1.07q Product and quotient rules: differentiation

6. A curve has the equation \(y = \mathrm { e } ^ { 3 x } \cos 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )\). The curve has a stationary point in the interval \([ 0,1 ]\).
3. Find the \(x\)-coordinate of the stationary point to 3 significant figures.
4. Determine whether the stationary point is a maximum or minimum point and justify your answer.

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

1. \(\sqrt { 1 - \cos x }\)
2. \(x ^ { 3 } \ln x\) (b) Given that $$x = \frac { y + 1 } { 3 - 2 y } ,$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).

8. A curve has the equation \(y = x ^ { 2 } - \sqrt { 4 + \ln x }\).

1. Show that the tangent to the curve at the point where \(x = 1\) has the equation $$7 x - 4 y = 11$$ The curve has a stationary point with \(x\)-coordinate \(\alpha\).
2. Show that \(0.3 < \alpha < 0.4\)
3. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
4. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with \(x _ { 0 } = 0.35\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 5 decimal places. END

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. \(\frac { 6 } { \sqrt { 2 x - 7 } }\)

5. (a) Show that \(( 2 x + 3 )\) is a factor of \(\left( 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\).
(b) Hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (c) Find the coordinates of the stationary points of the curve with equation $$y = \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$

6. (a) Use the derivative of \(\cos x\) to prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x$$ The curve \(C\) has the equation \(y = \mathrm { e } ^ { 2 x } \sec x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
(b) Find an equation for the tangent to \(C\) at the point where it crosses the \(y\)-axis.
(c) Find, to 2 decimal places, the \(x\)-coordinate of the stationary point of \(C\).

3. $$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$

1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

4. $$f ( x ) = x ^ { 4 } \sin ( 2 x )$$ Use Leibnitz's theorem to show that the coefficient of \(( x - \pi ) ^ { 8 }\) in the Taylor series expansion of \(\mathrm { f } ( x )\) about \(\pi\) is $$\frac { a \pi + b \pi ^ { 3 } } { 315 }$$ where \(a\) and \(b\) are integers to be determined. The Taylor series expansion of \(\mathrm { f } ( \mathrm { x } )\) about \(\mathrm { x } = \mathrm { k }\) is given by $$f ( x ) = f ( k ) + ( x - k ) f ^ { \prime } ( k ) + \frac { ( x - k ) ^ { 2 } } { 2 ! } f ^ { \prime \prime } ( k ) + \ldots + \frac { ( x - k ) ^ { r } } { r ! } f ^ { ( r ) } ( k ) + \ldots$$

5. $$y = \mathrm { e } ^ { 3 x } \sin x$$

1. Use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = 28 \mathrm { e } ^ { 3 x } \sin x + 96 \mathrm { e } ^ { 3 x } \cos x$$
2. Hence express \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) in the form $$\operatorname { Re } ^ { 3 \mathrm { x } } \sin ( \mathrm { x } + \alpha )$$ where \(R\) and \(\alpha\) are constants to be determined, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

1. Given \(k\) is a constant and that

$$y = x ^ { 3 } \mathrm { e } ^ { k x }$$ use Leibnitz theorem to show that $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = k ^ { n - 3 } \mathrm { e } ^ { k x } \left( k ^ { 3 } x ^ { 3 } + 3 n k ^ { 2 } x ^ { 2 } + 3 n ( n - 1 ) k x + n ( n - 1 ) ( n - 2 ) \right)$$

6 In this question you must show detailed reasoning. A curve has equation \(y = \frac { 2 x } { 3 x - 1 } + \sqrt { 5 x + 1 }\). Show that the equation of the tangent to the curve at the point where \(x = 3\) is \(19 x - 32 y + 95 = 0\).

9

1. Given that \(y = x ^ { - 2 } \ln x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 \ln x } { x ^ { 3 } }\).
2. Using integration by parts, find \(\int x ^ { - 2 } \ln x \mathrm {~d} x\).
3. The sketch shows the graph of \(y = x ^ { - 2 } \ln x\). \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-06_604_1045_687_536}
   1. Using the answer to part (a), find, in terms of e, the \(x\)-coordinate of the stationary point \(A\).
   2. The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x = 5\). Using your answer to part (b), show that the area of \(R\) is $$\frac { 1 } { 5 } ( 4 - \ln 5 )$$

6 A curve has equation \(y = \mathrm { e } ^ { 2 x } \left( x ^ { 2 } - 4 x - 2 \right)\).

1. Find the value of the \(x\)-coordinate of each of the stationary points of the curve.
2. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Determine the nature of each of the stationary points of the curve.

