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

9. $$f ^ { \prime } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) ^ { 2 } } { x ^ { 2 } } , \quad x \neq 0$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 9 x ^ { - 2 } + A + B x ^ { 2 }\),
   where \(A\) and \(B\) are constants to be found.
2. Find \(\mathrm { f } ^ { \prime \prime } ( x )\). Given that the point \(( - 3,10 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\),
3. find \(\mathrm { f } ( x )\).

4. Given that \(y = 2 x ^ { 5 } + \frac { 6 } { \sqrt { } x } , x > 0\), find in their simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

11. \begin{figure}[h] \end{figure} A sketch of part of the curve \(C\) with equation $$y = 20 - 4 x - \frac { 18 } { x } , \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has an \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = - 2 x + 7\) The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3 .
2. Use algebra to find the coordinates of \(B\).

Given that \(y = 4 x ^ { 3 } - \frac { 5 } { x ^ { 2 } } , x \neq 0\), find in their simplest form

1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. \(\int y \mathrm {~d} x\)

1. The curve \(C\) has equation

$$y = \frac { \left( x ^ { 2 } + 4 \right) ( x - 3 ) } { 2 x } , \quad x \neq 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form.
2. Find an equation of the tangent to \(C\) at the point where \(x = - 1\) Give your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

1. Given that

$$y = 3 x ^ { 2 } + 6 x ^ { \frac { 1 } { 3 } } + \frac { 2 x ^ { 3 } - 7 } { 3 \sqrt { } x } , \quad x > 0$$ find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Give each term in your answer in its simplified form.

11. The curve \(C\) has equation \(y = 2 x ^ { 3 } + k x ^ { 2 } + 5 x + 6\), where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\), where \(x = - 2\), lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the line with equation \(2 y - 17 x - 1 = 0\) Find
2. the value of \(k\),
3. the value of the \(y\) coordinate of \(P\),
4. the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2. Given $$y = \sqrt { x } + \frac { 4 } { \sqrt { x } } + 4 , \quad x > 0$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\), writing your answer in the form \(a \sqrt { 2 }\), where \(a\) is a rational number.
(5)

1. Given

$$y = 3 \sqrt { x } - 6 x + 4 , \quad x > 0$$

1. find \(\int y \mathrm {~d} x\), simplifying each term.
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. Hence find the value of \(x\) such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where

$$f ^ { \prime } ( x ) = ( x - 3 ) ( 3 x + 5 )$$ Given that the point \(P ( 1,20 )\) lies on \(C\),

1. find \(\mathrm { f } ( x )\), simplifying each term.
2. Show that $$f ( x ) = ( x - 3 ) ^ { 2 } ( x + A )$$ where \(A\) is a constant to be found.
3. Sketch the graph of \(C\). Show clearly the coordinates of the points where \(C\) cuts or meets the \(x\)-axis and where \(C\) cuts the \(y\)-axis.
   

6. $$f ( x ) = \frac { ( 2 x + 1 ) ( x + 4 ) } { \sqrt { x } } , \quad x > 0$$

1. Show that \(\mathrm { f } ( x )\) can be written in the form \(P x ^ { \frac { 3 } { 2 } } + Q x ^ { \frac { 1 } { 2 } } + R x ^ { - \frac { 1 } { 2 } }\), stating the values of the constants \(P , Q\) and \(R\).
2. Find f \({ } ^ { \prime } ( x )\).
3. Show that the tangent to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 1\) is parallel to the line with equation \(2 y = 11 x + 3\).
   (3)
   

   6. continued

   Leave blank

10. A curve \(C\) has equation $$y = 4 x ^ { 3 } - 9 x + \frac { k } { x } \quad x > 0$$ where \(k\) is a constant.
The point \(P\) with \(x\) coordinate \(\frac { 1 } { 2 }\) lies on \(C\).
Given that \(P\) is a stationary point of \(C\),

1. show that \(k = - \frac { 3 } { 2 }\)
2. Determine the nature of the stationary point at \(P\), justifying your answer. The curve \(C\) has a second stationary point.
3. Using algebra, find the \(x\) coordinate of this second stationary point. \includegraphics[max width=\textwidth, alt={}, center]{08aac50c-7317-4510-927a-7f5f2e00f485-26_2255_50_312_1980}

2. A curve has equation $$y = x ^ { 3 } - x ^ { 2 } - 16 x + 2$$

1. Using calculus, find the \(x\) coordinates of the stationary points of the curve.
2. Justify, by further calculus, the nature of all of the stationary points of the curve.

2. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 27 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } - 20 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving each term in simplest form.
2. Hence find the coordinates of the stationary point of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and hence determine the nature of the stationary point of \(C\).

