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

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

The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• the point \(P ( 2,8 \sqrt { 2 } )\) lies on \(C\)
• \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + \frac { k } { x ^ { 2 } }\) where \(k\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) at \(P\)
  1. find the exact value of \(k\),
  2. find \(\mathrm { f } ( x )\), giving your answer in simplest form.

1. The curve \(C\) with equation \(y = \mathrm { f } ( x ) , \quad x > 0\), passes through the point \(P ( 4,1 )\).

Given that \(\mathrm { f } ^ { \prime } ( x ) = 4 \sqrt { x } - 2 - \frac { 8 } { 3 x ^ { 2 } }\)

1. find the equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
   (4)
2. Find \(\mathrm { f } ( x )\).
   (5)
   \href{http://www.dynamicpapers.com}{www.dynamicpapers.com}

1. The curve \(C\) has equation

$$y = \frac { x ^ { 2 } } { 3 } + \frac { 4 } { \sqrt { x } } + \frac { 8 } { 3 x } - 5 \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in simplest form. The point \(P ( 4,3 )\) lies on \(C\).
2. Find the equation of the normal to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

6. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(C\) passes through the point \(P ( 8,2 )\)
• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 32 } { 3 x ^ { 2 } } + 3 - 2 ( \sqrt [ 3 ] { x } )\)
  1. find the equation of the tangent to \(C\) at \(P\). Write your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
     (3)
  2. Find, in simplest form, \(\mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{877d03f2-d62c-4060-bdd2-f0d5dfbe6203-21_2647_1840_118_111}

7. The curve \(C\) has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { \sqrt { x } } + \frac { A } { x ^ { 2 } } + 3\), where \(A\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 4\)
  1. find the value of \(A\).

Given also that

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C\) with equation $$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
2. show that the \(x\) coordinate of the point \(B\) is a solution of the equation $$12 x ^ { 2 } + 4 x - 33 = 0$$
3. Hence find the \(x\) coordinate of \(B\), justifying your answer. Given that the \(y\) intercept of \(l\) is - 1
4. find the value of \(k\).
   \includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}

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

Solutions relying entirely on calculator technology are not acceptable.

1. Find the equation of the tangent to the curve with equation $$y = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$ at the point \(P ( 4,12 )\) Give your answer in the form \(a x + b y + c = 0\) where \(a\), \(b\) and \(c\) are integers. The curve with equation \(y = \mathrm { f } ( x )\) also passes through the point \(P ( 4,12 )\) Given that $$f ^ { \prime } ( x ) = \frac { 1 } { 4 } x ^ { 3 } - 8 x ^ { - \frac { 1 } { 2 } }$$
2. find \(\mathrm { f } ( x )\) giving the coefficients in simplest form.

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

$$f ( x ) = 2 x ^ { 3 } - k x ^ { 2 } + 14 x + 24$$ and \(k\) is a constant.

1. Find, in simplest form,
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { f } ^ { \prime \prime } ( x )\) The curve with equation \(y = \mathrm { f } ^ { \prime } ( x )\) intersects the curve with equation \(y = \mathrm { f } ^ { \prime \prime } ( x )\) at the points \(A\) and \(B\). Given that the \(x\) coordinate of \(A\) is 5
2. find the value of \(k\).
3. Hence find the coordinates of \(B\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x > 0\)

Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 6 x - \frac { ( 2 x - 1 ) ( 3 x + 2 ) } { 2 \sqrt { x } }\)
• the point \(P ( 4,12 )\) lies on \(C\)
  1. find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found,
  2. find \(\mathrm { f } ( x )\), giving each term in simplest form.

5. A curve has equation $$y = \frac { x ^ { 3 } } { 6 } + 4 \sqrt { x } - 15 \quad x \geqslant 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving the answer in simplest form. The point \(P \left( 4 , \frac { 11 } { 3 } \right)\) lies on the curve.
2. Find the equation of the normal to the curve at \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.

   VIIIV SIHI NI III M I I N OC

   VIIV SIHI NI IM IMM ION OC

   VI4V SIHI NI JIIYM IONOO

10. \begin{figure}[h] \end{figure} Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$

1. Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\) The curve crosses the \(y\)-axis at the point \(P\), as shown.
2. Expand \(\mathrm { f } ( x )\) to the form $$a x ^ { 3 } + b x ^ { 2 } + c x + d$$ where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
3. Hence, or otherwise, find
   1. the coordinates of \(P\),
   2. the gradient of the curve at \(P\). The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
4. 1. Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
   2. Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).

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

The point \(P \left( 4 , \frac { 32 } { 3 } \right)\) lies on the curve.
Given that

• \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 4 } { \sqrt { x } } - 3\)
• \(\quad \mathrm { f } ^ { \prime } ( x ) = 5\) at \(P\) find
  1. the equation of the tangent to the curve at \(P\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found,
  2. \(\mathrm { f } ( x )\).
     

