1.07l Derivative of ln(x): and related functions

7. For the constant \(k\), where \(k > 1\), the functions f and g are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto \ln ( x + k ) , \quad x > - k , \\ & \mathrm {~g} : x \mapsto | 2 x - k | , \quad x \in \mathbb { R } . \end{aligned}$$

1. On separate axes, sketch the graph of f and the graph of g . On each sketch state, in terms of \(k\), the coordinates of points where the graph meets the coordinate axes.
2. Write down the range of f.
3. Find \(\mathrm { fg } \left( \frac { k } { 4 } \right)\) in terms of \(k\), giving your answer in its simplest form. The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The tangent to \(C\) at the point with \(x\)-coordinate 3 is parallel to the line with equation \(9 y = 2 x + 1\).
4. Find the value of \(k\).

6. (a) Differentiate with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
   (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
   (c) Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).

4. (i) Differentiate with respect to \(x\)

1. \(x ^ { 2 } \cos 3 x\)
2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\) (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

Differentiate with respect to \(x\)

1. \(\quad \ln \left( x ^ { 2 } + 3 x + 5 \right)\)
2. \(\frac { \cos x } { x ^ { 2 } }\)

7. $$f ( x ) = \frac { 4 x - 5 } { ( 2 x + 1 ) ( x - 3 ) } - \frac { 2 x } { x ^ { 2 } - 9 } , \quad x \neq \pm 3 , x \neq - \frac { 1 } { 2 }$$

1. Show that $$f ( x ) = \frac { 5 } { ( 2 x + 1 ) ( x + 3 ) }$$ The curve \(C\) has equation \(y = \mathrm { f } ( x )\). The point \(P \left( - 1 , - \frac { 5 } { 2 } \right)\) lies on \(C\).
2. Find an equation of the normal to \(C\) at \(P\).

1. (i) (a) Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { \frac { 1 } { 2 } } \ln x \right) = \frac { \ln x } { 2 \sqrt { } x } + \frac { 1 } { \sqrt { } x }\)

The curve with equation \(y = x ^ { \frac { 1 } { 2 } } \ln x , x > 0\) has one turning point at the point \(P\).
(b) Find the exact coordinates of \(P\). Give your answer in its simplest form.
(ii) A curve \(C\) has equation \(y = \frac { x - k } { x + k }\), where \(k\) is a positive constant. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), and show that \(C\) has no turning points.

6. $$f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 7 x - 6 } { x ^ { 2 } + x - 6 } , \quad x > 2 , x \in \mathbb { R }$$

1. Given that $$\frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 7 x - 6 } { x ^ { 2 } + x - 6 } \equiv x ^ { 2 } + A + \frac { B } { x - 2 }$$ find the values of the constants \(A\) and \(B\).
2. Hence or otherwise, using calculus, find an equation of the normal to the curve with equation \(y = \mathrm { f } ( x )\) at the point where \(x = 3\)

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 2 \ln ( 2 x + 5 ) - \frac { 3 x } { 2 } , \quad x > - 2.5$$ The point \(P\) with \(x\) coordinate - 2 lies on \(C\).

1. Find an equation of the normal to \(C\) at \(P\). Write your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. The normal to \(C\) at \(P\) cuts the curve again at the point \(Q\), as shown in Figure 2
2. Show that the \(x\) coordinate of \(Q\) is a solution of the equation $$x = \frac { 20 } { 11 } \ln ( 2 x + 5 ) - 2$$ The iteration formula $$x _ { n + 1 } = \frac { 20 } { 11 } \ln \left( 2 x _ { n } + 5 \right) - 2$$ can be used to find an approximation for the \(x\) coordinate of \(Q\).
3. Taking \(x _ { 1 } = 2\), find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving each answer to 4 decimal places.

8. \begin{figure}[h] \end{figure} The number of rabbits on an island is modelled by the equation $$P = \frac { 100 \mathrm { e } ^ { - 0.1 t } } { 1 + 3 \mathrm { e } ^ { - 0.9 t } } + 40 , \quad t \in \mathbb { R } , t \geqslant 0$$ where \(P\) is the number of rabbits, \(t\) years after they were introduced onto the island.
A sketch of the graph of \(P\) against \(t\) is shown in Figure 3.

1. Calculate the number of rabbits that were introduced onto the island.
2. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) The number of rabbits initially increases, reaching a maximum value \(P _ { T }\) when \(t = T\)
3. Using your answer from part (b), calculate
   1. the value of \(T\) to 2 decimal places,
   2. the value of \(P _ { T }\) to the nearest integer.
      (Solutions based entirely on graphical or numerical methods are not acceptable.) For \(t > T\), the number of rabbits decreases, as shown in Figure 3, but never falls below \(k\), where \(k\) is a positive constant.
4. Use the model to state the maximum value of \(k\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { - 2 x } + x ^ { 2 } - 3$$ The curve \(C\) crosses the \(y\)-axis at the point \(A\). The line \(l\) is the normal to \(C\) at the point \(A\).

