1.07i Differentiate x^n: for rational n and sums

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.

11 \includegraphics[max width=\textwidth, alt={}, center]{d3c76ceb-cff7-4155-9697-5c302a9d63a9-4_526_974_822_587} The diagram shows the curve with equation \(y = x ( x - 2 ) ^ { 2 }\). The minimum point on the curve has coordinates \(( a , 0 )\) and the \(x\)-coordinate of the maximum point is \(b\), where \(a\) and \(b\) are constants.

1. State the value of \(a\).
2. Find the value of \(b\).
3. Find the area of the shaded region.
4. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of the curve has a minimum value \(m\). Find the value of \(m\).

10 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-4_654_974_614_587} The diagram shows the curve \(y = ( 3 - 2 x ) ^ { 3 }\) and the tangent to the curve at the point \(\left( \frac { 1 } { 2 } , 8 \right)\).

1. Find the equation of this tangent, giving your answer in the form \(y = m x + c\).
2. Find the area of the shaded region.

9 A curve has equation \(y = \frac { k ^ { 2 } } { x + 2 } + x\), where \(k\) is a positive constant. Find, in terms of \(k\), the values of \(x\) for which the curve has stationary points and determine the nature of each stationary point.

11 \includegraphics[max width=\textwidth, alt={}, center]{77543862-ed95-42bf-b788-a9a43f039a89-4_995_905_260_621} The diagram shows parts of the curves \(y = ( 4 x + 1 ) ^ { \frac { 1 } { 2 } }\) and \(y = \frac { 1 } { 2 } x ^ { 2 } + 1\) intersecting at points \(P ( 0,1 )\) and \(Q ( 2,3 )\). The angle between the tangents to the two curves at \(Q\) is \(\alpha\).

1. Find \(\alpha\), giving your answer in degrees correct to 3 significant figures.
2. Find by integration the area of the shaded region.

5 A curve has equation \(y = \frac { 8 } { x } + 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary points and state, with a reason, the nature of each stationary point.

3

1. Express \(3 x ^ { 2 } - 6 x + 2\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a , b\) and \(c\) are constants.
2. The function f , where \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 7 x - 8\), is defined for \(x \in \mathbb { R }\). Find \(\mathrm { f } ^ { \prime } ( x )\) and state, with a reason, whether f is an increasing function, a decreasing function or neither.

10 The function f is defined by \(\mathrm { f } ( x ) = 2 x + ( x + 1 ) ^ { - 2 }\) for \(x > - 1\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\) and hence verify that the function f has a minimum value at \(x = 0\). \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-4_515_920_959_609} The points \(A \left( - \frac { 1 } { 2 } , 3 \right)\) and \(B \left( 1,2 \frac { 1 } { 4 } \right)\) lie on the curve \(y = 2 x + ( x + 1 ) ^ { - 2 }\), as shown in the diagram.
2. Find the distance \(A B\).
3. Find, showing all necessary working, the area of the shaded region. {www.cie.org.uk} after the live examination series. }

7 The equation of a curve is \(y = 2 + \frac { 3 } { 2 x - 1 }\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Explain why the curve has no stationary points. At the point \(P\) on the curve, \(x = 2\).
3. Show that the normal to the curve at \(P\) passes through the origin.
4. A point moves along the curve in such a way that its \(x\)-coordinate is decreasing at a constant rate of 0.06 units per second. Find the rate of change of the \(y\)-coordinate as the point passes through \(P\).

1 A curve has equation \(y = 2 x ^ { \frac { 3 } { 2 } } - 3 x - 4 x ^ { \frac { 1 } { 2 } } + 4\). Find the equation of the tangent to the curve at the point \(( 4,0 )\).

2 A function f is defined by \(\mathrm { f } : x \mapsto x ^ { 3 } - x ^ { 2 } - 8 x + 5\) for \(x < a\). It is given that f is an increasing function. Find the largest possible value of the constant \(a\).

4 The function f is such that \(\mathrm { f } ( x ) = ( 2 x - 1 ) ^ { \frac { 3 } { 2 } } - 6 x\) for \(\frac { 1 } { 2 } < x < k\), where \(k\) is a constant. Find the largest value of \(k\) for which f is a decreasing function.

10 A curve has equation \(y = \mathrm { f } ( x )\) and it is given that \(\mathrm { f } ^ { \prime } ( x ) = a x ^ { 2 } + b x\), where \(a\) and \(b\) are positive constants.

