Find derivative of product

OCR C3 2007 June Q1

5 marks Moderate -0.8

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

1. \(x ^ { 3 } ( x + 1 ) ^ { 5 }\)
2. \(\sqrt { 3 x ^ { 4 } + 1 }\)

OCR MEI C3 2007 June Q8

20 marks Standard +0.3

8 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-4_421_965_349_550} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find the exact coordinates of P .
2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.

OCR MEI C3 Q5

3 marks Standard +0.3

5 Differentiate \(x ^ { 2 } \tan 2 x\).

OCR MEI C3 Q3

7 marks Standard +0.3

3

1. Differentiate \(x \cos 2 x\) with respect to \(x\).
2. Integrate \(x \cos 2 x\) with respect to \(x\).

OCR MEI C3 Q1

20 marks Standard +0.3

1 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{aee8da6a-7d5c-442f-9729-55d81d9a606f-1_427_968_432_584} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find the exact coordinates of P .
2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.

OCR MEI C3 Q6

17 marks Standard +0.8

6 Fig. 8 shows part of the curve \(y = x \cos 3 x\). The curve crosses the \(x\)-axis at \(\mathrm { O } , \mathrm { P }\) and Q . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{11877196-83d9-4283-9eef-e617bea50c63-3_553_1178_622_529} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find the exact coordinates of P and Q .
2. Find the exact gradient of the curve at the point P . Show also that the turning points of the curve occur when \(x \tan 3 x = \frac { 1 } { 3 }\).
3. Find the area of the region enclosed by the curve and the \(x\)-axis between O and P , giving your answer in exact form.

OCR MEI C3 Q1

18 marks Standard +0.3

1 Fig. 8 shows a sketch of part of the curve \(y = x \sin 2 x\), where \(x\) is in radians.
The curve crosses the \(x\)-axis at the point P . The tangent to the curve at P crosses the \(y\)-axis at Q . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{35646966-3747-4f1d-bf94-60e9e3130afe-1_706_920_489_606} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that the \(x\)-coordinates of the turning points of the curve satisfy the equation \(\tan 2 x + 2 x = 0\).
2. Find, in terms of \(\pi\), the \(x\)-coordinate of the point P . Show that the tangent PQ has equation \(2 \pi x + 2 y = \pi ^ { 2 }\).
   Find the exact coordinates of Q.
3. Show that the exact value of the area shaded in Fig. 8 is \(\frac { 1 } { 8 } \pi \left( \pi ^ { 2 } - 2 \right)\).

OCR MEI C3 Q3

20 marks Standard +0.3

3 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{00c12cc4-f7ee-4219-8d34-a1854284f65d-2_425_974_478_591} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find the exact coordinates of P .
2. Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
5. Find the gradient of the curve at the origin. Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
6. Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.

OCR C1 2015 June Q7

8 marks Moderate -0.8

7

1. Given that \(\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 \right) ( 5 - x )\), find \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the gradient of the curve \(y = x ^ { - \frac { 1 } { 3 } }\) at the point where \(x = - 8\).

OCR C3 2011 January Q10

Moderate -0.8

10

8 (b) (i)
8 (b) (ii)
\multirow{29}{*}{}

{}

9 (a)
\multirow{12}{*}{}	D
	D
	□
	D
	□
	□
	D
	-
	□
	D
	□
	-
\multirow[t]{25}{*}{9 (b)}
	□

9

\section*{RECOGNISING ACHIEVEMENT} RECOGNISING ACHIEVEMENT

OCR MEI C3 2009 January Q2

7 marks Moderate -0.3

2

1. Differentiate \(x \cos 2 x\) with respect to \(x\).
2. Integrate \(x \cos 2 x\) with respect to \(x\).

OCR MEI C3 2010 January Q8

17 marks Standard +0.3

8 Fig. 8 shows part of the curve \(y = x \cos 3 x\).
The curve crosses the \(x\)-axis at \(\mathrm { O } , \mathrm { P }\) and Q . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3b3e20ee-457c-46be-b2e5-12573bee2fbf-3_551_1189_925_479} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find the exact coordinates of P and Q .
2. Find the exact gradient of the curve at the point P . Show also that the turning points of the curve occur when \(x \tan 3 x = \frac { 1 } { 3 }\).
3. Find the area of the region enclosed by the curve and the \(x\)-axis between O and P , giving your answer in exact form.

