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

1. The volume \(V \mathrm {~cm} ^ { 3 }\) of a box, of height \(x \mathrm {~cm}\), is given by

$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
2. Hence find the maximum volume of the box.
3. Use calculus to justify that the volume that you found in part (b) is a maximum.

8. The curve \(C\) has equation \(y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0\)

1. Use calculus to show that the curve has a turning point \(P\) when \(x = \sqrt { } 2\)
2. Find the \(x\)-coordinate of the other turning point \(Q\) on the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence or otherwise, state with justification, the nature of each of these turning points \(P\) and \(Q\).

3. The curve \(C\) has equation $$y = 2 \sqrt { } x + \frac { 18 } { \sqrt { } x } - 1 , \quad x > 0$$

1. Find
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
   2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
2. Use calculus to find the coordinates of the stationary point of \(C\).
3. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-09_138_154_2597_1804}

10. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).

1. Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum. The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
   The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\).
3. Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
4. Hence calculate the exact area of \(R\).

10. \begin{figure}[h] \end{figure} Figure 4 shows a solid brick in the shape of a cuboid measuring \(2 x \mathrm {~cm}\) by \(x \mathrm {~cm}\) by \(y \mathrm {~cm}\). The total surface area of the brick is \(600 \mathrm {~cm} ^ { 2 }\).

1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the brick is given by $$V = 200 x - \frac { 4 x ^ { 3 } } { 3 }$$ Given that \(x\) can vary,
2. use calculus to find the maximum value of \(V\), giving your answer to the nearest \(\mathrm { cm } ^ { 3 }\).
3. Justify that the value of \(V\) you have found is a maximum.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = 10 + 8 x + x ^ { 2 } - x ^ { 3 }\).
The curve has a maximum turning point \(A\).

1. Using calculus, show that the \(x\)-coordinate of \(A\) is 2 . The region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(y\)-axis and the line from \(O\) to \(A\), where \(O\) is the origin.
2. Using calculus, find the exact area of \(R\).

3. $$y = x ^ { 2 } - k \sqrt { } x , \text { where } k \text { is a constant. }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Given that \(y\) is decreasing at \(x = 4\), find the set of possible values of \(k\).

1. Using calculus, find the coordinates of the stationary point on the curve with equation

$$y = 2 x + 3 + \frac { 8 } { x ^ { 2 } } , \quad x > 0$$

1. The curve with equation

$$y = x ^ { 2 } - 32 \sqrt { } ( x ) + 20 , \quad x > 0$$ has a stationary point \(P\). Use calculus

1. to find the coordinates of \(P\),
2. to determine the nature of the stationary point \(P\).

1. 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 $$y = \frac { 16 } { 9 ( 3 x - k ) } \quad x \neq \frac { k } { 3 }$$ where \(k\) is a positive constant not equal to 3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving your answer in simplest form in terms of \(k\). The point \(P\) with \(x\) coordinate 1 lies on \(C\).
   Given that the gradient of the curve at \(P\) is - 12
2. find the two possible values of \(k\) Given also that \(k < 3\)
3. find the equation of the normal to \(C\) at \(P\), writing your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.
4. show, using algebraic integration that, $$\int _ { 1 } ^ { 3 } \frac { 16 } { 9 ( 3 x - k ) } d x = \lambda \ln 10$$ where \(\lambda\) is a constant to be found.

10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve with equation $$x = \frac { 2 y ^ { 2 } + 6 } { 3 y - 3 }$$

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) giving your answer as a fully simplified fraction. The tangents at points \(P\) and \(Q\) on the curve are parallel to the \(y\)-axis, as shown in Figure 4.
2. Use the answer to part (a) to find the equations of these two tangents.

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } ( x ) = 2 + 5 \ln x \quad x > 0 \\ & \mathrm {~g} ( x ) = \frac { 6 x - 2 } { 2 x + 1 } \quad x > \frac { 1 } { 3 } \end{aligned}$$

1. Find \(\mathrm { f } ^ { - 1 } ( 22 )\)
2. Use differentiation to prove that g is an increasing function.
3. Find \(\mathrm { g } ^ { - 1 }\)
4. Find the range of fg

6. \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 $$y = \sqrt { 4 x - 7 }$$ The line \(l\), shown in Figure 3, is the normal to the curve at the point \(P ( 8,5 )\)

1. Use calculus to show that an equation of \(l\) is $$5 x + 2 y - 50 = 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and \(l\).
2. Use algebraic integration to find the exact area of \(R\).

8. \begin{figure}[h] \end{figure} Figure 4 is a graph showing the path of a golf ball after the ball has been hit until it first hits the ground. The vertical height, \(h\) metres, of the ball above the ground has been plotted against the horizontal distance travelled, \(x\) metres, measured from where the ball was hit. The ball travels a horizontal distance of \(d\) metres before it first hits the ground.
The ball is modelled as a particle travelling in a vertical plane above horizontal ground.
The path of the ball is modelled by the equation $$h = 1.5 x - 0.5 x \mathrm { e } ^ { 0.02 x } \quad 0 \leqslant x \leqslant d$$ \section*{Use the model to answer parts (a), (b) and (c).}

1. Find the value of \(d\), giving your answer to 2 decimal places.
   (Solutions relying entirely on calculator technology are not acceptable.)
2. Show that the maximum value of \(h\) occurs when $$x = 50 \ln \left( \frac { 150 } { x + 50 } \right)$$ Using the iteration formula $$x _ { n + 1 } = 50 \ln \left( \frac { 150 } { x _ { n } + 50 } \right) \quad \text { with } x _ { 1 } = 30$$
3. 1. find the value of \(x _ { 2 }\) to 2 decimal places,
   2. find, by repeated iteration, the horizontal distance travelled by the golf ball before it reaches its maximum height. Give your answer to 2 decimal places. \includegraphics[max width=\textwidth, alt={}, center]{5a695b86-1660-4c06-ac96-4cdb07af9a2e-26_2270_56_309_1981}