1. Find, in terms of \(a\) and \(b\), the non-zero value of \(x\) for which the curve has a stationary point and determine, showing all necessary working, the nature of the stationary point.
2. It is now given that the curve has a stationary point at \(( - 2 , - 3 )\) and that the gradient of the curve at \(x = 1\) is 9 . Find \(\mathrm { f } ( x )\).

11 \includegraphics[max width=\textwidth, alt={}, center]{17ca6dd2-271b-4b06-8433-354493feaf06-18_428_857_260_644} The diagram shows the curve \(y = ( x - 1 ) ^ { \frac { 1 } { 2 } }\) and points \(A ( 1,0 )\) and \(B ( 5,2 )\) lying on the curve.

1. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
2. Find, showing all necessary working, the equation of the tangent to the curve which is parallel to \(A B\).
3. Find the perpendicular distance between the line \(A B\) and the tangent parallel to \(A B\). Give your answer correct to 2 decimal places.

6 A curve has a stationary point at \(\left( 3,9 \frac { 1 } { 2 } \right)\) and has an equation for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = a x ^ { 2 } + a ^ { 2 } x\), where \(a\) is a non-zero constant.

1. Find the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-08_67_1569_461_328}
2. Find the equation of the curve.
3. Determine, showing all necessary working, the nature of the stationary point.

10 A curve has equation \(y = \frac { 1 } { 2 } ( 4 x - 3 ) ^ { - 1 }\). The point \(A\) on the curve has coordinates \(\left( 1 , \frac { 1 } { 2 } \right)\).

1. (a) Find and simplify the equation of the normal through \(A\).
   (b) Find the \(x\)-coordinate of the point where this normal meets the curve again.
2. A point is moving along the curve in such a way that as it passes through \(A\) its \(x\)-coordinate is decreasing at the rate of 0.3 units per second. Find the rate of change of its \(y\)-coordinate at \(A\).

2 The function f is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 4 x + 7\) for \(x \geqslant - 2\). Determine, showing all necessary working, whether f is an increasing function, a decreasing function or neither.

10 \includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-16_648_823_262_660} The diagram shows part of the curve \(y = 2 ( 3 x - 1 ) ^ { - \frac { 1 } { 3 } }\) and the lines \(x = \frac { 2 } { 3 }\) and \(x = 3\). The curve and the line \(x = \frac { 2 } { 3 }\) intersect at the point \(A\).

1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. Find the equation of the normal to the curve at \(A\), giving your answer in the form \(y = m x + c\).

2 An increasing function, f , is defined for \(x > n\), where \(n\) is an integer. It is given that \(\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 6 x + 8\). Find the least possible value of \(n\).

3 The line \(y = a x + b\) is a tangent to the curve \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 3 x + c\) at the point \(( 2,6 )\). Find the values of the constants \(a , b\) and \(c\).

9 A curve for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 x - 1 ) ^ { \frac { 1 } { 2 } } - 2\) passes through the point ( 2,3 ).

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 on the curve and, showing all necessary working, determine the nature of this stationary point.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { \sqrt { } x }\), where \(k\) is a constant. The points \(P ( 1 , - 1 )\) and \(Q ( 4,4 )\) lie on the curve. Find the equation of the curve.

5 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-08_512_460_258_772} \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-08_462_85_260_1279} The diagram shows a solid cone which has a slant height of 15 cm and a vertical height of \(h \mathrm {~cm}\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the cone is given by \(V = \frac { 1 } { 3 } \pi \left( 225 h - h ^ { 3 } \right)\).
   [0pt] [The volume of a cone of radius \(r\) and vertical height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
2. Given that \(h\) can vary, find the value of \(h\) for which \(V\) has a stationary value. Determine, showing all necessary working, the nature of this stationary value.

10 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-18_979_679_262_731} The diagram shows part of the curve \(y = 1 - \frac { 4 } { ( 2 x + 1 ) ^ { 2 } }\). The curve intersects the \(x\)-axis at \(A\). The normal to the curve at \(A\) intersects the \(y\)-axis at \(B\).

1. Obtain expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\int y \mathrm {~d} x\).
2. Find the coordinates of \(B\).
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.

3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - 8 x + 7\). The curve has no stationary points in the interval \(a < x < b\). Find the least possible value of \(a\) and the greatest possible value of \(b\).