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

6. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { 2 x ^ { 2 } - x } { \sqrt { x } } - 2 \ln \left( \frac { x } { 2 } \right) , \quad x > 0$$ The curve has a minimum turning point at \(Q\), as shown in Figure 4.
a. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 6 x ^ { 2 } - x - 4 \sqrt { x } } { 2 x \sqrt { x } }\) b. Show that the \(x\)-coordinate of \(Q\) is the solution of $$x = \sqrt { \frac { x } { 6 } + \frac { 2 \sqrt { x } } { 3 } }$$ To find an approximation for the \(x\)-coordinate of \(Q\), the iteration formula $$x _ { n + 1 } = \sqrt { \frac { x _ { n } } { 6 } + \frac { 2 \sqrt { x _ { n } } } { 3 } }$$ is used.
c. Taking \(x _ { 0 } = 0.8\), find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give your answers to 3 decimal places.

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$f ( x ) = 4 \cos 2 x - 2 x + 1 \quad x > 0$$ and where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(A\), as shown in figure 1 .
Given that \(x\)-coordinate of \(A\) is \(\alpha\) a. show that \(\alpha\) lies between 0.7 and 0.8 Given that \(x\)-coordinates of \(B\) and \(C\) are \(\beta\) and \(\gamma\) respectively and they are two smallest values of \(x\) at which local maxima occur
b. find, using calculus, the value of \(\beta\) and the value of \(\gamma\), giving your answers to 3 significant figures.
c. taking \(x _ { 0 } = 0.7\) or 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show, your method and give your answer to 2 significant figures.

8. \begin{figure}[h] \end{figure} (5)

12. A curve has equation \(y = \frac { 2 x e ^ { x } } { x + k }\) where \(k\) is a positive constant.
i. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { e ^ { x } \left( 2 x ^ { 2 } + 2 k x + 2 k \right) } { ( x + k ) ^ { 2 } }\) ii. Given that the curve has exactly one stationary point find the value of \(k\).

1. A curve \(C\) has equation

$$y = x ^ { 2 } - 2 x - 24 \sqrt { x } , \quad x > 0$$

1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Verify that \(C\) has a stationary point when \(x = 4\)
3. Determine the nature of this stationary point, giving a reason for your answer.

10. \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 with equation $$y = 8 x - x ^ { \frac { 5 } { 2 } } \quad x \geqslant 0$$ The curve crosses the \(x\)-axis at the point \(A\).

1. Verify that the \(x\) coordinate of \(A\) is 4 The line \(l _ { 1 }\) is the tangent to the curve at \(A\).
2. Use calculus to show that an equation of line \(l _ { 1 }\) is $$12 x + y = 48$$ The line \(l _ { 2 }\) has equation \(y = 8 x\) The region \(R\), shown shaded in Figure 3, is bounded by the curve, the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Use algebraic integration to find the exact area of \(R\).

1. (i) Use proof by exhaustion to show that for \(n \in \mathbb { N } , n \leqslant 4\)

$$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that \(m ^ { 3 } + 5\) is odd, use proof by contradiction to show, using algebra, that \(m\) is even.

11. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$

1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \quad x > 3$$
2. Prove that \(\mathrm { f } ( x )\) is a decreasing function.

11. \begin{figure}[h] \end{figure} Figure 8 shows a sketch of the curve \(C\) with equation \(y = x ^ { x } , x > 0\)

1. Find, by firstly taking logarithms, the \(x\) coordinate of the turning point of \(C\).
   (Solutions based entirely on graphical or numerical methods are not acceptable.) The point \(P ( \alpha , 2 )\) lies on \(C\).
2. Show that \(1.5 < \alpha < 1.6\) A possible iteration formula that could be used in an attempt to find \(\alpha\) is $$x _ { n + 1 } = 2 x _ { n } ^ { 1 - x _ { n } }$$ Using this formula with \(x _ { 1 } = 1.5\)
3. find \(x _ { 4 }\) to 3 decimal places,
4. describe the long-term behaviour of \(x _ { n }\)

1. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 8 x + 5$$

1. Find \(f ^ { \prime \prime } ( x )\)
2. 1. Solve \(\mathrm { f } ^ { \prime \prime } ( x ) = 0\)
   2. Hence find the range of values of \(x\) for which \(\mathrm { f } ( x )\) is concave.

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

The curve

• passes through the point \(P ( 3 , - 10 )\)
• has a turning point at \(P\)

Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { 3 } - 9 x ^ { 2 } + 5 x + k$$ where \(k\) is a constant,

1. show that \(k = 12\)
2. Hence find the coordinates of the point where \(C\) crosses the \(y\)-axis.