1. Find the equation of \(l\), writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The line \(l\) meets \(C\) again at the point \(B\), as shown in Figure 1 .
2. Show that the \(x\) coordinate of \(B\) is a solution of $$x = \sqrt { 1 + \frac { 1 } { 2 } x - \mathrm { e } ^ { - 2 x } }$$ Using the iterative formula $$x _ { n + 1 } = \sqrt { 1 + \frac { 1 } { 2 } x _ { n } - \mathrm { e } ^ { - 2 x _ { n } } }$$ with \(x _ { 1 } = 1\)
3. find \(x _ { 2 }\) and \(x _ { 3 }\) to 3 decimal places.

1. (a) By writing \(\sec \theta = \frac { 1 } { \cos \theta }\), show that \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \sec \theta ) = \sec \theta \tan \theta\) (b) Given that

$$x = \mathrm { e } ^ { \sec y } \quad x > \mathrm { e } , \quad 0 < y < \frac { \pi } { 2 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \sqrt { \mathrm {~g} ( x ) } } , \quad x > \mathrm { e }$$ where \(\mathrm { g } ( x )\) is a function of \(\ln x\).

5. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve meets the \(x\)-axis at \(P ( p , 0 )\) and meets the \(y\)-axis at \(Q ( 0 , q )\).

1. On separate diagrams, sketch the curve with equation
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 3 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). In each case show, in terms of \(p\) or \(q\), the coordinates of points at which the curve meets the axes. Given that \(\mathrm { f } ( x ) = 3 \ln ( 2 x + 3 )\),
2. state the exact value of \(q\),
3. find the value of \(p\),
4. find an equation for the tangent to the curve at \(P\).

4. The rectangular hyperbola \(H\) has Cartesian equation \(x y = 4\) The point \(P \left( 2 t , \frac { 2 } { t } \right)\) lies on \(H\), where \(t \neq 0\)

1. Show that an equation of the normal to \(H\) at the point \(P\) is $$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$ The normal to \(H\) at the point where \(t = - \frac { 1 } { 2 }\) meets \(H\) again at the point \(Q\).
2. Find the coordinates of the point \(Q\).

5. The rectangular hyperbola \(H\) has equation \(x y = 9\) The point \(A\) on \(H\) has coordinates \(\left( 6 , \frac { 3 } { 2 } \right)\).

1. Show that the normal to \(H\) at the point \(A\) has equation $$2 y - 8 x + 45 = 0$$ The normal at \(A\) meets \(H\) again at the point \(B\).
2. Find the coordinates of \(B\).

9. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\)

1. Show that an equation of the normal to \(H\) at the point \(P \left( 5 p , \frac { 5 } { p } \right) , p \neq 0\), is $$y - p ^ { 2 } x = \frac { 5 } { p } - 5 p ^ { 3 }$$ This normal meets the line with equation \(y = - x\) at the point \(A\).
2. Show that the coordinates of \(A\) are $$\left( - \frac { 5 } { p } + 5 p , \frac { 5 } { p } - 5 p \right)$$ The point \(M\) is the midpoint of the line segment \(A P\).
   Given that \(M\) lies on the positive \(x\)-axis,
3. find the exact value of the \(x\) coordinate of point \(M\).
   

3. $$f ( x ) = \ln ( 1 + \sin k x )$$ where \(k\) is a constant, \(x \in \mathbb { R }\) and \(- \frac { \pi } { 2 } < k x < \frac { 3 \pi } { 2 }\)

1. Find f \({ } ^ { \prime } ( x )\)
2. Show that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { - k ^ { 2 } } { 1 + \sin k x }\)
3. Find the Maclaurin series of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).

4. $$y = \ln \left( \frac { 1 } { 1 - 2 x } \right) , \quad | x | < \frac { 1 } { 2 }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\)
2. Hence, or otherwise, find the series expansion of \(\ln \left( \frac { 1 } { 1 - 2 x } \right)\) about \(x = 0\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient in its simplest form.
3. Use your expansion to find an approximate value for \(\ln \left( \frac { 3 } { 2 } \right)\), giving your answer
   to 3 decimal places.

3. (a) Given that \(y = \ln ( 1 + 5 x ) , | x | < 0.2\), find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
(b) Hence obtain the M aclaurin series for \(\ln ( 1 + 5 x ) , | x | < 0.2\), up to and including the term in \(x ^ { 3 }\).

1. Given that

$$y = 3 x \arcsin 2 x \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$

1. determine an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence determine the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a \pi + b\) where \(a\) and \(b\) are fully simplified constants to be found.

Differentiate, with respect to \(x\),

1. \(\mathrm { e } ^ { 3 x } + \ln 2 x\),
2. \(\left( 5 + x ^ { 2 } \right) ^ { \frac { 3 } { 2 } }\).

8. The curve \(C\) has equation $$y = \ln \left( \frac { \mathrm { e } ^ { x } + 1 } { \mathrm { e } ^ { x } - 1 } \right) , \quad \ln 2 \leqslant x \leqslant \ln 3$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$$
2. Find the length of the curve \(C\), giving your answer in the form \(\ln a\), where \(a\) is a rational number.
   (6)