{4d8fcc3d-a2da-4d98-8500-075d10847be3-4_636_951_255_596} The diagram shows the line \(y = 1\) and part of the curve \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\).

1. Show that the equation \(y = \frac { 2 } { \sqrt { } ( x + 1 ) }\) can be written in the form \(x = \frac { 4 } { y ^ { 2 } } - 1\).
2. Find \(\int \left( \frac { 4 } { y ^ { 2 } } - 1 \right) \mathrm { d } y\). Hence find the area of the shaded region.
3. The shaded region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis. Find the exact value of the volume of revolution obtained.

9 A curve has equation \(y = \mathrm { f } ( x )\) and is such that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } + 3 x ^ { - \frac { 1 } { 2 } } - 10\).

1. By using the substitution \(u = x ^ { \frac { 1 } { 2 } }\), or otherwise, find the values of \(x\) for which the curve \(y = \mathrm { f } ( x )\) has stationary points.
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence, or otherwise, determine the nature of each stationary point.
3. It is given that the curve \(y = \mathrm { f } ( x )\) passes through the point \(( 4 , - 7 )\). Find \(\mathrm { f } ( x )\).

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { x ^ { 2 } }\) and \(( 2,9 )\) is a point on the curve. Find the equation of the curve.

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sqrt { } ( 2 x + 5 )\) and \(( 2,5 )\) is a point on the curve. Find the equation of the curve.

12 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { \frac { 1 } { 2 } } - x ^ { - \frac { 1 } { 2 } }\). The curve passes through the point \(\left( 4 , \frac { 2 } { 3 } \right)\).

1. Find the equation of the curve.
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. Find the coordinates of the stationary point and determine its nature.

8 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1\). Given that the curve has a minimum point at \(( 3 , - 10 )\), find the coordinates of the maximum point.

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.

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { \sqrt { } ( 4 x + a ) }\), where \(a\) is a constant. The point \(P ( 2,14 )\) lies on the curve and the normal to the curve at \(P\) is \(3 y + x = 5\).

1. Show that \(a = 8\).
2. Find the equation of the curve.

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) ^ { \frac { 1 } { 2 } }\) and the point (4,7) lies on the curve. Find the equation of the curve.

10 \includegraphics[max width=\textwidth, alt={}, center]{616a6177-0d5c-49f7-b0c1-9138a13c1963-4_687_488_262_826} The diagram shows the part of the curve \(y = \frac { 8 } { x } + 2 x\) for \(x > 0\), and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\int y ^ { 2 } \mathrm {~d} x\).
2. Find the coordinates of \(M\) and determine the coordinates and nature of the stationary point on the part of the curve for which \(x < 0\).
3. Find the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

7 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 - x ^ { 2 } - 6 x\) passes through the point \(( 3 , - 10 )\).

1. Find the equation of the curve.
2. Express \(7 - x ^ { 2 } - 6 x\) in the form \(a - ( x + b ) ^ { 2 }\), where \(a\) and \(b\) are constants.
3. Find the set of values of \(x\) for which the gradient of the curve is positive.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 } { ( 2 x + 1 ) ^ { 2 } }\). The point \(( 1,1 )\) lies on the curve. Find the coordinates of the point at which the curve intersects the \(x\)-axis.

10 A curve for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 5\) has a stationary point at \(( 3,6 )\).

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the other stationary point on the curve.
3. Determine the nature of each of the stationary points. \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-20_700_616_262_762} The diagram shows part of the curve \(y = \frac { 3 } { \sqrt { ( 1 + 4 x ) } }\) and a point \(P ( 2,1 )\) lying on the curve. The normal to the curve at \(P\) intersects the \(x\)-axis at \(Q\).
4. Show that the \(x\)-coordinate of \(Q\) is \(\frac { 16 } { 9 }\).
5. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - \frac { 4 } { x ^ { 2 } }\). The point \(P ( 2,9 )\) lies on the curve.

1. A point moves on the curve in such a way that the \(x\)-coordinate is decreasing at a constant rate of 0.05 units per second. Find the rate of change of the \(y\)-coordinate when the point is at \(P\). [2]
2. Find the equation of the curve.

11 \includegraphics[max width=\textwidth, alt={}, center]{ed5b77ae-6eac-4e73-bc43-613433abd3e1-16_723_942_260_598} The diagram shows part of the curve \(y = \sqrt { } ( 4 x + 1 ) + \frac { 9 } { \sqrt { } ( 4 x + 1 ) }\) and the minimum point \(M\).

1. Find expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(M\).
   The shaded region is bounded by the curve, the \(y\)-axis and the line through \(M\) parallel to the \(x\)-axis.
3. Find, showing all necessary working, the area of the shaded region.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 } { \sqrt { } ( 4 x - 3 ) }\) and \(P ( 3,3 )\) is a point on the curve.

1. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\).
2. Find the equation of the curve.

10 Functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 2 x + 1 , \quad x \in \mathbb { R } , \quad x > 0 \\ & \mathrm {~g} : x \mapsto \frac { 2 x - 1 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$

1. Solve the equation \(\operatorname { gf } ( x ) = x\).
2. Express \(\mathrm { f } ^ { - 1 } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) in terms of \(x\).
3. Show that the equation \(\mathrm { g } ^ { - 1 } ( x ) = x\) has no solutions.
4. Sketch in a single diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\), making clear the relationship between the graphs.

10

1. The diagram shows the line \(2 y = x + 5\) and the curve \(y = x ^ { 2 } - 4 x + 7\), which intersect at the points \(A\) and \(B\). Find
   1. the \(x\)-coordinates of \(A\) and \(B\),
   2. the equation of the tangent to the curve at \(B\),
   3. the acute angle, in degrees correct to 1 decimal place, between this tangent and the line \(2 y = x + 5\).
   4. Determine the set of values of \(k\) for which the line \(2 y = x + k\) does not intersect the curve \(y = x ^ { 2 } - 4 x + 7\).

2 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { x ^ { 3 } } - 1\) and the point \(( 2,4 )\) lies on the curve. Find the equation of the curve.

10 A curve is defined for \(x > 0\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x + \frac { 4 } { x ^ { 2 } }\). The point \(P ( 4,8 )\) lies on the curve.

1. Find the equation of the curve.
2. Show that the gradient of the curve has a minimum value when \(x = 2\) and state this minimum value.

8 A curve is such that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ( 3 x + 4 ) ^ { \frac { 3 } { 2 } } - 6 x - 8 .$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Verify that the curve has a stationary point when \(x = - 1\) and determine its nature.
3. It is now given that the stationary point on the curve has coordinates \(( - 1,5 )\). Find the equation of the curve.