OCR MEI C3 2012 June Q8

18 marks Standard +0.3

8 Fig. 8 shows a sketch of part of the curve \(y = x \sin 2 x\), where \(x\) is in radians.
The curve crosses the \(x\)-axis at the point P . The tangent to the curve at P crosses the \(y\)-axis at Q . \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7b77c646-2bc5-4166-b22e-3c1229abd722-4_712_923_463_571} \captionsetup{labelformat=empty} \caption{Fig. 8}

\end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). Hence show that the \(x\)-coordinates of the turning points of the curve satisfy the equation \(\tan 2 x + 2 x = 0\).
2. Find, in terms of \(\pi\), the \(x\)-coordinate of the point P . Show that the tangent PQ has equation \(2 \pi x + 2 y = \pi ^ { 2 }\).
   Find the exact coordinates of Q .
3. Show that the exact value of the area shaded in Fig. 8 is \(\frac { 1 } { 8 } \pi \left( \pi ^ { 2 } - 2 \right)\).

CAIE FP1 2009 November Q1

4 marks Standard +0.3

1 Given that $$y = x ^ { 2 } \sin x$$

1. show that the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) is \(\frac { 1 } { 2 } \pi\),
2. find the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).

OCR H240/01 2020 November Q8

7 marks Moderate -0.8

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 \(x\).

AQA C3 2007 January Q6

8 marks Moderate -0.3

6

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(y = \left( 4 x ^ { 2 } + 3 x + 2 \right) ^ { 10 }\);
   2. \(y = x ^ { 2 } \tan x\).
2. 1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(x = 2 y ^ { 3 } + \ln y\).
   2. Hence find an equation of the tangent to the curve \(x = 2 y ^ { 3 } + \ln y\) at the point \(( 2,1 )\).

AQA C3 2011 January Q1

7 marks Moderate -0.8

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 3 } - 1 \right) ^ { 6 }\).
2. A curve has equation \(y = x \ln x\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln x\) at the point on the curve where \(x = \mathrm { e }\).

AQA C3 2005 June Q1

8 marks Moderate -0.8

1

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = x \sin 2 x\).
2. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 2 } - 6 \right) ^ { 4 }\).
   2. Hence, or otherwise, find \(\int x \left( x ^ { 2 } - 6 \right) ^ { 3 } \mathrm {~d} x\).

AQA C3 2008 June Q1

7 marks Moderate -0.8

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:

1. \(y = ( 3 x + 1 ) ^ { 5 }\);
2. \(y = \ln ( 3 x + 1 )\);
3. \(y = ( 3 x + 1 ) ^ { 5 } \ln ( 3 x + 1 )\).

AQA C3 2008 June Q3

14 marks Standard +0.3

3 A curve is defined for \(0 \leqslant x \leqslant \frac { \pi } { 4 }\) by the equation \(y = x \cos 2 x\), and is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{6ce5aa0d-0a73-4bc4-aabc-314c0434e4f5-3_757_878_402_559}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(A\), where \(x = \alpha\), on the curve is a stationary point.
   1. Show that \(1 - 2 \alpha \tan 2 \alpha = 0\).
   2. Show that \(0.4 < \alpha < 0.5\).
   3. Show that the equation \(1 - 2 x \tan 2 x = 0\) can be rearranged to become \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\).
   4. Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find \(x _ { 3 }\), giving your answer to two significant figures.
3. Use integration by parts to find \(\int _ { 0 } ^ { 0.5 } x \cos 2 x \mathrm {~d} x\), giving your answer to three significant figures.

AQA C3 2012 June Q3

7 marks Moderate -0.3

3 A curve has equation \(y = x ^ { 3 } \ln x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find an equation of the tangent to the curve \(y = x ^ { 3 } \ln x\) at the point on the curve where \(x = \mathrm { e }\).
   2. This tangent intersects the \(x\)-axis at the point \(A\). Find the exact value of the \(x\)-coordinate of the point \(A\).

Edexcel C3 Q3

10 marks Moderate -0.8

3. Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\quad \ln ( \cos x )\)
2. \(x ^ { 2 } \sin 3 x\)
3. \(\frac { 6 } { \sqrt { 2 x - 7 } }\)

OCR C3 Q1

5 marks Moderate -0.8

Differentiate each of the following with respect to \(x\).

1. \(x^3(x + 1)^5\) [2]
2. \(\sqrt{3x^4 + 1}\) [3]

OCR MEI C3 2012 January Q1

3 marks Moderate -0.3

Differentiate \(x^2 \tan 2x\). [3]