9

1. Given that \(y = \frac { 4 x } { 4 x - 3 }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { ( 4 x - 3 ) ^ { 2 } }\), where \(k\) is an integer.
2. 1. Given that \(y = x \ln ( 4 x - 3 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the tangent to the curve \(y = x \ln ( 4 x - 3 )\) at the point where \(x = 1\).
3. 1. Use the substitution \(u = 4 x - 3\) to find \(\int \frac { 4 x } { 4 x - 3 } \mathrm {~d} x\), giving your answer in terms of \(x\).
   2. By using integration by parts, or otherwise, find \(\int \ln ( 4 x - 3 ) \mathrm { d } x\).

1 A curve has equation \(y = \mathrm { e } ^ { - 4 x } \left( x ^ { 2 } + 2 x - 2 \right)\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \mathrm { e } ^ { - 4 x } \left( 5 - 3 x - 2 x ^ { 2 } \right)\).
2. Find the exact values of the coordinates of the stationary points of the curve.

7 It is given that \(y = \tan 4 x\).

1. By writing \(\tan 4 x\) as \(\frac { \sin 4 x } { \cos 4 x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = p \left( 1 + \tan ^ { 2 } 4 x \right)\), where \(p\) is a number to be determined.
2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = q y \left( 1 + y ^ { 2 } \right)\), where \(q\) is a number to be determined.

1

1. Differentiate \(\ln x\) with respect to \(x\).
2. Given that \(y = ( x + 1 ) \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
3. Find an equation of the normal to the curve \(y = ( x + 1 ) \ln x\) at the point where \(x = 1\).

7

1. A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { x }\).
   1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. 1. Find the \(x\)-coordinate of each of the stationary points of the curve.
   2. Using your answer to part (a)(ii), determine the nature of each of the stationary points.

8

1. Write down \(\int \sec ^ { 2 } x \mathrm {~d} x\).
2. Given that \(y = \frac { \cos x } { \sin x }\), use the quotient rule to show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } ^ { 2 } x\).
3. Prove the identity \(( \tan x + \cot x ) ^ { 2 } = \sec ^ { 2 } x + \operatorname { cosec } ^ { 2 } x\).
4. Hence find \(\int _ { 0.5 } ^ { 1 } ( \tan x + \cot x ) ^ { 2 } \mathrm {~d} x\), giving your answer to two significant figures.

5

1. By writing \(\tan x\) as \(\frac { \sin x } { \cos x }\), use the quotient rule to show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \tan x ) = \sec ^ { 2 } x\).
   [0pt] [2 marks]
2. Use integration by parts to find \(\int x \sec ^ { 2 } x \mathrm {~d} x\).
   [0pt] [4 marks]
3. The region bounded by the curve \(y = ( 5 \sqrt { x } ) \sec x\), the \(x\)-axis from 0 to 1 and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the value of the volume of the solid generated, giving your answer to two significant figures.
   [0pt] [3 marks]

4

1. Differentiate \(x \tan ^ { - 1 } x\) with respect to \(x\).
2. Show that $$\int _ { 0 } ^ { 1 } \tan ^ { - 1 } x \mathrm {~d} x = \frac { \pi } { 4 } - \ln \sqrt { 2 }$$ (5 marks)

10

1. Given that $$y = \tan x$$ use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x$$ 10
2. The region enclosed by the curve \(y = \tan ^ { 2 } x\) and the horizontal line, which intersects the curve at \(x = - \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 4 }\), is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-17_1059_967_461_539} Show that the area of the shaded region is $$\pi - 2$$ Fully justify your answer.
   Do not write outside the box \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-19_2488_1716_219_153}

12 A curve, \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 5 x ^ { 2 } - 12 x + 12 } { x ^ { 2 } + 4 x - 4 }\) The line \(y = k\) intersects the curve, \(C _ { 1 }\) 12

1. 1. Show that \(( k + 3 ) ( k - 1 ) \geq 0\) [0pt] [5 marks]
      12
      1. (ii) Hence find the coordinates of the stationary point of \(C _ { 1 }\) that is a maximum point.
         [0pt] [4 marks] 12
   2. Show that the curve \(C _ { 2 }\) whose equation is \(y = \frac { 1 } { \mathrm { f } ( x ) }\), has no vertical asymptotes.
      [0pt] [2 marks]
      12
   3. State the equation of the line that is a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
      [0pt] [1 mark]