1.07q Product and quotient rules: differentiation

4

1. Differentiate \(\mathrm { e } ^ { x } ( \sin 2 x - 2 \cos 2 x )\), simplifying your answer.
2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { e } ^ { x } \sin 2 x \mathrm {~d} x\).

3 The equation of a curve is \(y = \mathrm { e } ^ { 2 x } \cos x\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any stationary points for which \(- \pi \leqslant x \leqslant \pi\). Give your answers correct to 3 significant figures.

6

1. Use the quotient rule to show that the derivative of \(\frac { \cos x } { \sin x }\) is \(\frac { - 1 } { \sin ^ { 2 } x }\).
2. Show that \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 4 } \pi } \frac { \sqrt { 1 + \cos 2 x } } { \sin x \sin 2 x } \mathrm {~d} x = \frac { 1 } { 2 } ( \sqrt { 6 } - \sqrt { 2 } )\).

5 You are given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } \sin x\).

1. Find \(f ( 0 )\) and \(f ^ { \prime } ( 0 )\).
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = - 2 \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime } ( 0 )\).
3. Find a similar expression for \(\mathrm { f } ^ { \prime \prime \prime } ( x )\) and hence, or otherwise, find \(\mathrm { f } ^ { \prime \prime \prime } ( 0 )\).
4. Find the Maclaurin series for \(\mathrm { f } ( x )\) up to and including the term in \(x ^ { 3 }\).

9 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 3 } \right) ^ { - n } \mathrm {~d} x$$ where \(n > 0\).

1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x \left( 1 + x ^ { 3 } \right) ^ { - n } \right] = - ( 3 n - 1 ) \left( 1 + x ^ { 3 } \right) ^ { - n } + 3 n \left( 1 + x ^ { 3 } \right) ^ { - n - 1 }$$ and hence, or otherwise, show that $$I _ { n + 1 } = \frac { 2 ^ { - n } } { 3 n } + \left( 1 - \frac { 1 } { 3 n } \right) I _ { n }$$
2. By considering the graph of \(y = \frac { 1 } { 1 + x ^ { 3 } }\), show that \(I _ { 1 } < 1\).
3. Deduce that \(I _ { 3 } < \frac { 53 } { 72 }\).

2 The integral \(I _ { n }\), where \(n\) is a non-negative integer, is defined by $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 3 } } \mathrm {~d} x$$ By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { n + 1 } \mathrm { e } ^ { - x ^ { 3 } } \right)\) or otherwise, show that $$3 I _ { n + 3 } = ( n + 1 ) I _ { n } - \mathrm { e } ^ { - 1 }$$ Hence find \(I _ { 6 }\) in terms of e and \(I _ { 0 }\).

7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 4 } \right) ^ { n } } \mathrm {~d} x\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \frac { x } { \left( 1 + x ^ { 4 } \right) ^ { n } } \right)\), show that $$4 n I _ { n + 1 } = \frac { 1 } { 2 ^ { n } } + ( 4 n - 1 ) I _ { n }$$ Given that \(I _ { 1 } = 0.86697\), correct to 5 decimal places, find \(I _ { 3 }\).

6 Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x ^ { n - 1 } \sqrt { } \left( 4 - x ^ { 2 } \right) \right] = \frac { 4 ( n - 1 ) x ^ { n - 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } - \frac { n x ^ { n } } { \sqrt { } \left( 4 - x ^ { 2 } \right) }$$ Let $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { x ^ { n } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x$$ where \(n \geqslant 0\). Prove that $$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } - \sqrt { } 3$$ for \(n \geq 2\). Given that \(I _ { 0 } = \frac { 1 } { 6 } \pi\), find \(I _ { 4 }\), leaving your answer in an exact form.

3 Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x \mathrm { e } ^ { a x } \right) = n a ^ { n - 1 } \mathrm { e } ^ { a x } + a ^ { n } x \mathrm { e } ^ { a x }$$

3 Prove by induction that, for all \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x ^ { 2 } } \right) = \mathrm { P } _ { n } ( x ) \mathrm { e } ^ { x ^ { 2 } } ,$$ where \(\mathrm { P } _ { n } ( x )\) is a polynomial in \(x\) of degree \(n\) with the coefficient of \(x ^ { n }\) equal to \(2 ^ { n }\).

3 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = 2 ^ { \frac { 1 } { 2 } n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$

12 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{1ba9fa5f-310f-4429-9bd1-4004852d5b3e-6_716_479_292_794} The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\).
[0pt] [10]

8

1. Differentiate \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) with respect to \(x\).
2. Hence show that \(\left( 2 + 3 x ^ { 2 } \right) \mathrm { e } ^ { 2 x }\) is increasing for all values of \(x\).

1

1. Differentiate the following.
   1. \(\frac { x ^ { 2 } } { 2 x + 1 }\)
   2. \(\tan \left( x ^ { 2 } - 3 x \right)\)
2. Use the substitution \(u = \sqrt { x } - 1\) to integrate \(\frac { 1 } { \sqrt { x } - 1 }\).
3. Integrate \(\frac { x - 2 } { 2 x ^ { 2 } - 8 x - 1 }\).

1 Differentiate the following with respect to \(x\).

1. \(\mathrm { e } ^ { - 4 x }\)
2. \(\frac { x ^ { 2 } } { x + 1 }\)

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$ The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\) c. Hence find the exact coordinates of the turning points of \(C\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation $$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$ The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\) Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).

