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

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

5. \(\mathrm { f } ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 4 x - 2\) a. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form \(x = \pm \sqrt { a + \frac { b } { x } }\), and state the values of the integers \(a\) and \(b\). \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
The iterative formula \(x _ { n + 1 } = \sqrt { a + \frac { b } { x _ { n } } } , \quad x _ { 0 } = 4\) is used to find an approximation value for \(\alpha\).
b. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to 4 decimal places.
c. Explain why for this question, the Newton-Raphson method cannot be used with \(x _ { 1 } = 2\).

11. \(\frac { - 6 x ^ { 2 } + 24 x - 9 } { ( x - 2 ) ( 1 - 3 x ) } \equiv A + \frac { B } { x - 2 } + \frac { C } { 1 - 3 x }\) a. Find the values of the constants \(A , B\) and \(C\).
b. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
c. Prove that \(\mathrm { f } ( x )\) is an increasing function.

1. (a) Express \(\lim _ { \delta x \rightarrow 0 } \sum _ { x = 2.1 } ^ { 6.3 } \frac { 2 } { x } \delta x\) as an integral.
   (b) Hence show that

$$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 2.1 } ^ { 6.3 } \frac { 2 } { x } \delta x = \ln k$$ where \(k\) is a constant to be found.

1. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by

$$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where \(k\) is an integer.
Given that \(u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0\)

1. show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
2. Find the value of \(k\), giving a reason for your answer.
3. Find the value of \(u _ { 3 }\)

13. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , x > 0\) The line \(l\) is the normal to \(C\) at the point \(P ( \mathrm { e } , \mathrm { e } )\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis.
Show that the exact area of \(R\) is \(A e ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
(10)

1. The curve \(C\) has parametric equations

$$x = t ^ { 2 } + 6 t - 16 \quad y = 6 \ln ( t + 3 ) \quad t > - 3$$

1. Show that a Cartesian equation for \(C\) is $$y = A \ln ( x + B ) \quad x > - B$$ where \(A\) and \(B\) are integers to be found. The curve \(C\) cuts the \(y\)-axis at the point \(P\)
2. Show that the equation of the tangent to \(C\) at \(P\) can be written in the form $$a x + b y = c \ln 5$$ where \(a\), \(b\) and \(c\) are integers to be found.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curves with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) where $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 4 x ^ { 2 } - 1 } & x > 0 \\ \mathrm {~g} ( x ) = 8 \ln x & x > 0 \end{array}$$

1. Find
   1. \(\mathrm { f } ^ { \prime } ( x )\)
   2. \(\mathrm { g } ^ { \prime } ( x )\) Given that \(\mathrm { f } ^ { \prime } ( x ) = \mathrm { g } ^ { \prime } ( x )\) at \(x = \alpha\)
2. show that \(\alpha\) satisfies the equation $$4 x ^ { 2 } + 2 \ln x - 1 = 0$$ The iterative formula $$x _ { n + 1 } = \sqrt { \frac { 1 - 2 \ln x _ { n } } { 4 } }$$ is used with \(x _ { 1 } = 0.6\) to find an approximate value for \(\alpha\)
3. Calculate, giving each answer to 4 decimal places,
   1. the value of \(x _ { 2 }\)
   2. the value of \(\alpha\)

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 4 x ^ { 2 } + x } { 2 \sqrt { x } } - 4 \ln x \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 12 x ^ { 2 } + x - 16 \sqrt { x } } { 4 x \sqrt { x } }$$ The point \(P\), shown in Figure 1, is the minimum turning point on \(C\).
2. Show that the \(x\) coordinate of \(P\) is a solution of $$x = \left( \frac { 4 } { 3 } - \frac { \sqrt { x } } { 12 } \right) ^ { \frac { 2 } { 3 } }$$
3. Use the iteration formula $$x _ { n + 1 } = \left( \frac { 4 } { 3 } - \frac { \sqrt { x _ { n } } } { 12 } \right) ^ { \frac { 2 } { 3 } } \quad \text { with } x _ { 1 } = 2$$ to find (i) the value of \(x _ { 2 }\) to 5 decimal places,
   (ii) the \(x\) coordinate of \(P\) to 5 decimal places.

1. The function \(g\) is defined by

$$g ( x ) = \frac { 3 \ln ( x ) - 7 } { \ln ( x ) - 2 } \quad x > 0 \quad x \neq k$$ where \(k\) is a constant.

1. Deduce the value of \(k\).
2. Prove that $$\mathrm { g } ^ { \prime } ( x ) > 0$$ for all values of \(x\) in the domain of g .
3. Find the range of values of \(a\) for which $$g ( a ) > 0$$

10

1. Sketch the graph of \(y = \frac { 1 } { x } + a\), where \(a\) is a positive constant.

13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).

1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
   \end{figure}
5. Explain why the trapezium rule gives an underestimate.

