Find derivative of polynomial

10. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\).
The curve meets the \(x\)-axis at the origin and at the point \(A\).
Given that $$f ^ { \prime } ( x ) = 3 x ^ { \frac { 1 } { 2 } } - 4 x ^ { - \frac { 1 } { 2 } } ,$$

1. find \(\mathrm { f } ( x )\),
2. find the coordinates of \(A\). The point \(B\) on the curve has \(x\)-coordinate 2 .
3. Find an equation for the tangent to the curve at \(B\) in the form \(y = m x + c\).

9 The diagram shows part of a curve crossing the \(x\)-axis at the origin \(O\) and at the point \(A ( 8,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{02e5dfac-18d7-480d-ac23-dfd2ca348cba-5_547_536_497_760} The curve has equation $$y = 12 x - 3 x ^ { \frac { 5 } { 3 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 8,0 )\) is \(y + 8 x = 64\).
3. Find \(\int \left( 12 x - 3 x ^ { \frac { 5 } { 3 } } \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve from \(O\) to \(A\) and the tangents \(O P\) and \(A P\).

7
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\hline \end{tabular} \end{center} 1. $$f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } - 2 x - 6$$ Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ).
2. The point \(A\) has coordinates \(( 2,5 )\) and the point \(B\) has coordinates \(( - 2,8 )\). Find, in cartesian form, an equation of the circle with diameter \(A B\).
3. $$f ( x ) = x ^ { 3 } - 19 x - 30$$

1. Show that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\).
2. Factorise \(\mathrm { f } ( x )\) completely.
   4. Express \(\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }\) as a single fraction in its simplest form.
   5. Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360 ^ { \circ }\) for which $$2 \cos ^ { 2 } \theta - \cos \theta - 1 = \sin ^ { 2 } \theta$$ Give your answers to 1 decimal place where appropriate.
   6. A geometric series is \(a + a r + a r ^ { 2 } + \ldots\)
3. Prove that the sum of the first \(n\) terms of this series is given by $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ The second and fourth terms of the series are 3 and 1.08 respectively.
   Given that all terms in the series are positive, find
4. the value of \(r\) and the value of \(a\),
5. the sum to infinity of the series.
   7. . \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-3_1141_1297_280_360}
   \end{figure} A rectangular sheet of metal measures 50 cm by 40 cm . Squares of side \(x \mathrm {~cm}\) are cut from each corner of the sheet and the remainder is folded along the dotted lines to make an open tray, as shown in Fig. 2.
6. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the tray is given by $$V = 4 x \left( x ^ { 2 } - 45 x + 500 \right) .$$
7. State the range of possible values of \(x\).
8. Find the value of \(x\) for which \(V\) is a maximum.
9. Hence find the maximum value of \(V\).
10. Justify that the value of \(V\) you found in part (d) is a maximum. \section*{8.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-4_556_554_276_840}
    \end{figure} Figure 1 shows the sector \(A O B\) of a circle, with centre \(O\) and radius 6.5 cm , and \(\angle A O B = 0.8\) radians.
11. Calculate, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(A O B\).
12. Show that the length of the chord \(A B\) is 5.06 cm , to 3 significant figures. The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
13. Calculate, in cm , the perimeter of \(R\). \section*{9.} \section*{Figure 2}
    \includegraphics[max width=\textwidth, alt={}]{8d60bedd-6496-4fc9-abc8-16fc5ac52e01-5_529_1205_324_269}
    Figure 2 shows the line with equation \(y = x + 1\) and the curve with equation \(y = 6 x - x ^ { 2 } - 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.
14. Calculate the coordinates of \(A\) and the coordinates of \(B\). The shaded region \(R\) is bounded by the line and the curve.
15. Calculate the area of \(R\).

5. (i) Differentiate \(2 x ^ { 3 } + \sqrt { } x + \frac { x ^ { 2 } + 2 x } { x ^ { 2 } }\) with respect to \(x\)
(ii) Evaluate \(\int _ { 1 } ^ { 4 } \left( \frac { x } { 2 } + \frac { 1 } { x ^ { 2 } } \right) \mathrm { d } x\).

9. For the curve \(C\) with equation \(y = x ^ { 4 } - 8 x ^ { 2 } + 3\),

1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
2. find the coordinates of each of the stationary points,
3. determine the nature of each stationary point. The point \(A\), on the curve \(C\), has \(x\)-coordinate 1 .
4. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

1. A curve has equation $$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.

7. A curve has equation \(y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }\).
i. Find \(\frac { d y } { d x }\).
ii. Hence sketch the gradient function for the curve.
iii. Find the equation of the tangent to the curve \(y = \frac { 1 } { 4 } x ^ { 4 } - x ^ { 3 } - 2 x ^ { 2 }\) at \(x = 4\).

13. Figure 1
\includegraphics[max width=\textwidth, alt={}, center]{d0c23635-3b9b-4666-9cb4-21b931fb3719-06_626_759_313_537} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 10 + \ln ( 3 x ) - \frac { 1 } { 2 } \mathrm { e } ^ { x } , 0.1 \leq x \leq 3.3$$ Given that \(\mathrm { f } ( k ) = 0\),

1. show, by calculation, that \(3.1 < k < 3.2\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\). The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).
3. 1. Find an equation of this tangent.
   2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\).

9 A curve has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ( 6 - x )$$ 9

1. \(\quad\) Find \(\mathrm { f } ^ { \prime } ( x )\)
   9
2. The diagram below shows the graph of \(y = \mathrm { f } ( x )\) On the same diagram sketch the gradient function for this curve, stating the coordinates of any points where the gradient function cuts the axes.
   \includegraphics[max width=\textwidth, alt={}, center]{f4f303a2-f029-42be-93a0-046b0c81e3c0-11_922_1198_1475_406} It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 2 ) ( 2 x - 1 ) ^ { 2 }$$ and when \(x = 6 , y = 900\)
   Find \(y\) in terms of \(x\)

2 Given that \(y = 2 x ^ { 3 }\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer.
[0pt] [1 mark]
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { 2 }\)
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 }\)
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 4 } } { 2 }\)
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 3 }\)

2 A curve has equation \(y = x ^ { 5 } + 4 x ^ { 3 } + 7 x + q\) where \(q\) is a positive constant.
Find the gradient of the curve at the point where \(x = 0\)
Circle your answer.
0
4
7
\(q\)

2 Simplify fully $$\frac { ( x + 3 ) ( 6 - 2 x ) } { ( x - 3 ) ( 3 + x ) } \quad \text { for } x \neq \pm 3$$ Circle your answer.
-2
2
\(\frac { ( 6 - 2 x ) } { ( x - 3 ) }\)
\(\frac { ( 2 x - 6 ) } { ( x - 3 ) }\)
\(3 \mathrm { f } ( x ) = 3 x ^ { 2 }\)
Obtain \(\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\)
Circle your answer. $$\frac { 3 h ^ { 2 } } { h } \quad x ^ { 3 } \quad \frac { 3 ( x + h ) ^ { 2 } - 3 x ^ { 2 } } { h } \quad 6 x$$