8. The curve \(C\) has equation $$y = ( x - 2 ) ( x - 4 ) ^ { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } - 20 x + 32$$ The line \(l _ { 1 }\) is the tangent to \(C\) at the point where \(x = 6\)
2. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The line \(l _ { 2 }\) is the tangent to \(C\) at the point where \(x = \alpha\) Given that \(l _ { 1 }\) and \(l _ { 2 }\) are parallel and distinct,
3. find the value of \(\alpha\)

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).

5. The line \(l _ { 1 }\) has equation \(3 y - 2 x = 30\) The line \(l _ { 2 }\) passes through the point \(A ( 24,0 )\) and is perpendicular to \(l _ { 1 }\) Lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. Find, using algebra and showing your working, the coordinates of \(P\). Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(B\),
2. find the area of triangle \(B P A\).

7. \begin{figure}[h] \end{figure} Figure 3 shows the design for a sign at a bird sanctuary.
The design consists of a kite \(O A B C\) joined to a sector \(O C X A\) of a circle centre \(O\).
In the design

• \(O A = O C = 0.6 \mathrm {~m}\)
• \(A B = C B = 1.4 \mathrm {~m}\)
• Angle \(O A B =\) Angle \(O C B = 2\) radians
• Angle \(A O C = \theta\) radians, as shown in Figure 3

Making your method clear,

1. show that \(\theta = 1.64\) radians to 3 significant figures,
2. find the perimeter of the sign, in metres to 2 significant figures,
3. find the area of the sign, in \(\mathrm { m } ^ { 2 }\) to 2 significant figures.

10. A curve has equation \(y = \mathrm { f } ( x ) , x > 0\) Given that

• \(\mathrm { f } ^ { \prime } ( x ) = a x - 12 x ^ { \frac { 1 } { 3 } }\), where \(a\) is a constant
• \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\) when \(x = 27\)
• the curve passes through the point \(( 1 , - 8 )\)
  1. find the value of \(a\).
  2. Hence find \(\mathrm { f } ( x )\).
     

8. \section*{Diagram NOT to scale} \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a design for a pond.
The design consists of a sector \(A O B X\) of a circle centre \(O\) joined to a quadrilateral \(A O B C\).

• \(B C = 8 \mathrm {~m}\)
• \(O A = O B = 3 \mathrm {~m}\)
• angle \(A O B\) is \(\frac { 2 \pi } { 3 }\) radians
• angle \(B C A\) is \(\frac { \pi } { 6 }\) radians
  1. Calculate (i) the exact area of the sector \(A O B X\),
     (ii) the exact perimeter of the sector \(A O B X\).
  2. Calculate the exact area of the triangle \(A O B\).
  3. Show that the length \(A B\) is \(3 \sqrt { 3 } \mathrm {~m}\).
  4. Find the total surface area of the pond. Give your answer in \(\mathrm { m } ^ { 2 }\) correct to 2 significant figures.

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } x ^ { 2 } - 10 x + 22$$

1. Write \(\frac { 1 } { 2 } x ^ { 2 } - 10 x + 22\) in the form $$a ( x + b ) ^ { 2 } + c$$ where \(a , b\) and \(c\) are constants to be found. The point \(M\) is the minimum turning point of \(C\), as shown in Figure 3.
2. Deduce the coordinates of \(M\) The line \(l\) is the normal to \(C\) at the point \(P\), as shown in Figure 3.
   Given that \(l\) has equation \(y = k - \frac { 1 } { 8 } x\), where \(k\) is a constant,
3. 1. find the coordinates of \(P\)
   2. find the value of \(k\) Question 9 continues on the next page \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db979349-3415-420f-a39f-8cc8c24a69d0-25_903_682_299_605} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 is a copy of Figure 3. The finite region \(R\), shown shaded in Figure 4, is bounded by \(l , C\) and the line through \(M\) parallel to the \(y\)-axis.
4. Identify the inequalities that define \(R\).

2. $$\mathrm { f } ( x ) = \frac { 8 } { x ^ { 2 } } - 4 \sqrt { x } + 3 x - 1 , \quad x > 0$$ Giving your answers in their simplest form, find

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

13. The curve \(C\) has equation $$y = \frac { ( x - 3 ) ( 3 x - 25 ) } { x } , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in a fully simplified form.
2. Hence find the coordinates of the turning point on the curve \(C\).
3. Determine whether this turning point is a minimum or maximum, justifying your answer. The point \(P\), with \(x\) coordinate \(2 \frac { 1 } { 2 }\), lies on the curve \(C\).
4. Find the equation of the normal at \(P\), in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{e878227b-d625-4ef2-ac49-a9dc05c5321a-35_90_72_2631_1873}

15. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$ where \(k\) is a constant.

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) The point \(P\) on the curve \(C\) is a minimum turning point.
   Given that the \(x\) coordinate of \(P\) is 4
2. show that \(k = - 78\) The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\).
   The finite region \(R\), shown shaded in Figure 5, is bounded by \(C\), the \(y\)-axis and \(P N\).
3. Use integration to find the area of \(R\).

10. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the coordinates of the stationary point on \(C\).
3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.