1. In this question you must show all stages of your working.

\section*{Solutions based entirely on calculator technology are not acceptable.} \begin{figure}[h] \end{figure} A brick is in the shape of a cuboid with width \(x \mathrm {~cm}\) ,length \(3 x \mathrm {~cm}\) and height \(h \mathrm {~cm}\) ,as shown in Figure 2. The volume of the brick is \(972 \mathrm {~cm} ^ { 3 }\)

1. Show that the surface area of the brick,\(S \mathrm {~cm} ^ { 2 }\) ,is given by $$S = 6 x ^ { 2 } + \frac { 2592 } { x }$$
2. Find \(\frac { \mathrm { d } S } { \mathrm {~d} x }\)
3. Hence find the value of \(x\) for which \(S\) is stationary.
4. Find \(\frac { \mathrm { d } ^ { 2 } S } { \mathrm {~d} x ^ { 2 } }\) and hence show that the value of \(x\) found in part(c)gives the minimum value of \(S\) .
5. Hence find the minimum surface area of the brick.

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point \(P\) is the only stationary point on the curve.

1. Use calculus to show that the \(x\) coordinate of \(P\) is 9 The line \(l\) passes through the point \(P\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(l\) and the line with equation \(x = 4\)
2. Use algebraic integration to find the exact area of \(R\).

10. The curve \(C\) has equation $$y = a x ^ { 3 } - 3 x ^ { 2 } + 3 x + b$$ where \(a\) and \(b\) are constants. Given that

• the point \(( 2,5 )\) lies on \(C\)
• the gradient of the curve at \(( 2,5 )\) is 7
  1. find the value of \(a\) and the value of \(b\).
  2. Prove that \(C\) has no turning points.
     

3. $$f ( x ) = a x ^ { 3 } - x ^ { 2 } + b x + 4$$ where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by ( \(x + 4\) ), the remainder is - 108

1. Use the remainder theorem to show that $$16 a + b = 24$$ Given also that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(a\) and the value of \(b\).
3. Find \(\mathrm { f } ^ { \prime } ( x )\).
4. Hence find the exact coordinates of the stationary points of the curve with equation \(y = \mathrm { f } ( x )\).

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8. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$ The point \(M\) is the maximum turning point of \(C\) and is shown in Figure 2.
Given that the \(x\) coordinate of \(M\) is 2

1. show that \(k = 28\)
2. Determine the range of values of \(x\) for which \(y\) is increasing. The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
   The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis.
3. Find, by algebraic integration, the exact area of \(R\).

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where

$$f ( x ) = ( 2 - k x ) ^ { 5 }$$ and \(k\) is a constant.
Given that when \(\mathrm { f } ( x )\) is divided by \(( 4 x - 5 )\) the remainder is \(\frac { 243 } { 32 }\)

1. show that \(k = \frac { 2 } { 5 }\)
2. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 2 - \frac { 2 } { 5 } x \right) ^ { 5 }$$ giving each term in simplest form. Using the solution to part (b) and making your method clear,
3. find the gradient of \(C\) at the point where \(x = 0\)

1. The curve \(C\) has equation

$$y = \frac { 12 x ^ { 3 } ( x - 7 ) + 14 x ( 13 x - 15 ) } { 21 \sqrt { x } } \quad x > 0$$

1. Write the equation of \(C\) in the form $$y = a x ^ { \frac { 7 } { 2 } } + b x ^ { \frac { 5 } { 2 } } + c x ^ { \frac { 3 } { 2 } } + d x ^ { \frac { 1 } { 2 } }$$ where \(a , b , c\) and \(d\) are fully simplified constants. The curve \(C\) has three turning points.
   Using calculus,
2. show that the \(x\) coordinates of the three turning points satisfy the equation $$2 x ^ { 3 } - 10 x ^ { 2 } + 13 x - 5 = 0$$ Given that the \(x\) coordinate of one of the turning points is 1
3. find, using algebra, the exact \(x\) coordinates of the other two turning points.
   (Solutions based entirely on calculator technology are not acceptable.)

9. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 2 } { 3 } x ^ { 2 } - 9 \sqrt { x } + 13 \quad x \geqslant 0$$

1. Find, using calculus, the range of values of \(x\) for which \(y\) is increasing. The point \(P\) lies on \(C\) and has coordinates (9, 40).
   The line \(l\) is the tangent to \(C\) at the point \(P\).
   The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(l\), the \(x\)-axis and the \(y\)-axis.
2. Find, using calculus, the exact area of \(R\).

7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Using the result from part (a), find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence, or otherwise, determine the nature of the turning points of \(C\).

1. $$f ( x ) = x ^ { 3 } + 3 x ^ { 2 } + 5$$ Find

1. \(\mathrm { f } ^ { \prime \prime } ( x )\),
2. \(\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\).

9. The curve \(C\) has equation \(y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0\)

1. Use calculus to find the coordinates of the turning point on \(C\).
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
3. State the nature of the turning point.