1. $$y = 4 x ^ { 3 } - 7 x ^ { 2 } + 5 x - 10$$

1. Find in simplest form
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Hence find the exact value of \(x\) when \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\)

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 4 x ^ { 2 } + x } { 2 \sqrt { x } } - 4 \ln x \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 x ^ { 2 } + x - 16 \sqrt { x } } { 4 x \sqrt { x } }$$ The point \(P\), shown in Figure 1, is the minimum turning point on \(C\).
2. Show that the \(x\) coordinate of \(P\) is a solution of $$x = \left( \frac { 4 } { 3 } - \frac { \sqrt { x } } { 12 } \right) ^ { \frac { 2 } { 3 } }$$
3. Use the iteration formula $$x _ { n + 1 } = \left( \frac { 4 } { 3 } - \frac { \sqrt { x _ { n } } } { 12 } \right) ^ { \frac { 2 } { 3 } } \quad \text { with } x _ { 1 } = 2$$ to find (i) the value of \(x _ { 2 }\) to 5 decimal places,
   (ii) the \(x\) coordinate of \(P\) to 5 decimal places.

1. The curve \(C\) has equation

$$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. 1. Verify that \(C\) has a stationary point at \(x = 1\)
   2. Show that this stationary point is a point of inflection, giving reasons for your answer.

1 It is given that \(\mathrm { f } ( x ) = 3 x - \frac { 5 } { x ^ { 3 } }\).
Find

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

1

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 3 } - 3 x + \frac { 5 } { x ^ { 2 } } \right)\).
2. Find \(\int \left( 6 x ^ { 2 } - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x\).

7 \includegraphics[max width=\textwidth, alt={}, center]{7fc02f90-8f8b-4153-bba1-dc0807124e96-5_421_944_251_242} The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the crosssection \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(a\) and \(b\) are integers to be determined.
2. Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
3. Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
4. Explain why, for this roof, \(x\) must be less than a certain value, which you should state.

8 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-06_823_588_260_242} The diagram shows the curve \(y = 1 - x + \frac { 6 } { \sqrt { x } }\) and the line \(l\), which is the normal to the curve at the point (1, 6).

1. Determine the equation of \(l\) in the form $$a x + b y = c$$ where \(a\), \(b\) and \(c\) are integers whose values are to be stated.
2. Verify that the curve intersects the \(x\)-axis at the point where \(x = 4\).
3. In this question you must show detailed reasoning. Determine the exact area of the shaded region enclosed between \(l\), the curve, the \(x\)-axis and the \(y\)-axis.

5 In this question you must show detailed reasoning.

1. Show that the gradient of the curve \(\mathrm { y } = \sqrt { \mathrm { x } } \left( \frac { 1 } { \mathrm { x } ^ { 2 } } - 2 \mathrm { x } \right)\) at the point \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) is \(- \frac { 99 } { 2 }\).
2. Find the equation of the tangent to the curve at \(\left( \frac { 1 } { 4 } , \frac { 31 } { 4 } \right)\) giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are integers.

5 Fig. 5.1 shows part of the curve \(y = x ^ { \frac { 1 } { 2 } }\). P is the point \(( 1,1 )\) and \(Q\) is the point on the curve with \(x\)-coordinate \(1 + h\). \begin{figure}[h] \end{figure} Table 5.2 shows, for different values of \(h\), the coordinates of P , the coordinates of Q , the change in \(y\) from P to Q and the gradient of the chord PQ . \begin{table}[h] \end{table}

1. Fill in the missing values for the case \(h = 1\) in the copy of Table 5.2 in the Printed Answer Booklet. Give your answers correct to 6 decimal places where necessary.
2. Explain how the sequence of values in the last column of Table 5.2 relates to the gradient of the curve \(y = x ^ { \frac { 1 } { 2 } }\) at the point \(P\).
3. Use calculus to find the gradient of the curve at the point P .

6 In this question you must show detailed reasoning.
A particle moves in a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t \mathrm {~s}\) is given by \(\mathrm { v } = \mathrm { t } ^ { 3 } - 5 \mathrm { t } ^ { 2 }\).

1. Find the times at which the particle is stationary.
2. Find the total distance travelled by the particle in the first 6 seconds.

7 In this question you must show detailed reasoning.
A curve has equation \(y = 4 x ^ { 3 } - 6 x ^ { 2 } - 9 x + 4\).

1. Sketch the gradient function for this curve, clearly indicating the points where the gradient is zero.
2. Find the set of values of \(x\) for which the gradient function is decreasing. Give your answer using set notation.

6 The displacement of a particle is modelled by the equation \(\mathrm { s } = 7 + 4 \mathrm { t } - \mathrm { t } ^ { 2 }\), where \(s\) metres is the displacement from the origin at time \(t\) seconds. The diagram shows part of the displacement-time graph for the particle. The point \(( 2,11 )\) is the maximum point on the graph. \includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-4_513_1381_422_255}

1. Kai argues that the point \(( 2,11 )\) is on the graph, so the particle has travelled a distance of 11 metres in the first 2 seconds. Comment on the validity of Kai's argument.
2. Determine the total distance the particle travels in the first 10 seconds.
3. Find an expression for the velocity of the particle at time \(t\).
4. Find the speed of the particle when \(t = 10\).

5 Particle P moves on a straight line that contains the point O .
At time \(t\) seconds the displacement of P from O is \(s\) metres, where \(s = t ^ { 3 } - 3 t ^ { 2 } + 3\).

1. Determine the times when the particle has zero velocity.
2. Find the distances of P from O at the times when it has zero velocity.