1. The function f is defined by

$$\mathrm { f } ( x ) = \frac { 5 x } { x ^ { 2 } + 7 x + 12 } + \frac { 5 x } { x + 4 } \quad x > 0$$

1. Show that \(\mathrm { f } ( x ) = \frac { 5 x } { x + 3 }\)
2. Find \(\mathrm { f } ^ { - 1 }\)
3. 1. Find, in simplest form, \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.
      (3)

14. The volume of a spherical balloon of radius \(r \mathrm {~cm}\) is \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\)

1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} r }\) The volume of the balloon increases with time \(t\) seconds according to the formula $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 9000 \pi } { ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$
2. Using the chain rule, or otherwise, show that $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$ where \(k\) and \(n\) are constants to be found. Initially, the radius of the balloon is 3 cm .
3. Using the values of \(k\) and \(n\) found in part (b), solve the differential equation $$\frac { \mathrm { d } r } { \mathrm {~d} t } = \frac { k } { r ^ { n } ( t + 81 ) ^ { \frac { 5 } { 4 } } } \quad t \geqslant 0$$ to obtain a formula for \(r\) in terms of \(t\).
4. Hence find the radius of the balloon when \(t = 175\), giving your answer to 3 significant figures.
   (1)
5. Find the rate of increase of the radius of the balloon when \(t = 175\). Give your answer to 3 significant figures.

   END

5. $$\frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \equiv A + \frac { B } { ( 2 - x ) } + \frac { C } { ( 1 + 2 x ) }$$

1. Find the values of the constants \(A , B\) and \(C\). $$f ( x ) = \frac { 6 - 5 x - 4 x ^ { 2 } } { ( 2 - x ) ( 1 + 2 x ) } \quad x > 2$$
2. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
3. Prove that \(\mathrm { f } ( x )\) is a decreasing function.

8. $$\mathrm { h } ( x ) = \frac { 2 } { x + 2 } + \frac { 4 } { x ^ { 2 } + 5 } - \frac { 18 } { \left( x ^ { 2 } + 5 \right) ( x + 2 ) } , \quad x \geqslant 0$$

1. Show that \(\mathrm { h } ( x ) = \frac { 2 x } { x ^ { 2 } + 5 }\)
2. Hence, or otherwise, find \(\mathrm { h } ^ { \prime } ( x )\) in its simplest form. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-26_679_1168_733_390} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a graph of the curve with equation \(y = \mathrm { h } ( x )\).
3. Calculate the range of \(\mathrm { h } ( x )\).

2. $$f ( x ) = \frac { 2 x + 2 } { x ^ { 2 } - 2 x - 3 } - \frac { x + 1 } { x - 3 }$$

1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
2. Hence show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)

2. (a) Express $$\frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) }$$ as a single fraction in its simplest form. Given that $$f ( x ) = \frac { 4 x - 1 } { 2 ( x - 1 ) } - \frac { 3 } { 2 ( x - 1 ) ( 2 x - 1 ) } - 2 , \quad x > 1$$ (b) show that $$f ( x ) = \frac { 3 } { 2 x - 1 }$$ (c) Hence differentiate \(\mathrm { f } ( x )\) and find \(\mathrm { f } ^ { \prime } ( 2 )\).

6. $$f ( x ) = x ^ { 2 } - 3 x + 2 \cos \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \pi$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a solution in the interval \(0.8 < x < 0.9\) The curve with equation \(y = \mathrm { f } ( x )\) has a minimum point \(P\).
2. Show that the \(x\)-coordinate of \(P\) is the solution of the equation $$x = \frac { 3 + \sin \left( \frac { 1 } { 2 } x \right) } { 2 }$$
3. Using the iteration formula $$x _ { n + 1 } = \frac { 3 + \sin \left( \frac { 1 } { 2 } x _ { n } \right) } { 2 } , \quad x _ { 0 } = 2$$ find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
4. By choosing a suitable interval, show that the \(x\)-coordinate of \(P\) is 1.9078 correct to 4 decimal places.

1. The curve \(C\) has equation

$$y = ( 2 x - 3 ) ^ { 5 }$$ The point \(P\) lies on \(C\) and has coordinates \(( w , - 32 )\).
Find

1. the value of \(w\),
2. the equation of the tangent to \(C\) at the point \(P\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

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

$$f ( x ) = \frac { 4 x + 1 } { x - 2 } , \quad x > 2$$

1. Show that $$f ^ { \prime } ( x ) = \frac { - 9 } { ( x - 2 ) ^ { 2 } }$$ Given that \(P\) is a point on \(C\) such that \(\mathrm { f } ^ { \prime } ( x ) = - 1\),
2. find the coordinates of \(P\).

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$ The curve crosses the \(x\)-axis at the point \(Q\) and has a minimum turning point at \(R\).

1. Show that the \(x\) coordinate of \(Q\) lies between 2.1 and 2.2
2. Show that the \(x\) coordinate of \(R\) is a solution of the equation $$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
3. find the values of \(x _ { 1 }\) and \(x _ { 2 }\) to 3 decimal places.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$\mathrm { g } ( x ) = x ^ { 2 } ( 1 - x ) \mathrm { e } ^ { - 2 x } , \quad x \geqslant 0$$

1. Show that \(\mathrm { g } ^ { \prime } ( x ) = \mathrm { f } ( x ) \mathrm { e } ^ { - 2 x }\), where \(\mathrm { f } ( x )\) is a cubic function to be found.
2. Hence find the range of g .
3. State a reason why the function \(\mathrm { g } ^ { - 1 } ( x )\) does not exist.