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

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.

3 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + \frac { k } { x ^ { 3 } }\) and passes through the point \(P ( 1,9 )\). The gradient of the curve at \(P\) is 2 .

1. Find the value of the constant \(k\).
2. Find the equation of the curve.

5 A curve has equation \(y = 8 x + ( 2 x - 1 ) ^ { - 1 }\). Find the values of \(x\) at which the curve has a stationary point and determine the nature of each stationary point, justifying your answers.

7 The point \(P ( x , y )\) is moving along the curve \(y = x ^ { 2 } - \frac { 10 } { 3 } x ^ { \frac { 3 } { 2 } } + 5 x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).

6 The horizontal base of a solid prism is an equilateral triangle of side \(x \mathrm {~cm}\). The sides of the prism are vertical. The height of the prism is \(h \mathrm {~cm}\) and the volume of the prism is \(2000 \mathrm {~cm} ^ { 3 }\).

1. Express \(h\) in terms of \(x\) and show that the total surface area of the prism, \(A \mathrm {~cm} ^ { 2 }\), is given by $$A = \frac { \sqrt { } 3 } { 2 } x ^ { 2 } + \frac { 24000 } { \sqrt { } 3 } x ^ { - 1 }$$
2. Given that \(x\) can vary, find the value of \(x\) for which \(A\) has a stationary value.
3. Determine, showing all necessary working, the nature of this stationary value.

9 The equation of a curve is \(y = 8 \sqrt { } x - 2 x\).

1. Find the coordinates of the stationary point of the curve.
2. Find an expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence, or otherwise, determine the nature of the stationary point.
3. Find the values of \(x\) at which the line \(y = 6\) meets the curve.
4. State the set of values of \(k\) for which the line \(y = k\) does not meet the curve.

10 The curve with equation \(y = x ^ { 3 } - 2 x ^ { 2 } + 5 x\) passes through the origin.

1. Show that the curve has no stationary points.
2. Denoting the gradient of the curve by \(m\), find the stationary value of \(m\) and determine its nature.
3. Showing all necessary working, find the area of the region enclosed by the curve, the \(x\)-axis and the line \(x = 6\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

6 A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius is \(r \mathrm {~cm}\) and the internal height is \(h \mathrm {~cm}\). The volume of the flask is \(1000 \mathrm {~cm} ^ { 3 }\). A flask is most efficient when the total internal surface area, \(A \mathrm {~cm} ^ { 2 }\), is a minimum.

1. Show that \(A = 2 \pi r ^ { 2 } + \frac { 2000 } { r }\).
2. Given that \(r\) can vary, find the value of \(r\), correct to 1 decimal place, for which \(A\) has a stationary value and verify that the flask is most efficient when \(r\) takes this value.

10 \includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-5_650_1038_260_550} The diagram shows part of the curve \(y = \frac { 1 } { 16 } ( 3 x - 1 ) ^ { 2 }\), which touches the \(x\)-axis at the point \(P\). The point \(Q ( 3,4 )\) lies on the curve and the tangent to the curve at \(Q\) crosses the \(x\)-axis at \(R\).

1. State the \(x\)-coordinate of \(P\). Showing all necessary working, find by calculation
2. the \(x\)-coordinate of \(R\),
3. the area of the shaded region \(P Q R\).

7 The function f is defined for \(x \geqslant 0\) by \(\mathrm { f } ( x ) = ( 4 x + 1 ) ^ { \frac { 3 } { 2 } }\).

1. Find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
   The first, second and third terms of a geometric progression are respectively \(\mathrm { f } ( 2 ) , \mathrm { f } ^ { \prime } ( 2 )\) and \(k \mathrm { f } ^ { \prime \prime } ( 2 )\).
2. Find the value of the constant \(k\).

9 The point \(A ( 2,2 )\) lies on the curve \(y = x ^ { 2 } - 2 x + 2\).

1. Find the equation of the tangent to the curve at \(A\).
   The normal to the curve at \(A\) intersects the curve again at \(B\).
2. Find the coordinates of \(B\).
   The tangents at \(A\) and \(B\) intersect each other at \(C\).
3. Find the coordinates of \(C\).

8 A curve has equation \(y = x ^ { 3 } + 3 x ^ { 2 } - 9 x + k\), where \(k\) is a constant.

1. Write down an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find the \(x\)-coordinates of the two stationary points on the curve.
3. Hence find the two values of \(k\) for which the curve has a stationary point on the \(x\)-axis.