11 The height \(h \mathrm {~cm}\) of a sunflower plant \(t\) days after planting the seed is modelled by \(\mathrm { h } = \mathrm { a } + \mathrm { b }\) Int for \(t \geqslant 9\), where \(a\) and \(b\) are constants. The sunflower is 10 cm tall 10 days after planting and 200 cm tall 85 days after planting.

1. 1. Show that the value of \(b\) which best models these values is 88.8 correct to \(\mathbf { 3 }\) significant figures.
   2. Find the corresponding value of \(a\).
2. 1. Explain why the model is not suitable for small positive values of \(t\).
   2. Explain why the model is not suitable for very large positive values of \(t\).
3. Show that the model indicates that the sunflower grows to 1 m in height in less than half the time it takes to grow to 2 m .
4. Find the value of \(t\) for which the rate of growth is 3 cm per day.

10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations: \(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\) \(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\) where \(t \geq 0\) and \(k\) is a positive constant.

1. 1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
   2. Give the equations of any asymptotes.
2. What does the model predict about the long term population of
   Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
3. Comment on the validity of the model given by the equations, giving a reason for your answer.
4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).

11 Fig. 11 shows the graph of \(y = x ^ { 2 } - 4 x + x \ln x\). \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinate of the stationary point on the curve may be found from the equation \(2 x - 3 + \ln x = 0\).
2. Use an iterative method to find the \(x\)-coordinate of the stationary point on the curve \(y = x ^ { 2 } - 4 x + x \ln x\), giving your answer correct to 4 decimal places.

4 In this question you must show detailed reasoning.
A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Determine the coordinates of the stationary points on the curve.
2. Determine the nature of each stationary point.
3. Write down the equation of the vertical asymptote.
4. Deduce the set of values of \(x\) for which the curve is concave upwards.

1. The curve \(y = \mathrm { f } ( x )\) passes through the point \(P ( - 1,3 )\) and is such that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { x ^ { 2 } } , \quad x \neq 0 .$$

1. Using integration, find \(\mathrm { f } ( x )\).
2. Sketch the curve \(y = \mathrm { f } ( x )\) and write down the equations of its asymptotes.

10. The curve \(C\) has the equation \(y = \mathrm { f } ( x )\). Given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \frac { 2 } { x ^ { 2 } } , \quad x \neq 0 ,$$ and that the point \(A\) on \(C\) has coordinates (2, 6),

1. find an equation for \(C\),
2. find an equation for the tangent to \(C\) at \(A\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers,
3. show that the line \(y = x + 3\) is also a tangent to \(C\).

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 \(x = \mathrm { e }\).

3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when $$y = \mathrm { e } ^ { 3 x } + \ln x$$
2. 1. Given that \(u = \frac { \sin x } { 1 + \cos x }\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 1 } { 1 + \cos x }\).
   2. Hence show that if \(y = \ln \left( \frac { \sin x } { 1 + \cos x } \right)\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosec } x\).

6

1. Use the mid-ordinate rule with four strips to find an estimate for \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\), giving your answer to three significant figures.
2. 1. Given that \(y = x \ln x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Hence, or otherwise, find \(\int \ln x \mathrm {~d} x\).
   3. Find the exact value of \(\int _ { 1 } ^ { 5 } \ln x \mathrm {~d} x\).

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

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

3

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when:
   1. \(\quad y = \ln ( 5 x - 2 )\);
   2. \(y = \sin 2 x\).
2. The functions f and g are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = \ln ( 5 x - 2 ) , & \text { for real values of } x \text { such that } x \geqslant \frac { 1 } { 2 } \\ \mathrm {~g} ( x ) = \sin 2 x , & \text { for real values of } x \text { in the interval } - \frac { \pi } { 4 } \leqslant x \leqslant \frac { \pi } { 4 } \end{array}$$
   1. Find the range of f .
   2. Find an expression for \(\operatorname { gf } ( x )\).
   3. Solve the equation \(\operatorname { gf } ( x ) = 0\).
   4. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).

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 \(A\).

2 A curve has equation \(y = 2 \ln ( 2 \mathrm { e } - x )\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find an equation of the normal to the curve \(y = 2 \ln ( 2 \mathrm { e } - x )\) at the point on the curve where \(x = \mathrm { e }\).
   [0pt] [4 marks]
3. The curve \(y = 2 \ln ( 2 \mathrm { e } - x )\) intersects the line \(y = x\) at a single point, where \(x = \alpha\).
   1. Show that \(\alpha\) lies between 1 and 3 .
   2. Use the recurrence relation $$x _ { n + 1 } = 2 \ln \left( 2 \mathrm { e } - x _ { n } \right)$$ with \(x _ { 1 } = 1\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
   3. Figure 1, on the opposite page, shows a sketch of parts of the graphs of \(y = 2 \ln ( 2 \mathrm { e } - x )\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      [0pt] [2 marks] \section*{(c)(iii)} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-05_864_1284_1802_386}
      \end{figure}