5. Given that $$y = \frac { 5 \cos \theta } { 4 \cos \theta + 4 \sin \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ Show that $$\frac { d y } { d \theta } = - \frac { 5 } { 4 ( 1 + \sin 2 \theta ) } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$

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. Given that

$$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { d y } { d \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(A\) is a rational constant to be found.

3. $$y = \frac { 5 x ^ { 2 } + 10 x } { ( x + 1 ) ^ { 2 } } \quad x \neq - 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A } { ( x + 1 ) ^ { n } }\) where \(A\) and \(n\) are constants to be found.
2. Hence deduce the range of values for \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\)

12. $$\mathrm { f } ( x ) = 10 \mathrm { e } ^ { - 0.25 x } \sin x , \quad x \geqslant 0$$

1. Show that the \(x\) coordinates of the turning points of the curve with equation \(y = \mathrm { f } ( x )\) satisfy the equation \(\tan x = 4\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-34_687_1029_495_518} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
2. Sketch the graph of \(H\) against \(t\) where $$\mathrm { H } ( t ) = \left| 10 \mathrm { e } ^ { - 0.25 t } \sin t \right| \quad t \geqslant 0$$ showing the long-term behaviour of this curve. The function \(\mathrm { H } ( t )\) is used to model the height, in metres, of a ball above the ground \(t\) seconds after it has been kicked. Using this model, find
3. the maximum height of the ball above the ground between the first and second bounce.
4. Explain why this model should not be used to predict the time of each bounce.

8. \begin{figure}[h] \end{figure} A car stops at two sets of traffic lights.
Figure 2 shows a graph of the speed of the car, \(v \mathrm {~ms} ^ { - 1 }\), as it travels between the two sets of traffic lights. The car takes \(T\) seconds to travel between the two sets of traffic lights.
The speed of the car is modelled by the equation $$v = ( 10 - 0.4 t ) \ln ( t + 1 ) \quad 0 \leqslant t \leqslant T$$ where \(t\) seconds is the time after the car leaves the first set of traffic lights.
According to the model,

1. find the value of \(T\)
2. show that the maximum speed of the car occurs when $$t = \frac { 26 } { 1 + \ln ( t + 1 ) } - 1$$ Using the iteration formula $$t _ { n + 1 } = \frac { 26 } { 1 + \ln \left( t _ { n } + 1 \right) } - 1$$ with \(t _ { 1 } = 7\)
3. 1. find the value of \(t _ { 3 }\) to 3 decimal places,
   2. find, by repeated iteration, the time taken for the car to reach maximum speed.

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

$$\mathrm { f } ( x ) = \frac { 7 x \mathrm { e } ^ { x } } { \sqrt { \mathrm { e } ^ { 3 x } - 2 } } \quad x > \ln \sqrt [ 3 ] { 2 }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 7 \mathrm { e } ^ { x } \left( \mathrm { e } ^ { 3 x } ( 2 - x ) + A x + B \right) } { 2 \left( \mathrm { e } ^ { 3 x } - 2 \right) ^ { \frac { 3 } { 2 } } }$$ where \(A\) and \(B\) are constants to be found.
2. Hence show that the \(x\) coordinates of the turning points of the curve are solutions of the equation $$x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }$$ The equation \(x = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) has two positive roots \(\alpha\) and \(\beta\) where \(\beta > \alpha\) A student uses the iteration formula $$x _ { n + 1 } = \frac { 2 \mathrm { e } ^ { 3 x _ { n } } - 4 } { \mathrm { e } ^ { 3 x _ { n } } + 4 }$$ in an attempt to find approximations for \(\alpha\) and \(\beta\) Diagram 1 shows a plot of part of the curve with equation \(y = \frac { 2 \mathrm { e } ^ { 3 x } - 4 } { \mathrm { e } ^ { 3 x } + 4 }\) and part of the line with equation \(y = x\) Using Diagram 1 on page 42
3. draw a staircase diagram to show that the iteration formula starting with \(x _ { 1 } = 1\) can be used to find an approximation for \(\beta\) Use the iteration formula with \(x _ { 1 } = 1\), to find, to 3 decimal places,
4. 1. the value of \(x _ { 2 }\)
   2. the value of \(\beta\) Using a suitable interval and a suitable function that should be stated
5. show that \(\alpha = 0.432\) to 3 decimal places. Only use the copy of Diagram 1 if you need to redraw your answer to part (c). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_736_812_372_143} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0839eb5f-2850-4d77-baf7-a6557d71076e-42_738_815_370_1114} \captionsetup{labelformat=empty} \caption{copy of Diagram 1}
   \end{figure}

1. The function f is defined by

$$f ( x ) = \frac { 2 x - 3 } { x ^ { 2 } + 4 } \quad x \in \mathbb { R }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { a x ^ { 2 } + b x + c } { \left( x ^ { 2 } + 4 \right) ^ { 2 } }$$ where \(a\), \(b\) and \(c\) are constants to be found.
2. Hence, using algebra, find the values of \(x\) for which f is decreasing. You must show each step in your working.

9. \section*{Figure 2} Figure 2 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = 4 \left( x ^ { 2 } - 2 \right) \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = 8 \left( 2 + x - x ^ { 2 } \right) \mathrm { e } ^ { - 2 x }\)
2. Hence find, in simplest form, the exact coordinates of the stationary points of \(C\). The function g and the function h are defined by $$\begin{array} { l l } \mathrm { g } ( x ) = 2 \mathrm { f } ( x ) & x \in \mathbb { R } \\ \mathrm {~h} ( x ) = 2 \mathrm { f } ( x ) - 3 & x \geqslant 0 \end{array}$$
3. Find (i) the range of \(g\) (ii) the range of h