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

1. Write down expressions for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. Find the coordinates of the stationary point on the curve and determine its nature.
3. Find the volume of the solid formed when the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated completely about the \(x\)-axis.

8 A function f is defined by \(\mathrm { f } : x \mapsto ( 2 x - 3 ) ^ { 3 } - 8\), for \(2 \leqslant x \leqslant 4\).

1. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { \prime } ( x )\) and show that f is an increasing function.
2. Find an expression, in terms of \(x\), for \(\mathrm { f } ^ { - 1 } ( x )\) and find the domain of \(\mathrm { f } ^ { - 1 }\).

7 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_490_665_1793_740} The diagram shows the curve \(y = x ( x - 1 ) ( x - 2 )\), which crosses the \(x\)-axis at the points \(O ( 0,0 )\), \(A ( 1,0 )\) and \(B ( 2,0 )\).

1. The tangents to the curve at the points \(A\) and \(B\) meet at the point \(C\). Find the \(x\)-coordinate of \(C\).
2. Show by integration that the area of the shaded region \(R _ { 1 }\) is the same as the area of the shaded region \(R _ { 2 }\).

8 The equation of a curve is \(y = \frac { 6 } { 5 - 2 x }\).

1. Calculate the gradient of the curve at the point where \(x = 1\).
2. A point with coordinates \(( x , y )\) moves along the curve in such a way that the rate of increase of \(y\) has a constant value of 0.02 units per second. Find the rate of increase of \(x\) when \(x = 1\).
3. The region between the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume obtained is \(\frac { 12 } { 5 } \pi\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-3_301_485_264_829} A piece of wire of length 50 cm is bent to form the perimeter of a sector \(P O Q\) of a circle. The radius of the circle is \(r \mathrm {~cm}\) and the angle \(P O Q\) is \(\theta\) radians (see diagram).

1. Express \(\theta\) in terms of \(r\) and show that the area, \(A \mathrm {~cm} ^ { 2 }\), of the sector is given by $$A = 25 r - r ^ { 2 } .$$
2. Given that \(r\) can vary, find the stationary value of \(A\) and determine its nature.

11 The equation of a curve is \(y = \frac { 9 } { 2 - x }\).

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and determine, with a reason, whether the curve has any stationary points.
2. Find the volume obtained when the region bounded by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
3. Find the set of values of \(k\) for which the line \(y = x + k\) intersects the curve at two distinct points.

3 The length, \(x\) metres, of a Green Anaconda snake which is \(t\) years old is given approximately by the formula $$x = 0.7 \sqrt { } ( 2 t - 1 ) ,$$ where \(1 \leqslant t \leqslant 10\). Using this formula, find

1. \(\frac { \mathrm { d } x } { \mathrm {~d} t }\),
2. the rate of growth of a Green Anaconda snake which is 5 years old.

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

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 maximum point \(A\) and the minimum point \(B\) on the curve.

11 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-5_710_931_255_607} The diagram shows parts of the curves \(y = 9 - x ^ { 3 }\) and \(y = \frac { 8 } { x ^ { 3 } }\) and their points of intersection \(P\) and \(Q\). The \(x\)-coordinates of \(P\) and \(Q\) are \(a\) and \(b\) respectively.

1. Show that \(x = a\) and \(x = b\) are roots of the equation \(x ^ { 6 } - 9 x ^ { 3 } + 8 = 0\). Solve this equation and hence state the value of \(a\) and the value of \(b\).
2. Find the area of the shaded region between the two curves.
3. The tangents to the two curves at \(x = c\) (where \(a < c < b\) ) are parallel to each other. Find the value of \(c\).

2 A curve has equation \(y = 3 x ^ { 3 } - 6 x ^ { 2 } + 4 x + 2\). Show that the gradient of the curve is never negative.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 - \frac { 8 } { x ^ { 2 } }\). The line \(3 y + x = 17\) is the normal to the curve at the point \(P\) on the curve. Given that the \(x\)-coordinate of \(P\) is positive, find

1. the coordinates of \(P\),
2. the equation of the curve.

2 It is given that \(\mathrm { f } ( x ) = \frac { 1 } { x ^ { 3 } } - x ^ { 3 }\), for \(x > 0\). Show that f is a decreasing function.